Writing Coupon-Bond Prices as Discount-Factor Equations

Why a Bond Price Is a Dated Cash-Flow Equation

A coupon bond is a contract for payments on specified dates. Its price depends not only on the amounts promised, but also on when each amount is paid. When interest rates vary by term, a payment due in one year and a payment due in four years need not have the same discount factor. The price equation must therefore pair each cash flow with the discount factor for its own date.

This tutorial considers a bond valued at time 0, immediately after any coupon due at that time has been paid. Assume the bond’s promised payments are certain, are in the same currency as the price, and have no embedded options. The discount factors are market values at time 0 for claims to one unit of currency on the corresponding future dates. In a practical valuation, the cash-flow dates and the discount factors must use the same time and settlement conventions.

Definition: Let $D(t)$ be the time-0 price of a claim that pays one unit of currency at time $t$. Thus $D(0)=1$. If a security pays known amounts $C_1,C_2,\ldots,C_n$ at dates $t_1,t_2,\ldots,t_n$, its value under the stated assumptions is $$ P_0=\sum_{k=1}^{n} C_kD(t_k). $$ Each term is the present value of one dated payment.

The discount factor is determined by both the valuation date and the payment date. It is not a coupon rate, and it is not automatically a single discount factor used for every payment. If the term structure is flat at an effective annual rate $i$, then $D(t)=(1+i)^{-t}$ for annual time units. With a non-flat term structure, use the appropriate $D(t)$ for each payment date instead.

Writing the Coupon-Bond Equation

Let a bond have face amount $F$, an annual coupon rate $c$, annual coupon payments, and maturity at the end of year $n$. Suppose the redemption amount at maturity is $R$. The bond pays $cF$ at the end of years $1$ through $n$, and pays the redemption amount $R$ at the end of year $n$. The final date therefore has two components: the last coupon and redemption.

P_0=\sum_{k=1}^{n-1} cF D(k)+(cF+R)D(n)

If redemption is at face amount, then $R=F$, so the final cash flow is $cF+F$. This expression is a cash-flow equation, not a formula for a yield rate. It values each dated payment separately. For any coupon schedule, including one that is not level, the general sum $P_0=\sum C_kD(t_k)$ is the useful starting point: list the actual amount and date of each payment before substituting discount factors.

Set the valuation date.
Measure every payment date from the same time-0 date used to define the discount factors.

List the contractual cash flows.
Write each coupon on its payment date and add redemption to the final payment. Do not discount the coupon and redemption as if they were separate dates when they are paid together.

Match dates and units.
Multiply each cash amount by its date-specific discount factor. Confirm whether the resulting price is for the stated face amount or for a standard quoted unit such as 100 of face amount.

Why the Sum of Discounted Payments Is the Price

The equation follows from valuing the promised payments as a portfolio of zero-coupon claims. A claim paying one dollar at date $t_k$ has price $D(t_k)$. A claim paying $C_k$ dollars on that date is $C_k$ one-dollar claims, so its value is $C_kD(t_k)$. The bond’s payment stream is the sum of those dated claims.

\text{Value of payment }C_k\text{ at }t_k=C_kD(t_k), \qquad P_0=\sum_{k=1}^{n}C_kD(t_k).

Under the assumed consistent, arbitrage-free valuation, the price of the combined stream is the sum of the prices of its components. If the bond had a different price from an otherwise identical portfolio of claims making exactly the same payments on exactly the same dates, the cheaper side could be bought and the more expensive side sold. That would create an arbitrage under the assumptions. This reasoning depends on the cash flows matching: different payment dates, currencies, or conditions on payment do not form the same stream.

The equation uses certain cash flows and discount factors appropriate to those claims. It does not, by itself, handle uncertain default payments or an embedded option that can change the bond’s cash flows. Those features require a valuation model that reflects the uncertainty or option.

Worked Example: Valuing a Three-Year Bond in Dollars

A hypothetical bond has face amount $F=\$1{,}000$, an annual coupon rate of $6\%$, and redemption at face amount after three years. Its annual discount factors are $D(1)=0.96$, $D(2)=0.91$, and $D(3)=0.85$. Find its time-0 value.

Method. List the promised cash flows first, then discount each one using the factor for its payment year. The discount factors are positive and decrease with term, as assumed for this example.

The annual coupon is $$ cF=0.06(\$1{,}000)=\$60. $$ Thus the bond pays $\$60$ at years 1 and 2, and $\$60+\$1{,}000=\$1{,}060$ at year 3.

The present values are $$ \$60(0.96)=\$57.60,\qquad \$60(0.91)=\$54.60,\qquad \$1{,}060(0.85)=\$901.00. $$ Adding them gives $$ P_0=\$57.60+\$54.60+\$901.00=\$1{,}013.20. $$

Check. The three discounted cash flows sum to $\$1{,}013.20$. The final term includes both the last coupon and the face amount; omitting either would value a different payment stream. No intermediate amount was rounded.

Keep Payment Dates and Price Units Aligned

Two independent checks prevent many errors. First, every cash flow must be attached to its contractual payment date, not merely to the nearest supplied discount factor. Second, the cash-flow amounts and the reported price must be expressed on the same face-amount basis. A price per 100 of face amount is not the total dollar price of a large holding.

Worked Example: A Per-100 Price With Four Annual Payments

A hypothetical bond has face amount 100, pays an annual coupon rate of $5\%$, and redeems at 100 at the end of year 4. The discount factors are $D(1)=0.975$, $D(2)=0.940$, $D(3)=0.900$, and $D(4)=0.840$. Find its price per 100 of face amount.

