ñòð. 124 
The area of normal distribution that is z standard deviations to the left or right of the mean may be found in
Appendix E. A value of z equal to 0.862 falls in the area between 0.1949 and 0.1922 in Appendix E. Therefore,
there is approximately a 19 percent chance that the projectâ€™s NPV will be zero or less. Putting it another way, there
is a 19 percent chance that the IRR of the project will be less than the riskfree rate.
261
CAPITAL BUDGETING UNDER RISK
CHAP. 91
9.6 PORTFOLIO RISK AND THE CAPITAL ASSET PRICING MODEL (CAPM)
Portfolio considerations play an important role in the overall capital budgeting process. Through
diversification, a firm can stabilize earnings, reduce risk, and thereby increase the market price of the
firmâ€™s stock.
Beta Coefficient
The capital asset pricing model (CAPM) can be used to determine the appropriate cost of capital.
The NPV method uses the cost of capital as the rate to discount future cash flows. The IRRmethod uses
the cost of capital as the cutoff rate. The required rate of return, or cost of capital according to the
CAPM, or security market line (SML), is equal to the riskfree rate of return (rf) plus a risk premium
equal to the firmâ€™s beta coefficient ( b )times the market risk premium (rm rf):
rj = rr + b(rm rf)
EXAMPLE 9.8 A project has the following projected cash flows:
Year 1
Year 0 Year 2 Year 3
$(400) $300 $200 $100
The estimated beta for the project is 1.5. The market return is 12 percent, and the riskfree rate is 6 percent.
Then the firmâ€™s cost of capital, or required rate of return, is:
rj = r f + b(rm rf) = 6% + 1.5(12%  6%) = 15%
The projectâ€™s NPV can be computed using 15 percent as the discount rate:
Cash Flow ($) PV at 15%
Year PV ($)
1.ooo
(400)
0 (400)
300 0.870 261
1
0.756
2 200 151
100 0.658 66
3
78â€œ
â€œNPV.
The project should be accepted since its NPV is positive, that is, $78. Also, the projectâ€™s IRR can be computed
by trial and error. It is almost 30 percent, which exceeds the cost of capital of 15percent. Therefore, by that standard
also the project should be accepted.
Calculation of Beta Coefficient
In measuring an assetâ€™s systematic risk, beta, an indication is needed of the relationship between
the assetâ€™s returns and the market returns (such as returns on the Standard & Poorâ€™s 500 Stock
Composite Index). This relationship can be statistically computed by determining the regression
coefficient between asset and market returns. The method is presented below.
Cov(rj,r m )
b= 2
gm
where COV(rj, r m ) is the covariance of the returns of the assets with the market returns, and g2,,,is the
variance (standard deviation squared) of the market returns.
An easier way to compute beta is to determine the slope of the leastsquares linear regression line
[CHAP. 9
CAPITAL BUDGETING UNDER RISK
262
(rj rf), where the excess return of the asset (rj  rf) is regressed against the excess return of the market
portfolio (rm rf).The formula for b is:
I K  nMR
:M
b=
XiWnP
where M = (rm rf)
K = (rj  r f )
n = number of years
ii$ = average of M
K = average of K
EXAMPLE 9.9 Compute the beta coefficient, b, using the following data for stock x and the market portfolio:
Historic Rates of Return
r,,, (Yo)
Year r, (Yo)
5 10
19x1
19x2 8
4
12
19x3 7
20
19x4 10
15
19x5 12
Assume that the riskfree rate is 6 percent. For easy computation, it is convenient to set up the
following table:
Stock Return, Market Return, Riskfree Rate,
(r, rJ)= K MK
A#
(r,,, 9)=M
r
J
Year 9 rm
0.11 0.04 0.0016 0.0044
0.05 0.10 0.06
19x1
0.02 0.m 0.m
0.02
0.0s 0.06
0.04
19x2
0.06 0.0036 0.OOO6
0.12 0.06 0.01
0.07
19x3
0.04 0.14 0.0196 0.0056
0.20 0.06
19x4 0.10
  

0.06
0.15 0.06 0.09 0.0081 0.0054
0.12
19x5
 0.35 0.0068
0.0333
0.02
M
R = 0.004 = 0.07
Therefore, beta is:
2 M K  ni$fR  0.0068 (5)( 0.OO4)(0.07)  0.0082
0.93
=
b=
0.0333 (5)(0.07)2 0.0088
XWnW
ñòð. 124 