On the impact of stochastic volatility, interest rates and ... · Résumé Lesrentesvariables,etplusparticulièrementlesgarantiesderachatviager(GRV),sontdeve-nuestrèsimportantesdansl’industriedelagestiondupatrimoine.Cesgaranties

On the impact of stochastic volatility, interest rates andmortality on the hedge efficiency of GLWB guarantees

Mémoire

Pierre-Alexandre Veilleux

Maîtrise en actuariatMaître ès sciences (M.Sc.)

Québec, Canada

© Pierre-Alexandre Veilleux, 2016

Résumé

Les rentes variables, et plus particulièrement les garanties de rachat viager (GRV), sont deve-nues très importantes dans l’industrie de la gestion du patrimoine. Ces garanties, qui offrentaux clients une protection de revenu tout en leur permettant de garder une participationdans les marchés boursiers, comprennent différents risques systématiques du point de vuede l’émetteur. La gestion des risques des GRV est donc une préoccupation majeure pour lescompagnies d’assurance, qui ont opté pour la couverture sur les marchés financiers commestratégie de gestion des risques simple et efficace.

Ce mémoire évalue l’impact de la modélisation du passif de la garantie sur l’efficacité de lacouverture des GRV par rapport à trois risques systématiques importants pour ces garanties,soient les risques de marchés boursiers, d’intérêt et de longévité. Le présent travail vise doncà étendre l’analyse effectuée par Kling et al. (2011), qui se concentre sur le risque de marchésboursiers. Ce mémoire montre que les taux d’intérêt stochastiques sont primordiaux dans lamodélisation du passif des GRV.

Ce mémoire analyse également l’impact de la modélisation de la mortalité utilisée dans laboucle externe sur l’efficacité de la couverture des GRV. Une allocation du risque entre lesrisques financiers et le risque de longévité est utilisée pour montrer que la longévité représenteune part importante du risque total des GRV couvertes. De plus, l’efficacité de la couver-ture dans des projections incluant une modélisation stochastique des risques financiers et durisque de longévité est comparée à l’efficacité dans des projections utilisant des marges pourécarts défavorables traditionnelles sur l’hypothèse d’amélioration de mortalité. La diversifica-tion entre les risques financiers et de longévité s’avère avoir un effet substantiel sur l’efficacitéde la couverture.

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Abstract

Variable annuity guarantees, and particularly guaranteed lifetime withdrawal benefit (GLWB)guarantees, have become very important in the wealth management industry. These guaran-tees, which provide clients with a revenue protection while allowing them to retain equitymarket participation, exhibit significant systematic risks from the issuer’s standpoint. Riskmanagement of GLWB guarantees thus is a main concern for insurance companies, which haveturned to capital market hedging as a straightforward and effective risk management method.

This thesis assesses the impact of the guarantee liability modeling on the hedge efficiencyof GLWB guarantees with respect to three significant systematic risks for these guarantees,namely, the stock market, interest rate and longevity risks. The present work thus aims toextend the hedge efficiency analysis performed in Kling et al. (2011), which focuses on the stockmarket risk. In this thesis, stochastic interest rates are shown to be of primary importance inthe guarantee liability modeling of GLWB guarantees.

This thesis also analyzes the impact of the outer loop modeling of mortality on the hedgeefficiency of GLWB guarantees. A risk allocation between financial and longevity risks is usedto show that longevity holds a significant share of the total risk of a hedged GLWB guarantee.The hedge efficiency in projections including both stochastic financial and mortality modelingis compared with the efficiency in projections using traditional actuarial margins for adversedeviations on the mortality improvement assumption. The diversification between financialand longevity risks is shown to have a substantial impact on hedge efficiency.

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Contents

Résumé iii

Abstract v

Contents vii

List of Tables ix

List of Figures xi

Remerciements xv

1 Introduction 11.1 Variable annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Variable annuity guarantees . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Guaranteed lifetime withdrawal benefit guarantees . . . . . . . . . . . . . . 51.4 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 GLWB guarantee valuation 152.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Guarantee model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Guarantee liability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Nested stochastic projection of GLWB guarantees 273.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Guarantee liability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.3 Guarantee liability greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Models 394.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.2 Financial market models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3 Mortality models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Risk assessment and hedging 575.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

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5.2 Outer loop model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 585.3 Real-world risk assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.4 Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.5 Hedge efficiency analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

6 Impact of the guarantee liability modeling on hedge efficiency 756.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.2 Inner loop model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 766.3 Hedge efficiency analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7 Longevity risk analysis 877.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877.2 Longevity risk impact on hedge efficiency . . . . . . . . . . . . . . . . . . . 887.3 Risk allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 907.4 Diversification between longevity and financial risks . . . . . . . . . . . . . . 957.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Conclusion 101

A Basic numerical procedures 103A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103A.2 Monte Carlo simulation for option valuation . . . . . . . . . . . . . . . . . . 103A.3 Finite difference techniques for greeks calculation . . . . . . . . . . . . . . . 104

B Basic probability theory 107B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107B.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107B.3 Change of measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

Bibliography 111

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List of Tables

1.1 Living benefit guarantees features comparison . . . . . . . . . . . . . . . . . . . 4

2.1 Illustration of a risk-neutral scenario used in the valuation of the GLWB guar-antee liability at time 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.1 Parameters for the regime-switching lognormal model . . . . . . . . . . . . . . 585.2 Quantiles of the stock index with SP

0 = 100 . . . . . . . . . . . . . . . . . . . . 595.3 Parameters of the two-factor extended Vasicek model . . . . . . . . . . . . . . . 605.4 Parameters of the Lee-Carter model . . . . . . . . . . . . . . . . . . . . . . . . 615.5 Contract holder parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.6 Contractual parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.7 Lifetime withdrawal rates by attained age . . . . . . . . . . . . . . . . . . . . . 635.8 Average tail hedged and unhedged losses (as a % of SP ) . . . . . . . . . . . . . 725.9 Impact of the hedging strategy on hedge efficiency . . . . . . . . . . . . . . . . 73

6.1 Parameter sets for the Hull-White model . . . . . . . . . . . . . . . . . . . . . . 776.2 One-year spot rate movement volatilities . . . . . . . . . . . . . . . . . . . . . . 776.3 Impact of the guarantee liability modeling on the initial guarantee liability . . . 806.4 Impact of the guarantee liability modeling on hedge efficiency for the 65-year-old

contract holder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.5 Impact of the guarantee liability modeling on hedge efficiency for the 50-year-old

contract holder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.1 Impact of mortality deviations on hedge efficiency for the 65-year-old contractholder using a 3-rho hedging strategy . . . . . . . . . . . . . . . . . . . . . . . . 88

7.2 Impact of mortality deviations on average tail losses for the 65-year-old contractholder without hedging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89



7.5 Allocation of risk with a 3-rho hedging strategy for the 65-year-old contractholder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92



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7.8 Extended risk allocation with a 3-rho hedging strategy for the 65-year-old con-tract holder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.9 Extended risk allocation with a 5-rho hedging strategy for the 65-year-old con-tract holder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.10 Assessment of the diversification between financial and longevity risks for the65-year-old contract holder using a 3-rho hedging strategy . . . . . . . . . . . . 97



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List of Figures

1.1 LIMRA’s annuity sales estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 GLWB guarantee illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Illustration of a triennial ratchet (left panel) and a yearly roll-up (right panel)

in the accumulation phase of a GLWB guarantee . . . . . . . . . . . . . . . . . 8

3.1 Key-rate rho shocks on the spot curve . . . . . . . . . . . . . . . . . . . . . . . 35

5.1 Quantiles of the 1-year (left panel) and 30-year (right panel) spot rates in thetwo-factor extended Vasicek model . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2 Estimated αx (left panel) and βx (right panel) values by age in the Lee-Cartermodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.3 Quantiles of the lifetime withdrawal amount (left panel) and distribution of thetime until the account value is exhausted (right panel) . . . . . . . . . . . . . . 64

5.4 Unhedged GLWB guarantee gains and losses distribution . . . . . . . . . . . . . 655.5 Smoothed distributions of the hedged (PV (k)

H ) and unhedged (PV (k)U + V

P,(k)0 )

GLWB guarantee gains and losses . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.1 Average paths of V Pti − V

P,S−ti

(left panel) and of the stock market index (rightpanel) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7.1 Quantiles of the future path of κt used as assumptions with margins for adversedeviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

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à ma tendre épouse, Marie-Eve

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Remerciements

Je voudrais d’abord remercier mon superviseur, M. Etienne Marceau, pour m’avoir accom-pagné dans ce projet qui m’a été cher. M. Marceau m’a à la fois grandement aidé dans larédaction de ce mémoire, m’a offert son support moral dans les moments de doute, et m’a offertla remarquable opportunité de présenter les résultats de mes travaux lors de colloques et ainside pouvoir échanger avec différents collègues des milieux académique et pratique. En somme,M. Marceau a fait de mes études aux cycles supérieurs une expérience très enrichissante.

Je me dois également de remercier les évaluateurs de mon mémoire, M. David Landriault etM. Patrice Gaillardetz, qui m’ont donné de judicieux commentaires sur ce travail.

J’aimerais remercier du fond du cœur les personnes qui ont gravité autour de moi au cours deces quelques dernières années. Tout d’abord, mon épouse, Marie-Eve, pour qui je décrocheraisvolontiers la lune, et aussi ma mère, Doris, la plus merveilleuse des mamans. Ces deux femmesont une générosité envers moi qui n’a pas d’égal, et je leur en serai toujours reconnaissant.Un immense merci également à mon père et à ma sœur, sur qui je peux toujours compter etqui m’ont encouragé à travers ce projet. Merci également à toutes ces autres personnes queje côtoie et que j’apprécie, et auxquelles je n’ai pas pu consacrer le temps que j’aurais voulu.

J’aimerais également remercier les personnes qui ont rendu ce projet possible en me permettantde conjuguer travail et étude, soient M. Mario Robitaille, M. Frédéric Tremblay et Mme KimGirard. Un merci particulier à mon collègue Maxime Turgeon-Rhéaume, qui a su me motiverà la fois à entreprendre ma maîtrise et à la poursuivre malgré l’engagement de temps que celapouvait représenter.

Enfin, j’aimerais exprimer ma gratitude à l’Industrielle Alliance pour son soutien financier enlien avec les coûts liés à la poursuite de mes études à la maîtrise. J’aimerais aussi remercierla Chaire d’actuariat de l’Université Laval et le Fonds de soutien à la réussite de l’UniversitéLaval pour leur soutien financier.

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Chapter 1

Introduction

1.1 Variable annuities

Wealth management and retirement planning have been growing concerns in the general pop-ulation as population ageing has pushed more people into the accumulation and retirementphases of their lives. Insurance companies and other players in the wealth management indus-try have responded to the wealth management and retirement planning needs with a full breathof products for potential clients. These products include mutual funds, annuities, guaranteedinterest contracts, bonds, variable annuities and other similar products.

Variable annuities are a relatively recent development in the wealth management industry.Indeed, although they were first introduced in 1952 in the United States, they have gainedmuch of their popularity in the 1990s (Poterba (1997)), and new product designs are stillemerging on the market.

Basic variable annuity contracts are often seen as being similar to mutual fund investments:these two investment vehicles allow the accumulation of money in various investment options,with little constraints regarding the timing of additional premiums or withdrawals from thecontract. Moreover, the contract holder is at risk regarding the investment performance, whichdepends on the chosen investment options.

However, variable annuity contracts also have characteristics that set them apart from mutualfund investments. First of all, variable annuities allow the accumulation of tax-deferred sav-ings, which may not always be the case for mutual funds. For variable annuities, income andinvestment gains are only taxed as money is withdrawn from the funds. Moreover, and oftenmost importantly, variable annuities offer guarantees to contract holders (U.S. Securities andExchange Commission (2011)). These guarantees provide contract holders with some kind ofprotection on their investment, thus reducing financial risks compared with traditional mutualfund investments. Finally, variable annuity guarantees are sold by insurance companies only.

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Although variable annuities share their name with traditional annuities, these two productshave significant dissimilarities. Indeed, variable annuities offer more flexibility than traditionalannuities in certain respects. First, they offer a more flexible range of investment options thantraditional annuity contracts. Investment options include money market funds, bond funds,stock market funds, diversified funds and more. Moreover, variable annuities allow contractholders to maintain their liquidity, meaning that they can withdraw their funds if they wishto lapse the variable annuity contract.

Variable annuities have attracted significant interest for several years now, even outperformingfixed annuities in sales. Figure 1.1 shows LIMRA’s annuity sales estimates since 2001 (Liu(2010) and LIMRA Secure Retirement Institute (2014)). LIMRA, formerly known as theLife Insurance and Market Research Association, is an insurance and financial service tradeassociation that provides research, learning and development programs to financial servicescompanies around the world.

Figure 1.1: LIMRA’s annuity sales estimates

Variable annuity sales are shown in Figure 1.1 to be consistently high for an extended periodof time. Macroeconomic conditions sure are an important cause for the variable annuities’popularity. With lowering fixed annuity payouts due to steadily decreasing interest rates overthe past three decades, variable annuities became an attractive alternative for retirement.Indeed, they allow contract holders to accumulate money while avoiding locking the currentlow payout rates. Moreover, as will be shown in section 1.2, some variable annuity guaranteesare dedicated to providing a retirement income.

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Compensation may be another factor for the variable annuities’ popularity, as these productsoften offer more generous commissions than mutual funds or traditional annuities.

Finally, guarantees offered in variable annuities are no doubt another explanation for theirpopularity, as they allow contract holders to have both a participation in equity markets andthe protection provided by the guarantees. They are thus suited to risk-averse investors.

Nowadays, variable annuities are at the heart of the wealth management, retirement plan-ning and estate planning business in the United States. In addition to being popular in theUnited States, variable annuities have counterparts in other countries. In Canada, segregatedfunds have much similarities with variable annuities. In Europe, the term “investment-linkedproduct” is used to denote products that include guarantees linked to market performance.

In all cases, the guarantees offered in these products set them apart from traditional invest-ments by providing protection to contract holders. Typical variable annuity guarantees aredescribed in more details in section 1.2.

1.2 Variable annuity guarantees

The common goal of all variable annuity guarantees is to provide protection on the investmentmade by a contract holder in a variable annuity contract. Typical variable annuity guaranteescan be classified in two broad categories: death benefit guarantees and living benefit guaran-tees. Death benefit guarantees only include the guaranteed minimum death benefit (GMDB)guarantee, which provides a guaranteed value to the contract holder at the time of death.This implies that no matter the account value of the contract holder at the time of death, theinsurance company pays the maximum between the account value and the guaranteed value tothe contract holder’s estate. The guaranteed value usually is a given percentage of the initialand subsequent investments in the variable annuity contract.

Living benefit guarantees include a variety of guarantees. First of all, the guaranteed minimumaccumulation benefit (GMAB) guarantee provides the contract holder with a guaranteed valueat maturity of the contract. Thus, the company pays the maximum between the accountvalue and the guaranteed value at a specified maturity date. The guaranteed value is oftena percentage of the initial and subsequent investments in the variable annuity contract. Theguarantee may or may not be renewable.

The guaranteed minimum income benefit (GMIB) guarantee provides the contract holder witha minimum annuity payout. At the time of annuitization, the contract holder is entitled toan annuity whose payout is the maximum between the guaranteed minimum annuity payoutand the annuity payout determined using the account value and the annuity rate at the timeof annuitization. The guarantee is only applicable if the contract holder annuitizes. Contractholders lose their liquidity after annuitization, as no withdrawals in excess of the annuity

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payout can be made.

The guaranteed minimum withdrawal benefit (GMWB) guarantee provides the contract holderwith a yearly guaranteed withdrawal amount over a given time period. A significant differencebetween GMWB guarantees and GMIB guarantees is that for GMWB guarantees, contractholders remain invested in the various investment options offered in variable annuities whilewithdrawing. Contract holders thus keep their liquidity at all time in a GMWB guarantee.If guaranteed withdrawals exhaust the contract holder’s account value, the contract holder isstill entitled to receive the guaranteed withdrawal amount for the remainder of the guaranteeperiod. Moreover, any remaining account value at maturity of the guarantee or at death ofthe contract holder is returned to the contract holder or to her estate.

The guaranteed lifetime withdrawal benefit (GLWB) guarantee is similar to the GMWB guar-antee. It provides the contract holder with a lifetime guaranteed withdrawal amount whilefunds remain invested in the various investment options offered in variable annuities. Onceagain, the company is liable to pay guaranteed withdrawal amounts once the contract holder’saccount value is exhausted. Any remaining account value at death is paid to the contractholder’s estate. The main difference between GLWB and GMWB guarantees is thus thelife-contingent nature of GLWB guarantees.

The important features of living benefit guarantees are compared in Table 1.1. First of all,the GMAB guarantee is the only one that does not provide a steady income, making it lessattractive as a retirement product. Moreover, the GMIB guarantee is the only guarantee inwhich liquidity has to be forsaken, thus providing less flexibility to retirees. Finally, amongthe two guarantees left, the GLWB guarantee is the only one providing a lifetime income,making it particularly attractive as a retirement product.

Feature GMAB GMIB GMWB GLWB

Guaranteed income No Yes Yes YesLiquidity Yes No Yes YesLifetime No Yes No Yes

Table 1.1: Living benefit guarantees features comparison

GMWB and GLWB guarantees hold an important market share of total variable annuity guar-antees. In the United States, a 2013 Towers Watson survey showed that the GMWB inforceof 14 of the 18 largest variable annuity writers represented 266 billion of dollars (see TowersWatson (2013)). The same survey revealed that 38% of the variable annuity inforce included aGMWB or GLWB guarantee. In Canada, GMWB and GLWB guarantees contributed to gen-erating very high segregated fund sales. In 2007 to 2010 alone, GMWB and GLWB guaranteesgenerated nearly 20 billion in net sales in the Canadian market (see Theriault (2011)).

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Lately, the decrease in interest rates and high capital requirements forced some insurers, par-ticularly in Canada, to stop offering GMWB and GLWB guarantees or significantly lower theguarantees offered. Nonetheless, insurers currently have large blocks of GMWB and GLWBguarantees to manage. Therefore, the risk management strategies and risk assessments as-sociated with GMWB and GLWB guarantees remain a hot topic in the insurance industry.Indeed, at the time of writing this thesis, the Canadian federal and Quebec’s provincial regu-lators are making progress on defining new capital requirement formulas for segregated fundguarantees, and place much emphasis on GMWB and GLWB guarantees.

GLWB guarantees will be described in more details in section 1.3.

1.3 Guaranteed lifetime withdrawal benefit guarantees

The focus of this thesis being GLWB guarantees, this section aims to describe these guaranteesin more details by presenting the relevant technical terminology, describing the guaranteephases and discussing the guarantee features.

1.3.1 Definitions

The description of GLWB guarantees laid out in this section is based on a few technical termsthat shall first be defined. First of all, the account value (AV) denotes the current value of thepremiums invested in the variable annuity. The account value is impacted by the underlyingfunds’ returns, contractual fees and withdrawal payments. Notwithstanding guarantees, theaccount value is the amount that the contract holder is entitled to at any time in the variableannuity contract.

The lifetime withdrawal amount (LWA) is the withdrawal amount that the insurance companyguarantees to the contract holder for the entire duration of the GLWB guarantee, that is tosay, for the contract holder’s lifetime. It is the maximum withdrawal that a contract holdercan make in a given year without incurring penalties on the guarantee.

Finally, the guaranteed withdrawal balance (GWB) is used to determine the lifetime with-drawal amount. The lifetime withdrawal amount is a contractual percentage of the guaran-teed withdrawal balance at the time the contract holder starts making withdrawals. Oncewithdrawals have started, the guaranteed withdrawal balance decreases on a dollar-for-dollarbasis with withdrawal payments. The guaranteed withdrawal balance is not affected by theunderlying funds’ returns and the contractual fees.

1.3.2 Guarantee phases

There are two phases in GLWB guarantees. The first of these phases is the accumulation phase.In the accumulation phase, no withdrawals are made by the contract holder. Investments in

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the variable annuity contract are rather allowed to evolve with fund returns and fees. Theaccumulation phase can be of particular interest for contract holders because of the variousfeatures that can increase the guaranteed withdrawal balance during this phase. These featuresare described in section 1.3.3.

The second phase of the GLWB guarantee is the withdrawal phase. During the withdrawalphase, periodic withdrawals of the lifetime withdrawal amount are made by the contractholder. As long as the contract holder’s account value is positive, the withdrawals are takenout of the account value. However, if the account value reaches zero because of guaranteedwithdrawals and poor market performance, the insurance company must pay the lifetimewithdrawal amount to the contract holder.

The two phases of GLWB guarantees are illustrated in Figure 1.2. The guarantee is assumedto be sold to a 65-year-old contract holder with a 6% withdrawal rate and to include a 5-yearaccumulation phase. Figure 1.2 shows that in the accumulation phase, no withdrawals aremade and the account value thus fluctuates based on market performance. Then, the contractholder transfers to the withdrawal phase, and lifetime withdrawals start being made. Theaccount value then still fluctuates with market returns, but also decreases as a result of with-drawals. When the account value eventually reaches zero, the company becomes responsiblefor paying the lifetime withdrawal amount to the contract holder until death.

65 68 71 74 77 80 83 86 89 92 95 98 101 104

Age

020

4060

8010

012

014

0

Account valueLifetime withdrawal amount

Figure 1.2: GLWB guarantee illustration

The GLWB guarantee being an addition to the variable annuity contract, it does not alter theunderlying contract. Therefore, at death of the contract holder, the contract holder’s estateis entitled to any remaining account value in the contract.

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1.3.3 Additional features

As mentioned above, GLWB guarantees contain additional features that are meant to increasetheir competitiveness. These features impact the guaranteed withdrawal balance.

The guaranteed withdrawal balance at inception of the contract is equal to the amount ini-tially invested in the variable annuity. Then, its evolution depends on the guarantee featuresincluded in the contract. Popular features in the Canadian and US markets include ratchetsand roll-ups of the guaranteed withdrawal balance.

The ratchet feature allows the guaranteed withdrawal balance to be increased to the accountvalue level periodically, for example every three years, if the latter is higher than the former.The ratchet feature thus allows contract holders to participate in strong market performancein the accumulation phase. Ratchets may also be allowed in the withdrawal phase. Then,ratchets increase either the guaranteed withdrawal balance, the lifetime withdrawal amount,or both. The lifetime withdrawal amount may never decrease following a ratchet. GLWBguarantees typically also include an additional automatic ratchet at the time of transfer fromthe accumulation phase to the withdrawal phase.

The roll-up feature allows the guaranteed withdrawal balance to be increased by a statedpercentage as long as no withdrawals are made in a given year. Hence, each year duringthe accumulation phase, the guaranteed withdrawal balance is increased by the contractualroll-up percentage. This feature encourages contract holders to accumulate money in theGLWB product, since a longer accumulation period implies more roll-ups of the guaranteedwithdrawal balance.

Neither of the ratchet feature nor the roll-up feature increase the contract holder’s accountvalue. However, both these features contribute in increasing the guaranteed withdrawal bal-ance at the time of transfer from the accumulation phase to the withdrawal phase, and as suchcontribute in increasing the lifetime withdrawal amount. The two features described aboveare illustrated in Figure 1.3.

Figure 1.3 shows how the guaranteed withdrawal balance may evolve in time based on theguarantee features considered. In this figure, the contract holder is 65 years old at inceptionof the contract. The x-axis must be interpreted as the age of the contract holder as time goesby in the variable annuity contract.

The left panel of Figure 1.3 shows that a triennial ratchet feature increases the guaranteedwithdrawal balance to the account value level every three years if the latter is higher than theformer. The ratchet feature is thus contingent on a strong market performance. Moreover,the right panel of Figure 1.3 shows that the roll-up feature causes a steady increase in theguaranteed withdrawal balance each year. The roll-up feature is applied regardless of theaccount value level.

7

65 66 67 68 69 70 71

100

110

120

130

Age

AVGWB

65 66 67 68 69 70 71

100

110

120

130

Age

AVGWB

Figure 1.3: Illustration of a triennial ratchet (left panel) and a yearly roll-up (right panel) inthe accumulation phase of a GLWB guarantee

1.4 Literature review

This thesis is interested in the risk assessment of GLWB guarantees. Hence, in this section,a review of the relevant literature on GLWB guarantees is presented. Because GMWB andGLWB guarantees are closely related to one another and that there is a larger body of literatureon GMWB guarantees, the relevant literature on GMWB guarantees is presented first, followedby the literature on GLWB guarantees.

1.4.1 GMWB guarantees

The problem of pricing and hedging GMWB guarantees has attracted growing interest in theacademic literature in the last few years. Milevsky and Salisbury (2006) is to our knowledge oneof the first articles dealing with the problem of pricing GMWB guarantees. Under the Black-Scholes framework and assuming a simple GMWB guarantee with continuous withdrawals,they find the fair guarantee charge under the passive and dynamic contract holder behaviourassumptions.

The passive (or static) and the dynamic (or optimal) behaviours are at the ends of the spectrumregarding contract holder sophistication. At one end, a passive contract holder behaviourimplies that the contract holder withdraws the lifetime withdrawal amount consistently untildeath. At the other end, a dynamic contract holder behaviour refers to the strategy in which

8

the contract holder makes withdrawals so as to maximize the expected cash flows from theGMWB or GLWB contract. A dynamic behaviour thus essentially becomes a loss-maximizingbehaviour from the insurance company’s standpoint.

Under the passive behaviour assumption, Milevsky and Salisbury (2006) shows that theGMWB guarantee can be decomposed into a Quanto Asian put plus a generic term-certainannuity. Under the dynamic behaviour assumption, they show that the valuation problem be-comes an optimal stopping problem similar to pricing an American put option. Their analysisleads them to conclude that GMWB guarantees are underpriced in the market.

Using a framework similar to the one presented in Milevsky and Salisbury (2006), Dai et al.(2008) develops a singular stochastic control model for pricing GMWB guarantees under thedynamic contract holder behaviour assumption and propose a finite difference algorithm usingthe penalty approximation approach to solve it. They also extend the valuation to the case ofdiscrete withdrawals.

Whereas Milevsky and Salisbury (2006) and Dai et al. (2008) focus on GMWB guarantees,Bauer et al. (2008) introduces a model meant to allow the consistent valuation of all variableannuity guarantees. They calculate the fair guarantee charge under the passive and dynamiccontract holder behaviour assumptions, using Monte Carlo simulation for the passive behaviourand a generalization of a finite mesh method for the dynamic behaviour. Their model is indiscrete time, considers the effect of mortality and allows for yearly ratchets of the guaranteedwithdrawal balance.

The pricing of GMWB guarantees under the dynamic contract holder behaviour assumptionis also studied in Chen et al. (2008). Their article generalizes some of the assumptions madein e.g. Dai et al. (2008) by introducing a split between mutual fund fees and the guaranteecharge and considering a jump diffusion model in addition to the geometric Brownian motion inpricing GMWB guarantees. Both of these generalizations increase the GMWB fair guaranteecharge. They also consider the effect of sub-optimal contract holder behaviour on the guaranteevalue and conclude that GMWB guarantees are underpriced in the market.

The framework of Milevsky and Salisbury (2006) under the passive contract holder behaviourassumption is extended in Peng et al. (2012), which models interest rates using the Vasicekmodel. Once again, writing the GMWB guarantee as the combination of a put option andan annuity, they are able to find lower and upper bounds for the GMWB guarantee value byusing Roger-Shi’s and Thompson’s approximation methods.

As in Bauer et al. (2008), Banicello et al. (2011) introduces a unifying framework for thevaluation of all variable annuity guarantees. Within this framework, they price a GMWBguarantee under the passive approach and the mixed approach using Monte Carlo simulationand least squares Monte Carlo simulation, respectively. The mixed approach is similar to

9

the passive approach, but includes the possibility of full surrender at an optimal time. Theirarticle goes beyond the Black-Scholes framework used in Bauer et al. (2008) by introducingstochastic volatility and interest rates through the use of the Heston model and the Vasicekmodel. They also introduce stochastic mortality in the modeling by the use of a square rootprocess for the force of mortality.

