A Two-Method Workshop for Level-Payment Loans

Why Use Two Methods?

An outstanding loan balance is the amount needed, at a specified date, to settle the loan under its stated payment terms and interest assumptions. For a level-payment loan, there are two standard ways to find that amount: value the remaining payments at the balance date, or accumulate the original loan and payments already made to that date. Both methods must give the same result when they use the same rate, timeline, and payment convention.

The prospective method is often quick when the remaining installments form an annuity. The retrospective method is a useful independent check, especially when the original principal and past payments are known. This workshop focuses on using both methods for a level-payment loan and explaining why their results agree.

Definition: Suppose a loan of amount $L$ is advanced at time 0 and repaid by $n$ equal payments of amount $R$, made at the end of each payment period. Let $i$ be the effective interest rate per payment period. The outstanding balance $B_k$, immediately after the $k$th payment, is the value at time $k$ of the payments and loan cash flows still relevant under the agreed schedule. For $0 \leq k \leq n$, the prospective and retrospective formulas are $$ B_k=R a_{\overline{n-k}|} $$ and $$ B_k=L(1+i)^k-Rs_{\overline{k}|}, $$ where the first formula values payments $k+1,\ldots,n$ at time $k$, and the second accumulates the original loan and the first $k$ payments to time $k$.

These formulas assume payments are level, are made at the end of each period, and occur on the same regular schedule used to define $i$. The balance is measured immediately after payment $k$. A balance immediately before that payment includes it among the future payments instead. For $k=n$, the prospective balance is zero after the final payment.

Set the Timeline and Rate First

Before using either formula, identify the loan date, each payment date, the balance date, and whether the balance is just before or just after a payment. Then express the interest rate per payment period. The effective annual rate $i$ can be used directly only when payments are annual. If the stated rate is nominal, convert it to an effective rate over the payment interval before calculating.

For example, if the nominal annual rate $i^{(m)}$ is convertible $m$ times per year and payments are made every $r$ months, with $r$ a whole-number multiple of the compounding interval, the effective rate per payment period is $$ \left(1+\frac{i^{(m)}}{m}\right)^{mr/12}-1. $$ This conversion assumes the stated nominal rate and compounding convention apply throughout the loan. If payment and compounding schedules do not align, the equivalent effective rate for the actual payment interval must still be found; using the nominal annual figure directly as the periodic rate is not valid.

Mark the balance date.
Count how many payments have occurred and state whether the balance is immediately before or after one of them.

Convert the rate.
Find the effective rate $i$ for one payment period, using the stated nominal compounding convention if applicable.

Calculate prospectively.
Discount only the payments that remain after the balance date.

Check retrospectively.
Accumulate the original principal to the balance date and subtract the accumulated value of payments already made.

Why the Two Formulas Agree

The prospective and retrospective methods are two ways of valuing the same scheduled cash flows. To see the connection, first write the original principal as the present value of all $n$ end-of-period payments. The payments already made are accumulated forward to time $k$; the remaining payments are valued at time $k$.

L=R a_{\overline{n}|} =R a_{\overline{k}|}+R v^k a_{\overline{n-k}|}, \qquad v=\frac{1}{1+i}.

Multiplying by $(1+i)^k$ moves the equation’s values to time $k$. Since $a_{\overline{k}|}(1+i)^k=s_{\overline{k}|}$, the result can be rearranged as follows:

L(1+i)^k =R s_{\overline{k}|}+R a_{\overline{n-k}|} \quad\Longrightarrow\quad L(1+i)^k-Rs_{\overline{k}|} =R a_{\overline{n-k}|}.

The left side is the retrospective balance; the right side is the prospective balance. The identity depends on using the same payment dates and rate in both calculations. It does not justify mixing, for example, a monthly rate in one method with an annual rate in the other.

Worked Example: A Balance After Two Semiannual Payments

A borrower receives a loan and agrees to make four semiannual payments of $1,000, each at the end of a half-year. The effective interest rate is 5% per half-year. Find the original loan amount and the balance immediately after the second payment using both methods. This example uses the prospective value of the remaining installments and checks it retrospectively.

Step 1: Set up the timing. Payments occur at times 1, 2, 3, and 4 half-years. The balance is at time 2, immediately after the second payment. Thus, payments at times 3 and 4 remain.

Step 2: Find the original loan amount. The four payments have a present value at time 0:

L=1000a_{\overline{4}|} =1000\left(\frac{1-(1.05)^{-4}}{0.05}\right) =1000(3.5459505) =\$3,545.95.

Step 3: Calculate prospectively. At time 2, the two remaining payments have present value

B_2=1000a_{\overline{2}|} =1000\left(\frac{1-(1.05)^{-2}}{0.05}\right) =1000(1.8594104) =\$1,859.41.

Step 4: Check retrospectively. Accumulate the original loan for two periods, then subtract the accumulated value at time 2 of the two payments already made:

B_2=3,545.9505(1.05)^2-1000s_{\overline{2}|} =3,545.9505(1.1025)-1000(2.05) =3,909.4104-2,050 =\$1,859.41.

