Risk Measurement, Dollar Duration, Convexity, and Macaulay Duration - 30 points The following discount factors are given:

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Risk Measurement, Dollar Duration, Convexity, and Macaulay Duration - 30 points The following discount factors are given: B(0, 1) = 0.92506 B(0, 2) = 0.88145 B(0, 3) = 0.83476 B(0, 4) = 0.79821 As a reminder, $ Duration, for a bond with continuously compounded yields is: ∆$ = X T t=1 tKte −yt . (a) Using continuous compounding, calculate the duration D(y), modified duration MD(y),the dollar duration ∆$, and the dollar convexity Γ$ of a 4-year 3% annual coupon bond with face value $1,000. (b) If there is a 300 basis point downward shift in the continuously compounded zeroyield curve (assume all shifts in the continuously compounded zero yield curve are uniform) calculate the following:

i. The actual price of this 4-year, 3% coupon bond;

ii. The price of the 4-year bond as estimated by ∆$ alone;

iii. The price of the 4-year bond as estimated by both ∆$ and Γ$.

2. Risk Measurement, Risk Management, Duration - 30 points Dollar duration and convexity are used to characterize and to control the riskiness of fixed income securities. However, several examples were presented in the lectures to exhibit the limitations of these measures. One way we might attempt to improve these measures is by making them more robust to non-uniform shifts in the term structure. For a fixed income security that makes annual payments over four years, P(R1, R2, R3, R4) = X 4 t=1 Kte −t·Rt , 1 where Rt is short for the spot rate R(0, t), we can perform a Taylor expansion to arrive at an expression relating the change in price of the bond to changes in the zero coupon term structure by ∆P ≈ X 4 t=1 ∂P ∂Rt ∆Rt . (1) Consider an 8% coupon bond maturing in four years with a face value of $1000. You are given the following information about the current state of the continuously compounded zero yield curve, and four possible scenarios that might occur instantaneously. Yield Yr. 1 Yield Yr. 2 Yield Yr. 3 Yield Yr. 4 Current 6.25% 7.00% 7.50% 8.00% Scenario 1 7.00% 7.75% 8.25% 8.75% Scenario 2 6.25% 7.50% 8.00% 8.50% Scenario 3 5.00% 6.00% 7.50% 7.00% Scenario 4 7.00% 6.50% 6.25% 6.00% Scenario 5 7.25% 7.25% 7.25% 8.20% For each scenario, compute the following:

(a) The actual price change of this bond.

(b) The price change of this bond as approximated by the duration method ∆$. This is by assuming that the bond price depends on a single risk factor, the yield-tomaturity. Comment on your findings.

(c) The price change of this bond as approximated by the first-order changes in the underlying five spot rates.

3. Risk Management, Duration, and Convexity - 40 points Consider the following three securities: • Security A is a 2-year zero which matures two years from today, and pays $1 at maturity; • Security B is a 3-year zero which matures three years from today, and pays $1 at maturity; • Security C is a 1-year forward contract which matures one year from today, and delivers a 1-year zero (with face value $1) at maturity. My portfolio currently consists of a long position of 5,000 units of security B and a short position (liability) of 3,000 units of security A. Suppose the only securities I can currently trade (buy/sell) are Security A and Security C. The current term structure based on zero coupon bonds (with 1-, 2-, 3-, and 4-year maturity from today) in annualized rates with continuous compounding is: R(0, 1) = 3% R(0, 2) = 5% R(0, 3) = 6% R(0, 4) = 6.5% 2

(a) What are the current value, the dollar duration and the convexity of my portfolio? Interpret the dollar duration and the convexity numbers (i.e. for a 100 basis point parallel increase in the term structure, what do the duration and convexity numbers say?).

(b) Without changing the value of my portfolio (found in part (a)), what do I have to do now (i.e., if anything, long/short what? how much?) to achieve a dollar duration of zero for my portfolio? (Suppose I don't care about convexity.)