A framework similar to the one presented in Milevsky and Salisbury (2006) is used in Liu (2010)to analyze the pricing and hedging of GMWB guarantees with discrete withdrawals under thepassive behaviour assumption. Liu (2010) first prices the GMWB guarantee from the contractholder and from the insurance company’s standpoint and shows that the two approaches arein fact equivalent. Moreover, in addition to investigating dynamic hedging strategies underthis framework, she proposes semi-static hedging strategies that improve the hedge efficiency,especially when jumps in prices are taken into account in measuring the efficiency. She alsoextends the proposed semi-static hedging strategies to the stochastic volatility case using theHeston model.

Similar ideas in terms of pricing and hedging in the Black-Scholes framework are presentedin Kolkiewicz and Liu (2012). Once again, a semi-static hedging strategy is proposed andcompared to dynamic hedging. The semi-static hedging strategy is shown to outperform thedynamic hedging strategy when there are random jumps in asset prices.

A flexible tree for the valuation of GMWB guarantees is proposed in Yang and Dai (2013).Their tree allows them to evaluate the GMWB fair guarantee charge without introducingsignificant numerical pricing errors, even when the guarantee includes more complex featuressuch as deferred withdrawals, roll-ups and ratchets. Mortality risk and the possibility of fullsurrender may also be incorporated into the pricing model. Their analysis is done in theBlack-Scholes framework.

By re-writing the GMWB pricing problem in the form of an Asian styled claim and a dimensionally-reduced partial differential equation, Donnelly et al. (2014) is able to price a GMWB guaranteeunder stochastic volatility and stochastic interest rates. They depart from the research workmade previously by modeling the fund value has a mix of equities and bonds. They solve thePDE using an Alternating Direction Implicit method.

Banicello et al. (2014) presents a dynamic algorithm for pricing GMWB guarantees under ageneral Lévy process framework. They consider the geometric Brownian motion, the Mertonjump diffusion, the variance-gamma and the Carr, Geman, Madan, Yor (CGMY) models asspecial cases of the Lévy process modeling. They also compare fair guarantee charges underpassive and dynamic contract holder behaviours.

Finally, Feng and Volkmer (2015) develops semi-analytical solutions for pricing GMWB guar-antees both from the insurer’s perspective and the contract holder’s perspective under the

10

Black-Scholes framework. They show that assuming no friction costs, the two approaches areequivalent. Their semi-analytic solutions lead to a fast and accurate algorithm for pricingGMWB guarantees.

1.4.2 GLWB guarantees

GLWB guarantees have attracted somewhat less attention in the literature than GMWBguarantees. Nonetheless, there is a growing body of literature on the subject.

Using a continuous-time GLWB guarantee under the Black-Scholes framework and a simpli-fying assumption for population mortality, Shah and Bertsimas (2008) presents an analyticalsolution for the GLWB fair guarantee charge under the passive contract holder behaviour.They then generalize their modeling to consider stochastic interest rates and volatility throughthe use of the two-factor Vasicek model and the Heston model, and also allow for an arbi-trary population mortality. They conclude that there is insufficient price discrimination in themarket and that model risk can be significant in determining the fair guarantee charge.

The pricing of GLWB guarantees under the Black-Scholes framework is also considered inPiscopo and Haberman (2011), which presents the calculation of the GLWB fair guaranteecharge under the passive contract holder behaviour using Monte Carlo simulations. They testthe sensitivity of the fair charge to key parameters and give special attention to mortality riskby using a stochastic mortality model. They also evaluate the cost of additional features, suchas roll-ups, ratchets and deferred withdrawals.

A different angle is taken by Ngai and Sherris (2011), which investigates the effectivenessof static hedging strategies on longevity risk using longevity bonds and derivatives. Theyconclude that the hedge is not effective for GLWB guarantees because of its exposure tomarket risk.

Holz et al. (2012) extends the framework presented in Banicello et al. (2011) to include GLWBguarantees. They price GLWB guarantees under the passive and dynamic contract holderbehaviours and evaluate the fair charge’s sensitivity to various assumptions and guaranteefeatures. They also compare GMWB and GLWB guarantees’ fair charges.

A similar framework to the one presented in Holz et al. (2012) is used in Kling et al. (2011) toprice and hedge GLWB guarantees. First using the Black-Scholes framework and then makingan extension to include stochastic volatility through the use of the Heston model, they find thefair guarantee charges and greeks of GLWB guarantees for various guarantee designs assuminga passive contract holder behaviour. They also analyze the hedge effectiveness for differentdynamic hedging strategies and examine the effects on hedge effectiveness if the hedging modelis different from the data-generating model.

Steinorth and Mitchell (2012) analyzes optimal behaviour from the contract holder’s stand-

11

point, which differs from the dynamic contract holder behaviour assumption most often usedin the literature. They argue that a risk-averse contract holder may not only consider the levelof future expected cash flows in making decisions, but also the negative effect of consump-tion fluctuations. They use the Money’s Worth Ratio and the Annuity Equivalent Wealth tocompare various investment opportunities, including a GLWB and a GLWB with ratchets.

The concept of utility is also used in Azimzadeh et al. (2014), which uses a method thatconsists in first determining the contract holder’s withdrawal strategy and then feeding itinto the pricing problem. Using a Markov regime-switching process, they show that whenconsumption is assumed to be governed by hyperbolic absolute risk aversion utility, optimalbehaviour leads to a cost similar to the one under the passive contract holder behaviour.

The impact of policyholder behaviour is further analyzed in Kling et al. (2014). They considervarious lapse strategies and study the impact on pricing, hedging and hedge effectiveness.

An optimal stochastic control framework allowing to price a GLWB guarantee under the Black-Scholes framework and under a regime-switching Markov process is developed in Forsyth andVetzal (2014). Various guarantee features and contract holder behaviour assumptions areconsidered.

Finally, Fung et al. (2014) is interested in the impact of systematic mortality risk on GLWBguarantees. They analyze the impact of mortality risk on the unhedged gains and lossesdistribution.

1.5 Motivation

As made clear by the literature review presented in section 1.4, much of the attention devotedto GLWB and GMWB guarantees has been focused on pricing. Risk management, on the otherhand, has drawn somewhat less attention. Nevertheless, risk management is very importantfor GLWB guarantees, as these guarantees contain significant systematic risks and representa large proportion of the insurance industry’s variable annuity inforce business.

Companies offering variable annuity guarantees have two opportunities at risk management.First of all, they can mitigate some of the risks associated with variable annuity guaranteesthrough an appropriate pricing and product design. Adequate pricing helps preventing short-falls in the value of future revenues versus future claims. In other words, it ensures that thecontract holder pays the fair price for the guarantee provided by the company. Guaranteedesign helps reducing the potential risks faced by insurers. For example, guarantee featuresthat can have an impact on risk include the size of the roll-up, the frequency of ratchets, thelifetime withdrawal rates, the fund offering and so on.

Product pricing and design can only do so much for companies in terms of risk management.

12

Indeed, GLWB guarantees by nature contain risks that cannot be fully mitigated by productdesign. Thus, the main part of risk management for variable annuity guarantees is carriedout after the guarantees are issued. Then, prices charged for the guarantees can seldom bechanged, and if so, only to a certain extent. Moreover, product design is usually carved in stoneby the contractual agreement that binds the company and the contract holder. Companiesthus have to find alternative ways to manage risks related to GLWB guarantees.

Options for risk management after issue of the guarantee are rather scarce for insurance com-panies. Self-insurance is certainly one of the least advisable risk management options. SinceGLWB guarantees contain mostly systematic risks, this strategy could prove very damaging toa company with a large exposure to these guarantees. Examples of the potential adverse im-pacts on companies have been seen during the last financial crisis on even the largest Canadianinsurers (see e.g. Tedesco (2010)).

Reinsurance also is a risk management option, but reinsurance opportunities for variableannuity guarantees have been rather rare since the last financial crisis. Thus, companies areoften left with capital market hedging as the most viable risk management alternative afterguarantee issue.

Since GLWB guarantees are complex, long-term, life-contingent options, it is not possibleto find instruments on the financial markets that completely offset the sensitivity of theseguarantees to their various systematic risk factors. The assessment of the residual risk ofhedged GLWB guarantees thus becomes a critical step in the valuation of these guarantees.In this assessment, modeling holds a crucial place. Indeed, modeling is of particular relevancefor variable annuity guarantees, as these guarantees are not marked-to-market like plain vanillaoptions, but rather marked-to-model.

The goal of this thesis is twofold. First of all, an analysis of how the GLWB guarantee liabilitymodeling affects the hedge efficiency of GLWB guarantees is carried out. This analysis is inline with the hedge efficiency analysis presented in Kling et al. (2011). However, whereasKling et al. (2011) focused on the stock market model, this thesis aims to expand the scopeto consider other very important systematic risks of GLWB guarantees, namely, interest rateand longevity risks. Interest rate risk is a major risk for GLWB guarantees as it affects boththe average return in the risk-neutral valuation of the guarantee liability and the discountingof the guarantee’s long-dated cash flows. Longevity risk also is a significant risk because claimpayments in GLWB guarantees are life-contingent. In this thesis, stochastic volatility for thestock market risk is introduced through the use of the regime-switching lognormal model,whereas it was done using the Heston model in Kling et al. (2011). Moreover, several interestrate models, including one-factor and two-factor models, are considered. Finally, longevityrisk is accounted for through the use of a stochastic mortality model.

Secondly, special attention is devoted to longevity risk in hedged GLWB guarantees. The

13

impacts on hedge efficiency of actual deviations in the mortality experience are assessed.Deviations in the mortality experience are obtained through the use of stochastic mortalityin the outer loop modeling of the hedge efficiency calculation. Moreover, a method thatallocates the residual risk of hedged GLWB guarantees between financial and longevity risksis proposed. The method is further extended to split longevity risk into two risk components.Finally, diversification between financial and longevity risks for hedged GLWB guarantees isinvestigated. Hedge efficiency in projections in which both the financial and longevity riskfactors are modeled in a single stochastic projection are compared to projections in whichlongevity risk is taken into account using an actuarial margin for adverse deviations on themortality improvement assumption.

This thesis is organized as follows. In chapter 2, the notation and formulas related to theGLWB guarantee model considered throughout this thesis are introduced. The valuation of theGLWB guarantee liability is also presented. Chapter 3 extends the GLWB guarantee liabilityvaluation to the valuation at future points in time and to the greeks calculation. The modelsconsidered for the stock market, the bond market, interest rates and longevity, both in theouter and inner loops, are then introduced in chapter 4. Chapter 5 first presents an unhedgedreal-world risk assessment of a GLWB guarantee, which leads naturally to introducing thehedging strategy considered in this thesis and illustrating how hedging can help mitigatingthe risks associated with GLWB guarantees. In chapter 6, the impact of the guarantee liabilitymodeling on hedge efficiency is thoroughly examined with respect to the stock market, interestrate and longevity risk factors. Finally, longevity risk in hedged GLWB guarantees is furtheranalyzed in chapter 7.

14

Chapter 2

GLWB guarantee valuation

2.1 Introduction

In chapter 1, GLWB guarantees were described qualitatively. In this chapter, the GLWBguarantee considered in this thesis is described in a more formal manner, taking into accountthe additional features included in the guarantee.

Then, through the use of generic notation for some of the risk factors of GLWB guarantees,the valuation of the GLWB guarantee liability is presented. The guarantee liability valuationconsists in determining, given the contractual charge set by the company for the guarantee,the expected present value of claims and revenues from the guarantee. Valuation is thusclosely related to pricing, as pricing consists in setting a charge that is such that the expectedrevenues equals the expected claims from the guarantee. It is the calculation that is mostlydealt with in the GLWB guarantee literature. Valuation of the GLWB guarantee liability isthe necessary foundation on which to build the risk assessment of hedged GLWB guarantees.

2.2 Guarantee model

Let SP be the single premium made in a variable annuity contract with a GLWB guaranteeissued at time t0 = 0. Assume that t0 falls on January 1st of a given year for the sake ofsimplicity. The contract holder’s age at inception of the contract is given by x. Let

ΩT = t0, t1, . . . , tW−1, tW , tW+1, . . . , t(ω−x)/∆t (2.1)

be the set of all ordered times at which events can occur, where tW is the time at which thecontract holder transfers from the accumulation phase to the withdrawal phase, ω is the maxi-mum age of the mortality table and ti+1−ti = ∆t ∈ (0, 1], ∀i. Events may include withdrawalsfrom the account value, ratchets and roll-ups of the guaranteed withdrawal balance, death ofthe contract holder or rebalancing of a hedge portfolio. Assume further that tW is an integernumber of years for the sake of simplicity.

15

The contract holder is assumed to invest in a combination of mutual funds whose total valueis given by the stochastic process

F = Fti , ti ∈ ΩT .

Flexibility in the contract holder’s asset mix is allowed by assuming that the fund is composedof two distinct mutual funds: a stock market mutual fund and a bond market mutual fund,each of which aims to track a given stock and bond index.

LetFS = FS,ti , ti ∈ ΩT

andFP = FP,ti , ti ∈ ΩT

be the stochastic processes for the mutual funds that track the stock market index and thebond market index, respectively. Moreover, let

S = Sti , ti ∈ ΩT

andP = Pti , ti ∈ ΩT

be the stochastic processes for the stock market index and the bond market index, respectively.

Rebalancing of the mutual funds between the stock market mutual fund and the bond marketmutual fund is assumed to be done yearly at a given fixed proportion ψ. The proportion ineach mutual fund within a given year depends on the performance of the stock market indexand the bond market index in that year. Thus, the mutual funds values, for ti ∈ ΩT , are givenby

FS,ti =

FS,ti−1

StiSti−1

e−mA∆t, ti 6∈ N

ψFti , ti ∈ N, (2.2)

FP,ti =

FP,ti−1

PtiPti−1

e−mA∆t, ti 6∈ N

(1− ψ)Fti , ti ∈ N, (2.3)

and

Fti =

FS,ti + FP,ti , ti 6∈ N(FS,ti−1

StiSti−1

+ FP,ti−1

PtiPti−1

)e−mA∆t, ti ∈ N

, (2.4)

where Ft0 = SP , FS,t0 = ψ SP , FP,t0 = (1 − ψ)SP and mA is the mutual fund managementfee, that is to say, the contractual fee that is meant to cover commissions, investment expenses,and all other company-related expenses that are not related to the GLWB guarantee. Therebalancing strategy described above aims to consider the fact that contract holders generally

16

have a target stock market proportion depending on their investment profile, but that theydo not rebalance their portfolio frequently during a given year.

Let also

A = Ati , ti ∈ ΩT ,

be the stochastic process of the contract holder’s account value, and

G = Gti , ti ∈ ΩT ,

be the stochastic process of the guaranteed withdrawal balance. As described in chapter 1,the guaranteed withdrawal balance is used to determine the lifetime withdrawal amount anddepends on guarantee features such as ratchets and roll-ups.

At inception of the contract, the account value and the guaranteed withdrawal balance areequal to the single premium such that

At0 = Gt0 = SP.

The dynamics of A and G depend on the GLWB guarantee phase. The dynamic in each ofthese phases are given in sections 2.2.1 and 2.2.2.

2.2.1 Accumulation phase

As described in chapter 1, the accumulation phase is the phase that precedes the withdrawalphase and in which no withdrawals are made by the contract holder. Thus, the account valuedynamic is given by

Ati = Ati−1

FtiFti−1

e−gA∆t, ti ∈ ΩT and t0 < ti ≤ tW , (2.5)

where gA is the GLWB guarantee fee. The guarantee fee is the additional charge that coversthe costs related with the GLWB guarantee. It is distinct from mA, the mutual fund fee,which only covers fund-related and company-related expenses. The guarantee fee can be seenas the revenue counterpart to the claims the company has to pay when the contract holder’saccount value is exhausted. It is the charge that is usually solved for in the pricing exercisesuch that the expected value of claims equals the expected value of revenues.

The evolution of the guaranteed withdrawal balance during the accumulation phase dependson the particular features of the GLWB guarantee. As discussed in chapter 1, popular GLWBfeatures in the Canadian and US markets include ratchets and roll-ups of the guaranteed with-drawal balance. Assume that the GLWB guarantee includes automatic ratchets every m yearsand yearly 100p% roll-ups of the guaranteed withdrawal balance. Then, the guaranteed with-drawal balance during the accumulation phase, that is to say, for ti ∈ ΩT and t0 < ti < tW , is

17

given by

Gti =

max(Gti−1(1 + p), Ati), ti/m ∈ N

Gti−1(1 + p), ti ∈ N and ti/m 6∈ N

Gti−1 , otherwise

. (2.6)

In (2.6), ti/m ∈ N implies that ti is an integer multiple of m years. Then, since ti is both atthe end of a year and at a ratchet time, both the roll-up and the ratchet features may alterthe guaranteed withdrawal balance. Moreover, ti ∈ N and ti/m 6∈ N implies that ti is at theend of a year, but not at an integer multiple of m years from the contract inception date. Inthis case, only the roll-up feature is applied to the guaranteed withdrawal balance.

Assume further that at the time of transfer from the accumulation phase to the withdrawalphase, an additional ratchet of the guaranteed withdrawal balance occurs. Then, since it isalso assumed that tW ∈ N, the guaranteed withdrawal balance at the time of transfer is givenby

GtW = max(GtW−1(1 + p), AtW ). (2.7)

Let R = Rti , ti ∈ ΩT denote the stochastic process of the company’s guarantee-relatedrevenues, that is to say, the future revenue from the guarantee fee. In the accumulation phase,the revenue process is given by

Rtj = Atj(1− e−gA∆t

), ti ∈ ΩT and t0 ≤ ti < tW . (2.8)

2.2.2 Withdrawal phase

In the withdrawal phase, the contract holder withdraws the lifetime withdrawal amount yearly.The dynamics of the account value and of the guaranteed withdrawal balance are thereforechanged to take withdrawals into account.

Let L = Lti , ti ∈ ΩT be the stochastic process of the annual lifetime withdrawal amount.At the time of transfer from the accumulation phase to the withdrawal phase, the lifetimewithdrawal amount is given by

LtW = GtW × lx+tW−t0 , (2.9)

where lx is the lifetime withdrawal rate at age x.

The contract holder generally can choose the withdrawal frequency, which can go from annualto monthly. Let n be the withdrawal frequency elected by the contract holder, with n = 1 forannual withdrawals and n = 12 for monthly withdrawals.

18

The account value dynamic for ti > tW and ti ∈ ΩT is then given by

Ati =

max

(Ati−1

FtiFti−1

e−gA∆t − 1

nLti−1 , 0

), nti ∈ N

Ati−1

FtiFti−1

e−gA∆t, otherwise. (2.10)

As shown in (2.10), the account value may never become negative despite withdrawals. More-over, since the lifetime withdrawal amount may change at time ti as a result of a ratchet, theamount withdrawn at time ti is based on the lifetime withdrawal amount determined at timeti−1.

The guaranteed withdrawal balance is decreased on a dollar-for-dollar basis when withdrawalsare made. However, there still is a possibility for a ratchet of the guaranteed withdrawalbalance every m years in the withdrawal phase. The guaranteed withdrawal balance dynamicfor ti > tW and ti ∈ ΩT is given by

Gti =

max

(Gti−1 −

1

nLti−1 , Ati

), ti/m ∈ N

max

(Gti−1 −

1

nLti−1 , 0

), nti ∈ N and ti/m 6∈ N

Gti−1 , otherwise

. (2.11)

Thus, the guaranteed withdrawal balance is increased in the accumulation phase as a result ofratchets and roll-ups, and then decreased in the withdrawal phase as a result of withdrawals,except for potential withdrawal phase ratchets. Like the account value, the guaranteed with-drawal balance may never become negative.

Although the lifetime withdrawal amount may never be decreased, hence providing protectionto the contract holder, it may be increased in the event of a ratchet. The withdrawal amountfor ti > tW and ti ∈ ΩT is given by

Lti =

max

(Lti−1 , lx+ti−t0Gti

), ti/m ∈ N

Lti−1 , otherwise. (2.12)

Note that the lifetime withdrawal rate used in determining the ratcheted lifetime withdrawalamount is based on the contract holder’s attained aged at the time of ratchet.

As time goes by in the withdrawal phase, withdrawals keep putting downward pressure onthe guaranteed withdrawal balance. Since the latter is used in the calculation of the ratch-eted lifetime withdrawal amount, its decrease reduces the likelihood of a ratchet late in thewithdrawal phase.

The revenue process in the withdrawal phase is very similar to the one in the accumulationphase. It is given by

Rtj = Atj(1− e−gA∆t

), ti ∈ ΩT and ti ≥ tW . (2.13)

19

Clearly, no revenues are collected once the account value is exhausted.

Let C = Cti , ti ∈ ΩT be the stochastic process of claim payments from the company’sperspective, that is to say, lifetime withdrawal payments to the contract holder when theaccount value is exhausted. The claim payments for ti ∈ ΩT are given by

Cti =

max

(1

nLtk∗−1

−Atk∗−1

FtiFti−1

e−gA∆t, 0

), ti = tk∗

1

nLti−1 , nti ∈ N and ti > tk∗

0, otherwise

, (2.14)

where

tk∗ =

inftk ∈ ΩT : Atk = 0, Aω−x = 0

ω − x, Aω−x > 0(2.15)

is the time at which the contract holder’s account value falls to zero if it does, or the timenecessary to reach age ω if it does not.

As made clear by (2.14), two events may lead to claims to the company. Firstly, the paymentthat reduces the contract holder’s account value to zero has to be partially paid by the com-pany. Second, all lifetime withdrawal payments after the account value is exhausted have tobe paid by the company.

The GLWB guarantee model presented above illustrates the optionality related with GLWBguarantees. Indeed, by issuing a GLWB guarantee, the company agrees to pay the maximumbetween zero and an unknown series of payments that is dependent on the account valueperformance, and thus on financial markets. GLWB guarantees are thus considered complexoptions.

2.3 Guarantee liability

Insurance companies make commitments to contract holders through GLWB guarantees. Assuch, they must maintain a liability to ensure that they can fulfill their obligations towardscontract holders. A liability in the general sense consists in the actuarial present value offuture claims minus the actuarial present value of future revenues.

Although the term liability is seldom used for plain vanilla options, the calculation of theGLWB guarantee liability is somewhat similar to the valuation of an option. For plain vanillaoptions, the option premium is paid as a lump sum, and so the liability associated with theoption consists only in expected future claims. The liability is then the expected discountedpayoffs of the option. GLWB guarantees, on the other hand, are not financed by a lump sumpremium. They are rather financed through a charge in percentage of the account value, theGLWB guarantee fee. Thus, the GLWB guarantee liability consists of two components. The

20

future claims component is the expected sum of the discounted lifetime withdrawal paymentsthat the company has to make once the contract holder’s account value is exhausted. Thefuture revenue component is the expected sum of the discounted revenues from the GLWBguarantee fee.

The dynamics of several important processes for the valuation of the GLWB guarantee, in-cluding A, G, L, R and C, were defined in section 2.2. In this section, these processes areused in the valuation of the GLWB guarantee liability.

2.3.1 Notation

In order to derive an expression for the guarantee liability, some generic notation regardingsome of the risk factors related with GLWB guarantees, that is the stock market, bond market,longevity and interest rate risks, shall be introduced. Specific models are not imposed at thispoint and will rather be the topic of chapter 4.

Stock market and bond market

The notation regarding the stock market index and the bond market index was already intro-duced in section 2.2, where

S = Sti , ti ∈ ΩT

andP = Pti , ti ∈ ΩT

were defined as the stochastic processes representing the stock market index and the bondmarket index, respectively. The notation used in this section is consistent with the notationdefined in section 2.2.

Mortality and longevity

In traditional actuarial valuations, mortality and longevity are often factored in through aconstant or projected deterministic mortality table. However, longevity risk, that is to say,the uncertainty around future mortality, is a systematic risk that is significant for GLWBguarantees. Hence, the notation introduced in this section takes into account the possibilityof stochastic mortality. Let

µ = µx,ti , ti ∈ ΩT

be the stochastic process of mortality forces, where µx,ti is the force of mortality that appliesbetween time ti and time ti+1 at age x.

Reflecting longevity risk through the introduction of stochastic mortality implies that thefuture values of µx,ti are not known with certainty. Longevity risk thus must be distinguishedfrom the mere randomness in future death times. Indeed, in this thesis, the fairly standard

21

actuarial assumption that the insurer’s portfolio is composed of a large population of similarrisks is made. Hence, conditional on µ, any remaining mortality risk is diversified away.

To sum up, assume that the random variable Z represents the present value of a fixed life-contingent stream of cash flows at a known interest rate. Then, using the assumption madeabove, the unconditional expectation of the present value of cash flows is given by

E[Z] = Eµ[E[Z|µ]

]= Eµ

[Z|µ

]. (2.16)

As shown in (2.16), the expectation is conditional on the mortality path, but once the mortalitypath is known then there is no further randomness in mortality.

Traditional actuarial notation can easily be modified to take into account stochastic mortality.For example, let tjp

(µ)

x,tibe the tj-year probability of survival for an individual aged x at time

ti given a mortality path µ such that

tjpx,ti = Eµ

[tjp

(µ)

x,ti

∣∣∣µ] .In this thesis, the terms mortality, mortality improvement and longevity will be used almostinterchangeably.

Interest rates

Interest rate risk also is a systematic risk that impacts the valuation of the GLWB guaranteeliability. The notation introduced thus once again considers the randomness in future interestrates. Let r(s), t0 < s < tω−x

∆tbe the short rate at time s. The short rate is such that, under

the risk-neutral Q-measure,

ZCM (t) = EQ[e−∫ t0 r(s)ds

],

where ZCM (t) is the t-year zero-coupon bond price observed on the market.

2.3.2 Guarantee liability valuation

As discussed previously, the guarantee liability can be split into two components: a futureclaims component and a future revenue component. Let Vt0 be the guarantee liability, V C

t0 bethe expected present value of future claims and V R

t0 be the expected present value of futurerevenues at time t0. Clearly, the relationship between these variables is given by

Vt0 = V Ct0 − V

Rt0 . (2.17)

Since the GLWB guarantee liability valuation is akin to the valuation of a complex option,the calculation of the guarantee liability is done under the risk-neutral Q-measure. However,because stochastic mortality is also included in the valuation, the usual single expectation is

22

replaced by a double expectation. The expression for the claims component of the guaranteeliability is given by

V Ct0 = Eµ

EQ

(ω−x)/∆t∑j=0

tjp(µ)x e

−tj∫t0

r(s)ds

Ctj

∣∣∣∣∣∣∣µ

= Eµ

EQ

(ω−x)/∆t∑j=k∗+1

tjp(µ)x e

−tj∫t0

r(s)ds 1

nLtj−11ntj∈N − tk∗p

(µ)x e

−tk∗∫t0

r(s)ds

Btk∗

∣∣∣∣∣∣∣µ ,(2.18)

where

Btk∗ = Atk∗−1

Ftk∗Ftk∗−1

e−gA∆t − 1

nLtk∗−1

(2.19)

and tk∗ is the time at which the account value is exhausted as defined in (2.15).

The three systematic risk factors considered are easily identified in (2.18). First of all, claimcash flows are related to the market performance through the GLWB guarantee model pre-sented in section 2.2. Secondly, the survival probabilities that are applied to claim cash flowsare random based on the future mortality path. Finally, future interest rates are also random,which affects both the average market return and the discount factor under the risk-neutralmeasure.

Moreover, as is shown in (2.18), there is no account for lapses in the claims component. In thepresent work, the guarantee is valued using the passive contract holder behaviour assumption,which implies that contract holders do not make any excess withdrawals and do not lapsethe guarantee. Qualitative arguments for the passive withdrawal behaviour assumption arepresented in Shah and Bertsimas (2008). Moreover, Azimzadeh et al. (2014) show that for alarge family of utility functions, the consumption-optimal strategy is close to withdrawing atthe contractual rate. In all cases, the modeling of lapses is in no way a trivial issue and is notthe focus of this thesis.

