Chapter 13. Risk & Return in Asset Pricing Models Portfolio Theory Managing Risk Asset Pricing...

Chapter 13. Risk & Return inChapter 13. Risk & Return inAsset Pricing ModelsAsset Pricing Models

Chapter 13. Risk & Return inChapter 13. Risk & Return inAsset Pricing ModelsAsset Pricing Models

• Portfolio Theory

• Managing Risk

• Asset Pricing Models

• Portfolio Theory

• Managing Risk

• Asset Pricing Models

I. Portfolio TheoryI. Portfolio TheoryI. Portfolio TheoryI. Portfolio Theory

• how does investor decide among group of assets?

• assume: investors are risk averse• additional compensation for risk• tradeoff between risk and expected

return

• how does investor decide among group of assets?

• assume: investors are risk averse• additional compensation for risk• tradeoff between risk and expected

return

goalgoalgoalgoal

• efficient or optimal portfolio• for a given risk, maximize exp.

return• OR• for a given exp. return, minimize

the risk

• efficient or optimal portfolio• for a given risk, maximize exp.

return• OR• for a given exp. return, minimize

the risk

toolstoolstoolstools

• measure risk, return

• quantify risk/return tradeoff• measure risk, return

• quantify risk/return tradeoff

return = R = change in asset value + income

initial value

Measuring ReturnMeasuring ReturnMeasuring ReturnMeasuring Return

• R is ex post• based on past data, and is known

• R is typically annualized

• R is ex post• based on past data, and is known

• R is typically annualized

example 1example 1example 1example 1

• Tbill, 1 month holding period

• buy for $9488, sell for $9528

• 1 month R:

• Tbill, 1 month holding period

• buy for $9488, sell for $9528

• 1 month R:

9528 - 9488

9488= .0042 = .42%

• annualized R:• annualized R:

(1.0042)12 - 1 = .052 = 5.2%

example 2example 2example 2example 2

• 100 shares IBM, 9 months

• buy for $62, sell for $101.50

• $.80 dividends

• 9 month R:

• 100 shares IBM, 9 months

• buy for $62, sell for $101.50

• $.80 dividends

• 9 month R:

101.50 - 62 + .80

62= .65 =65%

• annualized R:• annualized R:

(1.65)12/9 - 1 = .95 = 95%

Expected ReturnExpected ReturnExpected ReturnExpected Return

• measuring likely future return

• based on probability distribution

• random variable

• measuring likely future return

• based on probability distribution

• random variable

E(R) = SUM(Ri x Prob(Ri))

example 1example 1example 1example 1

R Prob(R)

10% .2 5% .4-5% .4

E(R) = (.2)10% + (.4)5% + (.4)(-5%)

= 2%

example 2example 2example 2example 2

R Prob(R)

1% .32% .43% .3

E(R) = (.3)1% + (.4)2% + (.3)(3%)

= 2%

examples 1 & 2examples 1 & 2examples 1 & 2examples 1 & 2

• same expected return

• but not same return structure• returns in example 1 are more

variable

• same expected return

• but not same return structure• returns in example 1 are more

variable

RiskRiskRiskRisk

• measure likely fluctuation in return• how much will R vary from E(R)• how likely is actual R to vary from

E(R)

• measured by• variance (• standard deviation

• measure likely fluctuation in return• how much will R vary from E(R)• how likely is actual R to vary from

E(R)

• measured by• variance (• standard deviation

= SUM[(Ri - E(R))2 x Prob(Ri)]

SQRT(

example 1example 1example 1example 1

= (.2)(10%-2%)2

= .0039

+ (.4)(5%-2%)2

+ (.4)(-5%-2%)2

= 6.24%

example 2example 2example 2example 2

= (.3)(1%-2%)2

= .00006

+ (.4)(2%-2%)2

+ (.3)(3%-2%)2

= .77%

• same expected return

• but example 2 has a lower risk• preferred by risk averse investors

• variance works best with symmetric distributions

• same expected return

• but example 2 has a lower risk• preferred by risk averse investors

• variance works best with symmetric distributions

symmetric asymmetric

E(R)R

prob(R)

