Why a Loan Balance Has a Recurrence
An outstanding loan balance is not fixed between payments: interest increases the amount owed, while a payment reduces it. The recurrence for the balance records those two changes in their time order. Once the balance at one payment date is known, the recurrence gives the balance at the next payment date without recalculating the entire loan.
The timing convention matters. In this tutorial, each payment is made at the end of a payment period, after interest for that period has been applied. The periodic rate must therefore correspond to the interval between payments. For example, a monthly payment requires a monthly effective rate, not an annual rate applied directly to each month.
When the payment is level, write \(R_k=R\) for every period, so the recurrence becomes \(B_k=B_{k-1}(1+j)-R\). A positive \(B_k\) means a balance remains after the payment; zero means the loan is exactly paid off. If a payment exceeds the amount due after interest, this simple recurrence produces a negative number. In practice, loan terms commonly limit the final payment to the amount owed, so a negative computed balance can signal an overpayment that needs to be capped or reconciled.
Deriving the Balance Update
The update follows directly from the order of events within one period. At the start of period \(k\), the amount owed is \(B_{k-1}\). Interest at rate \(j\) adds \(jB_{k-1}\), making the amount due immediately before the payment \(B_{k-1}+jB_{k-1}\). The payment \(R_k\) then reduces that amount.
This derivation also shows why the balance update is a recurrence: the next balance depends on the immediately preceding balance. The formula does not assume that payments are level; it works with changing payments as long as each \(R_k\) is specified at the end of its period. It does not apply unchanged to a beginning-of-period payment, a period with a different interest rate, or a loan where fees or other adjustments change the amount owed. Those situations require their timing or adjustment to be represented explicitly.
Use \(B_0\), the amount borrowed, or the known balance immediately after the previous payment.
Multiply the current balance by \(1+j\), where \(j\) is the effective rate for one payment period.
Deduct the end-of-period payment from the amount due after interest to obtain the new balance.
Worked Balance Updates
Keep the balance convention consistent throughout a calculation: \(B_k\) is measured after payment \(k\), while \(B_{k-1}(1+j)\) is the amount due before payment \(k\). Carry unrounded values through successive periods and round money only at the requested final step.
Worked Example: Monthly Payments at a Nominal Rate
A hypothetical borrower takes a loan of $10,000 at a nominal annual rate of 12% convertible monthly. The borrower pays $900 at the end of each month. Find the balances immediately after the first three payments. Assume no fees, no additional borrowing, and no rounding between periods.
Method. Convert the quoted nominal rate to the effective rate per monthly payment period, then apply the recurrence. The monthly rate is $$ j=\frac{0.12}{12}=0.01. $$ The initial balance is \(B_0=\$10{,}000\), and the level payment is \(R=\$900\).
After the first month, interest increases the balance to \(10{,}000(1.01)=\$10{,}100\). Subtract the payment: $$ B_1=10{,}000(1.01)-900=10{,}100-900=\$9{,}200. $$
For month 2, start with \(B_1=\$9{,}200\). The amount due before payment is \(9{,}200(1.01)=\$9{,}292\), so $$ B_2=9{,}200(1.01)-900=9{,}292-900=\$8{,}392. $$
For month 3, the amount due before payment is \(8{,}392(1.01)=\$8{,}475.92\). Therefore, $$ B_3=8{,}392(1.01)-900=8{,}475.92-900=\$7{,}575.92. $$
Check and common mistake avoided. Each new balance is the prior balance increased by 1% and then reduced by $900; the balances decline, as expected for these data. Using 12% as the monthly rate would incorrectly apply an annual nominal quote to a single month. The correct periodic rate is 1%.
Worked Example: Quarterly Payments and Successive Balances
A hypothetical business borrows $24,000 at a nominal annual rate of 8% convertible quarterly. It makes an end-of-quarter payment of $3,000 each quarter. Calculate the balance after the first three payments. Assume the rate and payment remain unchanged, with no fees or extra payments.
Method. Because the nominal rate is convertible quarterly, the rate per quarter is $$ j=\frac{0.08}{4}=0.02. $$ Starting from \(B_0=\$24{,}000\), update one quarter at a time. In quarter 1, interest gives \(24{,}000(1.02)=\$24{,}480\); after the payment, $$ B_1=24{,}000(1.02)-3{,}000=24{,}480-3{,}000=\$21{,}480. $$
In quarter 2, the amount due before payment is \(21{,}480(1.02)=\$21{,}909.60\). Thus $$ B_2=21{,}480(1.02)-3{,}000=21{,}909.60-3{,}000=\$18{,}909.60. $$
In quarter 3, the amount due before payment is \(18{,}909.60(1.02)=\$19{,}287.792\). Subtracting the payment gives $$ B_3=18{,}909.60(1.02)-3{,}000=\$16{,}287.792. $$ Rounding the final balance to the nearest cent gives $16,287.79.