The future revenue component of the guarantee liability at time t0 is given by

V Rt0 = Eµ

EQ

(ω−x)/∆t∑j=0

tjp(µ)x Rtj e

−tj∫t0

r(s)ds

∣∣∣∣∣∣∣µ

= Eµ

EQ

(k∗−1)∑j=0

tjp(µ)x Atj

(1− e−gA∆t

)e−tj∫t0

r(s)ds

∣∣∣∣∣∣∣µ . (2.20)

This expression once again emphasizes that the company has no revenues once the accountvalue is exhausted.

23

2.3.3 Guarantee liability valuation through simulation

The valuation of GLWB guarantees has been shown to be akin to that of complex options.Many types of options can be valued using closed-form formulas. However, as was shown inthis chapter, GLWB guarantees have characteristics that distinguish them from standard oreven mildly exotic options. These characteristics include contractual features such as ratchetsand roll-ups, but also flexibility in terms of fund choices, which can include bond funds anddiversified funds in addition to the traditional stock market funds. Therefore, in order tomodel realistic product designs, it is often necessary to use Monte Carlo simulation to valueGLWB guarantee liabilities. Monte Carlo simulation is used in most of the GLWB guaranteeliability valuation literature (see e.g. Kling et al. (2011), Holz et al. (2012) and Shah andBertsimas (2008)). For a brief overview of Monte Carlo simulation from a general optionvaluation perspective, see section A.2 in the appendix.

In section 2.2, several processes related with the dynamic of the GLWB guarantee were defined:S, P , µ, F , FS , FP A, G, L, R and C. For a given process denoted by X, let

X(k) = X(k)ti, ti ∈ ΩT

be the kth stochastic realization of the process under the risk-neutral Q-measure. Let also

r(k) = r(k)(s), t0 < s < ω − x

denote the kth realization under the risk-neutral Q-measure of the short rate path. Finally, letVC,(k)t0

, V R,(k)t0

and V (k)t0

denote the realizations in risk-neutral scenario k of the decrementedpresent value of claims, the decremented present value of revenues and the decremented presentvalue of claims minus revenues respectively.

The valuation of the GLWB guarantee liability at time t0 by Monte Carlo simulation usingNI risk-neutral scenarios is presented in Algorithm 2.1.

Algorithm 2.1. Valuation of the GLWB guarantee liability at time t0

1. Simulate, under the Q-measure, stochastic realizations of the following processes:

1.1. Stock market process: S(1), . . . , S(NI);

1.2. Bond market process: P (1), . . . , P (NI);

1.3. Short rate path: r(1), . . . , r(NI);

1.4. Mortality process: µ(1), . . . , µ(NI).

2. Using the realizations in step 1, compute realizations of the following processes:

2.1. Stock mutual fund value process (using (2.2)): FS(1), . . . , FS(NI);

24

2.2. Bond mutual fund value process (using (2.3)): FP (1), . . . , FP(NI);

2.3. Total mutual fund value process (using (2.4)): F (1), . . . , F (NI);

2.4. Account value process (using (2.5) and (2.10)): A(1), . . . , A(NI);

2.5. Guaranteed withdrawal balance process (using (2.6), (2.7) and (2.11)): G(1), . . . , G(NI);

2.6. Lifetime withdrawal amount process (using (2.9) and (2.12)): L(1), . . . , L(NI).

3. Using the realizations in step 2, (2.17), (2.18) and (2.20), compute realizations of thefollowing liability-related values:

3.1. Future claims at t0: VC,(1)t0

, . . . , VC,(NI)t0

;

3.2. Future revenues at t0: VR,(1)t0

, . . . , VR,(NI)t0

;

3.3. Guarantee liability at t0: V(1)t0, . . . , V

(NI)t0

.

4. Compute the estimator for the guarantee liability at time t0:

Vt0 =1

NI

NI∑j=1

V(j)t0.

Algorithm 2.1 thus consists in simulating several paths of the relevant risk factors and usingthe GLWB guarantee model and the expression for the guarantee liability to compute theliability in each of these paths.

The simulation process for a given risk-neutral scenario can be illustrated by an example.Assume that a contract holder elects a 5-year accumulation period and quarterly withdrawals,and that a single initial deposit of $100,000 is made in the contract. The contract specifiesthat 5% roll-ups are applied on the guaranteed withdrawal balance and that the withdrawalrate at age 70 is 5.5%. Then, for a given risk-neutral scenario, the projection may look likethe one presented in Table 2.1.

As shown in Table 2.1, the guaranteed withdrawal balance is increased yearly as a result ofroll-ups. Starting at time 5, the contract holder receives an annual lifetime payment of 7,020,computed as the product of the 5.5% withdrawal rate and of the guaranteed withdrawalbalance at time 5. Withdrawal payments reduce the account value to zero at time 18, time atwhich the company stops receiving revenues from the guarantee fee. The first claim paymentis lower than the following payments, since the company can use the remaining account valueto pay a part of this first payment. Then, the claim payments correspond to a fourth of thelifetime withdrawal amount since quarterly withdrawals are made. The claim and revenue cashflows illustrated in Table 2.1 would then be combined with the discount factors and survivalprobabilities to obtain one stochastic realization of the guarantee liability.

25

ti Sti Pti Fti Ati Gti Lti Rti Cti tipx,t0 e−ti∫0

r(s)ds

0.00 100.0 100.0 100,000 100,000 100,000 0 374.30 0.00 1.000 1.0000.25 104.3 96.9 100,591 100,214 100,000 0 375.10 0.00 0.996 0.9970.50 116.5 100.9 108,611 107,799 100,000 0 403.49 0.00 0.992 0.9920.75 106.8 97.5 100,763 99,636 100,000 0 372.94 0.00 0.987 0.9891.00 113.8 97.9 104,225 102,673 105,000 0 384.30 0.00 0.983 0.9861.25 113.4 100.0 104,136 102,202 105,000 0 382.54 0.00 0.979 0.9811.50 115.6 99.9 104,499 102,174 105,000 0 382.43 0.00 0.974 0.9761.75 110.1 101.3 101,376 98,749 105,000 0 369.62 0.00 0.970 0.9712.00 98.2 102.5 94,794 91,992 110,250 0 344.33 0.00 0.966 0.966...

......

......

......

......

......

4.75 120.0 104.7 99,585 92,736 121,551 0 347.11 0.00 0.912 0.9025.00 126.4 103.8 101,880 94,518 127,628 7,020 353.78 0.00 0.907 0.8985.25 123.4 106.1 100,522 91,155 125,873 7,020 341.19 0.00 0.901 0.8925.50 111.7 106.6 94,414 83,541 124,118 7,020 312.69 0.00 0.896 0.8845.75 122.0 108.8 99,425 85,890 122,363 7,020 321.49 0.00 0.891 0.8756.00 117.4 111.9 97,700 82,329 120,609 7,020 308.16 0.00 0.885 0.8666.25 115.4 113.1 96,386 79,163 118,854 7,020 296.31 0.00 0.880 0.8576.50 112.8 115.4 95,181 76,126 117,099 7,020 284.94 0.00 0.874 0.8496.75 113.6 115.8 94,998 73,940 115,344 7,020 276.76 0.00 0.868 0.8417.00 118.3 116.3 96,763 73,277 113,589 7,020 274.27 0.00 0.862 0.835...

......

......

......

......

......

17.50 234.8 269.3 152,311 1,778 39,884 7,020 6.66 0.00 0.562 0.32517.75 265.3 275.2 164,098 154 38,129 7,020 0.58 0.00 0.554 0.31418.00 276.1 289.4 170,182 0 36,374 7,020 0.00 1,595.97 0.545 0.30318.25 248.3 291.9 159,300 0 34,619 7,020 0.00 1,754.89 0.537 0.29218.50 239.1 299.6 156,568 0 32,864 7,020 0.00 1,754.89 0.528 0.28118.75 269.4 322.9 171,684 0 31,109 7,020 0.00 1,754.89 0.519 0.270...

......

......

......

......

......

Table 2.1: Illustration of a risk-neutral scenario used in the valuation of the GLWB guaranteeliability at time 0

2.4 Conclusion

In this chapter, the GLWB guarantee model considered in this thesis was presented. Thismodel, along with some generic notation about some of the risk factors of GLWB guarantees,leads to a generic expression for the GLWB guarantee liability, that is to say, an expressionthat is not based on the specific models used in the valuation. Finally, since closed-formformulas are usually not available to model realistic GLWB guarantee product designs, theapproximation of the GLWB guarantee liability valuation through Monte Carlo simulationwas presented and illustrated by an example.

26

Chapter 3

Nested stochastic projection of GLWBguarantees

3.1 Introduction

In chapter 2, the guarantee liability valuation at contract inception was presented. As men-tioned in section 2.1, the guarantee liability valuation is the foundation on which to build therisk assessment of hedged GLWB guarantees. However, the guarantee liability at time t0 doesnot suffice to evaluate the effectiveness of a hedging strategy. Indeed, in order to evaluate thequality of a risk mitigation strategy, the guarantee liability and the hedging strategy must beprojected at each point in time in many scenarios to obtain the residual gains and losses thatremain under the strategy.

Since the quality of the hedge is computed over many stochastic scenarios and the guaranteeliability is itself valued through simulation, the risk assessment of hedged GLWB guaranteesis a nested stochastic (or stochastic-on-stochastic) calculation. In this nested stochastic cal-culation, the scenarios over which the quality of the hedge is assessed are referred to as outerloop scenarios, whereas the scenarios used to value the guarantee liability are referred to asinner loop scenarios.

In this chapter, the first topic of interest is the projection of the GLWB guarantee liabilityat future points in time in a given outer loop scenario. This calculation is then extended tothe projection of sensitivities on the guarantee liability value at future points in time. Thesesensitivities are required in defining the hedging strategy considered in this thesis.

Note that in this chapter and throughout the present work, the word “projection” is used todenote the fact that the guarantee liability is computed at future points in time in an outerloop scenario. The word “projection” is differentiated from the word “simulation”, which willoften be used to refer to the calculation of the guarantee liability at a given point in time.

27

3.2 Guarantee liability

As for the GLWB guarantee liability at contract inception, three stochastic processes relatedto the guarantee liability value are defined.

LetV P = V P

ti , ti ∈ ΩT

be the stochastic process of the decremented guarantee liability, that is to say, the guaranteeliability required for the fraction of contract holders that survived up to the valuation time.Let also

V P,C = V P,Cti

, ti ∈ ΩT

be the stochastic process of the decremented present value of future claims and

V P,R = V P,Rti

, ti ∈ ΩT

be the stochastic process of the decremented present value of future revenues. Superscript P isused although the guarantee liability is calculated under the risk-neutral measure to stress thatthese processes are composed of liability values at all time steps over an outer loop scenario,and not of the individual realizations that are averaged to obtain the guarantee liability as inAlgorithm 2.1. This distinction will be helpful when presenting algorithms for the projectionof the guarantee liability in time.

Once again, the relationship between these random variables is given by

V Pti = V P,C

ti− V P,R

ti, ti ∈ ΩT . (3.1)

The guarantee liability at time t0 presented in chapter 2 is the starting point for the guaranteeliability process. However, as mentioned above, the assessment of a risk management strategyrequires going further than the initial guarantee liability. The formulas developed in chapter2 are thus extended to allow for the calculation of the liability at future points in time.

Let F be the filtration on financial data such that

F = Fti , ti ∈ ΩT ,

whereFti = σ

(Su, Pu, ZC(u, u+ s), u, s|u ∈ ΩT , u ≤ ti, s > 0

)and ZC(u, u+ s) is the s-year zero-coupon bond price generated from the interest rate modelat time u. For a review of basic probability theory, please refer to Appendix B.

Let also G be the filtration on mortality data such that

G = Gti , ti ∈ ΩT ,

28

whereGti = σ

(µx+u,u, u ∈ ΩT and u ≤ ti

).

The claims component of the guarantee liability at time ti is then given by

V P,Cti

=Eµ

tip(µ)

x,t0EQ

(ω−x)/∆t∑j=max(k∗,i)+1

tj−tip(µ)

x+ti,tie−tj∫ti

r(s)ds 1

nLtj−11ntj∈N

−k∗−tip(µ)

x+ti,tie−tk∗∫ti

r(s)ds

Btk∗1ti<tk∗

∣∣∣∣∣∣∣µ,Fti∣∣∣∣∣∣∣Gti

, ti ∈ ΩT , (3.2)

where Btk∗ is defined in (2.19). Notice in (3.2) that, as defined above, the guarantee liability

is decremented by the survival probability tip(µ)

x,t0, which is known with certainty given Gti .

The future revenue component at time ti ∈ ΩT is given by

V P,Rti

=

Eµ

tip(µ)

x,t0EQ

(k∗−1)∑j=i

tj−tip(µ)

x+ti,tiAtj

(1− e−gA∆t

)e−tj∫ti

r(s)ds

∣∣∣∣∣∣∣µ,Fti∣∣∣∣∣∣∣Gti

, ti < tk?

0, ti ≥ tk?

.

(3.3)When ti ≥ tk∗ , that is to say, when the account value is exhausted, the future claims componentof the guarantee liability reduces to

V P,Cti

= Eµ

tip(µ)

x,t0EQ

(ω−x)/∆t∑j=i+1

tj−tip(µ)

x+ti,tie−tj∫ti

r(s)ds 1

nLtj−11ntj∈N

∣∣∣∣∣∣∣µ,Fti∣∣∣∣∣∣∣Gti

,and the revenue component is equal to zero. Then, since the withdrawal amount can no longerbe increased through a ratchet, no further simulations of the stochastic processes A, G and Lare necessary.

The projection of the guarantee liability at future points in time can be summarized in algo-rithms, which will help in interpreting the link between outer and inner loop scenarios. For agiven stochastic process denoted by X, let

XP,(k) =X

P,(k)ti

, ti ∈ ΩT

be the kth stochastic realization of the process under the real-world P-measure. Let also

ZCP,(k) =ZCP,(k)(ti, ti + s), ti ∈ ΩT and s > 0

denote the kth stochastic realization of the zero-coupon bond prices process under the real-world P-measure.

The algorithm for the projection of the guarantee liability in the kth outer loop scenario ispresented next.

29

Algorithm 3.1. Projection of the guarantee liability in outer loop scenario k

1. Simulate, under the real-world P-measure, an outer loop stochastic realization of thefollowing processes:

1.1. Stock market process: SP,(k);

1.2. Bond market process: P P,(k);

1.3. Zero-coupon bonds process: ZCP,(k);

1.4. Mortality process: µP,(k).

2. Using the realizations in step 1, compute an outer loop realization of the followingprocesses:

2.1. Stock mutual fund value process (using (2.2)): FSP,(k);

2.2. Bond mutual fund value process (using (2.3)): FP P,(k);

2.3. Total mutual fund value process (using (2.4)): F P,(k);

2.4. Account value process (using (2.5) and (2.10)): AP,(k);

2.5. Guaranteed withdrawal balance process (using (2.6), (2.7) and (2.11)): GP,(k);

2.6. Lifetime withdrawal amount process (using (2.9) and (2.12)): LP,(k);

2.7. Claims process (using (2.14)): CP,(k);

2.8. Revenue process (using (2.8) and (2.13)): RP,(k).

3. Using the quantities in step 2 and Algorithm 3.2 presented below, compute a realizationof the guarantee liability process V P,(k).

Algorithm 3.1, along with (3.2) and (3.3), allows to make crucial interpretations about therelationship between the outer loop scenarios, which represent potential paths of the futureover which to assess the risk management strategy, and the inner loop scenarios, which areused to value the guarantee liability. At a given guarantee liability valuation date ti in outerloop scenario k, past information in the outer loop scenario is feeding σ-algebras Fti and Gti .More formally,

Fti = σ(SP,(k)u , P P,(k)

u , ZCP,(k)(u, u+ s), u, s|u ∈ ΩT , u ≤ ti, s > 0)

andGti = σ

(µP,(k)x+u,u, u ∈ ΩT and u ≤ ti

).

Thus, σ-algebras Fti and Gti contain all information about the stock index, the bond index,the zero-coupon bonds and the force of mortality up to time ti in the outer loop scenario. The

30

past values of these processes alone are sufficient to determine all other relevant quantities forthe valuation of the GLWB guarantee liability at time ti, that is to say, F P,(k)

S,ti, F P,(k)

P,ti, F P,(k)

ti,

AP,(k)ti

, GP,(k)ti

and LP,(k)ti

. The market and mortality data at time ti in the outer loop scenariomust therefore be seen as the starting point for the inner loop guarantee liability valuation.

The steps required to value the guarantee liability at time ti are presented in Algorithm 3.2.

Algorithm 3.2. Inner loop guarantee liability valuation at time ti

1. Simulate, under the risk-neutral Q-measure, inner loop stochastic realizations of thefollowing processes:

1.1. Stock market process: S(1)tj|Fti , . . . , S

(NI)tj|Fti , tj ∈ ΩT and tj > ti;

1.2. Bond market process: P (1)tj|Fti , . . . , P


1.3. Short rate paths: r(1)(s)|Fti , . . . , r(NI)(s)|Fti , ti < s < ω − x;

1.4. Mortality process: µ(1)x+tj ,tj

|Gti , . . . , µ(NI)x+tj ,tj

|Gti , tj ∈ ΩT and tj > ti.

2. Using the realizations in step 1, compute, for tj ∈ ΩT and tj > ti, inner loop realizationsof the following processes:

2.1. Stock mutual fund value process (using (2.2)): F (1)S,tj|Fti , . . . , F

(NI)S,tj|Fti ;

2.2. Bond mutual fund value process (using (2.3)): F (1)P,tj|Fti , . . . , F

(NI)P,tj|Fti ;

2.3. Total mutual fund value process (using (2.4)): F (1)tj|Fti , . . . , F

(NI)tj|Fti ;

2.4. Account value process (using (2.5) and (2.10)): A(1)tj|Fti , . . . , A

(NI)tj|Fti ;

2.5. Guaranteed withdrawal balance (using (2.6), (2.7) and (2.11)): G(1)tj|Fti , . . . , G

(NI)tj|Fti ;

2.6. Lifetime withdrawal amount process (using (2.9) and (2.12)): L(1)tj|Fti , . . . , L

(NI)tj|Fti .

3. Using the realizations in step 2, (3.1), (3.2) and (3.3), compute the following liability-related values:

3.1. Stochastic realizations of the guarantee liability V (1)ti, . . . , V

(NI)ti

;

3.2. Estimate for the guarantee liability: V P,(k)ti

= 1NI

NI∑j=1

V(j)ti

.

Algorithm 3.2 is used to compute the guarantee liability at a single point in time. Therefore,it must be used at each point in time in an outer loop scenario in order to obtain a realizationof the guarantee liability process V P,(k).

Now that the link between the outer and inner loop scenarios has been established throughthe calculation of the guarantee liability at future points in time, a natural extension that will

31

help in articulating the risk management strategy considered in this thesis is the calculationof the guarantee liability sensitivities. The calculation of sensitivities at future points in timeis the topic of the next section.

3.3 Guarantee liability greeks

The risk factors considered in this thesis are the stock market, the bond market, interestrates and longevity. The main risks hedged by North American insurers are the first orderof market and interest rate risk, which are denoted by delta and rho respectively (TowersWatson (2013)). Longevity risk, on the other hand, is not commonly hedged due to the lackof instruments allowing to do so in North America. Moreover, Ngai and Sherris (2011) showsthat the hedging of longevity risk may not be very efficient for GLWB guarantees because ofthe market risk involved. Hence, the hedging strategy considered in this thesis only aims tomitigate the risk related with the first-order sensitivity of the guarantee liability to the stockmarket, the bond market and interest rates.

In order to apply this risk mitigation strategy, the sensitivities of the guarantee liability tothe various risk factors must be computed at each point in time in the outer loop scenarios.These sensitivities are commonly known in the financial literature as “greeks”.

3.3.1 Liability delta

The first-order sensitivity of the guarantee liability V Pti to the stock market index is given by

∆LS,ti ≡

∂V Pti

∂SPti

=∂V P

ti

∂APti

∂APti

∂F Pti

∂F Pti

∂F PS,ti

∂F PS,ti

∂SPti

, ti ∈ ΩT .

Similarly, the first-order sensitivity of the guarantee liability to the bond market index is givenby

∆LP,ti ≡

∂V Pti

∂P Pti

=∂V P

ti

∂APti

∂APti

∂F Pti

∂F Pti

∂F PP,ti

∂F PP,ti

∂P Pti

, ti ∈ ΩT .

The calculation of sensitivities suffers from the same problem as the valuation of the guaranteeliability: the complexity of GLWB guarantees when considering realistic product designs usu-ally does not allow the computation of sensitivities using closed-form formulas. Approximationtechniques thus have to be relied upon.

Several methods can be used to evaluate the guarantee liability sensitivities. Among others,the bump and revalue approach, the pathwise estimator and the likelihood ratio estimator arepotential techniques to estimate these sensitivities (see e.g. Cathcart et al. (2015)). Cathcartet al. (2015) shows that the bump and revalue approach provides reliable estimates of the first

32

order greeks. Since only first order greeks are used in this thesis and the bump and revalueapproach is the most straightforward to implement, it is preferred to the other approaches.

The bump and revalue approach combines Monte Carlo simulation and finite difference tech-niques to estimate the guarantee liability sensitivities (see sections A.2 and A.3 in the appendixfor more details). As shown in section A.3, several estimators can be considered when usingfinite difference techniques. The most precise estimator of the first-order sensitivity is thecentral-difference estimator. However, each sensitivity calculation then requires two additionalguarantee liability valuations instead of only one with the forward and backward-differenceestimators. Given the large runtime involved in the hedging calculations, a one-sided finitedifference method is seen as more appropriate.

Kling et al. (2011) mentions choosing an estimator that is consistent with the direction of therisk. A similar approach is used in this thesis, and hence the backward-difference estimatoris selected. The first-order sensitivities to the stock market and the bond market indices arethus given by

∂V Pti

∂SPti

≈V Pti (S

Pti , P

Pti , . . . )− V

Pti (S

Pti − h, P

Pti , . . . )

h

and∂V P

ti

∂P Pti

≈V Pti (S

Pti , P

Pti , . . . )− V

Pti (S

Pti , P

Pti − h, . . . )

h,

where V Pti (S

Pti , P

Pti , . . . ) ≡ V

Pti , V

Pti (S

Pti − h, P

Pti , . . . ) is the guarantee liability when a downward

shock of size h is applied to the stock index at time ti and V Pti (S

Pti , P

Pti−h, . . . ) is the guarantee

liability when a downward shock of size h is applied to the bond index at time ti.

An appropriate value for h, the shock used in the computation of sensitivities, needs to bedetermined. Since the stock and bond index levels can change quite significantly with time,shocks that are proportional to their values are used. The proportional shock size is denotedby ε, such that h = εSti and h = εPti are used in the valuation of the stock index and thebond index deltas respectively at time ti.

The valuation of V Pti (S

Pti−εS

Pt , P

Pti , . . . ) and V

Pti (S

Pti , P

Pti−εP

Pt , . . . ) is thus required at each point

in time in outer loop scenarios. The steps involved are similar to the base guarantee liabilitycalculation. However, a shock must be applied on the initial value of the stock (or bond) indexand every related quantities. It is important to stress that despite a shock is applied on thestock or bond indices’ initial value, projected returns must be the exact same in the shockedliability valuation as in the base liability valuation. Indeed, using identical returns not onlyspares computation time, but also allows to obtain a convergent delta estimate with muchfewer scenarios.

The steps required for the calculation of the stock index delta at time ti are provided in Algo-rithm 3.3. The bond delta calculation is omitted as it is analog to the stock delta calculation.

33

Algorithm 3.3. Inner loop stock index delta valuation at time ti

1. Apply a shock such as to modify the relevant quantities at time ti:

1.1. The stock mutual fund value becomes (1− ε)F P,(k)S,ti

;

1.2. The total mutual fund value becomes (1− ε)F P,(k)S,ti

+ FP,(k)P,ti

;

1.3. The account value becomes(1−ε)F P,(k)

S,ti+F

P,(k)P,ti

FP,(k)ti

AP,(k)ti

.

2. Using the same stock market returns, bond market process, short rate paths and mortal-ity process as in step 1 of Algorithm 3.2 and the shocked quantities at time ti determinedin step 1, compute, for tj ∈ ΩT and tj > ti, inner loop realizations of the following pro-cesses:

2.1. Stock mutual fund value process (using (2.2)): F (1)S,tj|Fti , . . . , F

(NI)S,tj|Fti ;


(NI)P,tj|Fti ;


(NI)tj|Fti ;


(NI)tj|Fti ;

2.5. Guaranteed withdrawal balance process (using (2.6), (2.7) and (2.11)): G(1)tj|Fti , . . . , G

(NI)tj|Fti ;


(NI)tj|Fti .

3. Using the values computed in step 2, (3.1), (3.2) and (3.3), compute the following values:

3.1. Stochastic realizations of the shocked guarantee liability V (1),S−

ti, . . . , V

(NI),S−

ti;

3.2. Estimate for the shocked guarantee liability: V P,S−ti

= 1NI

NI∑j=1

V(j),S−

ti;

3.3. Estimate for the guarantee stock market delta: ∆LS,ti

=V Pti−V P,S−

ti

εSP,kti

.

3.3.2 Liability rho

The rho calculation is slightly more involved than the delta calculation, because the guaranteeliability valuation is done with a full interest rate curve rather than a single interest rate. Aninterest rate curve implies multiple sources of randomness in interest rate changes. Indeed,there is plenty of evidence that actual interest rate curve changes do not only include parallelshifts, but also increases and decreases in the steepness and curvature of interest rates. More-over, the interest rate models considered in this thesis allow the possibility for non-parallelshifts. A parallel shock on the interest rate curve is hence not sufficient to fully capture theinterest rate risk. The hedging strategy considered in this thesis is therefore more refined, andso must be the sensitivity calculations.

34

On the other hand, shocking every single point on the interest rate curve to obtain rho sen-sitivities is not appropriate either. First of all, run time becomes a major concern for sucha large number of shocks. Moreover, and most importantly, points on the interest rate curveare not independent. Indeed, principal component analysis can be used to show that most ofthe variability in interest rates can be explained by two or three independent factors on theyield curve (see e.g. Litterman and Scheinkman (1991)). Thus, a reasonably low number ofshocks can be expected to be required to successfully hedge the guarantee’s rho exposure.

Key-rate rhos are used to obtain sensitivities that consider the entire interest rate curvewithout shocking every single point. Each of the key-rate rhos is centered around a givenmaturity on the interest rate curve and aims to capture the sensitivity of one section of thecurve. For example, the key-rate rho shocks on the spot curve assuming a 3-rho hedgingstrategy at maturities 2, 10 and 30 years are illustrated in Figure 3.1.

0 10 20 30 40

Maturity on the spot curve

Sho

ck o

n th

e in

tere

st r

ate

curv

e

2−year rho10−year rho30−year rho

Figure 3.1: Key-rate rho shocks on the spot curve

The sum of all three shocks illustrated in Figure 3.1 is equal to a parallel shock on the spotcurve. The key-rate shocks thus cover both the entirety and independent subsections of theinterest rate curve, providing a more thorough assessment of the interest rate sensitivity.Moreover, this method does not overly rely on the model used in the guarantee liabilityvaluation to determine the sensitivity to interest rates. Indeed, sensitivities are calculatedindependently of the guarantee liability model, which allows a more flexible and realisticassessment of the risk.

35

As for the delta calculation, the shock size must be determined. Let cTj (u), j = 1, . . . , nc, u > 0

denote the key-rate rho shock centered around maturity Tj , where nc is the number of rhoshocks and c′ is the size of the rho shocks. The triangular rho shocks illustrated in Figure 3.1can be defined in a more formal way as

cTj (u) =

c′, j = 1, u ≤ Tj

max

(Tj+1 − uTj+1 − Tj

c′, 0

), j < nc, u > Tj

max

(u− Tj−1

Tj − Tj−1c′, 0

), j > 1, u ≤ Tj

c′, j = nc, u > Tj

.

