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