Check and common mistake avoided. The quoted rate is divided by four because it is nominal and convertible quarterly; the $3,000 payment is subtracted once per quarter. Rounding is postponed until the final balance, so intermediate rounding does not alter the successive updates.
Worked Example: A Loan with Changing Payments
A hypothetical equipment loan has an initial balance of $18,000 and an effective annual rate of 5%. Payments are made at each year-end: $2,500 after year 1, $4,000 after year 2, and $6,000 after year 3. Find the balance immediately after each of those payments. Assume there are no fees or other changes to the loan.
Method. The payment interval is one year, so the period rate is \(j=0.05\). Use the recurrence with the actual payment in each period; do not treat these changing payments as level.
After year 1, interest increases the balance to \(18{,}000(1.05)=\$18{,}900\). Subtract the first payment: $$ B_1=18{,}000(1.05)-2{,}500=18{,}900-2{,}500=\$16{,}400. $$
After year 2, the amount due before payment is \(16{,}400(1.05)=\$17{,}220\). Subtract the second payment: $$ B_2=16{,}400(1.05)-4{,}000=17{,}220-4{,}000=\$13{,}220. $$
After year 3, the amount due before payment is \(13{,}220(1.05)=\$13{,}881\). Subtract the third payment: $$ B_3=13{,}220(1.05)-6{,}000=13{,}881-6{,}000=\$7{,}881. $$
Check and common mistake avoided. The balances are positive, so a balance remains after each payment. Applying $2,500 to every period would ignore the stated second- and third-year payments; the recurrence requires the actual \(R_k\) for each period.
Payment Timing and Common Errors
The recurrence encodes a specific order: first interest for the period, then the end-of-period payment. A useful way to check an update is to calculate the amount due before payment separately. It must equal the prior balance multiplied by \(1+j\); the new balance must then equal that amount less the payment.
For example, suppose a balance is $10,000, the period rate is 1%, and the payment is $900 at the end of the period. The correct update is $$ 10{,}000(1.01)-900=\$9{,}200. $$ A common timing error is to subtract the payment first and then apply interest: $$ (10{,}000-900)(1.01)=\$9{,}191. $$ That calculation treats the payment as if it were made at the beginning of the period, so it is inconsistent with the stated end-of-period timing. The $9 difference is the 1% interest on the $900 that was incorrectly removed before the period began.
Other checks help catch errors quickly:
- Confirm the rate period matches the payment period before updating the balance.
- Confirm the balance being multiplied is the balance after the previous payment.
- Subtract the current period’s payment once, after applying the period’s interest.
- For a positive rate and positive balance, the amount due before payment should exceed the prior balance.
- Do not assume the balance must decline: if the payment is less than the period’s interest, the balance increases.
The recurrence is aligned with the SOA VEE Accounting and Finance syllabus because it provides a precise way to track how loan cash flows change the amount outstanding from one payment date to the next. It is a period-by-period update, not a substitute for stating the loan’s rate, payment timing, and payment amounts.
Check Your Understanding
For each problem, use the stated payment timing and keep full precision until the final answer.
- A hypothetical loan has \(B_0=\$5{,}000\), an effective monthly rate of 1%, and a $600 payment at each month-end. Find the balance after two payments. (Answer: $3,894.50)
- A loan of $12,000 has a nominal annual rate of 6% convertible monthly and a $800 end-of-month payment. Find the balance after two payments. (Answer: $10,516.30)
- A hypothetical quarterly loan starts at $20,000. Its nominal annual rate is 10% convertible quarterly, and its payment is $2,500 at each quarter-end. Find the balance after three payments. (Answer: $13,848.75)
- A loan has a $9,000 balance, an effective annual rate of 4%, and payments of $1,000 at the end of year 1 and $1,500 at the end of year 2. Find the balance after the second payment. (Answer: $7,194.40)
- A hypothetical loan has a balance of $8,000 immediately before a period with an effective rate of 3%. What end-of-period payment will leave a balance of exactly $7,200? (Answer: $1,040)