Rho shocks are applied on the continuously compounded spot curve. Let

sP(ti, ti + u), ti ∈ ΩT and u > 0

denote the continuously compounded spot rate of maturity u at time ti before any shock isapplied on the spot curve. Moreover, recall that the price of the zero-coupon bond of maturityu at time ti is denoted by ZCP (ti, ti + u) , ti ∈ ΩT , u > 0. The spot rate is then given by

sP(ti, ti + u) = −1

uln(ZCP (ti, ti + u)

), ti ∈ ΩT and u > 0.

Let alsosP,cTj (ti, ti + u), ti ∈ ΩT and u > 0

denote the continuously compounded spot rate of maturity u at time ti after a downwardkey-rate rho shock cTj is applied on the spot curve. The spot curve after the key-rate rhoshock is given by

sP,cTj (ti, ti + u) = sP(ti, ti + u)− cTj (u), ti ∈ ΩT and u > 0.

The price of the zero-coupon bond of maturity u at time ti after the key-rate rho shock,ZC

P,cTj (ti, ti + u), is then given by

ZCP,cTj (ti, ti + u) = e−u×s

P,cTj (ti,ti+u), ti ∈ ΩT and u > 0.

Finite difference techniques are once again used in the calculation of sensitivities. The key-raterho centered around maturity Tj is given by

ρti,Tj ≈V Pti (S

Pti , P

Pti , ZC

P (ti, ti + u))− V Pti (S

Pti , P

Pti , ZC

P,cTj (ti, ti + u))

c′,

where V Pti (S

Pti , P

Pti , ZC

P (ti, ti + u)) ≡ V Pti and V P

ti (SPti , P

Pti , ZC

P,cTj (ti, ti + u)) is the guaranteeliability calculated using the shocked zero-coupon bond curve.

36

An additional algorithm detailing the steps involved in the key-rate rho calculations is required.As for the delta calculations, care must be taken in determining the risk-neutral scenarios. Inthis case, a change in the initial zero-coupon bond prices implies a change in the resulting shortrate paths under the risk-neutral measure, which in turn affects the stock market returns, thebond market returns and the discounting factors. However, in order to help the convergenceof the rho calculation, the exact same random numbers must be used in the valuation of theshocked guarantee liability as in the valuation of the base guarantee liability, such that theonly difference between these liability values is related to the change in the initial zero-couponbond prices. Algorithm 3.4 details the necessary steps for the rho calculation.

Algorithm 3.4. Inner loop key-rate rho valuation at time ti

1. Simulate, under the risk-neutral Q-measure, using the same random numbers as instep 1 of Algorithm 3.2 and the shocked zero-coupon bond prices ZCP,cTj (ti, ti + u) ,

ti ∈ ΩT and u > 0, inner loop stochastic realizations of the following processes:

1.1. Stock market process: S(1)tj|Fti , . . . , S


1.2. Bond market process: P (1)tj|Fti , . . . , P


1.3. Short rate paths: r(1)(s)|Fti , . . . , r(NI)(s)|Fti , ti < s < ω − x.

2. Using the realizations in step 1 and the same mortality process as determined in step 1 ofAlgorithm 3.2, compute, for tj ∈ ΩT and tj > ti, inner loop realizations of the followingprocesses:

2.1. Stock mutual fund values process (using (2.2)): F (1)S,tj|Fti , . . . , F

(NI)S,tj|Fti ;


(NI)P,tj|Fti ;


(NI)tj|Fti ;


(NI)tj|Fti ;

2.5. Guaranteed withdrawal balance (using (2.6), (2.7) and (2.11)): G(1)tj|Fti , . . . , G

(NI)tj|Fti ;


(NI)tj|Fti .

3. Using the realizations in step 2, (3.1), (3.2) and (3.3), compute the following values:

3.1. Stochastic realizations of the shocked guarantee liability V(1),cTjti

, . . . , V(NI),cTjti

;

3.2. Estimate for the shocked guarantee liability: VP,cTjti

= 1NI

NI∑j=1

V(j),cTjti

;

3.3. Estimate for the key-rate rho: ρLcTj ,ti =V Pti−V

P,cTjtic′ .

37

The liability sensitivities are required to project the hedging strategy in the outer loop scenar-ios. Hence, step 3 of the outer loop projection algorithm (Algorithm 3.1) must be modified toinclude steps requiring to compute a realization of the stock delta process and the bond deltaprocess using Algorithm 3.3 and to compute a realization of each of the key-rate rho processesusing Algorithm 3.4.

3.4 Conclusion

In this chapter, the projection of the guarantee liability and of its sensitivities at futurepoints in time in an outer loop scenario was presented from a general perspective, that is tosay, without making any assumptions on particular models to use. Sensitivities to the stockmarket (delta), the bond market (delta) and interest rates (rho) were considered, as they willbe leveraged in the risk mitigation strategy assumed in the present work. Since no closed-formformulas are available, the bump and revalue approach is used to estimate the sensitivities.

This chapter also stressed some important considerations in the projection of the guaranteeliability and its sensitivities along outer loop scenarios. First of all, the information in theouter loop scenario must be seen as feeding the σ-algebras that become the starting point ofthe guarantee liability valuation. Moreover, ensuring consistency between the valuation of thebase guarantee liability and the shocked guarantee liabilities is crucial in obtaining convergentsensitivity estimates.

38

Chapter 4

Models

4.1 Introduction

Chapters 2 and 3 are important building blocks for the risk assessment of hedged GLWBguarantees, as they introduce the guarantee liability valuation and the calculation of theguarantee liability and greeks at future points in time. The formulas and algorithms presentedin these chapters are intentionally generic, avoiding reliance on particular models. However,in order to make a realistic assessment of the residual risk of hedged GLWB guarantees,appropriate models need to be selected for each of the relevant risk factors, for both the outerand inner loop valuations.

Moreover, one aspect of the GLWB risk assessment that is of particular interest in this thesisis how models used in the guarantee liability valuation impact the hedge effectiveness. Thisanalysis is carried out with respect to several systematic risks of GLWB guarantees, namelythe stock market, interest rate and longevity risks. Thus, for all risk factors considered in thisanalysis, at least two models are presented for the guarantee liability valuation, with simplermodels being specific cases of more complex models.

In this chapter, the financial market models, that is to say, the stock market, bond marketand interest rate models are presented first, followed by the mortality models.

4.2 Financial market models

The financial market is assumed to consist of the following assets:

– Risk-free asset Mt;

– Stock market index St;

– Bond market index Pt;

– Zero-coupon bonds of all maturities ZC(t, t+ u), u > 0.

39

Recall that in the risk assessment of hedged GLWB guarantees, models for both the outer andinner loop scenarios are required. The outer loop scenarios are the scenarios over which thehedging strategy is assessed, whereas the inner loop scenarios are the scenarios used for theguarantee liability valuation. These scenarios are respectively under the real-world P-measureand the risk-neutral Q-measure, and models under both these measures are thus consideredin this chapter.

Care is taken to maintain consistency between models used under the real-world and risk-neutral measures. Indeed, for each risk factor, the model used under the real-world measureis equivalent to the most complex risk-neutral model considered. Moreover, the other risk-neutral models considered are specific cases of the most complex risk-neutral model, thusensuring consistency throughout all models used.

For all risk factors, the change from one measure to the other is briefly discussed in this section.The models considered introduce stochastic interest rates and volatility in the market, whichgenerally implies that the market is incomplete (see e.g. Bjork (2009)). There are thereforean infinite number of equivalent martingale measures. As is usually the case, the change ofmeasure selected preserves the model form.

4.2.1 Outer loop

The real-world P-measure is used for outer loop scenarios, over which the efficiency of thehedging strategy is assessed. As shown in Algorithm 3.1, the real-world interest rate model isexpected to provide all zero-coupon bond prices at all future points in time in the outer loopscenarios. Moreover, the real-world stock and bond market models are expected to providethe stock market index and bond market index prices, respectively. These zero-coupon bond,stock market index and bond market index prices then become part of the filtration that isused as a starting point for the valuation of the guarantee liability and its sensitivities at eachpoint in time in the outer loop scenarios.

In this section, the real-world interest rate, stock market and bond market models consideredin this thesis are presented.

Interest rates: Two-factor extended Vasicek model

As mentioned above, the real-world interest rate model must provide the zero-coupon bondprices at each point in time in outer loop scenarios. Because the assessment of hedgingefficiency deals with both the real-world P-measure and the risk-neutral Q-measure, care mustbe taken in using the appropriate measure in a given calculation. Indeed, although the real-world dynamic must be used to simulate from the model in outer loop scenarios, bond pricesare the expectation under the risk-neutral measure of the future discount factor. Thus, in thissection, emphasis is put on the appropriate measure to use in each context.

40

The real-world interest rate model considered in this thesis is the well-known two-factor ex-tended Vasicek model. This model was selected because it contains more than one source ofrandomness in interest rates, it is analytically tractable, and it is relatively easy to implement.Moreover, Brigo and Mercurio (2006) deems a two-factor Gaussian model suitable for practi-cal applications. The dynamic of the two-factor extended Vasicek model under the real-worldmeasure is given by

r(t) = x(t) + y(t) + φ(t), r(0) = r0,

dx(t) = a(λ1 − x(t))dt+ σdW P1 (t), x(0) = 0,

dy(t) = b(λ2 − y(t))dt+ ηdW P2 (t), y(0) = 0,

where r(t) is the short rate, dW Pi (t) ∼ N(0, dt), i = 1, 2,

dW P1 (t) dW P

2 (t) = ρ dt,

and φ(t) is a deterministic function that is such that the initial bond prices generated bythe model match the bond prices observed on the market. This particular form for the two-factor extended Vasicek model is used in order to simplify the transition to the correspondingrisk-neutral model, the G2++ model.

Interest rates in the two-factor extended Vasicek model are thus based on the two latentGaussian random variables x(t) and y(t), and on the deterministic function φ(t). Randomvariables x(t) and y(t) must therefore be simulated in outer loop scenarios. Since processesx = x(t), t > 0 and y = y(t), t > 0 are Ornstein-Uhlenbeck processes, their exact solutionsare given by

x(T ) =λ1σ

a+

(x(t)− λ1σ

a

)e−a(T−t) +

∫ T

tσe−a(T−s)dW P

1 (s) (4.1)

y(T ) =λ2η

b+

(y(t)− λ2η

b

)e−b(T−t) +

∫ T

tηe−b(T−s)dW P

2 (s). (4.2)

Clearly, x(T )|x(t) and y(T )|y(t) have normal distributions (see e.g. Oksendal (2010)). Theconditional expectations of x(T )|x(t) and y(T )|y(t) are given by the deterministic componentsof (4.1) and (4.2) respectively, that is to say,

EP[x(T )|x(t)] =λ1σ

a+

(x(t)− λ1σ

a

)e−a(T−t)

EP[y(T )|y(t)] =λ2η

b+

(y(t)− λ2η

b

)e−b(T−t).

The conditional variances can also be found through the use of Ito’s isometry (see e.g. Oksendal(2010)). They are given by

Var(x(T )|x(t)) = σ2B(2a, T − t),

Var(y(T )|y(t)) = η2B(2b, T − t),

41

where

B(z, t) =1− e−zt

z.

Since dW P1 (t) and dW P

2 (t) are correlated normal random variables, a correlation betweenx(T )|x(t) and y(T )|y(t) is expected. From (4.1) and (4.2), it can be shown that

Cov(x(T ), y(T )|x(t), y(t)) = ρσηB(a+ b, T − t),

which leads to the following expression for the coefficient of correlation:

ρx,y = ρ(x(T ), y(T )|x(t), y(t))

=Cov(x(T ), y(T )|x(t), y(t))√Var(x(T )|x(t))Var(y(T )|y(t))

=ρB(a+ b, T − t)√

B(2a, T − t)B(2b, T − t). (4.3)

Using the set of times ΩT defined in (2.1), the exact solution of x(ti)|x(ti−1) and y(ti)|y(ti−1)

under the P-measure is given by the following system of equations:[x(ti)− λ1σ

a

y(ti)− λ2ηb

]=

[e−a∆t 0

0 e−b∆t

][x(ti−1)− λ1σ

a

y(ti−1)− λ2ηb

]

+

[σ√B(2a,∆t) 0

0 η√B(2b,∆t)

]Q

[ZPx,ti

ZPy,ti

], ti ∈ ΩT , (4.4)

where x(t0) = 0 and y(t0) = 0, ZPx,ti and ZP

y,ti are independent standard normal randomvariables under the P-measure, Q is the Cholesky decomposition of matrix

QQT =

[1 ρx,y

ρx,y 1

],

and ρx,y is given in (4.3).

Thus, (4.4) provides a means of simulating random components x(t) and y(t) under the real-world P-measure at all points in time of a given outer loop scenario.

The goal of the real-world interest rate model is to provide zero-coupon bond prices to be usedin the risk-neutral guarantee liability valuations at each point in time in outer loop scenarios.Fortunately, there is a closed-form formula for the zero-coupon bond price in the two-factorextended Vasicek model, which helps to speed the computation of these prices. As mentionedabove, note that although the two-factor extended Vasicek model is under the P-measure, thebond price calculation is obtained through an expectation under the risk-neutral Q-measure.

Recall that ZC(t, T ) is the price of a zero-coupon bond of maturity T at time t. Assuming thechange of measure used in this thesis, the zero-coupon bond price in the two-factor extended

42

Vasicek model is given by

ZC(t, T ) = EQ[e−∫ Tt r(s)ds

]=ZCM (T )

ZCM (t)e

12

(V (t,T )+V (0,t)−V (0,T ))e−(x(t)B(a,T−t)+y(t)B(b,T−t)), (4.5)

where ZCM (T ) is the zero-coupon bond price of maturity T observed on the market at time0, and

V (t, T ) =σ2

a2

(T − t+

2

ae−a(T−t) − 1

2ae−2a(T−t) − 3

2a

)+η2

b2

(T − t+

2

be−b(T−t) − 1

2be−2b(T−t) − 3

2b

)+ 2

σηρ

ab(T − t−B(a, T − t)−B(b, T − t) +B(a+ b, T − t)).

The change of measure that leads to the bond pricing formula in (4.5) will be discussed inmore details in section 4.2.2.

From (4.5), it is clear that random components x(t) and y(t) are the only random inputs inthe bond pricing formula. The simulation process of zero-coupon bond prices in outer loopscenarios can thus be summarized in Algorithm 4.1.

Algorithm 4.1. Simulation of zero-coupon bond prices in the two-factor extended Vasicekmodel under the P-measure

1. Simulate xP(ti) and yP(ti), ti ∈ ΩT using (4.4);

2. For each u|u ∈ ΩT and u > ti, compute zero-coupon bond prices ZCP(ti, u), ti ∈ ΩT

using (4.5) and the realizations of xP(ti) and yP(ti) simulated in step 1.

Stock market: Regime-switching lognormal model

The regime-switching lognormal model is a commonly used model for the valuation of long-term options such as variable annuity guarantees. The discrete version of the regime-switchingmodel was introduced by Hamilton (1989) and made popular in actuarial applications by Hardy(2001, 2003). The regime-switching lognormal model allows to capture some of the stylizedfacts of market returns, that is, volatility clustering, skewness and fat tails. As such, it iswidely used in the Canadian insurance industry. In the present work, the continuous versionof the regime-switching lognormal model is used to model the stock market index under thereal-world measure.

The dynamic of S in the continuous-time regime-switching lognormal model is given by

dSt = µSζtStdt+ σSζtStdWS,Pt , S0 = S(0),

43

where ζ = ζt, t ≥ 0 obeys to a continuous-time Markov chain with two discrete states andgenerator [

−g12 g12

g21 −g21

], (4.6)

with g12, g21 > 0. Clearly, ζt ∈ 1, 2, t ≥ 0 and the time required to transition from a givenregime i to the other regime j is exponentially distributed with mean 1

gij.

The initial regime is assumed to be determined from the stationary distribution of the Markovchain. The probability that the initial regime is the first regime is thus

π1 =g21

g12 + g21(4.7)

and the probability that it is the second regime is

π2 =g12

g12 + g21= 1− π1. (4.8)

Using the set of times ΩT , the exact solution to the stock market index dynamic is given by

Sti = Sti−1e∫ titi−1

µSζs−12(σSζs)

2ds+

∫ titi−1

σSζsdWS,Ps

= Sti−1e

(µS1−

12(σS1 )

2)R1,ti−1

+(µS2−

12(σS2 )

2)R2,ti−1

+√R1,ti(σ

S1 )

2+R2,ti(σ

S2 )

2ZS,Pti , (4.9)

where R1,ti and R2,ti are the times spent in regime 1 and 2 respectively during the time periodfrom ti to ti+1 and ZS,Pti

is a standard normal random variable under the P-measure.

The simulation process of the stock market index in outer loop scenarios is summarized inAlgorithm 4.2.

Algorithm 4.2. Simulation of the stock market index in the regime-switching lognormalmodel under the P-measure

1. Simulate a uniform variate U ;

2. Using (4.7),

2.1. if U ≤ π1, then the initial regime is 1;

2.2. otherwise, the initial regime is 2.

3. Simulate the times spent in each regime up to the chosen horizon using the initial regimefound in step 2 and the fact that the transition from one regime to the other follows acontinuous-time Markov chain with generator presented in (4.6);

4. Compute R1,ti and R2,ti , ti ∈ ΩT ;

5. Compute the stock market index price SPti , ti ∈ ΩT using (4.9).

44

Bond market: Lognormal model

The last financial market model required for outer loop scenarios is the bond market model.Bond market models usually link bond market returns to interest rate movements, for exampleby modeling the bond index as a rolling bond or a portfolio of bonds. However, in thepresent work, one of the risk-neutral interest rate models considered for the guarantee liabilityvaluation does not include interest rate volatility. In this model, since interest rates do notvary, it is not possible to model the bond index as a rolling bond.

Consistency among guarantee liability valuation models, and between guarantee liability val-uation models and real-world models, is important in the analysis of how modeling affectsthe hedge efficiency of GLWB guarantees. Hence, using a bond market model that introducesvolatility in bond market returns in a consistent way for all interest rate models consideredis preferred to changing the bond market model based on the interest rate model used. Aconsistent bond market model, however, implies having to forsake the relationship betweenbond returns and interest rate movements. Although not ideal from a theoretical standpoint,this concession has to be made for the sake of the analysis performed in this thesis.

The lognormal model is used for the bond index, as it can introduce volatility in bond marketreturns no matter whether interest rate volatility is modeled or not. Similar approaches arealso used in Steinorth and Mitchell (2012) and Shah and Bertsimas (2008) in the valuationof GLWB guarantees. The bond market index is introduced to consider the fact that GLWBguarantees are usually sold on diversified funds rather than on pure equity funds. Therefore,this index could alternatively be seen as a low volatility index.

The dynamic of P in the lognormal model is given by

dPt = µPt Ptdt+ σPPtdWP,Pt , P0 = P (0).

In order to have an appropriate value for µPt , the starting interest rate curve is leveraged. Theaverage return at time t is indeed given by

µPt = − ∂

∂tZCM (t),

that is to say, the instantaneous forward rate at time t determined using the starting zero-coupon bond curve.

Using the set of times ΩT , the exact solution to the bond market index dynamic is given by

Pti = Pti−1e∫ titi−1

µPs − 12(σP )

2ds+

∫ titi−1

σP dWP,Ps

= Pti−1eln

(ZCM (ti−1)

ZCM (ti)

)− 1

2(σP )2∆t+σP

√∆tZP,Pti , (4.10)

where ZP,Ptiis a standard normal random variable under the P-measure.

45

The simulation process of the bond market index in outer loop scenarios is summarized inAlgorithm 4.3.

Algorithm 4.3. Simulation of the bond market index in the lognormal model under theP-measure

1. Compute ln(ZCM (ti−1)ZCM (ti)

), ti ∈ ΩT , i ≥ 1;

2. Compute the bond market index P Pti , ti ∈ ΩT using (4.10).

4.2.2 Inner loop

Models for inner loop scenarios are used to value the guarantee liability at each point in timein outer loop scenarios. As mentioned previously, one of the analyses of this thesis pertainsto the evaluation of the impact of the guarantee liability modeling on the hedge efficiency ofGLWB guarantees. Therefore, several guarantee liability valuation models for each risk factorare considered in this section.

Interest rate models are the first building block of the modeling, as the average return anddiscount factors under the Q-measure are based on the random short rate. The selectedinterest rate models, that is, the G2++ model, the Hull-White model and the deterministicrates model, are thus introduced first. The stock market models, that is, the regime-switchinglognormal model and the lognormal model, are then presented.

Interest rates: G2++ model

The first risk-neutral interest rate model considered is the G2++ model. The G2++ model canbe shown to be the risk-neutral equivalent to the two-factor extended Vasicek model presentedin section 4.2.1. Indeed, using the multidimensional Girsanov theorem (see Appendix B) with

Θ(t) =

[σ 0

ηρ η√

1− ρ2

]−1 [aλ1

bλ2

],

the interest rate dynamic under the Q-measure for the two-factor extended Vasicek model canbe shown to be given by

r(t) = x(t) + y(t) + φ(t), r(0) = r0,

dx(t) = −ax(t)dt+ σdWQ1 (t), x(0) = 0,

dy(t) = −by(t)dt+ ηdWQ2 (t), y(0) = 0,

where r(t) is the short rate, dWQi (t) ∼ N(0, dt), i = 1, 2,

dWQ1 (t) dWQ

2 (t) = ρ dt

46

and φ(t) is a deterministic function that is such that the initial bond prices generated by themodel match the bond prices observed on the market. This dynamic is consistent with thatof the G2++ model presented in Brigo and Mercurio (2006) and leads to the bond pricingformula presented in (4.5).

First and foremost, because risk-neutral models are used in guarantee liability valuations attimes ti ∈ ΩT in outer loop scenarios, a slight change to the model dynamic is made. Indeed,the stochastic differential equations’ initial conditions are assumed in the present work to be attime ti, where ti is the given guarantee liability valuation time within the outer loop scenario,rather than at time 0. The initial conditions then become x(ti) = 0 and y(ti) = 0.

It must be stressed that the real-world values of xP(ti) and yP(ti) obtained from the two-factor extended Vasicek model are not used as the starting values for the guarantee liabilityvaluation, because the G2++ model is the only risk-neutral model considered in the presentwork whose dynamic is based on processes x and y. Thus, in order to ensure consistencyin the transition from the real-world model to the risk-neutral models, only the real-worldzero-coupon bond prices generated by the outer loop model at time ti are used as inputs forthe risk-neutral interest rate modeling.

In the inner loop, interest rate models are used to simulate short rate paths, which are in turnused to determine the discount rates and the average returns of the stock market and bondmarket indices. An analytical expression for∫ T

tr(s) ds,

the integral value of the short rate path over a given time horizon, must therefore be found.Let ∫ T

tr(s)ds = I(t, T ) +

∫ T

tφ(s) ds, (4.11)

where I(t, T ) =∫ Tt x(s) + y(s)ds. Following Brigo and Mercurio (2006), I(t, T )|x(t), y(t) is

normally distributed with expectation and variance respectively given by

M(t, T ) = B(a, T − t)x(t) +B(b, T − t)y(t)

and

V (t, T ) =σ2

a2

(T − t+

2

ae−a(T−t) − 1

2ae−2a(T−t) − 3

2a

)+η2

b2

(T − t+

2

be−b(T−t) − 1

2be−2b(T−t) − 3

2b

)+ 2

σηρ

ab(T − t−B(a, T − t)−B(b, T − t) +B(a+ b, T − t)).

47

Moreover, following Andersen and Piterbag (2010) and assuming a valuation time ti in theouter loop scenario, it can be shown that the random variables x(ti+j+1)|x(ti+j), y(ti+j+1)|y(ti+j)

and I(ti+j+1, ti+j)|x(ti+j), y(ti+j) are correlated normal random variables that evolve basedon the following system of equations: x(ti+j+1)

y(ti+j+1)

I(ti+j , ti+j+1)

=

e−a∆t 0

0 e−b∆t

B(a,∆t) B(b,∆t)

[ x(ti+j)

y(ti+j)

]

+

σ√B(2a,∆t) 0 0

0 η√B(2b,∆t) 0

0 0√V (ti+j , ti+j+1)

QZQx,ti+j+1

ZQy,ti+j+1

ZQI,ti+j+1

,(4.12)

j = 0, 1, . . . , ω−x∆t − i − 1, where ZQx,ti+j+1

, ZQy,ti+j+1

and ZQI,ti+j+1

are independent standardnormal random variables under the Q-measure, Q is the Cholesky decomposition of matrix

QQT =

1 ρx,y ρx,I

ρx,y 1 ρy,I

ρx,I ρy,I 1

,

ρx,y =Cov(x(ti+j+1), y(ti+j+1)|x(ti+j), y(ti+j))√Var(x(ti+j+1)|x(ti+j))Var(y(ti+j+1)|y(ti+j))

=ρB(a+ b,∆t)√

B(2a,∆t)B(2b,∆t),

ρx,I =Cov (x(ti+j+1), I(ti+j , ti+j+1)|x(ti+j), y(ti+j))√

Var(x(ti+j+1)|x(ti+j))√

Var(I(ti+j , ti+j+1)|x(ti+j), y(ti+j))

=σ2

2 (B(a,∆t))2 + σρηb (B(a,∆t)−B(a+ b,∆t))

σ√B(2a,∆t)

√V (ti+j , ti+j+1)

,

and

ρy,I =Cov (y(ti+j+1), I(ti+j , ti+j+1)|x(ti+j), y(ti+j))√

Var(y(ti+j+1)|y(ti+j))√

Var(I(ti+j , ti+j+1)|x(ti+j), y(ti+j))

=η2

2 (B(b,∆t))2 + σρηa (B(b,∆t)−B(a+ b,∆t))

η√B(2b,∆t)

√V (ti+j , ti+j+1)

.

The second part of (4.11) is deterministic. Again assuming a valuation time ti in the outerloop scenario, it is given (see Brigo and Mercurio (2006)) by∫ ti+j+1

ti+j

φ(s)ds = ln

(ZCP(ti, ti+j+1)

ZCP(ti, ti+j)

)− 1

2(V (ti, ti+j+1)− V (ti, ti+j)), (4.13)

48

j = 0, 1, . . . , ω−x∆t − i − 1, where ZCP(ti, ti+j) is the zero-coupon bond price with time tomaturity ti+j − ti at valuation time ti in a given outer loop scenario.

The simulation process for the inner loop risk-neutral G2++ model at a given time step ti inan outer loop scenario is summarized in Algorithm 4.4.

Algorithm 4.4. Inner loop simulation of the G2++ model under the risk-neutral measure attime ti in an outer loop scenario

For j = 0, 1, . . . , ω−x∆t − i− 1

1. Using the zero-coupon bond prices in the outer loop scenario at time ti derived fromAlgorithm 4.1 and (4.13), compute

∫ ti+j+1

ti+jφ(s)ds;

2. Using (4.12) and the fact that x(ti) = 0 and y(ti) = 0, compute I(ti+j , ti+j+1);

3. Compute∫ ti+j+1

ti+jr(s)ds using (4.11).

Interest rates: Hull-White model

The Hull-White model is closely related to the G2++ model in that it is its one-factor analog.Following Brigo and Mercurio (2006), the short rate dynamic is given by

r(t) = x(t) + α(t),

dx(t) = −ax(t)dt+ σdWQ(t), x(0) = 0, (4.14)

where, as for the G2++ model, α(t) is defined so as to fit the initial bond prices on themarket. Once again, a slight change to the model’s initial conditions is made to consider afuture valuation time ti within an outer loop scenario. The initial condition is x(ti) = 0. Let∫ T

tr(s)ds =

∫ T

t(x(s) + α(s)) ds

= I(t, T ) +

∫ T

tα(s)ds. (4.15)

It can be shown that I(t, T ) is normally distributed with mean and variance respectively givenby

M(t, T ) = B(a, T − t)x(t)

and

V (t, T ) =σ2

a2

(T − t+

2

ae−a(T−t) − 1

2ae−2a(T−t) − 3

2a

).

