Deriving the Fractional-Period Pricing Formula

Why the Fractional-Period Formula Needs a Derivation

A bond settlement date can fall between coupon dates. The next coupon is then less than one full coupon period away, while later payments are more than one period away. The fractional-period pricing formula handles this timing by discounting each payment over its actual remaining number of coupon periods.

The key is to distinguish elapsed time since the preceding coupon date from time remaining until a payment. If settlement occurs a fraction $f$ of a coupon period after the preceding coupon date, the first coupon is due in $1-f$ periods. The second is due in $2-f$ periods, and the $k$-th is due in $k-f$ periods. Using $f$ rather than $1-f$ as the time to the next coupon is a common timing error.

Definition: Suppose settlement lies between two regular coupon dates. Let $j$ be the effective yield per coupon period, let $f$ be the fraction of a coupon period elapsed between the preceding coupon date and settlement, and let $n$ be the number of remaining coupon payments, including the final redemption. Thus $0<f<1$. Let $C$ be the coupon paid on each remaining coupon date and $R$ the redemption amount paid with the final coupon.

This tutorial assumes regular coupon periods and the fractional-compounding convention: over a fraction $t$ of a coupon period, an amount accumulates by the factor $(1+j)^t$. The same convention discounts a future amount over $t$ periods by $(1+j)^{-t}$. The yield $j$ must be expressed per coupon period, not per year unless coupons are annual. The fraction $f$ must use the same coupon-period timeline.

The formula is a valuation convention based on fractional powers of the per-period accumulation factor. It is not a statement that cash is actually exchanged continuously between coupon dates. If the coupon schedule has an irregular period or a different convention is specified, this regular-period formula may not apply as written.

Deriving the Fractional-Period Price

First value the remaining cash flows on the preceding coupon date. On that date, the first remaining coupon is one full period away, the second is two periods away, and the final coupon and redemption are $n$ periods away. Write $v=1/(1+j)$. The value $V_0$ on that preceding coupon date is therefore

V_0=\sum_{k=1}^{n} C v^k+R v^n =\sum_{k=1}^{n} C(1+j)^{-k}+R(1+j)^{-n}.

Settlement occurs $f$ periods after that coupon date. Under the stated fractional-compounding convention, the earlier value $V_0$ accumulates forward to the settlement value $P$:

$$ P=V_0(1+j)^f. $$

Substitute the expression for $V_0$. Multiplying each term by $(1+j)^f$ changes a discount exponent of $-k$ to $-(k-f)$. This gives the fractional-period formula.

Theorem: For a regular coupon bond with $n$ remaining coupons, coupon amount $C$, redemption amount $R$, per-period yield $j$, and settlement $f$ periods after the preceding coupon date, $$ P=\sum_{k=1}^{n} C(1+j)^{-(k-f)}+R(1+j)^{-(n-f)}. $$ The redemption amount is discounted over the same remaining time as the final coupon because both are paid on the final coupon date.

This derivation also explains why the formula can be written in two equivalent ways. One may discount each cash flow directly over its remaining time, $k-f$, or first calculate $V_0$ on the preceding coupon date and then accumulate that value forward by $(1+j)^f$. The two approaches must agree.

Set the coupon-period timeline.
Identify the number of remaining payments $n$ and the elapsed fraction $f$ since the preceding coupon date.

Find each remaining time.
The $k$-th coupon is $k-f$ coupon periods after settlement. The final redemption is due at time $n-f$.

Discount the payments.
Use $C(1+j)^{-(k-f)}$ for each coupon and $R(1+j)^{-(n-f)}$ for redemption.

Check the direction.
Alternatively, calculate the value on the preceding coupon date and multiply it by $(1+j)^f$ to obtain the later settlement value.

Worked Applications

Worked Example: Pricing a Bond Halfway Through a Coupon Period

A hypothetical bond has two annual coupon payments remaining. Each coupon is $100, and the final payment includes $1,000 of redemption, so the final cash flow is $1,100. The effective yield per annual coupon period is $j=21\%$. Settlement is halfway through the period after the preceding coupon date, so $f=0.5$. Find the price.

Method. Discount each payment over the number of periods remaining from settlement. The next coupon is due in $1-0.5=0.5$ periods. The final coupon and redemption are due in $2-0.5=1.5$ periods.

P=100(1.21)^{-0.5}+1100(1.21)^{-1.5}.

Since $(1.21)^{0.5}=1.1$, the next coupon has present value $100/1.1=\$90.90909$. Also, $(1.21)^{1.5}=1.21(1.1)=1.331$, so the final cash flow has present value $1100/1.331=\$826.44628$. Thus

$$ P=90.90909+826.44628=917.35537. $$

Rounded to the nearest cent, the price is $917.36. As a check, the preceding coupon-date value is $100/1.21+1100/(1.21)^2=82.64463+751.31484=\$833.95947$. Accumulating forward half a period gives $833.95947(1.21)^{0.5}=833.95947(1.1)=\$917.35542$, which agrees before rounding. The small difference in the last decimal is from displayed intermediate rounding. This method avoids treating settlement as though it were a coupon date.

