AFM 371 Winter 2008Chapter 26 - Derivatives and Hedging RiskPart 2 - Interest Rate Risk Management

(26.4-26.7)

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Outline

Term Structure

Forward Contracts on Bonds

Interest Rate Futures Contracts

Duration

Immunization

Swaps

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The Term Structure of Interest Rates

the text coverage of this material is in Appendix 6Aalthough in almost all cases in this course we consider a flatterm structure, it is important to keep in mind that this is asimplificationwith a flat term structure, discount rates are the same for allmaturities, but this is rarely (if ever) the casefor Oct. 31, 2007, the Bank of Canada reported governmentzero coupon government bond yields as follows:

Maturity 1 yr 3 yr 5 yr 7 yr 10 yr 15 yrYield 4.18 4.16 4.18 4.21 4.28 4.37

this means, for example, that the price on Oct. 31 of a oneyear zero coupon government bond paying $1,000 at maturitywas $1,000/1.0418 = $959.88, while the price of a ten yearzero coupon government bond paying $1,000 at maturity was$1,000/1.042810 = $657.64the rates above, which can be used to determine prices atwhich bonds may be currently traded, are known as spot rates

Term Structure 3 / 30

Spot Rates and Govermnent Bond Prices

let spot risk free (semi-annual) rates be r1, r2, r3, . . . , etc.This means that we calculate present values as follows:

Cash Flow Date Received Present Value

C1 t = 0.5 years from now C1/(1 + r1)C2 t = 1.0 years from now C2/(1 + r2)

2

C3 t = 1.5 years from now C3/(1 + r3)3

......

...consider a government bond paying a semi-annual coupon of$C and a principal amount of F which matures in T years(note that there is a total of N = 2T payments)assuming the next coupon payment is in six months, thepresent value of these future cash flows is

C

(1 + r1)+

C

(1 + r2)2+

C

(1 + r3)3+ · · ·+ C + F

(1 + rN)N

if the term structure is flat, i.e. r1 = r2 = · · · = rN = r , thenthe above formula simplifies to the familiar CAN

r + F/(1 + r)N

Term Structure 4 / 30

Forward Contracts on Bonds

consider entering into a contract to purchase a governmentbond M periods from now which will have a remainingmaturity of T years when it is bought

the idea is that you will pay the forward price F0 (decidedupon today) M periods from now for a government bond thatwill mature M + N periods from now (recall N = 2T )

assuming that the date on which you buy the bond is exactlysix months before the next coupon payment, the present valueof the bond today is

S0 =M+N∑

i=M+1

C

(1 + ri )i+

F

(1 + rM+N)M+N

recall the spot forward parity relationship (assuming that theunderlying asset pays no income and there are no costs ofcarry and no convenience yield: Ft = St × (1 + rf )

T−t

Forward Contracts on Bonds 5 / 30

Pricing of Forward Contracts on Bonds

in this context, the spot-forward parity relationship impliesthat the forward price is

F0 = (1 + rM)M × S0

= (1 + rM)M ×

[M+N∑

i=M+1

C

(1 + ri )i+

F

(1 + rM+N)M+N

]

example: consider a 5 year (M = 10) forward contract to buya 30 year government bond which pays semi-annual couponsof $25, and assume the term structure is flat with asemi-annual yield of 2%. What is the value of the bondtoday? What is the forward price?

Forward Contracts on Bonds 6 / 30

Interest Rate Futures Contracts and Hedging

in practice, futures contracts on bonds are typically usedrather than forward contractsfutures contracts on bonds are referred to as interest ratefutures contractsthe pricing relationships derived above for forward contractswill only be an approximation in this context (recall thatforward and futures prices are identical if interest rates are notrandom; also the exact delivery date is determined by theshort party in a futures contract)suppose you own $10 million worth of 20 year 10% annualcoupon bonds (coupon payments are semi-annual). The termstructure is flat at 5% (semi-annual). These bonds aretherefore selling at par, i.e.

40∑i=1

$50

1.05i+

$1,000

1.0540= $1,000

Interest Rate Futures Contracts 7 / 30

Interest Rate Futures Contracts and Hedging (Cont’d)

if the term structure shifts up uniformly to 5.5%, the newprice per bond is

40∑i=1

$50

1.055i+

$1,000

1.05540= $919.77

since you have 10,000 of these bonds, you have lost

10,000× ($1,000− $919.77) = $802,300

suppose government bond futures contracts specify delivery of$100,000 par value of 20 year 8% coupon bondsbefore the term structure shift, what was the approximatefutures price for delivery in 6 months? What is it after theterm structure shift?

