AFM 371 Winter 2008 Chapter 26 - Derivatives and Hedging Risk Part 2 - Interest Rate Risk Management (26.4-26.7) 1 / 30
Outline Term Structure Forward Contracts on Bonds Interest Rate Futures Contracts Duration Immunization Swaps 2 / 30
The Term Structure of Interest Rates the text coverage of this material is in Appendix 6A although in almost all cases in this course we consider a flat term structure, it is important to keep in mind that this is a simplification with a flat term structure, discount rates are the same for all maturities, but this is rarely (if ever) the case for Oct. 31, 2007, the Bank of Canada reported government zero coupon government bond yields as follows: Maturity 1 yr 3 yr 5 yr 7 yr 10 yr 15 yr Yield 4.18 4.16 4.18 4.21 4.28 4.37 this means, for example, that the price on Oct. 31 of a one year zero coupon government bond paying $1,000 at maturity was $1,000/1.0418 = $959.88, while the price of a ten year zero coupon government bond paying $1,000 at maturity was $1,000/1.0428 10 = $657.64 the rates above, which can be used to determine prices at which bonds may be currently traded, are known as spot rates erm Structure 3 / 30
Spot Rates and Govermnent Bond Prices let spot risk free (semi-annual) rates be r 1, r 2, r 3,..., etc. This means that we calculate present values as follows: Cash Flow Date Received Present Value C 1 t = 0.5 years from now C 1 /(1 + r 1 ) C 2 t = 1.0 years from now C 2 /(1 + r 2 ) 2 C 3 t = 1.5 years from now C 3 /(1 + r 3 ) 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, the present value of these future cash flows is C (1 + r 1 ) + C (1 + r 2 ) 2 + C (1 + r 3 ) 3 + + C + F (1 + r N ) N if the term structure is flat, i.e. r 1 = r 2 = = r N = r, then the above formula simplifies to the familiar CA N r + F /(1 + r) N erm Structure 4 / 30
Forward Contracts on Bonds consider entering into a contract to purchase a government bond M periods from now which will have a remaining maturity of T years when it is bought the idea is that you will pay the forward price F 0 (decided upon today) M periods from now for a government bond that will mature M + N periods from now (recall N = 2T ) assuming that the date on which you buy the bond is exactly six months before the next coupon payment, the present value of the bond today is S 0 = M+N i=m+1 C F (1 + r i ) i + (1 + r M+N ) M+N recall the spot forward parity relationship (assuming that the underlying asset pays no income and there are no costs of carry and no convenience yield: F t = S t (1 + r f ) T t orward Contracts on Bonds 5 / 30
Pricing of Forward Contracts on Bonds in this context, the spot-forward parity relationship implies that the forward price is F 0 = (1 + r M ) M S 0 [ M+N = (1 + r M ) M i=m+1 ] C F (1 + r i ) i + (1 + r M+N ) M+N example: consider a 5 year (M = 10) forward contract to buy a 30 year government bond which pays semi-annual coupons of $25, and assume the term structure is flat with a semi-annual yield of 2%. What is the value of the bond today? What is the forward price? orward Contracts on Bonds 6 / 30
Interest Rate Futures Contracts and Hedging in practice, futures contracts on bonds are typically used rather than forward contracts futures contracts on bonds are referred to as interest rate futures contracts the pricing relationships derived above for forward contracts will only be an approximation in this context (recall that forward and futures prices are identical if interest rates are not random; also the exact delivery date is determined by the short party in a futures contract) suppose you own $10 million worth of 20 year 10% annual coupon bonds (coupon payments are semi-annual). The term structure is flat at 5% (semi-annual). These bonds are therefore selling at par, i.e. 40 i=1 $50 1.05 i + $1,000 = $1,000 1.0540 nterest Rate Futures Contracts 7 / 30
Interest Rate Futures Contracts and Hedging (Cont d) if the term structure shifts up uniformly to 5.5%, the new price per bond is 40 i=1 $50 1.055 i + $1,000 = $919.77 1.05540 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 bonds before the term structure shift, what was the approximate futures price for delivery in 6 months? What is it after the term structure shift? nterest 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 $100,000 futures contracts. Then you would have gained $691,000 on your short futures position, partially offsetting your loss of $802,300, and resulting in an overall loss of $111,300 reasons in practice why interest rate hedging using futures may not work perfectly: different maturities (bonds in portfolio vs. futures contract) different coupon rates different risk (e.g. corporate bonds in portfolio, government bonds in futures contract) see text pp. 748-751 for further examples nterest Rate Futures Contracts 9 / 30
