Coupon Reinvestment and Bond Realized Yield

From Bond Coupons to an Investor’s Realized Return

A bond’s coupon rate tells you how its periodic payments are set; it does not by itself tell you the return an investor will earn. The return over a holding period also depends on the price paid, the redemption proceeds, and what happens to coupons received before redemption. If coupons are reinvested, each one contributes both its original amount and the interest it earns before the horizon date.

This tutorial calculates the value at redemption of coupons reinvested at a specified rate, adds the redemption payment, and converts that total into an annual effective realized return. The calculations assume the bond is held until its stated redemption date. They are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: The horizon value is the total value at a selected future date of the bond’s redemption proceeds and all coupon payments accumulated to that date. The realized annual effective return is the annual effective rate that equates the original purchase price to the horizon value accumulated over the holding period.

Accumulating Coupons to the Redemption Date

Suppose a bond is purchased for price $P$, pays a level coupon $C$ at the end of each year, and is redeemed for amount $R$ at the end of year $n$. Assume each coupon received before the final year is reinvested at a constant effective annual rate $j$. The final coupon is paid at the horizon, so it earns no reinvestment interest before that date.

A coupon paid at the end of year $k$ has $n-k$ years to accumulate. Its horizon value is therefore $C(1+j)^{n-k}$. Adding the accumulated coupons to the redemption proceeds gives:

H=R+\sum_{k=1}^{n} C(1+j)^{n-k} =R+C\left(\frac{(1+j)^n-1}{j}\right), \qquad j\ne 0.

Here $H$ is the horizon value. If $j=0$, each coupon remains at its original amount, so $H=R+nC$. The formula assumes coupons are paid annually at year-end, the reinvestment rate is constant, and redemption occurs at the end of year $n$. It does not describe a sale before redemption, default, taxes, transaction costs, or changing reinvestment rates.

Realized-return equation: If the investment begins with price $P$ and ends with horizon value $H$ after $n$ years, its annual effective realized return $i_{\text{real}}$ satisfies $P(1+i_{\text{real}})^n=H$. Thus, $$ i_{\text{real}}=\left(\frac{H}{P}\right)^{1/n}-1. $$ This rate summarizes the growth from purchase price to total horizon wealth. It is not necessarily the bond’s yield to maturity.

Why the Horizon-Value Formula Works

The accumulation factor for a coupon depends on how long that particular payment remains invested. The coupon paid at year 1 earns interest for $n-1$ years; the coupon at year 2 earns interest for $n-2$ years; the coupon at year $n$ earns none. Consequently, the coupon portion of horizon value is the finite sum $C[1+(1+j)+\cdots+(1+j)^{n-1}]$.

For $j\ne 0$, multiply this sum by $1+j$ and subtract the original sum. All intermediate powers cancel:

(1+j)\sum_{m=0}^{n-1}(1+j)^m-\sum_{m=0}^{n-1}(1+j)^m =(1+j)^n-1.

The left side is $j$ times the original sum. Dividing by $j$ gives $\sum_{m=0}^{n-1}(1+j)^m=((1+j)^n-1)/j$, which proves the coupon accumulation expression. The redemption amount is added separately because it is paid at the horizon and is not reinvested beyond that date.

Worked Example: Annual Coupons Reinvested Below the Coupon Rate

A hypothetical bond has a redemption value of $1,000, pays annual coupons at a 6% coupon rate, and will be redeemed in three years. An investor pays $950 and reinvests each coupon at an effective annual rate of 4%. Find the horizon value and the annual effective realized return. Ignore taxes and transaction costs.

Method. First calculate the annual coupon: $C=1{,}000(0.06)=\$60$. Accumulate each coupon to year 3. The year-1 coupon earns two years of interest, the year-2 coupon earns one year, and the year-3 coupon earns none:

H=1{,}000+60(1.04)^2+60(1.04)+60

The intermediate amounts are $60(1.04)^2=60(1.0816)=\$64.896$, $60(1.04)=\$62.40$, and $\$60$ for the final coupon. Thus, accumulated coupons total $64.896+62.40+60=\$187.296$, and $H=\$1{,}187.296$.

There are three years between purchase and redemption. Therefore:

i_{\text{real}} =\left(\frac{1{,}187.296}{950}\right)^{1/3}-1 \approx 0.077156.

The realized annual effective return is 7.716% to three decimal places. As a check, $950(1.077156)^3\approx \$1{,}187.296$. A common mistake is to treat all three coupons as though they were paid at redemption, or to omit their reinvestment earnings. The date-by-date accumulation above avoids both errors.

Semiannual Coupons and Annualized Return

When coupons are paid more than once per year, count the coupon periods explicitly. If there are $N$ equal periods, each coupon is $C$, and the effective reinvestment rate per period is $j$, a coupon paid at the end of period $k$ accumulates for $N-k$ periods. The total horizon value is the redemption amount plus the accumulated coupons. If the investment lasts $T$ years, convert that total into an annual effective return using $T$, not $N$, in the annualization exponent.

The reinvestment rate per coupon period and the annualized realized return are different quantities. In particular, a rate of 2% per half-year is not an annual effective rate of 2%; the corresponding effective annual rate for that reinvestment process would be $(1.02)^2-1$. The realized return is determined by the entire investment from purchase to horizon value.

Worked Example: Semiannual Coupons Over Five Years

A hypothetical bond with redemption value $1,000 pays coupons at a 5% nominal annual coupon rate, convertible semiannually. It has ten half-year periods remaining, and its purchase price is $1,040. Each coupon is reinvested at an effective rate of 2% per half-year. Find the horizon value and annual effective realized return. Assume redemption at the end of the tenth period.

