Deriving the Coupon-Bond Present-Value Formula

From a Bond Timeline to a Present-Value Formula

A coupon bond is valued by matching each promised payment to its date and discounting it to the valuation date. The calculation has two components: the present value of the coupon stream and the present value of the redemption amount. The coupon payments form a level annuity-immediate when they are equal and are paid at the end of each period.

The formula developed here applies at a coupon date, with a fixed yield for each coupon period. It does not address valuation between coupon dates, where the time from the valuation date to the next payment must also be accounted for.

Model and assumptions: The valuation date is immediately before a coupon-period payment sequence begins. There are $n$ equal periods, each coupon $C$ is paid at the end of its period, and redemption value $R$ is paid with the final coupon at time $n$. The effective yield per coupon period is $j$, constant over all periods. Payments are treated as certain, and taxes, transaction costs, and default are ignored.

Write $v_j=(1+j)^{-1}$ for the discount factor for one coupon period. The symbol $j$ here means the effective yield per coupon period; it need not be an annual effective rate if the coupon period is not one year. The present value $P$ is the value at time 0 of all the bond’s promised payments.

Definition: Under the stated assumptions, a level-coupon bond pays $C$ at times $1,2,\ldots,n$, plus redemption value $R$ at time $n$. Its present value is the sum of the present values of those $n+1$ payment components, with the final coupon and redemption paid together.

Deriving the Coupon-Bond Formula

A payment $K$ made at the end of period $t$ has present value $K(1+j)^{-t}=Kv_j^t$. Therefore, the first coupon contributes $Cv_j$, the second contributes $Cv_j^2$, and the coupon at time $n$ contributes $Cv_j^n$. Redemption contributes a separate amount $Rv_j^n$, because it is also paid at time $n$.

P=Cv_j+C v_j^2+\cdots+C v_j^n+Rv_j^n

Factor $C$ from the coupon terms. The sum $v_j+v_j^2+\cdots+v_j^n$ is the present value of an $n$-payment annuity-immediate of 1 per period. Denote this sum by $a_{\overline{n}|j}$. Thus, the bond price can be written as the coupon-annuity value plus the discounted redemption value.

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

For $j\ne 0$, the annuity factor is a finite geometric series. Multiplying the sum by $1+j$ shifts each power down by one; subtracting the original sum leaves $1-v_j^n$. Since $1-v_j=j/(1+j)$, the same sum can be expressed using $j$ in the denominator:

\begin{aligned} a_{\overline{n}|j} &=v_j+v_j^2+\cdots+v_j^n\\ (1+j)a_{\overline{n}|j} &=1+v_j+\cdots+v_j^{n-1}\\ j a_{\overline{n}|j} &=1-v_j^n\\ a_{\overline{n}|j} &=\frac{1-v_j^n}{j} =\frac{1-(1+j)^{-n}}{j},\qquad j\ne 0. \end{aligned}

This division by $j$ is valid only when $j\ne 0$. If the yield is zero, no payment is discounted: each coupon is worth $C$ at time 0 and redemption is worth $R$. The annuity factor is then the sum of $n$ ones.

Zero-yield case: When $j=0$, define $a_{\overline{n}|0}=n$. The price is $P=nC+R$. Do not substitute $j=0$ into the quotient formula; its denominator would be zero.

Worked Examples

Worked Example: Pricing from the Individual Cash Flows

A hypothetical bond is valued immediately before its first of three annual coupon periods. It pays a coupon of $60 at the end of each year and redeems for $1,000 with the third coupon. The effective yield is 10% per year. Find the price, showing both the dated cash-flow calculation and the annuity form.

Method. Discount each payment by its number of annual periods. This makes the final combined payment explicit and avoids treating redemption as though it were paid earlier.

The first two cash flows are $60 at times 1 and 2. At time 3, the bond pays both the $60 coupon and the $1,000 redemption, for a total of $1,060. The discount factors are:

v_j=\frac{1}{1.10}=0.9090909091,\quad v_j^2=\frac{1}{1.10^2}=0.8264462810,\quad v_j^3=\frac{1}{1.10^3}=0.7513148009.

The present values are $60v_j=\$54.54545455$, $60v_j^2=\$49.58677686$, and $1{,}060v_j^3=\$796.3936890$. Therefore:

$$ P=54.54545455+49.58677686+796.3936890 =900.5259204. $$

As a check, the annuity factor is $a_{\overline{3}|j}=0.9090909091+0.8264462810+0.7513148009=2.486851991$. The coupon stream is worth $60(2.486851991)=\$149.2111195$, and redemption is worth $1{,}000(0.7513148009)=\$751.3148009$. These sum to $\$900.5259204$, agreeing with the cash-flow calculation. The price is $900.53 to the nearest cent.

Worked Example: Using the Formula with a Nonannual Coupon Period

A hypothetical bond pays $20 at the end of each of four coupon periods and $1,000 redemption with the fourth coupon. The effective yield is 25% per coupon period. Find the price.

