Coupon Bonds as Annuities Plus Face Value

Why Separate a Bond’s Cash Flows?

A level-coupon bond promises a repeated coupon payment and a face-value payment when it matures. These are two distinct cash-flow streams: the coupons form an annuity, while the face value is a single future payment. Valuing them separately makes the timing of each payment explicit and helps prevent a common error: discounting the maturity amount for the wrong number of periods.

The bond’s price is the present value of its promised future payments, given the required yield and the assumptions used for valuation. The method in this tutorial uses annual coupon payments, with the annual effective required yield applying to each year. It does not assume that a bond’s market price is necessarily its face value; the price depends on the size and timing of its promised cash flows and the required yield.

Definition: For a bond with face value $F$, level annual coupon payment $C$, and $n$ years remaining, the coupon stream consists of $C$ paid at the end of each year $1,2,\ldots,n$. The face-value payment is $F$, paid at the end of year $n$. Under an annual effective required yield $i$, with $v=1/(1+i)$, the bond price is the present value of the coupon annuity plus the present value of the face-value payment.

The Coupon Annuity and the Maturity Payment

The present-value annuity factor $a_{\overline{n}|}$ represents the present value of one dollar paid at the end of each of $n$ years, discounted at annual effective rate $i$. The coupons are level, so their present value is $C a_{\overline{n}|}$. The face value is paid only once, at the end of year $n$, so its present value is $Fv^n$.

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

Here, $P$ is the bond’s value at the valuation date, $C$ is the annual coupon payment, $F$ is the face value due at maturity, $n$ is the number of annual periods remaining, and $i$ is the annual effective required yield. The same discount rate applies to both components. The annuity factor includes coupons at the ends of years 1 through $n$; the face-value term places the principal payment at the end of year $n$.

This expression is appropriate when the bond has level annual coupons, the face amount is repaid in one payment at maturity, the next coupon is one full period away, and the yield used for discounting is an effective annual yield. If coupons occur more frequently than annually, or if the cash-flow timing is irregular, the periods and yield must be matched to those cash flows rather than applying this annual formula unchanged.

Why the Two-Part Formula Holds

Present value is additive: the present value of a combined set of cash flows equals the sum of the present values of its components, provided each cash flow is discounted to the same valuation date at the same yield. A coupon bond’s cash flows can be partitioned into its coupon payments and its face-value payment. The coupon payments are equal amounts at equally spaced year-ends, so their present value is the annuity value. The single face-value payment occurs at maturity, so its present value is the discounted lump sum.

Separate the promised payments.
List the $n$ annual coupons of $C$ and the single maturity payment of $F$. The final year includes both a coupon and the face value.

Value the coupons together.
Because the coupon amount is level and paid at each year-end, the coupon stream has present value $C a_{\overline{n}|}$.

Value the face amount.
Discount $F$ from year $n$ to the valuation date, giving $Fv^n$.

Add the component values.
The bond price is $P=C a_{\overline{n}|}+Fv^n$.

The formula can also be justified directly by writing the discounted cash flows. The coupon at the end of year $k$ has present value $Cv^k$, for $k=1,\ldots,n$, and the maturity payment has present value $Fv^n$. Thus

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

The second equality factors the constant coupon $C$ out of the sum. The final equality uses the definition of the annuity factor as the sum of the discount factors for year-end payments. Notice that the year-$n$ coupon appears in the annuity sum, while the year-$n$ face value appears separately. Both are due at maturity, but they are different promised amounts.

Worked Example: A Two-Year Bond with a 10% Required Yield

A hypothetical bond has face value $F=\$1{,}000$, pays an annual coupon of $C=\$100$, and matures in $n=2$ years. The annual effective required yield is $i=10\%$. Find its price by valuing the coupon annuity and face value separately. Assume coupons are paid at each year-end, including the maturity date, and the face value is paid at the end of year 2.

Method and assumptions. Use the annual coupon-bond formula. The yield is an effective annual rate, so $v=1/(1+i)$. The two coupon payments occur at the ends of years 1 and 2.

Step 1: Find the discount factors.

v=\frac{1}{1.10}=0.9090909,\qquad v^2=\frac{1}{(1.10)^2}=0.8264463

Step 2: Value the coupon stream. The two-period annuity factor is $a_{\overline{2}|}=v+v^2=0.9090909+0.8264463=1.7355372$. Therefore, the present value of the coupons is $C a_{\overline{2}|}=\$100(1.7355372)=\$173.5537$.

Step 3: Value the face amount. The $\$1{,}000$ is paid at the end of year 2, so its present value is $Fv^2=\$1{,}000(0.8264463)=\$826.4463$.

Step 4: Add and check. The price is $\$173.5537+\$826.4463=\$1{,}000.0000$, or $\$1{,}000.00$ to the nearest cent. Directly discounting all payments gives $\$100v+\$100v^2+\$1{,}000v^2=\$90.9091+\$82.6446+\$826.4463=\$1{,}000.0000$, which checks the split calculation. This method avoids treating the final coupon and the face value as a single payment when discounting the whole coupon stream.

Worked Valuations

Worked Example: Three Annual Coupons and a Separate Maturity Payment

A hypothetical bond has face value $\$1{,}000$, pays annual coupons of $\$60$, and has three years remaining. The annual effective required yield is $5\%$. Coupons are paid at the ends of years 1, 2, and 3, and the face value is repaid at the end of year 3. Calculate the price to the nearest cent.

