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Annuity loan

An annuity loan is paid with equal instalments. The loan today is the present value of all these amounts. The instalment can therefore be written as the loan times the interest, divided by 1 minus the growth factor raised to minus the number of periods. At the start a large part of the instalment is interest. Towards the end almost the whole amount is repayment. The interest in the formula is the interest per period, not necessarily per year.

a=L⋅r1−(1+r)−na = \frac{L \cdot r}{1 - (1 + r)^{-n}}instalment

Symbols

LLthe loanNOK
rrinterest per period
nnnumber of periods
aainstalmentNOK

Example

A loan of 1,000,000 NOK, 5% interest, 20 annual instalments.

a=1 000 000⋅0.051−1.05−20≈80 242.6a = \dfrac{1\,000\,000 \cdot 0.05}{1 - 1.05^{-20}} \approx 80\,242.6 NOK per year.

Rounded to the nearest krone this is 80,243 NOK, as in the course example.
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Part of Mathematics S2: Savings, loans and present value.