Question 2.S2.7: Mr X wishes to determine the present value of the annuity co......
Mr X wishes to determine the present value of the annuity consisting of cash inflows of Rs 1,000 per year for 5 years. The rate of interest he can earn from his investment is 10 per cent.
Table 2.8 shows the required calculations:
Table 2.8 shows the long way of determining the present value of annuity. This method is the same as the one adopted for mixed stream. This procedure yields a present value of Rs 3,791. However, calculations can be greatly cut short as the present value factor for each year is to be multiplied by the annual amount of Rs 1,000 . This method of calculating the present value of the annuity can also be expressed as an equation:
P=\text { Rs } 1,000(0.909)+\text { Rs } 1,000(0.826)+\text { Rs } 1,000(0.751)+\text { Rs } 1,000(0.683)+\text { Rs } 1,000(0.621)
= Rs 3,790.
Simplifying the equation by taking out 1,000 as common factor outside the equation,
P=\text { Rs } 1,000(0.909+0.826+0.751+0.683+0.621)=\text { Rs } 1,000(3.790)=\text { Rs } 3,790
Thus, the present value of an annuity can be found by multiplying the annuity amount by the sum of the present value factors for each year of the life of the annuity. Such ready-made calculations are available in Table A-4. This table presents the sum of present values for an annuity (PVIFA)/annuity discount factor (ADF) of Re 1 for wide ranges of interest rates, i, and number of years, n. From Table A-4 the sum ADF for five years at the rate of 10 per cent is found to be 3.791. Multiplying this factor by annuity amount (C) of Rs 1,000 in this example gives Rs 3,791. This answer is the same as the one obtained from the long method.
Now we can write the generalised formula to calculate the present value of an annuity:
\begin{aligned}\mathrm{P} & =\frac{C_{1}}{(1+i)}+\frac{C_{2}}{(1+i)^{2}}+\frac{C_{3}}{(1+i)^{3}}+\ldots+\frac{C_{n}}{(1+i)^{n}} \\& =C\left\{\frac{1}{(1+i)}+\frac{1}{(1+i)^{2}}+\frac{1}{(1+i)^{3}}+\ldots+\frac{1}{(1+i)^{n}}\right\} \\& =C\left\{\sum_{t=1}^{n} \frac{1}{(1+i)^{t}}\right\}\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad(2.7)\end{aligned}
The expression within brackets gives the appropriate annuity discount factor. Therefore, in more practical terms the method of determining present value is
P=C(A D F)=\operatorname{Rs} 1,000(3.791)=\text { Rs } 3,791
It may be noted that the interest factor for the present value of an annuity is always less than the number of years the annuity runs, whereas in case of compounding the relevant factor is larger than the number of years the annuity runs. The facts given in Example 2.7 can be shown graphically (Fig. 2.2).
Table A-4 can be easily applied to other problems relating to annuity also as shown in Example 2.8.
Table 2.8 Long Method for Finding Present Value of an Annuity of Rs 1,000 for Five Years
\begin{array}{cccc} \hline \text{Year end} & \text{Cash flows} & \text{Present value factor} & \text{Present value} (2) \times(3) \\ \hline 1 & 2 & 3 & 4 \\ \hline 1 & Rs 1,000 & 0.909 & 909.00 \\ 2 & 1,000 & 0.826 & 826.00 \\ 3 & 1,000 & 0.751 & 751.00 \\ 4 & 1,000 & 0.683 & 683.00 \\ 5 & 1,000 & 0.621 & 621.00 \\&&&3,790.00\\ \hline \end{array}