Worked Examples of Prospective Level-Loan Balances

Why the Payment Date Matters

A level-payment loan has equal installments made at regular intervals. Its outstanding balance changes with each installment: interest accrues between payments, and each payment covers interest and reduces principal. To calculate a balance by the prospective method, value the payments still due at the date being considered.

The key practical task is to identify that date precisely. Immediately after the $k$th payment, the first remaining installment is due one payment period later. If the loan has $n$ installments in total, there are $n-k$ installments still to be paid. Counting the original installments rather than the remaining ones is a common source of errors.

Definition: The outstanding balance immediately after a scheduled payment is the value, at that time, of the borrower’s remaining obligation under the loan. For a level-payment loan with end-of-period installments of amount $P$, periodic effective interest rate $i$, and $n-k$ installments remaining after payment $k$, the prospective balance is $P a_{\overline{n-k}|}$, where $a_{\overline{m}|}=(1-(1+i)^{-m})/i$ for $i\ne 0$. If $i=0$, the balance is $P(n-k)$.

Model and Timing Convention

Unless a problem states otherwise, use the following model: the installments are equal, paid at the end of each period; the effective interest rate $i$ is constant for each payment period; and there are no fees, skipped payments, or changes to the loan terms. The interest-rate period must match the payment period. For example, a monthly installment requires a monthly effective rate in the annuity factor.

Let $B_k$ denote the balance immediately after payment $k$. There are $n-k$ future payments, and their due dates are one, two, and then successive periods after the balance date. Therefore, the balance is their present value at that date:

B_k=P\left(v+v^2+\cdots+v^{n-k}\right) =P a_{\overline{n-k}|}, \qquad v=\frac{1}{1+i}.

This expression applies to a balance immediately after a scheduled payment. A date immediately before a payment has a different timing relationship: the payment due on that date has not yet been made. Here, keep the valuation date immediately after the specified installment unless the problem explicitly says otherwise.

Why the Prospective Formula Holds

At the balance date, each future installment is a contractual payment due at a known future time. Under the stated constant-rate model, a payment of $P$ due one period later has present value $Pv$; one due two periods later has present value $Pv^2$, and so on. Adding these values gives the obligation at the balance date. The geometric-series identity then gives the annuity factor.

B_k =P\sum_{j=1}^{n-k}v^j =P\frac{v(1-v^{n-k})}{1-v} =P\frac{1-v^{n-k}}{i} =P a_{\overline{n-k}|}.

For the third equality, $1-v=i/(1+i)=iv$, so $v/(1-v)=1/i$. This derivation assumes $i\ne 0$. When $i=0$, every future payment is already expressed in balance-date dollars, and their value is simply $P(n-k)$. The prospective formula does not directly apply if the installments are unequal, the rate changes, or payments are not at regular period ends; those cases require a cash-flow valuation consistent with their actual terms.

Worked Examples

Worked Example: Balance After the Sixth Monthly Payment

A loan is repaid by 14 monthly installments of $300. The effective monthly interest rate is 1%. Find the balance immediately after the sixth installment. State the method and avoid counting installments already paid.

Method. Use the prospective method. Six of the 14 installments have been made, so the number remaining is $14-6=8$. The balance date is immediately after payment six; consequently, the first of those eight payments is due one month later.

The annuity factor is

a_{\overline{8}|} =\frac{1-(1.01)^{-8}}{0.01} \approx 7.651678.

Thus,

B_6=300a_{\overline{8}|} \approx 300(7.651678) =\$2{,}295.5034.

Rounded to the nearest cent, the balance is $2,295.50. The calculation is consistent with the payment schedule: it values exactly eight future installments, not all 14 original installments or the six already paid.

Common mistake avoided: using $300a_{\overline{14}|}$ or $300a_{\overline{6}|}$. The first values the original series at its start; the second values only six installments. Neither counts the eight payments due after payment six.

Worked Example: Find the Installment, Then a Later Balance

Immediately after a payment, a loan has a balance of $10,000 and six level quarterly installments remaining. The effective quarterly interest rate is 2%. Find the installment to the nearest cent, then find the balance immediately after the second of those future installments. Keep full precision in intermediate calculations and round money only at the end.

Method. First treat the known balance as the present value of six future payments. Then, after two of those payments have been made, value the four payments still due.

The six-payment annuity factor and the level installment are

a_{\overline{6}|} =\frac{1-(1.02)^{-6}}{0.02} \approx 5.60143089, \qquad P=\frac{10{,}000}{5.60143089} \approx \$1{,}785.258.

The installment rounded to cents is $1,785.26. For the later balance, two installments have been paid from the original six, leaving $6-2=4$. The four-payment factor is

a_{\overline{4}|} =\frac{1-(1.02)^{-4}}{0.02} \approx 3.8077287.

Using the unrounded installment in the balance calculation,

B_2 =\left(\frac{10{,}000}{a_{\overline{6}|}}\right)a_{\overline{4}|} \approx 6{,}797.78.

