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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 bond price changes, we use **modified duration** and **modified convexity** when working with yield changes. ### Calculation: - **Current price**: $94.3138 - **Yield change**: Δy = 10.50% - 10.00% = 0.50% = 0.005 - **Modified duration**: 3.8653 - **Modified convexity**: 21.8945 **Price change approximation formula**: \[\frac{\Delta P}{P} \approx -D_m \cdot \Delta y + \frac{1}{2} \cdot C_m \cdot (\Delta y)^2\] Where: - D_m = Modified duration = 3.8653 - C_m = Modified convexity = 21.8945 - Δy = 0.005 **Calculation**: \[\frac{\Delta P}{94.3138} \approx -3.8653 \cdot 0.005 + \frac{1}{2} \cdot 21.8945 \cdot (0.005)^2\] \[\frac{\Delta P}{94.3138} \approx -0.0193265 + 0.00027368\] \[\frac{\Delta P}{94.3138} \approx -0.01905282\] **Dollar price change**: \[\Delta P \approx 94.3138 \cdot (-0.01905282) \approx -1.7969\] Therefore, the estimated **capital loss** is **$1.7969**. ### Why Option C is correct: - **Modified duration** is used for yield-based price sensitivity calculations - **Modified convexity** is the appropriate convexity measure when using modified duration - The calculation matches the result in Option C: $1.7969 ### Why other options are incorrect: - **Option A**: Uses Macaulay duration and regular convexity - incorrect pairing - **Option B**: Uses modified duration with regular convexity - inconsistent measures - **Option D**: Uses Macaulay duration with modified convexity - inconsistent measures
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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.
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