Chapter12 projectrateofreturn

© Centre for Financial Management, Bangalore
CHAPTER 12
PROJECT RATE OF RETURN

OUTLINE
• Pros and cons of multiple costs of capital
• Divisional WACC and project-specific WACC
• Hurdle rate and cost of capital
• Portfolio theory and capital budgeting

Pros and Cons of Multiple Costs of Capital
 Justification for Using Multiple Discount Rates.
 The appropriate discount rate for a project should reflect its opportunity cost of
capital.
 The bias on account of using a single cost of capital is shown in the following
diagram.

The Rationale for Using a Single Discount Rate
Despite the conceptual argument in favour of varying the discount rate to
reflect the project risk, most of the firms use a single discount rate for
evaluating all of their investment projects. There are several possible
reasons for this practice.
 Many firms engage in a small range of related activities that are
perhaps characterised by similar risks.
 It may be somewhat difficult to specify different discount rates for
different projects.
 The rationale for multiple discount rates may not be fully understood
by persons involved in capital budgeting.
 Opportunistic managers may underestimate project risk and reduce the
project cost of capital to get their projects approved.
 The use of a single discount rate reduces the incidence of what
economists call influence costs.

Weighing the Pros and Cons of Multiple Versus Single
Discount Rates
Taking into account its circumstances, each firm must properly weigh
the pros and cons of multiple versus single discount rates.
Multiple discount rates make more sense when the firm is engaged in
diverse businesses or when it operates in different geographies.
A single discount rate makes sense when managers have discretion in
specifying discount rates and incentive problems are real. Incentive
problems can be mitigated if the cost of capital is estimated, using an
objective procedure that relies on market data.

Equity Beta and Asset Beta
To explore the relationship between equity beta and asset beta, we will initially
ignore taxes. Look at Zenith Limited which has the following balance sheet :
Equity : 50 Assets : 100
Debt : 50
If you buy all the securities of Zenith (its entire equity as well as debt), you will
own all its assets. So the beta of your portfolio (bA) of Zenith’s securities is equal
to the beta of Zenith’s assets( bP )
bP =bA
(12.23)
Now, the beta of your portfolio is simply the weighted arithmetic average of the
betas of its components, viz., equity (E) and debt (D)
E D
bP = bE + bD (12.24)
E+D E+D

Hence
E D
bA = bE + bD (12.25)
E+D E+D
Juggling Eq.(12.25) a bit, you get
D
bE = bA + (bA - bD) (12.26)
E
If the beta of debt, bD, is assumed to be zero (this means that debt is considered to be risk-
free)
D D
bE = bA + bA = bA 1 +
E E
In a world of taxes, as Robert Hamada has shown.
D
bE = bA 1 + (1 – T)
E
This means
bE
bA =
[ 1 + D/ E (1-T) ]

Procedure for Calculating a Project’s Required
Rate of Return
1. Find a sample of firms engaged in the same line of business.
2. Obtain equity betas for the sample firms.
3. Derive asset betas:
bE
bA
[ 1 + D/E (1-T)]
4. Find the average of the asset betas.
5. Figure out the equity beta for the proposed project
bE = bA [ 1 + D/E (1 – T ) ]
6. Estimate the cost of equity for the proposed project, using the CAPM
rE = Rf + [ E (RM ) - Rf ] bE
7. Calculate the project’s required rate of return:
rA = WE rE + WD rD ( 1 – T)

Illustration
Diversified Limited is evaluating a granite project for which it proposes to use a
debt-equity ratio of 1.5:1. The pre-tax cost of debt is 15 percent and the tax rate is
expected to be 30 percent. The risk-free rate is 12 percent and the expected return
on the market portfolio is 16 percent. The project’s required rate of return may be
calculated as follows:
Step 1 : Find a sample of firms engaged in similar business According to the chief
executive of Diversified Limited the following firms are engaged wholly in the
same line of business :
Ankit Granites Limited
Bharath Granites Company
Modern Granites Limited
Step 2 : Obtain equity betas for the sample of comparable firms The equity betas
of the three firms, obtained by regressing their equity returns of the market portfolio
for the past 60 months, are as follows
Ankit Granites Limited : 1.20
Bharath Granites Company : 1.10
Modern Granites Limited : 1.05

Step 3 : Derive asset betas after adjusting equity betas for financial leverage
The debt-equity ratios for the three firms, namely, Ankit Granites Limited,
Bharath Granites Company, and Modern Granites Limited are 2.1, 1.8,
and 1.3 respectively. The effective tax rate for all of them is 40 percent.
Their asset betas are derived by using the formula :
bA = bE / [1 + D / E (1-T)]
1.20
Ankit Granites Ltd : = 0.53
[1 + 2.1 (0.6)]
1.10
Bharath Granites Co. : = 0.56
[1 + 1.6 (0.6)]
1.05
Modern Granites Ltd : = 0.59
[1 + 1.3 (0.6)]

