Valuing a Bond Between Coupon Dates

Why Settlement Timing Matters

A bond’s cash flows occur on scheduled coupon dates, but an investor may buy or value the bond on a different date. The settlement value must therefore discount each remaining payment for the actual time from settlement to that payment. If settlement is halfway through a regular coupon period, for example, the next coupon is only half a period away—not a full period away.

This tutorial assumes regular coupon periods and a settlement date strictly between two coupon dates. The bond makes its promised coupon payments and redemption payment, and a single yield applies to all remaining cash flows. There are no taxes, transaction costs, or default adjustments. The yield is compounded consistently with the coupon period, and fractional-period discounting uses that same yield convention.

Definition: The full value (also called the dirty price) is the present value, at settlement, of all remaining promised bond payments. If settlement is between coupon dates, the time to each payment is measured as a fraction of a coupon period. The clean price is the full value less accrued interest, when accrued interest is being quoted separately.

Let $n$ be the number of remaining coupon payments, including the next one. Let $f$ be the fraction of the current coupon period that has elapsed since the previous coupon date, where $0<f<1$. The next coupon is due in $1-f$ coupon periods; the following coupon is due in $2-f$ periods, and so on. If the yield per coupon period is $i_c$, a cash flow due in $t$ coupon periods is discounted by $(1+i_c)^{-t}$.

If the coupon amount is $C$ and the redemption amount is $R$, the settlement-date value can be written directly as the sum of the discounted cash flows:

P_{\text{full}} = \sum_{j=1}^{n-1} C(1+i_c)^{-(j-f)} + (C+R)(1+i_c)^{-(n-f)}

The final payment includes both the last coupon and redemption. This expression is a cash-flow valuation, not a formula for a special bond type: if a bond has different promised payments, discount each payment at its own fractional time instead.

Setting Up the Fractional Timeline

The most reliable approach is to mark the settlement date and each remaining cash-flow date before calculating. Count time in coupon periods, not in years or months unless you have converted those units consistently. For a regular schedule, the settlement time to payment $j$ is $j-f$ periods, with $j=1,\ldots,n$.

Find the period yield.
Use the rate per coupon period. For a nominal annual yield $i^{(m)}$ convertible $m$ times per year with matching coupon frequency, $i_c=i^{(m)}/m$. For an effective annual yield $i$ and $m$ regular coupon periods per year, $i_c=(1+i)^{1/m}-1$.

Measure the elapsed fraction.
Determine $f$, the fraction of the coupon period from the previous coupon date to settlement, using the fraction specified by the problem’s timing or day-count convention.

List the remaining payments and their times.
The next coupon is at $1-f$; subsequent coupons are at $2-f,3-f,\ldots$. Include redemption with the final coupon.

Discount to settlement.
Multiply each payment by $(1+i_c)^{-t}$, where $t$ is that payment’s time in coupon periods, and add the results.

Separate accrued interest if needed.
When the question requests a clean price, subtract the stated accrued interest from the full value. Do not subtract accrued interest when the requested amount is the full value.

Why Each Cash Flow Uses Its Own Fractional Time

The discounting rule follows from the meaning of a yield. Under a constant effective yield $i_c$ per coupon period, one unit invested for one full period grows to $1+i_c$. Thus a payment $X$ due one period from now has settlement value $X/(1+i_c)$. More generally, the yield convention assigns an accumulation factor $(1+i_c)^t$ over $t$ coupon periods, so the value now of $X$ due at that time is $X/(1+i_c)^t$. Applying this to each promised payment and adding their values gives the bond’s full value.

The fractional exponent is essential. It represents the portion of a coupon period remaining until a payment, under the stated fractional-period yield convention. Using a whole-period exponent for the next coupon would value it as though settlement occurred on the previous coupon date; using a whole-period exponent for every payment would shift every cash flow too far into the future.

