FRM Zvi Wiener Following P. Jorion, Financial Risk Manager Handbook Financial Risk Management

Ch. 1, HandbookZvi Wiener 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

Ch. 1, HandbookZvi Wiener slide 3 Quantitative Analysis Jorion, Value-at-Risk. Jorion, Financial Risk Manager Handbook 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.

Ch. 1, HandbookZvi Wiener slide 4 Quantitative Analysis Bond fundamentals Fundamentals of probability Fundamentals of Statistics Pricing Techniques

FRM Chapter 1 Quantitative Analysis Bond Fundamentals Following P. Jorion 2001 Financial Risk Manager Handbook

Ch. 1, HandbookZvi Wiener slide 6 Bond Fundamentals Discounting, Present Value Future Value

Ch. 1, HandbookZvi Wiener slide 7 Compounding US Treasuries market uses semi-annual compounding. Continuous compounding

Ch. 1, HandbookZvi Wiener slide 8 A bond pays $100 in ten years and its price is $ 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) )*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%.

Ch. 1, HandbookZvi Wiener 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.

Ch. 1, HandbookZvi Wiener 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%

Ch. 1, HandbookZvi Wiener slide 11 FRM-99, Question 17 (1 + y s /2) 2 = 1 + y ( /2) 2 = ==> 8.16%

Ch. 1, HandbookZvi Wiener 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 % per annum. B. 9.88% per annum. C. 9.76% per annum. D % per annum.

Ch. 1, HandbookZvi Wiener slide 13 FRM-99, Question 28 (1 + y s /2) 2 = e y (1 + y s /2) 2 = e y s /2 = e 0.05 y s = 2 (e ) = 10.25%

Ch. 1, HandbookZvi Wiener 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.

Ch. 1, HandbookZvi Wiener 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?

Ch. 1, HandbookZvi Wiener slide 16 y $ Price-yield Relationship Price of a straight bond as a function of yield

Ch. 1, HandbookZvi Wiener 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%

Ch. 1, HandbookZvi Wiener slide 18 FRM-98, Question 12 y s = 5%

Ch. 1, HandbookZvi Wiener 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:

Ch. 1, HandbookZvi Wiener slide 20 x F(x) Derivatives

Ch. 1, HandbookZvi Wiener slide 21 Properties of derivatives

Ch. 1, HandbookZvi Wiener slide 22 Bond Price Derivatives D* - modified duration, dollar duration is the negative of the first derivative: Dollar convexity = the second derivative, C - convexity.

Ch. 1, HandbookZvi Wiener slide 23 Duration of a portfolio

Ch. 1, HandbookZvi Wiener slide 24 Macaulay Duration Modified duration

Ch. 1, HandbookZvi Wiener slide 25 Bond Price Change

Ch. 1, HandbookZvi Wiener 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

Ch. 1, HandbookZvi Wiener slide 27 Example If the yield changes to 7% the price change is

Ch. 1, HandbookZvi Wiener slide 28 y $ Duration-Convexity Price of a straight bond as a function of yield

Ch. 1, HandbookZvi Wiener slide 29 Effective duration Effective convexity

Ch. 1, HandbookZvi Wiener 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 $ We can revalue this bond at 5% and 7%.

Ch. 1, HandbookZvi Wiener slide 31

Ch. 1, HandbookZvi Wiener slide 32 5%6%7%

Ch. 1, HandbookZvi Wiener 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.

Ch. 1, HandbookZvi Wiener 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 What is the estimated duration of the 7s? A B. 2.4 C D. 2.25

Ch. 1, HandbookZvi Wiener slide 35 FRM-98, Question 20

Ch. 1, HandbookZvi Wiener 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 What is the estimated convexity of the 7s? A. 53 B. 26 C. 13 D. -53

Ch. 1, HandbookZvi Wiener slide 37 FRM-98, Question 21

Ch. 1, HandbookZvi Wiener slide 38 Duration of a coupon bond

Ch. 1, HandbookZvi Wiener slide 39 Exercise Find the duration and convexity of a consol (perpetual bond). Answer: (1+y)/y.

Ch. 1, HandbookZvi Wiener slide 40 Convexity Exercise: compute duration and convexity of a 2-year, 6% semi-annual bond when IR=6%.

Ch. 1, HandbookZvi Wiener 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

Ch. 1, HandbookZvi Wiener 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 If the yield decreases by 1 bp, the price will increase to What is the modified duration of this bond? A. 5.0 B C. 4.5 D. -4.5

Ch. 1, HandbookZvi Wiener slide 43 FRM-98, Question 17

Ch. 1, HandbookZvi Wiener 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 B C D

Ch. 1, HandbookZvi Wiener slide 45 FRM-98, Question 22

Ch. 1, HandbookZvi Wiener 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

Ch. 1, HandbookZvi Wiener 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.

Ch. 1, HandbookZvi Wiener 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

Ch. 1, HandbookZvi Wiener slide 49 Portfolio Duration and Convexity Portfolio weights

Ch. 1, HandbookZvi Wiener 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

Ch. 1, HandbookZvi Wiener 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?

Ch. 1, HandbookZvi Wiener 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 B C D

Ch. 1, HandbookZvi Wiener slide 53 FRM-98, Question 18 Note that $100 means notional amount and can be misunderstood.

Ch. 1, HandbookZvi Wiener slide 54 Duration Gap A - L = C, assets - liabilities = capital

Ch. 1, HandbookZvi Wiener slide 55 Duration and Term Structure of IR

Ch. 1, HandbookZvi Wiener slide 56 Partial Duration Key rate duration

Ch. 1, HandbookZvi Wiener slide 57 Useful formulas

Ch. 1, HandbookZvi Wiener slide 58 UST example 8.75 UST 11/08 Security was purchased Security was sold Calculate the loss (10,000 units) …

Ch. 1, HandbookZvi Wiener slide 59 UST example Bought 11,096, Accrued 23 days 54, ,151, Sold 10,984, Accrued 115 days 273, ,257, Profit of $106,350.00