lecture 12-13 capm

8/17/2019 Lecture 12-13 CAPM

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Professor Sang Byung [email protected]

The Capital Asset Pricing Model

(CAPM)

8/17/2019 Lecture 12-13 CAPM


Last class

• The risk-return trade off • Return under uncertainty Expected return

• Risk Variance or Standard deviation

• Idiosyncratic risk vs. Systematic risk

• Diversification benefits

• Tendency to co-move Covariance or Correlation

8/17/2019 Lecture 12-13 CAPM


Today

•Population statistics vs. Sample statistics

• The Capital Asset Pricing Model (CAPM)

8/17/2019 Lecture 12-13 CAPM


Population statistics

• Last class we studied

• Mean (aka Expectation)

• Variance and Standard deviation

• Covariance and Correlation

• When we calculate them, you were given

• All possible outcomes

• The probability of each outcome

• Therefore, what we learned last class is also calledpopulation statistic.

• For example, population mean, population variance, and etc.

•Population: a set of entire objects of interest

8/17/2019 Lecture 12-13 CAPM


8/17/2019 Lecture 12-13 CAPM


Sample statistics

•Samples: {, , , … , } & {, , , … , }

Sample mean ( ) + + ⋯ +

1

=

Sample variance () 1 − 1

=

(− )

Samplevariance() 1 − 1

=

(− )(−)

8/17/2019 Lecture 12-13 CAPM


Recall: Not all risk is created equal

•

Total risk• The variance of an investment’s returns

• Firm-specific risk

• The risk that can be diversified away, only affects one firm

• Aka diversifiable risk, idiosyncratic risk, unique risk, or‘casino’ risk

• Systematic risk

• The risk that cannot be diversified away, affects every firm

• Aka non-diversifiable risk or market risk

8/17/2019 Lecture 12-13 CAPM


Risk premium

• Risk premium =

− • How much compensation should investors receive for

taking on risk?

• That is, how large is the “risk premium”?

• Depends on the type of risk

• Firm-specific risk can be diversified away, so there should be norisk premium for firm-specific risk.

• Systematic risk cannot be diversified away, so some risk premium isrequired.

• How much risk premium?

•

It depends on the amount of exposure to systematic risk.

8/17/2019 Lecture 12-13 CAPM


Proxy for systematic risk

• How do we measure systematic risk exposure?

• Use a portfolio that corresponds to systematic shocks.

• We usually use a broad market portfolio (e.g., S&P 500).

• Capital Asset Pricing Model (CAPM)

• A one-factor model in which a security’s risk premium

depends on the sensitivity of the security’s return to the

market portfolio’s return.• This sensitivity is the stock’s beta.

• CAPM is not perfect, but it remains widely used. There

are other models, but we won’t cover them in this class.

8/17/2019 Lecture 12-13 CAPM


CAPM formula

• An investment’s expected return:

()is a

function of,

1. The risk-free return: 2. The market risk premium: − 3. The investment’s sensitivity to market risk (or beta):

• What values should we use?

• Use U.S. Treasuries for the risk-free rate.

• The market risk premium is trickier.

• One option is to use the long-run average of the S&P500 less

the risk-free rate.

8/17/2019 Lecture 12-13 CAPM


CAPM formula

+ × [ − ]• What is the expected return for an investment

with a beta of 0.67 if the risk-free return is 1.5%and the market risk premium is 6.75%?

1.5% + 0.67 × 6.75% 6.0225%

8/17/2019 Lecture 12-13 CAPM


Example

•Compute expected returns for the below beta

values.

• Assume a risk-free rate of 2% and a market risk

premium of 7%.

1. Beta = 0.5

2. Beta = 1.25

3. Beta = 1.3386

8/17/2019 Lecture 12-13 CAPM


Answer

•Compute expected returns for the below beta

values.

• Assume a risk-free rate of 2% and a market risk

premium of 7%.

1. Beta = 0.5 => 5.5%2. Beta = 1.25 => 10.75%3. Beta = 1.3386 => 11.3702%

8/17/2019 Lecture 12-13 CAPM


How to compute beta

• The formula for beta:

( , )()

• Equivalently,

()() ()

8/17/2019 Lecture 12-13 CAPM


Examples

• What is the beta if,• , 0.048• 0.04

• What is the beta if,

• 0.5• 40%• 20%

8/17/2019 Lecture 12-13 CAPM


Quiz

• What is the beta on the market portfolio?

• What is the beta on the risk-free asset?

8/17/2019 Lecture 12-13 CAPM


How to compute beta

•In reality, we can’t calculate covariance and

variance in a population.

