Pricing Bonds with Nonpar Redemption Values

Why Face Value and Redemption Value Must Be Kept Separate

A bond’s face value is not necessarily the amount paid to the holder at maturity. The coupon amount may be based on face value, while the contract specifies a different redemption amount. To price such a bond correctly, value the coupon payments according to their own schedule and discount the maturity redemption amount as a separate cash flow.

The distinction matters even when the difference seems small: using face value instead of the specified redemption amount changes the final cash flow and therefore the price. This tutorial considers a bond valued on a coupon date, with equal coupon periods and a constant effective yield per coupon period. The method is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Let $F$ be a bond’s face value, $C$ its coupon payment at the end of each period, $R$ the amount redeemed at maturity, $n$ the number of remaining coupon periods, and $i$ the effective yield per coupon period. Let $v=1/(1+i)$. The bond is valued immediately after a coupon date, so its future coupons occur at times $1,2,\ldots,n$; the redemption $R$ is paid at time $n$, in addition to the final coupon.

If the coupon rate per period is $c$, then typically $C=Fc$. This determines the coupon amount, not the redemption amount. The contract may specify $R=F$, $R<F$, or $R>F$; use the stated $R$ in the price calculation. The model assumes the cash flows are known and that the same effective yield $i$ applies to every remaining period. If the yield is quoted on a different time basis, it must first be expressed on a compatible basis; rate conversion is a separate issue.

The Price Formula

The $n$ coupons are level payments at the ends of periods $1$ through $n$. Their present value is $C a_{\overline{n}|}$. The redemption payment is a single amount at time $n$, so its present value is $Rv^n$. Thus the price is

P=C a_{\overline{n}|}+Rv^n

For $i\ne 0$, $a_{\overline{n}|}=(1-v^n)/i$, giving the equivalent expression

P=C\frac{1-v^n}{i}+Rv^n,\qquad i\ne 0.

The annuity factor includes all $n$ coupons, including the coupon paid at maturity. The separate term $Rv^n$ adds only the redemption amount. Therefore, at time $n$, the total cash flow is $C+R$, not $C$ or $R$ alone.

Important zero-yield case: If $i=0$, then $v=1$ and $a_{\overline{n}|}=n$. The price is $P=Cn+R$. Do not use $(1-v^n)/i$ at $i=0$, because it would divide by zero.

Why the Formula Works

Each future payment is discounted from its payment time to the valuation date. A payment $C$ at time $k$ has present value $Cv^k$, and the redemption payment $R$ at time $n$ has present value $Rv^n$. Adding the values of the separate cash flows gives

P=\sum_{k=1}^{n}Cv^k+Rv^n =C\sum_{k=1}^{n}v^k+Rv^n =C a_{\overline{n}|}+Rv^n.

This argument also verifies the zero-yield case directly. When $i=0$, each discount factor is $v^k=1$, so the sum of the coupon values is $Cn$, and the redemption value is $R$. Hence $P=Cn+R$, agreeing with $a_{\overline{n}|}=n$.

A useful special case occurs when the coupon amount equals the face value times the yield per period, so $C=Fi$. For $i\ne 0$, substituting into the price formula gives

P=Fi\frac{1-v^n}{i}+Rv^n =F(1-v^n)+Rv^n =F+(R-F)v^n.

The result shows why a bond with coupons matched to the yield need not be priced at face value when its redemption is nonpar. The difference from face value is the discounted difference between redemption and face value. This identity was derived assuming $i\ne 0$. If $i=0$, then $C=Fi=0$; the direct price formula gives $P=R$, and $F+(R-F)v^n=F+(R-F)=R$ as well. The identity therefore also holds at zero yield in that special case, without dividing by zero.

Worked Example: Redemption Below Face Value

A bond has face value $\$1{,}000$, pays annual coupons at a rate of $5\%$ of face value, has four annual coupon payments remaining, and will redeem for $\$950$. The effective annual yield is $4\%$. Find its price immediately after a coupon date. Use the separate-present-values method, and do not substitute face value for redemption.

Method and cash flows. The coupon is $C=0.05(1{,}000)=\$50$. There are $n=4$ periods, $R=\$950$, and $i=0.04$. The final date includes both the fourth $\$50$ coupon and the $\$950$ redemption.

Coupon value. First, $1.04^4=1.16985856$, so $v^4=1/1.16985856=0.854804191$. Therefore,

a_{\overline{4}|}=\frac{1-0.854804191}{0.04}=3.629895225, \qquad 50(3.629895225)=181.494761.

Redemption value and price. The present value of the specified redemption is $950(0.854804191)=812.063982$. Add the two present values:

$$ P=181.494761+812.063982=993.558743. $$

The price is $\$993.56$, rounded to the nearest cent. A common error is to discount $\$1{,}000$ at maturity because it is the face value. The contract specifies redemption of $\$950$, so using face value would overstate the price by $50v^4=\$42.74$. The calculated price is below face value, consistent with redemption below face value and a coupon rate above the yield not being sufficient here to offset the discounted redemption shortfall.

Worked Example: Coupon Rate Equals Yield but Redemption Is Below Face

A bond has face value $\$1{,}000$, four annual periods remaining, and an annual coupon of $6\%$ of face value. It redeems for $\$900$, and its effective annual yield is $6\%$. Find the price in two ways: by discounting the cash flows directly and by using the special-case identity. State why the price is not $\$1{,}000$.