Interest Rate Futures Contracts 8 / 30

Interest Rate Futures Contracts and Hedging (Cont’d)

note that each short futures contract gains

100× [$828.41− $759.31] = $6,910

suppose you had hedged by shorting

K =size of exposure

size of futures contract=

$10,000,000

$100,000= 100

futures contracts. Then you would have gained $691,000 onyour short futures position, partially offsetting your loss of$802,300, and resulting in an overall loss of $111,300reasons in practice why interest rate hedging using futuresmay not work perfectly:

different maturities (bonds in portfolio vs. futures contract)different coupon ratesdifferent risk (e.g. corporate bonds in portfolio, governmentbonds in futures contract)

see text pp. 748-751 for further examples

Interest Rate Futures Contracts 9 / 30

Duration

assume for simplicity a flat term structure. Consider these fourbonds, each with $1,000 par value and coupons paid annually:

Price if Price if Price if %∆P for %∆P forBond Coupon T r = 9.9% r = 10% r = 10.1% −0.10%∆r +0.10%∆r

A 5% 3 years 877.93 875.66 873.39 +.2592% -.2592%B 12% 3 years 1,052.32 1,049.74 1,047.17 +.2458% -.2448%C 0% 3 years 753.37 751.31 749.27 +.2742% -.2715%D 0% 10 years 389.07 385.54 382.06 +.9156% -.9026%

Notes:

percentage price changes are calculated relative to the pricewhen r = 10%, e.g. (877.93-875.66)/875.66 = +.2592%.low coupon bond prices are more sensitive to changes in r ,given the same T (compare A, B, and C) (recall our earlierdiscussion about why callable bonds have relatively low interestrate risk because they tend to have high coupons tocompensate investors for granting the call option to the issuer)longer maturity bond prices are more sensitive to changes in r ,given the same coupon (compare C and D)

Duration 10 / 30

Duration (Cont’d)

how can we measure this sensitivity? Recall that:

P =C

1 + r+

C

(1 + r)2+ · · ·+

C

(1 + r)T+

F

(1 + r)T

⇒dP

dr=

−1

1 + r×

»C

1 + r+

2C

(1 + r)2+

3C

(1 + r)3+ · · ·+

TC

(1 + r)T+

TF

(1 + r)T

–

this means that the percentage change in price for a givenchange in r is:

dP

dr·

1

P=

−1

1 + r

×»

C

1 + r+

2C

(1 + r)2+

3C

(1 + r)3+ · · ·+

TC

(1 + r)T+

TF

(1 + r)T

–1

P

a bond’s duration is a weighted average of its cash flows (ameasure of the bond’s effective maturity given when its cashflows occur).

Duration 11 / 30

Duration (Cont’d)

duration is defined as:

D =

hC

1+r+ 2C

(1+r)2+ 3C

(1+r)3+ · · ·+ TC

(1+r)T+ TF

(1+r)T

iP

=

PTt=1

tC(1+r)t

+ TF(1+r)T

P

therefore:

dP

dr·

1

P=

−1

1 + r× D ⇒

dP

P=

−1

1 + r× D × dr

so we can use D to estimate the effect of a change in r onbond price

example: duration of Bond A (r = 10%)

DA =

»50

1.1+

(2× 50)

1.12+

(3× 1050)

1.13

–1

875.66

= .0519 + .0944 + 2.7027 = 2.849 years

Duration 12 / 30

Duration (Cont’d)

we can also calculate DB = 2.698 years (higher couponimplies shorter duration since more cash is paid earlier),DC = 3 years (duration of a zero coupon bond = T ), etc.

