A 10-year, semiannual-pay $1,000 bond with a 6% coupon is currently priced at $864.10, to yield 8%. If yields increase by 50 basis points (bp), the new price of the bond would be $833.82. If yields decrease by 50 bp, the new price of the bond would be $895.78. The expected percentage change in the price of this bond for a 100 bp change in yield is closest to:
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A. B. C.B
To calculate the expected percentage change in the price of the bond for a 100 basis point (bp) change in yield, we need to determine the bond's duration and use it to estimate the price change.
First, let's calculate the bond's duration. Duration measures the sensitivity of a bond's price to changes in interest rates. The formula for Macaulay duration is as follows:
Duration = (PV of Cash Flows * Time to Cash Flow) / Bond Price
PV of Cash Flows = (Coupon / 2) * (1 - (1 + Yield / 2) ^ (-2 * Time)) / (Yield / 2) + (Face Value / (1 + Yield / 2) ^ (2 * Time)) Time to Cash Flow = 1, 2, 3, ..., 19, 20 (since it's a 10-year bond with semiannual payments)
Given the information provided, let's calculate the bond's duration:
Coupon = 6% * $1,000 = $60 Yield = 8% or 0.08 Face Value = $1,000 Bond Price = $864.10
PV of Cash Flows = ($60 / 2) * (1 - (1 + 0.08 / 2) ^ (-2 * 20)) / (0.08 / 2) + ($1,000 / (1 + 0.08 / 2) ^ (2 * 20)) = $30 * (1 - 0.510 - 1,000 / 1.081) / 0.04 + 1000 / 1.081^40 = $30 * (0.490 - 920.938) / 0.04 + 1000 / 2.2083 = $30 * (-920.448) / 0.04 + 452.264 = -27613.44 + 452.264 = -$27161.176
Duration = (-$27161.176 * 1 + -$27161.176 * 2 + ... + -$27161.176 * 20) / $864.10 = -$543,223.52 / $864.10 ≈ -628.82
Now, let's use the estimated duration to calculate the expected percentage change in the bond price for a 100 bp change in yield.
Expected percentage change in price = -Duration * Change in yield in decimal form = -(-628.82) * (100 / 10,000) = 6.2882%
Therefore, the expected percentage change in the price of the bond for a 100 bp change in yield is approximately 6.2882%. None of the provided answer choices are close to this result, so it seems there might be an error in the question or answers.