1. Lecture 9: Derivatives and Hedging

2. Futures and forwards 2

3. Overview  Derivative securities have become increasingly important as FIs seek methods to hedge risk exposures. The growth of derivative usage is not without controversy since misuse can increase risk. This chapter explores the role of futures and forwards in risk management. 3

4. Futures and Forwards  Second largest group of interest rate derivatives in terms of notional value and largest group of FX derivatives.  Swaps are the largest. 4

5. Derivatives  Rapid growth of derivatives use has been controversial  Orange County, California  Bankers Trust  Allfirst Bank (Allied Irish)  As of 2000, FASB requires that derivatives be marked to market  Transparency of losses and gains on financial statements 5

6. Web Resources  For further information on the web, visit FASB www.fasb.orgwww.fasb.org 6

7. Spot and Forward Contracts  Spot Contract  Agreement at t=0 for immediate delivery and immediate payment.  Forward Contract  Agreement to exchange an asset at a specified future date for a price which is set at t=0.  Counterparty risk 7

8. Futures Contracts  Futures Contract similar to a forward contract except:  Marked to market  Exchange traded  Rapid growth of off market trading systems  Standardized contracts  Smaller denomination than forward  Lower default risk than forward contracts. 8

9. Hedging Interest Rate Risk  Example: 20-year $1 million face value bond. Duration of the bonds is 9 years. Current price = $970,000. Interest rates expected to increase from 8% to 10% over next 3 months.  From duration model, change in bond value:  P/P = -D   R/(1+R)  P/ $970,000 = -9  [.02/1.08]  P = -$161,666.67 9

10. Example continued: Naive hedge  Hedged by selling 3 months forward at forward price of $970,000.  Suppose interest rate rises from 8%to 10%. $970,000 - $808,333 = $161,667 (forward (spot price price)at t=3 months)  Exactly offsets the on-balance-sheet loss.  Immunized. 10

11. Hedging with futures  Futures more commonly used than forwards.  Microhedging  Individual assets.  Macrohedging  Hedging entire duration gap  Found more effective and generally lower cost.  Basis risk  Exact matching is uncommon  Standardized delivery dates of futures reduces likelihood of exact matching. 11

12. Routine versus Selective Hedging  Routine hedging: reduces interest rate risk to lowest possible level.  Low risk - low return.  Selective hedging: manager may selectively hedge based on expectations of future interest rates and risk preferences.  Partially hedge duration gap or individual assets or liabilities 12

13. Macrohedging with Futures  Number of futures contracts depends on interest rate exposure and risk-return tradeoff.  E = -[D A - kD L ] × A × [  R/(1+R)]  Suppose: D A = 5 years, D L = 3 years, Assets = $100 millions, L = $90 millions and interest rate expected to rise from 10% to 11%. A = $100 million.  E = -(5 - (.9)(3)) $100 (.01/1.1) = -$2.091 million. 13

14. Risk-Minimizing Futures Position  Sensitivity of the futures contract:   F/F = -D F [  R/(1+R)] Or,   F = -D F × [  R/(1+R)] × F and F = N F × P F Where  N F is the number of contracts bought or sold, and  P F is the price of each contract 14

15. Risk-Minimizing Futures Position  Fully hedged requires  F =  E D F (N F × P F ) = (D A - kD L ) × A Number of futures to sell: N F = (D A - kD L )A/(D F × P F )  Perfect hedge may be impossible since number of contracts must be rounded down. 15

16. Basis Risk  Spot and futures prices are not perfectly correlated.  We assumed in our example that  R/(1+R) =  R F /(1+R F )  Basis risk remains when this condition does not hold. Adjusting for basis risk, N F = (D A - kD L )A/(D F × P F × br) where br = [  R F /(1+R F )]/ [  R/(1+R)] 16

17. Hedging FX Risk  Hedging of FX exposure parallels hedging of interest rate risk.  If spot and futures prices are not perfectly correlated, then basis risk remains.  Tailing the hedge  Interest income effects of marking to market allows hedger to reduce number of futures contracts that must be sold to hedge 17

