## Present Value Calculations

An important economic application of sequences is the determination of the present value of a sum of money to be received at some poini in the future. This computation is the inverse of determining how much money one would have in the future upon investing a certain amount now. Suppose, for example, that one had S90.91 to invest currently at an annual interest rate of 10%. Then the amount of money received at the end of one year would be \$90.91 (I 4-0.1) = \$100. In general. investing \$X today at an annual rate of return r will generate V = X( I 4- r) at Ihe end of one year. Therefore ii is equivalent to say that the present value of amount V to be received in one year's time is X = V/( I 4- r), where r is the rate of return (or rate of interest).

The same rationale can be used to value an amount received any number of periods into the future. If one received \$Y now and could invest at 10% per year, with compounding (i.e., the accumulation and reinvestment of interest payments) at the end of each year, then the \$Y would be worth K(l + 0.1) at the end of the first year and, after reinvesting (including interest payments), would be worth | Y( 1 4- 0 1 )J( I -I- (). 1) = K( 1 + 0.1)2 at the end of the second year. Therefore the present value of \$100 to be received in two years' time is )', where Y( I 4- 0.1): = 100 or Y = \$100/(1 I 0.1 )2 = \$82.64 (approximately). Following this line of argument leads to the following formula, which determines ihe present value P V, of amount V to be received t periods from now when the interest rate is r per period and compounding occurs at the end of each period:

Notice that for r > 0 the denominator of (I + r)' becomes larger as t becomes larger, and thus PV, gets smaller. In other words, receiving a certain sum in the future has a lower present value the longer one has to wait for the payment. This is natural since the further in the future one receives the fixed amount V. the less one would need to invest now to replicate (hat future payment. For this reason economists refer to the discounting of future benefits and the value 1/(1 + n is referred to as the discount rate, or discount factor. Moreover (1 + r)' grows without bound as / —*■ oo. and so PV, —► 0 as t is left as an exercise at the end of this section.

oo. The proof of this statement

Example 3.5 Compute the present value of \$500 to be received in one year's time given the interest rate of 8%.

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