Answer:
-0.1402
Explanation:
Let's calculate the price impact of a 50-basis point (0.50%) interest rate change using a linear approximation for a 10-year, 7% semi-annual coupon bond priced at $97.50.
Step 1: Calculate the current yield:
Current yield = Coupon payment / Current price
Coupon payment = Face value * Coupon rate
Coupon payment = $100 * 0.07 = $7
Current yield = $7 / $97.50 = 0.0718 (or 7.18%)
Step 2: Calculate the Macaulay duration:
Macaulay duration = [(Time until Cash Flow) * (Cash Flow)] / [Current Price * (1 + Yield per period)]
The bond has 20 semi-annual periods, and the cash flow is the coupon payment of $7.
Macaulay duration = [(1 * $7) + (2 * $7) + ... + (20 * $7)] / [$97.50 * (1 + 0.0718)]
Using the formula for the sum of an arithmetic series, we can simplify the Macaulay duration calculation:
Macaulay duration = (20 * ($7 * 21)) / ($97.50 * 1.0718)
Macaulay duration = 2940 / 104.8375
Macaulay duration ≈ 28.0344
Step 3: Estimate the price impact:
Price impact ≈ -Macaulay duration * Interest rate change
Price impact ≈ -28.0344 * 0.005
Price impact ≈ -0.1402
Therefore, the estimated price impact of a 50-basis point interest rate change for the given bond is approximately -0.1402, implying a decrease in the bond price.