Constructing the Yield Equation from a Bond Price

From a Bond Price to a Yield Equation

A bond yield equation is the present-value equation that equates a bond’s known price with the discounted value of its future payments. The essential task is to put the right cash flow at the right time and discount it using the yield for the correct payment period. The equation may be ready to set up even when its unknown yield still needs a numerical method to solve.

Assume the bond’s stated price is on a coupon date, so there is no partial coupon period to account for. The bond pays its scheduled coupons at the ends of regular periods and pays its stated redemption amount with the final coupon. We use $j$ for the unknown effective yield per coupon period. If there are $m$ coupon periods per year, the corresponding effective annual yield is $i=(1+j)^m-1$. For a nominal annual yield convertible $m$ times per year, the nominal rate is $m j$.

Definition: A bond’s yield equation equates its price at the valuation date to the sum of the present values of all remaining contractual cash flows, discounted at the unknown yield per coupon period. If the cash flow at time $k$ is $C_k$, then for $n$ regular periods the equation is $P=\sum_{k=1}^{n} C_k(1+j)^{-k}$.

Here $P$ is the bond price, $n$ is the number of remaining coupon periods, and $C_k$ includes whatever payment is due at the end of period $k$. In particular, $C_n$ includes both the final coupon and redemption amount. The equation describes the valuation model; it does not by itself assert that every possible cash-flow pattern has a unique yield.

Build the Equation from the Timeline

Start at the valuation date and list each remaining payment in chronological order. A level-coupon bond with coupon $c$ per period, $n$ periods remaining, and redemption amount $R$ has coupons at times $1,\ldots,n$; the last cash flow is $c+R$. Its yield equation can be written as a sum or, using the annuity-immediate factor from bond pricing, as:

P=\sum_{k=1}^{n}\frac{c}{(1+j)^k}+\frac{R}{(1+j)^n} =c\,a_{\overline{n}|\,j}+R(1+j)^{-n}.

The notation $a_{\overline{n}|\,j}$ means the present-value annuity factor for $n$ payments of 1, discounted at the per-period rate $j$. The coupon at time $n$ is included in the annuity term, and the redemption amount is included separately in the final term. Equivalently, combine them into a final cash flow $c+R$; do not include the final coupon twice.

Set the valuation date and count periods.
Use the number of coupon periods remaining from the valuation date to maturity, not the number of years unless coupons are annual.

Write the cash flows.
List each coupon, then add the redemption amount to the final coupon when forming the maturity cash flow.

Define the unknown rate.
State that $j$ is the yield per coupon period. Convert any annual yield convention only after identifying the coupon frequency.

Discount and equate to price.
Discount the time-$k$ cash flow by $(1+j)^k$, then set the total equal to the known price.

The equation follows directly from present-value valuation. A dollar due after one period is worth $(1+j)^{-1}$ dollars at the valuation date; a dollar due after $k$ periods is worth $(1+j)^{-k}$. Multiplying each promised payment by its discount factor and adding the results gives the bond’s model price. Setting that model price equal to the observed price produces the yield equation.

Worked Equations

Worked Example: Three Annual Payments

A hypothetical bond is valued on a coupon date. It has three years remaining, pays a $100 coupon annually, and redeems for $1,000 at maturity. Its price is $707.20. Construct the yield equation, where $j$ is the annual effective yield.

Method: list the cash flows by time. The payments at years 1 and 2 are each $100. At year 3, the bond pays the $100 coupon and $1,000 redemption, for a total of $1,100. Thus $n=3$, $c=100$, and $R=1{,}000$.

Discount each payment at the unknown annual rate $j$, and set the sum equal to the price:

707.20=\frac{100}{1+j}+\frac{100}{(1+j)^2}+\frac{1{,}100}{(1+j)^3}.

In annuity notation, the same equation is $707.20=100a_{\overline{3}|\,j}+1{,}000(1+j)^{-3}$. As a check on the construction, substitute $j=0.25$: the discount factors are $1/1.25=0.8$, $1/1.25^2=0.64$, and $1/1.25^3=0.512$. The present values are $100(0.8)=80$, $100(0.64)=64$, and $1{,}100(0.512)=563.20$. Their total is $80+64+563.20=707.20$, the stated price. This check verifies the equation and avoids treating the coupon rate as the yield.

Worked Example: Semiannual Coupons and a Nominal Yield Convention

A hypothetical $1,000 face-amount bond has four semiannual periods remaining and pays a 6% annual coupon rate, with coupons paid semiannually. Its redemption amount is $1,000, and its price is $929.08. Construct the equation for the unknown yield $j$ per half-year. State the corresponding nominal annual yield convertible semiannually.

Method: convert the coupon rate to a payment amount and count periods. The annual coupon is $0.06(1{,}000)=\$60$. Since it is paid twice per year, each coupon is $60/2=\$30$. There are four coupon periods, so the final cash flow is $30+1{,}000=\$1{,}030$.

929.08= \frac{30}{1+j} +\frac{30}{(1+j)^2} +\frac{30}{(1+j)^3} +\frac{1{,}030}{(1+j)^4}.

Equivalently, $929.08=30a_{\overline{4}|\,j}+1{,}000(1+j)^{-4}$. The nominal annual yield convertible semiannually corresponding to $j$ is $2j$; the equation’s unknown is still the half-year rate $j$, not the annual nominal rate.

