1. Fall-01 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html FIBI Zvi Wiener 02-588-3049 mswiener@mscc.huji.ac.il Fixed Income Instruments 5

2. Zvi WienerFIFIBI - 5 slide 2 Fixed Income 5 Mortgage loans Pass-through securities Prepayments Agencies MBS CMO ABS

3. Zvi WienerFIFIBI - 5 slide 3 Bonds with Embedded Options (14) Traditional yield analysis compares yields of bonds with yield of on-the-run similar Treasuries. The static spread is a measure of the spread that should be added to the zero curve (Treasuries) to get the market value of a bond.

4. Zvi WienerFIFIBI - 5 slide 4 Active Bond Portfolio Management (17) Basic steps of investment management Active versus passive strategies Market consensus Different types of active strategies Bullet, barbell and ladder strategies Limitations of duration and convexity How to use leveraging and repo market

5. Zvi WienerFIFIBI - 5 slide 5 Investment Management Setting goals, idea of ALM or benchmark GAAP, FAS 133, AIMR - reporting standards passive or active strategy - views, not transactions available indexes mixed strategies

6. Zvi WienerFIFIBI - 5 slide 6 Major risk factors level of interest rates shape of the yield curve changes in spreads changes in OAS performance of a specific sector/asset currency/linkage

7. Zvi WienerFIFIBI - 5 slide 7 Parallel shift T r Current TS Downward move upward move

8. Zvi WienerFIFIBI - 5 slide 8 Twist T rr steepening flattening

9. Zvi WienerFIFIBI - 5 slide 9 Butterfly T rr

10. Zvi WienerFIFIBI - 5 slide 10 Yield curve strategies Bullet strategy: Maturities of securities are concentrated at some point on the yield curve. Barbel strategy: Maturities of securities are concentrated at two extreme maturities. Ladder strategy: Maturities of securities are distributed uniformly on the yield curve.

11. Zvi WienerFIFIBI - 5 slide 11 Example bondcouponmaturityyielddurationconvex. A8.5%58.54.00519.81 B9.5%209.58.882124.17 C9.25%109.256.43455.45 Bullet portfolio: 100% bond C Barbell portfolio: 50.2% bond A, 49.8% bond B

12. Zvi WienerFIFIBI - 5 slide 12 Dollar duration of barbell portfolio = 0.502*4.005 + 0.498*8.882 = 6.434 it has the same duration as bullet portfolio. Dollar convexity of barbell portfolio = 0.502*19.81 + 0.498*124.17 = 71.78 the convexity here is higher! Is this an arbitrage?

13. Zvi WienerFIFIBI - 5 slide 13 The yield of the bullet portfolio is 9.25% The yield of the barbell portfolio is 8.998% This is the cost of convexity!

14. Zvi WienerFIFIBI - 5 slide 14 Leverage Risk is not proportional to investment! This can be achieved in many ways: futures, options, repos (loans), etc. Duration of a levered portfolio is different form the average time of cashflow! Use of dollar duration!

15. Zvi WienerFIFIBI - 5 slide 15 Repo Market Repurachase agreement - a sale of a security with a commitment to buy the security back at a specified price at a specified date. Overnight repo (1 day), term repo (longer).

16. Zvi WienerFIFIBI - 5 slide 16 Repo Example You are a dealer and you need $10M to purchase some security. Your customer has $10M in his account with no use. You can offer your customer to buy the security for you and you will repurchase the security from him tomorrow. Repo rate 6.5% Then your customer will pay $9,998,195 for the security and you will return him $10M tomorrow.

17. Zvi WienerFIFIBI - 5 slide 17 Repo Example $9,998,195 0.065/360 = $1,805 This is the profit of your customer for offering the loan. Note that there is almost no risk in the loan since you get a safe security in exchange.

18. Zvi WienerFIFIBI - 5 slide 18 Reverse Repo You can buy a security with an attached agreement to sell them back after some time at a fixed price. Repo margin - an additional collateral. The repo rate varies among transactions and may be high for some hot (special) securities.

