Bond Pricing by Discounting Each Payment Separately

Why Discount Each Bond Payment Separately?

A coupon bond promises a sequence of cash flows: coupon payments during its term and, at maturity, the final coupon together with repayment of principal. Its price at a valuation date is the present value of those promised payments. Calculating each present value separately makes the timing and amount of every cash flow visible, which is useful for both checking a price and spotting a mismatched yield.

The central requirement is to match the discount rate to the time between payment dates. For an annual-coupon bond, use an effective annual yield to discount each year’s payment. For a semiannual-coupon bond, use a yield effective for each half-year. If a quoted yield uses a different compounding convention from the bond’s payment schedule, first convert it to a rate appropriate to those payment dates.

Definition: The payment-by-payment price of a bond is the sum of the present values, at the valuation date, of its promised coupon and principal payments. Each payment is discounted for the number of payment periods between the valuation date and its payment date, using the required yield per period.

For a bond valued just after a coupon date, let $F$ be its face value, $C$ its coupon per period, $n$ its number of remaining payment periods, and $r$ the required effective yield per payment period. The cash flows are $C$ at the end of each of periods 1 through $n-1$, followed by $C+F$ at the end of period $n$. The price $P$ is

P=\frac{C}{1+r}+\frac{C}{(1+r)^2}+\cdots+\frac{C}{(1+r)^{n-1}}+\frac{C+F}{(1+r)^n}.

The final term includes both the last coupon and the repayment of face value. It is not a separate extra coupon: the bond pays one final coupon, in addition to principal.

Set Up the Cash Flows and Discount Factors

Before calculating, establish the valuation date, payment schedule, remaining number of periods, and required yield for one such period. In the examples below, the bond is valued immediately after a coupon date, so there is no intervening accrued coupon payment to include. The price is calculated from promised cash flows; no default, tax, transaction-cost, or other adjustment is being modeled.

For annual payments discounted at an effective annual rate $i$, the discount factor for year $k$ is $v^k$, where $v=1/(1+i)$. For semiannual payments, if the quoted annual nominal yield is $i^{(2)}$, convertible semiannually, the yield per half-year is $i^{(2)}/2$; the period-$k$ discount factor is $(1+i^{(2)}/2)^{-k}$. Use the period yield with the matching payment period, not the annual quote as though it were already a rate per half-year.

List the remaining payment dates.
Count the periods from the valuation date to each coupon and to maturity.

Write each cash flow.
Use the coupon alone before maturity and coupon plus face value at maturity.

Match the yield to the period.
Convert a quoted yield if its compounding period differs from the bond’s payment period.

Discount and add.
Divide each payment by one plus the period yield raised to the number of periods until payment.

Check the result.
Compare with the coupon-annuity-plus-principal calculation when the payments are level and periodic.

Worked Examples

Worked Example: Annual Coupons and an Annual Required Yield

A hypothetical bond has face value $\$1{,}000$, an annual coupon rate of 5%, and three years remaining. It pays coupons annually, and the required yield is an effective annual rate of 4%. Find its price immediately after a coupon date.

Method. Calculate the annual coupon, list each dated payment, and discount each payment at 4% per year. The annual coupon is

C=0.05(1{,}000)=\$50.

The payments are $\$50$ at the end of year 1, $\$50$ at the end of year 2, and $\$1{,}050$ at the end of year 3. The last amount is the final $\$50$ coupon plus $\$1{,}000$ principal. Discount each one:

\begin{aligned} PV_1&=\frac{50}{1.04}=\$48.0769,\\ PV_2&=\frac{50}{1.04^2}=\frac{50}{1.0816}=\$46.2278,\\ PV_3&=\frac{1{,}050}{1.04^3}=\frac{1{,}050}{1.124864}=\$933.4462. \end{aligned}

Add the unrounded present values: $48.0769+46.2278+933.4462=\$1{,}027.7509$. Rounded to the nearest cent, the price is $\$1{,}027.75$.

Check and mistake avoided. The annuity check is $50a_{\overline{3}|}+1{,}000v^3$, using $i=0.04$. Since $a_{\overline{3}|}=v+v^2+v^3$, this gives the same sum of three discounted coupons plus discounted principal. A common mistake is to discount $\$1{,}000$ at maturity but omit the final $\$50$ coupon; the year-3 payment is $\$1{,}050$, not $\$1{,}000$.

Worked Example: Semiannual Coupons and a Semiannual Yield

A hypothetical bond has face value $\$4{,}000$, an annual coupon rate of 6%, and two years remaining. Coupons are paid semiannually. The required yield is a nominal annual rate of 4% convertible semiannually. Find the price immediately after a coupon date.

Method. Convert the annual coupon rate into a coupon per half-year, then discount the four payments at the yield per half-year. There are $2(2)=4$ half-year periods. The coupon each period is

C=\frac{0.06(4{,}000)}{2}=\$120.

The quoted yield is $i^{(2)}=0.04$, so the half-year yield is $r=0.04/2=0.02$. The payments are $\$120$ at periods 1, 2, and 3, and $\$4{,}120$ at period 4. Therefore,

\begin{aligned} P&=\frac{120}{1.02}+\frac{120}{1.02^2}+\frac{120}{1.02^3}+\frac{4{,}120}{1.02^4}\\ &=117.6471+115.3403+113.0787+3{,}806.2432\\ &=\$4{,}152.3093. \end{aligned}

Rounded to the nearest cent, the price is $\$4{,}152.31$. As a check, the three coupons have present value $120a_{\overline{4}|}$ at a 2% period rate, while principal has present value $4{,}000/(1.02)^4$. Their sum equals the payment-by-payment result.