Method. Use the yield stated for one coupon period directly. No annual-rate conversion is needed because the problem already gives the effective yield for each period.

Here $n=4$, $C=\$20$, $R=\$1{,}000$, and $j=0.25$. The successive discount factors are $0.8$, $0.64$, $0.512$, and $0.4096$, since each period’s factor is $1/1.25=0.8$. The coupon-annuity factor is:

a_{\overline{4}|j}=0.8+0.64+0.512+0.4096=2.3616.

The coupon stream is worth $20(2.3616)=\$47.232$. Redemption is worth $1{,}000(0.4096)=\$409.60$. Thus:

$$ P=47.232+409.60=456.832. $$

The direct cash-flow check gives $20(0.8)=\$16$, $20(0.64)=\$12.80$, $20(0.512)=\$10.24$, and $(20+1{,}000)(0.4096)=\$417.792$. Their sum is also $\$456.832$, or $456.83 to the nearest cent.

Worked Example: Finding the Coupon from a Quoted Price

A hypothetical two-period bond has redemption value $900. It is valued at $700, and the effective yield is 20% per period. Coupons are equal and paid at the end of each period, with the final coupon paid alongside redemption. Find the coupon amount.

Method. Separate the present value of redemption from the present value of the coupon stream, then solve the resulting equation for $C$. This keeps the final coupon in the annuity and avoids counting it twice.

The discount factors are $v_j=1/1.20=0.8333333333$ and $v_j^2=1/1.20^2=0.6944444444$. Redemption has present value $900(0.6944444444)=\$625$. The annuity factor is:

a_{\overline{2}|j}=0.8333333333+0.6944444444 =1.5277777777\ldots=\frac{55}{36}.

The coupon stream must therefore have present value $700-625=\$75$. Solve $C(55/36)=75$:

C=75\left(\frac{36}{55}\right) =\frac{2700}{55} =49.090909\ldots

Check the cash flows using the unrounded coupon: $C/1.20=\$40.9090909$, and $(C+900)/1.20^2=\$659.0909091$. The present values add to exactly $700. The coupon is $49.09 to the nearest cent.

What the Formula Does—and Does Not—Assume

The annuity factor works because each coupon is the same amount and each is discounted by the same per-period yield. Redemption is not part of the annuity: it is a separate cash flow at time $n$. If the coupon amounts differ, the coupon stream must instead be valued by discounting each amount at its own payment time; a single $C a_{\overline{n}|j}$ term is not valid unless all coupons equal $C$.

The zero-yield case is also a useful test of the underlying cash-flow logic. If $j=0$, every discount factor is 1, so the total must equal all payments added without discounting. That gives $nC+R$, not a formula obtained by dividing by zero. For nonzero $j$, the quotient expression and the sum of discount factors are equivalent.

Common Exam Traps

Double-counting the final coupon. The term $Ca_{\overline{n}|j}$ already includes the coupon at time $n$. Adding $(R+C)v_j^n$ to it counts that coupon twice. The correct expression is $Ca_{\overline{n}|j}+Rv_j^n$.
Discounting redemption for the wrong number of periods. Redemption occurs at time $n$, so its present value is $R(1+j)^{-n}$, not $R(1+j)^{-(n-1)}$.
Using an annual rate as though it were a coupon-period rate. The $j$ in this formula is the effective yield for one coupon period. The discount factor per period must correspond to that period.
Using the quotient at zero yield. The expression $(1-(1+j)^{-n})/j$ is for $j\ne0$. At $j=0$, use $a_{\overline{n}|0}=n$, giving $P=nC+R$.
Ignoring payment timing. An annuity-immediate has payments at the ends of periods. If the valuation date or payment dates differ from the assumptions here, the powers of the discount factor must reflect the actual times.

The central habit is to write the timeline first: identify each payment amount and its period, discount every payment to the valuation date, and only then compress equal coupon payments into an annuity factor. This provides a direct check on the formula and helps expose timing errors.

Check Your Understanding

For each problem, assume certain cash flows and a valuation date immediately before the first coupon period unless the problem states otherwise. Round dollar answers to the nearest cent.

A bond pays $40 at the end of each of three annual periods and redeems for $1,000 with the last coupon. The effective yield is 5% per year. Find its price. (Answer: $972.77)
A two-period bond pays a coupon of $30 each period and redeems for $500 with the second coupon. The effective yield is 10% per period. Find its price using the dated cash flows. (Answer: $465.29)
A five-period bond pays $18 at the end of every period and redeems for $900 with the final coupon. If the yield is zero, find its price and state why the quotient annuity formula is not used. (Answer: $990.00; the quotient has zero in its denominator)
A three-period bond pays level coupons and redeems for $800 with its third coupon. The effective yield is 25% per period. If each coupon is $12, find the price using the annuity factor and check the final combined payment. (Answer: $433.02)
A two-period bond is priced at $600 and redeems for $500 with its second coupon. Its effective yield is 25% per period. Find the level coupon $C$. (Answer: $194.44)

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