Why Irregular Coupon Dates Change the Yield Equation
A bond yield equation equates the price at settlement to the present value of all payments still due. When coupon dates are irregular, the payments do not necessarily occur at whole or equally spaced coupon periods from settlement. Each payment must therefore be discounted according to its own time to payment, rather than assigned a convenient regular-period count.
A stub period is a coupon period that is shorter or longer than the bond's regular coupon interval. It can affect both the amount of a coupon, if the contract prorates interest for the stub, and the time at which that coupon is paid. Use the coupon terms and date convention given in the problem; do not assume every stub coupon is a full coupon.
This equation assumes that the same effective annual yield is used to discount every payment and that fractional-year accumulation is represented by \((1+i)^t\). If yield is instead quoted as a nominal annual rate \(j^{(m)}\), convertible \(m\) times per year, the corresponding fractional-period convention is
The convention for measuring \(t_k\) matters. A problem may specify actual/actual, 30/360, or another day-count basis. There is no single year-fraction rule that applies to every bond. Once the problem gives the dates and convention, calculate each fraction from settlement to its payment date. Do not substitute the coupon number for the year fraction.
From Dates to Cash Flows
Before solving for a yield, make a dated cash-flow list. For each payment, record its date, its time from settlement, the coupon due on that date, and any redemption amount. If coupon interest accrues proportionally at an annual coupon rate \(c\) on face amount \(F\), and a coupon interval has year fraction \(\alpha\), its coupon is \(Fc\alpha\). This proportional rule applies only when the contract specifies it; some bond terms define stub coupons differently.
Use a dirty price in the present-value equation. If only a clean price is given, add stated accrued interest to obtain the dirty price.
Include each coupon and add redemption to the final payment. Calculate a stub coupon from its stated contractual rule.
Calculate the year fraction \(t_k\) from settlement to that payment date using the stated date convention.
Discount each cash flow using the selected yield basis. Use a numerical solver when the equation does not have a convenient algebraic solution.
Substitute the solved yield into the equation and confirm that the resulting present value agrees with the dirty price, allowing for rounding.
Why the Discount Exponent Is the Actual Time
Under an effective annual yield \(i\), one dollar invested for \(t\) years accumulates to \((1+i)^t\) dollars under the fractional-year accumulation convention. Consequently, one dollar payable at time \(t\) has present value \((1+i)^{-t}\). Multiplying by a payment \(C_k\) gives its value at settlement, \(C_k(1+i)^{-t_k}\). Present values are additive, so summing over all remaining payments gives the yield equation.
The same reasoning gives the nominal-rate form. A nominal rate \(j^{(m)}\) convertible \(m\) times per year has an accumulation factor \(1+j^{(m)}/m\) per conversion period. Over \(mt\) such periods, the accumulation factor is \((1+j^{(m)}/m)^{mt}\); its reciprocal discounts a payment at time \(t\). This fractional-period formula assumes that fractional periods are handled by extending the stated conversion-period accumulation rule. If the problem specifies a different convention, use that convention instead.
Worked Yield Calculations
Worked Example: Solving with Unequal Payment Intervals
A hypothetical bond has settlement price $848.71, stated as a dirty price. Its remaining payments are $40 in 0.5 years, $40 in 1.5 years, and $1,040 in 3 years; the last amount includes redemption. Find the effective annual yield. Use a constant effective annual yield and fractional-year discounting.
Method. Put each payment at its actual time from settlement. In particular, the cash flows are not equally spaced: the intervals are 1 year and 1.5 years after the first payment.
The yield equation is
A numerical solver gives \(i\approx 0.1025\), or an effective annual yield of 10.25% to two decimal places. To check, at \(i=0.1025\), \(1+i=1.1025=1.05^2\). Thus the three discount factors are \(1/1.05\), \(1/1.05^3\), and \(1/1.05^6\):
The present value rounds to $848.71, matching the stated dirty price. The common mistake is to discount the final payment by a regular coupon count that ignores its three-year actual time from settlement.