Step 4 : Find the average of asset betas The average of asset betas of Ankit
Granites Limited, Bharath Granites Company, and Modern Granites Limited is :
(0.53 + 0.56 + 0.59) / 3 = 0.56
Step 5 : Figure out the equity beta for the proposed project The equity beta for the
proposed project is :
bE = bA [1 + D / E (1-T)]
= 0.56 [1 + 1.5 (1 - .3)] = 1.14
Step 6 : Estimate the cost of equity for the proposed project. As per the capital
asset pricing model, the cost of equity for the proposed project is :
rE = Rf + (E(RM ) - Rf) bE
= .12 + (.16 - .12) 1.148 = .166
Step 7 : Calculate the project’s required rate of return The weighted average cost
of capital (the required rate of return) for the granite project of Diversified Limited
is :
rA = WE re + WD rD (1 – T)
= [ ( (1/2.5) 0.166 ) + ( (1.5/2.5) (0.15) (0.7) ) ]
= .1294 = 12.94 percent

Hurdle Rate and Cost of Capital
 In practice firms generally use discount rates, often called hurdle rates,
which are greater than the project cost of capital.
 There are several reasons for this:
 Constraints: Most firms believe that due to constraints, the WACC
is not the appropriate opportunity cost.
 Incentives: A high hurdle rate motivates project sponsors to find
better projects.
 Optimistic Forecasts: A higher-than-WACC hurdle rate is meant to
counter the optimistic projections and selection bias.

Survey Evidence on Project Discount Rates
In a survey, CFOs were asked: “How frequently would your company use the
following discount rates when evaluating a new project in an overseas market?”
The following list of discount rates was given.
 The discount rate for the company as a whole.
 A risk-adjusted discount rate for this particular project.
 A country discount rate.
 A different discount rate for each item of cash flow that has a different risk
characteristic (such as depreciation, tax shield, and so on).
The key findings of the survey were:
 59 percent of the CFOs use the discount rate for the company as a whole.
 51 percent of the CFOs use a risk adjusted discount rate.
 Large firms are more likely to use risk-adjusted discount rates, compared to
small firms.

Portfolio Theory: The Basic Ideas
 Portfolio theory developed by Harry Markowitz assumes that
investment decisions are based on a tradeoff between risk and return.
 Risk is measured in terms of variance or its square root the standard
deviation.

The Two-Asset Portfolio
The expected return and the standard deviation of return for a two- asset
portfolio are:
E(Rp) = WA E (RA) + WB E (RB) (12.6)
sp = (WA
2 sA
2 + WB
2 sB
2 + 2 WAWB sA,B )1/2 (12.7)
where WA and WB are the proportions of total funds invested in assets A
and B (WA + WB = 1). The other items are as defined earlier.
Since sA,B = rA,B . sA .sB, Eq (12.7) can be rewritten as :
sp = (WA
2 sA
2 + WB
2 sB
2 + 2 WAWB rA,B sA sB )1/2 (12.8)

The n-Asset Portfolio
When more than two assets are combined in a portfolio, the expected
portfolio return and the standard deviation of portfolio return are :
n
E (Rp) = S Wi E(Ri) (12.9)
i = 1
n n 1/2
sp = S S Wi Wj rij sisj (12.10)
j =1 i = 1

Determination of an Optimal Portfolio
According to portfolio theory, the optimal portfolio is determined as
follows:
1. Identify all feasible portfolios.
2. Determine the expected return and standard deviation of all the
feasible portfolios.
3. Define the efficient frontier.
4. Choose the portfolio on the efficient frontier that offers the
highest level of utility (satisfaction) to the decision maker.

Feasible Region
The investments available can be combined into any number of portfolios.
Each possible portfolio may be described in terms of its expected rate of
return and standard deviation of rate of return and plotted on a two-
dimensional graph like the one shown in exhibit. The collection of all
possible portfolios represents the feasible region which is the enclosed
region in exhibit.
•
•
•
•
•
Z
DB
A
C
Standard Deviation, sr
Expected
Return,
E(Rr)

Efficient Frontier
Given the feasible region, which portfolio should the investor choose? The
investor should choose the portfolio that maximises his utility function. This
choice involves two steps:
(i) Delineation of the set of efficient portfolios, and
(ii) Selection of optimal portfolio from the set of efficient
portfolios.
A portfolio is efficient if (and only if) there is no alternative with (i) the
same E(Rp) and a lower sp or (ii) the same sp and a higher E(Rp) or (iii) a
higher E(Rp) and a lower sp. Thus in exhibit, in which the available portfolios
are contained in the region ABCD, only the portfolios which lie along the
boundary BC are the portfolios which are efficient. The boundary BC may be
referred to as the efficient frontier. All other portfolios are inefficient. A
portfolio like Z is inefficient because a portfolio like B (among others)
dominates it. The efficient frontier is the same for all investors because the
portfolio theory is based on the assumption that investors have homogeneous
expectations.

