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Swaps
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Introduction
An agreement between two parties to exchange cashflows in the future.
The agreement specifies the dates that the cash flows
are to be paid and the way that they are to becalculated.
A forward contract is an example of a simple swap. Witha forward contract, the result is an exchange of cash
flows at a single given date in the future.In the case of a swap the cash flows occur at several
dates in the future. In other words, you can think of aswap as a portfolio of forward contracts.
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Mechanics of Swaps
The most commonly used swap agreement is anexchange of cash flows based upon a fixed andfloating rate.
Often referred to a plain vanilla swap, theagreement consists of one party paying a fixedinterest rate on a notional principal amount inexchange for the other party paying a floating
rate on the same notional principal amount for aset period of time.
In this case the currency of the agreement is thesame for both parties.
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Notional Principal
The term notional principal implies that theprincipal itself is not exchanged. If it was
exchanged at the end of the swap, the exactsame cash flows would result.
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An Example
Company B agrees to pay A 5% per annum on anotional principal of $100 million
Company A Agrees to pay B the 6 month LIBORrate prevailing 6 months prior to each paymentdate, on $100 million. (generally the floating rateis set at the beginning of the period for which it
is to be paid)
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The Fixed Side
We assume that the exchange of cash flowsshould occur each six months (using a fixed rate
of 5% compounded semi annually).Company B will pay:
($100M)(.025) = $2.5 Million
to Firm A each 6 months.
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Summary of Cash Flowsfor Firm B
Cash Flow Cash Flow Net
Date LIBOR Received Paid Cash Flow
3-1-98 4.2%
9-1-98 4.8% 2.10 2.5 -0.4
3-1-99 5.3% 2.40 2.5 -0.1
9-1-99 5.5% 2.65 2.5 0.15
3-1-00 5.6% 2.75 2.5 0.259-1-00 5.9% 2.80 2.5 0.30
3-1-01 6.4% 2.95 2.5 0.45
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Swap Diagram
LIBOR
Company A Company B
5%
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Offsetting Spot Position
Company A
Borrows (pays) 5.2%
Pays LIBOR
Receives 5%
Net LIBOR+.2%
Company B
Borrows (pays) LIBOR+.8%
Receives LIBOR
Pays 5%
Net 5.8%
Assume that A has a commitment to borrow at a fixed rate of
5.2% and that B has a commitment to borrow at a rate of
LIBOR + .8%
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Swap Diagram
Company A Company B
The swap in effect transforms a fixed rate liability
or asset to a floating rate liability or asset (andvice versa) for the firms respectively.
5.2% LIBOR+.8%
LIBOR +.2% 5%
LIBOR
5.8%
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Role of Intermediary
Usually a financial intermediary works toestablish the swap by bring the two parties
together.The intermediary then earns .03 to .04% perannum in exchange for arranging the swap.
The financial institution is actually entering into
two offsetting swap transactions, one with eachcompany.
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Swap Diagram
Co A FI Co B
A pays LIBOR+.215%B pays 5.815%
The FI makes .03%
5.2% LIBOR+.8%
5.015%
LIBOR
4.985%
LIBOR
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Day Count Conventions
The above example ignored the day countconventions on the short term rates.
For example the first floating payment was listedas 2.10. However since it is a money marketrate the six month LIBOR should be quoted onan actual /360 basis.
Assuming 184 days between payments the actualpayment should be
100(0.042)(184/360) = 2.1467
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Day Count Conventions II
The fixed side must also be adjusted and as aresult the payment may not actually be equal on
each payment date.The fixed rate is often based off of a longermaturity instrument and may therefore uses adifferent day count convention than the LIBOR.
If the fixed rate is based off of a treasury notefor example, the note is based on a different dayconvention.
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Role of the Intermediary
It is unlikely that a financial intermediary will becontacted by parties on both side of a swap atthe same time.
The intermediary must enter into the swapwithout the counter party. The intermediarythen hedges the interest rate risk using interestrate instruments while waiting for a counter
party to emerge.This practice is referred to as warehousingswaps.
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Why enter into a swap?
The Comparative Advantage Argument
Fixed Floating
A 10% 6 mo LIBOR+.3B 11.2% 6 mo LIBOR + 1.0%
Difference between fixed rates = 1.2%Difference between floating rates = 0.7%
B Has an advantage in the floating rate.
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Spread Differentials
Why do spread differentials exist?
Differences in business lines, credit history, asset
and liabilities, etc
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Valuation of Interest Rate Swaps
After the swap is entered into it can be valued aseither:
A long position in one bond combined with a shortposition in another bond or
A portfolio of forward rate agreements.
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Relationship of Swaps to Bonds
In the examples above the same relationshipcould have been written as
Company B lent company A $100 million at thesix month LIBOR rate
Company A lent company B $100 million at afixed 5% per annum
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Bond Valuation
Given the same floating rates as before the cashflow would be the same as in the swap example.