Duration assume for simplicity a flat term structure. Consider these four bonds, each with $1,000 par value and coupons paid annually: Price if Price if Price if % P for % P for Bond 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 price when 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 earlier discussion about why callable bonds have relatively low interest rate risk because they tend to have high coupons to compensate 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 = dp dr = C 1 + r + C (1 + r) 2 + + C 1 1 + r (1 + r) T + F (1 + r) T» 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 given change 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 (a measure of the bond s effective maturity given when its cash flows occur). uration 11 / 30
Duration (Cont d) duration is defined as: therefore: h C 1+r + 2C (1+r) 2 + D = P T t=1 = tc (1+r) t + TF (1+r) T P 3C (1+r) 3 + + P i TC (1+r) + TF T (1+r) T 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 on bond price example: duration of Bond A (r = 10%)» 50 (2 50) D A = + 1.1 1.1 2 + (3 1050) 1.1 3 1 875.66 =.0519 +.0944 + 2.7027 = 2.849 years Duration 12 / 30
Duration (Cont d) we can also calculate D B = 2.698 years (higher coupon implies shorter duration since more cash is paid earlier), D C = 3 years (duration of a zero coupon bond = T ), etc. The following table illustrates the use of duration as a measure of bond price sensitivity: dr = +.001 dr =.001 Bond Duration Estimated Change Actual Change Estimated Change Actual Change A 2.849 -.002590 -.002592.002590.002592 B 2.698 -.002453 -.002448.002453.002448 C 3 -.002727 -.002715.002727.002742 D 10 -.009091 -.009026.009091.009156 note that we get quite accurate estimates (for small changes in r) based on dp P = 1 1 + r D dr uration 13 / 30
Immunization immunization is a hedging strategy based on duration which is designed to protect against interest rate risk example: suppose a portfolio manager has to pay out $1 million in 2 years. Since there is only one cash outflow, the duration of this liability is 2 years. Suppose there are two different bonds available and r = 10%: Bond E: 8% annual coupon, T = 3 years, $1,000 par value Bond F: 7% annual coupon, T = 1 year, $1,000 par value it is easy to calculate that P E = $950.25, D E = 2.78 years, P F = $972.73, and D F = 1 year some possible strategies: buy F and then another 1 year bond after a year (but this runs the risk of lower rates available for the second year called reinvestment risk). buy E and sell after 2 years (but if rates rise before then, bond prices will fall, and so the investment may not be enough to cover the liability called price risk) invest in a combination of E and F Immunization 14 / 30
Immunization (Cont d) consider the last strategy. Let W 1 be the percentage invested in the 1 year bond and W 3 be the percentage invested in the 3 year bond. To immunize the liability, make the duration of the bond portfolio equal to the duration of the liability by solving: W 1 + W 3 = 1 W 1 1 + W 3 2.78 = 2 note that the second equation uses the property that the duration of a portfolio is a weighted average of the durations of the securities in it the solution is W 1 =.4382, W 3 =.5618 the total amount to be invested is the PV of the liability, which is $1,000,000/1.1 2 = $826,446 therefore, 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 mmunization 15 / 30
Immunization (Cont d) basic idea: if rates rise, the portfolio s losses on the 3 year bonds will be offset by gains on reinvested 1 year bonds if rates fall, the portfolio s losses on the 1 year bonds will be offset by gains on the 3 year bonds r after one year 9% 10% 11% Value at t = 2 from reinvesting 1 year bond proceeds: 1070 372.3 (1 + r) $434,213 $438,197 $442,181 Value at t = 2 of 3 year bonds: Value from reinvesting coupons received at t = 1: 80 488.6 (1 + r) $42,606 $42,997 $43,388 Coupons received at t = 2: 80 488.6 $39,088 $39,088 $39,088 Selling price at t = 2: 488.6 1080/(1 + r) $484,117 479,716 475,395 Total $1,000,024 $999,998 1,000,052 mmunization 16 / 30
