Download Systematic risk

Capital Asset Pricing
Models
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Market risk is the only risk left after diversification
Return that investors get in the market is rewarded for market
risk only, not total risk
Hence market risk is relevant risk, and specific risk is
irrelevant risk
In the market, higher beta gets higher return, not higher std
gets higher return.
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std
beta
amount invested
IBM
40%
0.95
2000
AT&T
20%
1.10
4000
1. What is the beta of market portfolio
2. Does IBM have more or less risk than the market
3. Which stock has more total risk
Which stock has more systematic risk
Which stock is expected to have higher return in the market
4. In the boom market, which stock do you choose
5. In the recession market, which stock do you choose
6. How do investors know whether the return they get in the market is high
enough to reward for the level of risk taken
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It is the equilibrium model that underlies all modern
financial theory.
Derived using principles of diversification with
simplified assumptions.
Markowitz, Sharpe, Lintner and Mossin are researchers
credited with its development.
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Investors care only about the mean-variance trade-off
of their portfolios in the next period
All investors are price-takers. i.e., no investor is
dominant such that her action alone will change
prices – perfect competition assumption
Investors have homogeneous beliefs and equal
investment opportunities
There is a risk-free asset and investors can borrow
and lend at the same risk-free rate
Markets are frictionless, i.e., with no taxes and
transaction costs. No limitation on the size of trading
and short sales
All of investors’ wealth is in market traded assets
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All investors will hold the same portfolio for risky
assets – market portfolio.
Market portfolio contains all securities and the
proportion of each security is its market value as a
percentage of total market value.
Risk premium on the the market depends on the
average risk aversion of all market participants.
Risk premium on an individual security is a function
of its covariance with the market.
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E(R p )
M
Rf
p
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The CML leads all investors to invest in the M
portfolio. The only difference is the location on the
CML depending on risk preferences
◦ Risk averse investors will lend part of the portfolio at the riskfree rate and invest the remainder in the market portfolio
◦ Investors preferring more risk might borrow funds at Rf and
invest everything in the market portfolio
◦ Two-fund separation theorem or “mutual fund theorem”
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Because Portfolio M lies at the point of tangency, it
has the highest portfolio possibility line
Everybody will want to invest in Portfolio M and
borrow or lend to be somewhere on the CML
Therefore this portfolio must include ALL RISKY
ASSETS
Because the market is in equilibrium, all assets are
included in this portfolio in proportion to their
market value
Therefore, Portfolio M must be the market
portfolio
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The tangency portfolio M is the market portfolio
◦ All assets included in this portfolio are weighted in
proportion to their market value
◦ Because portfolio M contains all risky assets, it is a
completely diversified portfolio. Only systematic risk
remains in the market portfolio.
◦ Systematic risk may be measured by the standard deviation
of returns on the market portfolio.
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Standard Deviation of Return
Unsystematic
(diversifiable)
Risk
Total
Risk
Standard Deviation of
the Market Portfolio
(systematic risk)
Systematic Risk
Number of Stocks in the Portfolio
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The
risk premium on the market portfolio
will be proportional to its risk and the
degree of risk aversion of the investor:
E (rM )  rf  A M2
where  M2 is the variance of the market portolio and
A is the average degree of risk aversion across investors
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The risk premium on individual securities
is a function of the individual security’s
contribution to the risk of the market
portfolio
An individual security’s risk premium is a
function of the covariance of returns with
the assets that make up the market
portfolio
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Covariance of GE return with the market
portfolio:
n

 n
Cov(rGE , rM )  Cov  rGE ,  wk rk    wk Cov(rk , rGE )
k 1

 k 1
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Therefore, the reward-to-risk ratio for
investments in GE would be:
GE's contribution to risk premium wGE  E (rGE )  rf  E (rGE )  rf
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GE's contribution to variance
wGE Cov(rGE , rM ) Cov(rGE , rM )
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Reward-to-risk ratio for investment in
market portfolio:
Market risk premium E (rM )  rf
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Market variance
 M2
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Reward-to-risk ratios of GE and the
E (rGE )  rf
E (rM (rf )
market portfolio:
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Cov(rGE , rM )
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 M2
And the risk premium for GE:
E (rGE )  rf 
Cov(rGE , rM )

