Settlement Dates and Bond Cash-Flow Timing

Why Settlement Dates Matter

A bond’s quoted price and the cash flows used to value it must refer to compatible dates. A quote may be given for settlement today, while a term-structure curve gives discount factors from an earlier valuation date. In addition, some of the bond’s scheduled payments may already have occurred by settlement. Using a payment that is no longer payable, or discounting a remaining payment from the wrong date, changes the bootstrap equation.

The key is to keep three decisions separate: identify the payments that remain payable, determine the bond’s full settlement value, and express that value and the payments at a common valuation date. The same method applies whether coupon dates are regular or irregular.

Definition: Let $s$ be the settlement date and $t$ a cash-flow date after settlement. $D(s,t)$ is the discount factor from $t$ back to $s$; $D(0,t)$ is the discount factor from $t$ back to the curve’s valuation date, time 0. A cash flow $C$ payable at $t$ has value $C D(s,t)$ at settlement or $C D(0,t)$ at time 0. A settlement-date bond price is the value at $s$ of cash flows payable after settlement.

First Determine Which Cash Flows Remain

Write the bond’s payment schedule against the settlement date before calculating. Include a coupon or redemption payment only if it is still payable to the buyer after settlement. Payments already made before settlement do not belong in the remaining-cash-flow equation. If settlement falls on a coupon date, the problem must establish whether settlement is before or after that payment; do not infer the entitlement from the date alone. In the examples below, any payment stated to have been made on settlement is treated as already paid.

For a bond with face amount $F$, annual coupon rate $c$, and $m$ coupon payments per year, each regular coupon is $Fc/m$. The final payment normally includes the last coupon and redemption amount, so for redemption at face value it is $Fc/m+F$. These amounts are contractual cash flows; the settlement date determines which of them remain to be valued.

If the market quote is a clean price, first account for accrued interest under the stated day-count convention. Accrued interest represents the portion of the current coupon earned since the last coupon date. The dirty price is the full amount paid at settlement, including accrued interest:

P_{\text{dirty}}(s)=P_{\text{clean}}(s)+AI(s).

The accrued-interest fraction depends on the problem’s day-count and coupon-period conventions. For example, if a coupon is $C$ and one-half of its coupon period has elapsed under the specified convention, accrued interest is $C/2$. Do not assume that fraction when a problem gives a different convention or dates.

Worked Example: Solving a Remaining Payment Factor from a Clean Quote

A hypothetical bond has face amount $1,000, pays 5% annual interest semiannually, and is settled halfway through a coupon period under a 30/360 convention. Its clean price is $997. The next three payments are $25, $25, and $1,025. The settlement-date discount factors for the first two payments are 0.980 and 0.950. Find the settlement-date factor for the final payment.

Method. Convert the clean quote to a dirty price, then equate that settlement value to the present values of the remaining payments. The stated schedule contains three payments after settlement, so all three are included.

The coupon is $1000(0.05)/2=\$25$. Half of the coupon period has elapsed, so accrued interest is $25(1/2)=\$12.50$. Thus

P_{\text{dirty}}(s)=997+12.50=1009.50.

The first two payments have settlement values $25(0.980)=\$24.50$ and $25(0.950)=\$23.75$. The amount of settlement value left for the final payment is

$$ 1009.50-24.50-23.75=961.25. $$

If $D(s,t_3)$ is the factor for the $1,025 final payment, then $1025D(s,t_3)=961.25$. Therefore

D(s,t_3)=\frac{961.25}{1025}=0.937804878\ldots\approx 0.937805.

Check and trap avoided. The three discounted payments sum to $24.50+23.75+961.25=\$1,009.50$, the dirty price. Using the clean price of $997 as the value of all remaining payments would omit the $12.50 accrued interest and produce the wrong factor.

Moving Between Settlement and Valuation Dates

A bootstrap curve is commonly expressed using factors measured from time 0, even when a bond price is quoted for settlement at a later date $s$. To use those curve factors, move both sides of the settlement-date price equation to time 0. Under a consistent discount-factor curve, for $0\leq s\leq t$,

$$ D(0,t)=D(0,s)D(s,t). $$

This identity follows by valuing one dollar payable at $t$. Its value at time 0 is $D(0,t)$. Alternatively, first value it at settlement as $D(s,t)$, then discount that settlement value to time 0 by $D(0,s)$. Consistency of the curve requires these two valuations to agree. If $D(0,s)>0$, rearranging gives $D(s,t)=D(0,t)/D(0,s)$.

