Solving For Redemption Value At A Target Yield

Why Solve for Redemption Value?

A bond’s price depends on more than its coupon payments. The amount repaid at maturity also affects the price, so if the price, coupon schedule, maturity, and yield are known, the redemption value can be the unknown. Solving for it is an application of the bond price equation: discount each promised payment at the specified yield, add the present values, and isolate the maturity payment.

The result is useful for interpreting the bond’s terms. The implied redemption value can be compared with the bond’s par value, but it is not the same thing as the bond’s market price. A price above par does not, on its own, establish that the redemption value is above par; coupon amounts and the target yield also matter.

Definition: The redemption value, denoted here by $R$, is the principal payment made at maturity. The par value, denoted by $F$, is the reference principal amount stated for the bond. A bond is redeemed at a premium to par if $R>F$, at par if $R=F$, and at a discount to par if $R<F$.

The Price Equation and Its Solution

Assume a conventional bond is valued on a coupon date, has $n$ remaining equal-length coupon periods, and pays a fixed coupon of $C$ at the end of each period. Assume also that the redemption payment $R$ is made with the final coupon. If the yield per coupon period is $j$, the price is the present value of all $n$ coupons plus the present value of redemption:

P=C a_{\overline{n}|\,j}+R(1+j)^{-n}.

Here $a_{\overline{n}|\,j}=\sum_{k=1}^{n}(1+j)^{-k}$ is the present-value annuity factor at rate $j$. If the coupon is not level, write the coupon in period $k$ as $C_k$ and use the sum of the individual present values instead.

Formula: For level coupons, the redemption value implied by price $P$ and period yield $j$ is

R=\frac{P-Ca_{\overline{n}|\,j}}{(1+j)^{-n}} =\bigl(P-Ca_{\overline{n}|\,j}\bigr)(1+j)^n.

The price and coupons must use the same monetary units, and $j$ must be the rate per coupon period.

If the target yield is a nominal annual rate $i^{(m)}$ convertible $m$-thly and the bond pays coupons $m$ times per year, then $j=i^{(m)}/m$ and $n$ is the number of remaining coupon periods. If the target is an effective annual rate $i$, the equivalent period rate is $j=(1+i)^{1/m}-1$. The rate conversion must match the coupon timing. This equation assumes valuation on a coupon date; a price between coupon dates requires appropriate treatment of the fractional period and accrued interest.

Why the Formula Works

The price equation separates the two kinds of payments. The coupons have known amounts and dates, so their present values can be calculated as an annuity. The redemption payment is a single amount paid at time $n$, so its present value is $R(1+j)^{-n}$. Subtract the coupon present value from the total price to leave only the present value of redemption:

P-Ca_{\overline{n}|\,j}=R(1+j)^{-n}.

Since $(1+j)^{-n}$ is positive for a valid yield $j>-1$, divide both sides by it. This gives $R=(P-Ca_{\overline{n}|\,j})(1+j)^n$. The result is unique: with the other inputs fixed, redemption has a positive present-value factor, so exactly one $R$ satisfies the price equation.

Set the period rate and period count.
Convert the target yield to the rate per coupon period, then count the remaining coupon periods.

Value the coupons.
Calculate the present value of the coupon schedule at the specified yield.

Isolate redemption.
Subtract coupon present value from price, then divide by the redemption discount factor.

Interpret and check.
Compare $R$ with par, and substitute the result into the original price equation.

Worked Examples

Worked Example: Semiannual Coupons and a Specified Nominal Yield

A six-year bond pays a coupon of $35 every six months. Its price on a coupon date is $990. The target yield is 8% nominal, convertible semiannually. Find the redemption value to the nearest cent and compare it with par value of $1,000.

Method. Use the level-coupon price equation, with the nominal yield converted to a semiannual rate. There are $n=6(2)=12$ periods, and $j=0.08/2=0.04$.

The coupon annuity factor is

a_{\overline{12}|\,0.04} =\frac{1-(1.04)^{-12}}{0.04} \approx 9.385074.

The present value of the coupons is $35(9.385074)\approx \$328.4776$. The portion of the price attributable to redemption is therefore $\$990-\$328.4776=\$661.5224$. The redemption discount factor is $(1.04)^{-12}$, so

R=\frac{661.5224}{(1.04)^{-12}} =661.5224(1.04)^{12} \approx \$1{,}059.1187.

Thus, the redemption value is $1,059.12 to the nearest cent. It exceeds the $1,000 par value by $59.12, so the bond is redeemed at a premium to par.

Check. The redemption present value is approximately $\$1{,}059.1187(1.04)^{-12}=\$661.5224$. Adding the coupon present value gives $\$661.5224+\$328.4776=\$990.0000$, as required. The common mistake avoided here is treating the residual price as the redemption amount; it is the present value of redemption.

Worked Example: A Two-Year Bond with a Clean Redemption Calculation

A two-year bond has annual coupons of $50 and a price of $1,000 on a coupon date. The target yield is 10% effective annually. Find the redemption value and compare it with par value of $1,000.

