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Present Worth, PW(i)
The present worth, PW(i), of a cash-flow stream shows how much that future stream is worth ("is equivalent to") right now at interest rate, i. The reference time frame for PW(i) is today, or, more precisely, the beginning of the first period (the end of period 0) of the cash-flow stream. Present worth is also called net present value, or NPV. The present worth is the answer to the question, "How much is this cash-flow stream worth today?"

The "present" can be any arbitrary point in time as appropriate for the decision being studied. In 1999, we could have talked about the PW(i) of a cash-flow stream that's planned to begin in 2009 and includes cash-flow instances through 2011.

The formula for calculating the PW(i) of a cash-flow stream is





Where Ft is the net cash-flow instance in period t.

The PW(i) formula essentially uses the single-payment present-worth (P/F,i,n) formula to translate each individual net cash-flow instance to its corresponding end-of-year-0 amount and then sums up all of those amounts. This is exactly the same approach as was used in Table 7.2 in the preceding chapter. Table 8.3 shows how PW(i) can be hand-calculated for Mr. Kinkaid's project. A spreadsheet that automates these calculations can be found at http://www.construx.com/returnonsw/. The table shows that Mr. Kinkaid's project is financially equivalent to his receiving a one-time net income of $4557 today.

Table 8.3. Manual Calculation of PW(i) for Mr. Kinkaid's Project Year
Net Cash Flow
Present-Worth Factor
Present Worth

P/F,9,0

0
–$10,000
(1.000)
–$10,000

P/F,9,1

1
–$850
(0.9174)
–$780

P/F,9,2

2
$650
(0.8417)
$547

P/F,9,3

3
$2900
(0.7722)
$2239

P/F,9,4

4
$8150
(0.7084)
$5774

P/F,9,5

5
$5900
(0.6499)
$3835

P/F,9,6

6
$3650
(0.5963)
$2176

P/F,9,7

7
$1400
(0.5470)
$766

PW(9%) = $4557





Notice that except for year 0 the individual present-worth values are always less than their original cash-flow instance. Some people refer to the process of translating cash-flow instances backward in time as "discounting," and also refer to the interest rate used in the calculations as the "discount rate."

Present worth is the second most widely used basis for comparison. Payback period, below, is the most widely used. The main advantages of PW(i) are that it's relatively simple to compute and it's easy to understand the meaning of the result. Any arbitrary cash-flow stream can be converted to a present worth and compared to any other arbitrary cash-flow stream. Expressed in PW(i) form, any two cash-flow streams can be easily compared.

On the other hand, PW(i) hides some possibly important information about the cash-flow stream. The amounts and the timings of the cash flows are hidden; so even though a given cash-flow stream may have a higher PW(i) than another, its initial investment may be more than the organization can afford. That proposal would be impossible to carry out regardless of how profitable it might be.

SOME ADDITIONAL COMMENTS ON PW(i)
For any given cash-flow stream, there will always be a single value of PW(i) for each unique interest rate. Generally speaking, as the interest rate goes up, the PW(i) of the cash-flow stream goes down. Figure 8.2 shows the PW(i) for Mr. Kinkaid's project over a range of interest rates. Whereas PW(i) calculations are actually meaningful over the range of –1 < i < , only the range 0 i < is important because negative interest rates are virtually impossible in practice—it's highly unlikely that anyone would ever be willing to pay you to borrow his money.


Figure 8.2. PW(i) over a range of i's for Mr. Kinkaid's project





If Mr. Kinkaid's estimates of all the cash flows are accurate, the graph in Figure 8.2 tells him several important things about his proposal. First, given some interest rate, the graph tells him how much profit (or loss if that interest rate is too high) he would get, in present-day terms, from his project. The graph also tells him what range of interest rates his project would be profitable in and in which ones it wouldn't. Finally, the picture tells him the "critical i" where PW(i) = 0. This critical i is discussed later in this chapter in the section "Internal Rate of Return, IRR."

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