Check the equation at a candidate rate. At $j=0.05$, the four discount factors, rounded here for the check, are $0.9523809524$, $0.9070294785$, $0.8638375985$, and $0.8227024748$. The first three cash flows have present values $30(0.9523809524)=28.5714286$, $30(0.9070294785)=27.2108844$, and $30(0.8638375985)=25.9151279$. The final present value is $1{,}030(0.8227024748)=847.3835490$. The total is approximately $929.0809899$, which rounds to $929.08. Thus the equation reproduces the quoted price to the nearest cent at $j=5\%$, corresponding to a 10% nominal annual yield convertible semiannually. The common mistake avoided here is using the full $60 annual coupon in each half-year period.

Worked Example: Redemption Differs from Face Amount

A hypothetical bond is valued on an annual coupon date. It pays a $80 coupon at each of the next three year-ends and redeems for $2,100 at the third year-end. Its price is $1,776.71. Construct the yield equation for the annual effective yield $j$.

Method: separate coupons from redemption. The first two cash flows are $80 each. The final payment is $80+2{,}100=\$2{,}180$. In particular, the redemption amount is $2,100, not the bond’s face amount or a presumed par value.

1{,}776.71= \frac{80}{1+j} +\frac{80}{(1+j)^2} +\frac{2{,}180}{(1+j)^3}.

The annuity form is $1{,}776.71=80a_{\overline{3}|\,j}+2{,}100(1+j)^{-3}$. To check, substitute $j=0.10$. The discount factors are $0.9090909091$, $0.8264462810$, and $0.7513148009$. The present values are $80(0.9090909091)=72.7272727$, $80(0.8264462810)=66.1157025$, and $2{,}180(0.7513148009)=1{,}637.8662660$. Their sum is $1{,}776.7092412$, or $1,776.71 to the nearest cent. The check confirms that the redemption amount is added to the last coupon before discounting.

Varying Payments and the Limits of the Formula

The sum-of-present-values form works whenever the payment dates are regular coupon periods, even if the payments are not level. If the payment at time $k$ is $C_k$, write:

P=\frac{C_1}{1+j}+\frac{C_2}{(1+j)^2}+\cdots+\frac{C_n}{(1+j)^n}.

This form is often clearer than annuity notation when coupons change over time or when the final redemption amount differs from the principal amount used to calculate coupons. The level-coupon annuity form is a convenient shortening, not a different valuation rule.

These equations assume a coupon-date valuation and regular periods. For a price between coupon dates, the time from settlement to each payment is not simply a whole number of coupon periods; a fractional-period valuation model is needed. Likewise, if payment dates are irregular, discount exponents must reflect the actual intervals under the stated yield convention. Do not apply the integer-period equation without checking the valuation date and payment schedule.

Common Exam Traps

Using the annual rate as the period rate. With $m$ coupon periods per year, the unknown $j$ is per period. A nominal annual yield convertible $m$ times is $mj$, while an effective annual yield is $(1+j)^m-1$.
Using the annual coupon at every payment date. Divide the annual coupon amount by the number of coupon payments per year to obtain the coupon cash flow.
Forgetting the final coupon or double-counting it. The last cash flow is coupon plus redemption. In the annuity form, coupons already include the final coupon, so add only redemption as the separate final term.
Replacing redemption with face amount automatically. Use the stated redemption amount, even when it differs from face amount.
Discounting from the wrong date. The exponent counts periods from the valuation date. If settlement is between coupon dates, the regular integer-period equation is not the correct setup.

Key takeaway: A correct yield equation is a timeline written algebraically: each known future cash flow is discounted for its number of coupon periods at the unknown per-period yield, and the present values add to the known bond price.

Check Your Understanding

For each problem, define $j$ as the yield per coupon period and write the yield equation. Use the stated price as given; do not solve for the yield.

A bond has three annual periods remaining, pays a $70 coupon each year, redeems for $1,000, and costs $1,035. (Answer: $1035=70/(1+j)+70/(1+j)^2+1070/(1+j)^3$.)
A $5,000 bond has five years remaining, pays coupons semiannually at a 4.8% annual coupon rate, redeems at $5,000, and costs $4,725. (Answer: $4725=\sum_{k=1}^{10}120/(1+j)^k+5000/(1+j)^{10}$.)
A bond has two years remaining with quarterly coupons. Its face amount is $2,000, its annual coupon rate is 3.2%, its redemption amount is $1,960, and its price is $1,905. (Answer: $1905=\sum_{k=1}^{8}16/(1+j)^k+1960/(1+j)^8$.)
A bond has three years remaining with semiannual coupons of $45, redeems for $3,000, and costs $2,850. State the equation and the corresponding nominal annual yield convertible semiannually. (Answer: $2850=45a_{\overline{6}|\,j}+3000(1+j)^{-6}$; the nominal annual yield is $2j$.)
A four-year bond pays annual cash flows of $90, $90, and $110 at the first three year-ends; at year 4 it pays a $90 coupon plus $2,000 redemption. Its price is $1,940. (Answer: $1940=90/(1+j)+90/(1+j)^2+110/(1+j)^3+2090/(1+j)^4$.)

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