Suppose you buy a 10-year $1,000 face-value bond. The annual coupon rate is 7% and interest payments are paid semi-annually. If the bond price is $932, what is the YTM on this bond?

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We start by clearly restating the problem and the available choices to anchor the discussion.
Question: Suppose you buy a 10-year $1,000 face-value bond. The annual coupon rate is 7% and interest payments are paid semi-annually. If the bond price is $932, what is the YTM on this bond?
Answer options: 4.00%, 3.50%, 8.00%, 7.68%

Now, let's evaluate each option step by step.
Option 1: 4.00%
- If the quoted YTM is 4% annually with semiannual payments, the per-half-year yield is 2%.
- The semiannual coupon is 7% of 1,000 annually, i.e., 70 per year, which is 35 every six months.
- The present value of the coupon stream at 2% per half-year is 35 × [1 − (1 + 0.02)^(−20)] / 0.02 ≈ 35 × 18.256 ≈ 639.0.
- The present value of the redemption 1,000 at 2% per half-year is 1,000 × (1 + 0.02)^(−20) ≈ 1,000 × 0.673 ≈ 673.0.
- Sum ≈ 639.0 + 673.0 = 1,312.0, which is far above the given price of 932. Therefore, 4.00% cannot be the correct YTM.
Option 2: 3.50%
- Per-half-year yield = 1.75%.
- Annuity factor: [1 − (1.0175)^(−20)] / 0.0175 ≈ [1 − 0.700] / 0.0175 ≈ 0.300 / 0.0175 ≈ 17.14.
- PV of coupons: 35 × 17.14 ≈ 599.9.
- PV of redemption: 1,000 × (1.0175)^(−20) ≈ 1,000 × 0.700 ≈ 700.
- Total ≈ 1,299.9, still much higher than 932. This indicates a yield lower than the market price would require, hence 3.50% is not consistent with a price of 932.
Option 3: 8.00%
- Per-half-year yield = 4%.
- Annuity factor: [1 − (1.04)^(−20)] / 0.04 ≈ [1 − 0.456] / 0.04 ≈ 0.544 / 0.04 ≈ 13.6.
- PV of coupons: 35 × 13.6 = 476.
- PV of redemption: 1,000 × (1.04)^(−20) ≈ 1,000 × 0.456 ≈ 456.
- Total ≈ 476 + 456 = 932, which matches the given price exactly.
Option 4: 7.68%
- Per-half-year yield ≈ 3.84%.
- Annuity factor: [1 − (1.0384)^(−20)] / 0.0384 ≈ approximately 13.7 (rough estimate).
- PV of coupons: about 35 × 13.7 ≈ 479.5.
- PV of redemption: 1,000 × (1.0384)^(−20) ≈ around 0.5 × 1,000 ≈ 500 (roughly).
- Total ≈ 979.5, which is still above 932; more precise calculation would likely differ, but the result would not align exactly with 932 as cleanly as the 8.00% case.

Summary of reasoning: The price of a coupon-bond is the present value of the coupon payments plus the present value of the principal repayment, both discounted at the per-period yield. When you test each option by converting the annual YTM to a semiannual rate and computing the PV of 20 semiannual coupons of 35 and the PV of 1,000 at the end, you see that only the 8.00% choice reproduces a present value very close to 932. The other rates produce prices far from 932 due to the discounting effects on the cash flows. The process demonstrates the sensitivity of bond pricing to yield and how semiannual compounding alters the calculation relative to the annual coupon rate.

MSB-250-300-002 Proctored Midcourse Exam 2

Suppose you buy a 10-year $1,000 face-value bond. The annual coupon rate is 7% and interest payments are paid semi-annually. If the bond price is $932, what is the YTM on this bond?

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MSB-250-300-002 Proctored Midcourse Exam 2

Suppose you buy a 10-year $1,000 face-value bond. The annual coupon rate is 7% and interest payments are paid semi-annually. If the bond price is $932, what is the YTM on this bond?

View Explanation

Log in for full answers

Similar Questions

What is the cost of bond?

A bond’s yield to maturity represents the investor’s realized return for any holding period, regardless of whether the bond is held to maturity.

A bond with a coupon rate of​ 8% will also have a yield to maturity of​ 8%.

A zero-coupon bond is currently priced at $456, has a face value of $1,000, and matures in 10 years. What is the yield to maturity of this bond?

More Practical Tools for Students Powered by AI Study Helper

Making Your Study Simpler

A bond with a coupon rate of 8% will also have a yield to maturity of 8%.