The following table illustrates the use of duration as ameasure of bond price sensitivity:

dr = +.001 dr = −.001Bond Duration Estimated Change Actual Change Estimated Change Actual Change

A 2.849 -.002590 -.002592 .002590 .002592B 2.698 -.002453 -.002448 .002453 .002448C 3 -.002727 -.002715 .002727 .002742D 10 -.009091 -.009026 .009091 .009156

note that we get quite accurate estimates (for small changesin r) based on

dP

P=

−1

1 + r× D × dr

Duration 13 / 30

Immunization

immunization is a hedging strategy based on duration which isdesigned to protect against interest rate riskexample: suppose a portfolio manager has to pay out $1million in 2 years. Since there is only one cash outflow, theduration of this liability is 2 years. Suppose there are twodifferent bonds available and r = 10%:

Bond E: 8% annual coupon, T = 3 years, $1,000 par valueBond F: 7% annual coupon, T = 1 year, $1,000 par value

it is easy to calculate that PE = $950.25, DE = 2.78 years,PF = $972.73, and DF = 1 yearsome possible strategies:

buy F and then another 1 year bond after a year (but this runsthe risk of lower rates available for the second year – calledreinvestment risk).buy E and sell after 2 years (but if rates rise before then, bondprices will fall, and so the investment may not be enough tocover the liability – called price risk)invest in a combination of E and F

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Immunization (Cont’d)

consider the last strategy. Let W1 be the percentage investedin the 1 year bond and W3 be the percentage invested in the 3year bond. To immunize the liability, make the duration of thebond portfolio equal to the duration of the liability by solving:

W1 + W3 = 1

W1 × 1 + W3 × 2.78 = 2

note that the second equation uses the property that theduration of a portfolio is a weighted average of the durationsof the securities in itthe solution is W1 = .4382, W3 = .5618the total amount to be invested is the PV of the liability,which is $1,000,000/1.12 = $826,446therefore, the manager should invest.4382($826,446) = $362,149 in 1 year bonds and.5618($826,446) = $464,297 in 3 year bonds.this means that 362,149/$972.73 = 372.3 1 year bonds and$464,297/$950.25 = 488.6 3 year bonds should be purchased

Immunization 15 / 30

Immunization (Cont’d)

basic idea:

if rates rise, the portfolio’s losses on the 3 year bonds will beoffset by gains on reinvested 1 year bondsif rates fall, the portfolio’s losses on the 1 year bonds will beoffset by gains on the 3 year bonds

r after one year9% 10% 11%

Value at t = 2 from reinvesting1 year bond proceeds:1070× 372.3× (1 + r) $434,213 $438,197 $442,181Value at t = 2 of 3 year bonds:Value from reinvesting couponsreceived at t = 1:80× 488.6× (1 + r) $42,606 $42,997 $43,388Coupons received at t = 2:80× 488.6 $39,088 $39,088 $39,088Selling price at t = 2:488.6× 1080/(1 + r) $484,117 479,716 475,395Total $1,000,024 $999,998 1,000,052

Immunization 16 / 30

Immunization (Cont’d)

as can be seen from the table above, the immunizationstrategy appears to perform fairly wellhowever, there are a number of assumptions needed for this towork. Some possible problems include:

the strategy assumes that there is no default risk or call riskfor the bonds in the portfoliothe strategy assumes that the term structure is flat and anyshifts in it are parallelduration will change over time (even if r does not), so themanager may have to rebalance the portfolio (note that thereis a tradeoff of accuracy from frequent rebalancing vs.transactions costs)

more complicated strategies exist to handle these types ofproblems, but immunization using duration is still a verywidely used tool in practice

duration can also be used to speculate, as well as to hedge:e.g. if bond portfolio managers want to bet on interest ratesfalling, they may increase the duration of their portfolio

Immunization 17 / 30

Immunization With Futures Contracts

let S be the spot price of the asset being hedged and DS beits durationlet F be the contract price of an interest rate futures contract,and DF be the duration of the asset underlying the futurescontractsuppose r changes by ∆r , implying

∆S ≈ −SDS∆r/(1 + r)

with K futures contracts, the change in the value of thefutures position is

K∆F = −KFDF∆r/(1 + r)

therefore, to offset the risk (i.e. make ∆S − K∆F ≈ 0), weshould pick K = SDS/FDF