18. Basis Risk  In order to adjust for basis risk, we require the hedge ratio, h =  S t /  f t Where: N f = (Long asset position × estimate of h)/(size of one contract). f t Futures price ($/£) for the contract S t Spot exchange rate ($/£) 18

19. Estimating the Hedge Ratio  The hedge ratio may be estimated using ordinary least squares regression:  S t =  +  f t + u t  The hedge ratio, h will be equal to the coefficient . The R 2 from the regression reveals the effectiveness of the hedge. 19

20. Hedging Credit Risk  More FIs fail due to credit-risk exposures than to either interest-rate or FX exposures.  In recent years, development of derivatives for hedging credit risk has accelerated.  Credit forwards, credit options and credit swaps. 20

21. Credit Forwards  Credit forwards hedge against decline in credit quality of borrower.  Common buyers are insurance companies.  Common sellers are banks.  Specifies a credit spread on a benchmark bond issued by a borrower.  Example: BBB bond at time of origination may have 2% spread over U.S. Treasury of same maturity. 21

22. Credit Forwards  CS F defines agreed forward credit spread at time contract written  CS T = actual credit spread at maturity of forward Credit Spread Credit Spread Credit Spread at EndSellerBuyer CS T > CS F ReceivesPays (CS T - CS F )MD(A) (CS T -C S F )MD(A) CS F >CS T PaysReceives (CS F - CS T )MD(A) 22

23. Futures and Catastrophe Risk  CBOT introduced futures and options for catastrophe insurance.  Contract volume is rising.  Catastrophe futures to allow PC insurers to hedge against extreme losses such as hurricanes.  Payoff linked to loss ratio (insured losses to premiums)  Example: Payoff = contract size × realized loss ratio – contract size × contracted futures loss ratio. $25,000 × 1.5 - $25,000 – 0.8 = $17,500 per contract. 23

24. Regulatory Policy  Three levels of regulation:  Permissible activities  Supervisory oversight of permissible activities  Overall integrity and compliance  Functional regulators  SEC and CFTC  As of 2000, derivative positions must be marked-to-market.  Exchange traded futures not subject to capital requirements: OTC forwards potentially subject to capital requirements 24

25. Regulatory Policy for Banks  Federal Reserve, FDIC and OCC require banks  Establish internal guidelines regarding hedging.  Establish trading limits.  Disclose large contract positions that materially affect bank risk to shareholders and outside investors.  Discourage speculation and encourage hedging  Allfirst/Allied Irish: Existing (and apparently inadequate) policies were circumvented via fraud and deceit. 25

26. Websites Federal Reserve www.federalreserve.govwww.federalreserve.gov Chicago Board of Trade www.cbot.orgwww.cbot.org Chicago Mercantile Exchange www.cme.comwww.cme.com CFTC www.cftc.govwww.cftc.gov FDIC www.fdic.govwww.fdic.gov FASB www.fasb.orgwww.fasb.org OCC www.ustreas.govwww.ustreas.gov SEC www.sec.govwww.sec.gov 26

27. Options, Caps, Floors and Collars 27

28. Overview  Derivative securities as a whole have become increasingly important in the management of risk and this chapter details the use of options in that vein. A review of basic options –puts and calls– is followed by a discussion of fixed-income, or interest rate options. The chapter also explains options that address foreign exchange risk, credit risks, and catastrophe risk. Caps, floors, and collars are also discussed. 28

29. Option Terms  Long position in an option is synonymous with: Holder, buyer, purchaser, the long  Holder of an option has the right, but not the obligation to exercise the option  Short position in an option is synonymous with: Writer, seller, the short  Obliged to fulfill terms of the option if the option holder chooses to exercise. 29

30. Call option  A call provides the holder (or long position) with the right, but not the obligation, to purchase an underlying security at a prespecified exercise or strike price.  Expiration date: American and European options  The purchaser of a call pays the writer of the call (or the short position) a fee, or call premium in exchange. 30