• It is not like our simple example with three possible

future states. In the example, you are given possible

outcomes and their probabilities.

• Instead,

• Use the time series of past returns.

• Estimate them using the sample covariance and sample

variance!

8/17/2019 Lecture 12-13 CAPM


Another way

• Run a regression!

I n v e s t m

e n t R e t u r n s

Return on market

, + , + ,

8/17/2019 Lecture 12-13 CAPM


Example: of L Brands

•Five step procedure to estimate beta:

1. Choose a period over which you want to

estimate .e.g. January 1999 – December 2004

8/17/2019 Lecture 12-13 CAPM



2. From finance.yahoo.com or some other

database, download time series of monthly

closing prices (incl. dividends) for:

• S&P 500 index (symbol: ^SPX)

• The proxy for the overall market portfolio

• Researchers prefer the CRSP value-weighted index, which

contains all publicly traded companies on the NYSE, NASDAQ,

and AMEX.

• L brands (symbol: LB)

8/17/2019 Lecture 12-13 CAPM



3. Import and merge the downloaded csv-files in

Excel, and calculate monthly returns for both

time-series of monthly closing prices.

8/17/2019 Lecture 12-13 CAPM



4. Draw a scatter plot: S&P 500 monthly returns

on x-axis, L Brand’s monthly returns on the y-

axis.

8/17/2019 Lecture 12-13 CAPM



5. Right-click on any data point in the scatter

plot, and select “Add trendline” and excel will

automatically fit a regression line:

y=1.3613x + 0.0163

8/17/2019 Lecture 12-13 CAPM



•Conclusion• The equity- of L brands over the period January 1999 –

December 2004, is 1.36, the slope of the regression line

in the scatter plot

• Another example: “CAPM example.xls”

spreadsheet for Southwest airlines

8/17/2019 Lecture 12-13 CAPM


Example: High beta

8/17/2019 Lecture 12-13 CAPM


Example: Low beta

8/17/2019 Lecture 12-13 CAPM


Betas of some stocks

•

Any trends in order?

• (Yahoo! Finance used

different time periodsso the betas may be

different than what I

computed in previous

slides.)

8/17/2019 Lecture 12-13 CAPM


What determines ?

•

What are the fundamental drivers of ? What isit about a firm’s operations that can cause a

high ?

• Two factors which are often put forward as

being the fundamental and strategic

determinants of :

• Correlation with the business cycle

• Operating leverage

8/17/2019 Lecture 12-13 CAPM


Correlation with business cycle

• Products that are highly correlated with the

business cycle should have high ′• e.g. consumer durables, such as new cars

• Products that people need all the time are less

likely to be cyclical, and will have lower ′• e.g. food, gasoline

• So you can think along these lines when it

comes to completely new projects

8/17/2019 Lecture 12-13 CAPM


Operating leverage

• A business that has a higher proportion of fixed

costs and a lower proportion of variable costs is

said to have used more operating leverage.

• A business that has a lower fixed costs and higher variable costs are said to employ less operating

leverage.

• Higher operating leverage higher beta (why?)

• Examples: wireless networking…

8/17/2019 Lecture 12-13 CAPM


Historical vs Future betas

•

All betas are historical in the sense that theyare estimated using historical data. Nothingwe can do about that.

• As a result, they may or may not be a goodpredictor of future betas.

•If you have a new firm, then you have no data.

• May want to think through fundamentals of thebusiness.

8/17/2019 Lecture 12-13 CAPM


Application to Capital Budgeting

• Previously, we calculated NPV given a discount

without discussion where came from.

• In fact, the main reason why we are interested

in the CAPM is that it tells us what should be!

• To see how this works,

1. Consider a firm with no debt.2. Consider a project in the same line of business as the

firm’s existing projects.

3. Assume that the CAPM is true.

( )

8/17/2019 Lecture 12-13 CAPM


NPV rule (revisited)

• Here, the cost of capital is

+ ( − )• This is the firm’s discount rate for future payoffs

• Beta for returns on the firm’s equity is

, /()• The firm should accept the project if NPV>0.

+ []1 + + []1 + + ⋯+ []1 +

+

=

= []1 +

R i b hi d NPV

8/17/2019 Lecture 12-13 CAPM


Reasoning behind NPV

• Situation 1: riskless project

• Suppose the firm has access to a project that costs $1000 (i.e. −$1000) and will pay for sure next year.

• What is the alternative to investing in this?

• Investing in (or equivalently, putting money in bank)

• If the firm invests at , it gets $1000(1+) next year.