The joint behaviour of x(ti+j+1)|x(ti+j) and I(ti+j , ti+j+1)|x(ti+j) is given by[x(ti+j+1)

I(ti+j , ti+j+1)

]=

[e−a∆t

B(a,∆t)

]x(ti+j)

+

[σ√B(2a,∆t) 0

0√V (ti+j , ti+j+1)

]Q

[ZQx,ti+j+1

ZQI,ti+j+1

], (4.16)

49

j = 0, 1, . . . , ω−x∆t −i−1, where ZQx,ti+j+1

and ZQI,ti+j+1

are independent standard normal randomvariables under the Q-measure, Q is the Cholesky decomposition of matrix

QQT =

[1 ρx,I

ρx,I 1

]

and ρx,I is the coefficient of linear correlation given by

ρx,I = ρ (x(ti+j+1), I(ti+j , ti+j+1)|x(ti+j))

=Cov (x(ti+j+1), I(ti+j , ti+j+1)|x(ti+j))√

Var(x(ti+j+1)|x(ti+j))√

Var(I(ti+j , ti+j+1)|x(ti+j))

=σ2

2 (B(a,∆t))2 + σρηb (B(a,∆t)−B(a+ b,∆t))

σ√B(2a,∆t)

√V (ti+j , ti+j+1)

.

Moreover, the deterministic component assuming a valuation time ti is given by∫ ti+j+1

ti+j

α(s)ds = ln

(ZCP(ti, ti+j+1)

ZCP(ti, ti+j)

)− 1

2(V (ti, ti+j+1)− V (ti, ti+j)), (4.17)

j = 0, 1, . . . , ω−x∆t − i− 1, where once again ZCP(ti, ti+j) denotes the zero-coupon bond pricesimulated in the outer loop scenario at valuation time ti for a bond maturing at time ti+j .

The simulation process for the inner loop risk-neutral Hull-White model at a given time stepti in an outer loop scenario is summarized in Algorithm 4.5.

Algorithm 4.5. Inner loop simulation of the Hull-White model under the risk-neutral mea-sure at time ti in an outer loop scenario

For j = 0, 1, . . . , ω−x∆t − i− 1

1. Using the zero-coupon bond prices in the outer loop scenario at time ti derived fromAlgorithm 4.1 and (4.17), compute

∫ ti+j+1

ti+jα(s)ds;

2. Using (4.16) and the fact that x(ti) = 0, compute I(ti+j , ti+j+1);

3. Compute∫ ti+j+1

ti+jr(s)ds using (4.15).

Interest rates: Deterministic rates model

The deterministic rates model assumes no volatility in interest rates. This model is thus suchthat the initial forward rates are realized when moving forward in time.

Under the risk-neutral Q-measure, the deterministic rates model implies that

r(t) = − ∂

∂tZCP(ti, t), t > ti,

50

where ZCP(ti, t) is the price of the zero-coupon bond maturing at time t at valuation time tiin the outer loop scenario.

The expression for∫ ti+j+1

ti+jr(s) ds is trivial in the deterministic rates model. It is given by∫ ti+j+1

ti+j

r(s)ds = ln

(ZCP(ti, ti+j)

ZCP(ti, ti+j+1)

), j = 0, 1, . . . ,

ω − x∆t

− i− 1.

Stock market: Regime-switching lognormal model

The regime-switching lognormal model is the most complex risk-neutral model considered forthe stock market index. It is obtained through a change of measure of the real-world stockmarket model presented in section 4.2.1. Following Bollen (1998) and Hardy (2001), regimerisk is assumed not to be priced for in the market under the risk-neutral Q-measure in thepresent work. As demonstrated in Yao et al. (2006), in which a Girsanov-like theorem is usedto find the equivalent martingale measure, this implies in the continuous-time regime-switchinglognormal model that the average return under the risk-neutral measure is the risk-free ratein both regimes, and that the transition probabilities and volatilities are unchanged comparedwith the real-world model.

The dynamic of S in the regime-switching lognormal model under the Q-measure is then givenby

dSt = r(t)Stdt+ σSζtStdWS,Qt ,

where ζt ∈ 1, 2 obeys a continuous-time Markov chain with two discrete states with thesame generator as under the real-world measure (presented in (4.6)).

Because flexibility in the modeling of interest rates is allowed through the use of stochasticinterest rate models, care must be taken regarding the average growth rate. Assuming aguarantee liability valuation time ti, the exact solution to the stock market index dynamic isgiven by

Sti+j+1 = Sti+je∫ ti+j+1ti+j

r(s)− 12

(σSζs )2ds+∫ ti+j+1ti+j

σSζsdWS,Qs

= Sti+je∫ ti+j+1ti+j

r(s)ds− 12

((σS1 )

2R1,ti+j

+(σS2 )2R2,ti+j

)+√R1,ti+j (σ

S1 )

2+R2,ti+j (σ

S2 )

2ZS,Qti+j ,

(4.18)

j = 0, 1, . . . , ω−x∆t −i−1, where R1,ti and R2,ti are the times spent in regime 1 and 2 respectivelyduring the time period from ti to ti+1, Z

S,Qti+j

is a standard normal random variable under theQ-measure and

∫ ti+j+1

ti+jr(s) ds is obtained from the chosen risk-neutral interest rate model.

Note that despite the fact that the current regime is known in the outer loop scenario attime ti, the initial regime in the inner loop simulations is assumed to be determined fromthe stationary distribution in (4.7) and (4.8). Using the regime obtained from the real-world

51

model would imply assuming that information on the current regime is available to marketparticipants, when in fact it is not. Alternatively, filtered probabilities could have been used toestimate the initial regime at time ti. The use of filtered probabilities is however less appealingwithin outer loop scenarios: the previous available prices are fairly limited, especially at thebeginning of the scenario. Given the additional complexity and run time implications relatedwith the implementation of a filtered probability computation at each point in time in outerloop scenarios, this method was not explored.

The simulation process for the inner loop risk-neutral regime-switching lognormal model at agiven time ti in an outer loop scenario is summarized in Algorithm 4.6.

Algorithm 4.6. Inner loop simulation of the regime-switching lognormal model under therisk-neutral measure at time ti in an outer loop scenario

1. Simulate a uniform variate U ;

2. Using (4.7),

2.1. if U ≤ π1, then the initial regime is 1;

2.2. otherwise, the initial regime is 2.

3. Simulate the times spent in each regime up to the chosen horizon using the initial regimefound in step 2 and the fact that the transition from one regime to the other follows acontinuous-time Markov chain with generator presented in (4.6);

4. For j = 0, 1, . . . , ω−x∆t − i− 1,

4.1. compute R1,ti+j and R2,ti+j ;

4.2. compute the stock market index Sti+j+1 using (4.18) and the fact that Sti = SPti .

Stock and bond market: Lognormal model

The lognormal model is of course a special case of the regime-switching lognormal model inwhich there is only one regime. Under the risk-neutral measure, the dynamic of S is given by

dSt = r(t)Stdt+ σSStdWS,Qt .

Thus, assuming a valuation time ti in an outer loop scenario, the exact solution to the stockindex process is given by

Sti+j+1 = Sti+je∫ ti+j+1ti+j

r(s)− 12(σS)

2ds+

∫ ti+j+1ti+j

σSdWS,Qs

= Sti+je∫ ti+j+1ti+j

r(s)ds− 12(σS)

2∆t+σS

√∆tZS,Qti+j , (4.19)

52

j = 0, 1, . . . , ω−x∆t − i − 1, where ZS,Qti+jis a standard normal random variable under the Q-

measure and∫ ti+j+1

ti+jr(s) ds is obtained from the selected risk-neutral interest rate model.

The stock market index can thus be simulated using (4.19) and the fact that Sti = SPti .

A similar model is used for the bond index under the risk-neutral measure. The exact solutionto the bond index process is given by

Pti+j+1 = Pti+je∫ ti+j+1ti+j

r(s)ds− 12(σP )

2∆t+σP

√∆tZP,Qti+j , (4.20)

j = 0, 1, . . . , ω−x∆t −i−1, where ZP,Qti+jis a standard normal random variable under theQ-measure

and Pti = P Pti .

4.3 Mortality models

Despite not being a financial risk, longevity is a significant systematic risk for GLWB guar-antees because of the life-contingent nature of these guarantees. In this thesis, longevity riskis addressed through the use of different mortality models, including a stochastic mortalitymodel.

In the present work, no distinction is made between the real-world and risk-neutral measuresfor mortality models. First of all, as shown in Ngai and Sherris (2011), longevity risk is fairlydifficult to hedge in GLWB guarantees. Secondly, the lack of a deep and liquid market forlongevity instruments makes the computation of a market price of risk a fairly complex task inpractice. Lastly, consistency between the models used in the outer and inner loop simulationsis important in the analysis of longevity risk presented in chapter 7. Thus, in this section, themortality models considered are presented irrespective of the measure used.

4.3.1 Lee-Carter model

The Lee-Carter model, proposed in Lee and Carter (1992), is a well-known stochastic mortalitymodel that introduces randomness in future mortality rates through a random time-varyingindex. Let mx,t be the central rate of death for a person aged x at time t. The Lee-Cartermodel states that

ln(mx,t) = αx + βxκt + εx,t, (4.21)

where κt is a time-varying index, αx and βx are age-specific constants, and εx,t is a normalrandom variable with zero mean that reflects age-specific historical influences not captured inthe model.

Mortality improvement in the Lee-Carter model is thus driven by a single mortality improve-ment index, κt, that evolves through time and affects all ages. The impact of the mortalityimprovement index on each age is modulated by the age-specific betas. Moreover, the alphascan be seen as being related to the base mortality rates.

53

The random variable that introduces stochastic mortality in the Lee-Carter model is κt. Letκ = κti , ti ∈ ΩT denote the process of the time-varying mortality improvement index. Asin Lee and Carter (1992), a random walk is used to model κ in the present work. Thus, usingthe set of times ΩT defined in (2.1), the dynamic of κ is given by

κti+j+1 = κti+j + θ∆t+ σµ√

∆tZµti+j+1, (4.22)

where Zµti+j+1is a standard normal random variable.

In Lee and Carter (1992), randomness in mortality rate forecasts is assumed to arise solelythrough the stochastic process κ. Randomness related with εx,t in (4.21) is ignored in thesimulation. This assumption is also used in the present work. Thus, the simulated centralrate of death is given by

ln(mx,ti+j ) = αx + βxκti+j , (4.23)

where κti+j is obtained using (4.22).

A popular assumption in the Lee-Carter model is the constant force of mortality over fractionalages, which consists in assuming that the force of mortality is constant between integer ages.However, because a time step of ∆t is assumed in the present work, the force of mortality µx,tis rather assumed to be constant within time intervals [ti, ti+1), i = 0, 1, . . . , ω−x∆t − 1. Then,the following relationship links the force of mortality to the central rate of death:

µx,ti+s = µx,ti = mx,ti+s, 0 ≤ s < ∆t, ti ∈ ΩT . (4.24)

The survival probability for an individual aged x from time ti to time tj , j > i given a mortalitypath µ is then given by

tj−tip(µ)

x,ti=

j−i−1∏k=0

e−∆t×µx+ti+k−ti,ti+k , (4.25)

that is to say, the product of the survival probabilities over time intervals [ti, ti+1), . . . , [tj−1, tj).

Although no distinction is made between the real-world and the risk-neutral measures formortality, both outer and inner loop simulations must be considered. The simulation processof survival probabilities using the Lee-Carter model in outer loop scenarios is summarized inAlgorithm 4.7. Note that although there is no distinction regarding the measure used, thesuperscript P is used for outer loop scenarios to simplify the notation.

Algorithm 4.7. Outer loop projection of survival probabilities in the Lee-Carter model

1. Compute κPti , ti ∈ ΩT using (4.22);

2. Compute µPx+ti,ti , ti ∈ ΩT using (4.23) and (4.24);

3. Compute tip(µP)

x,t0, ti ∈ ΩT using (4.25).

54

The simulation process for the inner loop Lee-Carter model at a given valuation time ti in anouter loop scenario is summarized in Algorithm 4.8.

Algorithm 4.8. Inner loop projection of survival probabilities in the Lee-Carter model attime ti in an outer loop scenario

For ti+j ∈ ΩT and ti+j > ti

1. Compute κti+j with κti = κPti using (4.22);

2. Compute µx+ti+j ,ti+j using (4.23) and (4.24);

3. Compute ti+j−tip(µ)

x+ti,tiusing (4.25).

4.3.2 Constant mortality improvement model

In the constant mortality improvement model, the same framework as in the Lee-Carter modelis used. However, randomness in the projection of κt is removed such that

κti+j+1 = κti+j + θ∆t. (4.26)

It is clear from (4.26) that the constant mortality improvement model is a special case of theLee-Carter model with σµ = 0. Once again, both the outer and inner loop simulations areconsidered, as the constant mortality improvement model is also used in outer loop simulationsin chapter 7.

The simulation process of survival probabilities using the constant mortality improvementmodel in outer loop scenarios is summarized in Algorithm 4.9.

Algorithm 4.9. Outer loop projection of survival probabilities in the constant mortalityimprovement model

1. Compute κPti , ti ∈ ΩT using (4.26);

2. Compute µPx+ti,ti , ti ∈ ΩT using (4.23) and (4.24);

3. Compute tip(µP)

x,t0, ti ∈ ΩT using (4.25).

The simulation process for the survival probabilities in the inner loop constant mortalityimprovement model at a given valuation time ti in an outer loop scenario is summarized inAlgorithm 4.10.

55

Algorithm 4.10. Inner loop projection of survival probabilities in the constant mortalityimprovement model at time ti in an outer loop scenario

For ti+j ∈ ΩT and ti+j > ti

1. Compute κti+j with κti = κPti using (4.26);

2. Compute µx+ti+j ,ti+j using (4.23) and (4.24);

3. Compute ti+j−tip(µ)

x+ti,tiusing (4.25).

4.4 Conclusion

In this chapter, specific models for the interest rate, stock market, bond market and longevityrisk factors were introduced. Financial market models were presented first and foremost.Both real-world outer loop and risk-neutral inner loop models were introduced, with a briefdiscussion on the change of measure. For inner loop models, several models per risk factorwere considered, as this thesis is interested in analyzing the impact of the inner loop modelingon the hedge efficiency of GLWB guarantees. Special attention was given to maintainingconsistency among the inner loop models and between the outer and inner loop models.

For mortality, both a stochastic and a deterministic model were presented. No distinction ismade between the real-world and the risk-neutral measure for mortality. However, simulationsof the survival probabilities in both the outer and inner loops were considered.

56

Chapter 5

Risk assessment and hedging

5.1 Introduction

GLWB guarantees have been shown in this thesis to contain several systematic risks, includingstock market, bond market, interest rate and longevity risks. Hence, so far, the need foran efficient risk management strategy for these guarantees has been considered self-evident.However, in order to fully justify the need for such a strategy, a risk assessment of unhedgedGLWB guarantees must be made.

In order to make an appropriate risk assessment, the modeling used in the assessment mustprovide an adequate description of reality. Appropriate models for the financial market andmortality were introduced in chapter 4. However, equally important are the parameters usedin these models, as parameters that are not representative of expected future experience maybias the risk appraisal and lead to poor risk management decisions. Therefore, this chapterfirst presents the outer loop model parameters used in the risk assessment and throughoutchapters 6 and 7.

Using these outer loop parameters, the real-world risk assessment of unhedged GLWB guar-antees is used to illustrate the significant tail risk associated with these guarantees, and tomotivate the need for an efficient hedging strategy for GLWB guarantees.

The hedging strategy considered in the present work is then introduced. After justifying thestrategy from a theoretical standpoint, the hedge portfolio is built and the hedge efficiencycalculation is detailed.

Hedge efficiency is then further analyzed, first by comparing the hedged and unhedged GLWBguarantee gains and losses distribution and then by determining how the hedging strategy canbe modified to improve the hedge efficiency.

57

5.2 Outer loop model parameters

In this section, outer loop model parameters for both the financial market and mortalitymodels are presented. Moreover, the contractual and contract holder parameters that areused repeatedly throughout this chapter and the following chapters are introduced. As men-tioned above, model parameters play an important role in ensuring a realistic risk assessment.Therefore, model parameters were determined so that they are representative of the Canadianfinancial market and population.

5.2.1 Stock market index

The stock market index under the real-world measure is modeled with the regime-switchinglognormal model described in section 4.2.1. The model parameters were determined so thatthey are representative of the Canadian stock market. More specifically, they were obtainedthrough a heuristic transformation of parameters for a monthly discrete-time regime-switchinglognormal model estimated on the TSX total return data from 1956 to 2014. The parametersare presented in Table 5.1.

i µSi σSi gi,3−i

1 0.15526 0.1163 0.45082 -0.17564 0.2548 2.2305

Table 5.1: Parameters for the regime-switching lognormal model

The parameters presented in Table 5.1 are in line with some of the stylized facts of financialmarkets. First of all, one of the regimes (regime 1) represents a bull market, while the otherregime (regime 2) represents a bear market. The bull market is characterized by high returnsand low volatility, while the bear market is characterized by poor returns and high volatility.Moreover, the bull markets have longer persistency than the bear markets. The mean timespent in a bull market is 2.2 years while is it 0.44 years in a bear market. The parameters alsolead to returns which exhibit volatility clustering and a negative correlation with volatility.

In Table 5.2, quantiles of SPt given SP

0 = 100 in the regime-switching lognormal model withthe parameters provided in Table 5.1 are presented. Notice that the model is able to includelong periods of poor returns, as shown by the 2.5th and 5th quantiles at time 10 years, whichare lower than the starting value of the stock market index.

As a check on the appropriateness of the parameters used in the present work, note that theaccumulation factors are such that the model meets the calibration criteria released by theCanadian Institute of Actuaries, which are based on the Canadian stock market experience(Committee on Life Insurance Financial Reporting (2012)).

58

α V aRα(SP1 ) V aRα(SP

5 ) V aRα(SP10) V aRα(SP

20)

0.025 71 62 69 1120.050 78 70 89 1590.100 87 87 115 2110.500 111 164 249 6000.900 134 257 500 1,7030.950 141 289 586 2,1610.975 148 319 662 2,721

Table 5.2: Quantiles of the stock index with SP0 = 100

5.2.2 Bond market index

The bond market index is modeled using the lognormal model under the real-world P-measure.The volatility parameter of the lognormal model is set at σP = 4% to represent a realisticvariability in bond fund returns.

5.2.3 Interest rates

The two-factor extended Vasicek model is used to model interest rates under the real-worldmeasure. Since the model is based on two unobservable processes, x and y, the parameterestimation from historical data in this model requires the use of Kalman filtering techniques(see e.g., Babbs and Nowman (1999), De Jong (2000) or Duan and Simonato (1999)).

The parameters derived in Babbs and Nowman (1999) are used for the real-world modeling. InBabbs and Nowman (1999), US interest rate data from 1987 to 1996 is used in the estimation.The main reason for using the parameters presented in Babbs and Nowman (1999) is thatthe article provides sets of parameters estimated in a consistent way for a one-factor modeland a two-factor model. Although the parameters for the one-factor model are not used inthe real-world interest rate model, they are used in the risk-neutral Hull-White model. Aslight departure from a model that is strictly representative of the Canadian market was thusaccepted.

Although the interest rate data used in the estimation of Babbs and Nowman (1999) is notCanadian interest rate data, the parameters derived are seen as broadly representative of theCanadian market. Interest rates in the United States are indeed fairly correlated with interestrates in Canada and exhibit similar volatility. Moreover, although the model parameters areestimated using data in the 80s and 90s, the initial interest rate curve used in the analysespresented in this thesis reflects the current interest rate environment. Indeed, the interest ratecurve at time 0 being an input to the extended Vasicek model, the Canadian swap curve asof December 31, 2014 is the initial curve used in the valuation.

59

The parameters for the two-factor extended Vasicek model from Babbs and Nowman (1999)are presented in Table 5.3. Note that since the market prices of risk were not statisticallyrelevant in the estimation of Babbs and Nowman (1999), they were forced to zero in the tableand in our valuation.

a 0.5529b 0.0652σ 0.0195η 0.0186ρ -0.836λ1 0λ2 0

Table 5.3: Parameters of the two-factor extended Vasicek model

The quantiles of the short term (1 year) and long term (30 year) spot rates using the parametersin Table 5.3 are plotted in Figure 5.1.

Figure 5.1: Quantiles of the 1-year (left panel) and 30-year (right panel) spot rates in thetwo-factor extended Vasicek model

The starting interest rates are shown in Figure 5.1 to reflect the current low interest rateenvironment. Moreover, the 1-year spot rate is shown to have a larger dispersion than the30-year rate, which is consistent with the stylized facts of historical interest rates. Finally,the two-factor extended Vasicek model does not prevent negative interest rates, which are

60

observed for the short term spot rate. The possibility of negative interest rates certainly is adrawback of this model.

5.2.4 Mortality

The parameters of the Lee-Carter model are estimated from data for Canadian males aged 30to 100 over the period from 1950 to 2011. The mortality data was obtained from the HumanMortality Database (University of California, Berkeley (USA), and Max Planck Institute forDemographic Research (Germany) (2014)).

The method used to estimate the parameters of the Lee-Carter model is consistent with theone described in Appendix A of Lee and Carter (1992). It consists in using singular valuedecomposition to estimate the values of αx and βx, and the historical values of κt. Thehistorical values of κt obtained from the estimation are denoted by κ−n, κ−n+1, . . . , κ0.

Since κt is modeled as a random walk, parameters θ and σµ in (4.22) must also be estimated.Estimated historical values of κt are used in the estimation of θ and σµ. They are given by

θ =1

n

−1∑t=−n

(κt+1 − κt)

and

σµ =1

n− 1

−1∑t=−n

((κt+1 − κt)− θ)2.

The parameters θ, σµ and κ0 of the Lee-Carter model are presented in Table 5.4 and thevalues of αx and βx are depicted in Figure 5.2.

θ -0.006957σµ 0.007333κ0 -0.2724

Table 5.4: Parameters of the Lee-Carter model

As discussed in section 4.3, the age-specific constants αx are related to the base mortality rates.The upward sloping relationship between αx and x was thus to be expected and indicates thatprobabilities of death increase with age. The increase is shown to be almost linear past age 30.Values of βx modulate the effect of the mortality improvement index on the force of mortality.The βx values are fairly low for older ages, indicating that there is less mortality improvementat these ages.

The constant mortality improvement model is also used in the outer loop in chapter 7. Thismodel has the same parameters as presented in Table 5.4 and in Figure 5.2, but with σµ = 0.

61

Figure 5.2: Estimated αx (left panel) and βx (right panel) values by age in the Lee-Cartermodel

5.2.5 Contract holder and contractual parameters

The contract holder and contractual parameters used in this chapter and throughout thisthesis are presented in Tables 5.5, 5.6 and 5.7.

Parameter Value Description

x 65 Age at inceptionSP 100,000 Single premiumtw 5 Accumulation period

Table 5.5: Contract holder parameters

In short, a 65-year-old contract holder is assumed to deposit $100,000 in a variable annuitycontract with a GLWB guarantee and to elect a five-year accumulation period. The diversifiedfund the contract holder invests in has a 60% stock market proportion. In the withdrawalphase, the contract holder makes quarterly withdrawals. The guarantee offers a yearly 5%

roll-up and triennial ratchets. Moreover, as discussed in chapter 2, an additional ratchet isapplied at the time of transition from the accumulation phase to the withdrawal phase.

62

Parameter Value Description

n 4 Withdrawal frequencygA 1.5% Guarantee feemA 3.0% Mutual fund feep 5.0% Roll-up ratem 3 Triennal ratchetsψ 0.6 Stock market proportion

Table 5.6: Contractual parameters

Age Rate

60− 64 4.5%65− 69 5.0%70− 74 5.5%75− 115 6.0%

Table 5.7: Lifetime withdrawal rates by attained age

The contractual parameters were chosen to be broadly representative of the Canadian GLWBguarantee inforce business.

5.3 Real-world risk assessment

The real-world model parameters presented in section 5.2 allow to make a relevant risk assess-ment of unhedged GLWB guarantees.

The risk related with GLWB guarantees under the real-world measure can be assessed fromdifferent perspectives. First of all, a quantity of interest is the distribution of the lifetimewithdrawal amount over time in the real-world modeling. Indeed, the lifetime withdrawalamount measures both the rate at which the contract holder’s account value gets depletedand the size of the claim payments when it is. As such, it has a close relationship with theinsurer’s liability in the contract. The left panel of Figure 5.3 presents quantiles of the lifetimewithdrawal amount in time.

The lifetime withdrawal amount distribution is shown in the left panel of Figure 5.3 to havea probability mass at 7,020. This withdrawal amount corresponds to the lifetime withdrawalrate at age 70 times the initial guaranteed withdrawal balance increased by 5% roll-ups for 5years. However, ratchets are such that the withdrawal amount can be quite higher. Indeed,when moving forward in time, ratchets increase the lifetime withdrawal amount such thatat the 95th percentile of the withdrawal amount distribution, withdrawals are up to 40%

higher than the median withdrawal. Moreover, at the very beginning of the withdrawal phase,

63

Figure 5.3: Quantiles of the lifetime withdrawal amount (left panel) and distribution of thetime until the account value is exhausted (right panel)

withdrawals at the 95th percentile are more than 25% higher than the median withdrawal.Larger withdrawals impose further downward pressure on the contract holder’s account value.Ratchets thus introduce an additional risk in GLWB guarantees: the risk of an increase in themarket in the accumulation phase, triggering ratchets, followed by a decrease in the marketin the withdrawal phase, yielding a quickly depleted account value.

The risk related with GLWB guarantees is also evidenced by the distribution of the timerequired for the contract holder’s account value to reach zero, presented in the right panel ofFigure 5.3. A very high variability in the time required to exhaust the account value can benoted in this plot, thus indicating variability in future revenues and future claims. Indeed,recall that market risk affects both revenues, which are only collected as long as the accountvalue is positive, and claims, which only start to get paid when the account value is exhausted.The positive skewness of the distribution implies that there is a significant weight on low timesuntil the account value is exhausted.

The ultimate assessment of risk arising from unhedged GLWB guarantees consists in calculat-ing the distribution of the present value of future revenues minus future claims, that is to say,the future gains or losses for the insurance company over the entire projection horizon of theGLWB guarantee. Indeed, this assessment considers not only the impact of market risk, butalso of interest rate risk through the discounting factors and mortality through the survival

64

probabilities. For a given outer loop scenario k, the present value of revenues minus claims isgiven by

PV(k)U = R

P,(k)t0

+

ω−x∆t∑i=1

tip(µP,(k))

x,t0

(R

P,(k)ti− CP,(k)

ti

) i−1∏j=0

ZCP,(k)(tj , tj+1)

, (5.1)

where ZCP,(k)(tj , tj+1) is the zero-coupon bond rate at time tj maturing at time tj+1 in outer

loop scenario k and tip(µP,(k))

x,t0is the survival probability given the mortality improvement

path in outer loop scenario k. The expression in (5.1) should be interpreted as the sum ofthe present value at the short term rate of the decremented net gains and losses from theGLWB guarantee. The resulting distribution using the parameters given in section 5.2 withNO = 1000 outer loop scenarios is plotted in Figure 5.4.

Figure 5.4: Unhedged GLWB guarantee gains and losses distribution

Although a large part of the distribution of unhedged gains and losses in Figure 5.4 is positive,indicating that the company makes a gain by offering the GLWB guarantee, a very fat lefttail implies huge potential losses for the company. Indeed, the average of the 5% worst lossesleads to a $35, 610 loss, which represents 36% of the initial single premium in the contract.Hence, an unhedged GLWB guarantee is no doubt a huge risk to an insurance company.

65

5.4 Hedging

The risk assessment carried out in section 5.3 highlights the importance of having an efficienthedging strategy to mitigate some of the risks related with GLWB guarantees. In addition, ithelps in setting an appropriate benchmark to which to compare the hedging strategy consid-ered.

In this section, the hedging strategy used in the present work is described in details. Thestrategy is first considered from a theoretical standpoint. Then, the hedge asset portfolio usedin the risk mitigation is introduced. Finally, the calculation of the present value of hedgedgains and losses is presented.