The value of the swap would then be thedifference between the value of the fixed ratebond and the floating rate bond.
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Fixed portion
The value of either bond can be found bydiscounting the cash flows from the bond (asalways). The fixed rate value is straight forward
it is given as:
where Q is the notional principal and k is thefixed interest payment
nnii trn
i
tr
fix QekeB
1
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Floating rate valuation
The floating rate is based on the fact that it is aseries of short term six months loans.
Immediately after a payment date Bfl is equal tothe notional principal Q. Allowing the time untilthe next payment to equal t1
where k* is the known next payment
1111 * trtr
fl ekQeB
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Swap Value
If the financial institution is paying fixed andreceiving floating the value of the swap is
Vswap = Bfl-Bfix
The other party will have a value of
Vswap = Bfix-Bfl
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Example
Pay 6 mo LIBOR & receive 8%
3 mo 10%
9 mo 10.5%15 mo 11%
Bfix = 4e.-1(.25)+4e-.105(.75)+104e-.11(1.25)=98.24M
Bfloat = 100e-.1(.25)
+ 5.1e-.1(.25)
=-102.5M-4.27 M
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A better valuation
Relationship of Swap value to Forward RteAgreements
Since the swap could be valued as a forward rateagreement (FRA) it is also possible to value theswap under the assumption that the forwardrates are realized.
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To do this you would need to:
Calculate the forward rates for each of the LIBORrates that will determine swap cash flows
Calculate swap cash flows using the forwardrates for the floating portion on the assumptionthat the LIBOR rates will equal the forward rates
Set the swap value equal to the present value of
these cash flows.
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Swap Rate
This works after you know the fixed rate.
When entering into the swap the value of theswap should be 0.
This implies that the PV of each of the two seriesof cash flows is equal. Each party is then willingto exchange the cash flows since they have thesame value.
The rate that makes the PV equal when used forthe fixed payments is the swap rate.
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Example
Assume that you are considering a swap wherethe party with the floating rate will pay the threemonth LIBOR on the $50 Million in principal.
The parties will swap quarterly payments eachquarter for the next year.
Both the fixed and floating rates are to be paid
on an actual/360 day basis.
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First floating payment
Assume that the current 3 month LIBOR rate is3.80% and that there are 93 days in the firstperiod.
The first floating payment would then be
3333.833,490000,000,50360
93038.
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Second floating payment
Assume that the three month futures price onthe Eurodollar futures is 96.05 implying aforward rate of 100-96.05 = 3.95
Given that there are 91 days in the period.
The second floating payment would then be
1111.236,499000,000,50360
910395.
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Example Floating side
PeriodDay
Count
FuturesPrice
Fwd
Rate
Floating
Cash flow
91 3.80
1 93 96.05 3.95 490,833.3333
2 91 95.55 4.45 499,236.1111
3 90 95.28 4.72 556,250.0000
4 91 596,555.5555
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PV of Floating cash flows
The PV of the floating cash flows is thencalculated using the same forward rates.
The first cash flow will have a PV of:
8263.061,486
360
93
038.1
3333.833,490
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PV of Floating cash flows
The PV of the floating cash flows is thencalculated using the same forward rates.
The second cash flow will have a PV of:
4412.495,489
360
910395.1360
93038.1
1111.236,499
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Example Floating side
PeriodDay
Count
Fwd
Rate
Floating
Cash flowPV of Floating CF
91 3.80
1 93 3.95 490,833.3333 486,061.8263
2 91 4.45 499,236.1111 489,495.4412
3 90 4.72 556,250.0000 539,396.1423
4 91 596,555.5555 525,668.5915
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PV of floating
The total PV of the floating cash flows is then thesum of the four PVs:
$2,040,622.0013
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Swap rate
The fixed rate is then the rate that using thesame procedure will cause the PV of the fixedcash flows to have a PV equal to the same
amount.The fixed cash flows are discounted by the samerates as the floating rates.
Note: the fixed cash flows are not the same eachtime due to the changes in the number of days ineach period.
The resulting rate is 4.1294686
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Example: Swap Cash Flows
PeriodDay
Count
Fwd
Rate
Floating
Cash flowFixed CF
91 3.80
1 93 3.95 490,833.3333 533,389.7003
2 91 4.45 499,236.1111 521,918.9541
3 90 4.72 556,250.0000 516,183.5810
4 91 596,555.5555 521,918.9541
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Swap Spread
The swap spread would then be the differencebetween the swap rate and the on the runtreasury of the same maturity.
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Swap valuation revisited
The value of the swap will change over time.
After the first payments are made, the futures
prices and corresponding interest rates havelikely changed.
The actual second payment will be based uponthe 3 month LIBOR at the end of the first period.
Therefore the value of the swap is recalculated.
