Question

Please answer the following questions in the space provided. As always, assume semiannual coupon payments. (I strongly suggest first completing these using a scientific or financial calculator, and then checking your answer using Excel.)

An investor is considering the purchase of a 20-year, 7% coupon bond selling for $816, with a par value of $1,000. The yield to maturity for this bond is currently 9%.

1. Why is this bond currently selling below par? (1 point)

2. What is the annual percentage total return the investor would earn if she purchased and held the bond for 20 years and was always able to reinvest the coupons at 9%? (1.5 points)

3. What is the annual percentage total return the investor would earn if she purchased and held the bond for 10 years, but was only able to reinvest the coupons at 7.5%? Assume she believes she can sell the bond after 10 years for a price that implies a YTM of 8%. (1.5 points)

Answer 1

The bond is currently selling below par as the coupon rate is less than the yield to maturity. As yields and price have inverse relationship , the price decreases below par if the yield increases above coupon rate.

If an investor holds the bonds till maturity and reinvests all the coupon amounts at YTM (yield to maturity), the realised total annual return is equal to the YTM. So, if the bond is purchased today and held till 20 years and all the coupons reinvested at 9%, the annual percentage total return will be 9%.

Price of bond after 10 years

P = 70/1.08 + 70/1.08^2+ ...+70/1.08^10+ 1000/1.08^10

=70/0.08*(1-1/1.08^10)+1000/1.08^10

=$932.90

If all the coupons are reinvested at 7.5%, the value of coupons at the end of 10 years

=70*1.075^9 + 70*1.075^8+ ... +70

= 70/0.075* (1.075^10-1)

=$990.30

So, total amount realised by investor after 10 years = $932.90+$990.30 = $1923.20 on an investment of $816 ten years ago, So the annual percentage total return (r) for the investor is given by

816*(1+r)^10 = 1923.20

=> (1+r)^10 = 2.3568

=> 1+r = 1.0895

=> r = 0.0895 or 8.95%