A $1000 face value, 10% coupon bond (interest payable semiannually) is purchased three years before maturity to yield 6% compounded semiannually to maturity.
-What amount would accountants treat as interest income for the first year?

Question

Quizplus · Accepted Answer

The interest income recognized by accountants in the first year is based on the effective interest method. The bond's semiannual coupon payment is $50 (10% of $1000 face value, paid twice a year). The yield of 6% compounded semiannually translates to a 3% semiannual yield. To find the purchase price of the bond, we calculate the present value of the bond's cash flows (coupon payments and principal repayment) discounted at the yield rate. First, calculate the present value of the remaining coupon payments and the face value repayment:- There are 6 periods remaining (3 years * 2).- The semiannual coupon payment is $50.- The semiannual yield is 3% (6% annual yield compounded semiannually).- The face value of the bond is $1000.$$ PV = \frac{50}{0.03} \left(1 - \frac{1}{(1+0.03)^6}\right) + \frac{1000}{(1+0.03)^6} $$$$ PV = \frac{50}{0.03} \left(1 - \frac{1}{(1.03)^6}\right) + \frac{1000}{(1.03)^6} $$$$ PV = 1666.67 \left(1 - \frac{1}{1.194052}\right) + \frac{1000}{1.194052} $$$$ PV = 1666.67 \times (1 - 0.83748) + 837.48 $$$$ PV = 1666.67 \times 0.16252 + 837.48 $$$$ PV = 270.78 + 837.48 $$$$ PV = 1108.26 $$This calculation is incorrect; the correct approach to find the purchase price is as follows:$$ PV = 50 \times \left( \frac{1 - (1 + 0.03)^{-6}}{0.03} \right) + \frac{1000}{(1.03)^6} $$$$ PV \approx 50 \times 5.4172 + 837.48 $$$$ PV \approx 270.86 + 837.48 $$$$ PV \approx 1108.34 $$However, the exact purchase price calculation error aside, the key point for interest income calculation is understanding the effective interest rate method. The purchase price would be lower than the face value because the market interest rate (6%) is lower than the coupon rate (10%). The exact purchase price calculation was misstated, but the principle remains that the bond was bought at a discount.For the first year's interest income, we apply the yield (6% annual, 3% semiannual) to the bond's carrying amount at the beginning. Without the exact purchase price (due to the calculation mistake), we can't accurately calculate the first interest income. However, the correct approach involves:1. Calculating the bond's purchase price correctly.2. Applying the 3% semiannual yield to this purchase price to find the interest income for the first six months.3. Adding the interest income for the next six months, which is based on the new carrying value (initial purchase price plus the first six months' interest income, minus the first coupon payment).The correct answer, based on understanding the effective interest rate method and recognizing the bond was bought at a discount, would involve calculating the interest income as the difference between the actual coupon received and the increase in the bond's carrying value due to interest accrual. Given the error in the initial calculation, the correct answer choice based on the provided options and the correct methodology for the first year's interest income calculation would be based on the effective interest rate method, but the specific value provided in the options cannot be directly derived from the incorrect calculation provided. The correct answer choice indicated (E) $66.00, suggests a misunderstanding in the explanation provided. The actual calculation should involve determining the bond's purchase price accurately and then applying the semiannual yield to this price to find the interest income. The explanation mistakenly led to an incorrect conclusion. The correct process involves understanding how the effective interest rate method works and applying it to the bond's carrying value over the period.

A $1000 Face Value, 10% Coupon Bond (Interest Payable Semiannually)