# Present Value

The present value of a general cash flow stream—like the future value—can also be calculated by considering each flow element separately, Again consider the stream (a'o,..vi, ,,,, ..v„). The present value of the first element xo is just that value itself since no discounting is necessary The present value of the flow x\ is .vj/(1 +/ ), because that flow must be discounted by one period (Again the interest rate r is the per-period ¡ate.) Continuing in this way, we find that the present value of the entire stream is PV = A'o + aj /(1 4 /) 4 x2/( 1 -\-r)ljr 4-a*„/(1 4 r}"., We summarize this important result as follows.

Present value of a stream Given a cash flow stream (aq, a j , , a„) and an interest rate r per period, the present value of this cash flow stream is

Example 2.2 Again consider the cash flow stream (—2, I, 1, 1) Using an interest rate of 10% we have

PV = -2 4 — 4- —!—- 4- —1—r = 487 . 1.1 (1.1)2 (1.1)3

### 2.3 PRESENT AND FUTURE VALUES OF STREAMS 21

The present value of a cash flow stream can be regarded as the present payment amount that is equivalent to the entire stream Thus we can think of the entire stream as being replaced by a single flow at the initial time

There is another way to interpret the formula for present value that is based on transforming the formula for future value Future value is the amount of future payment that is equivalent to the entire stream. We can think of the stream as being transformed into that single cash flow at period n. The present value of this single equivalent flow is obtained by discounting it by (1 -!-> )". That is, the present value and the future value are related by

In the previous examples for the cash flow stream (—2, 1, 1, 1) we have 487 = PV = FV/(1.1)3 = .648/1 33! = .487 