Worked Example: Annual Coupons and Half a Period to the Next Payment

A hypothetical bond has a face value of $\$1{,}000$, an annual coupon rate of $5\%$, and two coupon payments remaining. Settlement is halfway through the annual coupon period. The effective annual yield is $10\%$. Find the full value and, if accrued interest is one-half of the annual coupon, the clean price.

Method. There is one coupon of $\$50$ at each remaining coupon date. Redemption of $\$1{,}000$ is made with the second coupon, so the two cash flows are $\$50$ and $\$1{,}050$. Since the coupon period is one year, the period yield is $i_c=0.10$. Settlement is halfway through the period, so $f=0.5$; the payment times are $1-0.5=0.5$ and $2-0.5=1.5$ years.

The first discount factor and value are

(1.10)^{-0.5}=0.9534626, \qquad 50(1.10)^{-0.5}=47.6731.

For the second payment,

(1.10)^{-1.5}=0.8667842, \qquad 1{,}050(1.10)^{-1.5}=910.1234.

Therefore the full value is

P_{\text{full}}=47.6731+910.1234=957.7965.

Rounded to the nearest cent, the full value is $\$957.80$. The annual coupon is $\$1{,}000(0.05)=\$50$, so the stated half-period accrued interest is $\$50(0.5)=\$25$. The clean price is $\$957.80-\$25=$ $\$932.80$. The full value is the amount obtained by discounting the remaining payments; the clean price is a separate quotation after the stated accrued interest is removed.

Check. The next coupon is worth less than $\$50$ at settlement because it is due in half a year, and the later $\$1{,}050$ payment is discounted for one and a half years. Both discounted values are positive, and their sum is below the total undiscounted payments of $\$1{,}100$, as expected at a positive yield.

Worked Example: Semiannual Coupons with a Nominal Yield

A hypothetical bond has a face value of $\$2{,}000$, a $6\%$ nominal annual coupon rate payable semiannually, and four coupon payments remaining. Its nominal annual yield is $8\%$, convertible semiannually. Settlement is three months after the previous coupon date. Find the full value.

Method. The semiannual coupon is $\$2{,}000(0.06)/2=\$60$. The yield per semiannual period is $i_c=0.08/2=0.04$. A three-month elapsed time is half of a six-month coupon period, so $f=0.5$. The four payment times are $0.5,1.5,2.5,$ and $3.5$ coupon periods. The last payment is $\$2{,}060$, comprising the final $\$60$ coupon and the $\$2{,}000$ redemption.

The discount factors, calculated in order, are

(1.04)^{-0.5}=0.980580676,\quad (1.04)^{-1.5}=0.942866034,\quad (1.04)^{-2.5}=0.906601955,\quad (1.04)^{-3.5}=0.871732649.

The values of the first three coupons are

60(0.980580676)=58.834841, \quad 60(0.942866034)=56.571962, \quad 60(0.906601955)=54.396117.

The final coupon and redemption are worth

$$ 2{,}060(0.871732649)=1{,}796.769258. $$

Adding the four values gives

P_{\text{full}} =58.834841+56.571962+54.396117+1{,}796.769258 =1{,}966.572178.

The full value, rounded to the nearest cent, is $\$1{,}966.57$. As a check, the final cash flow is the largest payment but is also discounted the longest; it still contributes most of the value. The mistake to avoid is using $8\%$ as the semiannual yield: the matching yield per coupon period is $4\%$, not $8\%$.

Worked Example: Three Remaining Payments and a Clean Quotation

A hypothetical annual-coupon bond has a face value of $\$1{,}000$, an $8\%$ annual coupon rate, and three coupon payments remaining. Settlement is halfway through the annual period, and the effective annual yield is $21\%$. Assume accrued interest is the elapsed fraction of the annual coupon. Find the full value and clean price.

Method. The annual coupon is $\$1{,}000(0.08)=\$80$, and the final payment is $\$1{,}080$. The annual period yield is $i_c=0.21$, and $f=0.5$, so the three payment times are $0.5,1.5,$ and $2.5$ years. The square root of $1.21$ is $1.1$, which gives convenient discount factors:

(1.21)^{-0.5}=\frac{1}{1.1}=0.909090909, \quad (1.21)^{-1.5}=\frac{1}{1.331}=0.751314801, \quad (1.21)^{-2.5}=\frac{1}{1.61051}=0.620921323.

