Théorie Financière 2004-2005 Relation risque – rentabilité attendue (1) Professeur André Farber

Théorie Financière2004-2005Relation risque – rentabilité attendue (1)

Professeur André Farber

Tfin 2004 07 Risk and return (1) |2August 23, 2004

Introduction to risk

• Objectives for this session :

– 1. Review the problem of the opportunity cost of capital

– 2. Analyze return statistics

– 3. Introduce the variance or standard deviation as a measure of risk for a portfolio

– 4. See how to calculate the discount rate for a project with risk equal to that of the market

– 5. Give a preview of the implications of diversification


Setting the discount rate for a risky project

• Stockholders have a choice:

– either they invest in real investment projects of companies

– or they invest in financial assets (securities) traded on the capital market

• The cost of capital is the opportunity cost of investing in real assets

• It is defined as the forgone expected return on the capital market with the same risk as the investment in a real asset


Uncertainty: 1952 – 1973- the Golden Years

• 1952: Harry Markowitz*

– Portfolio selection in a mean –variance framework

• 1953: Kenneth Arrow*

– Complete markets and the law of one price

• 1958: Franco Modigliani* and Merton Miller*

– Value of company independant of financial structure

• 1963: Paul Samuelson* and Eugene Fama

– Efficient market hypothesis

• 1964: Bill Sharpe* and John Lintner

– Capital Asset Price Model

• 1973: Myron Scholes*, Fisher Black and Robert Merton*

– Option pricing model


Three key ideas

• 1. Returns are normally distributed random variables

• Markowitz 1952: portfolio theory, diversification

• 2. Efficient market hypothesis

• Movements of stock prices are random

• Kendall 1953

• 3. Capital Asset Pricing Model

• Sharpe 1964 Lintner 1965

• Expected returns are function of systematic risk


Preview of what follow

• First, we will analyze past markets returns.• We will:

– compare average returns on common stocks and Treasury bills

– define the variance (or standard deviation) as a measure of the risk of a portfolio of common stocks

– obtain an estimate of the historical risk premium (the excess return earned by investing in a risky asset as opposed to a risk-free asset)

• The discount rate to be used for a project with risk equal to that of the market will then be calculated as the expected return on the market:

Expected return on the market

Current risk-free rate

Historical risk premium

= +


Implications of diversification

• The next step will be to understand the implications of diversification.

• We will show that:

– diversification enables an investor to eliminate part of the risk of a stock held individually (the unsystematic - or idiosyncratic risk).

– only the remaining risk (the systematic risk) has to be compensated by a higher expected return

– the systematic risk of a security is measured by its beta (), a measure of the sensitivity of the actual return of a stock or a portfolio to the unanticipated return in the market portfolio

– the expected return on a security should be positively related to the security's beta

Normal distribution


Returns

• The primitive objects that we will manipulate are percentage returns over a period of time:

• The rate of return is a return per dollar (or £, DEM,...) invested in the asset, composed of

– a dividend yield

– a capital gain

• The period could be of any length: one day, one month, one quarter, one year.

• In what follow, we will consider yearly returns

1

1

1

t

tt

t

tt P

PP

P

divR


Ex post and ex ante returns

• Ex post returns are calculated using realized prices and dividends

• Ex ante, returns are random variables

– several values are possible

– each having a given probability of occurence

• The frequency distribution of past returns gives some indications on the probability distribution of future returns


Frequency distribution

• Suppose that we observe the following frequency distribution for past annual returns over 50 years. Assuming a stable probability distribution, past relative frequencies are estimates of probabilities of future possible returns .

Realized Return Absolutefrequency

Relativefrequency

-20% 2 4%

-10% 5 10%

0% 8 16%

+10% 20 40%

+20% 10 20%

+30% 5 10%

50 100%


Mean/expected return

• Arithmetic Average (mean)

– The average of the holding period returns for the individual years

• Expected return on asset A:

– A weighted average return : each possible return is multiplied or weighted by the probability of its occurence. Then, these products are summed to get the expected return.

N

RRRRMean N

...21

1...

return ofy probabilit with

...)(

21

2211

n

ii

nn

ppp

Rp

RpRpRpRE


Variance -Standard deviation

• Measures of variability (dispersion)

• Variance

• Ex post: average of the squared deviations from the mean

• Ex ante: the variance is calculated by multiplying each squared deviation from the expected return by the probability of occurrence and summing the products

• Unit of measurement : squared deviation units. Clumsy..

• Standard deviation : The square root of the variance

• Unit :return

VarR R R R R R

TT

2 12

22 2

1( ) ( ) ... ( )

Var R Expected RA A A( ) ) 2 2 val ue of (RA

Var R p R R p R R p R RA A A A A A N A N A( ) ( ) ( ) ... ( ), , , 21 1

22 2

2 2

SD R Var RA A A( ) ( )


Return Statistics - Example

Return Proba Squared Dev-20% 4% 0.08526-10% 10% 0.03686

0% 16% 0.0084610% 40% 0.0000620% 20% 0.0116630% 10% 0.04326

Exp.Return 9.20%Variance 0.01514Standard deviation 12.30%


Normal distribution

• Realized returns can take many, many different values (in fact, any real number > -100%)

• Specifying the probability distribution by listing:

– all possible values

– with associated probabilities

• as we did before wouldn't be simple.