Method and assumptions. Treat the coupons as a three-payment annuity and discount the face value for three full years at $i=0.05$. Thus $v=1/1.05$.

Step 1: Calculate the discount factors.

v=0.9523810,\qquad v^2=0.9070295,\qquad v^3=0.8638376

Step 2: Find the annuity factor and coupon value. The annuity factor is $a_{\overline{3}|}=v+v^2+v^3=0.9523810+0.9070295+0.8638376=2.7232481$. The coupon present value is $\$60(2.7232481)=\$163.3949$.

Step 3: Discount the face value. The face value is paid at the end of the third year, so its present value is $\$1{,}000(0.8638376)=\$863.8376$.

Step 4: Add the components. The price is $\$163.3949+\$863.8376=\$1{,}027.2325$, or $\$1{,}027.23$ to the nearest cent. The direct cash-flow check is $\$60v+\$60v^2+\$60v^3+\$1{,}000v^3=\$57.1429+\$54.4218+\$51.8303+\$863.8376=\$1{,}027.2326$. The small difference in the last decimal comes from rounding the displayed factors; the unrounded calculation gives the same price to the nearest cent.

Worked Example: A Longer Bond with a Lower Coupon Than Yield

A hypothetical bond has face value $\$5{,}000$, pays annual coupons equal to $6\%$ of face value, and matures in four years. The annual effective required yield is $8\%$. Find its price to the nearest cent. Assume each coupon is paid at year-end and the face value is paid with the fourth coupon.

Method and assumptions. First convert the stated coupon rate into the annual coupon amount: $C=0.06(\$5{,}000)=\$300$. Then value the four-coupon annuity and the maturity payment separately at $i=0.08$.

Step 1: Calculate the discount factors.

v=\frac{1}{1.08}=0.9259259,\quad v^2=0.8573388,\quad v^3=0.7938322,\quad v^4=0.7350299

Step 2: Value the coupons. The annuity factor is $a_{\overline{4}|}=0.9259259+0.8573388+0.7938322+0.7350299=3.3121268$. Thus the coupon present value is $\$300(3.3121268)=\$993.6380$.

Step 3: Value the face amount. The face value is due after four years, so its present value is $\$5{,}000(0.7350299)=\$3{,}675.1495$.

Step 4: Add and check. The price is $\$993.6380+\$3{,}675.1495=\$4{,}668.7875$, or $\$4{,}668.79$ to the nearest cent. As a direct check, the fourth-year cash flow is $\$300+\$5{,}000=\$5{,}300$, whose present value is $\$5{,}300(0.7350299)=\$3{,}895.6585$. Adding the first three coupon present values, $\$277.7778+\$257.2016+\$238.1497$, gives $\$4{,}668.7876$; the one-ten-thousandth difference is rounding. This method avoids discounting the face value as if it were an annuity payment.

Timing Traps and Practical Checks

The main timing question is: when is each payment made? Under the annual end-of-year assumption, the first coupon is one year away and the maturity payment is $n$ years away. The final coupon and face value share a payment date, but the coupon belongs to the annuity and the face value belongs to the lump-sum term.

For example, in the three-year bond above, a common incorrect calculation discounts the face value for only two years:

\text{Incorrect: } \$60a_{\overline{3}|}+\$1{,}000v^2 =\$163.3949+\$907.0295 =\$1{,}070.4244

The correction is to discount the face amount for all three years:

\text{Correct: } \$60a_{\overline{3}|}+\$1{,}000v^3 =\$163.3949+\$863.8376 =\$1{,}027.2325

The incorrect calculation advances the maturity payment by one year and therefore overstates its present value. Another error is to count the final coupon twice: once in $C a_{\overline{n}|}$, and again by including $C+F$ as the last cash flow while still adding the full annuity. When checking by listing payments, use either the split formula or a direct list in which the last payment is $C+F$, not both methods combined.

A useful check is to write down the payment dates before substituting numbers. Confirm that there are $n$ coupon payments, that the annuity includes the year-$n$ coupon, and that the face value is discounted by $v^n$. Keep full precision in intermediate calculations and round the final dollar price to the nearest cent. If the bond does not have annual year-end coupons, this particular annual setup needs adjustment to reflect the actual payment schedule and matching yield.

Key takeaway: Under the stated annual-payment assumptions, value the bond by adding the present value of its level coupon annuity and the present value of its face value at maturity: $P=C a_{\overline{n}|}+Fv^n$. The coupon annuity includes the final coupon; the separate maturity term is the face value only.

Check Your Understanding

For each hypothetical bond, use annual end-of-year coupons and an annual effective required yield. Show the coupon-annuity value and face-value value separately before adding.

A bond has face value $\$2{,}000$, an annual coupon rate of $5\%$, two years remaining, and a required yield of $4\%$. What is its price to the nearest cent? (Answer: $\$2{,}037.72$)
A bond has face value $\$5{,}000$, an annual coupon rate of $6\%$, three years remaining, and a required yield of $5\%$. What is its price to the nearest cent? (Answer: $\$5{,}136.16$)
A bond pays an annual coupon of $\$40$, has face value $\$1{,}000$, three years remaining, and a required yield of $6\%$. What is its price to the nearest cent? (Answer: $\$946.54$)
A bond has face value $\$2{,}500$, an annual coupon rate of $4\%$, five years remaining, and a required yield of $6\%$. Calculate its coupon present value, face-value present value, and total price to the nearest cent. (Answer: coupon present value $\$421.24$; face-value present value $\$1{,}868.15$; price $\$2{,}289.38$)

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