19. Zvi WienerFIFIBI - 5 slide 19 Example You manage $1M of your client. You wish to buy for her account an adjustable rate passthrough security backed by Fannie Mae. The coupon rate is reset every month according to LIBOR1M + 80 bp with a cap 9%. A repo rate is LIBOR + 10 bp and 5% margin is required. Then you can essentially borrow $19M and get 70 bp *19M. Is this risky?

20. Zvi WienerFIFIBI - 5 slide 20 Indexing The idea of a benchmark (liabilities, actuarial or artificial). Cellular approach, immunization, dynamic approach Tracking error Performance measurement, and attribution Optimization Risk measurement

21. Zvi WienerFIFIBI - 5 slide 21 Flattener T r Current TS Sell, Buy

22. Zvi WienerFIFIBI - 5 slide 22 Example of a flattener sell short, say 1 year buy long, say 5 years what amounts? In order to be duration neutral you have to buy 20% of the amount sold and invest the proceedings into money market. Sell 5M, buy 1M and invest 4M into MM.

23. Zvi WienerFIFIBI - 5 slide 23 Use of futures to take position Assume that you would like to be longer then your benchmark. This means that you expect that interest rates in the future will move down more than predicted by the forward rates. One possible way of doing this is by taking a future position. How to do this?

24. Zvi WienerFIFIBI - 5 slide 24 Use of futures to take position Your benchmark is 3 years, your current portfolio has duration of 3 years as well and value of $1M. You would like to have duration of 3.5 years since your expectation regarding 3 year interest rates for the next 2 months are different from the market. Each future contract will allow you to buy 5 years T-notes in 2 months for a fixed price.

25. Zvi WienerFIFIBI - 5 slide 25 Use of futures to take position Each future contract will allow you to buy 5 years T-notes in 2 months for a fixed price. If you are right and the IR will go down (relative to forward rates) then the value of the bonds that you will receive will be higher then the price that you will have to pay and your portfolio will earn more than the benchmark.

26. Zvi WienerFIFIBI - 5 slide 26 Use of futures to take position One should chose x such that the resulting duration will be 3.5 years. 02M3Y5Y -x(1+r 2M /6) (1+r 3Y ) 3 x(1+r 5Y ) 5

27. Fall-01 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html FIBI Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html Bond Risk Management

28. Zvi WienerFIFIBI - 5 slide 28 Duration Macauley duration Modified duration Effective duration Dollar duration

29. Zvi WienerFIFIBI - 5 slide 29 Fixed Income Risk Arises from potential movements in the level and volatility of bond yields. Factors affecting yields – inflationary expectations – term spread – higher volatility of the low end of TS

30. Zvi WienerFIFIBI - 5 slide 30 Volatilities of IR/bond prices Price volatility in %End 99End 96 Euro 30d0.220.05 Euro 180d0.300.19 Euro 360d0.520.58 Swap 2Y1.571.57 Swap 5Y4.234.70 Swap 10Y8.479.82 Zero 2Y1.551.64 Zero 5Y4.074.67 Zero 10Y7.769.31 Zero 30Y20.7523.53

31. Zvi WienerFIFIBI - 5 slide 31 Duration approximation What duration makes bond as volatile as FX? What duration makes bond as volatile as stocks? A 10 year bond has yearly price volatility of 8% which is similar to major FX. 30-year bonds have volatility similar to equities (20%).

32. Zvi WienerFIFIBI - 5 slide 32 Models of IR Normal model  (  y) is normally distributed. Lognormal model  (  y/y) is normally distributed. Note that:

33. Zvi WienerFIFIBI - 5 slide 33 Principal component analysis level risk factor 94% of changes slope risk factor (twist) 4% of changes curvature (bend or butterfly) See book by Golub and Tilman.

34. Zvi WienerFIFIBI - 5 slide 34 Forwards and Futures The forward or futures price on a stock. e -rt the present value in the base currency. e -yt the cost of carry (dividend rate). For a discrete dividend (individual stock) we can write the right hand side as S t - D, where D is the PV of the dividend.

