Capital Budgeting with Risk
Discount Rates, Expected Returns, and a Project‟s
Internal Rate of Return-- some clarification
The Capital Asset Pricing Model, and the SML in particular
give us an estimate of the normal rate of return that is expected
on assets with varying amounts of (beta) risk.
The expected return we get from the SML should be used as
the discount rate in valuing investment opportunities and
proposed projects. This discount rate is also referred to as the
cost of capital for the project, or the hurdle rate.
Once we have forecasted the cash flows from a proposed
project or investment, we can compute the project‟s Internal
Rate of Return.
IRR and NPV
Typically, a project will have a positive NPV
if the IRR exceeds the project‟s cost of capital
or hurdle rate.
That is, the NPV is positive if the IRR plots
above the Security Market Line on a
Project A would have a positive NPV,
while Project B would have a negative NPV
Two Complexities in Choosing Discount
Rates for Projects or Divisions.
If the firm has no debt in the capital structure and the project
being evaluated has the same (Beta) risk as the firm‟s existing
projects, then the expected return on the firm‟s equity is the
appropriate discount rate for the project.
If the project beta differs from the firm beta then use the project
How can this be estimated?
If the firm has debt in the capital structure, adjustments are
Interest is tax deductible. Either allow for in cash flows or alter
discount rate to reflect.
Financial leverage increases equity betas relative to the firm‟s beta
• Is an equity-beta or firm-beta estimated by
regression techniques using stock return data?
Project vs. Firm Risk (Unlevered Firm)
project cost SML
The Weighted Average Cost Of
• When a firm has both debt and equity in its capital
structure, the most frequent recommendation is to
work with the weighted average cost of capital
S B (13.5)
WACC rS rB (1 TC )
– S is the market value of the firm‟s stock
– B is the market value of the firm‟s debt
– rS is the required rate of return on the firm‟s stock
– rB is the required before tax rate of return on the firm‟s debt
– TC is the firm‟s marginal tax rate
The costs of debt and equity.
The before tax cost of debt can be calculated as the yield to
maturity on the firm‟s existing debt.
Can also be found from yields on companies with
comparable financial risk.
The after tax cost of debt is the before tax cost of debt
multiplied by (1-Tc), where Tc is the firm‟s effective
marginal tax rate.
• The cost of equity can be calculated using the Security
Market Line (SML) from the CAPM.
rS= Rf+ b (E[RM]- Rf)
• This can be using the firm‟s own beta, or that of another
firm that comprises a good surrogate for the project.
Gamma airlines is financed with 60% debt and 40%
equity. Currently the YTM on Gamma bonds is 9%, and
Gamma has estimated its cost of equity to be 14.5%.
Gamma‟s corporate tax rate is 40%. What is Gamma‟s
WACC = .40(14.5%) + .60(9%)(1-.40)
What if the risk of the project at hand differs from that
of Gamma‟s past projects?
What if risk of this project is similar to that of Delta
Airlines‟ projects, and the 14.5% cost of equity figure
was actually obtained for Delta. However, Delta‟s
capital structure differs from Gamma‟s.
Betas and Leverage
We noted earlier that the beta of a portfolio is the
average of the component betas. Also, we can
think of the firm‟s assets as a portfolio of the debt
and equity claims. From these insights it follows
(see pgs 565-566) that:
S B(1 TC ) See
Assets S B(1 T ) Debt
S B(1 T ) (13.3) for
C C no tax
• Where S is the market value of the stock (equity), B is the
market value of debt (borrowings), and Tc is the tax rate.
Adjusting beta for different
In this analysis it is often assumed that the debt has a
zero beta (a big simplification). Then:
S B(1 TC )
Example: Gamma airlines‟ equity beta is observed to be 1.31. It‟s
equity is worth 25.0 million while its debt is worth 15.0 million and
its tax rate is 40%. What is the beta of the underlying assets?
bAssets 131 0.96
Points to Note Regarding
Betas and Leverage
If the firm uses no debt (B=0) the equity beta and the asset
beta are equal.
If the firm uses debt, the equity beta is increased relative to
the asset beta:
b Equity 1 B (1 TC )
(1) equity holders will require a higher rate of return,
(2) when surrogate firms are used to estimate beta,
allowances for differing capital structures will be
Why does Beta increase with
Leverage Increases the volatility of the equity
Outcome Good Bad % Ch.
EBI 125 100 -20%
Interest 50 50 0%
Income 75 50 -33.3%
Aside on Leverage and Risk
A stock/margin example
Suppose you buy XOM at $70 per share with
no leverage (margin).
If stock price rises 10% to $77, your return is 10%
Now,suppose you buy XOM at $70 with 50%
margin (you pay $35 and borrow $35 at 5%).
If stock price is $77:
• You payback 35*1.05=36.75, and keep difference,
40.25. Your return is 15% (40.25/35-1)
But this is a double-edged sword…
Suppose you buy XOM at $70 per share with no
If stock price falls 10% to $63, your return is -10%
Now, suppose you buy XOM at $70 with 50% margin
(you pay $35 and borrow $35 at 5%).
