PROJECT ANALYSIS UNDER RISK
ANSWERS TO REVIEW QUESTIONS
7.1 The CapmBeta Company is considering a new capital investment proposal. This
project’s risk structure is very similar to that of the company’s existing business.
Returns for this company’s stocks for the past ten years are given in Table 7.4 together
with returns for a country’s stock-market index (e.g. the All Ordinaries Index in
Australia or the S&P Index in the United States). The government treasury bill rate was
around 5.6% per annum.
Table 7. 4 Stock Market Index and CapmBeta Company Stock Returns
Year Company’s Stock Returns ( rit ) Stock Market Index Returns ( rmt )
1992 0.09 0.07
1993 0.10 0.09
1994 0.10 0.10
1995 0.11 0.12
1996 0.10 0.11
1997 0.11 0.10
1998 0.11 0.10
1999 0.10 0.09
2000 0.09 0.08
2001 0.07 0.07
The total capital outlay of the proposed project is estimated as $3000 million and it is to be
incurred at the beginning of year 1. The forecasted after-tax net cash inflows of the project
are provided in Table 7.5.
Table 7. 5 CapmBeta Company- Forecasted Project Cash Flows
Year Net cash inflows ($ dollars)
(a) Compute the average stock market (index) return.
(b) Compute the average company stock return.
(c) Compute the variance and standard deviation of the stock-market returns: var (rm), or σ m
and SD (rm), or σ m .
(d) Compute the variance and standard deviation of the company stock returns - var (ri), or
σ i 2 and SD (ri), or σ i .
(e) Compute the covariance between company stock returns and stock-market index returns.
(f) Compute the correlation between company stock returns and stock-market index returns.
cov( ri , rm )
(g) Estimate Beta as β i =
var( rm )
ρ i , mσ i
(h) Estimate Beta as β i =
(i) Calculate the average risk premium, u, for the firm.
(j) Estimate the RADR to be used as the discount rate for this project.
(k) Compute the project’s NPV using this RADR.
(l) Compute the certainty equivalent coefficients using the relevant information from the
question under the condition that if risk adjustments are made correctly, the net present
value calculated from any given future cash flows must be identical in either the RADR
or CE method.
(m) Calculate the NPV using the RADR and CE methods to show the answer is same under
7.1 What are the relative merits and demerits of the RADR and CE method of
incorporating risk into project analysis.
7.2 Describe the relationship between the RADR and CE coefficient.
Answer to Q 7.1
See Excel file titled ‘Q 7.1 Excel Solutions.xls’ for the calculation details of the following
answers. Excel, Tools, Data Analysis has various functions. Averages (i.e. means), standard
deviations and variances are obtained using ‘Descriptive Statistics’ function; Covariance and
correlation are obtained by using ‘correlation’ and ‘covariance’ functions.
(a ) Average stock market (index) return: 0.093
(b) Average company stock return: 0.098
(c )Variance of the stock market returns var (rm), or σ m is:
mt − rm ) 2
= = .0002678 or .02678%
n −1 10 −1
Standard deviation of the stock market return can be calculated by taking the square root of
the variance. The answer is: 0.01636.
(d) Variance of the company stock returns: 0.0001511
Standard Deviation of the company stock returns: 0.012292
(e) Covariance between company’s stock returns and the stock market index returns is:
∑ (r it − rt )(rmt − rm )
cov( ri , rm ) =
= = .0001622
n −1 9
Unfortunately, however, the Excel formula uses ‘n’ , which is 10 and not ‘n-1’ which is 9.
Therefore, you will see in the Excel solution the covariance is 0.000146. This little
inconsistency carries over to the calculation of Beta, and you will see slightly different results
in the Excel version when Beta is calculated using (1) covariance-variance formula and (2)
correlation-standard deviations formula.
cov( ri , rm ) .0001622
ρ i,m = = = .8065638
σ mi σ i ( .01636)( .0122927 )
The Excel answer for this is 0.806445 . The insignificant difference is due to rounding and not
due to ‘n’ and ‘n-1’ problem, because the Excel routine for the calculation of correlation is
correct (i.e. it uses ‘n-1’ ).
(g) Beta from Covariance and variance:
cov( ri , rm ) ρ i , mσ i
βi = =
var( rm ) σm
cov( ri , rm ) .0001622
βi = = = .606
var( rm ) .0002677
The Excel answer is 0.545 and it is due to the ‘n’ and ‘n-1’ problem described above. In other
words, the Excel covariance is calculated by dividing by ‘n’, which is not strictly correct.
(h) Beta from Correlation and standard deviations:
ρ i , mσ i
If we use the alternative formula, β i = , we will still get the same correct answer (i.e.
0.606). The Excel answer also is same (after rounding to three decimal places). This is the
correct value for the Beta.
ρ i ,mσ i ( .806445)( .012292 )
βi = = = .606
(i) Average risk premium ‘u’ for the firm:
rm − r f β i = ( .093 − .056) 0.606 = ( .037 ) ( .606 ) = .022422
(j) Risk Adjusted Discount Rate:
Recall k = r + u + a.
Risk-free rate r can be approximated by the treasury bill rate which is 5.6%. Since there is no
difference between the risk structure of the existing business of the firm and the proposed
capital investment, ‘a’ is equal to 0. Therefore, k = .056 + .022422 + 0 = .078422. This is
the RROR or the RADR to be used for discounting the cash flows when using the RADR
technique to compute the NPV of the proposed project. In other words, 7.8422% is the
required rate of return or the appropriate discount rate to be used for this project.
(k) Computation of the project’s NPV using the estimated RADR:
The next step is to use the estimated k as the discount rate and compute the NPV.
The NPV formula under RADR technique is:
NPV radr = ∑ (1 + k )
25 2000 4000 6000
= + + +
(1.078422) (1.078422) (1.078422) (1.078422) 4
+ − 3000
= 23.18 + 1719.70 + 3189.29 + 4436.05 + 4456.25 − 3000
= 13824.47 − 3000
The project’s expected NPV is $10,824 million and therefore it is recommended for
consideration of acceptance.
Answer to Q 7.2
Relative merits and demerits of the RADR and CE methods are well explained in Chapter 7 of
the text book; see the section titled, ‘comparison of RADR and CE’.
Answer to Q 7.3
This relationship is well described in the chapter; See the section titled ‘The relationship
between CE and RADR’.