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A simple example
Assume that company A pays a fixed rate of 11% in sterling andreceives a fixed interest rate of 8% in dollars.
Let interest payments be made once a year and the principal amountsbe $15 million and L10 Million
Company A Dollar Cash Sterling CashFlow (millions) Flow (millions)
2/1/1999 -15.00 +10.002/1/2000 +1.20 -1.102/1/2001 +1.20 -1.10
2/1/2002 +1.20 -1.102/1/2003 +1.20 -1.102/1/2004 +16.20 -11.10
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Intuition
Suppose A could issue bonds in the US for 8%interest, the swap allows it to use the 15 millionto actually borrow 10million sterling at 11% (Acan invest L 10M @ 11% but is afraid that $ willstrength it wants US denominated investment)
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Comparative Advantage Again
The argument for this is very similar to thecomparative advantage argument presentedearlier for interest rate swaps.
It is likely that the domestic firm has anadvantage in borrowing in its home country.
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Example using comparativeadvantage
Dollars AUD (Australian $)
Company A 5% 12.6%
Company B 7% 13.0%2% difference in $US .4% difference in AUD
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The strategy
Company A borrows dollars at 5% per annum
Company B borrows AUD at 13% per annum
They enter into a swap
Result
Since the spread between the two companies isdifferent for each firm there is the ability of each
firm to benefit from the swap. We would expectthe gain to both parties to be 2 - 0.4 = 1.6%(the differences in the spreads).
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Swap Diagram
Co A FI Co B
A pays 11.9% AUD instead of 12.6% AUDB pays 6.3% $US instead of 7% $US
The FI makes .2%
5% AUD 13%
6.3%
AUD 11.9%
5%
AUD 13%
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Valuation of Currency Swaps
Using Bond Techniques
Assuming there is no default risk the currency
swap can be decomposed into a position in twobonds, just like an interest rate swap.
In the example above the company is long asterling bond and short a dollar bond. The value
of the swap would then be the value of the twobonds adjusted for the spot exchange rate.
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Swap valuation
Let S = the spot exchange rate at the beginningof the swap, BF is the present value of theforeign denominated bond and BD is the present
value of the domestic bond. Then the value isgiven as
Vswap = SBF BD
The correct discount rate would then depend uponthe term structure of interest rates in each
country
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Other swaps
Swaps can be constructed from a large number ofunderlying assets.
Instead of the above examples swaps for floating rates
on both sides of the transaction.The principal can vary through out the life of the swap.
They can also include options such as the ability toextend the swap or put (cancel the swap).
The cash flows could even extend from another assetsuch as exchanging the dividends and capital gainsrealized on an equity index for a fixed or floating rate.
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Beyond Plain Vanilla Swaps
Amortizing Swap -- The notional principal isreduced over time. This decreases the fixedpayment. Useful for managing mortgageportfolios and mortgage backed securities.
Accreting Swap The notional principal increasesover the life of the swap. Useful in construction
finances. For example is the builder draws downan amount of financing each period for a numberof periods.
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Beyond Plain Vanilla
You can combine amortizing and accreting swapsto allow the notional principal to both increaseand decrease.
Seasonal Swap -- Increase and decrease ofnotional principal based of f of designated plan
Roller Coaster Swap -- notional principal firstincreases the amortizes to zero.
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Off Market Swap
The interest rate is set at a rate above marketvalue.
For example the fixed rate may pay 9% whenthe yield curve implies it should pay 8%.
The PV of the extra payments is transferred as aone time fee at the beginning of the swap (thus
keeping the initial value equal to zero)
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Forward andExtension Swaps
Forward swap the payments are agreed tobegin at some point in time in the future
If the rates are based on the current forwardrate there should not be any exchange ofprincipal when the payments begin. Other wiseit is an off market swap and some form of
compensation is neededExtension Swap an agreement to extend thecurrent swap (a form of forward swap)
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Basis Swaps
Both parties pay floating rates based upondifferent indexes.
For example one party may pay the three monthLIBOR while the other pays the three month T-Bill.
The impact is that while the rates generally move
together the spread actually widens and narrows,Therefore the return on the swap is based uponthe spread.
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Yield Curve Swaps
Both parties pay floating but based off ofdifferent maturities. Is similar to a basis swapsince the effective result is based on the spreadbetween the two rates. A steepening curve thusbenefits the payer of the shorter maturity rate.
This is utilized by firms with a mismatch of
maturities in assets and liabilities (banks forexample). It can hedge against changes in theyield curve via the swap.
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Rate differential (diff) swap
Payments tied to rate indexes in differentcurrencies, but payments are made in only onecurrency.
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Corridor Swap
Payments obligation only occur in a given rangeof rates. For example if the LIBOR rate isbetween 5 and 7%.
The swap is basically a tool based on theuncertainty of rates.
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Flavored Currency Swaps
The basic currency swap can be modified similarto many of the modifications just discussed.