Method and direct calculation. The coupon is $C=0.06(1{,}000)=\$60$, $n=4$, $R=\$900$, and $i=0.06$. Compute $1.06^4=1.26247696$, hence $v^4=1/1.26247696=0.792093663$. The annuity factor and coupon present value are

a_{\overline{4}|}=\frac{1-0.792093663}{0.06}=3.465105617, \qquad 60(3.465105617)=207.906337.

The redemption present value is $900(0.792093663)=712.884297$. Thus the direct price is $207.906337+712.884297=\$920.790634$, or $\$920.79$ to the nearest cent.

Check with the identity. Since $C=Fi$, the special-case formula applies:

P=F+(R-F)v^n =1{,}000+(900-1{,}000)(0.792093663) =920.790634.

Both methods agree. The price is below face value because the bond pays $\$100$ less than face at maturity, and the present value of that shortfall is $\$79.21$. The trap is to apply the familiar par-price conclusion based only on coupon rate equaling yield; that conclusion requires redemption at face value.

Worked Example: Pricing at a Zero Yield

A bond has face value $\$1{,}200$, pays annual coupons of $6\%$ of face value for three remaining periods, and redeems for $\$1{,}080$. Its effective yield per period is $0\%$. Find the price and verify it using the cash flows without using a quotient formula.

Method and coupon amount. The coupon is $C=0.06(1{,}200)=\$72$, with $n=3$ and $R=\$1{,}080$. At zero yield, $v=1/(1+0)=1$, so no future payment is reduced when expressed as a present value.

The coupon present value is $72+72+72=\$216$. The redemption present value is $\$1{,}080$. Therefore,

$$ P=216+1{,}080=1{,}296. $$

The price is $\$1{,}296.00$. Using the zero-yield annuity value gives the same result: $a_{\overline{3}|}=3$, so $P=72(3)+1{,}080=\$1{,}296$. The expression $(1-v^n)/i$ is not usable here because both its numerator and denominator are zero; substituting $i=0$ into that quotient is not a valid calculation. The direct cash-flow sum is the safe method.

Interpreting Price and Avoiding Common Traps

The relation between price and face value depends on the whole cash-flow stream, not solely on whether $R$ is above or below $F$. A high coupon can raise price, while a redemption below face can lower it; the yield discounts both effects. In the special case $C=Fi$, the identity $P=F+(R-F)v^n$ makes the redemption effect explicit. In the general case, calculate the coupon and redemption present values and compare their sum with face value.

Identify the valuation date and remaining periods.
Count the future coupon payments from the valuation date. Under this tutorial’s model, valuation is immediately after a coupon date.

Find the coupon and the stated redemption.
If the coupon rate is applied to face value, calculate $C=Fc$. Record the contractual maturity amount $R$ separately.

Discount each cash-flow group.
Use $C a_{\overline{n}|}$ for the level coupons and $Rv^n$ for redemption. Use $P=Cn+R$ if $i=0$.

Check the maturity cash flow and price.
Confirm that the last date contains both $C$ and $R$. Check that the redemption used is the specified amount, not an assumed par amount.

Using face value at maturity. Face value may determine the coupon, but it is not a substitute for a separately stated redemption amount.
Omitting the last coupon. The annuity factor includes the coupon at time $n$; the redemption is an additional payment at that date.
Counting the final coupon twice. Use $C a_{\overline{n}|}+Rv^n$, not a coupon annuity plus a maturity amount that already includes the final coupon and then another coupon.
Dividing by zero. At $i=0$, use $a_{\overline{n}|}=n$ or sum the undiscounted payments directly.
Mixing period units. The yield used in $v^n$ must be effective for the same period represented by one step in the cash-flow timeline.

Key takeaway: A nonpar redemption bond is priced by discounting its coupons and its stated redemption amount separately. Face value may set the coupon, but only the contractual redemption amount belongs in the maturity-redemption term.

Check Your Understanding

For each problem, assume valuation immediately after a coupon date, equal periods, and a constant effective yield per period. Round monetary answers to the nearest cent.

A bond with face value $\$1{,}500$ pays annual coupons at $4\%$ of face value for three remaining years and redeems for $\$1{,}450$. Its effective annual yield is $5\%$. Find its price. (Answer: $\$1{,}415.96$)
A bond has face value $\$800$, annual coupons of $7.5\%$ of face value, four years remaining, redemption of $\$840$, and an effective annual yield of $6\%$. Find its price and identify the maturity cash flow. (Answer: $\$873.27$; maturity cash flow $=\$900$)
A bond has face value $\$2{,}500$, annual coupons of $3\%$ of face value, six annual periods remaining, and redemption of $\$2{,}350$. If the effective yield is zero, find its price using the undiscounted cash flows. (Answer: $\$2{,}800.00$)
A bond with face value $\$1{,}000$ pays annual coupons of $5\%$ of face value for two years and redeems for $\$1{,}100$. At an effective annual yield of $4\%$, calculate its price and explain why it exceeds face value. (Answer: $\$1{,}111.32$; the coupon stream and redemption premium together produce a price above face value.)
A bond has face value $\$1{,}200$, annual coupons of $4\%$ of face value for three years, and a $5\%$ effective annual yield. Find its price if redemption is $\$1{,}050$. Then find how much its price would increase if redemption were instead $\$1{,}120$, with all else unchanged. (Answer: $\$1{,}037.75$; increase $=\$60.47$)

What You'll Learn