стр. 123 
The last column shows the calculation of expected NPV, which is the weighted average of the individual path NPVs
where the weights are the path probabilities. In this example, the expected NPV of the project is $1,330, and the
project should be rejected.
9.4 CORRELATION OF CASH FLOWS OVER TIME
When cash inflows are independent from period to period, it is fairly easy to measure the overall
risk of an investment proposal. In some cases, however, especially with the introduction of a new
product, the cash flows experienced in early years affect the size of the cash flows in later years. This
is called the time dependence of cashjlows, and it has the effect of increasing the risk of the project
over time.
259
CHAP. 91 CAPITAL BUDGETING UNDER RISK
EXAMPLE 9.6 Janday CorporationвЂ™s aftertax cash inflows (ATCI) are timedependent, so that year 1 results
(ATCIl) affect the flows in year 2 (ATC12) as follows:
If ATCI, is $8,OOO with a 40 percent probability, the distribution for ATC12 is:
$ 5,OOO
0.3
0.5 $lO,OOO
0.2 $15,000
If ATCIl is $lS,OOO with a 50 percent probability, the distribution for ATC12 is:
$10,OOO
0.3
0.6 $20,000
0.1 $30,000
If ATCIl is $20,000 with a 10 percent chance, the distribution for ATC12 is:
0.1 $15,000
0.8 $40,000
$50,000
0.1
The project requires an initial investment of $20,000, and the riskfree rate of capital is 10 percent.
The company uses the expected NPV from decision tree analysis to determine whether the project should be
accepted. The analysis is as follows:
Joint Expected
Probability NPV
NPV at 10%
lime 1
TSme 0 lEme 2
$ 8,595вЂќ 0.12b $1,03 1
$5,000
$ 8,000 ˜ $ 1 0 , 0 0 0 $ 4,463 0.20 893
26
>$15,000 0.08
331
$
0.15 285
$1,901
$10,165 0.30 3,050
0.05
$18,429 921
$20,000 $15,000 $10,576 106
0.01
0.1
% $40,000 0.08
$31,238 2,499
0.01
0 . A $50,000 $39,502 395

$5,306
1.oo


 $20,000
= $8,000(0.9091) + $5,000(0.8264) $20,000 =  $8,595

Joint probability of the first path = (0.4)(0.3) = 0.12
Since the NPV is positive ($5,306), Janday Corporation should accept the project.
9.5 NORMAL DISTRIBUTION AND NPV ANALYSIS: STANDARDIZING THE
DISPERSION
With the assumption of independence of cash flows over time, the expected NPV would be
NPV=PVI
n t
CAPITAL BUDGETING UNDER RISK
260 [CHAP. 9
The standard deviation of NPVs is
The expected value (A) and the standard deviation (a) give a considerable amount of information by
which to assess the risk of an investment project. If the probability distribution is normal, some
probability statement regarding the projectвЂ™s NPV can be made. For example, the probability of a
projectвЂ™s NPV providing an NPV of less or greater than zero can be computed by standardizing the
normal variate x as follows:
 NPV
x
z=
a
x = the outcome to be found
where
NPV = the expected NPV
z = the standardized normal variate whose probability value can be found in Appendix E.
Assume an investment with the following data:
EXAMPLE 9.7
Period1 Period2 Period3
Expected cash inflow ( A ) $4,OOo $3,000
$5,000
$1,140 $1,140 $1,140
Standard deviation (U)
Assume that the firmвЂ™s cost of capital is 8 percent and the initial investment is $9,000. Then the expected
NPV is:
NPV=PVI
 $5,000 $4,000 $3,000
  $9,000
+
+
(1 + 0.08) (1 + 0.08)2 (1 + 0.08)3
= $5,000(PVIF8,˜+ $4,00O(PVIF8,z)+ $3,00O(PVIF8,3)  $9,000
)
= $5,000(0.9259) + $4,000(0.8573) + $3,000(0.7938)  $9,000
= $4,630 + $3,429 + $2,381  $9,000 = $1,440
The standard deviation about the expected NPV is
$1,1402 $1,1402
=
t (1 + 0.08)4+ (1 + 0.08)6
= d$2,888,411 = $1,670
The probability that the NPV is less than zero is then:
z =  x  NPV
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