Worked Example: Using a Different Fractional Power

A hypothetical bond has two semiannual coupon payments remaining. Each coupon is $50, and the final payment includes a $1,000 redemption amount. The effective yield per semiannual coupon period is $10.25\%$, and settlement is halfway through a coupon period. Calculate the price to the nearest cent.

Here $j=0.1025$, $f=0.5$, and the two cash flows are $50 and $1,050. Since $1.1025=(1.05)^2$, the required fractional powers are $(1.1025)^{0.5}=1.05$ and $(1.1025)^{1.5}=(1.05)^3=1.157625$. Therefore,

P=\frac{50}{1.05}+\frac{1050}{1.157625}.

The first present value is $50/1.05=47.6190476$. For the second, $1.157625=9261/8000$, so

\frac{1050}{1.157625} =\frac{8{,}400{,}000}{9{,}261} =907.0294785.

Adding the present values gives $P=47.6190476+907.0294785=954.6485261$, or $954.65 to the nearest cent. As a check, the value one-half period earlier is $50/1.1025+1050/(1.1025)^2=\$909.1890725$. Accumulating it forward by $1.05$ gives $\$954.6485261$. The method uses the per-period yield $10.25\%$ and fractional powers of its accumulation factor; it does not treat $10.25\%$ as an annual yield.

Worked Example: Checking the Direction of Accumulation

A hypothetical zero-coupon bond pays $1,771.561 exactly three annual coupon periods after a preceding coupon date. The effective yield per period is $21\%$. Settlement occurs half a period after that preceding date. Find the settlement value, and verify it using both date-based approaches.

Method 1: value at the preceding coupon date, then accumulate forward. The value there is

V_0=\frac{1771.561}{(1.21)^3} =\frac{1771.561}{1.771561} =1000.

The value at settlement is later, so the preceding-date value accumulates for half a period: $P=1000(1.21)^{0.5}=1000(1.1)=\$1,100$.

Method 2: discount from the payment date to settlement. The payment is $3-0.5=2.5$ periods away. Since $(1.21)^{2.5}=(1.21)^2(1.1)=1.61051$,

P=\frac{1771.561}{1.61051}=1100.

Both methods give $1,100. This check avoids the common mistake of discounting the earlier value to obtain a later value. In particular, $1000/(1.21)^{0.5}=\$909.09$ is not the settlement value: it moves the value backward in time, not forward.

Boundary Checks and Common Traps

A useful formula check is to examine its behavior at the edges of a coupon period. If settlement is on the preceding coupon date, $f=0$, and the formula reduces to the ordinary coupon-date present value: the first coupon is one full period away. As settlement moves through the period, $f$ increases and each payment's remaining time $k-f$ decreases. With a positive yield, that makes each future payment's present value larger, all else equal.

As $f$ approaches 1, the settlement date approaches the next coupon date from before it. The first coupon's remaining time $1-f$ approaches zero, so its discounted value approaches the coupon amount. This limiting check helps catch a formula that mistakenly uses $k+f$ or $f$ in place of $k-f$.

Reversing the accumulation direction. The preceding-date value $V_0$ accumulates forward to settlement: $P=V_0(1+j)^f$. It is not divided by $(1+j)^f$.
Using elapsed time as time remaining. If $f=0.4$, the next coupon is $0.6$ periods away, not $0.4$ periods away.
Using a yield with the wrong period. The exponent is measured in coupon periods, so $j$ must be the effective yield per coupon period.
Discounting redemption from the wrong date. Redemption is paid with the final coupon and has the same remaining time $n-f$.

For example, if $V_0=100$, $j=21\%$, and $f=0.5$, the correct later settlement value is $100(1.21)^{0.5}=\$110$. The incorrect calculation $100/(1.21)^{0.5}=\$90.91$ discounts backward. Before calculating a bond price, state which date each value belongs to and whether the calculation moves forward or backward.

Key takeaway: For regular coupon periods, settlement $f$ periods after the preceding coupon date leaves $k-f$ periods until the $k$-th remaining coupon. Discount each cash flow over that remaining time, or calculate the preceding-date value and accumulate it forward by $(1+j)^f$. This treatment of timing is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, use the stated fractional-period convention and round monetary answers to the nearest cent.

A hypothetical bond has two annual coupons remaining, each $40, and a $1,000 redemption payment with the second coupon. The annual effective yield is $21\%$, and settlement is halfway through the coupon period. Find the price. (Answer: $817.73)
A hypothetical zero-coupon bond pays $1,210 two coupon periods after the preceding coupon date. The effective yield per period is $10.25\%$, and settlement is halfway through the period. Find the settlement price. (Answer: $1,045.24)
A hypothetical bond has two semiannual coupons remaining. Each coupon is $25, the redemption amount is $1,000, the effective yield per semiannual period is $10.25\%$, and settlement is halfway through a period. Find the price. (Answer: $909.24)
On the preceding coupon date, a hypothetical bond's value under a yield model is $700. The effective yield per coupon period is $21\%$, and the settlement value one-half period later is to be found. Calculate that settlement value and state whether the earlier value is accumulated or discounted. (Answer: $770.00; accumulated forward)

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