Swaps may also be combined to produce desiredoutcomes.
CIRCUS Swap (Combined interest rate andcurrency swap). Combines two basic swaps
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Circus Swap Diagram
LIBOR
Company A Company B
5% US$6% German Marks
Company A Company C
LIBOR
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Circus Swap Diagram
Company B
Company A 5% US$
6% German Marks
Company C
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Swapation
An option on a swap that specifies the tenor,notional principal fixed rate and floating rate
Price is usually set a a % of notional principal
Receiver Swapation
The holder has the right to enter into a swap asthe fixed rate receiver
Payer SwapationThe holder has the right to enter into a particularswap as the fixed rate payer.
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Swapation ascall (or put) Options
Receiver swapation similar to a call option on abond. The owner receives a fixed payment (likea coupon payment) and pays a floating rate (the
exercise price)
Payer swapation if exercised the owner ispaying a stream similar to the issue of a bond.
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In-the-Money Swapations
A receiver swapation is in the money if interestrates fall. The owner is paying a lower fixed ratein exchange for the fixed rate specified in the
contract.
Similarly a payer swapation is generally in themoney if interest rates increase since the owner
will receive a higher floating rate.
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When to Exercise
The owner of the receiver swapation shouldexercise if the fixed rate on the swap underlyingthe swapation is greater than the market fixed
rate on a similar swap. In this case the swap ispaying a higher rate than that which is availablein the market.
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A fixed incomeswapation example
Consider a firm that has issued a corporate bondwith a call option at a given date in the future.
The firm has paid for the call option by being
forced to pay a higher coupon on the bond thanon a similar noncallable bond.
Assume that the firm has determined that it doesnot want to call the bond at its first call date at
some point in the future.The call option is worthless to the firm, but itshould theoretically have value.
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Capturing thevalue of the call
The firm can sell a receiver swapation with termsthat match the call feature of the bond.
The firm would receive for this a premium that isequal to the value of the call option.
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Example
Assume the firm has previously issued a 9%coupon bond that makes semiannual paymentsand matures in 7 years with a face value of $150
Million.
The bond has a call option for one year fromtoday.
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Example continued
The firm can sell a European Receiver Swapation with anexpiration in one year. The Swapation terms are forsemiannual payments at a fixed rate of 9% in exchange forfloating payments at LIBOR.
The firm receives a premium for the swapation equal to afixed percentage of the $150 Million notional value (equalto the value of the call option).
The firm can keep the premium but has a potentialobligation in one year if the counter party exercises theswap.
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Example Continued
In one year the fixed rate for this swap is 11%
The option will expire worthless since the ownercan earn a fixed 11% on a similar swap.
The firm gets to keep the premium.
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Example Continued
If in one year the fixed rate of interest on asimilar swap is 7% the owner will exercise theswap since it calls for a 9% fixed rate.
The firm can call the bond since rates havedecreased. It can finance the call by issuing afloating rate note at LIBOR for the term of theswap.
The floating rate side of the swap pays for thenote and the firm is still paying the original 9%fixed, but it has also received the premium on theswapation
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Extendible and Cancelableswaps
Similar to extension swaps except extensionswaps represent a firm commitment to extendthe swap. An extendible swap has the option to
extend the agreement.
Arranged via a plain vanilla swap an a swapation.
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Extendible and Cancelable
Extendible pay fixed swap
= plain vanilla pay fixed plus payer swapation
Extendible Receive-Fixed Swap
= plain vanilla receive fixed swap + receiverswapation
Cancelable Pay Fixed Swap
= plain vanilla pay fixed swap + receiver swapation
Cancelable Receive Fixed Swap
=plain vanilla receive fixed swap + payableswapation
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DAUVERGNECreating synthetic
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Creating syntheticsecurities using swaps
The origins of the swap market are based in thedebt market.
Previously there had been restrictions on the flowof currency.
A parallel loan market developed to get aroundrestrictions on the flow of currency from one
country to another, Especially restrictionsimposed by the Bank of England.
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The Parallel Loan Market
Consider two firms, one British and oneAmerican, each with subsidiaries in bothcountries.
Assume that the free-market value of the poundis L1=$1.60 and the officially required exchangerate is L1=$1.44.
Assume the British Firm wants to undertake aproject in the US requiring an outlay of$100,000,000.
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Parallel Loan Market
The cost of the project at the official exchangerate is 100,000,0000/1.44 = L69,444,000
The cost of the project at the free marketexchange rate is 100,000,0000/1.60 =L62,500,000
The firm is paying an extra L7,000,000
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Parallel Loan Market
The British firm lends L62,500,000 pounds to theUS subsidiary operating in England at a floatingrate based on LIBOR and The US firm lends
$100,000,000 to the British firm at a fixed rate of7% in the US the official exchange rate isavoided.