Risk – Return Preferences
Standard Deviation
of Return,sp
Expected
Return,
E(Rp)
l4
l3
l2
l1

Optimal Portfolio
Standard Deviation
of Return,sr
Expected
Return,
E(Rr)
•
•
•
••
D
X*
A
C
X1
X2
X3
Y2
Y3
Y1
B
Y*

Portfolio Theory and Capital Budgeting
In applying portfolio theory to capital budgeting we face some special
problems. These relate to :
• Indivisibility of assets
• Holding period
• Data requirements

Simpler Applications
A full blown application of portfolio theory to capital budgeting seems
impractical. The basic ideas of portfolio analysis, however, may be
applied in somewhat simpler ways:
 When a large project is being considered, it may be viewed as one asset
and the existing firm as a second asset.
 The mean-variance portfolio model may be used to allocate funds across
major business divisions which may be few in number, say three to five
for most companies. Data requirement of such analysis is quite
manageable,
 When a small project is being considered, the beta between the project
and the company’s existing portfolio may be regarded as the risk
measure for incremental decision making.

Developing the Inputs for Portfolio Analysis
Estimates of expected return, variance, and covariance are required to
apply the mean-variance portfolio model. For security portfolios,
historical values can be used as proxies for future values. This is,
however, not feasible for proposed capital investments. What is the way
out? One method is to identify various states of nature , assign
probabilities to them, and estimate the return of each investment in each
state of nature. Based on this information, the required inputs for
portfolio analysis, viz., expected returns, variances, and covariances, can
be generated.

SUMMARY
 According to finance theory, the appropriate discount rate for a project should
reflect is opportunity cost of capital.
 When a firm uses a single discount rate (WACC) for all its projects, it will tend to
reject a relatively safe project, even though its expected rate of return exceeds its
opportunity cost of capital (or risk-adjusted cost of capital) and accept a relatively
risky project, even though its expected rate of return is less than its opportunity cost
of capital (risk-adjusted cost of capital).
 Despite the conceptual argument in favour of varying the discount rate to reflect
the project risk, most firms use a single discount rate for evaluating all of their
investment projects. This practice is meant to cope with measurement problems, to
mitigate influence costs, and check incentive behaviour.
 The relationship between equity beta and asset beta is as follows:
βE = βA 1 + (1 – T)
 The two key determinants of asset betas are cyclicality and operating leverage.
D
E

 The procedure for calculating the divisional WACC, as per the CAPM, involves the
following steps: (i) Find a sample of firms engaged in the same line of business. (ii)
Obtain equity betas for the sample firms. (iii) Derive asset betas after adjusting equity
betas for financial leverage. (iv) Find the average of the asset betas. (v) Figure out the
equity beta for the division. (vi) Estimate the cost of equity for the division. (vii)
Calculate the divisional WACC.
 The procedure for calculating the project-specific WACC would be the same as that for
the divisional WACC. However, implementing this procedure for individual projects is
difficult because it is not easy to find market proxies for the project risk and to define
capital structure weights since equity values of individual projects are not observable.
 Firms typically set a hurdle rate that is higher than the WACC. There are three main
reasons for this: (i) Most firms believe that the WACC is not the appropriate
opportunity cost. (ii) A higher-than-WACC hurdle rate is meant to incentivise project
sponsors to find better projects. (iii) A higher-than-WACC hurdle rate is aimed at
countering the optimistic projections and selection bias.
 Although there are various portfolio risk measures, the mean-variance portfolio model
is the most widely used. This model assumes that the risk of a portfolio is defined by
the variance or standard deviation of the probability distribution of portfolio returns.

 The expected return and the standard deviation of return for a two-asset portfolio are:
E (Rp) = WA E (RA) + WBE (RB)
sp = (WA
2 sA
2 + WB
2 sB
2 + 2WAWBPA,B sA sB)1/2
 The expected return and the standard deviation of return for an n-asset portfolio are:
E (Rp) = Wi E (Ri)
sp = Wi Wj Pij si sj
 A portfolio is efficient if and only if there is no alternative with (i) the same E(Rp) and
a lower sp, or (ii) the same sp and a higher E(Rp), or (iii) a higher E(Rp) and a lower sp.
The efficient frontier contains all the efficient portfolios.
n
i = 1
S
n
t = 1
S
n
i = 1
S

Chapter12 projectrateofreturn

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