Method. The coupon per half-year is $1{,}000(0.05)/2=\$25$. Each coupon is accumulated to period 10 at 2% per period. The final coupon earns no interest, so the coupon accumulation is:

25\sum_{m=0}^{9}(1.02)^m =25\left(\frac{(1.02)^{10}-1}{0.02}\right).

The tenth power is $(1.02)^{10}=1.21899442$, so the accumulated coupon value is $25(0.21899442/0.02)=25(10.949721)=\$273.7430$, to four decimal places. Adding the $1,000 redemption payment gives $H=\$1{,}273.7430$.

The holding period is five years. Thus, the annual effective realized return is:

i_{\text{real}} =\left(\frac{1{,}273.7430}{1{,}040}\right)^{1/5}-1 \approx 0.041381.

The return is 4.138% to three decimal places. As a check, $1{,}040(1.041381)^5\approx \$1{,}273.743$. The exponent is $1/5$ because the investment lasts five years; using $1/10$ would instead calculate a half-year rate and would not answer for the annual effective return.

Coupon Rate, Reinvestment Rate, and Realized Return

These rates play different roles. The coupon rate determines the bond’s scheduled payments from its redemption value. The reinvestment rate determines how much earlier coupons grow before the horizon date. The realized return combines the purchase price, reinvested coupons, and redemption proceeds into one annual effective rate.

For example, the first worked example has a 6% coupon rate and a 4% annual reinvestment rate, yet its realized return is 7.716%. The realized return exceeds both rates because the bond was bought below its $1,000 redemption value. In another situation, a purchase at a premium or a different reinvestment rate could change the comparison. There is no general rule that the realized return must equal the coupon rate or the reinvestment rate.

Worked Example: A Premium Bond with Reinvested Coupons

A hypothetical bond with a redemption value of $1,000 pays an annual coupon of $70 and is redeemed after three years. An investor pays $1,025 and reinvests coupons at an effective annual rate of 5%. Calculate the horizon value and the annual effective realized return. Assume no taxes, fees, or default.

Method. Accumulate each coupon to year 3. The first coupon earns two years of interest, the second earns one year, and the third is paid at the horizon:

H=1{,}000+70(1.05)^2+70(1.05)+70.

The accumulated coupon amounts are $70(1.1025)=\$77.175$, $70(1.05)=\$73.50$, and $\$70$. Their sum is $\$220.675$, so the horizon value is $H=\$1{,}220.675$.

Now equate the initial price accumulated for three years to that horizon value:

i_{\text{real}} =\left(\frac{1{,}220.675}{1{,}025}\right)^{1/3}-1 \approx 0.059966.

The annual effective realized return is 5.997% to three decimal places. Checking with the rounded rate, $1{,}025(1.05997)^3\approx \$1{,}220.69$, consistent with the horizon value after rounding. The main trap is to infer the realized return directly from the 7% coupon rate. That ignores both the premium purchase price and the interest earned on reinvested coupons.

Common Errors to Avoid

Accumulating every coupon for the same number of periods. Coupons arrive at different dates; the first coupon generally earns more reinvestment interest than later ones.
Giving the final coupon an extra period of interest. A coupon paid at the horizon is already at the valuation date and is not accumulated further.
Using the number of coupon periods as the annualization exponent. For an annual effective return, use the number of years in $(H/P)^{1/T}-1$.
Confusing the coupon rate with the realized return. Price, reinvestment, and redemption all affect the realized return.
Assuming yield and realized return are interchangeable. A yield calculation uses a specified yield assumption for valuing bond cash flows; the realized-return calculation here uses the stated reinvestment rate to find actual horizon wealth.

Key takeaway: For a bond held to redemption, accumulate each coupon from its payment date to the horizon, add the redemption proceeds, and solve $P(1+i_{\text{real}})^T=H$. Correct timing is essential: each coupon has its own accumulation period, and the final coupon has none.

Check Your Understanding

For each problem, assume the bond is held to redemption, coupons are reinvested at the stated constant rate, and there are no taxes, fees, or default. Round realized returns to three decimal places.

A bond with redemption value $1,000 pays annual coupons at a 4% rate and is redeemed in two years. It costs $970, and coupons are reinvested at 3% annually. Find the horizon value and annual effective realized return. (Answer: horizon value $1,081.20; realized return 5.576%)
A bond with redemption value $5,000 pays annual coupons at a 6% rate and is redeemed in three years. Its price is $4,920, and coupons are reinvested at 2% annually. Find the horizon value and annual effective realized return. (Answer: horizon value $5,918.12; realized return 6.351%)
A bond with redemption value $2,000 pays a 4% nominal annual coupon rate convertible semiannually and is redeemed in three years. Its price is $1,950, and each coupon is reinvested at 1.5% per half-year. Find the horizon value and annual effective realized return. (Answer: horizon value approximately $2,249.18; realized return approximately 4.873%)
A bond with redemption value $1,000 pays annual coupons of $80 and is redeemed in two years. Its price is $960, and coupons earn no reinvestment interest. Find the horizon value and annual effective realized return. (Answer: horizon value $1,160; realized return 9.924%)
A bond with redemption value $1,000 pays annual coupons of $50 and is redeemed in four years. Its price is $1,010, and coupons are reinvested at 4% annually. Find the horizon value and annual effective realized return. (Answer: horizon value $1,212.32; realized return 4.671%)

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