1. QA-1 FRM-GARP Sep-2001 Zvi Wiener 02-588-3049 http://pluto.mscc.huji.ac.il/~mswiener/zvi.html Quantitative Analysis 1

2. http://www.tfii.orgZvi Wiener - QA1 slide 2 FRM 2000 Capital Markets Risk Management20 Legal, Accounting and Tax6 Credit Risk Management36 Operational Risk Management8 Market Risk Management35 Quantitative Analysis23 Regulation and Compliance12

3. http://www.tfii.orgZvi Wiener - QA1 slide 3 Quantitative Analysis Jorion, Value-at-Risk. Jorion, … Hull, Options, Futures and Other Derivatives. Fabozzi F., Bond Markets: Analysis and Strategies. Fabozzi F., Fixed Income Mathematics. Golub B., Risk Management. Crouchy, Galai, Mark, Risk Management.

4. http://www.tfii.orgZvi Wiener - QA1 slide 4 Quantitative Analysis Bond fundamentals Fundamentals of probability Fundamentals of Statistics Pricing Techniques

5. QA-1 FRM-GARP Sep-2001 Bond Fundamentals Following Jorion 2001, Chapter 1 Financial Risk Manager Handbook

6. http://www.tfii.orgZvi Wiener - QA1 slide 6 Bond Fundamentals Discounting, Present Value Future Value

7. http://www.tfii.orgZvi Wiener - QA1 slide 7 Compounding US Treasuries market uses semi-annual compounding. Continuous compounding

8. http://www.tfii.orgZvi Wiener - QA1 slide 8 A bond pays $100 in ten years and its price is $55.9126. This corresponds to an annually compounded rate of 6% using PV=C T /(1+y) 10, or (1+y) = (C T /PV) 0.1. This rate can be transformed into semiannual compounded rate, using (1+y s /2) 2 = (1+y), or y s = ((1+0.06) 0.5 -1)*2 = 5.91%. It can be transformed into a continuously compounded rate exp(y c ) = 1+y, or y c = ln(1+0.06) = 5.83%.

9. http://www.tfii.orgZvi Wiener - QA1 slide 9 Note that as we increase the frequency of the compounding the resulting rate decreases. Intuitively, since our money works harder with more frequent compounding, a lower rate will achieve the same payoff. Key concept: For a fixed present and final values, increasing the frequency of the compounding will decrease the associated yield.

10. http://www.tfii.orgZvi Wiener - QA1 slide 10 FRM-99, Question 17 Assume a semi-annual compounded rate of 8% per annum. What is the equivalent annually compounded rate? A. 9.2% B. 8.16% C. 7.45% D. 8%

11. http://www.tfii.orgZvi Wiener - QA1 slide 11 FRM-99, Question 17 (1 + y s /2) 2 = 1 + y (1 + 0.08/2) 2 = 1.0816 ==> 8.16%

12. http://www.tfii.orgZvi Wiener - QA1 slide 12 FRM-99, Question 28 Assume a continuously compounded interest rate is 10% per annum. What is the equivalent semi-annual compounded rate? A. 10.25% per annum. B. 9.88% per annum. C. 9.76% per annum. D. 10.52% per annum.

13. http://www.tfii.orgZvi Wiener - QA1 slide 13 FRM-99, Question 28 (1 + y s /2) 2 = e y (1 + y s /2) 2 = e 0.1 1 + y s /2 = e 0.05 y s = 2 (e 0.05 - 1) = 10.25%

14. http://www.tfii.orgZvi Wiener - QA1 slide 14 Price-Yield Relationship Here C t is the cashflow t - number of periods to each payment T number of periods to maturity y - the discount factor.

15. http://www.tfii.orgZvi Wiener - QA1 slide 15 Face value, nominal. Bond that sells at face value is called par bond. A bond has a 8% annual coupon and IRR of 8%. What is the price of the bond? Is this always true?