Immunization 18 / 30

Immunization With Futures Contracts (Cont’d)

example: It is May 20. A firm will receive $3.3M on August 5.The funds are needed for a major capital investment nextFebruary, so they will be invested in 6-month T-bills whenreceivedthis implies that S = $3,300,000, DS = 0.5the firm is concerned that T-bill yields will fall by August 5(i.e. T-bill prices will rise), so the hedge should payoff whenT-bill prices rise (i.e. it should be a long hedge)the quoted price for September 3-month T-bill futures is0.9736. Each contract calls for delivery of $1 million of T-bills,so the contract price is $973,600 (F = $973, 600,DF = 0.25)the firm should take a long position of

($3,300,000× 0.5)

($973,600× 0.25)= 6.78

contracts

Immunization 19 / 30

Interest Rate Swaps

swaps are private agreements to exchange future cash flowsaccording to a predetermined formulathe global market size has increased from zero in 1980 to anotional amount of $271.8 trillion (as reported by the Bankfor International Settlements) as of June 2007as a point of comparison, the notional market size was $163.7trillion as of June 2005as a further point of comparison, as of December 2006 theNYSE had 2,021 listed stocks with a total market value ofabout $20.2 trillionthere are many different kinds of swaps, we will concentrateon plain vanilla interest rate swaps

party A makes fixed rate payments to party B; in return Bmakes floating rate payments to Apayment size is based on notional principalfloating rate is usually 6 month LIBORthe London Interbank Offer Rate is an interest rate fortransactions between banks on Eurodollar deposits; a referencerate similar in some respects to the prime rate

Swaps 20 / 30

Example of Plain Vanilla Interest Rate Swap

companies A and B agree to a 3 year swap on October 1, 2007

the notional principal is $100 million

A pays 5.40% semi-annually to B; B pays LIBOR + 10 bps toA

on October 1, 2007 LIBOR is 5%

the first payments are exchanged on April 1, 2008

A pays B:

$100,000,000×(

183

365

)(.054) = $2,707,397

B pays A:

$100,000,000×(

183

360

)(.051) = $2,592,500

note the quoting conventions (365 for fixed rate, 360 forfloating rate)

the payments are netted, so that A pays B $114,897

Swaps 21 / 30

Example of Plain Vanilla Interest Rate Swap (Cont’d)

total cash flows over the swap’s life might be:

Floating FixedDay Payment Payment Net payment

Date Count LIBOR (B pays) (B receives) by BOctober 1, 2007 5.00%

April 1, 2008 183 5.25% $2,592,500 $2,707,397 ($114,897)October 1, 2008 183 4.75% $2,719,583 $2,707,397 $12,186

April 1, 2009 182 4.95% $2,451,944 $2,692,603 ($240,658)October 1, 2009 183 5.35% $2,567,083 $2,707,397 ($140,314)

April 1, 2010 182 5.60% $2,755,278 $2,692,603 $62,675October 1, 2010 183 $2,897,500 $2,707,397 $190,103

note that it makes no difference if principal is exchanged atthe end since payments are netted

the swap can be viewed as an exchange of a floating ratebond for a fixed rate bond

in this case, B has given a floating rate bond to A in returnfor a fixed rate bond

Swaps 22 / 30

The Role of Banks in Swaps

normally parties do not negotiate directly with each other; abank serves as an intermediarya typical pricing schedule looks like:

Maturity Bank Pays Bank Receives Current TNYears Fixed Fixed Rate

2 2 yr TN + 31 bps 2 yr TN + 36 bps 6.795 5 yr TN + 41 bps 5 yr TN + 50 bps 7.067 7 yr TN + 48 bps 7 yr TN + 60 bps 7.10

(All rates quoted against 6 month LIBOR)

this schedule indicates that for a 7 year swap, the bank willpay 7.58% on the fixed side in return for 6 month LIBOR, andthe bank will pay 6 month LIBOR in return for 7.70% fixedthe bank’s profits are 12 bps if it can negotiate offsettingtransactions (if it cannot, the swap will be “warehoused” andthe interest rate risk hedged, e.g. using futures contracts)typical swap spreads are now about 2-3 bps (they were up to100 bps in the early days of the market)

Swaps 23 / 30

Reasons for Using Swapsone reason is to transform a liability:

suppose B has a $35 million loan on which it pays a fixed rateof 7.5%assume B enters into a swap in which it pays LIBOR + 30 bpsand receives 7.19%B’s net position after the swap is

