A bond with a face value of $1,000 has had a 3% rate of return over the last year. If there had been a $10 capital gain over this period, what is the bond's coupon rate?

A bond with a face value of $1,000 has had a 3% rate of return over the last year. If there had been a $10 capital gain over this period, what is the bond's coupon rate?

• A. 1%
• B. 3%
• C. 4%
• D. 2%

Answer: (D)

Total return on a bond over a year comes from two pieces: the income from the coupon and any change in price (capital gain or loss), all divided by the price at the start. Here, the bond has a face value of 1000, so the coupon payment equals the coupon rate times 1000. The year’s capital gain is 10, and the overall return is 3% of the initial investment. Assuming the initial price is 1000 (purchased at par is the standard assumption unless stated otherwise), 3% of 1000 is 30. Therefore, coupon income plus the capital gain must add up to 30: coupon + 10 = 30, so coupon = 20. The coupon rate is 20 divided by 1000, which is 0.02 or 2%. If the coupon were 3%, you'd have 30 in coupons plus a 10 capital gain for a total of 40, which would be a 4% return, not 3%. Similarly, a 1% coupon would yield 10 in coupons plus 10 gain = 20 total, i.e., a 2% return, not 3%.

Total return on a bond over a year comes from two pieces: the income from the coupon and any change in price (capital gain or loss), all divided by the price at the start. Here, the bond has a face value of 1000, so the coupon payment equals the coupon rate times 1000. The year’s capital gain is 10, and the overall return is 3% of the initial investment. Assuming the initial price is 1000 (purchased at par is the standard assumption unless stated otherwise), 3% of 1000 is 30. Therefore, coupon income plus the capital gain must add up to 30: coupon + 10 = 30, so coupon = 20. The coupon rate is 20 divided by 1000, which is 0.02 or 2%.

If the coupon were 3%, you'd have 30 in coupons plus a 10 capital gain for a total of 40, which would be a 4% return, not 3%. Similarly, a 1% coupon would yield 10 in coupons plus 10 gain = 20 total, i.e., a 2% return, not 3%.