Sinking Fund and EMI

Two of the most useful applications of annuities are the sinking fund (saving up for a future lump sum) and the EMI (repaying a loan) — one is an accumulated-value problem solved for $C$, the other a present-value problem solved for $C$.

Sinking fund

A sinking fund is a fund built up by equal periodic deposits so as to accumulate a required sum $A$ (to replace an asset, redeem a debt, etc.) by a target date. Setting the accumulated value equal to $A$ and solving for the deposit $C$:

$C=A\left[\frac{i}{(1+i)^{n}-1}\right].$

EMI (Equated Monthly / periodic Instalment)

An EMI is the fixed equal instalment that repays a loan of principal $P$ (interest $+$ principal) over $n$ periods. Since the loan equals the present value of the instalments, solve the present-value formula for $C$:

$\text{EMI}=P\left[\frac{i(1+i)^{n}}{(1+i)^{n}-1}\right].$

Here $i$ is the rate per instalment period — for a monthly EMI at an annual rate $R\%$, use $i=\dfrac{R}{100\times12}$ and $n=$ number of months.