The balance immediately after the second future installment is $6,797.78. If the loan is actually administered with the installment rounded to $1,785.26, the final scheduled balance can differ slightly from an unrounded theoretical schedule. Here the requested balance uses the unrounded installment implied by the stated $10,000 balance.

Common mistake avoided: rounding the installment before using it in the next calculation can introduce avoidable drift. Also, the later balance uses four payments remaining, not the six payments that were remaining at the first valuation date.

Worked Example: Compare Balances at Two Payment Dates

A hypothetical loan is repaid by 12 monthly installments of $500. The effective monthly rate is 1%. Find the balance immediately after payment eight and immediately after payment ten. Then check that the change between the two balances is consistent with the payment schedule.

Method. Apply the prospective formula separately at each date. After payment eight, four installments remain; after payment ten, two remain.

For payment eight, the factor and balance are

a_{\overline{4}|} =\frac{1-(1.01)^{-4}}{0.01} \approx 3.90196555, \qquad B_8=500(3.90196555) \approx \$1{,}950.98.

For payment ten,

a_{\overline{2}|} =\frac{1-(1.01)^{-2}}{0.01} \approx 1.97039506, \qquad B_{10}=500(1.97039506) \approx \$985.20.

To check the schedule, grow the balance after payment eight for one month, then subtract payment nine:

B_9=1{,}950.9828(1.01)-500 \approx \$1{,}470.49.

Grow that balance for another month and subtract payment ten:

B_{10}=1{,}470.4926(1.01)-500 \approx \$985.20.

The recurrence check agrees with the prospective result after rounding to cents. The balances decrease as payments are made, but the check also accounts for interest accruing between payment dates.

Common mistake avoided: comparing balances by subtracting two installments without allowing for interest. A balance first accumulates interest for the period; only then is the next installment deducted.

Choosing the Correct Annuity Factor

For a level-payment loan, the prospective calculation is short, but its inputs must all refer to the same valuation date and payment schedule. Before calculating, identify the payment amount, the effective rate per payment period, and the number of installments remaining after the payment at the balance date. Then use the annuity-immediate factor for that remaining count.

Mark the balance date.
Write whether the date is immediately after a particular scheduled payment. Do not leave the payment timing implicit.

Count future installments.
If the date is immediately after payment $k$ of $n$, calculate $m=n-k$. The first remaining payment is one period after the date.

Match rate and period.
Use the effective rate for one interval between installments. Convert a quoted rate if needed before substituting it into the annuity factor.

Value the remaining series.
Calculate $P a_{\overline{m}|}$, or solve $P=B/a_{\overline{m}|}$ if the balance and remaining count are given.

Check the result.
Confirm that the factor has $m$ payments, that the balance is in the expected range, and, where useful, that one payment-period update is consistent with the schedule.

A balance should be less than the undiscounted total of the remaining installments when $i>0$, because those payments are due in the future. It should also be positive when the loan has positive installments remaining. These checks do not replace calculation, but they can reveal a wrong payment count or a misplaced decimal.

Common Exam Traps

Counting paid installments as future payments. After payment $k$, there are $n-k$ installments left. Use that count in the annuity factor.
Using the wrong date convention. The balance immediately after a payment excludes that payment from future cash flows. The next installment is one full period away.
Mixing rate and payment periods. A nominal annual rate convertible monthly is not itself the monthly effective rate. Convert to the appropriate period rate before calculating.
Rounding too early. Keep the installment and annuity factor unrounded through intermediate steps unless the problem explicitly says actual payments are rounded.
Forgetting interest between payments in a check. To move a balance forward one period, multiply by $1+i$, then subtract the payment made at the end of that period.

Key takeaway: Under a constant periodic rate and level end-of-period installments, the balance immediately after a scheduled payment is the present value of the installments still due: $B_k=P a_{\overline{n-k}|}$. Correct date placement and installment counting are essential to using the formula correctly.

Check Your Understanding

For each problem, identify the balance date and rate period before calculating.

A loan has 11 monthly installments of $240, and the effective monthly rate is 0.5%. Find the balance immediately after the fifth payment. (Answer: $240a_{\overline{6}|}\approx \$1{,}415.13$.)
A loan has nine quarterly installments of $500 at an effective quarterly rate of 2%. Find the balance immediately after the fourth payment. (Answer: $500a_{\overline{5}|}\approx \$2{,}356.73$.)
Immediately after a payment, a loan balance is $15,000, with six annual installments remaining. The effective annual rate is 5%. Find the level annual installment to the nearest cent. (Answer: $15{,}000/a_{\overline{6}|}\approx \$2{,}955.25$.)
A loan is repaid by 20 monthly installments of $180. Its stated rate is 12% nominal, convertible monthly. Find the balance immediately after the 12th payment. (Answer: $180a_{\overline{8}|}$ at 1% monthly, approximately $\$1{,}377.30$.)
Immediately after the third payment, a loan has seven monthly installments remaining. The effective monthly rate is 1%, and each installment is $1,200. Find the balance immediately after the fifth payment. (Answer: $1{,}200a_{\overline{5}|}\approx \$5{,}824.12$.)

What You'll Learn