Risk, returns and WACC
CAPM and the capital
budgeting
FIN 351: lecture 7
Today’s plan

Review what we have learned in the last
lecture
•
•
•
•
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Risk
Portfolio
CAPM
The security market line
Portfolio rules
The application of CAPM in capital budgeting
WACC (Weighted Average Cost of Capital)
What have we learned in the
last lecture?
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How to measure investment performance?
How to measure risk?
Two kinds of risk?
How to measure systematic risk?
What is the heuristic meaning of the Beta?
What is a portfolio?
How to calculate a portfolio weight?
What is the CAPM?
What is the basic idea behind CAPM?
What is the security market line?
Measuring Market Risk

Market Portfolio
• It is a portfolio of all assets in the economy.
In
practice a broad stock market index, such as the
S&P 500 is used to represent the market portfolio.
The market return is denoted by Rm

Beta (β)
• Sensitivity of a stock’s return to the return on the
•
market portfolio,
Mathematically,
Cov(ri , Rm )
i 
Var ( Rm )
An intuitive example for Beta
Turbo Charged Seafood has the following
% returns on its stock, relative to the
listed changes in the % return on the
market portfolio. The beta of Turbo
Charged Seafood can be derived from
this information.
Measuring Market Risk
(example, continue)
Month Market Return % Turbo Return %
1
+ 1
+ 0.8
2
+ 1
+ 1.8
3
+ 1
- 0.2
4
-1
- 1.8
5
-1
+ 0.2
6
-1
- 0.8
Measuring Market Risk
(continue)



When the market was up 1%, Turbo
average % change was +0.8%
When the market was down 1%, Turbo
average % change was -0.8%
The average change of 1.6 % (-0.8 to
0.8) divided by the 2% (-1.0 to 1.0)
change in the market produces a beta of
0.8. β=1.6/2=0.8
Another example

Suppose we have following information:
Market
Stock A
bad
-8%
-10%
-6%
good
32%
38%
24%
State
Stock B
a. What is the beta for each stock?
b. What is the expected return for each stock if each scenario is
equally likely?
c. What is the expected return for each stock if the probability
for good economy is 20%?
Solution
a.
b.
0.38  ( 0.1) 0.48

 1 .2
0.32  ( 0.08) 0.40
0.24  ( 0.06) 0.30
B 

 0.75
0.32  ( 0.08) 0.40
A 
rA  0.5 * 0.38  0.5 * (0.1)  0.14
rB  0.5 * 0.24  0.5 * (0.06)  0.09
c.
rA  0.2 * 0.38  0.8 * (0.1)  0.004
rB  0.2 * 0.24  0.8 * (0.06)  0
Betas for the market portfolio
and risk-free investment

What is the beta of the market portfolio?

What is the beta of the risk-free security?
Market risk and risk premium

Risk premium for bearing market risk
• The difference between the expected return
•
•
required by investors and the risk-free asset.
Example, the expected return on IBM is 10%,
the risk-free rate is 5%, and the risk premium
is 10% -5%=5%
If a security ( an individual security or a
portfolio) has market or systematic risk, riskaverse investors will require a risk premium.
CAPM (Capital Asset Pricing
Model)

The risk premium on each security is
proportional to the market risk premium
and the beta of the security.
• That is,
ri  r f  i ( Rm  r f )
ri  r f  risk premium for sec urity i
Rm  r f  risk premium for the market portfolio
Security market line (SML)
The graphic representation of CAPM in
the expected return and Beta plane
Security Market Line
16
14
Expected Return (%) .

12
Rm
10
8
6
4
rf
2
0
0
0.2
0.4
0.6
Beta
0.8
1
1.2
Some true or false questions
1.A market index is used to measure performance of a
broad-based portfolio of stocks.
2. Long-term corporate bonds are riskier than common
stocks.
3.If one portfolio's variance exceeds that of another
portfolio, its standard deviation will also be greater
than that of the other portfolio.
4. Portfolio weights are always positive.
Some true or false questions
5. Standard deviation can be calculated as the square
of the variance.
6. Market risk can be eliminated in a stock portfolio
through diversification.
7. Macro risks are faced by all common stock investors.
8. The risk that remains in a stock portfolio after efforts
to diversify is known as unique risk.
9. We use the standard deviation or variance of stock
prices to measure the risk of a stock.
Portfolio rules
Rule 1: The realized return of a portfolio will be an weighted
average of the realized returns of the securities in the portfolio.
rp 
n
xr
i 1
i i
Rule 2: The expected return of a portfolio will be an weighted
average of the expected returns of the securities in the portfolio.
rp 
n
xr
i 1
i i
Rule 3: The Beta of a portfolio will be an weighted average of the
Betas of the securities in the portfolio.
p 
n
x 
i 1
i
i
Example

Suppose you have a portfolio of IBM and
Dell with a beta of 1.2 and 2.2,
respectively. If you put 50% of your
money in IBM, and the other in Dell,
what is the beta of your portfolio
Beta of your portfolio =0.5*1.2 +0.5*2.2=1.7
Project Risk and cost of the capital


In capital budgeting, in order to calculate the
NPV of the project, we need to measure the
risk of the project and thus find out the
discount rate (the cost of capital)
We can use Beta of the project cash flows to
measure the risk of the project and use CAPM
to get the expected return required by
investors
• rproject  r f   project ( Rm  r f )
Example 1

Based on the CAPM, ABC Company has a
cost of capital of 17%. (4 + 1.3(10)). A
breakdown of the company’s investment
projects is listed below.
• 1/3 Nuclear Parts: β=2.0
• 1/3 Computer Hard Drive:
• 1/3 Dog Food Production:

β =1.3
β =0.6
When evaluating a new dog food production
investment, which cost of capital should be
used and how much?
Solution

Since dog food projects may have
similar systematic risk to the dog food
division, we use a beta of 0.6 to measure
the risk of the projects to be taken.

