Accruing Interest on Missed Installments

Why the Rules for an Unpaid Installment Matter

A missed loan installment does not have one universal value at a later date. Its value depends on the contract’s specified treatment of arrears: the interest rate, the accrual period, whether interest is capitalized, any grace period, and whether a payment is made before or after interest is assessed. A calculation can use the correct rate and still be wrong if it counts the wrong number of periods.

This tutorial treats an unpaid installment as an amount due under the loan’s stated rules. It does not assume that the amount is automatically treated as unpaid principal in the ordinary amortization schedule. The loan agreement determines how arrears accrue; that rule may differ from the interest calculation on the remaining loan principal.

Definition: An unpaid installment is a scheduled payment that remains unpaid after its due date. Its later obligation is the amount required under the specified arrears rules to settle that installment at a stated later date. The later obligation may include interest on the unpaid amount, but only if the stated rules provide for that interest.

Specify the Accrual Model Before Calculating

For a correct valuation, first write down what happens during each period. In particular, determine when the unpaid installment starts accruing interest, whether interest is added to the arrears balance, and whether a payment occurs before or after the period’s interest is assessed. Do not infer these details from the loan’s ordinary payment schedule.

If the applicable rate is nominal annual interest $i^{(m)}$, convertible $m$ times per year, the rate per conversion period is $i^{(m)}/m$. If that rate is applied to the overdue balance and interest is capitalized each period, the balance after $n$ periods is the original installment multiplied by the accumulation factor for those periods.

j=\frac{i^{(m)}}{m}, \qquad B_n=P(1+j)^n

Here, $P$ is the unpaid installment at the start of the accrual, $j$ is the rate per accrual period, $n$ is the number of completed periods, and $B_n$ is the resulting obligation. This formula assumes a constant periodic rate, no intervening payments, and interest capitalized at the end of each period. It does not apply unchanged if the contract specifies simple interest, a grace period, a different rate basis, or payments during the interval.

With simple interest and no capitalization, interest is calculated only on the original unpaid amount. If the simple annual rate is $r$ and the overdue interval is $t$ years, the amount due is $P(1+rt)$. This is not the same as compounding at a periodic rate. The stated convention—not a preference for one method—determines which calculation is appropriate.

Why Period-by-Period Capitalization Gives the Formula

Suppose interest at rate $j$ is assessed at each period-end and added to the overdue balance. At the first period-end, the balance is $P+jP=P(1+j)$. At the next period-end, interest is assessed on that new balance, giving $P(1+j)(1+j)=P(1+j)^2$. Repeating the same step once per completed period gives the accumulation formula.

B_0=P,\qquad B_{k+1}=B_k(1+j)

By repeated substitution, $B_1=P(1+j)$, $B_2=P(1+j)^2$, and, after $n$ periods, $B_n=P(1+j)^n$. The exponent counts periods for which the overdue balance earns interest. A period that has not yet ended contributes no period-end interest under this model.

Timing rule: Count completed accrual periods from the date interest begins to the valuation or payment date. If payment is made at the period-end after interest is assessed, include that period’s interest. If payment is made before the period-end assessment, do not include it.

Worked Examples

Worked Example: Monthly Capitalization on a Missed Installment

A hypothetical loan installment of $1,200 is due at the end of month 0 and is not paid. Under the stated arrears rule, interest is charged at a nominal annual rate of 12% convertible monthly and capitalized at each month-end. The installment is paid at the end of month 5, immediately after the month-end interest assessment. Find the payment required to settle this installment. Ignore fees.

Method. Convert the nominal annual rate to a monthly rate, count the completed accrual periods, and accumulate the installment. Since the installment is due at month 0 and payment follows the month-5 assessment, there are five monthly periods.

The monthly rate is

j=\frac{0.12}{12}=0.01.

Thus, 1% is added to the overdue balance at each of months 1, 2, 3, 4, and 5. The obligation is

$$ B_5=1200(1.01)^5 =1200(1.0510100501) =1261.2120601. $$

Rounded to the nearest cent, the required payment is $1,261.21. The calculation uses five periods, not six: the due date at month 0 is the starting point, not an additional month of accrual.

Check. The interest added is $1,261.21 − $1,200.00 = $61.21. The first month’s interest is $12.00, and later months charge interest on the capitalized balance, so total interest should exceed the $60.00 that five months of simple interest would produce. It does.

Common mistake avoided: Using 12% as the monthly rate would treat a nominal annual rate as though it were a monthly rate. The correct monthly rate is $12\%/12=1\%$.

Worked Example: A Grace Period Before Interest Begins

A hypothetical installment of $800 is due at the end of month 2. The agreement gives a grace period of two full months after the due date; no interest accrues during those two months. Beginning at the end of month 4, interest is charged at 1% per month on the unpaid amount and capitalized monthly. The installment is paid at the end of month 9, after that month’s interest assessment. Find the amount required to settle it.

