## Present Value of a Stream of Payments

Earlier we saw how the sequence P V, = V/( 1 +r)' (see equation 3.3) represents the present value of an amount of money V received t periods into the future. In many economic settings we need to compute the equivalent present value of a series (i.e., the sum total) of such amounts. For example, a mortgage or other long-term loan represents a current sum of money loaned to ar, individual or institution in return for a stream of future payments. Thus, if an individual makes annual payments at the end of each year in amount V for T years, with the interest rate being r, then the present value of this stream of payments is

The relationships among the variables in equation (3.6) are worked out in mortgage tables. They are generally computed on a monthly basis, in which case the appropriate interest rate is r/12, where r is the annual rate and T refers to the number of monthly rather than yearly payments. However, let us consider the following example based on annual payments. The yearly payment required to compensate a lender for a loan of amount $ 100,000 at an interest rate of 8% (r = 0.08) spread over 25 years is V = $9,367.88. while if payments are spread over 50 years the annual payments would be "$8,174.28. Note that if the payments were to be made in perpetuity (i.e., 7 oo). we could use the formula for an infinite geometric series with a = V/(l + r|, p — 1/(1 4- r). since

Thus we get V = r P = $8,000 for our example, which is fairly close to the value when payments are made for 50 years. The reason for this is illustrated by the fact that the present value of $8,000 received 50 years from now, when the interest rate is 8%. is only $.8.000/11 + 0.08)50 = $170.57. The present value of all payments of amount V = $8,000 received after 50 years is which is only $2,132.13 in our example (100,000 - $97,867.87 ), a difference of only 2.1% (approximately) Thus we see that using the formula lor an infinite series is often a good approximation for evaluating the present value of a finite series, provided that the number of periods or the interest rate is not too small. The reason for the latter requirement is that if r is close to zero, the discount t'actur, p = 1/(1 + r), is close to one. and so future payments are not discounted very much and thus should not be ignored. In fact, if the interest rate is only 1/2 of 1%, then we find that the present value of an infinite stream of payments of $8,000 per year is $1.6 million and the present value of the payments received after the

50th year is

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