31. Payoff to Buyer of a Call Option  If the price of the bond underlying the call option rises above the exercise price, by more than the amount of the premium, then exercising the call generates a profit for the holder of the call.  Since bond prices and interest rates move in opposite directions, the purchaser of a call profits if interest rates fall. 31

32. The Short Call Position  Zero-sum game:  The writer of a call (short call position) profits when the call is not exercised (or if the bond price is not far enough above the exercise price to erode the entire call premium).  Gains for the short call position are losses for the long call position.  Gains for the long call position are losses for the short call position. 32

33. Writing a Call  Since the price of the bond could rise to equal the sum of the principal and interest payments (zero rate of interest), the writer of a call is exposed to the risk of very large losses.  Recall that losses to the writer are gains to the purchaser of the call. Therefore, potential profit to call purchaser could be very large. (Note that call options on stocks have no theoretical payoff limit at all).  Maximum gain for the writer occurs if bond price falls below exercise price. 33

34. Call Options on Bonds Buy a callWrite a call X X 34

35. Put Option  A put provides the holder (or long position) with the right, but not the obligation, to sell an underlying security at a prespecified exercise or strike price.  Expiration date: American and European options  The purchaser of a put pays the writer of the put (or the short position) a fee, or put premium in exchange. 35

36. Payoff to Buyer of a Put Option  If the price of the bond underlying the put option falls below the exercise price, by more than the amount of the premium, then exercising the put generates a profit for the holder of the put.  Since bond prices and interest rates move in opposite directions, the purchaser of a put profits if interest rates rise. 36

37. The Short Put Position  Zero-sum game:  The writer of a put (short put position) profits when the put is not exercised (or if the bond price is not far enough below the exercise price to erode the entire put premium).  Gains for the short position are losses for the long position. Gains for the long position are losses for the short position. 37

38. Writing a Put  Since the bond price cannot be negative, the maximum loss for the writer of a put occurs when the bond price falls to zero.  Maximum loss = exercise price minus the premium 38

39. Put Options on Bonds Buy a PutWrite a Put (Long Put)(Short Put) X X 39

40. Writing versus Buying Options  Many smaller FIs constrained to buying rather than writing options.  Economic reasons  Potentially large downside losses for calls.  Potentially large losses for puts  Gains can be no greater than the premiums so less satisfactory as a hedge against losses in bond positions  Regulatory reasons  Risk associated with writing naked options. 40

41. Combining Long and Short Option Positions  The overall cost of hedging can be custom tailored by combining long and short option positions in combination with (or alternative to) adjusting the exercise price.  Example: Suppose the necessary hedge requires a long call option but the hedger wishes to lower the cost. A higher exercise price would lower the premium but provides less protection. Alternatively, the hedger could buy the desired call and simultaneously sell a put (with a lower exercise price). The put premium offsets the call premium. Presumably any losses on the short put would be offset by gains in the bond portfolio being hedged. 41

42. Hedging Downside with Long Put  Payoffs to Bond + Put X X Put Bond Net 42

43. Tips for plotting payoffs  Students often find it helpful to tabulate the payoffs at critical values of the underlying security:  Value of the position when bond price equals zero  Value of the position when bond price equals X  Value of position when bond price exceeds X  Value of net position equals sum of individual payoffs 43

44. Tips for plotting payoffs 44

45. Futures versus Options Hedging  Hedging with futures eliminates both upside and downside  Hedging with options eliminates risk in one direction only 45

46. Hedging with Futures Bond Portfolio Bond Price Purchased Futures Contract X 0 Gain Loss 46

47. Hedging Bonds  Weaknesses of Black-Scholes model.  Assumes short-term interest rate constant  Assumes constant variance of returns on underlying asset.  Behavior of bond prices between issuance and maturity  Pull-to-par. 47

