Yield To Maturity With Unequal Coupon Payments

Why Unequal Coupons Need Their Own Yield Equation

A bond’s coupon payments are not always level. A hypothetical bond might pay a smaller coupon early in its term and a larger coupon later, while still repaying its principal at maturity. To determine its yield to maturity, value each promised payment at the time it is due. Replacing the schedule with a level-coupon annuity would change the bond’s cash flows and produce the wrong yield.

The method is the same present-value principle used for ordinary bonds, but the payment schedule must be entered payment by payment. This tutorial assumes the bond is held to maturity, all promised payments are made in full, and there are no call or default effects. The yield is the effective annual rate that equates the bond’s price with the present value of its promised cash flows. A realized return equals this yield only under additional conditions, including reinvestment of interim coupons at the yield.

Definition: For a bond with price $P$, unequal coupon amounts $C_k$ paid at times $t_k$, and redemption amount $R$ paid at maturity $T$, the yield to maturity $i$ is the rate satisfying $$ P=\sum_{k=1}^{n}\frac{C_k}{(1+i)^{t_k}}+\frac{R}{(1+i)^T}. $$ Here time is measured in years and $i$ is an effective annual yield. If the final coupon is paid at maturity, it may instead be combined with redemption in that final cash flow. For annual payment dates $1,2,\ldots,n$, this becomes $P=\sum_{k=1}^{n}C_k(1+i)^{-k}+R(1+i)^{-n}$.

The schedule is defined by the amount and timing of each payment, not by a single coupon rate applied as if every coupon were equal. If payments occur at regular intervals shorter than a year, use a yield rate consistent with those periods. For example, a nominal annual yield $i^{(m)}$, convertible $m$ times per year, discounts a payment at period $k$ by $(1+i^{(m)}/m)^{-k}$. With a stated effective annual yield and fractional-year payment times, the annual compounding model uses the exponents $t_k$ shown above.

Building the Equation From the Cash-Flow Schedule

Start by writing each date and the total cash flow due on that date. Add the redemption amount to the final coupon if both are paid together. Then discount each total by the yield factor for its own time. The price is the sum of these present values.

List the promised payments.
Record each coupon amount, its payment time, and the redemption amount and date.

Combine amounts due on the same date.
If a coupon and principal are both paid at maturity, discount their sum using the maturity period.

Set present value equal to price.
Use one yield variable throughout and assign each cash flow its correct exponent.

Solve and verify.
Substitute the candidate yield into the original equation and check that the resulting price agrees with the stated price.

Worked Example: Solving a Two-Year Yield Algebraically

A hypothetical two-year bond costs $1,000. It pays a coupon of $50 at the end of year 1 and a coupon of $100 at the end of year 2, together with $1,000 redemption. Find its effective annual yield to three decimal places as a percentage.

Method. Write the present-value equation, combining the final coupon and redemption because they are paid at the same time:

1000=\frac{50}{1+i}+\frac{1100}{(1+i)^2}.

Let $x=1+i$. Multiplying through by $x^2$ gives

1000x^2=50x+1100,\qquad 20x^2-x-22=0.

The quadratic formula gives $x=(1\pm\sqrt{1761})/40$. Since $x=1+i$ must be positive, use the positive root:

x=\frac{1+\sqrt{1761}}{40}\approx 1.0741068,\qquad i=x-1\approx 0.0741068.

Thus the yield is $7.41068\%$, which rounds to 7.411% to three decimal places. As a check, substituting $i\approx0.0741068$ into the original equation gives a present value of approximately $1,000.00.

Common mistake avoided: The final-year cash flow is $1,100, not $1,000: the final $100 coupon is due in addition to the redemption.

Why the Equation Gives One Yield

For a conventional bond with positive promised payments, the price decreases as the yield increases. This helps explain why a numerical yield search has a single solution when the price is positive and the cash flows are all positive.

For fixed payment times $t_k>0$, each present-value term is $C_k(1+i)^{-t_k}$. Its derivative with respect to $i$, for $i>-1$, is

\frac{d}{di}\left[C_k(1+i)^{-t_k}\right] =-t_kC_k(1+i)^{-t_k-1}<0

when $C_k>0$. The redemption term has the same property. Therefore the total present value is strictly decreasing in $i$. If a yield is found that makes the equation hold, no other yield greater than $-1$ can make it hold for the same positive cash flows. This conclusion depends on the cash flows being positive; cash-flow schedules containing negative payments need not have the same uniqueness property.

