Level-Payment Loan Calculation Workshop

Introduction

A level-payment loan has the same scheduled installment at each payment date. Calculating that installment requires more than dividing the principal by the number of payments: each installment must also compensate for interest accumulated between payment dates. The central task is to value all installments at the loan date using a rate that matches the payment interval.

This workshop applies the present-value method to fixed-rate loans with different payment frequencies. It is aligned with the SOA VEE Accounting and Finance syllabus. The calculation assumes a fully amortizing loan: after the final scheduled payment, no balance or balloon payment remains.

Definition: A level-payment loan has an initial principal $P$, a fixed interest rate, and $n$ equal payments $R$ made at regular intervals. Unless stated otherwise, payments occur at the end of each interval, the first payment is one interval after the loan date, and there are no fees, additional principal payments, or final balloon payment.

Match the Rate to the Payment Interval

Let $j$ denote the effective interest rate for one payment interval. If the quoted rate is an effective annual rate $i$, and there are $m$ payment intervals per year, the equivalent interval rate is

j=(1+i)^{1/m}-1.

The exponent is needed because $m$ intervals at rate $j$ must accumulate to the same annual factor as the effective annual rate: $(1+j)^m=1+i$. In general, dividing an effective annual rate by $m$ does not produce the equivalent interval rate.

If a nominal annual rate $i^{(k)}$ is compounded $k$ times per year, each compounding period has rate $i^{(k)}/k$. When payments occur $m$ times per year, the equivalent rate per payment interval is

j=\left(1+\frac{i^{(k)}}{k}\right)^{k/m}-1.

When compounding and payment frequencies are the same, this simplifies to $j=i^{(k)}/k$. If they differ, first find the accumulation over one payment interval; do not use the compounding-period rate as though it were already the payment-period rate.

The Present-Value Equation

With end-of-interval payments, the first $R$ is paid one interval after the loan is advanced. At the loan date, the payments therefore have present values $R/(1+j), R/(1+j)^2,\ldots,R/(1+j)^n$. Their sum must equal the principal $P$.

P=R\sum_{t=1}^{n}(1+j)^{-t} =R\frac{1-(1+j)^{-n}}{j}, \qquad R=P\frac{j}{1-(1+j)^{-n}}.

The fraction $[1-(1+j)^{-n}]/j$ is the present value of an annuity-immediate of $n$ payments of one unit each, at rate $j$ per payment interval. The formula follows from the geometric series: the discounted payment factors have first term $(1+j)^{-1}$ and common ratio $(1+j)^{-1}$. Summing those $n$ terms gives $\{1-(1+j)^{-n}\}/j$. Multiplying by $R$ gives the present value of the actual installments, which must equal the amount borrowed.

The formula assumes that payments are level, regular, and made at the end of each interval. If payments are made at the beginning of each interval, their dates differ and the end-of-interval formula does not apply unchanged. Likewise, a fee added to the loan, a variable rate, or a balloon payment changes the cash flows that must be valued.

Count the payments.
Multiply the term in years by the number of payments per year to obtain $n$.

Find the interval rate.
Convert the quoted rate to an effective rate $j$ over one payment interval.

Set present value equal to principal.
Use $P=R[1-(1+j)^{-n}]/j$, then solve for $R$.

Round and check.
Round the calculated payment as requested. If payments are rounded to cents, their present value may differ slightly from the exact principal.

Worked Calculations

Each example treats the loan as a hypothetical fixed-rate agreement. Calculations retain extra precision until the final payment is rounded to the nearest cent.

Worked Example: Monthly Payments at an Effective Annual Rate

A hypothetical loan advances $18,000 and is repaid by equal monthly payments over two years. The effective annual interest rate is 6.0%. Find the monthly payment, rounded to the nearest cent.

Method. Convert the effective annual rate to an effective monthly rate, count the monthly installments, and equate their present value to the principal. This avoids treating 6.0% as a monthly rate or simply dividing it by 12.

Step 1: Count payments. There are $m=12$ monthly intervals per year over two years, so $n=12(2)=24$.

Step 2: Convert the rate. The monthly rate $j$ satisfies $(1+j)^{12}=1.06$, hence $j=1.06^{1/12}-1\approx0.00486755$, or approximately 0.4868% per month.

Step 3: Set up and solve the present-value equation.

18{,}000 =R\frac{1-(1.00486755)^{-24}}{0.00486755} \approx R(22.59937), \qquad R\approx\frac{18{,}000}{22.59937} \approx796.4825.

The monthly payment rounded to the nearest cent is $796.48. Using the unrounded payment, the present value is $18,000. Using 24 payments of $796.48 instead gives a present value of approximately $17,999.94; the small difference is caused by rounding the payment to cents, not by a different rate conversion.