• We will, instead, rely on a theoretical distribution function (the Normal distribution) that is widely used in many applications.

• The frequency distribution for a normal distribution is a bellshaped curve.

• It is a symetric distribution entirely defined by two parameters

• – the expected value (mean)

• – the standard deviation


Belgium - Monthly returns 1951 - 1999

Bourse de Bruxelles 1951-1999

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Normal distribution illustrated

Normal distribution

0.0000

0.0050

0.0100

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0.0200

0.0250

68.26%

95.44%

Standard deviation from mean


Risk premium on a risky asset

• The excess return earned by investing in a risky asset as opposed to a risk-free asset

•

• U.S.Treasury bills, which are a short-term, default-free asset, will be used a the proxy for a risk-free asset.

• The ex post (after the fact) or realized risk premium is calculated by substracting the average risk-free return from the average risk return.

• Risk-free return = return on 1-year Treasury bills

• Risk premium = Average excess return on a risky asset


Total returns US 1926-1999

Arithmetic Mean

Standard Deviation

Risk Premium

Common Stocks 13.3% 20.1% 9.5%

Small Company Stocks 17.6 33.6 13.8

Long-term Corporate Bonds 5.9 8.7 2.1

Long-term government bonds 5.5 9.3 1.7

Intermediate-term government bond

5.4 5.8 1.6

U.S. Treasury bills 3.8 3.2

Inflation 3.2 4.5

Source: Ross, Westerfield, Jaffee (2002) Table 9.2


Market Risk Premium: The Very Long Run

1802-1870 1871-1925 1926-1999 1802-1999

Common Stock 6.8 8.5 13.3 9.7

Treasury Bills 5.4 4.1 3.8 4.4

Risk premium 1.4 4.4 9.5 5.3

Source: Ross, Westerfield, Jaffee (2002) Table 9A.1

The equity premium puzzle:

Was the 20th century an anomaly?

Diversification


Covariance and correlation

• Statistical measures of the degree to which random variables move together

• Covariance

• Like variance figure, the covariance is in squared deviation units.• Not too friendly ...

• Correlation

• covariance divided by product of standard deviations• Covariance and correlation have the same sign

– Positive : variables are positively correlated– Zero : variables are independant– Negative : variables are negatively correlated

• The correlation is always between –1 and + 1

)])([(),cov( BBAABAAB RRRRERR

BA

BABAAB

RRCovRRCorr

),(

),(


Risk and expected returns for porfolios

• In order to better understand the driving force explaining the benefits from diversification, let us consider a portfolio of two stocks (A,B)

• Characteristics:

– Expected returns :

– Standard deviations :

– Covariance :

• Portfolio: defined by fractions invested in each stock XA , XB XA+ XB= 1

• Expected return on portfolio:

• Variance of the portfolio's return:

BA RR ,

BA ,

BAABAB

BBAAP RXRXR

22222 2 BBABBAAAP XXXX


Example

• Invest $ 100 m in two stocks:

• A $ 60 m XA = 0.6

• B $ 40 m XB = 0.4

• Characteristics (% per year) A B

• • Expected return 20% 15%

• • Standard deviation 30% 20%

• Correlation 0.5

• Expected return = 0.6 × 20% + 0.4 × 15% = 18%

• Variance = (0.6)²(.30)² + (0.4)²(.20)²+2(0.6)(0.4)(0.30)(0.20)(0.5)

²p = 0.0532 Standard deviation = 23.07 %

• Less than the average of individual standard deviations:

• 0.6 x0.30 + 0.4 x 0.20 = 26%


Diversification effect

• Let us vary the correlation coefficient

• Correlationcoefficient Expected return Standard deviation

• -1 18 10.00

• -0.5 18 15.62

• 0 18 19.7

• 0.5 18 23.07

• 1 18 26.00

• Conclusion:

– As long as the correlation coefficient is less than one, the standard deviation of a portfolio of two securities is less than the weighted average of the standard deviations of the individual securities


The efficient set for two assets: correlation = +1

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Risk (standard deviation)


The efficient set for two assets: correlation = -1

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The efficient set for two assets: correlation = 0

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Choosing portfolios from many stocks

• Porfolio composition :

• (X1, X2, ... , Xi, ... , XN)

• X1 + X2 + ... + Xi + ... + XN = 1

• Expected return:

• Risk:

• Note:

• N terms for variances

• N(N-1) terms for covariances

• Covariances dominate

NNP RXRXRXR ...2211

i ij i j

ijjiijjijj

jP XXXXX 222


Some intuition

Var Cov Cov Cov CovCov Var Cov Cov CovCov Cov Var Cov CovCov Cov Cov Var CovCov Cov Cov Cov Var