Method. Annual coupon periods and an effective annual yield use the same period, so $j=0.10$ and $n=2$. The coupon present value is

50a_{\overline{2}|\,0.10} =50\left(\frac{1-(1.10)^{-2}}{0.10}\right) =50(1.73553719) \approx \$86.77686.

The residual price attributable to redemption is $\$1{,}000-\$86.77686=\$913.22314$. Since the redemption is paid at the end of period 2,

R=\frac{913.22314}{(1.10)^{-2}} =913.22314(1.10)^2 =\$1{,}105.00.

The redemption value is $1,105.00, which is $105.00 above par. Substitution checks the calculation: the present value of redemption is $\$1{,}105/(1.10)^2=\$913.22314$, and adding $\$86.77686$ gives the $1,000 price.

Common mistake avoided. Multiplying the residual price by the discount factor would give $913.22314(1.10)^{-2}\approx \$754.73$, which discounts an amount that is already a present value. The redemption amount must be recovered by dividing by the discount factor, equivalently multiplying by $(1.10)^2$.

Worked Example: Interpreting Redemption Above Par

A four-year bond has an annual coupon of $70 and is priced at $1,080 on a coupon date. Its target yield is 6% effective annually. Find the redemption value to the nearest cent and explain its relation to par value of $1,000.

Method. There are four annual periods, so $j=0.06$ and $n=4$. First calculate the coupon annuity factor and coupon present value:

a_{\overline{4}|\,0.06} =\frac{1-(1.06)^{-4}}{0.06} \approx 3.4651056, \qquad 70(3.4651056)\approx \$242.5574.

The residual price is $\$1{,}080-\$242.5574=\$837.4426$. Accumulating this residual for four years at 6% gives

R=837.4426(1.06)^4 =837.4426(1.26247696) \approx \$1{,}057.2520.

To the nearest cent, $R=\$1{,}057.25$, or $57.25 above par. As a check using accumulated values, the price accumulated four years at 6% is $\$1{,}080(1.26247696)=\$1{,}363.4751$. The coupons accumulated to year 4 are $70[(1.06)^3+(1.06)^2+1.06+1]=70(4.374616)=\$306.2231$. Their difference is $\$1{,}057.2520$, matching the redemption calculation.

Common mistake avoided. Do not label the bond’s $1,080 price as its redemption value. The price includes the present value of all coupons as well as redemption; the inferred maturity payment is $1,057.25.

Reading the Result and Avoiding Traps

A solved redemption value should be interpreted in the context of the specified price, coupons, and yield. Compare $R$ with the stated par amount $F$: $R>F$ means redemption above par, $R=F$ means redemption at par, and $R<F$ means redemption below par. This comparison concerns the maturity payment. It does not classify the bond’s price relative to par, and it does not replace the yield calculation.

The most consequential calculation errors usually come from the timing or the direction of the final step:

Using an annual rate for semiannual periods. A nominal yield convertible semiannually must be divided by 2 to obtain the semiannual rate; the number of periods must also be counted semiannually.
Discounting the residual again. After coupons are subtracted, the residual is the present value of redemption, not the future payment. Divide by $(1+j)^{-n}$, or multiply by $(1+j)^n$.
Omitting the final coupon. When the bond pays a coupon at maturity, that coupon belongs in the coupon present value. Redemption is a separate payment at the same date.
Mixing price units. If price and coupons are given per $100 of par, solve in those units; do not compare the resulting redemption amount with a $1,000 par value until the units have been aligned.

The level-coupon formula applies when the stated coupon amount and period timing are fixed and the target yield uses those same periods. With unequal coupons, the same present-value principle applies, but calculate the coupon present value as a sum: $P=\sum_{k=1}^{n}C_k(1+j)^{-k}+R(1+j)^{-n}$. If valuation is not on a coupon date, or the cash-flow timing differs from the standard schedule, do not use the integer-period formula without adjusting the timing assumptions.

Key takeaway: Subtract the present value of the coupon schedule from the bond price to obtain the present value of redemption. Then divide by the redemption discount factor, check the original price equation, and compare the resulting maturity payment—not the price—with par.

Check Your Understanding

For each problem, assume the bond is priced on a coupon date and redemption is paid with the final coupon. Round monetary answers to the nearest cent.

A three-year bond pays annual coupons of $40. Its price is $920 and its target yield is 5% effective annually. Find the redemption value. (Answer: $938.92)
A four-year bond pays $30 every six months. Its price is $960 and its target yield is 8% nominal, convertible semiannually. Find the redemption value. (Answer: $1,037.40)
A three-year bond pays $28 every six months. Its price is $980 and its target yield is 10% nominal, convertible semiannually. Find the redemption value and compare it with par of $1,000. (Answer: $1,122.84; $122.84 above par)
A two-year bond pays quarterly coupons of $18. Its price is $1,005 and its target yield is 8% nominal, convertible quarterly. Find the redemption value. (Answer: $1,023.02)
A three-year bond has annual coupon payments of $20, $30, and $40 at the ends of years 1, 2, and 3, respectively. Its price is $900 and its target yield is 5% effective annually. Find the redemption value. (Answer: $948.31)

What You'll Learn