Pays 7.5% to outside lendersPays LIBOR + 30 bps in swapReceives 7.19% in swapPays LIBOR + 61 bps

of course, it is also possible to go the other way and transforma floating rate liability into a fixed rate liability

another reason is to transform an asset:

suppose B has a $35 million asset earning LIBOR - 20 bpsassuming B enters into the same swap as above, its netposition after the swap is

Receives LIBOR - 20 bps on assetPays LIBOR + 30 bps in swapReceives 7.19% in swapReceives 6.69%

Swaps 24 / 30

The Comparative Advantage Argument for Swaps

the situations described on the previous slide are for caseswhere a firm uses a swap to transform an already existingasset or liability

note that this could also be done through renegotiation (e.g.to transform an existing floating rate loan into a fixed rateloan, repurchase the existing loan and issue a new one), butthis is more costly than using a swap

we can also consider cases where a firm doesn’t have analready existing asset or liability, yet still wants to use a swapmarket

in this context, we can think in terms of comparativeadvantage arguments (just like international trade ineconomics)

in this case potential gains arise from relative differences infixed and floating rates

Swaps 25 / 30

Comparative Advantage (Cont’d)

example:Fixed Floating

A 8% 6 month LIBOR + 40 bpsB 9% 6 month LIBOR + 70 bps

B is less credit worthy than A (it pays higher rates for eitherfixed or floating), but it has a comparative advantage infloating (since it pays only 30 bps more in floating than Adoes but 100 bps more in fixed)let A borrow fixed, B borrow floating, and suppose they entera swap where A pays 6 month LIBOR + 10 bps and receives8.05%

Swaps 26 / 30

Comparative Advantage (Cont’d)however, in floating rate markets lenders have the option toreview terms every 6 months whereas fixed rates are usuallyon a 5-10 year term ⇒ the greater differential reflects agreater chance of a default by B over the longer termthe apparent gain of 35 bps to B assumes that B cancontinue to pay LIBOR + 70 bps outside, if its credit ratingworsens this amount could increase (e.g. to LIBOR + 250bps, in which case its net position including the swap wouldbe 10.45% fixed)A does lock in LIBOR + 5 bps for the length of the swap, butit is also taking on the risk of default by a counterparty (eitherB or a financial institution) and ignoring the possibility thatits credit rating might improvenote that the swap can be viewed as a portfolio of forwardcontracts: in the above example B has agreed to pay a fixedamount (8.05% of the principal) in return for a cash flow ofLIBOR + 10 bps times the principal every 6 months

Swaps 27 / 30

Cross-Currency Interest Rate Swaps

one simple variation is a plain vanilla currency swap whichinvolves exchanging fixed rate payments in different currenciesand principal

example:

$ £

A 6% 8.7%B 7.5% 9%

suppose A borrows at 6% in $ outside and enters into a swapwhere it pays 8.25% in £ and receives 6% in $ from a bank

B borrows at 9% in £ outside and enters into a swap where itreceives 9% in £ from the bank and pays 7.05% in $

A trades a 6% $ loan for an 8.25% £ loanB trades a 9% £ loan for a 7.05% $ loanBank gains 1.05% on $, loses 0.75% on £

Swaps 28 / 30

Cross-Currency Interest Rate Swaps (Cont’d)

the principal exchange is roughly of equal value at the start,e.g. $50 million and £25 million (i.e. A pays $50 million andreceives £25 million at the start of the swap, at the end Apays £25 million and receives $50 million)

this exchange may not be of equal value at the end

the bank also has foreign exchange rate risk, but it can hedgeusing forward or futures contracts

Swaps 29 / 30

Other Variations

fixed in one currency, floating in anotheramortizing/accreting swaps (notional principal changes overtime depending on interest rates)constant yield swaps (both parts floating, but differentmaturities)rate capped swaps (floating rate is capped)deferred swaps (rates set now, contract starts later)extendable/puttable swaps (one party has the option tochange the maturity of the swap contract)commodity swaps (e.g. a portfolio of forward contracts to buya commodity such as oil)equity swaps (many variants, e.g. floating payments dependon return on a stock index, both parts float (e.g. receive S&P500, pay Nikkei), one receives Ford, pays GM, etc.)options on swaps (“swaptions” - in plain vanilla interest ratecase, an option to exchange a fixed rate bond for a floatingrate bond, which is equivalent to an option to buy a fixed ratebond for its par value)

Swaps 30 / 30