Both calculations give $1,859.41. The common mistake this example avoids is including the payment at time 2 as a remaining payment: the balance is measured immediately after that installment.

Worked Examples With Rate Conversion and Reconciliation

Worked Example: Annual Payments With Monthly Compounding

A two-year loan is repaid by two equal payments of $1,000, each made at the end of a year. The nominal annual interest rate is 12% convertible monthly. Find the original loan amount and the balance immediately after the first payment. Use the prospective method and verify by the retrospective method. The key method is to convert the nominal rate to the annual effective rate before valuing annual payments.

Step 1: Convert the rate. The monthly rate is $0.12/12=0.01$. There are 12 monthly compounding periods in one annual payment period, so

i=(1.01)^{12}-1=0.1268250.

Step 2: Find the principal. The two annual payments form an annuity-immediate at this effective annual rate:

L=1000a_{\overline{2}|} =1000\left(\frac{1-(1.1268250)^{-2}}{0.1268250}\right) \approx 1000(1.6750154) =\$1,675.02.

Step 3: Value the remaining payment at time 1. Immediately after the first payment, only the second payment remains:

B_1=1000v=\frac{1000}{1.1268250} \approx \$887.45.

Step 4: Verify retrospectively. Accumulate the original principal by one year and subtract the first payment:

B_1=L(1+i)-1000 \approx 1,675.0154(1.1268250)-1,000 =\$887.45.

The two methods reconcile. The common mistake avoided here is treating 12% nominal convertible monthly as a 12% effective annual rate; the annual effective rate is approximately 12.6825%.

Worked Example: Solve for the Payment, Then Reconcile the Balance

A loan of $10,000 is repaid by three equal annual payments at the end of each year. The effective annual interest rate is 6%. Find the level payment and the balance immediately after the first payment, using both balance methods. Carry unrounded values through the calculations and report money to the nearest cent.

Step 1: Find the payment. The principal equals the present value of the three payments:

10,000=R a_{\overline{3}|}, \qquad a_{\overline{3}|} =\frac{1-(1.06)^{-3}}{0.06} =2.6730119.

Therefore,

R=\frac{10,000}{2.6730119}\approx \$3,741.10.

Step 2: Calculate prospectively. Immediately after the first payment, two annual payments remain. Using the unrounded payment, approximately $3,741.0981:

B_1=R a_{\overline{2}|} =3,741.0981\left(\frac{1-(1.06)^{-2}}{0.06}\right) =3,741.0981(1.8333927) \approx \$6,858.90.

Step 3: Check retrospectively. The original loan accumulates to $10,600 at time 1. Subtract the first payment:

B_1=10,000(1.06)-3,741.0981 =10,600-3,741.0981 \approx \$6,858.90.

The balance agrees to the cent. The common mistake avoided is rounding the installment before using it in one method but not the other. Here the exact level payment is used throughout; a contract that requires installments rounded to cents may need a separate rounding convention.

Common Timing and Rate Traps

A correct formula with the wrong date or rate still gives the wrong balance. For the first worked example, the balance is immediately after payment 2. A mistaken prospective calculation might include the payment at time 2 and value payments at times 2, 3, and 4:

\text{Wrong: }1000+1000v+1000v^2 =1000(1+a_{\overline{2}|}) =\$2,859.41.

That is not the balance after payment 2, because it includes an installment that has already been paid. The correction is to value only payments at times 3 and 4:

\text{Correct: }1000v+1000v^2 =1000a_{\overline{2}|} =\$1,859.41.

Other useful checks are straightforward: a balance immediately after the final payment should be zero, and a balance just after an earlier payment should generally be positive when future payments remain. If the prospective and retrospective results disagree, inspect the balance date, rate conversion, and payment count before changing the formulas.

Key takeaway: For a level-payment loan with end-of-period payments, value the remaining installments prospectively or accumulate the original loan and subtract accumulated past payments retrospectively. Both methods agree when the rate, payment timing, and balance date are consistent.

Check Your Understanding

For each problem, identify the rate per payment period and balance date before calculating. Report dollar amounts to the nearest cent.

A loan is repaid by three annual end-of-year payments of $500. The effective annual rate is 4%. Find the original principal and the balance immediately after the first payment. (Answer: Principal $1,387.55; balance $943.05.)
A loan is repaid by two annual payments of $2,000, at each year-end. The nominal annual rate is 12% convertible monthly. Find the original principal and the balance immediately after the first payment. (Answer: Effective annual rate 12.6825%; principal $3,350.03; balance $1,774.90.)
A $8,000 loan has an effective annual rate of 5% and is repaid by four equal annual end-of-year payments. Find the payment and the balance immediately after the second payment. (Answer: Payment $2,256.09; balance $4,195.01.)
A loan is repaid by six quarterly end-of-quarter payments of $900. The nominal annual rate is 8% convertible quarterly. Find the balance immediately after the fourth payment using the prospective method, and state the retrospective expression that should give the same result. (Answer: Quarterly rate 2%; balance $1,747.40; retrospective expression $L(1.02)^4-900s_{\overline{4}|}$, with $L=900a_{\overline{6}|}$.)

What You'll Learn