## R 106006

(in as of the beginning of the 50th year the present value of \$10,000 per year in perpetuity is \$166,666.67, as computed in part (i). Since this is in effect received at the end of the 50th year, its current ( i.e.. as of now) present value must be discounted so that it becomes

\$166,666 67

(iii) The present value of the first 50 years' worth of payments is simply the answer in (i) less that in (ii):

The examples above all deal with the problem of determining the present value of a series of equal pay ments. In general, however, one can evaluate the present value of any pattern of payments. Suppose, for example, that a business firm is considering the possibility of making a current (and immediate) investment of \$C, the payoff of which will be the sales revenue of a product whose sales will increase over time. Le( us assume that the production process will begin at the end of one year and thai net profit from sales of the product is \$tt( 1 + g) the first year and will grow at a rate of 8 each subsequent year. Thus the profit for period / will be a, = 7r(i + g)', and the (undiscounted) value of the stream of benefits (gross benefits) will be

i=i r=i which is a divergent series if n and g are positive. The discounted or present value of the stream of benefits is

which is just a geometric series, with a — n( \ +£)/(l H-r) and p = ( 1 +£)/ (1 +r), so that PVB = 7r(l g)/(r - g) and is finite valued if and only if g < r (i.e., \p\ < 1). To decide whether the investment is profitable, one merely needs to determine whether PVB>C or PVB <C. This illustrates how discounting is used in assessing the net benefits of a project. In most instances the costs are heavily concentrated in the early periods with the benefits spread out over a longer time horizon.

Example 3.20 Suppose that a stream of payments arising from some business venture begins with an amount of \$20,000 immediately and grows at the rate of 4% per annum thereafter, forever. Given an interest rate of 8%, find the present value of this stream of payments.

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