Deltacorp Ltd has issued bonds with semi-annual coupon payments. The bonds are currently trading at a price of $87.12 in the market and are priced using a yield of 12% p.a. compounded semi-annually. The bonds have a face value of $100 and mature in 5 years’ time. What is the bond’s nominal annual coupon rate?

Question

BlackTom AI · Accepted Answer

The task provides a bond valuation scenario and asks for the nominal annual coupon rate. We'll restate the details and then examine each answer option.

Restated information:
- Bond price P = 87.12
- Face value F = 100
- Maturity = 5 years, semiannual coupons → 10 periods
- Yield is 12% per year compounded semiannually → per-period yield i = 0.12/2 = 0.06
- Coupon per half-year = F * (annual coupon rate)/2 = 100 * (r/2), where r is the nominal annual coupon rate we want to find

Option-by-option analysis:
Option A: 8.68%
- If the nominal annual coupon rate were 8.68%, the semiannual coupon would be 100 * (8.68%)/2 = 4.34 per period.
- The present value of the coupon stream would be 4.34 times the annuity factor a-angle-10 at i = 6%. The annuity factor is (1 − v^10)/i, where v = 1/1.06. Numerically, v^10 ≈ 0.558, so a-angle-10 ≈ (1 − 0.558)/0.06 ≈ 7.37. Thus the PV of coupons ≈ 4.34 * 7.37 ≈ 31.96.
- PV of redemption = 100 * v^10 ≈ 100 * 0.558 ≈ 55.8. Total ≈ 31.96 + 55.8 ≈ 87.76.
- This is close to but slightly higher than the given price 87.12, indicating 8.68% is a plausible but not exact match. The precise calculation would show a small mismatch, suggesting this option may be slightly off.

Option B: 8.50%
- Semiannual coupon = 100 * (8.50%)/2 = 4.25 per period.
- PV of coupons = 4.25 * a-angle-10 ≈ 4.25 * 7.37 ≈ 31.33.
- PV of redemption ≈ 55.8 (as above).
- Total ≈ 31.33 + 55.8 ≈ 87.13, which is extremely close to the given price 87.12. This indicates a near-perfect match within typical rounding.

Option C: There is not enough information to determine the coupon rate
- The problem provides price, yield, maturity, and face value, which is sufficient to solve for the coupon rate in a standard yield-to-maturity framework. Therefore, this statement is not correct because the data are enough to determine the coupon rate.

Option D: 8.19%
- Semiannual coupon = 100 * (8.19%)/2 = 4.095 per period.
- PV of coupons = 4.095 * 7.37 ≈ 30.19.
- PV of redemption ≈ 55.8.
- Total ≈ 86.0, which is noticeably below the given price 87.12. This would imply a higher price is needed, so 8.19% is inconsistent with the data.

Option E: 8.36%
- Semiannual coupon = 100 * (8.36%)/2 = 4.18 per period.
- PV of coupons = 4.18 * 7.37 ≈ 30.79.
- PV of redemption ≈ 55.8.
- Total ≈ 86.59, which is still below 87.12, though closer than 8.19%. This remains somewhat inconsistent with the exact price given.

Summary of reasoning across options:
- By equating the price to the present value of coupons plus redemption, we derive a linear relation in the unknown coupon rate. The option that makes the calculated price align most closely with 87.12 is the correct one. Among the options, 8.50% yields a total price (~87.13) that matches the given price very tightly, whereas the other rates produce totals that diverge noticeably from 87.12. The option asserting insufficient information is invalid because the data provided are sufficient to solve for the coupon rate. The other alternating rates produce totals that are either too high or too low relative to the observed price, indicating they are incorrect.

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