48. Hedging With Bond Options Using Binomial Model  Example: FI purchases zero-coupon bond with 2 years to maturity, at BP 0 = $80.45. This means YTM = 11.5%.  Assume FI may have to sell at t=1. Current yield on 1-year bonds is 10% and forecast for next year ’ s 1-year rate is that rates will rise to either 13.82% or 12.18%.  If r 1 =13.82%, BP 1 = 100/1.1382 = $87.86  If r 1 =12.18%, BP 1 = 100/1.1218 = $89.14 48

49. Example (continued)  If the 1-year rates of 13.82% and 12.18% are equally likely, expected 1-year rate = 13% and E(BP 1 ) = 100/1.13 = $88.50.  To ensure that the FI receives at least $88.50 at end of 1 year, buy put with X = $88.50. 49

50. Value of the Put  At t = 1, equally likely outcomes that bond with 1 year to maturity trading at $87.86 or $89.14.  Value of put at t=1: Max[88.5-87.86, 0] =.64 Or, Max[88.5-89.14, 0] = 0.  Value at t=0: P = [.5(.64) +.5(0)]/1.10 = $0.29. 50

51. Actual Bond Options  Most pure bond options trade over-the- counter.  Open interest on CBOE relatively small  Preferred method of hedging is an option on an interest rate futures contract.  Combines best features of futures contracts with asymmetric payoff features of options. 51

52. Web Resources Visit: Chicago Board Options Exchange www.cboe.com www.cboe.com 52

53. Hedging with Put Options  To hedge net worth exposure,  P = -  E N p = [(D A -kD L )  A]  [   D  B] Where:  is the delta of the option. D is the duration of the bond  Adjustment for basis risk: N p = [(D A -kD L )  A]  [   D  B  br] 53

54. Macrohedge of Interest Rate Risk Using a Put Option 54

55. Using Options to Hedge FX Risk  Example: FI is long in 1-month T-bill paying £100 million. FIs liabilities are in dollars. Suppose they hedge with put options, with X=$1.60 /£1. Contract size = £31,250.  FI needs to buy £100,000,000/£31,250 = 3,200 contracts. If cost of put = 0.20 cents per £, then each contract costs $62.50. Total cost = $200,000 = (62.50 × 3,200). 55

56. Hedging Credit Risk With Options  Credit spread call option  Payoff increases as (default) yield spread on a specified benchmark bond on the borrower increases above some exercise spread S.  Digital default option  Pays a stated amount in the event of a loan default. 56

57. Hedging Catastrophe Risk  Catastrophe (CAT) call spread options to hedge unexpectedly high loss events such as hurricanes, faced by PC insurers.  Provides coverage within a bracket of loss-ratios. Example: Increasing payoff if loss-ratio between 50% and 80%. No payoff if below 50%. Capped at 80%. 57

58. Caps, Floors, Collars  Cap: buy call (or succession of calls) on interest rates.  Floor: buy a put on interest rates.  Collar: Cap + Floor.  Caps, Floors and Collars create exposure to counterparty credit risk since they involve multiple exercise over-the-counter contracts. 58

59. Fair Cap Premium  Two period cap: Fair premium = P = PV of year 1 option + PV of year 2 option  Cost of a cap (C) Cost = Notional Value of cap × fair cap premium (as percent of notional face value) C = NV c  pc 59

60. Collars: Buy a Cap and Sell a Floor  Net cost of long cap and short floor: Cost = (NV c × pc) - (NV f × pf ) = Cost of cap - Revenue from floor  Counterparty credit risk is an issue 60

61. Pertinent websites Chicago Board of Trade www.cbot.comwww.cbot.com CBOE www.cboe.comwww.cboe.com Chicago Mercantile Exchange www.cme.comwww.cme.com Wall Street Journal www.wsj.comwww.wsj.com 61

62. Swaps 62

63. Overview The market for swaps has grown enormously and this has raised serious regulatory concerns regarding credit risk exposures. Such concerns motivated the BIS risk-based capital reforms. At the same time, the growth in exotic swaps such as inverse floater have also generated controversy (e.g., Orange County, CA). Generic swaps in order of quantitative importance: interest rate, currency, credit, commodity and equity swaps. 63

