If a bond's yield to maturity does not change, the return on the bond each year will be equal to the yield to maturity. Confirm this for both a premium and a discount bond using a 4-year 4.6 percent coupon bond with annual coupon payments and a face value of $1,000.Assume the yield to maturity is 3.6 percent. (Do not round intermediate calculations. Enter "Bond price" answers to 2 decimal places and "Rate of return" rounded to 1 decimal place.)

Question

BlackTom AI · Accepted Answer

The bond price equals the present value of its coupons plus the discounted face value: P = C·[1 − (1+r)^−n]/r + FV·(1+r)^−n. Here C = 0.046·1000 = $46, r = 0.036, n = 4.

1) Price today (t = 0)
Compute (1+r)^4 = 1.036^4 = 1.151964303616, so discount factor v^4 = (1.036)^−4 = 1 / 1.151964303616 ≈ 0.8680824544.
Annuity factor = [1 − v^4]/r = (1 − 0.8680824544)/0.036 = 0.1319175456/0.036 = 3.66437682222.
PV of coupons = 46 × 3.66437682222 = 168.56133382212.
PV of face = 1000 × 0.8680824544 = 868.0824544.
P0 = 168.56133382212 + 868.0824544 = 1,036.64378822212 → rounded to $1,036.64.

2) Price in one year (t = 1), with 3 years remaining
(1+r)^3 = 1.036^3 = 1.111934656, so v^3 = (1.036)^−3 = 1 / 1.111934656 ≈ 0.899333422.
Annuity factor (3 years) = (1 − v^3)/r = (1 − 0.899333422)/0.036 = 0.100666578/0.036 = 2.7962943889.
PV of remaining coupons = 46 × 2.7962943889 = 128.6295418894.
PV of face in one year = 1000 × 0.899333422 = 899.333422.
P1 = 128.6295418894 + 899.333422 = 1,027.9629638894 → rounded to $1,027.96.

3) One‑year rate of return
Return = (Coupon + P1 − P0) / P0 = (46 + 1,027.9629638894 − 1,036.64378822212) / 1,036.64378822212
= 37.31917566728 / 1,036.64378822212 = 0.036 = 3.6%.

Why this is correct: when YTM stays constant, the expected holding-period return equals the YTM because price changes between periods reflect the same discounting at that yield; coupon income plus the capital change combine to the yield. Other answer choices are incorrect because their presented P0, P1 or returns do not satisfy the present-value calculations at r = 3.6% (their implied returns differ from the YTM).

Final confirmation: Bond price today = $1,036.64, bond price in one year = $1,027.96, rate of return = 3.6% — matching the provided (choice b).

类似问题

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In November 2017, Treasury 4 3/4s of 2043 offered a semiannually compounded yield to maturity of 2.72%. Recognizing that coupons are paid semiannually, calculate the bond's price. Assume face value is $1,000. (Do not round intermediate calculations. Round your answer to 2 decimal places.)

A 21-year German government bond (bund) has a face value of €550 and a coupon rate of 4% paid annually. Assume that the interest rate (in euros) is equal to 7.10% per year. What is the bond's PV? (Do not round intermediate calculations. Round your answer to 2 decimal places.)

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