Question 2.S2.3: Mr X deposits Rs 2,000 at the end of every year for 5 years ......
Mr X deposits Rs 2,000 at the end of every year for 5 years in his saving account paying 5 per cent interest compounded annually. He wants to determine how much sum of money he will have at the end of the 5 th year.
Table 2.6 presents the relevant calculations:
The calculations in this case can be cut short and simplified since the compound interest factor is to be multiplied by the same rupee amount (Rs 2,000) each year as shown in the following calculations:
Amount at the end of 5 years = Rs 2,000(1.216)+ Rs 2,000(1.158)+ Rs 2,000(1.103)+ Rs 2,000(1.050)
+ Rs 2,000(1.000)
Taking out the common factor Rs 2,000,= Rs 2,000(1.216+1.158+1.103+1.050+1.000)
= Rs 2,000(5.527)= Rs 11,054 .
From the above, it follows that in order to find the sum of the annuity, the annual amount must be multiplied by the sum of the appropriate compound interest factor annuity (CVIFA). Such calculations are available for a wide range of i and n. They are given in Table A-2, labelled as the sum of an annuity table. To find the answer to the annuity question of Example 2.3, we are required to look for the 5 per cent column and the row for the fifth year and multiply the factor by the annuity amount of Rs 2,000. From the table we find that the sum of annuity of Re 1 deposited at the end of each year for 5 years is 5.526 (CVIFA). Thus, when multiplied by Rs 2,000 annuity (A) we find the total sum as Rs 11,052.
Symbolically, S n=C V I F A \times A
where A is the value of annuity, and CVIFA represents the appropriate factor for the sum of the annuity of Re 1 and S n represents the compound sum of an annuity. The answer which we get from the long method was Rs 11,054. This discrepancy can be attributed to the rounding off of values in Table A-2. Moreover, it may be noted that the sum of an annuity is always larger than the number of years the annuity runs, unless of course, when interest rate is zero; in the latter case it will equal the number of years.
Annuity tables are of great help in the field of investment banking as they guide the depositors and investors as to what sum an amount (\mathrm{X}) paid for number of years, n, will accumulate to at a stated rate of
compound interest. Let us illustrate. Mr \mathrm{X} wishes to know the sum of money he will have in his saving account which pays 5 per cent interest at the end of 12 years if he deposits Rs .1,000, at the end of each year for the next twelve years. The appropriate factor for the sum of a twelve-year annuity at 5 per cent as given in Table \mathrm{A}-2 is 15.917. Multiplying this factor by Rs 1,000 deposit, we find the resultant sum to be Rs 15,917.
Table 2.6 Annual Compounding of Annuity
\begin{array}{ccccc} \hline \text{End of year} & \text{Amount } & \text{Number of years } & \text{Compounded interest factor } & \text{Future value} \\ &\text{deposited} & \text{compounded} & \text{from Table A-1} & (2) \times(4) \\ \hline 1 & 2 & 3 & 4 & 5 \\ \hline 1 & Rs 2,000 & 4 & 1.216 & Rs 2,432 \\ 2 & 2,000 & 3 & 1.158 & 2,316 \\ 3 & 2,000 & 2 & 1.103 & 2,206 \\ 4 & 2,000 & 1 & 1.050 & 2,100 \\ 5 & 2,000 & 0 & 1.000 & 2,000 \\ &&&&11,054 \\ \hline \end{array}