Immunization (Cont d) as can be seen from the table above, the immunization strategy appears to perform fairly well however, there are a number of assumptions needed for this to work. Some possible problems include: the strategy assumes that there is no default risk or call risk for the bonds in the portfolio the strategy assumes that the term structure is flat and any shifts in it are parallel duration will change over time (even if r does not), so the manager may have to rebalance the portfolio (note that there is a tradeoff of accuracy from frequent rebalancing vs. transactions costs) more complicated strategies exist to handle these types of problems, but immunization using duration is still a very widely 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 rates falling, they may increase the duration of their portfolio mmunization 17 / 30
Immunization With Futures Contracts let S be the spot price of the asset being hedged and D S be its duration let F be the contract price of an interest rate futures contract, and D F be the duration of the asset underlying the futures contract suppose r changes by r, implying S SD S r/(1 + r) with K futures contracts, the change in the value of the futures position is K F = KFD F r/(1 + r) therefore, to offset the risk (i.e. make S K F 0), we should pick K = SD S /FD F mmunization 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 next February, so they will be invested in 6-month T-bills when received this implies that S = $3,300,000, D S = 0.5 the firm is concerned that T-bill yields will fall by August 5 (i.e. T-bill prices will rise), so the hedge should payoff when T-bill prices rise (i.e. it should be a long hedge) the quoted price for September 3-month T-bill futures is 0.9736. Each contract calls for delivery of $1 million of T-bills, so the contract price is $973,600 (F = $973, 600, D F = 0.25) the firm should take a long position of contracts ($3,300,000 0.5) ($973,600 0.25) = 6.78 mmunization 19 / 30
Interest Rate Swaps swaps are private agreements to exchange future cash flows according to a predetermined formula the global market size has increased from zero in 1980 to a notional amount of $271.8 trillion (as reported by the Bank for International Settlements) as of June 2007 as a point of comparison, the notional market size was $163.7 trillion as of June 2005 as a further point of comparison, as of December 2006 the NYSE had 2,021 listed stocks with a total market value of about $20.2 trillion there are many different kinds of swaps, we will concentrate on plain vanilla interest rate swaps party A makes fixed rate payments to party B; in return B makes floating rate payments to A payment size is based on notional principal floating rate is usually 6 month LIBOR the London Interbank Offer Rate is an interest rate for transactions between banks on Eurodollar deposits; a reference rate similar in some respects to the prime rate waps 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 to A on October 1, 2007 LIBOR is 5% the first payments are exchanged on April 1, 2008 A pays B: ( ) 183 $100,000,000 (.054) = $2,707,397 365 B pays A: $100,000,000 ( ) 183 (.051) = $2,592,500 360 note the quoting conventions (365 for fixed rate, 360 for floating rate) the payments are netted, so that A pays B $114,897 waps 21 / 30
Example of Plain Vanilla Interest Rate Swap (Cont d) total cash flows over the swap s life might be: Floating Fixed Day Payment Payment Net payment Date Count LIBOR (B pays) (B receives) by B October 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 $62,675 October 1, 2010 183 $2,897,500 $2,707,397 $190,103 note that it makes no difference if principal is exchanged at the end since payments are netted the swap can be viewed as an exchange of a floating rate bond for a fixed rate bond in this case, B has given a floating rate bond to A in return for a fixed rate bond waps 22 / 30
The Role of Banks in Swaps normally parties do not negotiate directly with each other; a bank serves as an intermediary a typical pricing schedule looks like: Maturity Bank Pays Bank Receives Current TN Years Fixed Fixed Rate 2 2 yr TN + 31 bps 2 yr TN + 36 bps 6.79 5 5 yr TN + 41 bps 5 yr TN + 50 bps 7.06 7 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 will pay 7.58% on the fixed side in return for 6 month LIBOR, and the bank will pay 6 month LIBOR in return for 7.70% fixed the bank s profits are 12 bps if it can negotiate offsetting transactions (if it cannot, the swap will be warehoused and the interest rate risk hedged, e.g. using futures contracts) typical swap spreads are now about 2-3 bps (they were up to 100 bps in the early days of the market) waps 23 / 30
Reasons for Using Swaps one reason is to transform a liability: suppose B has a $35 million loan on which it pays a fixed rate of 7.5% assume B enters into a swap in which it pays LIBOR + 30 bps and receives 7.19% B s net position after the swap is Pays 7.5% to outside lenders Pays LIBOR + 30 bps in swap Receives 7.19% in swap Pays LIBOR + 61 bps of course, it is also possible to go the other way and transform a 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 bps assuming B enters into the same swap as above, its net position after the swap is Receives LIBOR - 20 bps on asset Pays LIBOR + 30 bps in swap Receives 7.19% in swap Receives 6.69% waps 24 / 30
The Comparative Advantage Argument for Swaps the situations described on the previous slide are for cases where a firm uses a swap to transform an already existing asset or liability note that this could also be done through renegotiation (e.g. to transform an existing floating rate loan into a fixed rate loan, repurchase the existing loan and issue a new one), but this is more costly than using a swap we can also consider cases where a firm doesn t have an already existing asset or liability, yet still wants to use a swap market in this context, we can think in terms of comparative advantage arguments (just like international trade in economics) in this case potential gains arise from relative differences in fixed and floating rates waps 25 / 30
Comparative Advantage (Cont d) example: Fixed Floating A 8% 6 month LIBOR + 40 bps B 9% 6 month LIBOR + 70 bps B is less credit worthy than A (it pays higher rates for either fixed or floating), but it has a comparative advantage in floating (since it pays only 30 bps more in floating than A does but 100 bps more in fixed) let A borrow fixed, B borrow floating, and suppose they enter a swap where A pays 6 month LIBOR + 10 bps and receives 8.05% waps 26 / 30
Comparative Advantage (Cont d) however, in floating rate markets lenders have the option to review terms every 6 months whereas fixed rates are usually on a 5-10 year term the greater differential reflects a greater chance of a default by B over the longer term the apparent gain of 35 bps to B assumes that B can continue to pay LIBOR + 70 bps outside, if its credit rating worsens this amount could increase (e.g. to LIBOR + 250 bps, in which case its net position including the swap would be 10.45% fixed) A does lock in LIBOR + 5 bps for the length of the swap, but it is also taking on the risk of default by a counterparty (either B or a financial institution) and ignoring the possibility that its credit rating might improve note that the swap can be viewed as a portfolio of forward contracts: in the above example B has agreed to pay a fixed amount (8.05% of the principal) in return for a cash flow of LIBOR + 10 bps times the principal every 6 months waps 27 / 30
Cross-Currency Interest Rate Swaps one simple variation is a plain vanilla currency swap which involves exchanging fixed rate payments in different currencies and principal example: $ A 6% 8.7% B 7.5% 9% suppose A borrows at 6% in $ outside and enters into a swap where it pays 8.25% in and receives 6% in $ from a bank B borrows at 9% in outside and enters into a swap where it receives 9% in from the bank and pays 7.05% in $ A trades a 6% $ loan for an 8.25% loan B trades a 9% loan for a 7.05% $ loan Bank gains 1.05% on $, loses 0.75% on waps 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 and receives 25 million at the start of the swap, at the end A pays 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 hedge using forward or futures contracts waps 29 / 30
Other Variations fixed in one currency, floating in another amortizing/accreting swaps (notional principal changes over time depending on interest rates) constant yield swaps (both parts floating, but different maturities) rate capped swaps (floating rate is capped) deferred swaps (rates set now, contract starts later) extendable/puttable swaps (one party has the option to change the maturity of the swap contract) commodity swaps (e.g. a portfolio of forward contracts to buy a commodity such as oil) equity swaps (many variants, e.g. floating payments depend on return on a stock index, both parts float (e.g. receive S&P 500, pay Nikkei), one receives Ford, pays GM, etc.) options on swaps ( swaptions - in plain vanilla interest rate case, an option to exchange a fixed rate bond for a floating rate bond, which is equivalent to an option to buy a fixed rate bond for its par value) waps 30 / 30