Consider a five-year, $1,000 face value bond with a 4% Coupon Rate and semi-annual coupons.

An investor purchased this bond today when the market YTM for it was 7%. Then, right after buying the bond, the YTM suddenly increased to 9%.

What is the percentage change in the bond price due to this change in the YTM?

Question

Consider a five-year, $1,000 face value bond with a 4% Coupon Rate and semi-annual coupons.

An investor purchased this bond today when the market YTM for it was 7%. Then, right after buying the bond, the YTM suddenly increased to 9%.

What is the percentage change in the bond price due to this change in the YTM?

BlackTom AI · Accepted Answer

We start by identifying the bond details and how price responds to yield changes. The bond has a 4% annual coupon on a 1000 face value, paid semi-annually, so each half-year coupon is 20. There are 5 years to maturity, which means 10 semi-annual periods. The purchase YTM is 7% annual, i.e., 3.5% per half-year. After purchase, the YTM rises to 9% annual, i.e., 4.5% per half-year. The price of a bond is the present value of all future coupons plus the present value of the face value at the applicable per-period discount rate.

Option evaluation:
- Option a: -8.35%
This option implies a price drop of about 8.35% when YTM rises from 7% to 9%. To check, compute initial price P0 using i = 3.5%: P0 = 20 * a-angle-10 at 3.5% + 1000/(1.035)^10. The annuity factor a-angle-10 at 3.5% is (1 - (1.035)^-10)/0.035 ≈ (1 - 0.707)/0.035 ≈ 0.293/0.035 ≈ 8.37. So P0 ≈ 20*8.37 + 1000*0.707 ≈ 167.4 + 707 ≈ 874.4. After the yield jumps to i = 4.5%, the price P1 = 20 * a-angle-10 at 4.5% + 1000/(1.045)^10. Now (1.045)^10 ≈ 1.55, so (1.045)^-10 ≈ 0.645. The annuity factor is (1 - 0.645)/0.045 ≈ 0.355/0.045 ≈ 7.89 (rounded). Then P1 ≈ 20*7.89 + 1000*0.645 ≈ 157.8 + 645 ≈ 802.8. The percentage change: (802.8 - 874.4)/874.4 ≈ -71.6/874.4 ≈ -0.0819 ≈ -8.19%. With refined rounding this aligns with about -8.3%. Therefore, this option reflects a plausible price drop close to the calculated value.

- Option b: -2.00%
A price drop of only 2% would correspond to a much smaller increase in yield or a longer duration impact. Given the move from 7% to 9% is a large jump, the price would drop more than 2% for a 5-year semiannual coupon bond, whose approximate modified duration is around 8-9, so a 2% move is inconsistent with standard duration-based estimates for this instrument.

- Option c: 2.00%
A positive price change when yield increases is counterintuitive for a plain-vanilla bond. Since yields and prices move inversely, an increase in yield from 7% to 9% would reduce price, not raise it. This makes a +2% price change illogical in this context.

- Option d: 9.11%
A price drop of about 9% would be somewhat larger than the approximate -8.2% to -8.4% calculated above. While not wildly off, the precise calculation based on the given cash flows and yield levels yields a figure around -8.3%. A -9.11% change would imply a larger discounting effect than the actual figures indicate, given the 5-year maturity and coupon structure.

In summary, option a (-8.35%) is consistent with the mechanics of bond pricing under a shift from a 7% to a 9% YTM for a 5-year, semiannual coupon bond, while the other options conflict with the direction of price movement or with the magnitude expected from a yield change of this size.

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