R

prob(R)

E(R)

II. Managing riskII. Managing riskII. Managing riskII. Managing risk

• Diversification• holding a group of assets• lower risk w/out lowering E(R)

• Diversification• holding a group of assets• lower risk w/out lowering E(R)

• Why?• individual assets do not have same

return pattern• combining assets reduces overall

return variation

• Why?• individual assets do not have same

return pattern• combining assets reduces overall

return variation

two types of risktwo types of risktwo types of risktwo types of risk

• unsystematic risk• specific to a firm• can be eliminated through

diversification• examples:

-- Safeway and a strike

-- Microsoft and antitrust cases

• unsystematic risk• specific to a firm• can be eliminated through

diversification• examples:

-- Safeway and a strike

-- Microsoft and antitrust cases

• systematic risk• market risk• cannot be eliminated through

diversification• due to factors affecting all assets

-- energy prices, interest rates, inflation, business cycles

• systematic risk• market risk• cannot be eliminated through

diversification• due to factors affecting all assets

-- energy prices, interest rates, inflation, business cycles

exampleexampleexampleexample

• choose stocks from NYSE listings

• go from 1 stock to 20 stocks• reduce risk by 40-50%

• choose stocks from NYSE listings

• go from 1 stock to 20 stocks• reduce risk by 40-50%

# assets

systematicrisk

unsystematic risk

totalrisk

measuring relative riskmeasuring relative riskmeasuring relative riskmeasuring relative risk

• if some risk is diversifiable,• then is not the best measure of

risk • σ is an absolute measure of risk

• need a measure just for the systematic component

• if some risk is diversifiable,• then is not the best measure of

risk • σ is an absolute measure of risk

• need a measure just for the systematic component

Beta, Beta, Beta, Beta,

• variation in asset/portfolio return

relative to return of market portfolio• mkt. portfolio = mkt. index

-- S&P 500 or NYSE index

• variation in asset/portfolio return

relative to return of market portfolio• mkt. portfolio = mkt. index

-- S&P 500 or NYSE index

= % change in asset return

% change in market return

interpreting interpreting interpreting interpreting • if

• asset is risk free

• if • asset return = market return

• if • asset is riskier than market index

• asset is less risky than market index

• if • asset is risk free

• if • asset return = market return

• if • asset is riskier than market index

• asset is less risky than market index

Sample betas Sample betas Sample betas Sample betas

Amazon 2.23

Anheuser Busch -.107

Microsoft 1.62

Ford 1.31

General Electric 1.10

Wal Mart .80

(monthly returns, 5 years back)

measuring measuring measuring measuring

• estimated by regression• data on returns of assets• data on returns of market index• estimate

• estimated by regression• data on returns of assets• data on returns of market index• estimate

mRR

problemsproblemsproblemsproblems

• what length for return interval?• weekly? monthly? annually?

• choice of market index?• NYSE, S&P 500• survivor bias

• what length for return interval?• weekly? monthly? annually?

• choice of market index?• NYSE, S&P 500• survivor bias

• # of observations (how far back?)• 5 years?• 50 years?

• time period?• 1970-1980?• 1990-2000?

• # of observations (how far back?)• 5 years?• 50 years?

• time period?• 1970-1980?• 1990-2000?

III. Asset Pricing ModelsIII. Asset Pricing ModelsIII. Asset Pricing ModelsIII. Asset Pricing Models

• CAPM• Capital Asset Pricing Model• 1964, Sharpe, Linter• quantifies the risk/return tradeoff