35. Fall-01 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html FIBI Hedging Linear Risk Following Jorion 2001, Chapter 14 Financial Risk Manager Handbook

36. Zvi WienerFIFIBI - 5 slide 36 Hedging Taking positions that lower the risk profile of the portfolio. Static hedging Dynamic hedging

37. Zvi WienerFIFIBI - 5 slide 37 Unit Hedging with Currencies A US exporter will receive Y125M in 7 months. The perfect hedge is to enter a 7-months forward contract. Such a contract is OTC and illiquid. Instead one can use traded futures. CME lists yen contract with face value Y12.5M and 9 months to maturity. Sell 10 contracts and revert in 7 months.

38. Zvi WienerFIFIBI - 5 slide 38 Market data07mP&L time to maturity92 US interest rate6%6% Yen interest rate5%2% Spot Y/$125.00150.00 Futures Y/$124.07149.00

39. Zvi WienerFIFIBI - 5 slide 39 Stacked hedge - to use a longer horizon and to revert the position at maturity. Strip hedge - rolling over short hedge.

40. Zvi WienerFIFIBI - 5 slide 40 Basis Risk Basis risk arises when the characteristics of the futures contract differ from those of the underlying. For example quality of agricultural product, types of oil, Cheapest to Deliver bond, etc. Basis = Spot - Future

41. Zvi WienerFIFIBI - 5 slide 41 Cross hedging Hedging with a correlated (but different) asset. In order to hedge an exposure to Norwegian Krone one can use Euro futures. Hedging a portfolio of stocks with index future.

42. Zvi WienerFIFIBI - 5 slide 42 The Optimal Hedge Ratio  S - change in $ value of the inventory  F - change in $ value of the one futures N - number of futures you buy/sell

43. Zvi WienerFIFIBI - 5 slide 43 The Optimal Hedge Ratio Minimum variance hedge ratio

44. Zvi WienerFIFIBI - 5 slide 44 Hedge Ratio as Regression Coefficient The optimal amount can also be derived as the slope coefficient of a regression  s/s on  f/f:

45. Zvi WienerFIFIBI - 5 slide 45 Optimal Hedge One can measure the quality of the optimal hedge ratio in terms of the amount by which we have decreased the variance of the original portfolio. If R is low the hedge is not effective!

46. Zvi WienerFIFIBI - 5 slide 46 Optimal Hedge At the optimum the variance is

47. Zvi WienerFIFIBI - 5 slide 47 Example Airline company needs to purchase 10,000 tons of jet fuel in 3 months. One can use heating oil futures traded on NYMEX. Notional for each contract is 42,000 gallons. We need to check whether this hedge can be efficient.

48. Zvi WienerFIFIBI - 5 slide 48 Example Spot price of jet fuel $277/ton. Futures price of heating oil $0.6903/gallon. The standard deviation of jet fuel price rate of changes over 3 months is 21.17%, that of futures 18.59%, and the correlation is 0.8243.

49. Zvi WienerFIFIBI - 5 slide 49 Compute The notional and standard deviation f the unhedged fuel cost in $. The optimal number of futures contracts to buy/sell, rounded to the closest integer. The standard deviation of the hedged fuel cost in dollars.

50. Zvi WienerFIFIBI - 5 slide 50 Solution The notional is Qs=$2,770,000, the SD in $ is  (  s/s)sQ s =0.2117  $277  10,000 = $586,409 the SD of one futures contract is  (  f/f)fQ f =0.1859  $0.6903  42,000 = $5,390 with a futures notional fQ f = $0.6903  42,000 = $28,993.

51. Zvi WienerFIFIBI - 5 slide 51 Solution The cash position corresponds to a liability (payment), hence we have to buy futures as a protection.  sf = 0.8243  0.2117/0.1859 = 0.9387  sf = 0.8243  0.2117  0.1859 = 0.03244 The optimal hedge ratio is N* =  sf Q s  s/Q f  f = 89.7, or 90 contracts.