Method. Identify the date interest starts, then count only the monthly periods in which interest is assessed. The two full grace months run from month 2 to month 4. Accrual begins after the grace period, so the interest-bearing periods are month 4 to 5, 5 to 6, 6 to 7, 7 to 8, and 8 to 9: five periods.

The monthly rate is already specified as $j=0.01$. The obligation at month 9 is

$$ B=800(1.01)^5 =800(1.0510100501) =840.8080401. $$

Rounded to the nearest cent, the required payment is $840.81.

Check. There are five, not seven, interest-bearing months. The amount is greater than $800 because interest accrues after the grace period, but it does not include interest for the two grace months.

Common mistake avoided: Counting from the original due date as if interest began immediately ignores the stated grace period. A grace period changes the start date for accrual; it does not change the installment’s due date.

Worked Example: Valuing Two Missed Installments at One Date

A hypothetical borrower misses two installments of $500 each. The first is due at the end of month 2 and the second at the end of month 4. Under the agreement, each unpaid installment earns interest at 1% per month, capitalized at month-end starting immediately after its own due date. Neither installment is paid before the end of month 7. Find the combined obligation just after the month-7 interest assessments.

Method. Accumulate each installment separately from its own due date to the common valuation date. The first installment accrues for five periods, from month 2 to month 7. The second accrues for three periods, from month 4 to month 7.

For the first installment,

$$ B_1=500(1.01)^5 =500(1.0510100501) =525.5050251. $$

For the second installment,

$$ B_2=500(1.01)^3 =500(1.030301) =515.1505. $$

The combined obligation is

$$ B_1+B_2 =525.5050251+515.1505 =1040.6555251. $$

Rounded to the nearest cent, the total is $1,040.66.

Check. The total exceeds the original $1,000 by $40.66. The first installment has accumulated for two more months than the second, so its accumulated value ($525.51) is greater. Adding the original installments first and applying one common exponent would incorrectly give them the same accrual duration.

Common mistake avoided: Do not treat installments due on different dates as one amount accruing from a single date. Each installment has its own start date and its own number of periods.

Interest Accrual Is Not Automatically the Same as Capitalization

The words “interest accrues” do not, by themselves, specify whether unpaid interest earns interest in later periods. Capitalization means that assessed interest is added to the balance on which subsequent interest is calculated. If a contract instead specifies simple interest on the original installment, the interest does not itself earn interest.

For example, consider a $1,200 installment overdue for five months at a simple annual rate of 12%. With time measured in years, the simple-interest amount is

1200\left(1+0.12\frac{5}{12}\right) =1200(1.05) =1260.

That gives $1,260.00, compared with $1,261.21 when interest is capitalized monthly at 1%. Both calculations can be correct under their respective rules. It is incorrect to choose between them without reading the stated capitalization convention.

Common Exam Traps

Using the loan rate without checking the arrears rule. The interest charged on missed installments may be specified separately from the rate used to calculate scheduled loan payments.
Miscounting periods. Mark the due date, accrual start date, and valuation or payment date. Count only complete periods during which the stated rule applies.
Confusing simple interest with capitalization. Simple interest applies to the original installment; capitalization applies interest to the growing balance.
Ignoring payment timing. A payment made after a period-end assessment includes that period’s interest; a payment made before the assessment does not.
Combining installments too early. If due dates differ, accumulate each amount over its own overdue interval before adding the resulting values.

Key takeaway: For each unpaid installment, identify its amount, accrual start date, periodic rate, capitalization rule, and settlement timing. Calculate its value to the target date using only the periods and interest treatment specified, then combine amounts that are valued at that same date.

Check Your Understanding

For each problem, use only the arrears rules stated. Round money to the nearest cent.

A $900 installment is due now and is paid after six completed quarterly periods. Interest is nominal 8% per year convertible quarterly and is capitalized at each quarter-end. Find the settlement amount. (Answer: $1,013.55)
A $1,500 installment is overdue for four months. The agreement specifies simple interest at 9% per year, with no capitalization. Find the amount due. (Answer: $1,545.00)
A $600 installment is due at the end of month 1. No interest accrues during the first two full months after its due date. Thereafter, interest is 1% per month, capitalized at month-end. The installment is paid just after the month-7 assessment. Find the amount due. (Answer: $624.36)
Two $400 installments are due at the ends of months 0 and 3. Each earns 0.5% per month, capitalized monthly from immediately after its due date. Find the combined obligation just after the month-8 assessment. (Answer: $826.38)
A $1,000 installment is due at the end of month 2 and remains unpaid until just after the end-of-month-10 assessment. Interest is nominal 6% per year convertible monthly and capitalized monthly, beginning immediately after the due date. Find the amount due. (Answer: $1,040.71)

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