You own a bond that pays $64 in interest annually. The face value is $1,000 and the current market price is $1,021.61. The bond matures in 11 years. What is the yield to maturity?单项选择题
A
A. 6.12 percent
B
B. 6.22 percent
C
C. 6.46 percent
D
D. 6.71 percent

Question

You own a bond that pays $64 in interest annually. The face value is $1,000 and the current market price is $1,021.61. The bond matures in 11 years. What is the yield to maturity?

BlackTom AI · Accepted Answer

Yield to maturity (YTM) is the discount rate that sets the present value of all future bond cash flows equal to the bond’s current market price. For a bond paying annual coupons, the pricing equation is:
Price = C * [1 − (1 + r)^−n] / r + FV * (1 + r)^−n,
where C = annual coupon, r = YTM, n = years to maturity, FV = face value.

1) Relevant knowledge points:
- Coupon C = $64, FV = $1,000, Price = $1,021.61, n = 11.
- We must solve 1,021.61 = 64 * [1 − (1 + r)^−11] / r + 1000 * (1 + r)^−11 for r (YTM).
- Numeric (iterative) solution or financial calculator is normally used.

2) Complete reasoning and 3) intermediate calculations (trial at r = 6.12%):
Compute (1 + r)^11 for r = 0.0612:
ln(1.0612) ≈ 0.059388 → 11 * 0.059388 = 0.65327 → (1.0612)^11 ≈ e^{0.65327} ≈ 1.9219.
Discount factor (1 + r)^−11 ≈ 1 / 1.9219 = 0.5205.

Annuity factor = [1 − 0.5205] / 0.0612 = 0.4795 / 0.0612 ≈ 7.837.
PV of coupons = 64 * 7.837 ≈ $501.57.
PV of face = 1000 * 0.5205 ≈ $520.50.
Total PV ≈ 501.57 + 520.50 = $1,022.07.

The computed PV at r = 6.12% is $1,022.07, which is extremely close to the market price $1,021.61 (difference ≈ $0.46). A slightly higher r would reduce PV marginally; therefore the YTM is very near 6.12%.

4) Why the correct answer is correct:
Because plugging r = 6.12% into the bond price formula yields a present value essentially equal to the quoted market price, that r is the bond’s yield to maturity. The small residual error is due to rounding in intermediate steps; numerically solving more precisely would converge to the value given (6.12%).

5) Why other options are incorrect:
- 6.22% (B): gives a noticeably lower PV (further from $1,021.61), so r is too high.
- 6.46% (C) and 6.71% (D): these higher yields produce even lower PVs, so they cannot match the market price.

6) Conclusion:
The YTM that equates the bond’s cash flows to its current price is 6.12%, matching answer A.

You own a bond that pays $64 in interest annually. The face value is $1,000 and the current market price is $1,021.61. The bond matures in 11 years. What is the yield to maturity?单项选择题

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