Why Irregular Dates Require Care
A coupon bond does not have to pay on a regular semiannual schedule. Its first coupon may follow a short or long stub period, or its payment dates may simply fall between the standard dates used by other instruments. The bootstrap principle remains the same: value each known payment using the discount factor for its own date. The extra work is to construct the dates and cash amounts correctly before solving.
A frequent error is to place an irregular payment on the nearest familiar date, such as replacing a payment at 0.48 years with one at 0.50 years. That changes the valuation equation. A second error is to assume that every bond quote automatically reveals a new discount factor. A price gives an equation; it identifies an unknown factor only if the other factors in that equation are known, or if enough independent equations are available to solve for the unknowns.
The model assumes that the bond's contractual payments are known, the valuation date and payment dates are specified, and the same discount curve is used to value all the payments. There is no default or other contingent payment in these examples. The quoted price is treated as the full value at the valuation date, so no conversion from another price convention is needed here.
Construct the Coupon Schedule First
Let \(F\) be the bond's face amount and \(c\) its annual coupon rate. If the contract specifies simple coupon accrual at rate \(c\), and the year fraction for coupon period \(j\) is \(\alpha_j\), then that period's coupon is
The fraction \(\alpha_j\) comes from the bond's stated accrual or day-count convention. It is not necessarily the time from valuation to payment. For example, if valuation occurs before the start of a short first coupon period, the payment time \(t_1\) measures the discounting interval, while \(\alpha_1\) measures the contractual coupon accrual interval. These may coincide in a particular setup, but they represent different quantities.
At maturity, the final payment ordinarily includes both the last coupon and redemption of face amount. If redemption is at face, that payment is \(F+K_k\). Always use the actual contractual redemption amount and coupon schedule supplied by the problem; the prorated coupon formula applies only when the contract specifies that accrual rule.
Measure every date from the valuation date and retain the given time convention.
Apply the contract's coupon rule and accrual fraction to each period, including any stub.
Combine the final coupon with the redemption payment when they are paid on the same date.
Multiply each payment by \(D(t)\) at that payment's actual date, then assess which factors are known.
Worked Example: Solving for a Stub-Date Maturity Factor
A hypothetical bond has face amount $100 and an annual coupon rate of 6%. Valuation is at time 0. The first coupon is due at \(t=0.25\), followed by coupons at \(t=0.75\) and maturity at \(t=1.25\). The accrual fractions are \(0.25\), \(0.50\), and \(0.50\). Redemption is $100, the full price is $101.22, and \(D(0.25)=0.99\), \(D(0.75)=0.97\). Find \(D(1.25)\).
Method. Construct the stub coupon and the later regular coupons, then use the dated price equation. The common mistake avoided is treating the first coupon as a regular half-year coupon.
The first coupon is \(100(0.06)(0.25)=\$1.50\). Each later coupon is \(100(0.06)(0.50)=\$3.00\). Thus the payments are $1.50 at \(t=0.25\), $3.00 at \(t=0.75\), and $103.00 at \(t=1.25\), where the last amount includes the $100 redemption.
Substitute the known factors and the quoted price:
The known present-value contributions are \(1.50(0.99)=1.485\) and \(3.00(0.97)=2.91\), for a total of \(4.395\). Therefore,
Rounded to four decimal places, \(D(1.25)=0.9391\). Check using the unrounded result: \(4.395+103(0.9390776699\ldots)=101.22\), as required.
Why the Price Equation Holds
The pricing equation follows from valuing each payment separately. By the definition of \(D(t_j)\), an amount \(C_j\) paid at \(t_j\) is worth \(C_jD(t_j)\) at time 0. Adding the values of all payments gives the value of the complete stream. Equivalently, a portfolio of zero-coupon payments matching the bond's cash flows has the same payments on every date; under no-arbitrage pricing, the two portfolios have the same value.
This argument does not require coupon dates to be equally spaced. It does require that each \(C_j\) be assigned to its actual \(t_j\). The formula itself does not determine every discount factor in the equation: that depends on which factors are already known and how many independent price equations are available.
Worked Example: A Nonstandard Schedule with Known Earlier Factors
A hypothetical $100 bond has an annual coupon rate of 8%, with accrual fractions \(0.40\), \(0.50\), \(0.50\), and \(0.50\). Its payments occur at \(t=0.40\), \(0.90\), \(1.40\), and \(1.90\); the last payment includes $100 redemption. The full price is $103.20. Given \(D(0.40)=0.98\), \(D(0.90)=0.95\), and \(D(1.40)=0.925\), find \(D(1.90)\).
Method. Use the accrual fractions to construct the coupon amounts, then isolate the factor at the final date. The common mistake avoided is assigning the maturity payment to a standard two-year date just because it is close to one.
The coupons are \(100(0.08)(0.40)=\$3.20\) for the first period and \(100(0.08)(0.50)=\$4.00\) for each of the next three periods. The final cash flow is therefore $104.00. The price equation is
The known contributions are \(3.20(0.98)=3.136\), \(4(0.95)=3.80\), and \(4(0.925)=3.70\). Their sum is \(10.636\). Hence
To four decimal places, \(D(1.90)=0.8891\). Substitution of the unrounded value gives \(10.636+104(0.8890769231\ldots)=103.20\), checking the calculation.
Identification: What Can the Available Prices Determine?