The result is a basic fixed for floating currency
swap. (In this case each loan is separatedefault on one loan does not constitute defaulton the other).
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Synthetic Fixed Rate Debt
A firm with an existing floating rate debt caneasily transform it into a fixed rate debt via aninterest rate swap.
By receiving floating and paying fixed, the firmnets just a spread on the floating transactioncreating a fixed rate debt (the rate paid on the
swap plus the spread)
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Synthetic Floating Rate Debt
Combining a fixed rate debt with a pay floating /receive fixed rate swap easily transforms thefixed rate. Again the fixed rates cancel out (or
result in a spread) leaving just a floating rate.
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Synthetic Callable Debt
Consider a firm with an outstanding fixed ratedebt without any call option.
It can create a call option. If it had a call optionin place it would retire the debt if called. Look atthis as creating a new financing need (you needto finance the retirement of the debt.)
You want the ability to call the bond but not theobligation to do so.
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h ll bl b
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Synthetic Callable Debt
Buying a receiver swapation allows the firm toreceive a fixed rate, canceling out its currentfixed rate obligation.
It will pay a new floating rate as part of the swap(similar to financing the call with new floatingrate debt).
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h ll bl b
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Synthetic non callable Debt
Basically the earlier example swapations.
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S h i D l C D b
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Synthetic Dual Currency Debt
Dual Currency bond principal payments aredenominated in one currency and couponpayments denominated in another currency.
Assume you own a bond that makes itspayments in US dollars, but you would prefer thecoupon payments to be in another currency with
the principal repayment in dollars.A fixed for fixed currency swap would allow thisto happen
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S h i D l C D b
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Synthetic Dual Currency Debt
Combine a receive fixed German marks and payUS dollars swap with the bond.
The dollars received from the bond are used topay the dollar commitment on the swap. Youthen just receive the German Marks.
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All i C t
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All in Cost
The IRR for a given financing alternative, itincludes all costs including administration,flotation , and actual cash flows.
The cost is simply the rate that makes the PV ofthe cash flows equal to the current value of theborrowing.
UNIVERSITEDAUVERGNECompare two
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Compare twoalternative proposals
A 10 year semiannual 7% coupon bond with aprincipal of $40 million priced at par
A loan of $40 million for 10 years at a floatingrate of LIBOR + 30 Bps reset every six monthswith the current LIBOR rate of 6.5%. Plus a swaptransforming the loan to a fixed rate
commitment. The swap will require the firm topay 6.5% fixed and receive floating.
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All i t
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All in cost
The bond has a all in cost equal to its yield tomaturity, 7%
Assuming the firm must pay $400,000 to enter
into the swap so it only nest $39,600,000.Today. The net interest rate it pays is 6.8%implying semiannual payments of(.068/2)(40,000,000) = $1,360,000 plus a final
payment of 40,000,000. This implies a rate of.034703 every six months or .069406 every year.
UNIVERSITEDAUVERGNEBF Goodrich and Rabobank
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BF Goodrich and RabobankAn early swap example*
In the early 1980s BF Goodrich needed to raisenew funds, but its credit rating had beendowngraded to BBB-. The firm needed
$50,000,000 to fund continuing operations.They wanted long term debt in the range of 8 to10 years and a fixed rate. Treasury rates wereat 10.1 % and BF Goodrich anticipated paying
approximately 12 to 12.5%
* taken from Kolb - Futures Options and Swaps
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R b b k
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Rabobank
Rabobank was a large Dutch bankingorganization consisting of more than 1,000 smallagricultural banks. The bank was interested in
securing floating rate financing on approximately$50,000,000 in the Eurobond market.
With a AAA rating Rabobank could issue fixed
rate in the Eurobond market for approximately11% and for a floating rate of LIBOR plus .25%
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Th I t di
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The Intermediary
Salomon Brothers suggested a swap agreementto each party.
This would require BF Goodrich to issue the firstpublic debt tied to LIBOR in the United States.Salomon Brothers felt that there would be amarket for the debt because of the increase in
deposits paying a floating rate due toderegulation.
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P bl
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Problems
Rabobank was interested in the deal,but fearfulof credit risk. A direct swap would expose it tocredit risk. Without an active swap market it was
common for swaps to be arranged between thetwo counter parties.
The two finally reached an agreement to use
Morgan Guaranty as an intermediary.
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The ag eement
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The agreement
BF Goodrich issued a noncallable 8 year floatingrate note with a principal value of $50,000,000paying the 3 month LIBOR rate plus .5%
semiannually. The bond was underwritten bySalomon.
