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Finance maths

Loan repayment formula

The fixed payment that pays off a loan, interest included, in N equal instalments.

M=Pi1−(1+i)−N, i=rk

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05101520253020000400006000080000100000120000140000160000180000200000half repaid
  • Monthly payment 1,199.10; total interest 231,676.38.
  • Half the balance is repaid after 21 years: early payments are mostly interest.

What the symbols mean

M
the payment each period
P
the amount borrowed (principal)
r
the annual interest rate as a decimal (6% = 0.06)
k
payments per year (12 for monthly)
N
the total number of payments (30 years monthly is 360)
i
the interest rate per payment period, r/k

When to use it

Use it for mortgages, car loans and other amortising loans with equal payments and a fixed rate. Solving for N shows how long a given payment takes to clear a debt; solving for r finds the rate implied by a quoted payment. The rate has no algebraic rearrangement, so the calculator finds it numerically. Real loans may add fees and use their own rate conventions, so treat results as a check, not a quote.

Worked example

A mortgage of 200,000 is repaid monthly (k = 12) over 360 payments at 6% a year. What is the monthly payment?

  1. Start from the formula.

    M=P⁢rk1−(1+rk)−N

  2. The rate per payment period:

    i=rk
    =0.0612
    =0.005

  3. Substitute and work it out:

    M=P⁢i1−(1+i)−N
    =200,000.00·0.0051−(1+0.005)−360
    =1,000.001−1.005−360
    ≈1,199.10

  4. Total paid over the loan:

    total=M⁢N
    ≈1,199.10·360
    ≈431,676.38

  5. Total interest:

    interest=M⁢N−P
    ≈1,199.10·360−200,000.00
    ≈231,676.38

Answer M ≈ 1,199.10 a month; total interest ≈ 231,676.38

Why it works

The loan equals the present value of all the payments: P=M1−(1+i)−Ni, the formula for the present value of an annuity. Solving that for M gives this formula. With a 0% rate it becomes M=P/N.

Common mistakes

  • Using the annual rate as i: monthly payments need i=0.06/12=0.005.
  • Using years for N: a 30-year monthly loan has N = 360 payments.
  • Thinking each payment reduces the balance equally: early payments are mostly interest (see the graph).
  • Dropping the minus sign in (1+i)−N, which gives a nonsensical payment.