When algebra does not give a convenient solution, calculate the price at trial yields. If the trial present value is above the bond’s price, the trial yield is too low; if it is below the price, the yield is too high. A numerical solver can then narrow the interval until the desired precision is reached. Always check the final rounded yield in the original cash-flow equation.

Worked Applications

Worked Example: Pricing Unequal Annual Coupons

A hypothetical three-year bond pays coupons of $15, $45, and $30 at the ends of years 1, 2, and 3, respectively. It redeems for $1,000 at the end of year 3. Find its price if the effective annual yield is 5%, to the nearest cent.

Method. Discount the first two coupons separately. Combine the year-3 coupon with redemption, giving a final payment of $30+1{,}000=1{,}030$:

P=\frac{15}{1.05}+\frac{45}{1.05^2}+\frac{1030}{1.05^3}.

The periods in the denominators match the payment dates:

\frac{15}{1.05}=14.285714,\qquad \frac{45}{1.1025}=40.816327,\qquad \frac{1030}{1.157625}=889.752726.

Adding the present values gives $P=944.854767$, so the price is $944.85 to the nearest cent. Substituting the three present values back into the sum confirms the unrounded price.

Common mistake avoided: Do not apply a level-coupon annuity factor to these unequal amounts. Each coupon has its own date and amount.

Worked Example: Finding Yield Numerically From a Rounded Price

A hypothetical three-year bond pays coupons of $25, $75, and $40 at the ends of years 1, 2, and 3, respectively, plus $1,000 redemption at the end of year 3. Its observed price is $1,017.94. Find the effective annual yield to three decimal places as a percentage.

Method. The equation must use the first coupon at time 1, the second at time 2, and the combined final payment of $40+1{,}000=1{,}040$ at time 3:

1017.94=\frac{25}{1+i}+\frac{75}{(1+i)^2}+\frac{1040}{(1+i)^3}.

There is no need to force this cubic equation into a closed-form solution. Evaluate the right-hand side at trial rates. At $i=3.900\%$, the present value is approximately $1,020.77, which is above the observed price, so the yield is higher than 3.900%. At $i=4.000\%$, the present value is

\frac{25}{1.04}+\frac{75}{1.04^2}+\frac{1040}{1.04^3} \approx 1017.9364,

which rounds to $1,017.94, the stated price. Solving with the unrounded equation gives a yield of approximately $3.9999\%$. The answer is therefore 4.000% to three decimal places. The price check agrees to the nearest cent.

Common mistake avoided: The final coupon is discounted for three years, not two. The exponent is determined by the payment date, not by the coupon’s position in a list of unequal amounts.

Common Traps and Practical Checks

An unequal coupon schedule invites a few predictable errors. A timeline is a simple way to catch them before solving:

Using an annuity formula for unequal coupons. The level-coupon annuity factors $a_{\overline{n}|}$ and $\ddot{a}_{\overline{n}|}$ apply to level payments under their stated timing conventions. They cannot replace a sum of unequal coupon present values.
Using the wrong exponent. A payment at the end of year $k$ is discounted by $(1+i)^k$, regardless of whether its amount is larger or smaller than earlier coupons.
Omitting or double-counting the final coupon. If the last cash flow is written as coupon plus redemption, do not add the coupon again as a separate term.
Mixing yield conventions. An effective annual yield is not the same as a nominal annual yield convertible more than once per year. Convert or use the matching period rate before discounting.
Rounding too early. Keep intermediate present values and numerical yield estimates unrounded until the final answer. Then check using the original equation.

Unequal coupon amounts change the present-value equation, but not the central discipline: identify every promised payment, place it at its correct time, and discount it consistently. This payment-by-payment approach is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, use the payment dates stated and keep full precision until the final answer.

A two-year bond pays $20 at the end of year 1 and $60 plus $1,000 redemption at the end of year 2. Find its price at an effective annual yield of 6%, to the nearest cent. (Answer: $962.26)
A two-year bond pays $25 at the end of year 1 and $75 plus $1,000 redemption at the end of year 2. Its price is $998.87. Find the effective annual yield to three decimal places as a percentage. (Answer: 5.000%)
A bond pays $40, $80, and $50 at times 0.5, 1.5, and 2.5 years, respectively, and pays $1,000 redemption with the last coupon. Find its price using an effective annual yield of 4%, to the nearest cent. (Answer: $1,066.58)
A three-year bond pays coupons of $10, $70, and $30 at the ends of years 1, 2, and 3, respectively, and redeems for $1,000 at year 3. Its price is exactly the present value of these payments at a 3% effective annual yield. What is its yield to maturity? (Answer: 3.000%)

What You'll Learn