a. The 7.6%, twenty-year bond yields 5.6%. If this yield to maturity remains unchanged, what will be its price one year hence? Assume annual coupon payments and a face value of $1,000. (Do not round intermediate calculations. Round your answer to 2 decimal places.)b. What is the total return to an investor who held the bond over this year? (Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.)

Question

BlackTom AI · Accepted Answer

Price one year hence = present value of the remaining 19 annual $76 coupons plus $1,000 principal, discounted at the unchanged YTM 5.6%.

1) Relevant formulas and concepts
- Coupon C = 0.076 × $1,000 = $76.
- Discount factor v = 1/(1 + y) with y = 0.056.
- Price for m years remaining: P(m) = C × (1 − v^m)/y + F × v^m.

2) Compute the price one year hence (m = 19)
- v^19 = (1.056)^−19 ≈ 0.3549858 (computed from (1.056)^19 ≈ 2.8163848).
- Annuity factor for 19 years = (1 − v^19)/y = (1 − 0.3549858)/0.056 = 0.6450142/0.056 = 11.517388.
- PV of coupons = 76 × 11.517388 = 875.3236.
- PV of principal = 1,000 × 0.3549858 = 354.9858.
- Price in one year = 875.3236 + 354.9858 = $1,230.3094 ≈ $1,230.31.

3) Total return over the year
Two approaches produce the same exact result:

A. Numeric check (using P0 for 20 years and P1 for 19 years)
- Compute P0 (20-year price) similarly: v^20 = (1.056)^−20 ≈ 0.336338 (so P0 ≈ $1,237.285).
- Holding period dollar gain = coupon + (P1 − P0) = 76 + (1,230.3094 − 1,237.285) = 69.0244.
- HPR = 69.0244 / 1,237.285 ≈ 0.055999 ≈ 5.60%.

B. Algebraic identity (exact)
Using P0 = C(1 − v^n)/y + Fv^n and P1 = C(1 − v^{n−1})/y + Fv^{n−1}, one can show
(C + P1 − P0) = y × P0, so HPR = (C + P1 − P0)/P0 = y = 5.6% exactly. This identity proves the holding-period return equals the initial YTM when the yield remains unchanged.

4) Why the correct answer is correct
- The computed P1 equals $1,230.31 (matches option a).
- The algebra shows the one-year total return equals the initial YTM 5.6% when yields are unchanged, so the total return is 5.60%.

5) Why the other options are incorrect
- Option b ($1,130.31 and 4.60%) and option c ($1,220.31 and 5.50%) give wrong price calculations and incorrect returns; they do not satisfy the bond-pricing formulas or the algebraic identity that HPR = initial YTM when yields are unchanged.

Conclusion: The correct answers are a. $1,230.31 and b. 5.60%, matching the provided answer.

类似问题

What is the yield of a 2-year zero coupon bond which has a face value of $1000, and which is trading at a Price of $900? (please answer in %. If the answer is 8.05%, then in the box, write 8.05)

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.)

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单