A bootstrap step is possible when an instrument equation has one unknown discount factor after all other required factors have been established. If several factors are unknown, count the independent equations and examine their coefficients. One equation in two unknown factors generally identifies only a relationship between them, not each factor separately. Additional instruments can resolve the unknowns only if their equations are independent.
This is an identification check, not an invitation to fill gaps by assumption. In particular, a discount factor at one date cannot be substituted for a factor at a nearby date unless the problem explicitly supplies a rule for doing so. No interpolation rule is implied by the dated cash-flow equation.
Worked Example: Two Irregular-Date Equations Identify Two Factors
Two hypothetical bonds each have face amount $100. Both pay at \(t=0.75\) and \(t=1.25\), with accrual fractions \(0.75\) for the first coupon and \(0.50\) for the second. Bond A has a 4% annual coupon and price $96.75. Bond B has an 8% annual coupon and price $101.50. Redemption is at face. Find \(D(0.75)\) and \(D(1.25)\).
Method. Write one equation for each bond and solve the two-equation system. The common mistake avoided is trying to bootstrap either factor from just one of these equations while both are unknown.
For Bond A, the first coupon is \(100(0.04)(0.75)=\$3.00\); the final coupon is \(100(0.04)(0.50)=\$2.00\), so the final payment is $102.00. For Bond B, the corresponding coupons are $6.00 and $4.00, making its final payment $104.00. Let \(x=D(0.75)\) and \(y=D(1.25)\). The price equations are
Multiply the first equation by 2: \(6x+204y=193.50\). Subtract the second equation to eliminate \(x\): \(100y=92.00\), so \(y=0.92\). Substitute into the first equation:
Thus \(D(0.75)=0.9700\) and \(D(1.25)=0.9200\). Check: Bond A is worth \(3(0.97)+102(0.92)=2.91+93.84=96.75\); Bond B is worth \(6(0.97)+104(0.92)=5.82+95.68=101.50\).
Worked Example: One Equation Does Not Identify Two Factors
A hypothetical $100 bond has a 4% annual coupon, with accrual fractions \(0.25\), \(0.50\), and \(0.50\). It pays at \(t=0.25\), \(0.75\), and \(1.25\); the last payment includes redemption. Its full price is $96.00, and \(D(0.25)=0.99\). Determine whether the price identifies both later discount factors.
Method. Write the one available price equation and check its number of independent restrictions. The common mistake avoided is reporting a unique pair of factors from one equation.
The coupons are $1.00, $2.00, and $2.00, so the final cash flow is $102.00. The equation is
or \(2D(0.75)+102D(1.25)=95.01\). This is one equation in two unknowns. For instance, setting \(D(0.75)=0.95\) gives \(D(1.25)=(95.01-1.90)/102=0.912843\ldots\). Setting \(D(0.75)=0.90\) instead gives \(D(1.25)=(95.01-1.80)/102=0.913824\ldots\). Both pairs satisfy the price equation, so the available price does not uniquely identify either factor. More independent information is required.
Common Exam Trap: Misplacing a Payment Date
Suppose a payment is made at \(t=0.48\), and a known factor is supplied at \(t=0.50\). The two dates are not interchangeable. Substituting \(D(0.50)\) for \(D(0.48)\) silently changes the cash-flow schedule. The correct equation retains \(D(0.48)\); if it is unknown, the equation must be solved for that factor or left as part of a system.
Check Your Understanding
For each problem, construct the dated cash flows before solving. Assume default-free bonds, the stated coupon accrual fractions, and full prices at time 0.
- A $100 bond pays coupons at \(t=0.30\), \(0.80\), and \(1.30\), with accrual fractions \(0.30\), \(0.50\), and \(0.50\). Its annual coupon rate is 5%, redemption is $100, price is $100.00, \(D(0.30)=0.99\), and \(D(0.80)=0.96\). Find \(D(1.30)\). (Answer: \(0.9377\), to four decimal places.)
- A $100 bond with a 6% annual coupon has payments at \(t=0.20\), \(0.80\), and \(1.40\), with accrual fractions \(0.20\), \(0.60\), and \(0.60\). Redemption is $100, price is $101.034, \(D(0.20)=0.995\), and \(D(0.80)=0.97\). Find \(D(1.40)\). (Answer: \(0.9300\), to four decimal places.)
- Two $100 bonds pay at \(t=0.75\) and \(t=1.25\), with accrual fractions \(0.75\) and \(0.50\). Their annual coupon rates are 4% and 8%, and their prices are $96.75 and $101.50, respectively. Redemption is at face. Find the two discount factors. (Answer: \(D(0.75)=0.9700\) and \(D(1.25)=0.9200\).)
- A bond has a known first payment at \(t=0.25\), with a known present value of $0.99, followed by payments of $2 at \(t=0.75\) and $102 at \(t=1.25\). Its price is $96.00. How many independent equations are available for the two unknown later discount factors, and are they uniquely identified? (Answer: One equation for two unknowns; they are not uniquely identified.)
- A $100 bond has a 4% annual coupon, with accrual fractions \(0.48\) and \(0.52\), and pays at \(t=0.48\) and \(t=1.00\). The final payment includes redemption, the price is $72.992, and \(D(1.00)=0.70\). Find \(D(0.48)\). (Answer: \(0.8000\), to four decimal places; use the factor at 0.48, not a factor at a nearby date.)