• The firm should take the project only if it offers a greater

payoff than investing in the risk-free asset. That is, > $1000(1+ ) ⇔ −$1000 + /(1 + ) > 0

• This is exactly equivalent to asking if NPV>0 because

−$1000 + /(1 + )

R i b hi d NPV

8/17/2019 Lecture 12-13 CAPM



• Situation 2: extension to risky projects

• What if instead the payoff next year is risky and its expected value is E[C]?

• The firm should only invest in the project if it offers a higherreturn than the alternative given the level of risk.

• What is the alternative to investing in this?

• A stock portfolio with the same as the firm.

•

The portfolio’s risk would then equal the project’s risk.• According to the CAPM, the portfolio’s rate of return must be:

+ ( − )• If the firm puts $1000 into this portfolio, the expected payoff

is $1000 1 + .

R i b hi d NPV

8/17/2019 Lecture 12-13 CAPM



•

The firm should take the project only if it offers a greaterpayoff than investing in the stock portfolio. That is,

[] > $1000(1 + ) ⇔ −$1000 + []/(1+ ) > 0• This is exactly equivalent to asking if NPV>0 because

−$1000 + []/(1 + )• Notes:

1. The multi-payoff case works similarly to the single-

payoff case.2. measures the project’s amount of risk. Since

shareholders hold well-diversified portfolios, only

systematic risk matters. An increase in idiosyncratic

risk will be diversified away.

T i t t ti

8/17/2019 Lecture 12-13 CAPM


Two important assumptions

• For this result, we have made two important

simplifying assumptions:1. The project is in the same line of business as the firm.

2. The firm has no debt.

• This allowed to use the beta estimated from thefirm’s stock.

• If either assumption does not hold, it is incorrect to discountusing the stock’s required rate of return.

• Similar reasoning tells us that if a firm is considering aproject in a new line of business, the firm should use thebeta for the new line

• The beta estimated from the firm’s stock represents only the riskof the existing businesses.

E l

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Example

•

Bell Computers is a computer hardwarecompany. It has a project with cost of $1M and

expected cash flow at year 1 of $1.24M.

• Suppose Bell is an all-equity firm, and the project’s risk

is comparable to that of Bell’s other projects.

• Bell stock’s (like others in the computer industry) is

1.83. Should Bell accept this project?

•

The risk-free rate is 5% and the market risk premium is13%.

A

8/17/2019 Lecture 12-13 CAPM


Answer

•

Calculate the cost of capital + −

5% + 1.83 13% − 5% 20%

• Calculate the NPV

−1 + 1.24

1.2 0.033 > 0

• The NPV rule says

• Bell should accept the project.

E l ( t’d)

8/17/2019 Lecture 12-13 CAPM


Example (cont’d)

•

Macrosoft (MS) is an all-equity softwarecompany but want to invest in a project to build

computer hardware (like Bell).

• Macrosoft’s stock’s is 1.25, implying a cost of capital of:

+ − 5% + 1.25 13% − 5% 15%

• Should Macrosoft use its own cost of capital () or

Bell’s ()?

E l ( t’d)

8/17/2019 Lecture 12-13 CAPM


Example (cont’d)

• Macrosoft should use

because:

• Investing in hardware is a riskier project, so the cash

flows will be riskier.

• Macrosoft’s investors need to be compensated for therisk. If they used the discount rate 15%, they might

accept projects that do not sufficiently compensate them

for the risk they are taking.

• For example, a project with −1 and E[] 1.7 would be accepted if it were a software project, but not if

it were a hardware project.

E l ( t’d)

8/17/2019 Lecture 12-13 CAPM


Example (cont’d)

•

Suppose Macrosoft invested in a hardwareproject, so 50% of Macrosoft’s equity were

accounted for by hardware, and 50% by

software. What is Macrosoft’s ?

0.5 1.83 + 0.5 1.25 1.54

A th l

8/17/2019 Lecture 12-13 CAPM


Another example

• Conglomerate Corp. is a 100% equity-financed

company, with three equally large divisions.The risk-free rate is 5% and the market riskpremium is 6%

1/3 in car dealership division 2.51/3 in dental equipment division 1.51/3 in utility division 0.5

• What is the overall risk of the company?• just the weighted average of its divisions

• weight by the fraction of firm value attached to eachproject

A th l ( t’d)

8/17/2019 Lecture 12-13 CAPM


Another example (cont’d)

• You are a financial analyst at Conglomerate’s

headquarter and have been asked to evaluate aproject involving “electricity generation” in the

utility division.

•

What opportunity cost of capital should you use todiscount the expected cash flows from the project?

lecture 12-13 capm

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