Rabobank issued a $50,000,000 non callable 8
year Eurobond with annual payments of 11%Both entered into a swap with Morgan Guaranty
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The swaps
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The swaps
BF Goodrich promised to pay Morgan Guaranty5,500,000 each year for eight years (matchingthe coupon on the Rabobanks debt). Morgan
agreed to pay BF Goodrich a semi annual ratetied to the 3 month LIBOR equal to:.5(50,000,000)(3 mo LIBOR-x)
x represents an undisclosed discountRabobank received $5,500,000 each year for 8years and paid semi annul payments of LIBOR-x
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The intermediary role
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The intermediary role
The two swap agreements were independent ofeach other eliminating the credit risk concerns ofRabobank.
Morgan received a one time fee of $125,000 paidby BF Goodrich plus an annual fee of 8 to 37 Bp($40,000 to $185,000) also paid by BF Goodrich.
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BF Goodrich
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BF Goodrich
Assuming that the discount from LIBOR was 50Bp and that the service fee was 22.5 BP (themidpoint of the range). BF Goodrich paid an all
in cost of 11.9488 % annually compared to 12 to12.5% if they had issued the debt on their own.
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D kRabobanks Position
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Rabobank s Position
At the time of financing it would have paid LIBORplus 25 to 50 Bp. Given that it paid no fees andthe fixed rate canceled out it ended up paying
LIBOR - x.
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D kSecuring financing
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Securing financing
BF Goodrich was able to secure financing via itsuse of the swaps market, this is a common useof the market.
The example provides a good illustration of theidea of the comparative advantage argumentswe discussed earlier.
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D kA Second Example of securing
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p gfinancing*
It is possible for swaps to increases accessibilitytwo the debt market
Mexcobre (Mexicana de Corbre) is the copper
exporting subsidiary of Grupo Mexico. In the late1980s it would have had a difficult timeborrowing in international credit markets due toconcerns or default risk
However it was able to borrow $210 million for38 months from a group of banks led by Paribas
* from Managing Financial Risk by Smithson, Smith and Wilford
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D kThe original loan
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The original loan
The banks lent the firm $210 Million at a fixedrate of 11.48%. The debt replaced borrowingfrom the Mexican government which had cost the
firm 23%.A Belgian company Sogem agreed to buy 4,000tons of copper per month at the prevailing spotrate from Mexcobre making payments into an
escrow account in New York that was used toservice the debt with any extra funds returned toMexcobre.
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D k
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Banks Escrow
Mexcobre SOGEM4,000 tons of copper
per month
Cash based
on Spot Price
Quarterly payments of 11.48%
interest plus principal
$210
million
loan
Excess cash
if it builds
up
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D kSwaps
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Swaps
Swaps were added between Paribas and theescrow account to hedge the price risk of copperand between Paribas and the banks to change
the banks position to a floating rate
Paribas
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Banks Escrow
Mexcobre SOGEM4,000 tons of copper
per month
Cash per
ton based
on Spot Price
Quarterly payments of 11.48%
interest plus principal
$210
million
loan
Excess cash
if it builds
up
Paribas
Spot
Priceper ton
FloatingFixed $2,000
per ton
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D kDuration of Interest
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A plain vanilla swap can be valued as a portfolioof two bonds, therefore the duration of the swapshould equal the duration of the bond portfolio.
The duration can be either positive or negativedepending on the side of the swap
* Kolb, Futures Options and Swaps
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D kDuration of Swaps
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Duration of Swaps
Duration of Receive Fixed Swap =
Duration of Underlying coupon bond
- Duration of underlying floating Rate Bond
>0
Duration of Pay Fixed Swap =
Duration of underlying floating Rate Bond
- Duration of Underlying coupon bond
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Example
Consider a swap with a semiannual fixed rate of7% and a floating rate that resets each sixmonths.
The duration of the fixed rate side (assuming a100 notional principal) is 5.65139 years
Duration of Receive Fixed Swap
=5.65139-0.5=5.15369Duration of Pay Fixed Swap
=0.5-5.65139=-5.15369
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DrakeCalculating Duration
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Calculating Duration
Duration of floating rate security is equal to thetime between resetting of the rate.
Therefore the duration of the swap actually
depends upon the duration of the fixed rate side.
Receive Fixed rate swaps will then usuallylengthen the duration of an existing position
while pay fixed swaps will shorten the duration ofan existing position.
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Immunization with Swaps
Swaps can be used to hedge interest rate risk byimpacting the duration of the assets andliabilities on the balance sheet.
