Balances Immediately Before and After a Payment

Why the Side of the Payment Matters

A loan balance is not fully specified by an amount alone: its timing matters. The balance immediately before a scheduled payment includes the interest accrued since the preceding payment and still includes the payment due at that instant. The balance immediately after the payment reflects both that interest and the reduction from the payment. Confusing these two values shifts the balance by an entire payment-period interest charge or by the payment itself.

This tutorial focuses on that distinction for payments made at the end of each interest period. The same distinction is useful when reading an amortization schedule, checking a payoff amount at a payment date, or interpreting an outstanding balance stated as “before” or “after” a payment. The calculations are aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Let $B_0$ be the loan balance immediately after the loan is advanced at time 0. Let $B_k$ be the balance immediately after payment $k$, and let $C_k$ be the balance immediately before payment $k$. If the effective interest rate per payment period is $i$, and payment $k$ is $P_k$, then, under end-of-period payment timing, $$ C_k=B_{k-1}(1+i), \qquad B_k=C_k-P_k. $$

The first equation accounts for one period of interest on the balance left after payment $k-1$. The second subtracts payment $k$, which is made at the end of that period. A stated annual rate must first be converted to the effective rate for the payment period if payments are more frequent than once a year. For example, if the effective annual rate is $i_{\text{annual}}$ and there are $m$ equal periods in the year, the equivalent effective rate per period is $(1+i_{\text{annual}})^{1/m}-1$, not generally $i_{\text{annual}}/m$.

The Order of the Calculation

For any scheduled payment, use the same sequence: begin with the post-payment balance from the prior period, accrue that period’s interest, then subtract the payment due. The balance just before payment $k$ is not the previous post-payment balance, because interest has accrued in the intervening period. Nor is it the new balance after payment $k$, because the scheduled payment has not yet been subtracted.

Start with the prior post-payment balance.
For payment $k$, the starting balance is $B_{k-1}$, immediately after payment $k-1$.

Accrue one period of interest.
Multiply by $1+i$ to obtain $C_k$, the balance immediately before payment $k$.

Subtract the scheduled payment.
Subtract $P_k$ from $C_k$ to obtain $B_k$, the balance immediately after payment $k$.

Worked Example: Finding Both Balances at the Third Payment

A hypothetical equipment loan has a balance of $12,000 immediately after the loan is advanced. The effective interest rate is 3% per quarter, and the scheduled payment is $2,500 at the end of each quarter. Find the balances immediately before and immediately after payment 3. Ignore fees and do not round intermediate amounts.

Method. Apply the balance recurrence period by period. Carry each post-payment balance forward, accrue 3% interest, and then subtract the next payment.

Payment 1. The pre-payment balance is $$ C_1=12{,}000(1.03)=12{,}360. $$ After the $2,500 payment, $$ B_1=12{,}360-2{,}500=9{,}860. $$

Payment 2. The pre-payment balance is $$ C_2=9{,}860(1.03)=10{,}155.80. $$ After payment 2, $$ B_2=10{,}155.80-2{,}500=7{,}655.80. $$

Payment 3. Interest for the third quarter is $0.03(7{,}655.80)=229.674$. Therefore, $$ C_3=7{,}655.80+229.674=7{,}885.474, $$ and $$ B_3=7{,}885.474-2{,}500=5{,}385.474. $$ Rounded to the nearest cent, the balance immediately before payment 3 is $7,885.47, and the balance immediately after it is $5,385.47.

Check. The pre-payment balance exceeds the prior post-payment balance of $7,655.80 by $229.674, exactly 3% of that prior balance. Subtracting the $2,500 payment gives the stated post-payment balance. A common mistake is to report $7,655.80 as the balance before payment 3; that is the balance after payment 2, before the third quarter’s interest accrues.

