If money is wroth i per period, then present value P of amount S due n periods hence is given by P = S(1 + i)⁻ⁿ.

Annuity: An annuity is a series of payments made at equal intervals of time. Examples of annuities are regular deposits to a savings account, monthly insurance payments, and pension payments.

In an ordinary annuity, the first payment is made at the end of the first payment period.

Amount of an Annuity:- The future value of an annuity is the value of a group of recurring payments at a certain date in the future, assuming a particular rate of return, or discount rate.

The future value S of an ordinary annuity of ₹R per period for n periods at the rate i per period is given by 

$S=R\left\{\frac{{\left(1+i\right)}^n-1}{i}\right\}$

Present Value of an Annuity:- The present value of an annuity is the current value of future payments from an annuity, given a specified rate of return, or discount rate.

The present value P of an ordinary annuity of ₹R per payment period for n periods at the rate i per period is given by 

$P=R\left\{\frac{1-{\left(1+i\right)}^{-n}}{i}\right\}$

Sinking Fund:-

A sinking fund is a fund created by a corporation or business organisation by putting money aside over time to meet a future capital outlay or the repayment of long-term debt. It is a fund that is built up with the intention of paying off a financial obligation at a later date.

It is an annuity created for accumulating money that can be used for paying off a financial obligation at some future predecided date.

For example, sometimes an individual or a company accumulates money, probably by periodic deposits, either to repay the principal of a loan in one installment or for the expansion of a business, etc.

Amount in a Sinking Fund:-

The amount in a sinking fund at any time is the amount of the annuity formed by the payments. Thus, the amount S in a sinking fund at any time is given by 

$$\begin{gathered} S=R\left\{\frac{{\left(1+i\right)}^n-1}{i}\right\} \\ R=\mathrm{Periodic} \mathrm{deposit} \mathrm{or} \mathrm{payment} \\ n=\mathrm{Number} \mathrm{of} \mathrm{periodic} \mathrm{deposits} \\ i=\mathrm{Interest} \mathrm{per} \mathrm{period} \end{gathered}$$

Sinking Fund Payment:-

The periodic payment of ₹R is required to accumulate a sum of ₹ S over n periods with interest charged at the rate i per period is given by
$R=\frac{iS}{\left\{{\left(1+i\right)}^n-1\right\}}$

Let us understand these topics with examples.

Example 1: How much should a company set aside at the end of each year, if it has to buy a machine expected to cost ₹100,000 at the end of 3 years and the rate of interest is 10% per annum compounded annually? ( Given ${\left(1.1\right)}^3=1.331$)

Solution: Let ₹ R be set aside at the end of each year. Since the company wants ₹100,000 at the end of 3 years. Therefore,
$$\begin{gathered} S=100,000, n=3, i=\frac{10}{100}=0.1 \\ \therefore R=\frac{iS}{{\left(1+i\right)}^n-1} \\ \Rightarrow R=\frac{0.1\times 100,000}{{\left(1.1\right)}^3-1} \\ \Rightarrow R=\frac{10000}{{\left(1.1\right)}^3-1} \\ \Rightarrow R=\frac{10000}{0.331}=30,211.48 \end{gathered}$$

Thus, to accumulate ₹100,000 after 3 years the company should keep aside ₹30,211.48 every year at 10% per annum compounded annually.

Example 2: A sinking fund is created for the redemption of debentures of ₹200,000 at the end of 25 years. How much money should be provided out of profits each year for the sinking fund, if the investment can earn interest 6% per annum? ( Given ${\left(1.06\right)}^{25}=4.2918$)

Solution: Suppose ₹R are provided out of profits each year for the sinking fund. Then,
$$\begin{gathered} S=200,000, n=25 \mathrm{and} i=\frac{6}{100}=0.06 \\ R=\frac{iS}{{\left(1+i\right)}^n-1} \\ \Rightarrow \frac{0.\mathrm{0\dots }}{} \end{gathered}$$