Worked Example: Pricing Stub Coupons Before Solving Yield
A hypothetical $1,000 bond has a 6% annual coupon rate. The contract says that coupons accrue proportionally to the year fraction in each interval. From settlement, the remaining payment dates are 0.25, 0.75, and 1.50 years away. The intervals between the settlement date and successive coupon dates are therefore 0.25, 0.50, and 0.75 years. The final payment includes $1,000 redemption. The dirty price is $945.23. Find the effective annual yield.
Method. Calculate the coupon for each interval using the stated proportional accrual rule. The coupons are \(1000(0.06)(0.25)=$15\), \(1000(0.06)(0.50)=$30\), and \(1000(0.06)(0.75)=$45\). The final cash flow is therefore $1,045.
The equation uses the times from settlement, not the lengths of the coupon intervals:
A numerical solution is approximately \(i=10.25\%\). To verify, at this rate the discount factors are approximately 0.975900 for 0.25 years, 0.929429 for 0.75 years, and 0.863838 for 1.50 years. Hence
The check agrees with the price to the nearest cent. A common mistake is to use a full $30 coupon at the first date, even though the stated stub interval is only one quarter of a year.
Worked Example: Using a Nominal Yield and a Clean Price
A hypothetical bond has the same $1,000 face amount, 6% annual coupon rate, and proportional coupon rule. Its remaining payments are $15 at 0.25 years, $30 at 0.75 years, and $1,045 at 1.50 years. The clean price is $960.00 and accrued interest is $12.00. Find the nominal annual yield convertible semiannually, \(j^{(2)}\).
Method. First convert the clean price to the dirty price used in the yield equation:
For a nominal annual yield convertible twice a year, the periodic rate is \(j^{(2)}/2\). The exponents are twice the year fractions, so the equation is
A numerical solver gives \(j^{(2)}\approx 8.000\%\). At 8%, the periodic rate is 0.04. The discount factors are approximately 0.980581, 0.942866, and 0.888996, respectively. Repricing gives
The small difference before rounding is due to using the stated price to the nearest cent. The common mistake is to put the clean price, $960.00, on the left side of the equation. The present value of payments must be compared with the dirty price.
Common Traps and a Reliable Check
Irregular-date yield problems often contain several time conventions at once: coupon intervals determine coupon amounts, settlement-to-payment intervals determine discount exponents, and the price quotation may require accrued interest to be added. Keep those roles separate. A useful calculation table has one row per payment and columns for the payment date, coupon interval, coupon amount, time from settlement, redemption amount, and total cash flow.
| Quantity | Used for | Common error |
|---|---|---|
| Coupon interval length | Calculating a prorated stub coupon when the contract requires it | Using it as the discount exponent |
| Time from settlement to payment | Discounting each cash flow | Replacing it with the next coupon number |
| Dirty price | Left side of the present-value equation | Using a clean price without adding accrued interest |
| Yield compounding basis | Choosing the correct discount factor | Mixing an effective annual rate with a nominal-period formula |
After solving, reprice with the same cash flows, date fractions, price basis, and rate convention. If the result does not reproduce the dirty price to the expected rounding accuracy, check the stub coupon, each exponent, and the clean-to-dirty conversion before changing the solver method. This approach is aligned with the SOA Exam FM (Financial Mathematics) syllabus.
Check Your Understanding
For each question, use the stated discounting convention and keep coupon intervals distinct from settlement-to-payment times.
- A bond pays $100 in 0.5 years and $1,050 in 1.5 years. Find its dirty price at a 6% effective annual yield. (Answer: $1,059.25)
- A bond pays $20 in 0.25 years and $1,020 in 1 year. Write its dirty-price equation at a nominal annual yield of 6% convertible semiannually. (Answer: \(P_{\mathrm{dirty}}=20(1.03)^{-0.5}+1020(1.03)^{-2}\))
- A zero-coupon bond pays $1,000 in 2 years and has a dirty price of $900. Find the effective annual yield, to four decimal places as a percentage. (Answer: 5.4093%)
- A bond's clean price is $985.00 and its accrued interest is $7.50. What price belongs on the left side of the yield equation? (Answer: $992.50)