Example

• Consider the risk of an equally weighted portfolio of N "identical« stocks:

• Equally weighted:

• Variance of portfolio:

• If we increase the number of securities ?:

• Variance of portfolio:

NX j

1

cov)1

1(1 22

NNP

NP cov2

cov),(,, jijj RRCovRR


Diversification

Risk Reduction of Equally Weighted Portfolios

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5.00%

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15.00%

20.00%

25.00%

30.00%

35.00%

# stocks in portfolio

Po

rtfo

lio

sta

nd

ard

de

via

tio

n

Market risk

Unique risk


Conclusion

• 1. Diversification pays - adding securities to the portfolio decreases risk. This is because securities are not perfectly positively correlated

• 2. There is a limit to the benefit of diversification : the risk of the portfolio can't be less than the average covariance (cov) between the stocks

• The variance of a security's return can be broken down in the following way:

• The proper definition of the risk of an individual security in a portfolio M is the covariance of the security with the portfolio:

Total risk of individual security

Portfolio risk

Unsystematic or diversifiable risk

Efficient markets


Notions of Market Efficiency

• An Efficient market is one in which:

– Arbitrage is disallowed: rules out free lunches

– Purchase or sale of a security at the prevailing market price is never a positive NPV transaction.

– Prices reveal information

• Three forms of Market Efficiency

• (a) Weak Form Efficiency

• Prices reflect all information in the past record of stock prices

• (b) Semi-strong Form Efficiency

• Prices reflect all publicly available information

• (c) Strong-form Efficiency

• Price reflect all information


Efficient markets: intuition

Expectation

Time

Price

Realization

Price change is unexpected


Weak Form Efficiency

• Random-walk model:

– Pt -Pt-1 = Pt-1 * (Expected return) + Random error

– Expected value (Random error) = 0

– Random error of period t unrelated to random component of any past period

• Implication:

– Expected value (Pt) = Pt-1 * (1 + Expected return)

– Technical analysis: useless

• Empirical evidence: serial correlation

– Correlation coefficient between current return and some past return

– Serial correlation = Cor (Rt, Rt-s)


Random walk - illustration

Bourse de Bruxelles 1980-1999

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-30.00 -25.00 -20.00 -15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 20.00 25.00

Rentabilité mois t

Re

nta

bili

té m

ois

t+

1


Semi-strong Form Efficiency

• Prices reflect all publicly available information

• Empirical evidence: Event studies

• Test whether the release of information influences returns and when this influence takes place.

• Abnormal return AR : ARt = Rt - Rmt

• Cumulative abnormal return:

• CARt = ARt0 + ARt0+1 + ARt0+2 +... + ARt0+1


Strong-form Efficiency

• How do professional portfolio managers perform?

• Jensen 1969: Mutual funds do not generate abnormal returns

• Rfund - Rf = + (RM - Rf)

• Insider trading

• Insiders do seem to generate abnormal returns

• (should cover their information acquisition activities)

Portfolio selection

Professeur André Farber


Portfolio selection

• Objectives for this session

– 1. Gain a better understanding of the rational for benefit of diversification

– 2. Identify measures of systematic risk : covariance and beta

– 3. Analyse the choice of an optimal portfolio


Combining the Riskless Asset and a single Risky Asset

• Consider the following portfolio P:

• Fraction invested

– in the riskless asset 1-x (40%)

– in the risky asset x (60%)

• Expected return on portfolio P:

• Standard deviation of portfolio :

Riskless asset

Risky asset

Expected return

6% 12%

Standard deviation

0% 20%

SFP RxRxR )1(

%60.912.060.006.040.0 PR

SP x

%1220.060.0 P


Relationship between expected return and risk

• Combining the expressions obtained for :

• the expected return

• the standard deviation

• leads to

SFP RxRxR )1(

SP x

PS

FSFP

RRRR

SSPR 30.006.020.0

06.012.006.0

P

PR

S

SR

FR


Risk aversion

• Risk aversion :

• For a given risk, investor prefers more expected return

• For a given expected return, investor prefers less risk

Expected return

Risk

Indifference curve

P


Utility function

• Mathematical representation of preferences

• a: risk aversion coefficient

• u = certainty equivalent risk-free rate

• Example: a = 2

• A 6% 0 0.06

• B 10% 10% 0.08 = 0.10 - 2×(0.10)²

• C 15% 20% 0.07 = 0.15 - 2×(0.20)²

• B is preferred

2),( PPPP aRRU

PR P Utility


Optimal choice with a single risky asset

• Risk-free asset : RF Proportion = 1-x

• Risky portfolio S: Proportion = x

• Utility:

• Optimum:

• Solution:

• Example: a = 2

SSR ,

22 ²])1[( SSFPP axRxRxaRu

02)( 2 SFS axRRdx

du

22

1

S

FS RR

ax

375.0)20.0(

06.012.0

22

1

2

122

S

FS RR

ax

Documents

Théorie Financière 2004-2005 Relation risque – rentabilité attendue (1) Professeur André Farber