Suppose the remaining cash flows are $C_1,\ldots,C_k$ at dates $t_1,\ldots,t_k$, all strictly after settlement. Their settlement-date value is $\sum_{j=1}^k C_jD(s,t_j)$. Multiplying this equation by $D(0,s)$ gives the common-date equation:

P_{\text{dirty}}(s)D(0,s) =\sum_{j=1}^{k} C_jD(0,t_j).

This is an accounting change of valuation date, not a new bond-pricing model. It requires discount factors from a single consistent curve and the same cash-flow and settlement assumptions on both sides. If the supplied price is already stated at time 0, do not multiply it by $D(0,s)$ again. If a price is quoted clean, convert it to dirty before applying the equation.

Worked Example: Converting a Settlement Price to a Common-Date Equation

A hypothetical bond has dirty price $1,025 at settlement date $s$. The curve gives $D(0,s)=0.992$. The two remaining payments are $40 at $t_1$ and $1,040 at $t_2$, with $D(0,t_1)=0.970$. Find $D(0,t_2)$.

Method. Express the settlement price and both future cash flows at time 0. Because the quote is already dirty, no accrued-interest adjustment is needed.

The common-date equation is

$$ 1025(0.992)=40(0.970)+1040D(0,t_2). $$

The left side is $1025(0.992)=1016.80$. The known first payment contributes $40(0.970)=38.80$. The time-0 value left for the final payment is therefore $1016.80-38.80=978.00$. Solving,

D(0,t_2)=\frac{978.00}{1040}=0.940384615\ldots\approx 0.940385.

Check and trap avoided. Substitution gives $38.80+1040(0.940384615\ldots)=1016.80$, which equals $1025(0.992)$. A common-date equation must discount the price from settlement to time 0 as well as discounting the cash flows from their payment dates. Equating $1,025 directly to time-0 present values would mix dates.

Putting the Timing Decisions Together

For each bond in a bootstrap, a short timeline prevents most settlement errors. Mark time 0, settlement $s$, every coupon date, and redemption. Cross out payments already made by settlement. Then calculate the dirty settlement price and write the value equation using either settlement-date factors throughout or time-0 factors throughout. Do not mix the two forms within a single equation.

Mark the dates.
Place settlement and all scheduled coupon and redemption dates in chronological order. Resolve explicitly whether a payment on the settlement date has already been made.

List remaining payments.
Include only amounts payable after settlement. Combine the final coupon and redemption if they are due on the same date.

Find the full settlement price.
If a clean price is given, add accrued interest calculated using the stated convention. If the price is already dirty, use it as given.

Choose one valuation date.
Use $P_{\text{dirty}}(s)=\sum C_jD(s,t_j)$ at settlement, or multiply through by $D(0,s)$ to use $P_{\text{dirty}}(s)D(0,s)=\sum C_jD(0,t_j)$ at time 0.

Solve only for the unknown factor.
After all known cash flows have been valued at the same date, isolate the unknown factor multiplying the cash flow at its payment date.

Worked Example: A Settlement Between Coupon Dates

A hypothetical $1,000 bond pays 6% annual interest semiannually, with coupon dates at times 0.5, 1.0, 1.5, and 2.0 years. Settlement is at time 0.75, after the coupon at 0.5 has been paid. Under the stated day-count convention, one-half of the current coupon period has elapsed. The clean price is $1,010. The curve gives $D(0,0.75)=0.985$, $D(0,1.0)=0.970$, and $D(0,1.5)=0.945$. Find $D(0,2.0)$.

Method. The 0.5-year coupon is already paid and is excluded. The remaining cash flows are $30 at 1.0, $30 at 1.5, and $1,030 at 2.0. The last amount includes the final $30 coupon and $1,000 redemption.