5.4.1 Hedging strategy

Different kinds of hedging strategies exist for variable annuities. For example, Liu (2010)describes a semi-static hedging strategy on a GMWB guarantee, and Coleman et al. (2006)introduces a locally risk-minimizing hedging strategy using standard options. Despite thevariety of hedging strategies, the most common strategy used by North American insurancecompanies is dynamic hedging. Dynamic hedging consists in rebalancing a portfolio of liquidmarket instruments such that the sensitivity of the asset portfolio matches the sensitivity ofthe liability generated by the variable annuity guarantees. Dynamic hedging of plain vanillaoptions or of simple variable annuity guarantees is well documented in the literature (see e.g.Bjork (2009) for more details on plain vanilla options hedging and Hardy (2003) for moredetails on variable annuity guarantees hedging). However, since several models are consideredfor the guarantee liability valuation in the present work, this strategy must be defined for eachof them. The liability sensitivities presented in chapter 3, that is to say, the stock index delta,the bond index delta and the key-rate rho buckets, are leveraged in defining the strategy.

One of the models considered for the stock market index and the bond market index in thepresent work is the lognormal model. The Black-Scholes framework, which uses the lognormalmodel as its equity model, results in a perfect delta-hedge when rebalancing of the hedgeportfolio is performed in continuous time. In this thesis, hedging is not in continuous timeand interest rates are stochastic, thus preventing a perfect hedge. Nonetheless, a similar deltacalculation as in the Black-Scholes framework is used, and so the deltas obtained in chapter 3can be used directly in determining the hedge positions.

More attention needs to be paid to the hedging strategy once stochastic volatility is introducedin the modeling with the regime-switching lognormal model. Indeed, as presented in Klinget al. (2011), the delta that results in a locally risk-minimizing strategy in the Heston model isnot obtained by simply taking the partial derivative of the guarantee liability value with respectto the stock market. However, as shown in Di Masi et al. (1994), the partial derivative of theguarantee liability with respect to the stock process does result in a locally risk-minimizing

66

strategy when using a regime-switching lognormal model. The delta obtained from Algorithm3.3 can thus be used to determine the stock market positions in the hedge portfolio.

Finally, the hedging strategy for interest rates is not based on the models’ specific risk factors.First of all, in practice, insurance companies often want to cover more sections of the interestrate curve than the number of risk factors used in their modeling. Secondly, implementing ahedging strategy based on the models’ specific risk factors may introduce rather significantmodel risk. Therefore, the hedging strategy is based on the rhos obtained from Algorithm 3.4.

5.4.2 Hedge asset portfolio

The hedging strategy described in section 5.4.1 leads to defining the hedge asset portfolio.The financial instruments used in the hedge portfolio are positions in the stock market indexand the bond market index as well as zero-coupon bonds with the same maturities as thekey-rate rho shocks and short-term zero-coupon bonds. The positions in the stock market andthe bond market indices are used to cover the guarantee’s delta exposures. The positions inthe zero-coupon bonds that match the maturities of the key-rate rhos are used to cover thekey-rate rho exposures. Finally, the difference between the guarantee liability value and thehedge asset portfolio value is invested in short-term zero-coupon bonds.

The hedge portfolio can be defined in a more formal way. Let

HS = HSti , ti ∈ ΩT ,

HP = HPti , ti ∈ ΩT ,

HZCcTj = H

ZCcTjti

, ti ∈ ΩT , j = 1, . . . , nc,

HM = HMti , ti ∈ ΩT

be the processes for the market value of the positions in the stock market index, the bondmarket index, zero-coupon bonds and short-term zero-coupon bonds in the hedge portfolio,respectively. The positions are determined after rebalancing of the hedge portfolio, that is tosay, after the positions have been modified to match the liability sensitivities at a given pointin time.

In order to determine the hedge positions, both the liability and the asset greeks must becomputed. The guarantee liability greeks are determined using the algorithms presented inchapter 3. Because simple financial instruments are used, closed-form formulas are availablefor the asset greeks. They are given by

∆ASti

= 1, ti ∈ ΩT ,

∆APti

= 1, ti ∈ ΩT ,

ρAcTj ,ti= −Tj , ti ∈ ΩT , j = 1, . . . , nc,

67

where nc is the number of zero-coupon bonds.

The market values of the positions in the hedge asset portfolio after rebalancing for ti ∈ ΩT

are then given by

HSti = (∆L

Sti/∆A

Sti)SPti ,

HPti = (∆L

Pti/∆A

Pti)P P

ti ,

HZCcTjti

= ρLcTj ,ti/ρAcTj ,ti

, j = 1, . . . , nc,

HMti = V P

ti −HSti −H

Pti −

∑Tj

HZCcTjti

.

As mentioned above, the part of the hedge portfolio not invested in the stock market, the bondmarket or the zero-coupon bonds that match the key-rate rho shock maturities is invested inshort-term zero-coupon bonds, that is to say, zero-coupon bonds of maturity ∆t. These zero-coupon bonds have no sensitivity to any risk factors.

The process for the total hedge portfolio value after rebalancing is given byH = Hti , ti ∈ ΩT ,where

Hti = HSti +HP

ti +∑Tj

HZCcTjti

+HMti .

Clearly, the total hedge portfolio after rebalancing is such thatHti = V Pti . This equality implies

that any surplus or shortfall in the hedge portfolio is compensated by short-term zero-couponbonds at each point in time. As will be shown, these surpluses and shortfalls are gains andlosses from the company’s perspective.

5.4.3 Hedge efficiency calculation

The hedge portfolio’s composition presented in section 5.4.2 completes the definition of thehedging strategy used in the present work. However, in order to evaluate the efficiency of thehedging strategy, the hedged GLWB guarantee gains and losses need to be measured.

Let

HS∗ = HS

t−i, ti ∈ ΩT ,

HP∗ = HP

t−i, ti ∈ ΩT ,

HZCcTj∗ = H

ZCcTj

t−i, ti ∈ ΩT , j = 1, . . . , nc,

HM∗ = HM

t−i, ti ∈ ΩT

be the processes for the market value of the positions in the stock market index, the bondmarket index, the zero-coupon bonds and the short-term zero-coupon bonds in the hedgeportfolio before rebalancing of the hedge portfolio. The market values of the positions before

68

rebalancing at time ti are obtained by valuing at time ti the hedge positions taken at timeti−1. Note that these processes differentiate themselves from the processes defined in section5.4.2 by looking at market values before rather than after rebalancing of the hedge portfolio.

The market values of the positions in the hedge portfolio before rebalancing are given by

HSt−i+1

= HSti

(SPti+1

SPti

− 1

),

HPt−i+1

= HPti

(P Pti+1

P Pti

− 1

),

HZCcTj

t−i+1

= HZCcTjti

(ZCP(ti+1, ti+1 + Tj −∆t)

ZCP(ti, ti + Tj)− 1

), j = 1, . . . , nc,

HMt−i+1

= HMti

1

ZCP(ti, ti+1).

The process for the total hedge asset portfolio value before rebalancing, H∗ = Ht−i, ti ∈ ΩT ,

is therefore such that

Ht−i+1= HS

t−i+1+HP

t−i+1+∑Tj

HZCcTj

t−i+1

+HMt−i+1

.

The hedge asset portfolio value before rebalancing at time ti will generally not be equal tothe guarantee liability (and hence to the hedge asset portfolio value after rebalancing) at timeti. Differences between the hedge asset portfolio value before and after rebalancing may bepositive or negative. If the hedge asset portfolio value before rebalancing is higher than thevalue after rebalancing, then the company has more than enough money to rebalance theportfolio and thus makes a gain. Otherwise, the company has to finance the hedge portfoliorebalancing and makes a loss. The gains and losses associated with the rebalancing of thehedge portfolio, also known as the hedging errors, have to be assessed in determining theefficiency of a hedging strategy.

Let HE = HE ti , ti ∈ ΩT be the process of hedging errors. Further assume that HE (k), H(k)

and H(k)∗ are realizations of the corresponding processes for outer loop scenario k. Once the

guarantee liability value and the hedge portfolios before and after rebalancing are determinedat all points in time in a given outer loop scenario k, the hedging errors can be computed.They are given by

HE(k)ti+1

= H(k)

t−i+1

−H(k)ti

+tip(µP,(k))

x,t0R

P,(k)ti

1

ZCP,(k)(ti, ti+1)−ti+1p

(µP,(k))

x,t0C

P,(k)ti+1−(V

P,(k)ti+1

− V P,(k)ti

),

i = 0, . . . , ω−x∆t − 1. Recall that V Pti is the decremented guarantee liability value, which is why

no survival probability is applied to it. Moreover, notice that the revenue received at thebeginning of the period is itself invested in short-term zero-coupon bonds.

69

The hedging error in a given time period from ti to ti+1 is thus the difference between thechange in the hedge portfolio value and the change in the guarantee liability value, taking intoaccount any cash flows that are received or paid by the company during that time period.

In order to discount back the hedging errors to time 0, the same discount factors as for thepresent value of unhedged gains and losses presented in (5.1) are used, that is to say, thepathwise short-term discount factors. For a given real-world scenario k, the present value ofhedged gains and losses is given by

PV(k)H =

ω−x∆t∑i=1

HE(k)ti

i−1∏j=0

ZCP,(k)(tj , tj+1). (5.2)

The distribution of the present value of hedged gains and losses obtained from (5.2) is usedin determining the residual risk associated with hedged GLWB guarantees, or equivalently,the hedge efficiency of GLWB guarantees. This distribution is thus paramount for companieswanting to assess the quality of the risk mitigation brought by their hedging strategy.

So far in the present work, two measures of the present value of gains and losses have beenintroduced. First, the unhedged gains and losses measure was presented in (5.1). Then, thehedged gains and losses measure was presented in (5.2). These two measures are in fact closelyrelated to one another. Indeed, for a given outer loop scenario k,

PV(k)H − V P,(k)

0

=

ω−x∆t∑i=1

HE(k)ti

i−1∏j=0

ZCP,(k)(tj , tj+1)

− V P,(k)0

=

ω−x∆t∑i=1

(H

(k)

t−i−H(k)

ti−1+ ti−1p

(µP,(k))

x,t0R

P,(k)ti−1

1

ZCP,(k)(ti−1, ti)− tip

(µP,(k))

x,t0C

P,(k)ti

−(V

P,(k)ti

− V P,(k)ti−1

)) i−1∏j=0

ZCP,(k)(tj , tj+1)− V P,(k)0

=RP,(k)t0

+

ω−x∆t∑i=1

tip(µP,(k))

x,t0

(R

P,(k)ti− CP,(k)

ti

) i−1∏j=0

ZCP,(k)(tj , tj+1)

+

ω−x∆t∑i=1

(H

(k)

t−i−H(k)

ti−1−(V

P,(k)ti

− V P,(k)ti−1

)) i−1∏j=0

ZCP,(k)(tj , tj+1)− V P,(k)0

=PV(k)U +

ω−x∆t∑i=1

(H

(k)

t−i+1

−H(k)ti− V P,(k)

ti−1

(1

ZCP,(k)(ti−1, ti)− 1

)) i−1∏j=0

ZCP,(k)(tj , tj+1). (5.3)

Hence, the present value of hedged gains and losses, after adjusting for the initial guaranteeliability, can be split into the present value of unhedged gains and losses and the present

70

value of the difference between investing in the hedge portfolio and investing in short-termzero-coupon bonds.

5.5 Hedge efficiency analysis

The real-world risk assessment presented in section 5.3 gave a powerful illustration of the riskrelated with unhedged GLWB guarantees. This assessment motivated the need for a hedgingstrategy, which was presented in section 5.4. In this section, the efficiency of the hedgingstrategy is assessed by comparing the hedged and unhedged gains and losses distribution.Moreover, improvements to the hedging strategy are considered.

5.5.1 Risk mitigation through hedging

In order to assess the hedging strategy’s efficiency, the example presented in Figure 5.4 isrevisited, but now considering both hedged and unhedged gains and losses. The contractholder, contractual and outer loop model parameters presented in section 5.2 are still used inthis revisited example. For the inner loop modeling, the same financial market models as inthe real-world measure are used, but with the change of measure described in section 4.2.2.For mortality, the constant mortality improvement model with the parameters presented insection 5.2.4 is assumed. The key-rate rho buckets used for rho hedging are centered aroundmaturities 2, 10 and 30 years. The proportional delta shock sizes are ε = 1% for both thestock market and the bond market, and the rho shock size is c′ = 0.1%. The numbers of outerand inner loop scenarios are NO = 1,000 and NI = 1,000, respectively.

Hedged gains and losses are measured using PV (k)H , presented in (5.2), whereas unhedged gains

and losses are measured using PV (k)U +V

P,(k)0 . As can be inferred from (5.3), comparing PV (k)

H

to PV (k)U + V

P,(k)0 ensures an appropriate assessment of the impact of the hedge portfolio on

the gains and losses distribution while removing focus on the initial guarantee liability value.The resulting smoothed empirical gains and losses distributions are presented in Figure 5.5.

Figure 5.5 shows that hedging reduces expected gains from offering the GLWB guarantee.However, it also shows that the left tail of the gains and losses distribution is significantlyreduced, indicating that the company has greatly mitigated the risk of being in jeopardybecause of its exposure to variable annuity guarantees.

The assessment of the risk mitigation brought by hedging can be formalized by looking ataverage losses in the left tail. The measure of risk in the left tail for hedged gains and lossesis given by

EPVHα =1

NO(1− α)

NO∑k=1

PV(k)H 1

PV(k)H ≤

l∈PV (k)

H ,k=1,...,NO:#PV(k)H ≤l=NO(1−α)

, (5.4)

71

Figure 5.5: Smoothed distributions of the hedged (PV (k)H ) and unhedged (PV (k)

U + VP,(k)

0 )GLWB guarantee gains and losses

where NO is the number of outer loop scenarios. Similarly, the measure of risk in the left tailfor unhedged gains and losses is given by

EPVUα = VP,(k)

0 +1

NO(1− α)

NO∑k=1

PV(k)U 1

PV(k)U ≤

l∈PV (k)

U ,k=1,...,NO:#PV(k)U ≤l=NO(1−α)

.(5.5)

Note that the initial guarantee liability is included in the risk measure for unhedged gains andlosses in (5.5) to ensure a consistent evaluation of risk. Also note that the left-tail scenariosselected may differ depending on whether the guarantee is hedged or not, because a givenscenario may be unfavourable when the guarantee is not hedged and favourable when it is orvice-versa. The resulting left-tail risk measures for various confidence levels are presented inTable 5.8. All results are presented as a percentage of the initial single premium.

α 0 0.6 0.8 0.9 0.95

EPVHα 0.02 -1.26 -1.94 -2.56 -3.10EPVUα 16.42 1.53 -7.73 -16.69 -25.89

Table 5.8: Average tail hedged and unhedged losses (as a % of SP )

As expected, average gains are shown in Table 5.8 to be much higher when the GLWB guar-

72

antee is not hedged. However, losses at the 95% confidence level are also shown to be morethan 8 times smaller when a hedging strategy is in place, which shows the efficiency of hedgingin the tail of the distribution.

5.5.2 Improving the hedging strategy

The hedging strategy used in this thesis has been shown in section 5.5.1 to be successful atreducing the size of the left tail of the GLWB guarantee gains and losses distribution. However,hedging is also shown not to be perfect, and so it is worthwhile to assess potential ways toenhance the dynamic hedging strategy considered. Indeed, in the analysis of longevity riskpresented in chapter 7, two versions of the dynamic hedging strategy are considered to evaluatethe impact of the strategy on the conclusions reached.

The dynamic hedging strategy considered in this thesis only aims to hedge the guaranteeliability’s first order sensitivities. Therefore, within this limit, the stock market and bondmarket hedges cannot be improved. However, in section 5.5.1, only three rho buckets are usedto hedge interest rates. By increasing the number of key-rate rho buckets, a more thoroughhedge of the interest rate risk associated with GLWB guarantees may be obtained.

Table 5.9 presents the hedged gains and losses risk measure defined in (5.4), with standarderrors in brackets, for two additional hedging strategies:

• 5 rho buckets : 1, 5, 10, 20 and 30 years;

• 7 rho buckets : 1, 5, 10, 15, 20, 25 and 30 years.

Hedgingstrategy

EPVHα (as a % of SP )

0 0.6 0.8 0.9 0.95

3 rho buckets 0.02 -1.26 -1.94 -2.56 -3.10(0.04) (0.06) (0.08) (0.11) (0.14)

5 rho buckets 0.01 -1.09 -1.64 -2.15 -2.65(0.04) (0.05) (0.07) (0.10) (0.12)

7 rho buckets 0.01 -1.09 -1.62 -2.13 -2.62(0.04) (0.05) (0.07) (0.09) (0.12)

Table 5.9: Impact of the hedging strategy on hedge efficiency

As shown in Table 5.9, significant gains can be made in the left tail by going from 3 to 5 rhobuckets. However, only negligible differences are seen when changing to a strategy using 7 rhobuckets. Moreover, on average, all strategies lead to similar gains and losses. Thus, in chapter7, hedging strategies with 3 and 5 rho buckets are considered.

73

5.6 Conclusion

In this chapter, a real-world risk assessment of an unhedged GLWB guarantee was used toillustrate the significant tail risk of GLWB guarantees and to motivate the need for an efficienthedging strategy. The dynamic hedging strategy used in this thesis was described in detailsand tail measures of the gains and losses distribution with and without hedging were com-pared. This comparison shows that hedging does reduce the left tail of the gains and lossesdistribution, that is to say, the large tail losses. Improvements to the hedging strategy werealso considered by increasing the number of rho buckets. Improvements in efficiency are ob-served when increasing the number of rho buckets from 3 to 5, but only negligible differencesare seen when further increasing it from 5 to 7.

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Chapter 6

Impact of the guarantee liabilitymodeling on hedge efficiency

6.1 Introduction

The real-world risk assessment and the hedge efficiency analysis presented in chapter 5 makea strong case for hedging GLWB guarantees. Indeed, hedging was shown in section 5.5 tosignificantly reduce the fatness of the left tail of the GLWB guarantee gains and losses distri-bution. Thus, since GLWB guarantees contain mostly systematic risks, hedging is importantfrom a risk management perspective.

The decision to hedge, however, is not an end in itself. In hedging their GLWB guarantees,companies are constantly looking for ways to improve their hedge efficiency. For example, itwas shown in section 5.5.2 that the hedge efficiency can be improved by increasing the numberof rho buckets in the hedging strategy. In this chapter, hedge efficiency is assessed with respectto a more subtle but still important aspect of hedging: the guarantee liability modeling.

In section 5.5, only one combination of models is considered for the guarantee liability valua-tion. However, companies do have to make modeling choices for the hedging of their GLWBguarantees and many considerations can actually influence these choices. Considerations in-clude of course the impact on the efficiency of the hedging strategy, but also the practicabilityand maintainability of the models and computation time. These latter considerations mayprompt companies to choose simpler models for the guarantee liability valuation. Thus, it isimportant to assess how the modeling of the guarantee liability impacts the hedge efficiencyin order to identify the potential benefits of using more complex models.

The parameters used in the inner loop for the various models considered are presented firstin this chapter. Then, the impact of the guarantee liability modeling on hedge efficiency isanalyzed for various combinations of inner loop GLWB guarantee valuation models.

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6.2 Inner loop model parameters

In this section, the parameters used in the various inner loop models are presented. Moreover,although they are not inner loop model parameters per say, the contractual and contractholder parameters considered in this chapter are given.

6.2.1 Stock index

Two models for the stock index under the risk-neutral measure are considered in the GLWBguarantee liability valuation. First of all, the regime-switching lognormal model is considered.The parameters for this model under the risk-neutral measure are obtained through the changeof measure described in section 4.2.2. The volatilities and transition probabilities are thus thesame as in Table 5.1, but the mean return is related to the short-term interest rate as presentedin (4.18).

The second model considered for the risk-neutral modeling of the stock index is the lognormalmodel. In order to maintain consistency between the regime-switching lognormal model andthe lognormal model, the volatility assumption is based on the same set of data as for theregime-switching lognormal model. The volatility for the lognormal model is σS = 0.1541.The mean return in the lognormal model under the risk-neutral measure is related to theshort-term rate as presented in (4.19).

6.2.2 Bond index

As discussed in section 4.2.2, the lognormal model is used both for the stock index and forthe bond index under the risk-neutral measure. The volatility parameter used is the same asunder the real-world measure, that is, σP = 0.04.

6.2.3 Interest rates

Three models are considered for interest rates under the risk-neutral measure. First of all,the G2++ model parameters are obtained through the change of measure described in section4.2.2. Because market prices of risk were already set to 0 since they were not relevant inthe estimation of Babbs and Nowman (1999), the model parameters under the risk-neutralmeasure are the same as the one presented in Table 5.3.

The Hull-White model is also used in the risk-neutral modeling. A parameter estimation ofthe Hull-White model is done in Babbs and Nowman (1999) on the same set of data as forthe G2++ model. These parameters are thus used in the risk-neutral modeling.

The parameters found in Babbs and Nowman (1999) lead to large differences in the volatilityof spot rate movements for long maturities (10 to 30 years) when compared with the G2++model. Since GLWB guarantees are very long-term guarantees, one may presume that volatil-

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ity at these long maturities is an important assumption in the guarantee liability valuation.Therefore, a second set of parameters for the Hull-White model was found heuristically so thatthe spot rate movement volatilities using these parameters are similar to the G2++ model’sfor long-term maturities. This second parameter set is referred to as the “Adjusted” set. Bothsets of parameters are presented in Table 6.1.

Parameter Babbs and Nowman (1999) Adjusted

a 0.1908 0.055σ 0.0132 0.015

Table 6.1: Parameter sets for the Hull-White model

The volatilities of spot rate movements for the two Hull-White model parameter sets can becompared to the G2++ volatilities. For the Hull-White model, the volatility of spot ratemovements is given by

(Var(s(t+ 1, T + 1)− s(t, T ))

) 12 =

(1

T − t

)[(B(a, T − t))2σ2B(2a, 1)

] 12.

For the G2++ model, it is given by

(Var(s(t+ 1, T + 1)− s(t, T ))

) 12 =

(1

T − t

)[(B(a, T − t))2σ2B(2a, 1)

+ (B(b, T − t))2η2B(2b, 1)

+ 2B(a, T − t)B(b, T − t)σηρB(a+ b, 1)] 1

2.

Table 6.2 shows the volatility of spot rate movements for the two Hull-White model parametersets as well as for the G2++ model parameter set.

Maturity G2++Hull-White

Babbs and Nowman (1999) Adjusted

1 year 0.0102 0.0110 0.01425 year 0.0115 0.0078 0.012810 year 0.0111 0.0054 0.011220 year 0.0090 0.0031 0.008930 year 0.0072 0.0021 0.0071

Table 6.2: One-year spot rate movement volatilities

As shown in Table 6.2, the adjusted parameter set leads to spot rate movement volatilities forthe 10, 20 and 30 year maturities that are much more in line with the G2++ model’s.

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Finally, the last risk-neutral interest rate model considered is the deterministic rates model.The deterministic rates model simply consists in rolling down the forward curve, and thusdoes not have any related parameters. Recall that the Canadian swap curve as of December31, 2014 is used as the initial interest rate curve for all interest rate models.

6.2.4 Mortality

As discussed in section 4.3, there is no distinction between the risk-neutral and the real-worldmeasures for longevity risk in this thesis. Thus, the Lee-Carter model parameters used in theguarantee liability valuation are the same as the one presented in Table 5.4 and Figure 5.2.The constant mortality improvement model also uses the same parameters, but with σµ = 0,thereby eliminating any volatility in future mortality improvement.

6.2.5 Contractual and contract holder parameters

Two typical contract holders are considered in the analysis of hedge efficiency presented in thischapter. For the first contract holder, the same contractual and contract holder parametersas presented in Tables 5.5, 5.6 and 5.7 are used. For the second contract holder, the sameparameters are still used, except that x = 50, tw = 15 and p = 0.03. Hence, the secondcontract holder has the same characteristics as the first, but is aged 50 at inception of thecontract, waits 15 years before withdrawing and has a 3% contractual roll-up rate. For easeof presentation, these contract holders will frequently be referred to in the remainder of thisthesis by their ages at inception of the contract, that is “Age 65” and “Age 50”, respectively.

6.3 Hedge efficiency analysis

6.3.1 Hedge efficiency measure

In order to assess the impact of the guarantee liability modeling on hedge efficiency, an ap-propriate risk measure must be defined. The measure of the average hedged tail losses givenin (5.4) is the starting point for the measure considered in this chapter. However, an adjust-ment to this measure is made to improve the comparability of the results when using differentguarantee liability valuation models.

When issuing and hedging a GLWB guarantee, a company must first set the initial guaranteeliability, and then pay for future hedge inefficiencies. The initial guarantee liability may not beequal to zero, because revenues in the guarantee liability valuation come from the guaranteefee, which is contractual. In the present work, it is assumed that the company cannot repricethe GLWB guarantee such as to obtain a zero guarantee liability.

The company’s total gain and loss associated with offering a GLWB guarantee in a given outer

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loop scenario k is hence given by

PV(k)H − V P

0 . (6.1)

where PV (k)H is the present value of future hedged gains and losses defined in (5.2).

The total hedged gain and loss in (6.1) considers both the risk of misestimation of the guaranteeliability and the future hedge inefficiencies. As such, this measure is preferable to PV (k)

H forthe assessment of the impact of the guarantee liability modeling on hedge efficiency. Indeed,PV

(k)H only considers future hedge inefficiencies, and thus ignores that the initial guarantee

liability varies based on the inner loop modeling used.

However, the total hedged gain and loss in (6.1) is fairly dependent on the pricing of theGLWB guarantee. Indeed, if the company has underpriced the GLWB guarantee, the initialguarantee liability is high, and so the total hedged gains and losses distribution is shiftedto the left. Conversely, if the guarantee is overpriced, then the initial guarantee liability isnegative, and so the distribution is shifted to the right. In the present work, it is assumed thatthe company cannot change its pricing. Thus, in order for results not to be overly dependenton the pricing of the guarantee and rather to focus on hedge efficiency, an adjustment is madeto the total hedged gain and loss in (6.1).

The adjustment consists in adding the true initial guarantee liability to the total gain andloss. The true initial guarantee liability is the guarantee liability determined using inner loopmodels that are consistent with the outer loop modeling. The measure of hedged gain andloss from the GLWB guarantee in a given outer loop scenario k considered in this chapter isthus given by

PV(k)H − (V P

0 − VP,∗

0 ),

where V P0 is the initial guarantee liability calculated with the same inner loop modeling as

PV(k)H and V P,∗

0 is the initial guarantee liability calculated using inner loop models that areconsistent with the outer loop models.

The adjustment described above allows to determine the hedge efficiency measure consideredin this chapter. It is given by

EPVHα =1

NO(1− α)

NO∑k=1

(PV

(k)H 1

PV(k)H ≤l∈PV (k)

H ,k=1,...,NO:#PV(k)H ≤l=NO(1−α)

)−(V P0 −V

P,∗0 ),

(6.2)where NO is the number of outer loop scenarios. This measure improves on (5.4) by allowinga better comparability of hedge losses when different guarantee liability modeling choices arecompared.

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6.3.2 Initial guarantee liability

As mentioned in section 6.3.1, the modeling used for the guarantee liability valuation hasa direct impact on the initial guarantee liability value. Table 6.3 thus presents the initialguarantee liabilities for the different combinations of guarantee liability valuation models con-sidered. Initial guarantee liabilities are computed using NI = 1,000 scenarios. Note that thetwo contract holders presented in section 6.2.5 are referred to by their ages at inception of thecontract.

