Fractional-Period Balance Calculations

Why Fractional-Period Balances Need a Convention

Loan payments are often scheduled at regular dates, but a balance may be needed on a date between them: for example, to settle a loan early or to compare a payment with a balance quoted on another date. The scheduled payment dates alone do not determine how interest accrues over a fraction of a payment period. A calculation must use the fractional-period convention specified in the problem or loan terms.

The central rule is to use that convention consistently. Under compound fractional-year accumulation at effective annual rate $i$, a balance grows for $s$ years by the factor $(1+i)^s$. Other conventions can define the within-year growth differently. In particular, do not substitute a fractional-period factor into only one side of a prospective or retrospective calculation while leaving the other cash flows under a different rule.

Definition: A fractional-period balance is the outstanding loan balance at a date between scheduled payment dates. It is calculated by valuing the loan cash flows at that date under a specified accumulation convention. The balance includes interest accrued or allowed for the fraction of a period, but excludes any payment not yet made.

Fractional Compounding Under an Effective Annual Rate

Suppose a loan has an effective annual rate $i$, and the specified convention is compound fractional-year accumulation. The accumulation factor over $t$ years is $(1+i)^t$, including when $t$ is not an integer. If $B_k$ is the balance immediately after the $k$th payment, and no payment occurs during the next $s$ years, then the balance at time $k+s$ is $B_k(1+i)^s$, for $0<s<1$.

B_{k+s}=B_k(1+i)^s

This shortcut is for moving a known balance forward over an interval with no intervening payments. If the requested date is before a payment, the upcoming payment is not subtracted early. The same result can be obtained prospectively by discounting the future payments to the fractional date.

Condition: The factor $(1+i)^s$ applies when the stated fractional-period convention is compound accumulation based on the effective annual rate. If a problem specifies a different within-period rule, use that rule instead; do not assume fractional compounding automatically.

Worked Example: A Half-Year Balance on a Two-Payment Loan

A loan is issued at an effective annual rate of 21%. It is repaid by two payments of $12,100, at the ends of years 1 and 2. Use compound fractional-year accumulation. Find the original loan amount and the balance at time 1.5, immediately before any payment on that date.

Method. First find the loan amount as the present value of the two payments. Then value the remaining payment at time 1.5. The fractional-year accumulation factor for half a year is $(1.21)^{0.5}=1.1$, since $1.1^2=1.21$.

Original loan amount. Discount the payment at time 1 by one year and the payment at time 2 by two years:

L=\frac{12{,}100}{1.21}+\frac{12{,}100}{1.21^2} =10{,}000+8{,}264.4628 =18{,}264.4628

The original loan amount is $18,264.46, rounded to the nearest cent.

Balance at time 1.5. The only remaining payment is $12,100 at time 2, which is half a year after the balance date. Discount it by the half-year factor:

B_{1.5}=\frac{12{,}100}{(1.21)^{0.5}} =\frac{12{,}100}{1.1} =11{,}000

As a check, the balance immediately after the first payment at time 1 is $12{,}100/1.21=10{,}000$. Accumulating that balance for half a year gives $10{,}000(1.1)=11{,}000$, the same result. This check avoids treating the future payment as if it were due at time 1.5.

Prospective and Retrospective Valuation at a Fractional Date

A prospective calculation values payments still to come at the balance date. A retrospective calculation accumulates the original loan and subtracts past payments accumulated to that date. Both methods can be used at a fractional date, provided every cash flow is moved to that date under the same accumulation convention.

To state this precisely, let $A(t)$ denote the accumulation factor from time 0 to time $t$, with $A(0)=1$. Under compound fractional-year accumulation at effective annual rate $i$, $A(t)=(1+i)^t$. The accumulation factor from time $u$ to a later time $t$ is $A(t)/A(u)$. Therefore, if $L$ is the original loan and payments $P_j$ have been made at times $t_j$, the retrospective balance at time $t$ is:

B(t)=L A(t)-\sum_{t_j\le t}P_j\frac{A(t)}{A(t_j)}

For payments still to come at times $t_j>t$, the prospective balance is:

B(t)=\sum_{t_j>t}P_j\frac{A(t)}{A(t_j)}

These formulas describe the same balance when the loan cash flows satisfy the original loan equation and the same $A(t)$ is used throughout. A payment exactly at the balance date requires care about whether the balance is requested immediately before or immediately after that payment; the timing convention must be stated. The calculations here use dates strictly between scheduled payments.

Derivation: A payment $P_j$ made at time $t_j$ has value $P_j/A(t_j)$ at time 0. Its value at time $t$, under the same accumulation law, is $P_jA(t)/A(t_j)$. Thus the original loan accumulated to $t$, less the accumulated values of payments already made, gives the retrospective balance. Equivalently, the remaining payments valued at $t$ give the prospective balance. The ratio $A(t)/A(t_j)$ ensures that cash flows are moved between dates using the same accumulation rule.

Worked Example: Retrospective Balance with an Unequal Payment

A loan of $5,000 accrues at an effective annual rate of 21%, using compound fractional-year accumulation. The borrower pays $1,000 at time 1 and will pay the remaining amount at time 2. Find the balance at time 1.5 using past cash flows, then verify it using the remaining payment.

