FIN 351: lecture 7

Risk, returns and WACC

CAPM and the capital budgeting

Today’s plan

Review what we have learned in the last lecture • 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? 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, )(

),(

m

mii RVar

RrCov

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:

State Market Stock A Stock B

bad

good

-8% -10%

38%

-6%

24%32%

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.

c.

09.0)06.0(*5.024.0*5.0

14.0)1.0(*5.038.0*5.0

B

A

r

r

75.040.0

30.0

)08.0(32.0

)06.0(24.0

2.140.0

48.0

)08.0(32.0

)1.0(38.0

B

A

0)06.0(*8.024.0*2.0

004.0)1.0(*8.038.0*2.0

B

A

r

r

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, risk-averse 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,

)( fmifi rRrr

portfoliomarkettheforpremiumriskrR

iurityforpremiumriskrr

fm

fi

sec

Security market line (SML)

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2

Beta

Ex

pe

cte

d R

etu

rn (

%)

. The graphic representation of CAPM in

the expected return and Beta plane

rf

Security Market Line

Rm

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.

Rule 2: The expected return of a portfolio will be an weighted average of the expected returns of the securities in the portfolio.

Rule 3: The Beta of a portfolio will be an weighted average of the Betas of the securities in the portfolio.

i

n

iip rxr

1

i

n

iip rxr

1

i

n

iip x

1

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 • )( fmprojectfproject rRrr

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: β =1.3

• 1/3 Dog Food Production: 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

%9

%1

)(*5.113

)(*5.05

m

f

fmf

fmf

R

r

rRr

rRr

Example 3

You have $1 million of your own money and borrow another $1 million at a risk-free 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

%2012*24*1

21

2;1

1

1

2;1;1$

p

mf

mf

R

xx

WWW

%208*24

21*20*1

21

2;1

1

1

2;1;1$

p

p

mf

mf

R

xx

WWW

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=

eVE

dVD r +Tc)r-(1 =WACC

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)

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 by

the investors.

• That is

• The expected return on a bond can also be calculated by using CAPM

tddd r

principalcpn

r

cpnr

cpn

111

P2bond

)( fmdfd rRrr

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)

(b)

%3.9

)1(

2000

)1(

111601900

55

YTM

YTMYTMYTMYTM

922,1$09.1

2000

09.1*09.0

1

09.0

1160

55

bondP

The cost of capital for a stock

The cost of capital for a stock is calculated by using • CAPM

• Dividend growth model

)r-(R+r=r fmfe i

gP

DIVr

gr

DIVP e

e

0

110

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

%12.802.0

02.1*5.125

rr

Using the dividend discount model, we have

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

%2.1214.0*647

45308.0*

647

194WACC

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

%13

18.0*12

612.0*

12

206.0*

12

4

12624

WACC

V

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

%1114.0*647

4535.0*08.0*

647

194WACC