Crossfade Corporation has a bond with a par value of $2,000 that sells for $1,917.12. The bond has a coupon rate of 7.02 percent and matures in 20 years. If the bond makes semiannual coupon payments, what is the YTM of the bond?单项选择题
A
7.42%
B
5.57%
C
3.71%
D
7.05%
E
6.68%

Question

Crossfade Corporation has a bond with a par value of $2,000 that sells for $1,917.12. The bond has a coupon rate of 7.02 percent and matures in 20 years. If the bond makes semiannual coupon payments, what is the YTM of the bond?单项选择题

A

7.42%

B

5.57%

C

3.71%

D

7.05%

E

6.68%

BlackTom AI · Accepted Answer

YTM is the discount rate that makes the bond’s price equal to the present value of its semiannual coupon payments plus the redemption (par) value. Set up the bond-price equation and solve for the per-period rate, then annualize.

1) Relevant knowledge and concepts
- Coupon (annual) = coupon rate × par. For semiannual payments, each period’s coupon = (annual coupon)/2.
- Number of periods N = years × 2.
- Price = PV of an annuity (coupons) + PV of a lump sum (par).
- YTM (quoted) for semiannual-pay bond is the nominal annual rate = 2 × (semiannual rate).

2) Step-by-step reasoning and setup
- Par = $2,000.
- Annual coupon = 0.0702 × 2,000 = $140.40 → semiannual coupon = 140.40/2 = $70.20.
- N = 20 × 2 = 40 semiannual periods.
- Price P = $1,917.12.
Form equation for semiannual yield i:
1917.12 = 70.20 × [1 − (1 + i)^−40]/i + 2000 × (1 + i)^−40

3) Calculations (intermediate steps)
This equation must be solved numerically. Using trial/financial-calculator methods:
- Try i ≈ 0.03710 (3.710% per half-year):
  (1 + i)^40 ≈ 4.295 → (1 + i)^−40 ≈ 0.2327
  Annuity factor = [1 − 0.2327]/0.03710 ≈ 20.68
  PV(coupons) ≈ 70.20 × 20.68 ≈ 1,452.6
  PV(par) ≈ 2000 × 0.2327 ≈ 465.4
  Total PV ≈ 1,918.0 (very close to 1,917.12)
Refining i slightly to ≈ 0.037095 gives PV essentially equal to 1,917.12. Therefore semiannual rate ≈ 3.7095%.

4) Why the correct answer is correct
Annual nominal YTM (with semiannual compounding) = 2 × semiannual rate ≈ 2 × 3.7095% ≈ 7.419% → rounded to 7.42%. This is the rate that equates the given price to the bond’s cash flows.

5) Why other choices are incorrect
- 3.71%: This is approximately the semiannual rate, not the quoted annual YTM (the quoted YTM doubles the semiannual rate).
- 5.57%, 7.05%, 6.68%: These annual rates would produce different present values and do not satisfy the price equation for the given coupons and maturity.

6) Conclusion
Solving the bond price equation yields a semiannual rate ≈ 3.7095%, so the nominal annual YTM = 2 × 3.7095% ≈ 7.42%, which matches the provided correct answer (7.42%).

Crossfade Corporation has a bond with a par value of $2,000 that sells for $1,917.12. The bond has a coupon rate of 7.02 percent and matures in 20 years. If the bond makes semiannual coupon payments, what is the YTM of the bond?单项选择题

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