Many transactions involve making or receiving cash payments at various future dates. A person who takes a house loan trades a promise to make monthly payments for say, fifteen or twenty years for a large amount of cash now to pay for a home. This case and other similar cases relate to the time value of money. The time value of money refers to the fact that a rupee to be received in the future is not worth a rupee today. Therefore, it is necessary to have techniques for measuring the value today (i.e., the present value) of rupees to be received or paid at different points in the future. This section outlines the approach to analyzing problems that involve payment and/or receipt of money at one or more points in time.
One may ask how much money today would be equivalent to Rs. 100 a year from now if the rate of interest is 5%. This involves determining the present value of Rs. 100 to be received after one year. Applying the formula –
PV1 = 100/(1.05)1
We will obtain Rs. 95.24
Rs. 95.24 will accumulate to an amount exactly equal to Rs. 100 in one year at the interest rate of 5 per cent. Looked at another way, you will be willing to pay maximum of Rs. 95.24 for the benefit of receiving Rs. 100 after one year from now if the prevailing interest rate is 5 per cent.
The same analysis can be extended to any number of periods. A sum of Rs. 100 two years from now is worth:
PV2 = 100/(1.05)2
=Rs. 90.70 of today
In general, the present value of a sum to be received at any future date can be found by the following formula:
PV = present value, Rn = amount to be received in future, i = rate of interest, n = number of years lapsing between the receipt of R. If the receipts are made available over a number of years, the formula becomes:
In the above formula if R1 = R2= R3 etc., it becomes an ‘annuity’. An annuity has been defined as series of periodic equal payments. Although the term is often thought of in terms of a retirement pension, there are many other examples of annuities. The repayment schedule for a home loan is an annuity. A father’s agreement to send his son Rs. 1000 each month while he is in college is another example. Usually, the number of periods is specified, but not always. Sometimes retirement benefits are paid monthly as long as a person is alive. In other case, the annuity is paid forever and is called ‘perpetuity.’
It must be emphasized that the strict definition of an annuity implies equal payments. A contract to make 20 annual payments, which increase each year by, say, 10 per cent, would not be an annuity. As some financial arrangements provide for payments with periodic increase, care must be taken not to apply an annuity formula if the flow of payments is not a true annuity.
The present value of an annuity can be thought of as the sum of the present values of each of several amounts. Consider an annuity of three Rs. 100 payments at the end of each of the next three years at 10 percent interest. The present value of each payment is
Although this approach works, it clearly would be cumbersome for annuities of more than a few periods. For example, consider using this method to find the present value of a monthly payment for forty years if the monthly interest rate is i per cent. That would require evaluating the present value of each of 480 amounts! In general, the formula for the present value of an annuity of A rupees per period for n periods and a discount rate of is: