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Chapter 13. Risk & Return in
Asset Pricing Models
Portfolio Theory
Managing Risk
Asset Pricing Models
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I. Portfolio Theory
how does investor decide amonggroup of assets?
assume: investors are risk averse additional compensation for risk
tradeoff between risk and expected
return
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goal
efficient or optimal portfolio
for a given risk, maximize exp.
return OR
for a given exp. return, minimize
the risk
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tools
measure risk, return
quantify risk/return tradeoff
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return = R =change in asset value + income
initial value
Measuring Return
R is ex post
based on past data, and is known R is typically annualized
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example 1
Tbill, 1 month holding period
buy for $9488, sell for $9528
1 month R:
9528 - 9488
9488
= .0042 = .42%
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annualized R:
(1.0042)12
- 1 = .052 = 5.2%
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example 2
100 shares IBM, 9 months
buy for $62, sell for $101.50
$.80 dividends 9 month R:
101.50 - 62 + .80
62= .65 =65%
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annualized R:
(1.65)12/9
- 1 = .95 = 95%
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Expected Return
measuring likely future return
based on probability distribution
random variable
E(R) = SUM(Ri x Prob(Ri))
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example 1
R Prob(R)
10% .2
5% .4
-5% .4
E(R) = (.2)10% + (.4)5% + (.4)(-5%)
= 2%
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example 2
R Prob(R)
1% .3
2% .4
3% .3
E(R) = (.3)1% + (.4)2% + (.3)(3%)
= 2%
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examples 1 & 2
same expected return
but not same return structure
returns in example 1 are morevariable
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Risk
measure likely fluctuation in return how much will R vary from E(R)
how likely is actual R to vary fromE(R)
measured by
variance (s2) standard deviation (s)
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s2 = SUM[(Ri - E(R))2 x Prob(Ri)]
s = SQRT(s2)
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example 1
s2 = (.2)(10%-2%)2
= .0039
+ (.4)(5%-2%)2
+ (.4)(-5%-2%)2
s = 6.24%
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example 2
s2 = (.3)(1%-2%)2
= .00006
+ (.4)(2%-2%)2
+ (.3)(3%-2%)2
s = .77%
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same expected return
but example 2 has a lower risk
preferred by risk averse investors variance works best with symmetric
distributions
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symmetric asymmetric
E(R)R
prob(R)
R
prob(R)
E(R)
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II. Managing risk
Diversification
holding a group of assets
lower risk w/out lowering E(R)
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Why?
individual assets do not have
same return pattern combining assets reduces overall
return variation
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two types of risk
unsystematic risk
specific to a firm
can be eliminated throughdiversification
examples:
-- Safeway and a strike-- Microsoft and antitrust cases
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systematic risk
market risk cannot be eliminated through
diversification
due to factors affecting all assets
-- energy prices, interest rates,inflation, business cycles
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example
choose stocks from NYSE listings
go from 1 stock to 20 stocks
reduce risk by 40-50%
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s
# assets
systematic
risk
unsystematic
risktotal
risk
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measuring relative risk
if some risk is diversifiable,
then sis not the best measure ofrisk
is an absolute measure of risk
need a measure just for the
systematic component
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Beta, b variation in asset/portfolio return
relative to return of market portfolio
mkt. portfolio = mkt. index
-- S&P 500 or NYSE index
b =% change in asset return
% change in market return
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interpreting b if b = 0
asset is risk free
if b = 1 asset return = market return
if b > 1 asset is riskier than market index
b < 1 asset is less risky than market index
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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)
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measuring b estimated by regression
data on returns of assets
data on returns of market index estimate
b= mRR
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problems
what length for return interval?
weekly? monthly? annually?
choice of market index? NYSE, S&P 500
survivor bias
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# of observations (how far back?)
5 years? 50 years?
time period?
1970-1980?
1990-2000?
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III. Asset Pricing Models
CAPM
Capital Asset Pricing Model
1964, Sharpe, Linter quantifies the risk/return tradeoff
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assume
investors choose risky and risk-freeasset
no transactions costs, taxes
same expectations, time horizon
risk averse investors
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implication
expected return is a function of
beta
risk free return market return
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]R)R(E[R)R(E fmf b=
or
]R)R(E[R)R(E fmf b=
fR)R(E is the portfolio risk premium
where
fm R)R(E is the market risk premium
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so if b >1,
portfolio exp. return is larger thanexp. market return
riskier portfolio has larger exp.return
fR)R(E fm R)R(E
)R(E )R(Em
>
>
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so if b
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so if b =1,
portfolio exp. return is same thanexp. market return
equal risk portfolio means equal exp.return
fR)R(E fm R)R(E
)R(E )R(Em
=
=
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so if b = 0,
portfolio exp. return is equal to riskfree return
fR)R(E
)R(Ef
R
= 0
=
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example
Rm = 10%, Rf = 3%, b = 2.5]R)R(E[R)R(E
fmf
b=
%]3%10[5.2%3)R(E =
%5.17%3)R(E =
%5.20)R(E =
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CAPM tells us size of risk/returntradeoff
CAPM tells use the price of risk
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Testing the CAPM
CAPM overpredicts returns
return under CAPM > actual return
relationship between and return? some studies it is positive
some recent studies argue no
relationship (1992 Fama & French)
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other factors important in
determining returns January effect
firm size effect
day-of-the-week effect ratio of book value to market value
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problems 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
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APT
Arbitrage Pricing Theory
1976, Ross
assume: several factors affect E(R)
does not specify factors
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implications
E(R) is a function of severalfactors, F
each with its own bNN332211f F....FFFR)R(E bbbb=
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APT 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
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testing the APT
how many factors?
what are the factors?
1980 Chen, Roll, and Ross industrial production
inflation
yield curve slope other yield spreads
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summary
known risk/return tradeoff
how to measure risk?
how to price risk?
neither CAPM or APT are perfect orfree of testing problems
both have shown value in assetpricing