16. http://www.tfii.orgZvi Wiener - QA1 slide 16 y $ Price-yield Relationship Price of a straight bond as a function of yield

17. http://www.tfii.orgZvi Wiener - QA1 slide 17 FRM-98, Question 12 A fixed rate bond, currently priced at 102.9, has one year remaining to maturity and is paying an 8% coupon. Assuming that the coupon is paid semiannually, what is the yield of the bond? A. 8% B. 7% C. 6% D. 5%

18. http://www.tfii.orgZvi Wiener - QA1 slide 18 FRM-98, Question 12 y s = 5%

19. http://www.tfii.orgZvi Wiener - QA1 slide 19 Taylor Expansion To measure the price response to a small change in risk factor we use the Taylor expansion. Initial value y 0, new value y 1, change  y:

20. http://www.tfii.orgZvi Wiener - QA1 slide 20 x F(x) Derivatives

21. http://www.tfii.orgZvi Wiener - QA1 slide 21 Properties of derivatives

22. http://www.tfii.orgZvi Wiener - QA1 slide 22 Bond Price Derivatives D* - modified duration, dollar duration is the negative of the first derivative: Dollar convexity = the second derivative, C - convexity.

23. http://www.tfii.orgZvi Wiener - QA1 slide 23 Duration of a portfolio

24. http://www.tfii.orgZvi Wiener - QA1 slide 24 Macaulay Duration Modified duration

25. http://www.tfii.orgZvi Wiener - QA1 slide 25 Bond Price Change

26. http://www.tfii.orgZvi Wiener - QA1 slide 26 Example 10 year zero coupon bond with a semiannual yield of 6% The duration is 10 years, the modified duration is: The convexity is

27. http://www.tfii.orgZvi Wiener - QA1 slide 27 Example If the yield changes to 7% the price change is

28. http://www.tfii.orgZvi Wiener - QA1 slide 28 y $ Duration-Convexity Price of a straight bond as a function of yield

29. http://www.tfii.orgZvi Wiener - QA1 slide 29 Effective duration Effective convexity

30. http://www.tfii.orgZvi Wiener - QA1 slide 30 Effective Duration and Convexity Consider a 30-year zero-coupon bond with a yield of 6%. With semi-annual compounding its price is $16.9733. We can revalue this bond at 5% and 7%.

31. http://www.tfii.orgZvi Wiener - QA1 slide 31

32. http://www.tfii.orgZvi Wiener - QA1 slide 32 5%6%7%

33. http://www.tfii.orgZvi Wiener - QA1 slide 33 Coupon Curve Duration If IR decrease by 100bp, the market price of a 6% 30 year bond will go up close to the price of a 30 years 7% coupon bond. Thus we associate a higher coupon with a drop in yield equal to the difference in coupons. This approach is useful for mortgages.

34. http://www.tfii.orgZvi Wiener - QA1 slide 34 FRM-98, Question 20 Coupon curve duration is a useful method to estimate duration from market prices of MBS. Assume that the coupon curve of prices for Ginnie Maes is as follows: 6% at 92, 7% at 94, 8% at 96.5. What is the estimated duration of the 7s? A. 2.45 B. 2.4 C. 2.33 D. 2.25

35. http://www.tfii.orgZvi Wiener - QA1 slide 35 FRM-98, Question 20

36. http://www.tfii.orgZvi Wiener - QA1 slide 36 FRM-98, Question 21 Coupon curve duration is a useful method to estimate duration from market prices of MBS. Assume that the coupon curve of prices for Ginnie Maes is as follows: 6% at 92, 7% at 94, 8% at 96.5. What is the estimated convexity of the 7s? A. 53 B. 26 C. 13 D. -53

37. http://www.tfii.orgZvi Wiener - QA1 slide 37 FRM-98, Question 21

38. http://www.tfii.orgZvi Wiener - QA1 slide 38 Duration of a coupon bond

39. http://www.tfii.orgZvi Wiener - QA1 slide 39 Exercise Find the duration and convexity of a consol (perpetual bond). Answer: (1+y)/y.

40. http://www.tfii.orgZvi Wiener - QA1 slide 40 Convexity Exercise: compute duration and convexity of a 2-year, 6% semi-annual bond when IR=6%.