If stock price is $63:
• You payback 35*1.05=36.75, and keep difference, 26.25. Your
return is -25% (26.25/35-1)
Stock/margin example illustrated
Buy stock at Price = $70, r=5%
Return Return 1
Price Margin=0 Margin=.5
50 -0.28571 -0.62143
55 -0.21429 -0.47857 0.6
60 -0.14286 -0.33571
65 -0.07143 -0.19286 0.4
70 0 -0.05 0.2
75 0.071429 0.092857
80 0.142857 0.235714 0 Margin=.5
85 0.214286 0.378571
90 0.285714 0.521429
95 0.357143 0.664286 -0.4
100 0.428571 0.807143
50 60 70 80 90 100
Back to Notes
How to use the tools developed here to select
discount rates for capital budgeting.
The cost of capital for each project (or division) should
reflect the systematic risk of that project and the capital
structure of the firm (or division) taking the project. So,
Select a publicly traded company that is comparable in
terms of the risk of the underlying business.
Obtain the unlevered (asset) beta of the comparable.
Obtain the corresponding project equity beta for your
firm, reflecting your firm‟s capital structure.
Obtain the cost of equity and cost of debt for this
project at your firm.
Calculate the WACC for the project and perform NPV
How to use the tools, continued…
Why is this so hard?
Because your comparable need not have the same capital
structure as your own firm.
What we‟re going to do is start with a levered comparable,
figure out the beta of the equivalent unlevered firm, and
find the corresponding equity beta for your project at your
Then, figure out the WACC and solve NPV...
It‟s just ugly.
Example: BK Industries
BK Industries is a conglomerate company with
operations in marine power, pleasure boating,
defense, and fishing tackle. BK‟s equity beta is
1.0. BK has and will maintain a debt/equity ratio
Can we use the company cost of capital to
value an investment in text editing?
Latec Inc. is a firm that makes only text editing
systems. Latec‟s equity beta is 1.35. Latec has a
debt to equity ratio of 0.75, and a marginal tax rate
Delevered Betas with debt/equity ratios
The formulas for obtaining asset betas from equity betas and
vice versa provided earlier required dollars values for debt (B)
and equity (S). What if you are only given the leverage ratio,
L = B/S? The formulas are restated as:
1 L(1 TC )
bEquity bAssets (1 L(1 TC ))
Step 1: Delever Latec‟s Beta to obtain the
Beta of text-editing assets:
Latec has L =0.75, TC = .45, and an equity beta of
bAssets 135 0.955
1 0.75(1 0.45)
Step 2: Relever the asset Beta to
reflect BK‟s capital structure:
Recalling that BK will keep its debt/equity
ratio equal to one, we can get:
bEquity 0.9551 1(1.45) 148
•This is the beta for a BK equity position in a text
•Why is this equity beta greater than Latec‟s?
BK Industries, Cont.
Assume that the risk free rate is 8% and that BK‟s
cost of debt is also 8%. The market risk premium
is 7%. The required return on BK‟s equity is:
rS RF bEquity ( RM RF ) 8% 148 * 7% 18.36%
The weighted average cost of capital for the text editing
venture (using the fact that B/S = 1 here) is:
WACC rS rB (1 TC )
= (0.5) *18 .36 % (0.5) * 8%(1 0.45 ) 11 .38 %
Finally, we can evaluate the NPV of the text editing
venture using the WACC that reflects the risk
associated with this particular business. Using the cash
flow estimates obtained earlier:
3.980 5.419 6.685 5.990 22.465
(1.1138 ) (1.1138 ) 2 (1.1138 ) 3 (1.1138 ) 4 (1.1138 ) 5
• The NPV is positive, so proceed with the text editing business.
• Notice that the selected discount rate of 11.38% reflects:
The risk (beta) of text editing businesses, not BK‟s existing
BK‟s capital structure, not that of the surrogate firm.
Why the firm (or division‟s) capital
In the long run, each individual project is
funded from a “pool” of capital. That pool
includes both debt and equity.
If we evaluated projects based on the specific
financing used then those that were debt
financed would tend to look better than those
that are equity financed. But are they really
Some practical observations
(useful on cases and in real applications)
Leverage is some times measured as,
the debt to equity ratio (which we denoted L)
the debt to total capital (equity + debt) ratio,
which is the weighting on debt (W).
You can convert these measures as
• L = W/(1-W)
• W = L/(1+L)
The debt and equity numbers used should, in
principle, be based on market values.
• In practice, the market value of equity is typically
used, along with the book value of debt.
Summary: Risk, Return, and Discount Rates
Some risk can be diversified, some cannot.
„Beta‟ coefficients provide a measure of non-
We expect that non-diversifiable risk will earn a risk
premium in equilibrium but that diversifiable risk will
The CAPM (and particularly the SML) is a simple model
capturing important insights regarding risk and
Discount rates for projects should reflect the systematic
risk of the project (not necessarily that of the firm) and
the capital structure of the firm or division (not
necessarily the financing of the project).