Stockmarket

Interestrates Longevity

V0 (as a % of SP )

Age 65 Age 50

LN Cst Cst 8.73 16.58RSLN Cst Cst 8.53 16.33RSLN HW Cst 8.58 16.82RSLN HW (Adj) Cst 9.13 18.92RSLN G2 Cst 9.72 19.01RSLN G2 LC 9.72 19.02

Table 6.3: Impact of the guarantee liability modeling on the initial guarantee liability

The first three columns of Table 6.3 present the guarantee liability models used. For thestock market, both the lognormal (LN) and regime-switching lognormal (RSLN) models areconsidered. For interest rates, the deterministic rates model (Cst), the Hull-White model(HW), the Hull-White model with adjusted parameters (HW (Adj)) and the G2++ (G2)model are considered. Finally, the constant mortality improvement model (Cst) and Lee-Carter model (LC) are considered for longevity. The guarantee liability modeling is shown inTable 6.3 to become increasingly complex when moving down the table.

Table 6.3 shows that going from the lognormal model to the regime-switching lognormal modelslightly decreases the initial guarantee liability. This is caused by a slightly lower volatility inthe regime-switching lognormal model scenarios compared with the lognormal model scenarios.Stochastic interest rate models, on the other hand, incrementally increase the initial guaranteeliability, as going from one model to the other increases volatility. Finally, the stochasticmortality model has a very small effect on the initial guarantee liability.

6.3.3 Hedge efficiency results

In this section, hedge efficiency is analyzed using the risk measure defined in (6.2) for allcombinations of guarantee liability models presented in Table 6.3.

The outer loop modeling is constant for all hedge efficiency calculations. It is a combinationof the regime-switching lognormal model for the stock index, the two-factor extended Vasicek

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model for interest rates, and the Lee-Carter model for mortality. Hence, the last combination ofguarantee liability models used in Table 6.3 is equivalent under the risk-neutral measure to thereal-world models used. The initial guarantee liability V P,∗

0 used in the adjustment presentedin (6.2) is thus the one found using the regime-switching lognormal model, the G2++ modeland the Lee-Carter model. These guarantee liabilities for the two contract holders consideredare presented on the last line of Table 6.3.

All hedge efficiency calculations are done with 1,000 outer loop scenarios and 1,000 inner loopscenarios. Moreover, the hedging strategy used is a 3-rho bucket strategy centered aroundmaturites 2, 10 and 30 years. The proportional delta shock sizes are ε = 1% for both thestock index and the bond index, and the rho shock size is c′ = 0.1%.

Average tail losses for the 65-year-old contract holder are presented in Table 6.4 (with standarderrors in brackets). Note that the results are presented as a percentage of the initial singlepremium. Hence, for a company with a $1 billion block of GLWB guarantees, the 3.65 lossseen at α = 0.95 for the lognormal model implies a $36.5 million loss.

Stockmarket



0 0.6 0.8 0.9 0.95

LN Cst Cst -0.38 -1.65 -2.38 -3.04 -3.65(0.04) (0.06) (0.09) (0.12) (0.17)

RSLN Cst Cst -0.51 -1.80 -2.54 -3.25 -3.91(0.04) (0.07) (0.09) (0.13) (0.18)

RSLN HW Cst -0.32 -1.59 -2.35 -3.04 -3.69(0.04) (0.07) (0.09) (0.13) (0.17)

RSLN HW (Adj) Cst 0.07 -1.27 -2.06 -2.76 -3.37(0.04) (0.07) (0.09) (0.12) (0.16)

RSLN G2 Cst 0.02 -1.26 -1.94 -2.56 -3.10(0.04) (0.06) (0.08) (0.11) (0.14)

RSLN G2 LC 0.02 -1.26 -1.94 -2.56 -3.10(0.04) (0.06) (0.08) (0.11) (0.14)

Table 6.4: Impact of the guarantee liability modeling on hedge efficiency for the 65-year-oldcontract holder

Table 6.4 shows that the simplest modeling, which ignores stochastic volatility, interest ratesand mortality, leads to losses on average (-0.38). Because the risk measure used in thischapter considers both the misestimation of the initial guarantee liability and future hedgeinefficiencies, a loss on average implies hedge inefficiencies in addition to those related with theinitial guarantee liability underestimation. The lognormal model without stochastic interestrates or mortality is therefore insufficiently complex for the guarantee liability valuation.

The transition from the lognormal model to the regime-switching lognormal model actually

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increases observed losses, both on average and in tail scenarios. This increase is certainlycounterintuitive given that the regime-switching lognormal model is also used in the outer loop.In order to explain this result, the average evolution of V P

ti −VP,S−ti

, a quantity proportional tothe stock index dollar delta, is plotted in the left panel of Figure 6.1 for three of the guaranteeliability models considered: the lognormal model, the regime-switching lognormal model andthe combination of the G2++ model and the regime-switching lognormal model.

Figure 6.1: Average paths of V Pti − V

P,S−ti

(left panel) and of the stock market index (rightpanel)

As illustrated in Figure 6.1, going from the lognormal model to the regime-switching lognormalmodel increases the delta’s magnitude, moving it further away from the one calculated usingthe model combining the G2++ model and the regime-switching lognormal model. A deltathat is more negative implies overhedging the guarantee liability, that is to say, shorting morepositions than required for hedging the stock market delta.

Overhedging does explain higher losses on average, as upward market scenarios lead to losseswhen the company has taken too many short positions. However, as presented in Table 6.4,using the regime-switching lognormal model in the inner loop actually worsens the tail of theloss distribution as well. Losses in the tail may not have been expected, because in poormarket scenarios, overhedging leads to gains.

The reason for the worsening tail when using the regime-switching lognormal model is thatscenarios generating tail losses when the guarantee is hedged are not scenarios with poor

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market returns. Indeed, the right panel of Figure 6.1 shows that, when modeling the guaranteeliability with either the lognormal or regime-switching lognormal models, the average stockmarket index in scenarios that generate tail hedge losses is even higher than in the average ofall scenarios. It is also much higher than the average of the stock index over the unhedged tailscenarios. Thus, scenarios generating tail losses when the guarantee liability is modeled withthe regime-switching lognormal model are scenarios showing rather good market performance.Hence, the heavier left tail when using the regime-switching lognormal model in the guaranteeliability valuation also seems to be caused by losses related with overhedging.

Despite the fact that it has little impact on the initial guarantee liability, introducing stochasticinterest rates through the Hull-White model using the parameters of Babbs and Nowman(1999) helps reducing hedge losses. Gains made are fairly similar across all α values. Sincethe real-world modeling includes significant interest rate volatility, gains from adding interestrate volatility in the guarantee liability valuation were to be expected.

The use of the adjusted parameters for the Hull-White model further reduces losses both inthe tail and on average. The adjusted parameters were determined so that they match moreclosely the G2++ model’s volatility of spot rate movements for long-term maturities. Hence,this result seems to confirm that the volatility of long-term spot rates is important in theGLWB guarantee liability valuation.

The use of the G2++ model still improves results. However, improvements are more pro-nounced in the tail of the distribution, with little or no improvement for α = 0 and α = 0.6.

Finally, adding stochastic mortality in the inner loop modeling has very little impact on theresulting gains and losses. In fact, the liability and greeks are very similar no matter whetherstochastic mortality is included or not. This conclusion may appear at odds with initialexpectations regarding longevity risk. Indeed, longevity risk is a significant systematic riskfor GLWB guarantees because of the life-contingent nature of these guarantees. Therefore,introducing volatility in future longevity was expected to impact the guarantee liability andits sensitivities.

The reason for the negligible difference is twofold. First of all, as mortality is not correlatedwith the stock market, the bond market or interest rates, the expected value, and not the wholedistribution, of the survival probability is the main driver of the liability value. Furthermore,it can be shown that with the parameters used in this thesis, the mean and the median ofthe survival probabilities are very close to each other in the Lee-Carter model. Since theconstant mortality improvement model leads to survival probabilities that correspond to themedian survival probabilities in the Lee-Carter model, the liability values are barely affectedby stochastic mortality.

All in all, modeling reduces inefficiencies for confidence levels greater or equal to 0.6 by 15 to

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25%. Given that companies strive to find ways to improve their hedge efficiency, this level ofreduction may be deemed non-negligible from an insurance company’s perspective.

Table 6.5 presents a similar analysis, but now for the 50-year-old contract holder presented insection 6.2.5.

Stockmarket



0 0.6 0.8 0.9 0.95

LN Cst Cst -1.39 -3.98 -5.22 -6.60 -8.16(0.10) (0.13) (0.21) (0.35) (0.60)

RSLN Cst Cst -1.70 -4.41 -5.78 -7.29 -9.08(0.10) (0.14) (0.23) (0.39) (0.67)

RSLN HW Cst -1.21 -3.73 -5.05 -6.56 -8.42(0.10) (0.14) (0.24) (0.41) (0.71)

RSLN HW (Adj) Cst -0.18 -2.76 -4.12 -5.65 -7.68(0.10) (0.14) (0.24) (0.42) (0.73)

RSLN G2 Cst -0.01 -2.65 -3.64 -4.52 -5.44(0.10) (0.10) (0.14) (0.21) (0.35)

RSLN G2 LC -0.01 -2.65 -3.64 -4.50 -5.40(0.10) (0.10) (0.13) (0.20) (0.34)

Table 6.5: Impact of the guarantee liability modeling on hedge efficiency for the 50-year-oldcontract holder

Results for the 50-year-old contract holder lead to similar conclusions as for the 65-year-old contract holder: the regime-switching lognormal model deteriorates the hedge efficiency,the Hull-White model with the parameters of Babbs and Nowman (1999) and the adjustedparameters both lead to improvements in the hedge efficiency that are mostly parallel acrossall confidence levels, the G2++ model improves the hedge efficiency especially for higherconfidence levels and the Lee-Carter model has only a negligible impact. However, losses tendto be higher for the 50-year-old contract holder. The number of rho buckets in the hedgingstrategy may be a cause for this, as the 50-year-old contract holder is particularly sensitive tolong-term interest rates.

The guarantee liability modeling has larger impacts on hedge efficiency for the 50-year-oldcontract holder than for the 65-year-old contract holder. Indeed, decreases in average lossesof 30 to 33% are observed for confidence levels greater or equal to 0.6. Therefore, once again,modeling can play an important role in improving the hedge efficiency of GLWB guarantees.

6.3.4 Computation efficiency

As discussed in chapter 3, the assessment of hedge efficiency requires a nested stochasticcalculation, which is very computation-intensive. Thus, before concluding this chapter, a

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discussion on computation efficiency with respect to hedge efficiency calculations is made.

The hedge efficiency calculations were coded in R and C++, with much attention devotedto code efficiency. Despite using a native language for the most computation-intensive partsof the calculation, hedge efficiency calculations could take as much as one or two days on an8-core personal computer.

In the present work, efficiency is a concern only to the extent that it does not prevent doingthe illustrative examples presented in chapters 5 to 7. However, the focus is not primarily onefficiency. It is still worth noting some of the techniques that could have been used to furtherreduce run time.

First of all, the hardware involved in the assessment could have been substantially improved.As mentioned above, a single personal computer was used in the calculations, which clearly isnot representative of the hardware actually used by insurance companies. Indeed, insurancecompanies have large grids of several hundred cores (see e.g. Briere-Giroux, G. (2014)).Moreover, it is worth noting that although insurance companies have much more than onecontract holder’s guarantee to value, they often use compression techniques such as groupingor clustering to reduce the number of contracts in their hedge efficiency valuations and thusreduce the run time involved (see e.g. Briere-Giroux, G. (2014)).

Secondly, control variate techniques could have been used to reduce the number of innerloop scenarios required, for example by using the work of Forsyth and Vetzal (2014) as acontrol variate. Inner loop valuations being the most computation-intensive part of the hedgeefficiency calculation, reducing the number of inner loop scenarios while maintaining a goodprecision could have been fairly beneficial.

Finally, other standard techniques such as reducing, as time goes by in outer loop scenarios, thefrequency at which the guarantee liability is computed or the number of inner loop scenarioscould also have been used to reduce the run time.

Efficiency considerations did prevent the assessment of how different parameters may affect theconclusions reached in this chapter. Indeed, results for two contract holders were presented,but other sensitivities may have been considered as well. These sensitivities could be thesubject of future work.

6.4 Conclusion

In this chapter, the impact of the guarantee liability modeling on the hedge efficiency ofGLWB guarantees was assessed. This assessment allowed one to conclude that companiescan make rather substantial gains from an appropriate modeling of the financial variables inthe guarantee liability. In particular, stochastic interest rates, and the volatility structure

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implied by the parameters used, appear to be very important in the GLWB guarantee liabilitymodeling. Hence, although Kling et al. (2011) concluded that the equity market model usedin the guarantee liability valuation has a fairly small impact on hedge efficiency, consideringmore of the relevant systematic risks of GLWB guarantees reveals that this conclusion cannotbe extended to all other risk factors and that the interest rate model used in the guaranteeliability valuation does have a significant impact on hedge efficiency.

Moreover, in complexifying their modeling, companies have to be mindful of what they chooseto model and what they choose not to. Indeed, complexifying the stock market modelingwithout including stochastic interest rates may actually cause additional tail losses because ofthe opposite effect that these two modeling choices have on the stock market delta. Properconsideration of the effect of each modeling choice must therefore be given in order to avoidunintended worsening of the hedge efficiency.

Finally, it appears that the modeling of stochastic mortality in the guarantee liability valuationdoes not have a material impact on hedge efficiency and can be ignored when using the modeland parameters considered in the present work. However, one should not infer from thisconclusion that longevity risk is not significant for GLWB guarantees. Indeed, as will beshown in the next chapter, the impact of longevity risk on hedge efficiency, when approachedfrom another angle, can be shown to be quite significant.

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Chapter 7

Longevity risk analysis

7.1 Introduction

In chapter 6, the modeling of stochastic mortality in the guarantee liability valuation of GLWBguarantees was shown to have a fairly negligible impact on the overall hedge efficiency for themodel and parameters considered.

Longevity risk, however, is perceived as a significant systematic risk of GLWB guaranteesbecause of the life-contingent nature of these guarantees. Therefore, in this chapter, longevityrisk in GLWB guarantees is analyzed from a different perspective. Instead of determining howthe guarantee liability modeling can impact the hedge efficiency as in chapter 6, the impacton hedge efficiency of actual deviations in the mortality experience are assessed.

Outer loop scenarios are meant to model potential paths of the future over which to assessthe hedge efficiency. Stochastic mortality in the outer loop modeling is thus used to analyzethe impact of deviations in the mortality experience on hedge efficiency.

In this chapter, projections with and without stochastic mortality are first compared to as-sess the materiality of longevity risk on average tail hedge losses at various confidence levels.Then, a risk allocation method that allows to compare the relative importance of financialand longevity risks is presented. This method is further extended to consider two componentsof longevity risk from the perspective of a hedged GLWB guarantee, namely a component re-lated to the liability cash flows and a component related to the hedge portfolio’s performance.Finally, diversification between financial and longevity risks is assessed. Projections in whichboth financial and longevity risks are modeled stochastically are compared with projectionsin which financial risks are modeled stochastically, but longevity risk is considered through anactuarial margin for adverse deviations on the deterministic mortality improvement assump-tion.

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7.2 Longevity risk impact on hedge efficiency

In this section, the impact on hedge efficiency of stochastic mortality in the outer loop modelingof the hedge efficiency calculation is assessed. Recall that longevity risk is assumed not to behedged because of the lack of instruments allowing to do so for variable annuities in Canada,and that as such the insurance company is at risk of an adverse deviation in longevity in thepresent work.

Table 7.1 shows the impact of mortality deviations on hedge efficiency by comparing averagetail losses as a percentage of the single premium when stochastic mortality is either ignoredor included in the outer loop modeling. It must be stressed that although Table 7.1 has aformat that is similar to Tables 6.4 and 6.5, the outer loop mortality model, and not the innerloop mortality model, is presented in the first column of the table. Thus, both the constantmortality improvement model (Cst) and the Lee-Carter model (LC) are used in the outer loop.Moreover, the regime-switching lognormal model and the two-factor extended Vasicek modelare used for the stock market and interest rates respectively in the outer loop.

Inner loop models are constant throughout all calculations in this chapter. As such, the hedgeefficiency measure used is the one presented in (5.4). In other words, no initial guaranteeliability adjustment is necessary in this chapter. The inner loop models used are the regime-switching lognormal model for the stock index, the lognormal model for the bond index, theG2++ model for interest rates and the constant mortality improvement model.

The contract holders used in the examples of this chapter are the same as those describedin section 6.2.5 and used in chapter 6. They are referred to by their age at inception of thecontract, that is to say, the 65-year-old contract holder and the 50-year-old contract holder.The numbers of scenarios used in the outer and inner loops are consistent with other chapters,that is, 1,000 scenarios for each loop. The 3-rho hedging strategy is centered around maturities2, 10 and 30 years. The proportional delta shock sizes are ε = 1% for both the stock marketand the bond market, and the rho shock size is c′ = 0.1%. Standard errors are presented inbrackets.

LongevityEPVHα (as a % of SP )

0 0.6 0.8 0.9 0.95

Cst 0.02 -1.13 -1.75 -2.32 -2.81(0.04) (0.06) (0.08) (0.11) (0.14)

LC 0.02 -1.26 -1.94 -2.56 -3.10(0.04) (0.06) (0.08) (0.11) (0.14)

Table 7.1: Impact of mortality deviations on hedge efficiency for the 65-year-old contractholder using a 3-rho hedging strategy

88

As shown in Table 7.1, including mortality deviations through the use of a stochastic mortalitymodel in the outer loop has a fairly significant impact for high confidence levels. For example,losses are increased by 10% at the 95% confidence level when including stochastic mortality.Mortality deviations are also shown to have a negligible impact on average. This result can beexplained by the same argument as the one made in chapter 6 to explain the negligible impactof stochastic mortality on the guarantee liability: in both cases, the mean of the survivalprobabilities, which is very close to the median, is the main driver of the resulting value.

A similar example is considered next, but this time without hedging. Results are presented inTable 7.2.

LongevityEPVUα (as a % of SP )

0 0.6 0.8 0.9 0.95

Cst 16.43 1.52 -7.78 -16.70 -25.91(0.5) (0.9) (1.2) (1.9) (2.6)

LC 16.42 1.53 -7.73 -16.69 -25.89(0.5) (0.9) (1.2) (1.8) (2.6)

Table 7.2: Impact of mortality deviations on average tail losses for the 65-year-old contractholder without hedging

As evidenced by Table 7.2, it is much more difficult to obtain a reasonable convergence onan unhedged basis, because the resulting gains and losses are much more widely spread. It isstill possible to conclude that, for an unhedged GLWB guarantee, mortality deviations havevery negligible impacts for all confidence levels. The negligible impacts indicate that, whenthe guarantee is not hedged, financial risks are so predominant that longevity risk does notcreate any significant differences in tail losses.

Given the impact that removing hedging has on the conclusions reached regarding mortalitydeviations, the impact of improving the hedging strategy can also be analyzed. Using the65-year-old contract holder once again, Table 7.3 shows the tail losses when the 5-rho hedgingstrategy introduced in section 5.5.2 is used.

Despite smaller overall losses with a 5-rho strategy than with a 3-rho strategy, the impact ofintroducing stochastic mortality is slightly larger when the hedging strategy is improved. Thisindicates that once financial risks have been mitigated through a hedging strategy, other risks,such as contract holder-related risks, come out as being more important on a relative basis.

Finally, results for the 50-year-old contract holder are presented in Table 7.4.

Stochastic mortality is shown in Table 7.4 to have larger impacts on tail losses for the 50-year-old contract holder than for the 65-year-old contract holder. Larger impacts for a youngercontract holder are in line with expectations: mortality deviations have more time to occur

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0 0.6 0.8 0.9 0.95

Cst 0.02 -0.90 -1.39 -1.87 -2.31(0.03) (0.04) (0.06) (0.09) (0.13)

LC 0.01 -1.09 -1.64 -2.15 -2.65(0.04) (0.05) (0.07) (0.10) (0.12)



0 0.6 0.8 0.9 0.95

Cst 0.01 -2.46 -3.37 -4.21 -5.08(0.09) (0.09) (0.12) (0.18) (0.29)

LC -0.01 -2.65 -3.65 -4.53 -5.44(0.10) (0.10) (0.14) (0.21) (0.35)


for a younger contract holder, thus increasing the variability in future survival probabilities.

7.3 Risk allocation

Section 7.2 allowed one to make two conclusions regarding longevity risk in hedged GLWBguarantees. First of all, mortality deviations in the outer loop modeling have a significantimpact on the tail losses of a hedged GLWB guarantee. Moreover, risk mitigation of financialrisks may cause other risks, such as longevity risk, to stand out on a relative basis. Inthis section, the last statement is formalized by allocating the global residual risk of hedgedGLWB guarantees between financial and longevity risks. The allocation method used andthe resulting allocations for the examples considered in section 7.2 are first presented. Theallocation method is then extended to split the longevity risk into two risk components, andthe examples are revisited once again.

7.3.1 Risk allocation method

The method used to allocate risk between financial and longevity risks is based on Euler’scapital allocation method. Let Y , X1 and X2 be random variables such that

Y = X1 +X2.

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Then, Euler’s capital allocation method allows to determine the contribution of each randomvariable to the tail value at risk. Indeed, the contribution of factor Xi is given by

CTV aRκ (Xi) =1

1− κ

∫ ∞V aRκ(X1+X2)

E[Xi × 1Y=y] dy, i = 1, 2, (7.1)

such thatTV aRκ(Y ) = CTV aRκ (X1) + CTV aRκ (X2).

In the present work, since the left tail of the gains and losses distribution is of interest, amodified version of Euler’s capital allocation method is used. In this modified version, thecontribution of each risk factor is given by

CTV aR′

κ (Xi) =1

1− κ

∫ V aR1−κ(X1+X2)

−∞E[Xi × 1Y=y]dy, i = 1, 2, (7.2)

such thatTV aR′κ(Y ) = CTV aR

′κ (X1) + CTV aR

′κ (X2),

where TV aR′κ(Y ) is defined as

TV aR′κ(Y ) = −TV aRκ(−Y ),

that is to say, the average of tail losses.

The problem of allocating the residual risk of hedged GLWB guarantees between longevityand financial risks does not fit nicely in the modified Euler’s capital allocation method: thepresent value of hedged gains and losses is not merely a sum of random variables, but rathera non-linear function of several risk factors. In order to express the present value of hedgedgains and losses PVH as a sum of random variables, the Hoeffding decomposition is used.Following Karabey et al. (2014), let

PVH = g(X1, X2),

where X1 represents the financial risk factors and X2 represents the longevity risk factor.Then, the Hoeffding decomposition allows to write PVH as a sum of random variables in thefollowing way:

PVH = EX2 [PVH |X1] + (PVH − EX2 [PVH |X1]) . (7.3)

The first part of (7.3) can be interpreted as the random variable representing the hedgedgains and losses over financial risks alone, whereas the second part can be seen as the randomvariable representing the additional hedged gain or loss contributed by longevity risk.

The interpretation of the risk allocation method presented above in the context of hedgedGLWB guarantees is of primary importance to understand the risk allocation results presentedin section 7.3.2. In this thesis, risk measures are based on average simulated tail scenarios.However, unlike section 7.2, where tail scenarios for calculations with and without stochastic

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mortality were determined independently, tail scenarios used to determine the risk allocationsare the same for the financial risks and for the longevity risk allocations. Tail loss scenariosused are based on the gains and losses distribution in which both financial and longevity risksare modeled stochastically. More formally, using a notation similar to the one used for therisk measure presented in (5.4), the financial risks allocation is given by

1

NO(1− α)

NO∑k=1

E[PVH |X(k)

1

]1

PV(k)H ≤

l∈PV (k)

H ,k=1,...,NO:#PV(k)H ≤l=NO(1−α)

,and the longevity risk allocation is given by

1

NO(1− α)

NO∑k=1

(PV

(k)H − E

[PVH |X(k)

1

])1

PV(k)H ≤

l∈PV (k)

H ,k=1,...,NO:#PV(k)H ≤l=NO(1−α)

.Thus, the risk allocation method truly helps to determine how much of the total risk, asmeasured by the tail value at risk of the loss distribution, can be attributed to the financialrisks and to the longevity risk.

7.3.2 Risk allocation results

In this section, the risk allocation method described in section 7.3.1 is used to revisit theexamples of section 7.2. Risk allocations are presented both as a percentage of the initialsingle premium and as a percentage of the total risk. The latter is referred to as the relativerisk allocation.

The results for the 65-year-old contract holder using the 3-rho hedging strategy are presentedin Table 7.5.

Risk Allocation (as a % of SP ) Relative allocation (in %)

0.6 0.8 0.9 0.95 0.6 0.8 0.9 0.95

Financial -0.99 -1.55 -1.97 -2.30 79 80 77 74Longevity -0.26 -0.38 -0.59 -0.80 21 20 23 26

Total -1.26 -1.94 -2.56 -3.10 100 100 100 100

Table 7.5: Allocation of risk with a 3-rho hedging strategy for the 65-year-old contract holder

The percentage of the total risk allocated to longevity is shown Table 7.5 to be quite significantfor a hedged GLWB guarantee: between 20% and 26% of the total residual risk is accountedfor by longevity risk. The percentage allocation is fairly stable across confidence levels, witha slight increase for higher confidence levels.

It was shown in section 7.2 that a better hedging strategy leads to slightly larger mortalityimpacts. In this section, the consequences of these larger impacts on the risk allocation areinvestigated. The risk allocation for the 5-rho hedging strategy is presented in Table 7.6.

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0.6 0.8 0.9 0.95 0.6 0.8 0.9 0.95


Total -1.09 -1.64 -2.15 -2.65 100 100 100 100


The risk allocated to longevity is shown in Table 7.6 to be greater or equal when using the5-rho hedging strategy than when using the 3-rho hedging strategy both as a percentage of theinitial single premium and as a percentage of the total risk. This result tends to confirm thatthe better the financial risk mitigation, the more important the mortality risk on a relativebasis. Moreover, the relative allocation is shown to be fairly stable across confidence levels.

Finally, the results for the 50-year-old contract holder are presented in Table 7.7.


0.6 0.8 0.9 0.95 0.6 0.8 0.9 0.95


Total -2.65 -3.65 -4.53 -5.44 100 100 100 100


A similar pattern is observed for the 50-year-old contract holder in Table 7.7, that is to say,fairly stable allocations as a percentage of the total risk. The mortality allocation is alsoshown to be slightly larger as a percentage of the single premium than for the 65-year-oldcontract holder. This larger allocation supports the conclusion reached in section 7.2 wherebya younger contract holder allows for more mortality deviations and thus a larger longevity riskimpact.

However, the longevity risk allocations as a percentage of the total risk are shown to be fairlylower than for the 65-year-old contract holder. The lower allocations may be due to thehedging strategy used, which is not as effective for the 50-year-old contract holder as for the65-year-old contract holder, thus giving a larger weight to financial risks.

7.3.3 Extended risk allocation

So far in this section, the risk associated with hedged GLWB guarantees has been allocatedto two types of risks: financial risks and longevity risk. Longevity risk, however, may itselfbe split into two risk components. Indeed, recall from (5.3) that the present value of hedged

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gains and losses in a given outer loop scenario k can be split into component PV (k)U , defined

in (5.1), and componentω−x∆t∑i=1

(H

(k)

t−i+1

−H(k)ti− V P,(k)

ti−1

(1

ZCP,(k)(ti−1, ti)− 1

)) i−1∏j=0

ZCP,(k)(tj , tj+1).

In section 7.3.1, the Hoeffding decomposition is used to split the present value of hedged gainsand losses in a component related with financial risks and a component related with longevityrisk. Using the relationship between hedged and unhedged gains and losses presented in(5.3), the part of the decomposition associated with longevity risk, PVH − E[PVH |X1], canbe rewritten as

PVH − E[PVH |X1]

= (PVU − E[PVU |X1])

+

ω−x∆t∑i=1

(Ht−i+1

−Hti − V Pti−1

(1

ZCP(ti−1, ti)− 1

)) i−1∏j=0

ZCP(tj , tj+1)

−E

ω−x∆t∑i=1

(Ht−i+1

−Hti − V Pti−1

(1

ZCP(ti−1, ti)− 1

)) i−1∏j=0

ZCP(tj , tj+1)

∣∣∣∣∣∣X1

. (7.4)

Thus, longevity risk for hedged GLWB guarantees is itself composed of two risks. First ofall, there is a risk that mortality deviations increase the insurer’s liability from an unhegdedperspective, that is to say, that the actual liability cash flows come out higher than expected.Secondly, there is a risk that mortality deviations alter the hedge portfolio’s performance.