52. Zvi WienerFIFIBI - 5 slide 52 Solution  2 unhedged = ($586,409) 2 = 343,875,515,281 -  2 SF /  2 F = -(2,605,268,452/5,390) 2  hedged = $331,997 The hedge has reduced the SD from $586,409 to $331,997. R 2 = 67.95%(= 0.8243 2 )

53. Zvi WienerFIFIBI - 5 slide 53 Duration Hedging Dollar duration

54. Zvi WienerFIFIBI - 5 slide 54 Duration Hedging If we have a target duration D V * we can get it by using

55. Zvi WienerFIFIBI - 5 slide 55 Example 1 A portfolio manager has a bond portfolio worth $10M with a modified duration of 6.8 years, to be hedged for 3 months. The current futures prices is 93-02, with a notional of $100,000. We assume that the duration can be measured by CTD, which is 9.2 years. Compute: a. The notional of the futures contract b.The number of contracts to by/sell for optimal protection.

56. Zvi WienerFIFIBI - 5 slide 56 Example 1 The notional is: (93+2/32)/100  $100,000 =$93,062.5 The optimal number to sell is: Note that DVBP of the futures is 9.2  $93,062  0.01%=$85

57. Zvi WienerFIFIBI - 5 slide 57 Example 2 On February 2, a corporate treasurer wants to hedge a July 17 issue of $5M of CP with a maturity of 180 days, leading to anticipated proceeds of $4.52M. The September Eurodollar futures trades at 92, and has a notional amount of $1M. Compute a. The current dollar value of the futures contract. b. The number of futures to buy/sell for optimal hedge.

58. Zvi WienerFIFIBI - 5 slide 58 Example 2 The current dollar value is given by $10,000  (100-0.25(100-92)) = $980,000 Note that duration of futures is 3 months, since this contract refers to 3-month LIBOR.

59. Zvi WienerFIFIBI - 5 slide 59 Example 2 If Rates increase, the cost of borrowing will be higher. We need to offset this by a gain, or a short position in the futures. The optimal number of contracts is: Note that DVBP of the futures is 0.25  $1,000,000  0.01%=$25

60. Zvi WienerFIFIBI - 5 slide 60 Beta Hedging  represents the systematic risk,  - the intercept (not a source of risk) and  - residual. A stock index futures contract

61. Zvi WienerFIFIBI - 5 slide 61 Beta Hedging The optimal N is The optimal hedge with a stock index futures is given by beta of the cash position times its value divided by the notional of the futures contract.

62. Zvi WienerFIFIBI - 5 slide 62 Example A portfolio manager holds a stock portfolio worth $10M, with a beta of 1.5 relative to S&P500. The current S&P index futures price is 1400, with a multiplier of $250. Compute: a. The notional of the futures contract b. The optimal number of contracts for hedge.

63. Zvi WienerFIFIBI - 5 slide 63 Example The notional of the futures contract is $250  1,400 = $350,000 The optimal number of contracts for hedge is The quality of the hedge will depend on the size of the residual risk in the portfolio.

64. Zvi WienerFIFIBI - 5 slide 64 A typical US stock has correlation of 50% with S&P. Using the regression effectiveness we find that the volatility of the hedged portfolio is still about (1-0.5 2 ) 0.5 = 87% of the unhedged volatility for a typical stock. If we wish to hedge an industry index with S&P futures, the correlation is about 75% and the unhedged volatility is 66% of its original level. The lower number shows that stock market hedging is more effective for diversified portfolios.

65. Fall-01 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html FIBI FRM-GARP type question Zvi Wiener

66. FIFIBI - 5 slide 66 FRM-GARP 98:18 A portfolio consists of two positions: One position is long $100M of a two year bond priced at 101 with duration of 1.7, the other position is short $50M of a five year bond priced at 99 with a duration of 4.1. What is the duration of the portfolio?

67. Zvi WienerFIFIBI - 5 slide 67 FRM-GARP 98:18 The dollar duration is sum of dollar durations, so $100M 101/100 1.7 = $171.7M -$50M 99/100 4.1 = -$202.95M total dollar duration is -$31.25M, portfolio’s value is $50M, thus its duration is -0.61.