Method. Accumulate the original loan to time 1.5, then subtract the first payment accumulated from time 1 to time 1.5. The half-year factor is $1.1$, and the factor from time 0 to 1.5 is $1.21(1.1)=1.331$.

Retrospective calculation. The original loan accumulated to time 1.5 is $5{,}000(1.331)=6{,}655$. The $1,000 payment at time 1 accumulates for half a year to $1{,}000(1.1)=1{,}100$. Subtract:

$$ B(1.5)=5{,}000(1.331)-1{,}000(1.1) =6{,}655-1{,}100 =5{,}555 $$

Prospective check. The payment due at time 2 is the amount required to bring the balance at time 1 forward one-half year to time 2:

P_2=5{,}555(1.1)=6{,}110.50

Discounting this payment by the same half-year factor gives $6{,}110.50/1.1=5{,}555$, confirming the balance. The result is $5,555.00. The check avoids discounting the final payment by a full year when it is only half a year after the balance date.

When the Stated Convention Is Not Fractional Compounding

A problem may specify that interest within a payment period is allocated by a rule other than fractional compounding. For example, it may define the accumulated value $s$ years after the start of an annual period as $1+si$, where $i$ is the effective annual rate. This rule gives the balance at that point in the period, but the factor from that fractional date to the period end is not generally $1+(1-s)i$. Instead, if the accumulation from the start of the year is $1+si$ and the full-year factor is $1+i$, the factor from the fractional date to year-end must be their ratio, $(1+i)/(1+si)$.

The ratio matters because accumulation between two dates must be consistent with the stated values at both dates. Multiplying $1+si$ by $1+(1-s)i$ generally does not give $1+i$. When a convention is given, represent it with its accumulation factors and move each cash flow using the ratio between the factor at the destination and the factor at the starting date.

Worked Example: A Stated Simple Allocation Within the Year

A $1,000 loan is due in one payment of $1,100 at the end of one year. The annual effective rate is 10%. The contract specifies that the balance $s$ years after the start of the year is the original amount multiplied by $1+0.10s$. Find the balance at half a year and verify it from the year-end payment.

Method. Apply the contract’s specified within-year balance rule. Do not replace it with compound fractional-year accumulation.

Balance at half a year. With $s=0.5$, the factor from the start of the year is $1+0.10(0.5)=1.05$. Thus:

$$ B(0.5)=1{,}000(1.05)=1{,}050 $$

Prospective check. The accumulation factor from time 0 to time 1 is $1.10$, while the factor from time 0 to time 0.5 is $1.05$. Therefore, the factor from time 0.5 to time 1 is $1.10/1.05$. Discount the $1,100 due at time 1 by this ratio:

\frac{1{,}100}{1.10/1.05} =1{,}100\left(\frac{1.05}{1.10}\right) =1{,}050

The check agrees, so the balance is $1,050.00. Using a separate half-year discount factor of $1/1.05$ would be wrong here: the specified rule gives a factor of $1.05$ from time 0 to 0.5, not from time 0.5 to time 1.

Common Fractional-Period Traps

The most reliable way to avoid errors is to mark the balance date and each payment date on a timeline before calculating. Then identify the time from each cash flow to the balance date and use one consistent accumulation convention.

Using the wrong time unit. If the rate is effective annual and the date is three months later, the fractional time is $0.25$ years, not 3 years or 3 periods.
Mixing conventions. Do not grow a past balance using compound fractional-year accumulation and discount a future payment using a separately assumed simple-interest factor.
Counting a payment too soon. A payment scheduled after the balance date remains a future payment. It is discounted to the balance date; it is not subtracted as though already paid.
Using the wrong interval. From time $k$ to $k+s$ the interval is $s$, while from $k+s$ to $k+1$ it is $1-s$. A payment at $k+1$ is only $1-s$ years after the balance date.
Rounding too early. Keep unrounded factors and balances through the calculation. Round money only at the final step unless the problem specifies otherwise.

Fractional-period balance calculations are useful whenever a loan must be valued away from a scheduled payment date. The key is not to memorize a single factor for every situation, but to identify the stated accumulation law, measure each time interval correctly, and use the same law for past and future cash flows. This approach is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each question, use the convention stated in the problem and round money to the nearest cent.

At an effective annual rate of 21%, a balance immediately after a payment is $2,500. Under compound fractional-year accumulation, find the balance one quarter of a year later. (Answer: $2,622.02)
A monthly nominal rate of 9% is convertible monthly. Under compound fractional-month accumulation, a balance is $4,200 immediately after a monthly date. Find the balance half a month later. (Answer: $4,215.72)
A $4,000 loan accrues at an effective annual rate of 21%, with compound fractional-year accumulation. A $700 payment is made at time 1. Find the retrospective balance at time 1.5. (Answer: $4,554.00)
An $8,000 loan has an effective annual rate of 8%. The contract specifies that the balance $s$ years after the start of the year is the opening balance multiplied by $1+0.08s$. Find the balance six months into the year. (Answer: $8,320.00)
A single payment of $10,800 is due at the end of a year. The contract specifies an accumulation factor of $1+0.08s$ from the year’s start to $s$ years into the year. Find the balance at time 0.5 by prospective valuation. (Answer: $10,400.00)

What You'll Learn