Each coupon is $1000(0.06)/2=\$30$. Accrued interest is $30(1/2)=\$15$, so the dirty settlement price is $1010+15=\$1,025$. Move this price to time 0:

$$ 1025(0.985)=1009.625. $$

The known cash flows contribute $30(0.970)=\$29.10$ and $30(0.945)=\$28.35$. The remaining time-0 value is $1009.625-29.10-28.35=\$952.175$. Hence

D(0,2.0)=\frac{952.175}{1030} =0.924441747\ldots\approx 0.924442.

Check and trap avoided. The time-0 cash-flow values sum to $29.10+28.35+952.175=\$1,009.625$, equal to the time-0 value of the dirty settlement price. Including the coupon at 0.5 would count a payment already made; using $1,010 rather than $1,025 would omit accrued interest.

Common Timing Errors and Their Corrections

Settlement-date mistakes are often easy to spot if every term in the equation is labelled by its valuation date. In particular, a coupon schedule alone does not tell you which payments remain due; settlement and payment entitlement do.

Including a past coupon. A coupon paid before settlement is not a remaining asset of the buyer. Remove it from the remaining-cash-flow list rather than discounting it over a negative or zero interval.
Using a clean quote as the full value. If accrued interest is positive, $P_{\text{clean}}(s)$ is not the cash amount representing all remaining payments. Add accrued interest to obtain $P_{\text{dirty}}(s)$.
Mixing valuation dates. Do not equate a settlement-date price directly to cash flows discounted to time 0. Either use all settlement-date factors or multiply the dirty price by $D(0,s)$ and use time-0 factors for every payment.
Discounting from the wrong starting date. If $D(0,t)$ is supplied, it discounts a payment from $t$ to time 0, not from $t$ to settlement. The corresponding settlement factor is $D(0,t)/D(0,s)$ under a consistent curve.
Assuming a coupon-date entitlement. When settlement coincides with a payment date, follow the problem’s explicit timing convention. Whether that coupon has been paid determines whether it belongs in the remaining cash flows.

A useful final check is to add the valued remaining cash flows and compare with the bond price expressed at exactly the same date. If the two sides use different dates, the equation is not yet ready to solve.

Key takeaway: A correct settlement-date bootstrap starts with the cash flows still payable after settlement, uses the dirty settlement price, and values both sides at one common date. The relation $D(0,t)=D(0,s)D(s,t)$ lets you convert consistently between settlement-date and time-0 equations.

Check Your Understanding

For each problem, identify the remaining payments and keep all terms at a common valuation date.

A hypothetical $2,000 bond pays 4% annual interest semiannually. Settlement is one-quarter of a coupon period after the previous coupon date, under a convention that gives accrued interest of one-quarter of the coupon. The clean price is $1,985. The two remaining payments are $40 and $2,040. Given $D(s,t_1)=0.990$, find $D(s,t_2)$. (Answer: $0.958529$, approximately.)
A hypothetical bond has dirty settlement price $800, with $D(0,s)=0.990$. Its remaining payments are $24 at $t_1$ and $1,024 at $t_2$. Given $D(0,t_1)=0.980$, find $D(0,t_2)$. (Answer: $0.750469$, approximately.)
A hypothetical bond settles immediately after its $30 coupon has been paid. Its dirty settlement price is $1,000; its remaining payments are $30 at $t_1$ and $1,030 at $t_2$. Given $D(0,s)=0.990$ and $D(0,t_1)=0.970$, find $D(0,t_2)$. (Answer: $0.932913$, approximately.)
A hypothetical bond has dirty settlement price $1,010 at time $s$, with $D(0,s)=0.996$. Its remaining payments are $20 at $t_1$, $20 at $t_2$, and $1,020 at $t_3$. Given $D(0,t_1)=0.985$ and $D(0,t_2)=0.960$, find $D(0,t_3)$. (Answer: $0.948098$, approximately.)
A hypothetical bond’s dirty settlement price is $1,100, and $D(0,s)=0.980$. Its remaining payments are $50 at $t_1$ and $1,050 at $t_2$. Given $D(0,t_1)=0.970$, write the common-date equation and find $D(0,t_2)$. (Answer: $1100(0.980)=50(0.970)+1050D(0,t_2)$; $D(0,t_2)=0.980476$, approximately.)

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