Worked Example: Semiannual Payments with Matching Compounding

A hypothetical $10,000 loan is to be repaid in two equal installments, one at the end of each half-year. The quoted rate is 20% nominal per year, compounded semiannually. Calculate the installment to the nearest cent.

Method. Because compounding and payment intervals are both semiannual, use the semiannual compounding-period rate directly. This avoids mistakenly using the annual nominal quote as the rate for each installment period.

Step 1: Count payments. The term is one year, with two payments per year, so $n=2$.

Step 2: Find the payment-period rate. The nominal rate is compounded twice per year, so $j=0.20/2=0.10$, or 10% per half-year.

Step 3: Value the two payments. The present-value factor is $1/1.10+1/1.10^2=0.9090909+0.8264463=1.7355372$. Therefore,

10{,}000=R(1.7355372), \qquad R=\frac{10{,}000}{1.7355372} \approx5{,}761.9048.

The installment is $5,761.90. As a check, the exact rate and unrounded installment give a present value of $10,000. The rounded installments have a present value of approximately $9,999.99, a cent-level rounding difference.

Worked Example: Quarterly Payments with Monthly Compounding

A hypothetical loan of $10,000 is repaid by four equal quarterly payments over one year. The quoted rate is 12% nominal per year, compounded monthly. Find the quarterly payment to the nearest cent.

Method. First accumulate the monthly rate over the three months in one payment interval. Applying the monthly rate directly as a quarterly rate would undervalue the loan.

Step 1: Count payments. One year of quarterly payments gives $n=4$.

Step 2: Convert the rate. The monthly compounding rate is $0.12/12=0.01$. Three monthly periods occur in each quarter, so the effective quarterly rate is $j=(1.01)^3-1=0.030301$, or 3.0301% per quarter.

Step 3: Calculate the annuity factor. The annual accumulation over the four quarters is $(1.01)^{12}\approx1.12682503$, so $(1.01)^{-12}\approx0.88744923$. Thus the present-value factor is $(1-0.88744923)/0.030301\approx3.7144244$.

Step 4: Solve and check.

10{,}000=R(3.7144244), \qquad R\approx\frac{10{,}000}{3.7144244} \approx2{,}692.207.

The quarterly payment is $2,692.21. Multiplying the rounded payment by the annuity factor gives a present value of approximately $10,000.01, consistent with the cent rounding. The rate used is quarterly, not monthly: each quarterly discount period contains three monthly compounding periods.

Common Mistakes and Checks

A fast way to detect an incorrect result is to ask whether the rate, number of payments, and payment dates describe the same timeline. The principal is valued at the loan date; the first ordinary installment is one payment interval later. Both $j$ and $n$ must refer to that interval.

Wrong calculation: divide an effective annual rate by 12. For a 6% effective annual rate, this gives $0.06/12=0.005$. That rate accumulates to $(1.005)^{12}$, not to 1.06. Correction: use $j=1.06^{1/12}-1$ for monthly payments.
Wrong calculation: use the compounding-period rate when payment periods differ. In the quarterly-payment example, using 1% per quarter ignores the three monthly compounding periods in each quarter. Correction: convert to $(1.01)^3-1=0.030301$ per quarter.
Wrong timing: discount the first ordinary payment for zero periods. That treats it as due immediately, not at the end of the first interval. Correction: discount the first payment by $(1+j)^{-1}$. A payment made at the start requires a different timing model.
Wrong check: expect the rounded payment to reproduce the principal exactly. A payment rounded to cents can cause a small present-value difference. Correction: use the unrounded payment to verify the equation, then explain any small residual from cent rounding.

For a quick reasonableness check, a positive-rate loan with end-of-period payments should have a payment larger than $P/n$: the installments must repay principal and cover interest. Also, increasing the rate while holding principal, term, and payment frequency fixed increases the required payment; increasing the number of payments while holding the term and rate fixed usually reduces each installment.

Key takeaway: A level-payment calculation is a present-value equation. Count the installments, convert the quoted rate to the installment interval, value the end-of-interval payments, and solve for $R$. Keep rate conversion and payment timing consistent before rounding.

Check Your Understanding

For each problem, assume a fully amortizing loan with equal payments at the end of each stated interval, no fees, and no balloon payment. Round money to the nearest cent.

A $6,000 loan is repaid monthly over one year at a 12% nominal annual rate compounded monthly. Find the payment. (Answer: $533.09 per month.)
A $10,000 loan is repaid in two equal semiannual installments over one year. The nominal annual rate is 20%, compounded semiannually. Find the payment. (Answer: $5,761.90 per installment.)
A $12,000 loan is repaid quarterly over one year at a 12% nominal annual rate compounded monthly. Find the quarterly payment. (Answer: $3,230.65 per quarter.)
A $18,000 loan is repaid monthly over two years at an effective annual rate of 6.0%. Find the monthly payment. (Answer: $796.48 per month.)

What You'll Learn