Going to look at a fictional financial services firmFSF
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DrakeBalance Sheet for FSF
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Balance Sheet for FSF
AssetsCash $7,000,000
Marketable Sec $18,000,000
(6 mo mat Yield 7%)
Amortizing loans $130,467,133
(10 yr avg mat
semiannual
8% avg yield)
Total Assets $1555,467,133
Liabilities6mo money mkt $75,000,000
(avg yield 6%)
Floating Rate Notes $40,000,000(5 yr mat7.3% yld semi)
Coupon Bond $24,111,725
(10 yr semi 6.5% coup
$25,000,000 par, 7% YTM
Net worth $16,355,408
Total Liab & NW $155,467,133
UNIVERSITEDAUVERGNE
DrakeBasic Duration
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Basic Duration
iassetofDurationMacaulayDa
AssetsAllofValueMarket
Assetwwhere
DawDAPortfolioAssetofDurationWeighted$
i
ii
i
N
1ii
jLiabilityofDurationMacaulayDl
sLiabilitieAllofValueMarket
Assetwwhere
DlwDLPortfolioLiabilityof
DurationWeighted$
j
j
j
j
N
1j
j
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DrakeDuration
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Duration
Assets
Duration
Cash 0.00
Marketable Sec 0.500
Amortizing loans 4.604562
Total Duration
(7,000,000/155,467,133)0.000
+(18,000,000/155,467,133)0.500+(130,467,133/155,467,133)4.605
3.922013
Liabilities
Duration
6mo money mkt 0.5000
Floating Rate Notes 0.5000
Coupon Bond 7.453369
Total Duration
(75,000,000/155,467,133)0.500
+(40,000,000/155,467,133)0.500+(24,111,725/155,467,133)7.45337
1.705202
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DrakeHedging the
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DrakeDrake Universityportfolios separately
It is easy to use duration to hedge the interestrate risk of the portfolio.
The idea is to construct a portfolio with a
duration of zero.Let MVi be the market value and Di be theDuration of the assets (A), liabilities (L) orhedge vehicle (H) then
MVA(DA)+MVH(DH) = 0and
MVL(DL)+MVH(DH) = 0
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DrakeSwap notional value
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Swap notional value
Given the duration of the hedge (a swap) it isthen possible to solve for a notional value (ormarket value) of the swap that would make the
portfolio duration zero.Previously we found the duration of a swap:
Duration of Receive Fixed Swap
=5.65139-0.5=5.15369Duration of Pay Fixed Swap
=0.5-5.65139=-5.15369
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Asset Hedge
The asset can then be hedged by solving for thenotional value (MVH) of the pay fixed swap
MVA(DA)+MVH(DH) = 0
155,467,133(3.922)+(-5.15369)(MVH) =0
MVH=$118,365,451
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Liability Hedge
The liabilities can then be hedged by solving forthe notional value (MVH) of the receive fixedswap
MVL(DL)+MVH(DH) = 0
(-139,111,725)(1.705202)+(5.15369)(MVH) =0
MVH=$46,048,651
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The entire balance sheet can be hedged with oneinterest rate swap by using GAP analysis.
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(Th i i d l)
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Repricing GAP
The difference between the value of interestsensitive assets and interest sensitive liabilities of
a given maturity.Measures the amount of rate sensitive (asset orliability will be repriced to reflect changes ininterest rates) assets and liabilities for a given
time frame.
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GAP Analysis
Static GAP-- Goal is to manage interest rateincome in the short run (over a given period oftime)
Measuring Interest rate risk calculating GAPover a broad range of time intervals provides a
better measure of long term interest rate risk.
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Interest Sensitive GAP
Given the Gap it is easy to investigate thechange in the net interest income (NII) of the
financial institution.
sLiabilitieSensistiveRate-AssetsSensistiveRateGAP
R)(GAP)(NII
Rates)inge(GAP)(ChanNIIinChange
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DrakeExample
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Example
Over next 6 Months:
Rate Sensitive Liabilities = $120 million
Rate Sensitive Assets = $100 Million
GAP = 100M 120M = - 20 Million
If rate are expected to decline by 1%
Change in net interest income
= (-20M)(-.01)= $200,000
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GAP Analysis
Asset sensitive GAP (Positive GAP)
RSA RSL > 0
If interest rates h NII will h
If interest rates i NII will iLiability sensitive GAP (Negative GAP)
RSA RSL < 0
If interest rates h NII will i
If interest rates i NII will h
Would you expect a commercial bank to beasset or liability sensitive for 6 mos? 5 years?
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DrakeImportant things to note:
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Important things to note:
Assuming book value accounting is used -- onlythe income statement is impacted, the bookvalue on the balance sheet remains the same.
The GAP varies based on the bucket or timeframe calculated.
It assumes that all rates move together.
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p g
Select time Interval
Develop Interest Rate Forecast
Group Assets and Liabilities by the time interval(according to first repricing)
Forecast the change in net interest income.
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Cumulative GAP
Totals the GAP over a range of of possiblematurities (all maturities less than one year for
example).Total GAP including all maturities
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DrakeOther useful measures using
GAP
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Relative Interest sensitivity GAP (GAP ratio)
GAP / Bank Size
The higher the number the higher the risk that is
present
Interest Sensitivity Ratio
SensitiveAsset1
SensitiveLiability1
sLiabilitieSensitiveRate
AssetsSensitiveRate
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Any Asset or Liability that matures during thetime frame
Any principal payment on a loan is rate sensitive
if it is to be recorded during the time period
Assets or liabilities linked to an index
Interest rates applied to outstanding principal
changes during the interval
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rates
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So far we have assumed that the change thelevel of interest rates will be the same for bothassets and liabilities.