Common mistake avoided. Using 4% as the half-year rate would discount each cash flow too heavily: 4% is the nominal annual quote, not the yield per half-year. The rate for each of the four periods is $2\%$.

Worked Example: Annual Payments with a Semiannual-Convertible Yield Quote

A hypothetical bond has face value $\$1{,}000$, pays annual coupons of $\$60$, and has three years remaining. The required yield is quoted as a nominal annual rate of 8% convertible semiannually. Since the bond pays annually, find the equivalent effective annual yield and price.

Method. Convert the quoted yield to its equivalent effective annual rate, then discount the annual payments at that annual rate. A half-year accumulation factor at the quoted yield is $1+0.08/2=1.04$. Over one year, there are two such periods, so the effective annual yield is

i=(1.04)^2-1=0.0816=8.16\%.

The annual payments are $\$60$, $\$60$, and $\$1{,}060$. Their present values at 8.16% are

\begin{aligned} PV_1&=\frac{60}{1.0816}=\$55.4734,\\ PV_2&=\frac{60}{1.0816^2}=\frac{60}{1.16985856}=\$51.2883,\\ PV_3&=\frac{1{,}060}{1.0816^3}=\frac{1{,}060}{1.265319018496}=\$837.7334. \end{aligned}

Adding the unrounded values gives $P=\$944.4950$, so the price rounded to the nearest cent is $\$944.50$.

Check and mistake avoided. The coupon-annuity check is $60a_{\overline{3}|}+1{,}000v^3$, with $i=0.0816$, and it produces the same price. Using 8% directly as the annual effective rate would instead calculate $60/1.08+60/1.08^2+1{,}060/1.08^3$, approximately $\$948.46$. That is the wrong discount basis: 8% is nominal convertible semiannually, while these payments are annual. The equivalent effective annual yield is 8.16%.

Why the Annuity Check Works

For level coupons $C$ paid at the end of each period, the present value of the coupons is the sum of $C$ times the discount factor for each payment date. The principal $F$ is paid only at the final date, so its present value is $F$ times the final discount factor. This directly justifies the bond-price decomposition.

\begin{aligned} P &=Cv+Cv^2+\cdots+Cv^n+Fv^n\\ &=C\left(v+v^2+\cdots+v^n\right)+Fv^n\\ &=Ca_{\overline{n}|}+Fv^n, \qquad v=\frac{1}{1+r}. \end{aligned}

Here $a_{\overline{n}|}$ is the present value of one unit paid at the end of each of $n$ periods at effective period rate $r$. The equality follows because the annuity factor is precisely the sum of the individual period discount factors. This provides a useful arithmetic check; it does not remove the need to use the correct yield period or include every promised payment.

Interpretation and Common Errors

Discounting separately is especially transparent when checking a cash-flow timeline: each payment has a stated amount, a stated date, and a corresponding discount exponent. The method also makes clear why the final payment differs from earlier payments: it contains principal as well as the final coupon. The annuity form is more compact when coupons are level, but both methods value the same dated cash flows under the same yield assumption.

Using the wrong period rate. A nominal annual yield convertible semiannually is not automatically the annual effective yield, nor is the annual quote itself the half-year rate.
Using the wrong exponent. A payment at the end of period $k$ is discounted by $(1+r)^k$, not by one period regardless of its date.
Leaving out the final coupon. The maturity cash flow is face value plus the coupon due at maturity.
Mixing annual and semiannual counts. A two-year bond with semiannual coupons has four remaining payment periods, not two.
Rounding too early. Keep full precision through the discounted cash flows and round the total price only at the end.

Key takeaway: A coupon bond’s price is the sum of the present values of all its promised payments. Write the cash flows and dates first, match the yield to the payment period, discount each payment by its own period count, and use the annuity-plus-principal expression as a check.

Check Your Understanding

For each problem, assume valuation immediately after a coupon date, use the stated required yield, and round the price to the nearest cent.

A hypothetical bond has face value $\$2{,}000$, an annual coupon rate of 5%, and two years remaining. It pays annually, and the required yield is an effective annual rate of 4%. Find its price. (Answer: $\$2{,}037.72$)
A hypothetical bond has face value $\$1{,}000$, an annual coupon rate of 8%, and 1.5 years remaining. It pays semiannually, and the required yield is a nominal annual rate of 6% convertible semiannually. Find its price. (Answer: $\$1{,}028.29$)
A hypothetical bond has face value $\$3{,}000$, pays annual coupons at a 7% rate, and has two years remaining. The required yield is an effective annual rate of 8%. Find its price by discounting each payment separately. (Answer: $\$2{,}946.50$)
A hypothetical bond has face value $\$1{,}000$, pays annual coupons at a 4% rate, and has two years remaining. Its required yield is an effective annual rate of 5%. Write the payment-by-payment price and verify it using the annuity-plus-principal form. (Answer: $\$40/1.05+\$1{,}040/1.05^2=\$981.41$; equivalently, $40a_{\overline{2}|}+1{,}000v^2=\$981.41$)

What You'll Learn