Since the two components of longevity risk are summed to obtain the total longevity risk, themodified Euler’s capital allocation method introduced in (7.2) can be used to allocate riskbetween the two components.

Allocation results for the 65-year-old contract holder using the 3-rho and 5-rho hedging strate-gies are presented in Tables 7.8 and 7.9 respectively. Longevity risk associated with the un-hedged gains and losses distribution, the first part of (7.4), is denoted by “Longevity – PVU ”,whereas longevity risk associated with the hedge portfolio, the second part of (7.4), is denotedby “Longevity – Hedge ptf”.

As shown in Tables 7.8 and 7.9, the impact of the adverse mortality experience on claims andrevenues is the main driver of mortality-related losses. Nevertheless, the impact of mortalitydeviations on the hedge portfolio’s performance holds a significant share of the longevity risk.The impacts related with the hedge portfolio seem to be fairly stable across all confidencelevels as a percentage of the single premium, and to decrease as a percentage of the total risk.

Hence, one can conclude that the impact of mortality deviations on the hedge portfolio’sperformance cannot be overlooked when assessing longevity risk in hedged GLWB guarantees.

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0.6 0.8 0.9 0.95 0.6 0.8 0.9 0.95

Financial -0.99 -1.55 -1.97 -2.30 79 80 77 74Longevity – PVU -0.17 -0.29 -0.50 -0.68 13 15 19 22

Longevity – Hedge ptf -0.09 -0.09 -0.09 -0.12 8 5 4 4Total -1.26 -1.94 -2.56 -3.10 100 100 100 100

Table 7.8: Extended risk allocation with a 3-rho hedging strategy for the 65-year-old contractholder


0.6 0.8 0.9 0.95 0.6 0.8 0.9 0.95

Financial -0.77 -1.15 -1.49 -1.85 71 71 69 70Longevity – PVU -0.22 -0.33 -0.52 -0.65 20 20 24 25

Longevity – Hedge ptf -0.10 -0.15 -0.15 -0.15 9 9 7 5Total -1.09 -1.64 -2.15 -2.65 100 100 100 100

Table 7.9: Extended risk allocation with a 5-rho hedging strategy for the 65-year-old contractholder

As such, hedged GLWB guarantees hold an additional longevity risk component compared withunhedged GLWB guarantees. Moreover, this additional risk component seems to exacerbatethe impact of longevity risk.

7.4 Diversification between longevity and financial risks

Although longevity risk is a systematic risk, the actuarial method commonly used to dealwith the uncertainty associated with the mortality improvement assumption in the Canadianinsurance industry involves adding a margin for adverse deviations on the best estimate as-sumption. Adding a margin for adverse deviations implies using an assumption in the hedgeefficiency calculation that is more conservative than the best estimate assumption.

However, for a hedged GLWB guarantee, since financial risks and longevity risk are often seenas not exhibiting correlation with one another, margins for adverse deviations that do notrecognize diversification can introduce undue conservatism in the valuation. In this section,diversification between financial and longevity risks is assessed by comparing projections inwhich both financial and longevity risks are modeled stochastically to projections in whichlongevity risk is modeled deterministically and conservatism is introduced through a marginfor adverse deviations on the mortality improvement assumption.

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7.4.1 Margins for adverse deviations

Margins for adverse deviations in Canada are usually presented as a multiple of the bestestimate assumption for non-economic risks. However, in the present work, the stochasticmortality model used allows to compute quantiles of the future mortality distribution. Hence,in order to be consistent with the chosen models and thus allow a more relevant analysis,quantiles of the mortality index in the Lee-Carter model are used to determine assumptionswith margins for adverse deviations.

The deterministic mortality index with margin for adverse deviations is given by

κti = κti−1 + θ∆t+ σµ√

∆tΦ−1(q), ti ∈ ΩT , (7.5)

where 1 − q is the desired confidence level of the margin for adverse deviations. Marginsgenerate an increase in tail losses for q < 0.5. Indeed, the lower the value of κt, the lowerthe force of mortality, and thus the higher the resulting survival probabilities. High survivalprobabilities imply more claim payments for the insurance company in the GLWB guarantee.

The median as well as various quantiles of the future path of κt are presented in Figure7.1. Note that using the median of the future path of κt is equivalent to using the constantmortality improvement model since Φ−1(0.5) = 0.

Figure 7.1: Quantiles of the future path of κt used as assumptions with margins for adversedeviations

Since hedge efficiency calculations involve both outer loop and inner loop simulations, it is

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important to stress that the margin for adverse deviations is only added to the outer loopmortality assumption. The inner loop mortality model remains the constant mortality im-provement model, which is equivalent to projecting at the median of the future path of κt.

7.4.2 Results

Results for the 65-year-old contract holder using a 3-rho hedging strategy are presented inTable 7.10. Note that the same assumptions as in section 7.2 are used regarding the models,the number of scenarios, the hedging strategies and the shock sizes.

The outer loop mortality models used are given in the first column. When mortality im-provement is deterministic with a margin for adverse deviations as presented in (7.5), theconfidence level q used to determine the quantile of the future path of κt is presented in thesecond column.

Longevity Quantileof κt (q)


0 0.6 0.8 0.9 0.95

LC n/a 0.02 -1.26 -1.94 -2.56 -3.10(0.04) (0.06) (0.08) (0.11) (0.14)

Cst 50%0.02 -1.13 -1.75 -2.32 -2.81(0.04) (0.06) (0.08) (0.11) (0.14)

Cst 40%-0.13 -1.31 -1.95 -2.54 -3.05(0.04) (0.06) (0.08) (0.11) (0.14)

Cst 30%-0.30 -1.50 -2.17 -2.78 -3.30(0.04) (0.06) (0.08) (0.11) (0.14)

Cst 5%-1.00 -2.33 -3.11 -3.83 -4.44(0.04) (0.07) (0.10) (0.12) (0.16)

Table 7.10: Assessment of the diversification between financial and longevity risks for the65-year-old contract holder using a 3-rho hedging strategy

Thus, in Table 7.10, the first line shows the results obtained when both financial risks andlongevity risk are modeled stochastically. The second line gives the simulation results using thebest estimate mortality improvement assumption, that is to say, using the constant mortalityimprovement model. Finally, the following lines present the simulation results at variousquantiles of the distribution of κt, or equivalently, the simulation results determined usingassumptions with margins for adverse deviations.

The results obtained using the constant mortality improvement model can be seen as thebaseline results. Impacts of either introducing stochastic mortality or using assumptions withmargins for adverse deviations are thus measured with respect to these results.

The impacts of margins for adverse deviations on average tail losses are shown in Table 7.10 to

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be more stable across all confidence levels than the impacts of introducing stochastic mortalityin the modeling. Indeed, the differences in average tail losses between results obtained usingstochastic mortality and baseline results are shown to increase significantly with the valueof α, whereas they are more stable when assumptions with margins for adverse deviationsare used. This was expected since stochastic mortality introduces a distribution in futuremortality improvement, whereas a margin for adverse deviations merely shifts the mean, thusaffecting all scenarios.

Because margins for adverse deviations have more stable impacts across all confidence levels,margins at a given quantile of κt may lead to average tail losses that are higher than whenusing stochastic mortality for low α values, but that are lower for high alpha values. Forexample, using the 40th percentile of the future path of κt is sufficient to cover the tail lossescalculated with stochastic mortality for α = 0.8, but not for α = 0.95. In all cases, usingthe 30th percentile of the future path of κt is more than enough to cover the tail losses withstochastic mortality at all confidence levels, illustrating the important diversification betweenfinancial and longevity risks.

Table 7.10 also allows to make an important inference regarding diversification benefits. Di-versification benefits measure how much recognizing the interaction between risks can reduceaverage tail losses compared with adding conservatism on each assumption independently. Inthis chapter, results using a stochastic projection of both financial and longevity risks properlyrecognize the interaction between these risks, whereas results using a margin for adverse devi-ations on the mortality improvement assumption introduce conservatism on both the financialand longevity risks. Thus, a large diversification benefit implies that the stochastic projectionof both financial and longevity risks leads to a large reduction in average tail losses comparedwith a projection using a margin for adverse deviations.

It is shown in Table 7.10 that when using high values of α and low values of q, not recognizingthe interaction between risks leads to average tail losses that are much more overestimatedthan when using low values of α and values of q closer to 0.5. For example, using α = 0.9 andq = 0.05, the diversification benefit is 1.27, whereas it is only 0.24 when using α = 0.6 andq = 0.3. This result suggests that the level of diversification depends on the overall confidencelevel considered in the simulation. Properly recognizing diversification thus appears veryimportant for calculations using high confidence levels such as required capital calculations.

In order to assess the sensitivity of the diversification benefits to the hedging strategy, a similarexample using the 5-rho hedging strategy is presented in Table 7.11.

The hedging strategy is shown in Table 7.11 to have an effect on the level of diversification.Indeed, results with stochastic mortality shift from being fairly close to the 40th percentile ofthe simulated path of κt in Table 7.10 to being closer to the 30th percentile in Table 7.11.Moreover, diversification benefits are smaller using the 5-rho hedging strategy than using the

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0 0.6 0.8 0.9 0.95

LC n/a 0.01 -1.09 -1.64 -2.15 -2.65(0.04) (0.05) (0.07) (0.10) (0.12)

Cst 50%0.02 -0.90 -1.39 -1.87 -2.31(0.03) (0.04) (0.06) (0.09) (0.13)

Cst 40%-0.14 -1.06 -1.56 -2.05 -2.50(0.03) (0.04) (0.06) (0.09) (0.13)

Cst 30%-0.31 -1.24 -1.75 -2.24 -2.71(0.03) (0.05) (0.07) (0.09) (0.13)

Cst 5%-1.00 -2.05 -2.62 -3.15 -3.68(0.04) (0.05) (0.07) (0.10) (0.14)


3-rho hedging strategy. This decrease in diversification suggests that the greater the mitigationof financial risks, the smaller the diversification benefits between financial and longevity risks.

The negative relationship between the quality of the risk mitigation strategy and the size ofdiversification benefits is in line with expectations: improving the financial risk mitigationstrategy reduces the residual financial risk of hedged GLWB guarantees. Since diversificationbenefits measure how much recognizing the interaction between risks can reduce average taillosses, a smaller residual financial risk implies a lower potential for a decrease in total riskthrough diversification.

Significant diversification can still be observed using the 5-rho hedging strategy, as the 30th

percentile of the distribution of κt is still sufficient to cover losses calculated with stochasticmortality at very high confidence levels (α = 0.95).

Finally, results for the 50-year-old contract holder are presented in Table 7.12.

Similar conclusions as for the 65-year-old contract holder can be reached for the 50-year-oldcontract holder: the results of the projection including stochastic mortality are between theresults obtained with margins for adverse deviations at the 30th and the 40th percentile ofκt, and diversification benefits increase with the overall confidence level of the calculation.Furthermore, Table 7.12 shows larger diversification benefits than for the 65-year-old contractholder. As mentioned in section 7.3.2, the 3-rho hedging strategy is not as effective for the50-year-old contract holder than for the 65-year-old contract holder. The residual financialrisk is thus higher for the 50-year-old contract holder, which may explain larger diversificationbenefits.

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0 0.6 0.8 0.9 0.95

LC n/a -0.01 -2.65 -3.65 -4.53 -5.44(0.10) (0.10) (0.14) (0.21) (0.35)

Cst 50%0.01 -2.46 -3.37 -4.21 -5.08(0.09) (0.09) (0.12) (0.18) (0.29)

Cst 40%-0.27 -2.69 -3.61 -4.47 -5.40(0.09) (0.09) (0.12) (0.19) (0.31)

Cst 30%-0.56 -2.94 -3.87 -4.76 -5.75(0.09) (0.09) (0.13) (0.20) (0.33)

Cst 5%-1.78 -4.08 -5.13 -6.26 -7.58(0.09) (0.11) (0.16) (0.27) (0.44)


7.5 Conclusion

In this chapter, mortality deviations were shown to impact the hedge efficiency of GLWBguarantees through a comparison of average tail hedge losses with and without stochasticmortality in the outer loop modeling. Moreover, using a modified version of Euler’s capitalallocation method and the Hoeffding decomposition, it was possible to allocate the global riskof hedged GLWB guarantees between financial risks and longevity risk. It was shown thatlongevity risk holds a significant share of the total risk for a hedged GLWB guarantee andthat this share increases when the hedging strategy is improved.

The risk allocation method discussed above was further extended to consider two componentsof longevity risk. It was shown that the increase in liability cash flows caused by an adverselongevity experience holds the largest share of a hedged GLWB guarantee’s longevity risk,but that the impact of mortality deviations on the hedge portfolio’s performance also holdsa significant share of the longevity risk. Hedged GLWB guarantees thus have an additionallongevity risk component compared with unhedged GLWB guarantees that exacerbates thetotal longevity risk.

Finally, the issue of diversification between financial and longevity risks for a hedged GLWBguarantee was investigated by considering how fixed margins for adverse deviations on themortality improvement assumption compared with a stochastic projection of both financialand longevity risks. Significant diversification was shown to exist between financial risks andlongevity risk. Diversification benefits were shown to increase with the overall confidence levelof the calculation, and to decrease with the quality of the risk mitigation of financial risks.

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Conclusion

Variable annuity guarantees, and particularly guaranteed lifetime withdrawal benefit (GLWB)guarantees, have become very important in the wealth management industry. These guaran-tees, which provide to clients a guaranteed stream of revenue while allowing them to retainequity market participation, exhibit significant systematic risks from the issuer’s standpoint.Risk management of GLWB guarantees thus is a main concern for insurance companies, whichhave turned to capital market hedging as the most straightforward and effective risk manage-ment method.

From a risk management perspective, the assessment of the distribution of the residual hedgedGLWB guarantee gains and losses is of primary importance. Indeed, because a perfect hedgeof GLWB guarantees is unattainable, hedge inefficiencies imply potential financial liabilitiesfor insurance companies. However, although a relatively substantial body of literature ex-ists regarding the pricing of GLWB guarantees, the assessment of hedge efficiency for theseguarantees has attracted somewhat less attention in the academic literature.

In this thesis, a typical GLWB guarantee including popular features such as ratchets androll-ups is considered. Algorithms for the guarantee liability valuation and the calculationof the guarantee liability and its sensitivities at future points in time in outer loop scenariosare developed. A risk assessment of GLWB guarantees from an unhedged perspective is thenused to motivate the need for an efficient hedging strategy for these guarantees. The hedgingstrategy considered is a dynamic hedging strategy on the first order sensitivity of the guaranteeliability to the stock market, the bond market and interest rates.

This thesis is interested in the impact of the guarantee liability modeling on the hedge efficiencyof GLWB guarantees with respect to three significant systematic risks for these guarantees,namely the stock market, interest rate and longevity risks. The present work thus aims toextend the hedge efficiency analysis presented in Kling et al. (2011), which focuses on thestock market risk. The analysis carried out in this thesis shows that although Kling et al.(2011) concluded that the stock market model used in the guarantee liability valuation doesnot have a large impact on hedge efficiency, the conclusion cannot be extended when more ofthe relevant risk factors of GLWB guarantees are considered. Indeed, interest rate volatilityhas a significant impact on hedge efficiency and seems to be of particular importance for the

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modeling of the GLWB guarantee liability. Moreover, one can conclude that care must betaken in selecting guarantee liability valuation models, as complexifying the modeling of oneof the risk factors may not lead to improvements in the hedge efficiency. For example, in thepresent work, going from the lognormal model to the regime-switching lognormal model doesnot improve hedge efficiency when stochastic interest rates are ignored. Finally, stochasticmortality in the guarantee liability modeling seems to have only a negligible impact on hedgeefficiency for the particular model considered.

Longevity risk is further analyzed through an assessment of the impact of deviations in themortality experience on the hedge efficiency of GLWB guarantees. Stochastic mortality in theouter loop modeling is shown to have a significant impact on the tail losses of hedged GLWBguarantees. Moreover, a risk allocation method based on Euler’s capital allocation methodand the Hoeffding decomposition is used to allocate the total risk of hedged GLWB guaran-tees between financial and longevity risks. The allocation method leads one to conclude thatlongevity risk holds a significant share of the total residual risk of hedged GLWB guarantees,and that this share increases with the quality of the financial risk mitigation. Longevity riskis further allocated to the increase in the unhedged liability and to the impact of mortalitydeviations on the hedge portfolio’s performance. The former is shown to hold the largest shareof longevity risk, but the latter is also shown to hold a significant share of the total longevityrisk, indicating that hedged GLWB guarantees have an additional longevity risk componentcompared with unhedged GLWB guarantees. Finally, diversification benefits between financialand longevity risks are assessed through a comparison of hedge efficiency when both financialand longevity risks are modeled stochastically and when traditional actuarial margins for ad-verse deviations are used for longevity risk. Diversification benefits between these systematicrisks are shown to be significant and to increase with the overall confidence level of the calcula-tion. Diversification is also shown to decrease with the quality of the financial risk mitigationstrategy.

Several areas of future research may be considered. First of all, additional sensitivities to pa-rameters in the analysis of how guarantee liability models impact the hedge efficiency may beperformed. Moreover, other systematic risks of GLWB guarantees, such as risks related withcontract holder behaviour like lapse or fund mix, may be added to the analysis. Alternativehedging strategies may also be considered to assess whether the conclusions reached in thisthesis are robust to the risk mitigation strategy used. These areas of future research mayprompt the need for more efficient computations, which may be attained through the tech-niques discussed in section 6.3.4. Finally, in the analysis of the impact of mortality deviationson hedge efficiency, more complex stochastic mortality models may be considered to assessthe robustness of the conclusions reached regarding risk allocation and diversification to themortality model used.

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Appendix A

Basic numerical procedures

A.1 Introduction

This appendix details basic numerical procedures related with option valuation through MonteCarlo simulation and greeks calculation using finite difference techniques. These proceduresare covered extensively in the option valuation literature (see e.g. Glasserman (2003)).

A.2 Monte Carlo simulation for option valuation

Following Glasserman (2003), let Y (θ1, θ2, . . . , θm) be a random variable representing thediscounted payoff of an option, θ1, θ2, . . . , θm be market parameters that influence the option’sprice and α(θ1, θ2, . . . , θm) be the actual price of the option.

The option price is given by

α(θ1, θ2, . . . , θm) = E[Y (θ1, θ2, . . . , θm)]

= limn→∞

1

n

n∑i=1

Yi(θ1, θ2, . . . , θm)

= limn→∞

Yn(θ1, θ2, . . . , θm),

where Y1(θ1, θ2, . . . , θm), . . . , Yn(θ1, θ2, . . . , θm) are independent simulations of the discountedoption payoff and Yn(θ1, θ2, . . . , θm) is the empirical mean of the simulations.

When no closed-form formula is available for E[Y (θ1, θ2, . . . , θm)], Monte Carlo simulation canbe used to approximate the expectation. Indeed, the option price can be estimated as

α(θ1, θ2, . . . , θm) = Yn(θ1, θ2, . . . , θm) =1

n

n∑i=1

Yi(θ1, θ2, . . . , θm), n ∈ N.

Thus, the valuation of an option can be done by simulating a large number of random pathsof the relevant risk factors and taking the empirical mean of the resulting discounted payoffs.

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A.3 Finite difference techniques for greeks calculation

Building on the notation defined in section A.2 and letting α ≡ α(θ1, θ2, . . . , θm) for ease ofpresentation, finite difference techniques can be used in the approximation of the option’ssensitivities. By definition, the partial derivative of the option with respect to one of its riskfactors is given by

∂α

∂θi= lim

h→0

α(θ1, θ2, . . . , θi + h, . . . , θm)− α(θ1, θ2, . . . , θi, . . . , θm)

h

= limh→0

α(θ1, θ2, . . . , θi + h, . . . , θm)− αh

.

Furthermore, using the Taylor series expansion around α, the shocked option price can berewritten as

α(θ1, θ2, . . . , θi + h, . . . , θm) = α+ h∂α

∂θi+

1

2h2∂

2α

∂θ2i

+1

6h3∂

3α

∂θ3i

+ · · · (A.1)

Thus,α(θ1, θ2, . . . , θi + h, . . . , θm)− α

h=∂α

∂θi+

1

2h∂2α

∂θ2i

+1

6h2∂

3α

∂θ3i

+ · · ·

and

∂α

∂θi=α(θ1, θ2, . . . , θi + h, . . . , θm)− α

h− 1

2h∂2α

∂θ2i

− · · ·

=α(θ1, θ2, . . . , θi + h, . . . , θm)− α

h+ o(h).

When no closed-form formula exists for the partial derivative, it can be approximated as

∂α

∂θi≈ α(θ1, θ2, . . . , θi + h, . . . , θm)− α(θ1, θ2, . . . , θi, . . . , θm)

h,

the forward-difference estimator of the sensitivity of α with respect to θi. This approximationimplies that both the baseline option price and a price with a shocked risk factor must becalculated. The error in the estimation of the partial derivative is then proportional to theshock used on risk factor θi in the calculation of the greek.

A similar development would lead one to conclude that

α− α(θ1, θ2, . . . , θi − h, . . . , θm)

h,

the backward-difference estimator of the sensitivity of α with respect to θi, is an equally goodcandidate to estimate the partial derivative. Indeed,

α(θ1, θ2, . . . , θi − h, . . . , θm) = α− h ∂α∂θi

+1

2h2∂

2α

∂θ2i

− 1

6h3∂

3α

∂θ3i

+ · · · (A.2)

104

and so

∂α

∂θi=α− α(θ1, θ2, . . . , θi − h, . . . , θm)

h+

1

2h∂2α

∂θ2i

+ · · ·

=α− α(θ1, θ2, . . . , θi − h, . . . , θm)

h+ o(h).

Finally, another candidate for approximating the partial derivative is the central-differenceestimator, given by

α(θ1, θ2, . . . , θi + h, . . . , θm)− α(θ1, θ2, . . . , θi − h, . . . , θm)

2h.

The Taylor expansions in (A.1) and (A.2) presented above lead to

α(θ1, θ2, . . . , θi + h, . . . , θm)− α(θ1, θ2, . . . , θi − h, . . . , θm) = 2h∂α

∂θi+ 0 +

2

6h3∂

3α

∂θ3i

+ · · ·

and

∂α

∂θi=α(θ1, θ2, . . . , θi + h, . . . , θm)− α(θ1, θ2, . . . , θi − h, . . . , θm)

2h− 1

6h2∂

3α

∂θ3i

+ · · ·

=α(θ1, θ2, . . . , θi + h, . . . , θm)− α(θ1, θ2, . . . , θi − h, . . . , θm)

2h+ o(h2).

Approximating the partial derivative using the central-difference estimator thus leads to anerror proportional to h2. Since the value of h used in computing the finite difference is small,the central-difference estimator usually results in a more precise approximation of the partialderivative than the forward-difference or the backward-difference estimators.

105

Appendix B

Basic probability theory

B.1 Introduction

The aim of this appendix is to review the basics of probability theory and to formalize someof the concepts related with the change of measure. The definitions and theorems presentedin this appendix are from Shreve (2004).

B.2 Definitions

Definition 1. σ-algebra

Let Ω be a nonempty set, and let F be a collection of subsets of Ω. F is said to be a σ-algebraprovided that:

1. the empty set ∅ belongs to F

2. whenever a set A belongs to F , its complement Ac also belongs to F , and

3. whenever a sequence of sets A1, A2, . . . belongs to F , their union⋃∞n=1An also belongs

to F .

A σ-algebra contains all relevant events to which a probability can be assigned. A σ-algebrathus can be interpreted as containing all of the available information on the events of a givenprocess.

Definition 2. Probability measure and probability space

Let Ω be a nonempty set, and let F be a σ-algebra of subsets of Ω. A probability measure Pis a function that, for every set A ∈ F , assigns a number in [0, 1], called the probability of Aand written P(A). It is required that:

107

1. P(Ω) = 1

2. (countable additivity) whenever A1, A2, . . . is a sequence of disjoint sets in F , then

P

( ∞⋃n=1

An

)=∞∑n=1

P(An).

The triple (Ω,F ,P) is called a probability space.

Moreover, if no probability measure is defined, that is, if only the double (Ω,F) is considered,then the double is a measurable space. Thus, a probability measure simply assigns probabilitiesto the events in the σ-algebra. Notice that as long as the conditions are met, many probabilitymeasures can be defined for a given measurable space.

Definition 3. Random variable

Let (Ω,F ,P) be a probability space. A random variable is a real-valued function X definedon Ω such that, for all x ∈ R,

ω ∈ Ω : X(ω) ≤ x ∈ F .

The random variable is then said to be F-measurable.

Definition 4. σ-algebra generated by a random variable

A σ-algebra F is generated by a random variable X when F is the smallest σ-algebra suchthat X is F-measurable. This is denoted by F = σ(X).

Definition 5. Filtration

Let Ω be a nonempty set. Let T be a fixed positive number, and assume that for each t ∈ [0, T ]

there is a σ-algebra F(t). Assume further that if s ≤ t, then every set in F(s) is also in F(t).Then the collection of σ-algebras F(t), 0 ≤ t ≤ T, is called a filtration.

The filtration is denoted by FT = F(t), 0 ≤ t ≤ T . A filtration is a collection of σ-algebras inwhich a given σ-algebra contains at least as much information as the σ-algebras that precededit. Thus, a filtration can be interpreted as containing all the past and present informationabout a process. For example, FT contains all information for time 0 ≤ t ≤ T .

Definition 6. Equivalent probability measure

Let Ω be a nonempty set, and let F be a σ-algebra of subsets of Ω. Two probability measures Pand P on (Ω,F) are said to be equivalent if they agree on which sets in F have zero probability.

In other words, probability measures P and P are equivalent if and only if

P(A) = 0 ⇐⇒ P(A) = 0, ∀A ∈ F .

108

B.3 Change of measure

Theorem 1. Radon-Nikodým

Let P and P be equivalent probability measures defined on (Ω,F). Then there exists an almostsurely positive random variable Z such that E[Z] = 1 and

P(A) =

∫AZ(ω)dP(ω) for every A ∈ F .

Theorem 2. Girsanov in one dimension

Let W (t), 0 ≤ t ≤ T, be a Brownian motion on a probability space (Ω,F ,P), and also letF(t), 0 ≤ t ≤ T , be a filtration for this Brownian motion. Let Θ(t), 0 ≤ t ≤ T, be an adaptedprocess. Define

Z(t) = e−∫ t0 Θ(u)dW (u)− 1

2

∫ t0 Θ2(u)du,

W (t) = W (t) +

∫ t

0Θ(u)du,

and assume that

E[∫ T

0Θ2(u)Z2(u)du

]<∞.

Set Z = Z(T ). Then E[Z] = 1 and under the probability measure P given by

P(A) =

∫AZ(ω)dP(ω) for every A ∈ F ,

the process W (t), 0 ≤ t ≤ T, is a Brownian motion.

Theorem 3. Girsanov in multiple dimensions

Let W (t) = (W1(t), . . . ,Wd(t)), be a multidimensional Brownian motion on a probability space(Ω,F ,P), and let F(t), 0 ≤ t ≤ T, be a filtration for this Brownian motion. Let T be a fixedpositive time and let Θ(t) = (Θ1(t), . . . ,Θd(t)) be a d-dimensional adapted process. Define

Z(t) = e−∫ t0 Θ(u)dW (u)− 1

2

∫ t0 ||Θ(u)||2du,

W (t) = W (t) +

∫ t

0Θ(u)du,

and assume that

E[∫ T

0||Θ(u)||2Z2(u)du

]<∞.

Set Z = Z(T ). Then E[Z] = 1 and under the probability measure P given by

P(A) =

∫AZ(ω)dP(ω) for every A ∈ F ,

the process W (t) is a d-dimensional Brownian motion.

109

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