If it isnt you need to calculate GAP using therespective change.
Spread effect The spread between assets and
liabilities may change as rates rise or decrease
)R(RSL)(-)R(RSA)(NII liabiltiesassets
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g
Easy to understand and calculate
Allows you to identify specific balance sheet
items that are responsible for risk
Provides analysis based on different time frames.
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Market Value Effects
Basic repricing model the changes in marketvalue. The PV of the future cash flows should
change as the level of interest rates change.(ignores TVM)
Over aggregation
Repricing may occur at different times within thebucket (assets may be early and liabilities latewithin the time frame)
Many large banks look at daily buckets.
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Runoffs
Periodic payment of principal and interest that canbe reinvested and is itself rate sensitive.
You can include runoff in your measure of ratesensitive assets and rate sensitive liabilities.
Note: the amount of runoffs may be sensitive torate changes also (prepayments on mortgages forexample)
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Off Balance Sheet Activities
Basic GAP ignores changes in off balance sheetactivities that may also be sensitive to changes in
the level of interest rates.Ignores changes in the level of demand deposits
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p
Duration Gap
DLDADGAPBasic
PortfolioLibailityofDurationWeighted$
PortfolioAssetofDurationWeighted$DGAPBasic
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If the Basic DGAP is +
If Rates h
i in the value of assets > i in value of liab
Owners equity will decrease
If Rate i
h in the value of assets > h in value of liab
Owners equity will increase
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If the Basic DGAP is (-)
If Rates h
i in the value of assets < i in value of liab
Owners equity will increase
If Rate i
h in the value of assets < h in value of liab
Owners equity will decrease
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Does that imply that if DA = DL the financialinstitution has hedged its interest rate risk?
No, because the $ amount of assets > $ amountof liabilities otherwise the institution would beinsolvent.
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Let MVL = market value of liabilities and MVA =market value of assets
Then to immunize the balance sheet we can use
the following identity:
MVA
MVLDLDADGAP
MVA
MVLDLDA
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396201.2
133,467,155
725,111,139705202.1922013.3DGAP
MVA
MVLDLDADGAP
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The net cash flows represented on the balancesheet have the same properties as a longposition in a bond with a duration of 2.396201.
We can hedge using our equation from beforeand the duration of the interest rate swap.
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Hedging with DGAP
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Since the duration of our position is positive wewant the duration of the hedge to be negative.This requires the pay fixed swap from before
with a notional value equal to MVH below.
MVi(Di)+MVH(DH) = 0
$155,467,725(2.396201)+(-5.151369)MVH=0MVH=$72,316,800
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DrakeD k U i it
DGAP and owners equity
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Let MVE = MVAMVL
We can find MVA &MVL using duration
From our definition of duration:
MVLy1
y-DLMVL
MVAy1
y
-DAMVA
formulatheApplyingPi)(1
iDP
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MVAy1
y-DGAPMVE
MVAy1
y
MVA
MVL(DL)-(DA)-
y1y(DL)MVL-(DA)MVA-
MVLy1
yDL--MVA
y1
y-DA
MVL-MVAMVE
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DGAP Analysis
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If DGAP is (+)
An h in rates will cause MVE to i
An i in rates will cause MVE to h
If DGAP is (-)
An h in rates will cause MVE to h
An i in rates will cause MVE to i
The closer DGAP is to zero the smaller thepotential change in the market value of equity.
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Weaknesses of DGAP
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It is difficult to calculate duration accurately(especially accounting for options)
Each CF needs to be discounted at a distinct rate
can use the forward rates from treasury spotcurve
Must continually monitor and adjust duration
It is difficult to measure duration for non interestearning assets.
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More General Problems
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Interest rate forecasts are often wrong
To be effective management must beat the abilityof the market to forecast rates
Varying GAP and DGAP can come at the expenseof yield
Offer a range of products, customers may notprefer the ones that help GAP or DGAP Need tooffer more attractive yields to entice thisdecreases profitability.
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Changing Duration
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You can also manipulate the duration of yourcash flows. This allows you to lower yourinterest rate sensitivity instead of eliminating it.
Let DG* be the desired duration gap, DG be thecurrent duration gap, DS be the duration of theSwap, and MVH* be the notional value ofrequired for the swap.
AssetsTotal
MVDDD
*H
SG*G
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Decreasing Duration GAPto One year
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Drake Universityto One year
The negative sign just indicate that we need a payfi d (th d ti ld th b ti
025,137,42MV
33$155,467,1
MV
15369.5396201.20.1
AssetsTotal
MVDDD
*H
*H
*H
SG*G