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Answer: The risk analyst should use modified duration and modified convexity, and the dollar amount of the capital loss is $1.7969.
## Explanation For duration-convexity approximation of price changes, we use **modified duration** and **modified convexity** because: - **Modified duration** measures the percentage price change for a 1% change in yield - **Modified convexity** measures the second-order price sensitivity - The formula for price change approximation is: $$\Delta P \approx -MD \times \Delta y \times P + \frac{1}{2} \times MC \times (\Delta y)^2 \times P$$ Where: - MD = Modified Duration = 3.8653 - MC = Modified Convexity = 21.8945 - P = Current Price = $94.3138 - Δy = Change in yield = 0.50% = 0.005 **Calculation:** 1. **Duration effect:** $$-3.8653 \times 0.005 \times 94.3138 = -1.8228$$ 2. **Convexity effect:** $$\frac{1}{2} \times 21.8945 \times (0.005)^2 \times 94.3138 = 0.0258$$ 3. **Total price change:** $$-1.8228 + 0.0258 = -1.7969$$ Therefore, the estimated capital loss is **$1.7969**, which matches option C. **Why not the other options:** - **Option A & D**: Use Macaulay duration, which is not appropriate for direct price change calculations - **Option B**: Uses modified duration but regular convexity instead of modified convexity - **Option C**: Uses the correct pair - modified duration and modified convexity
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Consider a 5-year 8.5% coupon bond (assume annual coupons) priced to yield a 10% per annum, with a par value of $100 and a price of $94.3138. A risk analyst has computed the following information:
| Macaulay Duration | 4.2518 |
|---|---|
| Modified Duration | 3.8653 |
| Convexity | 24.0839 |
| Modified Convexity | 21.8945 |
Which pair of duration and convexity should the risk analyst use in computing the duration-convexity approximation for the capital loss if the yield were to change to 10.50% per annum? And what is the estimated dollar amount of the capital loss?
A
The risk analyst should use Macaulay duration and convexity, and the dollar amount of the capital loss is $1.9766.
B
The risk analyst should use modified duration and convexity, and the dollar amount of the capital loss is $1.7944.
C
The risk analyst should use modified duration and modified convexity, and the dollar amount of the capital loss is $1.7969.
D
The risk analyst should use Macaulay duration and modified convexity, and the dollar amount of the capital loss is $1.9792.