68. Zvi WienerFIFIBI - 5 slide 68 Cap and Floor Cap: Max[i T -i C, 0] Floor: Max[i F -i T, 0] What is Long Cap and Short Floor position? Cap - Floor = Max[i T -i C, 0] - Max[i C -i T, 0] = i T -i C pay fixed swap

69. Zvi WienerFIFIBI - 5 slide 69 FRM-GARP 98:50 A hedge fund leverages its 100M of investor capital by a factor of 3 and invests it into a portfolio of junk bonds yielding 14%. If its borrowing costs are 8%, what is the yield on investor capital?

70. Zvi WienerFIFIBI - 5 slide 70 FRM-GARP 98:50 300M invested at 14% yield 42M, borrowing costs are 200 at 8% or 16M, the difference of 26M provides 26% yield on equity of 100M.

71. Zvi WienerFIFIBI - 5 slide 71 FRM-GARP 98:51 A portfolio consists of two long assets $100 each. The probability of default over the next year is 10% for the first asset, 20% for the second asset, and the joint probability of default is 3%. What is the expected loss on this portfolio due to credit risk over the next year assuming 40% recovery rate for both assets.

72. Zvi WienerFIFIBI - 5 slide 72 FRM-GARP 98:51 0.1  (1-0.2) - default probability of A 0.2  (1-0.1) - default probability of B 0.03 - default probability of both Expected losses are 0.1  (1-0.2)  100  (1-0.4) 0.2  (1-0.1)  100  (1-0.4) 0.03  200  (1-0.4) 4.8 + 10.8 + 3.6 = 19.2M

73. Zvi WienerFIFIBI - 5 slide 73 Example Assume a 1-year US Treasury yield is 5.5% and a Eurodollar deposit rate is 6%. What is the probability of the Eurodollar deposit to default (assuming zero recovery rate)?

74. Zvi WienerFIFIBI - 5 slide 74 FRM-GARP 97:24 Assume the 1-year US Treasury yield is 5.5% and a default probability of a one year Commercial Paper is 1%. What should be the yield on the CP assuming 50% recovery ratio?

75. Zvi WienerFIFIBI - 5 slide 75 FRM-GARP 00:47 Which one of the following deals has the largest credit exposure for a $1,000,000 deal size. Assume that the counterparty in each deal is a AAA-rated bank and there is no settlement risk. A. Pay fixed in an interest rate swap for 1 year B. Sell USD against DEM in a 1 year forward contract. C. Sell a 1-year DEM Cap D. Purchase a 1-year Certificate of Deposit

76. Zvi WienerFIFIBI - 5 slide 76 FRM-GARP 00:47 Which one of the following deals has the largest credit exposure for a $1,000,000 deal size. Assume that the counterparty in each deal is a AAA-rated bank and there is no settlement risk. A. Pay fixed in an interest rate swap for 1 year B. Sell USD against DEM in a 1 year forward contract. C. Sell a 1-year DEM Cap D. Purchase a 1-year Certificate of Deposit

77. Zvi WienerFIFIBI - 5 slide 77 FRM-GARP 98 A step-up coupon bond pays LIBOR for 2 years, 2  LIBOR for the next two years and 3  LIBOR for the last two years. The principal amount is paid at the end of year 6. Prices of zero coupon bonds maturing in 2, 4, and 6 years are Z 2, Z 4, Z 6. What is the price of the step-up bond?

78. Zvi WienerFIFIBI - 5 slide 78 FRM-GARP 98 01234560123456 ?LL2L2L3L3L+100

79. Zvi WienerFIFIBI - 5 slide 79 01234560123456 ?LL2L2L3L3L+100 100LLLLLL+100 3003L3L3L3L3L3L+300 100LLLL+100 100LL+100 Z 2 100 Z 4 100 2Z 6 200 ? = 300 - 100 - 100 + Z 2 + Z 4 - 2Z 6