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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

$$P=\frac{L}{a_{\overline{n}|i}}$$

Interest in payment t

$$I_t=P\left(1-v^{n-t+1}\right)$$

Principal in payment t

$$P_t=P{v}^{n-t+1}$$

Principal repayments grow

$$P_{t+1}=P_t(1+i)$$

Totals

$$\sum _{t=1}^n{P}_t=L,\sum _{t=1}^n{I}_t=nP-L$$

Where it comes from

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

Worked example

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

• Outstanding loan balance
• Sinking funds
• Annuity-immediate

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