The cost of capital
The application of the
portfolio theory and CAPM
FIN351: lecture 6
Today’s plan

What have we accomplished in the last
lecture?
Cost of capital

The market efficiency concept

• Capital structure and WACC
• Calculate WACC without tax
• Calculate WACC with tax
FIN351: lecture 6
What have we learned in the
last lecture?
Beta (β): Measuring market risk or
systematic risk of a security
• Which is defined as the contribution
•
of a
security to the risk on the market portfolio,
Mathematically,
~
Cov (~
ri , Rm )
i 
~
Var ( Rm )
• How to understand β intuitively?
FIN351: lecture 6
Portfolio standard deviation
What have we learned in the
last lecture? (1)
Unique
risk
Market risk
0
5
10
Number of Securities
FIN351: lecture 6
15
What have we learned in the
last lecture? (2)

The beta of a portfolio
• The beta of a portfolio will be an weighted
average of the betas of the securities in the
portfolio.

The expected return of a portfolio
• The expected return of a portfolio will be an
weighted average of the expected returns of
the securities in the portfolio.
FIN351: lecture 6
What have we learned in the
last lecture? (3)

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
FIN351: lecture 6
What have we learned in the
last lecture? (4)
The graphic representation of CAPM in
the expected return and Beta plane
Security Market Line
16
14
Expected Return (%) .

12
10
Rm
8
6
4
rf
2
0
0
0.2
0.4
0.6
Beta
FIN351: lecture 6
0.8
1
1.2
What have we learned in the
last lecture? (5)

We can use the Beta of the project
cash flows to measure the risk of the
project and use CAPM to get the
discount rate or the expected return
required by investors
•
rproject  r f   project ( Rm  r f )
FIN351: lecture 6
What have we learned in the
last lecture? (6)

Some results
• The Beta of the market portfolio is 1.
• The Beta of the risk-free asset is 0.
• The portfolio weights can be negative, but
their sum must be 1.
FIN351: lecture 6
Example 1


You have $1 million of your own money to
invest in stock A and B. You short-sell $1
million stock A and then invest in stock B.
The expected returns for stock A and B are
15% and 20%, respectively. The risk-free
rate is 5%. Stock A has a beta of 1.5 and
stock B has a beta of 2.0.The expected rate
of return for the market portfolio is 13%.
Please use two approaches to calculate the
expected rate of return on your portfolio?
FIN351: lecture 6
Solution
First, calculate the portfolio weight as follows:
x A  1; xB  2
(1) The expected rate of return of a portfolio is the weighted average of the
rates of return of the securities in the portfolio.
R p  1* 0.15  2 * 0.2  25%
(2) The beta of a portfolio is the weighted average of the betas of the
securities in the portfolio. Then use CAPM to calculate the expected return
 p  1*1.5  2 * 2  2.5
R p  5%  2.5 * 8%  25%
FIN351: lecture 6
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.
FIN351: lecture 6
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 )
FIN351: lecture 6
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
$1600.
(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?
FIN351: lecture 6
Solution
(a)
 1

1
2000

1600  160


5 
5
 YTM YTM (1  YTM )  (1  YTM )
YTM  13.8%
(b)
1
 1
 2000
Pbond  160


 $1,922
5 
5
 0.09 0.09 *1.09  1.09
FIN351: lecture 6
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
FIN351: lecture 6
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?
FIN351: lecture 6
Solution
Using the dividend discount model, we have
1.5 *1.02
25 
 r  8.12%
r  0.02
FIN351: lecture 6
Capital structure

Capital Structure
• The firm’s mix of debt financing and equity
•
•
•
financing.
If we use D , E and V to denote the values of
debt, equity and firm (asset), respectively.
Then V=D+E
The ratio of D/V or D/E can be used to
measure the capital structure of the firm.
FIN351: lecture 6
Weighted Average Cost of Capital
(WACC)




If you invest in a stock, you require a an expected rate of
return, or the cost of capital for the stock
If you invest in a bond, you require a an expected rate of
return, or the cost of capital for the bond
Then, if you invest in a a portfolio of all the firm’s securities
(debt and stock), what is the expected rate of return you
require?
The expected rate of return on a portfolio of all the firm’s
securities is called the weighted average cost of capital
(WACC). It is also called the expected return on the asset of
the firm.
FIN351: lecture 6
WACC (continues)


According to portfolio theory, we know
the expected return on a portfolio is the
weighted average of the expected
returns on the securities of the portfolio.
Thus, WACC for a firm with debt and
equity is
E
D
WACC 
V
re 
V
rd
where V  D  E
FIN351: lecture 6
WACC (continue)



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?
FIN351: lecture 6
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?
FIN351: lecture 6
Solution
194
453
WACC 
* 0.08 
* 0.14  12.2%
647
647
FIN351: lecture 6
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.?
FIN351: lecture 6
Solution
V  4  2  6  12
4
2
6
WACC  * 0.06  * 0.12  * 0.18
12
12
12
 13%
FIN351: lecture 6
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
protfolio.
FIN351: lecture 6
The impact of tax on WACC

Taxes are an important consideration in the
company cost of capital because interest
payments are deducted from income before
tax is calculated.
After - tax cost of debt = (pretax cost of debt) times (1 - tax rate)
= rd (1 - Tc)
FIN351: lecture 6
WACC with tax
Weighted -average cost of capital=
D (1 - Tc)r + E r
WACC = V
d V e
FIN351: lecture 6
A summary example

