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Loan amortisation

A level-payment loan is an annuity seen from the lender's side; the payment is L/aₙ and every later question is about splitting one payment into interest and principal.

The formulas

Level payment
P=Lan‾∣iP = \dfrac{L}{a_{\overline{n}|i}}
Interest in payment t
It=P(1−v n−t+1)I_t = P\left(1 - v^{\,n-t+1}\right)
Principal in payment t
Pt=P v n−t+1P_t = P\,v^{\,n-t+1}
Principal repayments grow
Pt+1=Pt(1+i)P_{t+1} = P_t(1+i)
Totals
∑t=1nPt=L,∑t=1nIt=nP−L\sum_{t=1}^{n} P_t = L, \qquad \sum_{t=1}^{n} I_t = nP - L

Where it comes from

  1. The loan is the present value of the payments: L=P an‾∣L = P\,a_{\overline{n}|}, which rearranges to P=L/an‾∣P = L/a_{\overline{n}|}.
  2. Immediately after payment t−1t-1 the outstanding balance is P an−t+1‾∣P\,a_{\overline{n-t+1}|}, so the interest due next is iP an−t+1‾∣=P(1−vn−t+1)iP\,a_{\overline{n-t+1}|} = P(1 - v^{n-t+1}).
  3. The rest of the payment is principal: Pt=P−It=Pvn−t+1P_t = P - I_t = P v^{n-t+1}, which grows by exactly (1+i)(1+i) each period.

Worked example

A 250,000 loan is repaid with level annual payments over 20 years at 5.5% effective. Find the interest portion of the 8th payment.

  1. a20‾∣0.055=11.950382a_{\overline{20}|0.055} = 11.950382, so P=250,000/11.950382=20,919.83P = 250{,}000/11.950382 = 20{,}919.83.
  2. I8=P(1−v20−8+1)=P(1−1.055−13)I_8 = P\left(1 - v^{20-8+1}\right) = P\left(1 - 1.055^{-13}\right).
  3. 1.055−13=0.4985611.055^{-13} = 0.498561, so I8=20,919.83×0.501439=10,490.03I_8 = 20{,}919.83 \times 0.501439 = 10{,}490.03.
  4. The remaining 10,429.8010{,}429.80 of that payment reduces the balance.

Answer: 10,490.03

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

  • Principal repayments form a geometric series with ratio (1 + i). Find one, and you have them all.
  • Iₜ + Pₜ = P always. If you can get one, subtract.

Traps

  • Using n − t instead of n − t + 1 in the exponent.
  • Computing interest on the ORIGINAL balance rather than the outstanding balance.

Related

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