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Fin 500Q - Quantitative Risk Management

Homework #8 Solutions

1. Consider a fixed income product that pays a coupon of $25 at the end of each year for the next 5 years, and at the end of the 5 years also pays back the principal of $1000. Suppose the current yield on the bond is 3%.

Find the following quantities:

(a) The current price of the bond, and the (actual) price if the yield rises instantaneously to 4%. Answer: The current bond price is

£ 25 e"⁰-⁰³-^ + 1000 e^-0-⁰³'⁵ = 975.05.

i=1

Using a similar computation, we find that if the yield rises to 4%, the price falls to 929.77. Thus, the price change is -45.28.

(b) The approximate duration-based price change if the yield increases instantaneously to 4%.

Answer: The duration of the bond is 4.76. The approximate duration-based price change is

-D - B - Ay = -4.76 - 975.05 - 0.01 = -46.40.

(c) The approximate duration and convexity-based price change if the yield increases instantaneously to 4%.

Answer: The convexity of the bond is 23.32. Then, the duration and convexity-based approximate price change is

c

-D - B - Ay +^ - B - (Ay)² = -46.40 + 0.5 - 23.32 - 975.05 - (0.01)² = -45.26.

(d) If the annual volatility of yield changes is 0.025, then find the monthly VaR°.05 using the durationbased as well as the convexity-based approximation. You may assume that the average monthly change in yields is zero.

Answer: For the duration-based VaR, we get

^VaR0.05 = ^-975.⁰⁵ -翌 ⁺ L⁶⁴⁵ - ⁴.⁷⁶ - ⁹⁷⁵.⁰⁵ -穿= ⁵².⁶⁴.

For the duration and convexity-based VaR, we get

1.645² 0.025²

VaR0.05 = 52.64 — ——23.32 - 975.05 . ‘患 =51.04.

(e) Explain in words and with a diagram why the two VaRs that you calculated differ.

Answer: The diagram is given in Figure 1 below. It is quite similar to the delta hedging error figure in Module 8, “Risk Management using Options.” As can be seen from the plot, if the yield increases from y to y‘，then the duration approximation of the new price is along the tangency line — the approximate price falls to PHowever, the bond price function is a convex function of yields. Therefore, duration itself falls as the yield rises, and the tangency line overestimates the price drop. The actual new price is P〃. So, the price decline (loss) is smaller than the durationbased price decline. Since the VaR is a measure of the worst 5% of the losses, the duration-based VaR will be larger (in absolute value) than the more accurate duration and covexity-based VaR.

Figure 1: Duration Error in VaR