John Cox, a recent MBA student of SFSU, was asked by his
boss in Geothermal to decide whether the firm should take an
expansion project: the cost of the project is $30 million, and the
project is expected to generate a perpetual incremental cash
flow of $4.5 million. Currently, Geothermal has 20 million
shares of common stocks outstanding, with a market price of
$22.65 per share. The Beta of the firm’s equity is 1.1. The risk
free rate is 4% and the market risk premium is 5.6%. The firm
also has long-term debt, with the YTM of 9%. John also got the
following information from the firm’s balance sheet:
•
•

Debt (12 years maturity, 8% coupon): $200 million
Common stocks:$110 million
If the tax rate is 35%, should John suggest to his boss to take
the project or not?
FIN351: lecture 6
Solution
rd  9%
re  4%  1.1* 5.6%  10.16%
E  20 * 22.65  453
1
1
200
D  16 * (

)
 185.7
12
12
0.09 0.09 *1.09
1.09
D
E
WACC 
(1  t )rd 
re  8.9%
DE
DE
FIN351: lecture 6
Investment vs. Financing
Asset
V
Liabilities and equity
Debt: D
Equity: E


Investment decisions or capital budgeting is
about how to take projects to maximize V.
Financing decisions are about how to raise
capital (E or D) to finance the projects to be
taken
FIN351: lecture 6
Financing and market Efficiency
Market Efficiency
Market efficiency is concerned about
whether capital markets have all
information about the cash flows and
risk of projects.
FIN351: lecture 6
Efficient capital markets
Efficient Capital Markets – If capital markets are
efficient, then security prices reflect all relevant
information about asset values ( cash flows and
risk)
FIN351: lecture 6
Market efficiency and random
walk


Market efficiency concepts are very
abstract.
How can we use a simple way to check
whether the stock market (one of the
capital markets) is efficient or not?
• If the stock price follows a random walk, then
the stock market is efficient.
FIN351: lecture 6
What is a random walk of stock
prices?


The movement of stock prices from day
to day DO NOT reflect any pattern.
Statistically speaking, the movement of
stock prices is random.
FIN351: lecture 6
A Random Walk example
Heads
Heads
$106.09
$103.00
Tails
$100.43
$100.00
Heads
Tails
$100.43
$97.50
Tails
Coin Toss Game
FIN351: lecture 6
$95.06
Three forms of market
efficiency


The random walk concept is still abstract
Financial economists have used three
more specific forms to characterize or
judge market efficiency.
• Weak-form
• Semi-strong form
• Strong form
FIN351: lecture 6
Weak-form of market efficiency
Weak Form Efficiency - Market prices reflect
all information contained in the history of past
prices, or you cannot use past stock prices to
predict future prices
Technical Analysts - Investors who attempt to
identify over- or undervalued stocks by searching
for patterns in past prices.
FIN351: lecture 6
Efficient Market Theory
$90
EI’s Stock
Price
70
50
Cycles
disappear
once
identified
Last
Month
FIN351: lecture 6
This
Month
Next
Month
Semi-strong form of market
efficiency

Semi-Strong Form Efficiency - Market
prices reflect all publicly available information
such as earnings, price-to-earnings
ratios,etc.
Fundamental Analysts - Analysts who attempt to
fund under- or overvalued securities by analyzing
fundamental information, such as earnings, asset
values, and business prospects.
FIN351: lecture 6
Cumulative Abnormal Return
(%)
Efficient Market Theory
39
34
29
24
19
14
9
4
-1
-6
-11
-16
Announcement Date
Days Relative to annoncement date
FIN351: lecture 6
Market Efficiency
Fama & French
Return vs. Book-Market
Average return, percent
25
20
15
10
5
0
Highest
FIN351: lecture 6
Book-Market Ratio
Strong form of market efficiency
Strong Form Efficiency - Market prices reflect
all information that could in principle be used
to determine true value.

Inside trading
• Investors use private information to predict
future price movements
FIN351: lecture 6
Cumulative Abnormal Return
(%)
Efficient Market Theory
Announcement Date
39
34
29
24
19
14
9
4
-1
-6
-11
-16
Days Relative to annoncement date
FIN351: lecture 6
Some exercises
1.
2.
If stock markets are efficient, what should the
correlation between stock returns for two
non-overlapping periods?
Which is the most likely to contradict the
weak-form of efficiency
a. Over 25% of mutual funds outperform the market on
b.
c.
average
Insiders can make abnormal profits
Every January, the stock market earns abnormal
return
FIN351: lecture 6
Several types of securities

Three types of securities
• Common Stock
• Preferred stock
• Corporate debt
FIN351: lecture 6
Common Stock

Common stocks have the following
forms:
• Treasury stock
• Issued shares
• Outstanding shares
• Authorized share capital
• Par value

Ownership of the corporation
FIN351: lecture 6
Corporate debt

Corporate bonds
• Primary rate
• Funded debt
• Sink fund
• Callable bond
• Subordinate debt
• Secure debt
FIN351: lecture 6
Preferred stock

Preferred stock and common stock

Preferred stock and bond
• Priority and voting rights
• Obligation and bankruptcy
FIN351: lecture 6