Thus the expected return on the project
or the cost of capital is
0.04+0.6*(0.1)=0.l or 10%
Example 2

Stock A has a beta of .5 and investors
expect it to return 5%. Stock B has a
beta of 1.5 and investors expect it to
return 13%. What is the market risk
premium and the expected rate of return
on the market portfolio?
Solution

According to the CAPM
5  r f  0.5 * ( Rm  r f )
13  r f  1.5 * ( Rm  r f )
r f  1%
Rm  9%
Example 3

You have $1 million of your own money
and borrow another $1 million at a riskfree rate of 4% to invest in the market
portfolio. The expected return for the
market portfolio is 12%, what is the
expected return on your portfolio?
Solution

We can use two approaches to solve it:
• First, the expected rate of return of a portfolio
•
is the weighed average of the expected rates
of return of the securities in the portfolio.
Second , the beta of a portfolio is the weighed
average of the betas of the securities in the
portfolio. Then use the CAPM to get the
expected rate of return.
Solution (continue)

First approach

Second approach
W  $1; W f  1; Wm  2
1
2
xf 
 1; xm   2
1
1
R p  1* 4  2 *12  20%
W  $1; W f  1; Wm  2
1
2
xf 
 1; xm   2
1
1
 p  1* 0  2 *1  2
R p  4  2 * 8  20%
The cost of capital
Cost of Capital
• The expected return the firm’s investors
require if they invest in securities or projects
with comparable degrees of risk.
WACC to approximate the cost
of capital or discount rate
Weighted -average cost of capital=
D (1 - Tc)r + E r
WACC = V
d V e
Summary of WACC calculation

Three steps in calculating WACC
• First step: Calculate the portfolio weight using
•
•
the market value.
Second step: Determine the required rate of
return on each security in the portfolio.
Third step: Calculate a weighted average of
these returns, or the expected return on the
portfolio.
WACC calculation(continue)
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

In calculating WACC, we have to use
market values of debt and equity.
Even if you are given the book value of
debt, you may convert this book value to
market debt value to calculate WACC
Why do we use market values of debt
and equity, but not book values of debt
and equity, in calculating WACC?
The cost of capital for the bond

The cost of capital for the bond
• It is the YTM, the expected return required
•
the investors.
That is
Pbond 
cpn
cpn
cpn  principal


1  rd 1  rd 2
1  rd t
• The expected return on a bond can also be
calculated by using CAPM
rd  r f   d ( Rm  r f )
by
Example 2

A bond with a face value of $2000
matures in 5 years. The coupon rate is
8%. If the market price for this bond is
$1900.
(a) What is the expected return on this bond or
what is the cost of debt or interest rate for this
bond?
(b) Suppose that the YTM is 9%, what is the
market value of this bond?
Solution
(a)
 1

1
2000

1900  160


5 
5
 YTM YTM (1  YTM )  (1  YTM )
YTM  9.3%
(b)
 1
 2000
1
Pbond  160

 $1,922
 
5
5
 0.09 0.09 *1.09  1.09
The cost of capital for a stock

The cost of capital for a stock is
calculated by using
• CAPM
re = rf + i (R m - rf )
• Dividend growth model
DIV1
DIV1
P0 
 re 
g
re  g
P0
Example 3

Sock A now pays a dividend of $1.5 per
share annually, It is expected that
dividend is going to grow at a constant
rate of 2%. The current price for stock A
is $25 per share. What is the expected
return or the cost of capital by investing
in this stock?
Solution
Using the dividend discount model, we have
1.5 *1.02
25 
 r  8.12%
r  0.02
Example 4

Geothermal Inc. has two securities:
debt and stocks. The market debt value
is $194 million, but the firm’s market
value is $647 million. Given that
geothermal pays 8% for debt and 14%
for equity, what is the Company Cost of
Capital (There is no corporate tax)?
Solution
194
453
WACC 
* 0.08 
* 0.14  12.2%
647
647
Example 5

Executive Fruit has issued debt,
preferred stock and common stock.
The market value of these securities
are $4mil, $2mil, and $6mil,
respectively. The required returns are
6%, 12%, and 18%, respectively.
•
What is the WACC for Executive Fruit, Inc.?
Solution
V  4  2  6  12
4
2
6
WACC  * 0.06  * 0.12  * 0.18
12
12
12
 13%
Example 6 (with tax)

Geothermal Inc. has two securities:
debt and stocks. The market debt value
is $194 million, but the firm’s market
value is $647 million. Given that
geothermal pays 8% for debt and 14%
for equity, what is the Company Cost of
Capital if the tax rate is 50%?
Solution
194
453
WACC 
* 0.08 * 0.5 
* 0.14  11%
647
647