Method. Work first in the quoted unit, which is 100 of face amount. The annual coupon is $0.05(100)=5$, so the cash flows at years 1, 2, and 3 are each 5. At year 4 the cash flow is $5+100=105$.

Discount each flow at its own date: $$ 5(0.975)=4.875,\qquad 5(0.940)=4.700,\qquad 5(0.900)=4.500,\qquad 105(0.840)=88.200. $$ Therefore, $$ P_0=4.875+4.700+4.500+88.200=102.275. $$ Rounded to the nearest cent per 100 of face amount, the price is $102.28$.

Check. The last discounted amount is $105(0.840)$, not $100(0.840)$, because the final coupon and redemption are both payable at year 4. The unrounded price $102.275$ rounds to $102.28$.

Worked Example: Rejecting a Discount Factor for the Wrong Date

A hypothetical bond has face amount $\$250$, pays a $4\%$ annual coupon, and redeems at face amount after two years. It pays coupons at the ends of years 1 and 2. The supplied factors are $D(0.5)=0.985$, $D(1)=0.960$, $D(1.5)=0.930$, and $D(2)=0.900$. Find the time-0 value.

Method. Match the actual annual payment dates to the corresponding factors. The half-year factors are not relevant because the bond makes no payment at half-year dates.

The annual coupon is $0.04(\$250)=\$10$. The cash flows are $\$10$ at year 1 and $\$260$ at year 2. Their present values are $$ \$10(0.960)=\$9.60,\qquad \$260(0.900)=\$234.00. $$ Thus $$ P_0=\$9.60+\$234.00=\$243.60. $$

Common mistake and correction. Using the half-year factors just because they are listed would give $\$10(0.985)+\$260(0.930)=\$9.85+\$241.80=\$251.65$. That calculation assigns the year-1 payment to $D(0.5)$ and the year-2 payment to $D(1.5)$, shifting both dates earlier. The correct dates are years 1 and 2, so the correct value is $\$243.60$.

Check. The two cash flows occur at the two stated annual coupon dates, and the year-2 cash flow includes both the last coupon and redemption.

Worked Example: Converting a Per-100 Price to a Holding Value

A hypothetical investor holds $\$25{,}000$ face amount of a bond with a $3.2\%$ annual coupon and three years remaining. It redeems at face amount. The discount factors are $D(1)=0.970$, $D(2)=0.920$, and $D(3)=0.860$. Find the price per 100 of face amount and the total value of the holding.

Method. Calculate the bond price per 100 first, then multiply by the number of 100-unit face blocks held. The coupon per 100 is $0.032(100)=3.20$. Cash flows per 100 are 3.20 at years 1 and 2, and $3.20+100=103.20$ at year 3.

The discounted amounts per 100 are $$ 3.20(0.970)=3.104,\qquad 3.20(0.920)=2.944,\qquad 103.20(0.860)=88.752. $$ So the unrounded price per 100 is $$ 3.104+2.944+88.752=94.800. $$ The holding contains $\$25{,}000/100=250$ blocks of 100 face amount. Its total value is therefore $$ 250(94.800)=\$23{,}700.00. $$

Check. The total holding value also equals the unrounded per-100 value multiplied by 250. The result is reported in dollars to the cent. Multiplying the per-100 quote by the face amount directly would overstate the value by a factor of 100.

Common Exam Traps

Discounting by maturity instead of by payment date. A coupon due before maturity is not discounted using the maturity factor. Use the factor for the coupon’s own date.
Leaving out the final coupon or redemption. The maturity cash flow normally combines the last coupon and the redemption amount.
Using a single yield factor for every payment. A single effective annual rate can generate discount factors when the model assumes a flat rate. With given term-specific discount factors, use each factor as supplied for its matching date.
Mixing quote units and total-value units. A price of 94.80 per 100 face amount must be scaled by the number of 100-unit blocks, not by the unscaled face amount.
Assuming dates that the contract does not specify. The cash-flow schedule comes from the bond terms. A listed discount factor is not evidence that a payment occurs on that date.

Key takeaway: Write the bond’s payment amounts and dates before calculating. The price is the sum of each amount multiplied by the discount factor for that exact date, with all amounts expressed on a consistent face-value basis.

Check Your Understanding

For each problem, write the dated cash flows before calculating or identifying the price equation.

A hypothetical bond has face amount $\$500$, an annual coupon rate of $5\%$, and redemption at face amount after two years. Given $D(1)=0.96$ and $D(2)=0.90$, find its time-0 value. (Answer: $\$496.50$)
A bond is quoted per 100 face amount. It pays an annual coupon rate of $6\%$, redeems at 100 after three years, and has $D(1)=0.98$, $D(2)=0.94$, and $D(3)=0.88$. Find its price per 100. (Answer: $104.80$)
A hypothetical holding has $\$20{,}000$ face amount of a bond paying a $3\%$ annual coupon, redeemable at face after two years. Given $D(1)=0.95$ and $D(2)=0.89$, find the total holding value. (Answer: $\$18{,}904.00$)
A bond with face amount $\$1{,}000$ pays $\$40$ annually for three years and redeems for $\$1{,}020$ at the end of year 3. Given $D(1)=0.97$, $D(2)=0.92$, and $D(3)=0.86$, write and evaluate its price. (Answer: $\$987.20$)
A bond quoted per 100 face amount has a $7\%$ annual coupon, pays annually for four years, and redeems at 98. Given $D(1)=0.96$, $D(2)=0.93$, $D(3)=0.89$, and $D(4)=0.84$, find its price per 100. (Answer: $107.66$)

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