What if payments are made at the beginning of the period?

This is an annuity due. Calculate the standard ordinary annuity payment, then divide the result by $(1+r)$.

How do I handle interest-only loans?

For an interest-only loan, the payment is simply $P imes r$, and the principal is paid in a single lump sum at the end of the term.

What if the compounding frequency differs from the payment frequency?

Calculate the effective interest rate for the payment period using the formula $(1 + r_{comp})^{m/k} - 1$, where $m$ is the compounding periods per year and $k$ is the payments per year.

How to calculate a loan payment

To calculate a loan payment, use the present value of an ordinary annuity formula solved for the payment amount. This method applies to any amortizing loan with fixed periodic payments, a fixed interest rate, and a set term, such as standard mortgages and car loans.

The setup

Define the variables: $P$ is the principal loan amount, $r$ is the periodic interest rate, $n$ is the total number of payments, and $PMT$ is the periodic payment. The governing equation is $P = PMT \left[ \frac{1 - (1+r)^{-n}}{r} \right]$ . Rearranging for the payment yields $PMT = P \left[ \frac{r}{1 - (1+r)^{-n}} \right]$ .

The steps

Identify the principal $P$ . 2. Determine the periodic interest rate $r$ by dividing the annual rate by the number of compounding periods per year. 3. Calculate total periods $n$ by multiplying the loan term in years by the number of periods per year. 4. Substitute $P$ , $r$ , and $n$ into the rearranged formula $PMT = P \left[ \frac{r}{1 - (1+r)^{-n}} \right]$ . 5. Compute the final value for $PMT$ .

Checking the result

Multiply the calculated payment $PMT$ by the total number of periods $n$ . The resulting total amount paid must be greater than the original principal $P$ . The difference $(PMT imes n) - P$ represents the total interest paid over the life of the loan.

Common errors

The most frequent error is using the annual interest rate instead of the periodic rate for $r$ . Another common mistake is omitting the negative sign on the exponent $-n$ . Finally, rounding intermediate steps rather than keeping full precision until the final step will cause cent-level inaccuracies.

Worked example

Calculate the monthly payment for a 5-year, $20,000 car loan with an annual interest rate of 6%.

The principal $P = 20000$ . 2. The annual rate is 0.06, so the monthly rate $r = \frac{0.06}{12} = 0.005$ . 3. The term is 5 years with monthly payments, so $n = 5 imes 12 = 60$ . 4. The formula is $PMT = 20000 \left[ \frac{0.005}{1 - (1+0.005)^{-60}} \right]$ . 5. Compute the denominator: $1 - (1.005)^{-60} \approx 1 - 0.74137 = 0.25863$ . 6. Compute the payment: $PMT = 20000 \left[ \frac{0.005}{0.25863} \right] \approx 386.66$ . The monthly payment is $386.66.

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References: Principles of Corporate Finance, Brealey, Myers, and Allen · OpenStax Principles of Finance, Chapter 4: Time Value of Money