41. http://www.tfii.orgZvi Wiener - QA1 slide 41 FRM-99, Question 9 A number of terms in finance are related to the derivative of the price of a security with respect to some other variable. Which pair of terms is defined using second derivatives? A. Modified duration and volatility B. Vega and delta C. Convexity and gamma D. PV01 and yield to maturity

42. http://www.tfii.orgZvi Wiener - QA1 slide 42 FRM-98, Question 17 A bond is trading at a price of 100 with a yield of 8%. If the yield increases by 1 bp, the price of the bond will decrease to 99.95. If the yield decreases by 1 bp, the price will increase to 100.04. What is the modified duration of this bond? A. 5.0 B. -5.0 C. 4.5 D. -4.5

43. http://www.tfii.orgZvi Wiener - QA1 slide 43 FRM-98, Question 17

44. http://www.tfii.orgZvi Wiener - QA1 slide 44 FRM-98, Question 22 What is the price of a 10 bp increase in yield on a 10-year par bond with a modified duration of 7 and convexity of 50? A. -0.705 B. -0.700 C. -0.698 D. -0.690

45. http://www.tfii.orgZvi Wiener - QA1 slide 45 FRM-98, Question 22

46. http://www.tfii.orgZvi Wiener - QA1 slide 46 FRM-98, Question 29 A and B are perpetual bonds. A has 4% coupon, and B has 8% coupon. Assume that both bonds are trading at the same yield, what can be said about duration of these bonds? A. The duration of A is greater than of B B. The duration of A is less than of B C. They have the same duration D. None of the above

47. http://www.tfii.orgZvi Wiener - QA1 slide 47 FRM-97, Question 24 Which of the following is NOT a property of bond duration? A. For zero-coupon bonds Macaulay duration of the bond equals to time to maturity. B. Duration is usually inversely related to the coupon of a bond. C. Duration is usually higher for higher yields to maturity. D. Duration is higher as the number of years to maturity for a bond selling at par or above increases.

48. http://www.tfii.orgZvi Wiener - QA1 slide 48 FRM-99, Question 75 You have a large short position in two bonds with similar credit risk. Bond A is priced at par yielding 6% with 20 years to maturity. Bond B has 20 years to maturity, coupon 6.5% and yield of 6%. Which bond contributes more to the risk of the portfolio? A. Bond A B. Bond B C. A and B have similar risk D. None of the above

49. http://www.tfii.orgZvi Wiener - QA1 slide 49 Portfolio Duration and Convexity Portfolio weights

50. http://www.tfii.orgZvi Wiener - QA1 slide 50 Example Construct a portfolio of two bonds: A and B to match the value and duration of a 10-years, 6% bond with value $100 and modified duration of 7.44 years. A. 1 year zero bond - price $94.26 B. 30 year zero - price $16.97

51. http://www.tfii.orgZvi Wiener - QA1 slide 51 Barbel portfolio consists of very short and very long bonds. Bullet portfolio consists of bonds with similar maturities. Which of them has higher convexity?

52. http://www.tfii.orgZvi Wiener - QA1 slide 52 FRM-98, Question 18 A portfolio consists of two positions. One is long $100 of a two year bond priced at 101 with a duration of 1.7; the other position is short $50 of a five year bond priced at 99 with a duration of 4.1. What is the duration of the portfolio? A. 0.68 B. 0.61 C. -0.68 D. -0.61

53. http://www.tfii.orgZvi Wiener - QA1 slide 53 FRM-98, Question 18 Note that $100 means notional amount and can be misunderstood.

54. http://www.tfii.orgZvi Wiener - QA1 slide 54 Duration Gap A - L = C, assets - liabilities = capital

55. http://www.tfii.orgZvi Wiener - QA1 slide 55 Duration and Term Structure of IR

56. http://www.tfii.orgZvi Wiener - QA1 slide 56 Partial Duration Key rate duration

57. http://www.tfii.orgZvi Wiener - QA1 slide 57 Useful formulas

58. http://www.tfii.orgZvi Wiener - QA1 slide 58 UST example 8.75 UST 11/08 Security was purchased 06-Jun-01 @ 110-31 Security was sold 06-Sep-01 @ 109-27+ Calculate the loss (10,000 units) …

59. http://www.tfii.orgZvi Wiener - QA1 slide 59 UST example Bought 11,096,875.00 Accrued 23 days 54,687.50 11,151,562.50 Sold 10,984,375.00 Accrued 115 days 273,437.50 11,257,812.50 Profit of $106,350.00