Reinvestment of Coupons

Study guide: Exam FM-2

Reinvestment of coupons refers to the practice of investing each periodic coupon payment at a reinvestment rate $r_i$ , which may differ from the original Yield Rate $j$ .

$AV = Fr \cdot s_{\overline{n}|r_i} + C$

When coupons are reinvested at a rate $r_i$ per period, the accumulated value at the end of $n$ periods equals the future value of the coupon annuity at the reinvestment rate plus the redemption value $C$ .
Here $Fr$ is the coupon per period, $s_{\overline{n}|r_i}$ is the accumulated value factor for an annuity-immediate at rate $r_i$ , and $C$ is the redemption payment received at time $n$ .
If the reinvestment rate equals the original yield ( $r_i = j$ ), the total accumulated value equals $P(1+j)^n$ , confirming that the investor earns exactly the yield rate.
If $r_i < j$ , the accumulated value is lower and the realized return falls below $j$ ; if $r_i > j$ , the investor does better than the promised yield.
This analysis motivates Reinvestment Risk as an important practical concern for coupon bonds.

Accumulated Value with Coupon Reinvestment A $ $1{,}000$ face value 4-year bond pays 7% annual coupons and is redeemed at par. It is bought at par, for $ $1{,}000$ (a 7% yield). Coupons are reinvested at 4% per year. Find the total accumulated value at the end of 4 years and the realized annual yield.

Answer Coupon $= Fr = 1000 \times 0.07 = \$70$ per year. $s_{\overline{4}|4\%} = \dfrac{(1.04)^4 - 1}{0.04} = \dfrac{1.16985856 - 1}{0.04} = 4.246464$ .

$AV = 70 \times 4.246464 + 1000 = 297.25 + 1000 = \$1{,}297.25$

The purchase price is $P = \$1{,}000$ . The realized yield $r$ satisfies $1000(1+r)^4 = 1297.25$ , giving $(1+r)^4 = 1.29725$ , so $r = 1.29725^{1/4} - 1 \approx 6.72\%$ . This is below the 7% yield the bond was bought at because the coupons were reinvested at only 4%.