• CAPM• Capital Asset Pricing Model• 1964, Sharpe, Linter• quantifies the risk/return tradeoff

assumeassumeassumeassume

• investors choose risky and risk-free asset

• no transactions costs, taxes

• same expectations, time horizon

• risk averse investors

• investors choose risky and risk-free asset

• no transactions costs, taxes

• same expectations, time horizon

• risk averse investors

implicationimplicationimplicationimplication

• expected return is a function of• beta• risk free return• market return

• expected return is a function of• beta• risk free return• market return

]R)R(E[R)R(E fmf or

]R)R(E[R)R(E fmf

fR)R(E is the portfolio risk premium

where

fm R)R(E is the market risk premium

so if so if so if so if

• portfolio exp. return is larger than exp. market return

• riskier portfolio has larger exp. return

• portfolio exp. return is larger than exp. market return

• riskier portfolio has larger exp. return

fR)R(E fm R)R(E

)R(E )R(E m

>

>

so if so if so if so if

• portfolio exp. return is smaller than exp. market return

• less risky portfolio has smaller exp. return

• portfolio exp. return is smaller than exp. market return

• less risky portfolio has smaller exp. return

fR)R(E fm R)R(E

)R(E )R(E m

<

<

so if so if so if so if

• portfolio exp. return is same than exp. market return

• equal risk portfolio means equal exp. return

• portfolio exp. return is same than exp. market return

• equal risk portfolio means equal exp. return

fR)R(E fm R)R(E

)R(E )R(E m

=

=

so if so if so if so if

• portfolio exp. return is equal to risk free return

• portfolio exp. return is equal to risk free return

fR)R(E

)R(E fR

= 0

=

exampleexampleexampleexample

• Rm = 10%, Rf = 3%, = 2.5• Rm = 10%, Rf = 3%, = 2.5

]R)R(E[R)R(E fmf %]3%10[5.2%3)R(E

%5.17%3)R(E %5.20)R(E

• CAPM tells us size of risk/return tradeoff

• CAPM tells use the price of risk

• CAPM tells us size of risk/return tradeoff

• CAPM tells use the price of risk

Testing the CAPMTesting the CAPMTesting the CAPMTesting the CAPM

• CAPM overpredicts returns• return under CAPM > actual return

• relationship between β and return?• some studies it is positive• some recent studies argue no

relationship (1992 Fama & French)

• CAPM overpredicts returns• return under CAPM > actual return

• relationship between β and return?• some studies it is positive• some recent studies argue no

relationship (1992 Fama & French)

• other factors important in determining returns• January effect• firm size effect• day-of-the-week effect• ratio of book value to market value

• other factors important in determining returns• January effect• firm size effect• day-of-the-week effect• ratio of book value to market value

problems w/ testing CAPMproblems w/ testing CAPMproblems w/ testing CAPMproblems w/ testing CAPM

• Roll critique (1977)• CAPM not testable

• do not observe E(R), only R

• do not observe true Rm

• do not observe true Rf

• results are sensitive to the sample period

• Roll critique (1977)• CAPM not testable

• do not observe E(R), only R

• do not observe true Rm

• do not observe true Rf

• results are sensitive to the sample period

APTAPTAPTAPT

• Arbitrage Pricing Theory

• 1976, Ross

• assume:• several factors affect E(R)• does not specify factors

• Arbitrage Pricing Theory

• 1976, Ross

• assume:• several factors affect E(R)• does not specify factors

• implications• E(R) is a function of several

factors, F

each with its own

• implications• E(R) is a function of several

factors, F

each with its own

NN332211f F....FFFR)R(E

APT vs. CAPMAPT vs. CAPMAPT vs. CAPMAPT vs. CAPM

• APT is more general• many factors• unspecified factors

• CAPM is a special case of the APT• 1 factor• factor is market risk premium

• APT is more general• many factors• unspecified factors

• CAPM is a special case of the APT• 1 factor• factor is market risk premium

testing the APTtesting the APTtesting the APTtesting the APT

• how many factors?

• what are the factors?

• 1980 Chen, Roll, and Ross• industrial production• inflation• yield curve slope• other yield spreads

• how many factors?

• what are the factors?

• 1980 Chen, Roll, and Ross• industrial production• inflation• yield curve slope• other yield spreads

summarysummarysummarysummary

• known risk/return tradeoff• how to measure risk?• how to price risk?

• neither CAPM or APT are perfect or free of testing problems

• both have shown value in asset pricing

• known risk/return tradeoff• how to measure risk?• how to price risk?

• neither CAPM or APT are perfect or free of testing problems

• both have shown value in asset pricing