Question

An investor purchases a nine-year, 7% annual coupon payment bond at a price equal to par value. After the bond is purchased and before the first coupon is received, interest rates increase to 8%. The investor sells the bond after five years. Assume that interest rates remain unchanged at 8% over the five-year holding period.

Assuming that all coupons are reinvested over the holding period, the investor's five-year horizon yield is closest to:

1. 5.66%

2. 6.62%

3. 7.12%

Answer 1

First let us understand the bond. It is a 7% coupon bond purchased at par value.

Let us say it is purchased at $100 then the investor receives annual coupon payment of 7. If the investor holds the bond till maturity, he receives the principal of $ 100 and all the coupon payments. The graph below summarizes the bond.

Chart Title 1 2 3 4 5 6 7 8 9

But in this case, the investor holds the bond till 5 years, reinvests the coupon payments at a market rate of 8% till five years and sells the bond after 5 years. So the period of relevance is 5 years for us.

Chart Title 1 2 3 4 5 6 7 8 9

Step 1 : Arrive at the reinvested coupon values till 5 years

Return from reinvestment of coupons @ 8% till year 5
Year 1	7	7*( 1.08)^ 4	9.5234
Year 2	7	7*( 1.08)^ 3	8.8179
Year 3	7	7*( 1.08)^ 2	8.1648
Year 4	7	7*( 1.08)^ 1	7.5600
Year 5	7	7*( 1.08)^ 0	7.0000
			41.0661

Step 2: Arrive at the bond price. As this bond is not held till maturity, bond price has to be arrived at market rate. Market price factor the remaining cash flows from the bond i.e. the 4 coupon payments of $ 7 annually and final principal payment of $100

Chart Title 1 2 3 4

This can be arrived at by taking the present values of all the remaining cash flows from the bond till year 5

Bond cash flows for the remaining 4 years
Year 6	7	7/(1.08) 1	6.4815
Year 7	7	7/(1.08) 2	6.0014
Year 8	7	7/(1.08)3	5.56
Year 9	107	107/(1.08)4	78.6187
			96.6616

Step 3 : Now we can calculate the horizon yield:

At year 5, the bondholder receives an amount of $41.0661 from coupon reinvestment and $96.6616 from the sale of bond. This amounts to $ 137.7277.

Horizon yield is the bond holder made by staying invested till 5 years. This can be viewed as below

Principal * (1 + Horizon yield) ⁵= Amount received from coupon reinvestment + Amount received from sale of bond.

i.e. 100* (1+r)⁵ = 41.0661 +96.6616

By calculating further, holding period yield = 6.62% i.e. option 2