Why the Recurrence Holds

The recurrence follows directly from the timing of interest and payment. Under the standard model, the balance $B_{k-1}$ earns interest at the effective rate $i$ for one complete period. The interest charge is $iB_{k-1}$, so the amount owed just before the next payment is the original balance plus that charge. The payment then reduces the debt dollar for dollar.

C_k=B_{k-1}+iB_{k-1}=B_{k-1}(1+i), \qquad B_k=C_k-P_k.

This argument assumes the balance is subject to the stated rate for the entire period, the payment is made at the period’s end, and no other transactions occur in the period. If a fee, extra payment, rate change, or payment-date adjustment occurs, it must be included at its actual time. If the payment exceeds the amount due, the recurrence produces a negative mathematical balance; an actual lender would generally limit the payment to the amount needed to discharge the debt, subject to the contract.

Checking the Timing With Remaining Payments

For a fully amortizing loan with level end-of-period payments, the balances at a payment date can also be checked by valuing the payments still due. Suppose there are $n$ payments of amount $P$, at effective rate $i$ per payment period. Immediately before payment $k$, the payment due at time $k$ is still included, as are payments $k+1$ through $n$. Immediately after payment $k$, only payments $k+1$ through $n$ remain.

Result: For $1\leq k\leq n$, the pre-payment and post-payment balances are $$ C_k=P\ddot{a}_{\overline{n-k+1}|}, \qquad B_k=Pa_{\overline{n-k}|}, $$ where the annuity values use the effective rate $i$ per payment period. If $k=n$, there are no payments after the final payment, so $B_n=0$.

To justify the pre-payment expression, discount the payment due immediately at time $k$ by no periods, the next payment by one period, and each later payment by its number of periods back to time $k$. The resulting values are $P(1+v+\cdots+v^{n-k})$, where $v=1/(1+i)$, which is $P\ddot{a}_{\overline{n-k+1}|}$. Immediately after payment $k$, the first remaining payment is one period away. Its value, together with the values of all later payments, is $P(v+\cdots+v^{n-k})=Pa_{\overline{n-k}|}$. These formulas rely on level payments, a constant effective rate, and full amortization; the recurrence itself also works for varying scheduled payments.

Worked Example: Verifying the Balance From Future Payments

A hypothetical loan is repaid by three annual payments of $1,000 at each year-end. The effective annual rate is 10%, and the loan is fully amortized by payment 3. Find the balances immediately before and after payment 2, and check them using both the recurrence and the remaining-payment values. Round only the final amounts to the nearest cent.

Method. First find the initial loan amount as the present value of the three payments. Then use the recurrence to reach payment 2 and compare with the value of the payments remaining at that date.

The discount factor is $v=1/1.10$. Thus $v=0.9090909$, $v^2=0.8264463$, and $v^3=0.7513148$, with additional digits retained in the calculations. The original loan amount is $$ L=1{,}000(v+v^2+v^3) =1{,}000(0.9090909+0.8264463+0.7513148) \approx 2{,}486.85199. $$ So the initial loan amount is $2,486.85 to the nearest cent.

Before payment 1, the balance is $2{,}486.85199(1.10)\approx 2{,}735.53719$. After payment 1, $$ B_1=2{,}735.53719-1{,}000\approx 1{,}735.53719. $$ Before payment 2, $$ C_2=1{,}735.53719(1.10)\approx 1{,}909.09091. $$ After payment 2, $$ B_2=1{,}909.09091-1{,}000\approx 909.09091. $$ Therefore, the balances are $1,909.09 immediately before payment 2 and $909.09 immediately after payment 2.

Check from remaining payments. Immediately before payment 2, the $1,000 payment due now and the $1,000 payment due one year later have values $1{,}000+1{,}000v=1{,}909.09091$. Immediately after payment 2, only the final $1,000 payment remains, whose value one year earlier is $1{,}000v=909.09091$. These agree with the recurrence. The common timing error is to treat the post-payment value as though it still includes the payment just made.