The three discounted payments are

80(0.909090909)=72.727273, \quad 80(0.751314801)=60.105184, \quad 1{,}080(0.620921323)=670.595029.

Therefore,

P_{\text{full}}=72.727273+60.105184+670.595029=803.427486.

The full value is $\$803.43$, rounded to the nearest cent. Accrued interest is $\$80(0.5)=\$40$, so the clean price is $\$803.43-\$40=$ $\$763.43$. The check is that clean price is lower than full value by exactly the stated accrued interest, not by an additional discounting adjustment.

Common Timing and Price-Quote Traps

Between-coupon valuation is particularly sensitive to timing conventions. The full value is formed from the settlement date, not from the previous coupon date. A useful check is to make sure that every remaining payment has a positive time that increases by one coupon period from one payment to the next.

Using whole-period times. If settlement is halfway through the period, discounting the next coupon for one full period is wrong. Its time is $0.5$ periods, and the following payment is $1.5$ periods away.
Using the quoted annual yield as the period yield. A nominal yield convertible twice per year must be divided by two when coupons are semiannual. An effective annual yield must instead be converted using the effective-rate relationship.
Forgetting redemption in the last payment. The final cash flow is the last coupon plus redemption, not the coupon alone.
Confusing full and clean value. Discounting the promised payments gives full value. Subtract accrued interest only when asked for a clean price and only using the specified accrual convention.
Assuming every fraction is a calendar fraction. Use the problem’s stated day-count convention to determine $f$. A number of elapsed days is not automatically the same fraction under every convention.

A wrong calculation illustrates the first trap. For the first worked example, discounting the next coupon for a full year and the final payment for two years would give $50(1.10)^{-1}+1{,}050(1.10)^{-2}$. Those are the times from the previous coupon date, not from settlement. The correction is to use $0.5$ and $1.5$ years, producing the full value of $\$957.80$.

Key takeaway: At an between-coupon settlement, first locate each payment in fractional coupon periods from settlement. Then discount using the yield per coupon period. The resulting present value is the full value; any clean-price adjustment is a separate accrued-interest step.

Check Your Understanding

For each problem, use the settlement date as time zero and keep yield periods consistent with coupon periods.

A hypothetical annual-coupon bond has face value $\$1{,}000$, a $10\%$ annual coupon rate, two payments remaining, and settlement halfway through the annual period. Its effective annual yield is $21\%$. Find the full value. (Answer: $\$917.36$)
A hypothetical semiannual bond has face value $\$1{,}000$, a $6\%$ nominal annual coupon rate payable semiannually, and two payments remaining. Its nominal annual yield is $42\%$ convertible semiannually, and settlement is halfway through the coupon period. Find the full value. (Answer: $\$801.13$)
A hypothetical annual-coupon bond has face value $\$2{,}000$, a $6\%$ annual coupon rate, and three payments remaining. Settlement is halfway through the annual period, and the effective annual yield is $21\%$. Find the full value. (Answer: $\$1{,}515.60$)
A hypothetical annual-coupon bond has face value $\$1{,}000$, an $8\%$ annual coupon rate, and three payments remaining. Settlement is halfway through the period, the effective annual yield is $21\%$, and accrued interest is defined as one-half of the annual coupon. Find the clean price. (Answer: $\$763.43$)
A hypothetical semiannual-coupon bond has face value $\$1{,}000$, an $8\%$ nominal annual coupon rate, and two payments remaining. Its nominal annual yield is $10\%$ convertible semiannually. Settlement is one month after the previous coupon date in a six-month period. State the yield per coupon period, the two payment times in coupon periods, and the full-value equation. (Answer: $i_c=0.05$; payment times $5/6$ and $11/6$; $P_{\text{full}}=40(1.05)^{-5/6}+1{,}040(1.05)^{-11/6}$)

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