# Market value of a bond between coupon payment

A 10,000 par value bond with coupons at 8%, convertible semiannually, is being sold 3 years and 4 months before the bond matures. The purchase will yield 6% convertible semiannually to the buyer. The price at the most recent coupon date, immediately after the coupon payment, was 5640. Calculate the market price of the bond, assuming compound interest through- out.

Correct answer: 5563

I'm confused about what's going on. Since coupons are payed every 6 months, the last coupon payment was 3 years and 6months from maturity date, and the bond had a market price of 5640 then. This means the coupon has accrued 2 months worth. This gives flat price at 3 years and 4 months before maturity date as $$5640(1.03)^{\frac{2}{12}}$$, and an accrued coupon of $$1000(.04)\left[\frac{(1.03)^{2/12}-1}{.03}\right]$$, and difference between flat price and accrued coupon = bond price, but it's slightly off answer.

Other solutions on forums use $$k=4/12$$ not $$2/12$$, which I don't get why. clearly the bond matures 2 months from time where price wasa 5640?

• I must be missing something. If the bond pays 10,000 in a little over 3 years, the value now needs to be close to that, not half of it. The coupon is higher than the interest rate you are promising the buyer, so it should trade at a premium. Where can I buy this bond? Sep 8 '20 at 2:18

## 1 Answer

The price of the bond $$3$$ years and $$4$$ months before the bond matures is

$$5640 \cdot 1.03^{2/6} \approx 5695.84$$

Here the exponent refers to two months out of six (remind that purchase will yield $$6\%$$ convertible semiannually, i.e. $$3\%$$ every $$6$$ months).

The accrued coupon is given by

$$400 \left[\frac{1.03^{2/6} -1}{0.03} \right]\approx 132.02$$

So we get

$$5695.84 - 132.02\approx 5563$$