Worked Example: A Balance Given After Payment 2

Immediately after payment 2 on a hypothetical monthly loan, the balance is $5,000. The effective interest rate is 4% per month. Payment 3 is $1,200 and payment 4 is $900. Find the balance immediately before and immediately after payment 4. Assume there are no intervening fees or extra payments, and round the final answers to the nearest cent.

Method. First advance the post-payment-2 balance through month 3 and subtract payment 3. Then advance the resulting balance through month 4 and subtract payment 4.

Immediately before payment 3, the balance is $$ C_3=5{,}000(1.04)=5{,}200. $$ After payment 3, $$ B_3=5{,}200-1{,}200=4{,}000. $$ The balance immediately before payment 4 is therefore $$ C_4=4{,}000(1.04)=4{,}160. $$ After the $900 payment, $$ B_4=4{,}160-900=3{,}260. $$ Thus the balance immediately before payment 4 is $4,160.00, and immediately after it is $3,260.00.

Check. The difference between the balance after payment 3 and the balance before payment 4 is $160, which is 4% of $4,000. Subtracting payment 4 then reduces the balance by exactly $900. A common mistake is to report $4,000 as the balance before payment 4. That amount is the balance immediately after payment 3; the fourth month’s interest must still be added.

Common Timing Errors and Practical Checks

The phrase “balance at payment $k$” can be ambiguous unless the convention is stated. In a schedule, label the two values separately or say explicitly “immediately before payment $k$” and “immediately after payment $k$.” The following checks help detect a timing error:

Interest check: $C_k-B_{k-1}=iB_{k-1}$. If this fails, the interest period or rate may be wrong.
Payment check: $C_k-B_k=P_k$. If this fails, the payment may have been added, omitted, or assigned to the wrong date.
Direction check: For a positive balance and positive rate, $C_k$ should exceed $B_{k-1}$. A positive payment should then make $B_k$ lower than $C_k$.
Final-payment check: If the loan is fully amortized and the payment is the final payment, the balance immediately after it should be zero, apart from any specified rounding adjustment.

These checks do not replace the loan contract’s terms. They apply to the standard model of periodic interest followed by a scheduled payment, with no other balance changes in the period. A payoff quoted on a different date, a payment in advance, or an amount that includes fees must be handled according to its stated timing and terms.

Key takeaway: At the end of period $k$, calculate interest first to obtain the balance immediately before payment $k$, $C_k=B_{k-1}(1+i)$. Then subtract the payment to obtain the balance immediately after it, $B_k=C_k-P_k$. Keeping those two moments distinct is the essential timing discipline in an amortization schedule.

Check Your Understanding

For each question, identify the requested balance as immediately before or after the stated payment. Use the effective rate per payment period and assume no unmentioned transactions.

A loan has a balance of $9,000 immediately after the advance. The effective quarterly rate is 2%, and each quarterly payment is $1,500. Find the balances immediately before and after payment 2. (Answer: Before payment 2: $7,833.60; after payment 2: $6,333.60.)
Immediately after payment 3, a loan balance is $4,200. The effective monthly rate is 3%, and payment 4 is $800. Find the balances immediately before and after payment 4. (Answer: Before payment 4: $4,326.00; after payment 4: $3,526.00.)
A loan balance immediately after payment 2 is $8,614. The effective rate is 6% per semiannual period, and payment 3 is $4,000. Find the balances immediately before and after payment 3. (Answer: Before payment 3: $9,130.84; after payment 3: $5,130.84.)
A fully amortized loan is repaid with five annual payments of $700. The effective annual rate is 5%. What are the balances immediately before and after the fifth payment? (Answer: Before payment 5: $700.00; after payment 5: $0.00.)
Immediately after payment 2 on a quarterly loan, the balance is $5,000. The effective quarterly rate is 4%; payment 3 is $1,200, and payment 4 is $900. Find the balances immediately before and after payment 4. (Answer: Before payment 4: $4,160.00; after payment 4: $3,260.00.)

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