64. Interest Rate Swaps  Interest rate swap as succession of forwards.  Swap buyer agrees to pay fixed-rate  Swap seller agrees to pay floating-rate.  Purpose of interest rate swap  Allows FIs to economically convert variable-rate instruments into fixed-rate (or vice versa) in order to better match the duration of assets and liabilities.  Off-balance-sheet transaction. 64

65. Plain Vanilla Interest Rate Swap Example  Consider money center bank that has raised $100 million by issuing 4-year notes with 10% fixed coupons. On asset side: loans linked to LIBOR. Duration gap is negative. D A - kD L < 0  Second party is savings bank with $100 million in fixed-rate mortgages of long duration funded with CDs having duration of 1 year. D A - kD L > 0 65

66. Example (continued)  Savings bank can reduce duration gap by buying a swap (taking fixed-payment side).  Notional value of the swap is $100 million.  Maturity is 4 years with 10% fixed- payments.  Suppose that LIBOR currently equals 8% and bank agrees to pay LIBOR + 2%. 66

67. Realized Cash Flows on Swap  Suppose realized rates are as follows End of YearLIBOR 1 9% 2 9% 3 7% 4 6% 67

68. Swap Payments End of LIBORMCBSavings MCB Year+ 2%PaymentBank Net 111%$11$10 +1 211 11 10 +1 39 9 10 - 1 48 8 10 - 2 Total 39 40 - 1 68

69. Off-market Swaps  Swaps can be molded to suit needs  Special interest terms  Varying notional value  Increasing or decreasing over life of swap.  Structured-note inverse floater  Example: Government agency issues note with coupon equal to 7 percent minus LIBOR and converts it into a LIBOR liability through a swap. 69

70. Macrohedging with Swaps  Assume a thrift has positive gap such that  E = -(D A - kD L )A [  R/(1+R)] >0 if rates rise. Suppose choose to hedge with 10-year swaps. Fixed-rate payments are equivalent to payments on a 10-year T-bond. Floating-rate payments repriced to LIBOR every year. Changes in swap value  S, depend on duration difference (D 10 - D 1 ).  S = -(D Fixed - D Float ) × N S × [  R/(1+R)] 70

71. Macrohedging (continued)  Optimal notional value requires  S =  E -(D Fixed - D Float ) × N S × [  R/(1+R)] = -(D A - kD L ) × A × [  R/(1+R)] N S = [(D A - kD L ) × A]/(D Fixed - D Float ) 71

72. Currency Swaps  Fixed-Fixed  Example: U.S. bank with fixed-rate assets denominated in dollars, partly financed with £50 million in 4-year 10 percent (fixed) notes. By comparison, U.K. bank has assets partly funded by $100 million 4-year 10 percent notes.  Solution: Enter into currency swap. 72

73. Cash Flows from Swap 73

74. Fixed-Floating + Currency  Fixed-Floating currency swaps.  Allows hedging of interest rate and currency exposures simultaneously  Combined Interest Rate and Currency (CIRCUS) Swap 74

75. Credit Swaps  Credit swaps designed to hedge credit risk.  Involvement of other FIs in the credit risk shift  Total return swap  Hedge possible change in credit risk exposure  Pure credit swap  Interest-rate sensitive element stripped out leaving only the credit risk. 75

76. Swaps and Credit Risk Concerns  Credit risk concerns partly mitigated by netting of swap payments.  Netting by novation  When there are many contracts between parties.  Payment flows are interest and not principal.  Standby letters of credit may be required.  Greenspan stated that credit swap market has helped strengthen the banking system ’ s ability to deal with recession 76

77. Websites BIS www.bis.orgwww.bis.org Federal Reserve www.federalreserve.govwww.federalreserve.gov FDIC www.fdic.govwww.fdic.gov International Swaps and Derivatives Association www.isda.orgwww.isda.org Moody ’ s Investor Services www.moodys.com www.moodys.com 77