2
M
 E (rM )  rf 
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CAPM holds for the overall portfolio
because:
E (rP )  wk E (rk ) and

k
 P   wk  k
k
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This also holds for the market portfolio:
E (rM )  rf   M  E (rM )  rf 
Stock
Beta
A
B
C
D
E
0.70
1.00
1.15
1.40
-0.30
RFR = 6%
RM = 12%
Implied market risk premium = 6%
Assume:
E(R i )  RFR   i (R M - RFR)
E(RA) = 0.06 + 0.70 (0.12-0.06) = 0.102 = 10.2%
E(RB) = 0.06 + 1.00 (0.12-0.06) = 0.120 = 12.0%
E(RC) = 0.06 + 1.15 (0.12-0.06) = 0.129 = 12.9%
E(RD) = 0.06 + 1.40 (0.12-0.06) = 0.144 = 14.4%
E(RE) = 0.06 + (-0.30) (0.12-0.06) = 0.042 = 4.2%
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CAPM gives the relationship between risk and return.
It gives the minimum return required by investors in order for
them to buy stock
What is the meaning of E(R) calculated in the previous slide?
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Remember earlier, we have
n
E ri    pi ri
i 1
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In CAPM, we have
E ri   rf  i Erm   rf 
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What is the difference in meaning between the two expected
return?
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When forecasted E(R) > required E(R), stock is undervalued
or the price is too low
•
When forecasted E(R) < required E(R), stock is undervalued
or the price is too high
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In equilibrium, forecasted E(R) = required E(R)
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1.
2.
Example: E(Rm) = 14%, Rf = 6%
Stocks Beta
E(R) (forecasted)
IBM
1.2
17%
ATT
1.5
14%
According to CAPM, what is the required E(R) for IBM and
ATT
Which stock is undervalued, which stock is overvalued
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Let alpha (α) be the difference between the actual
(forecasted) E(R) and the required E(R)
In equilibrium, all assets and all portfolios of assets
should plot on the SML ( i.e., α = 0)
Any security with an estimated return that plots above
the SML is underpriced (α > 0 )
Any security with an estimated return that plots below
the SML is overpriced ( α < 0 )
Previous example:
◦ αIBM= 17 – 15.6 = 1.4 > 0. Actual E(R) is above the SML
◦ αATT= 14-18 = -4 > 0. Actual E(R) is below the SML
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Expected return of a portfolio
n
n
i 1
i 1
E ( R p )   wi E ( Ri )   wi Rf   i RM  Rf
 n
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 Rf    wi  i RM  Rf
 i 1
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

n
  p   wi  i
i 1
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Q: Suppose that the risk premium on the market portfolio
is estimated at 8% with a standard deviation of 22%.
What is the risk premium on a portfolio invested 25%
in GM and 75% in Ford, if they have betas of 1.10 and
1.25, respectively?
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Q: Two investment advisors:
A: return=19%; beta=1.5
B: return=16%; beta=1
a) Who was better?
b) If T-bill rate were 6% and market return were 14%, who would
be better?
c) What if T-bill rate were 3% and market return were 15%?
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To move from expected to realized
returns—use the index model in excess
return form:
Ri  i  i RM  ei
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The index model beta coefficient turns
out to be the same beta as that of the
CAPM expected return-beta relationship
Ri,t  ai  βi RM,t  ε
Ri,t  R f,t   i  βi RM,t  R f,t   ε
where:
Ri,t = rate of return for asset i during period t
RM,t = rate of return for the market portfolio M during t
R ft  riskfree rate;   random error term
•Adjustments to 
Merrill-Lynch: Adjusted Beta = 2/3*(unadjusted. ) + 1/3
•Time interval problems
•Different holding periods produce different beta
•More pronounced for small company and illiquid stocks
•Weekly and monthly returns better for estimation, not daily data.
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Risk-free rate
◦ Most use short-term Treasury bill returns
◦ Notice that bill returns are variable, not truly risk-free.
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Market risk premium
◦ Historical data of excess return on market index
◦ But expected return on market index may change over
time
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Proxies for market portfolio
◦ S&P (U.S. equity only)
◦ World indices (ignore other assets, like real estates, etc.)
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ri  rf   i   i rm  rf  ei
ri : return on asset i
ri  rf : exess return or risk premium of asset i
 i ,  i : are intercept and slope of the regression
rm : return on market
r
m
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 rf : exess return or risk premium of the market
ei : residual which measures firm specfic effects.
• alpha is the abnormal return = actual return – return
predicted by CAPM
•According to CAPM, alpha should be = 0
•Beta is the systematic risk
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All the points are actual values
Line is the predicted relationship
If there are a lot of specific risk, there will be a wide scatter of
points around the line. Hence, using market risk only in this
case does not produce a precise estimate of expected return
If the points are close to the line, there is only small specific
risk. Using market risk can explain most of the company
return.
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R-square: 0.2866
ANOVA table
Total risk = systematic risk + unsystematic risk
7449.17
= 2224.696 +
5224.45
(100%)
= 29.87%
+
70.13%
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Alpha = 0.8890 > 0 (positive alpha, undervalued or overvalued?)
During the period Jan 99-Dec03: the risk-adjusted or abnormal return of
GM = 0.8990% or actual return is higher than CAPM predicted
Is this value statistically different from 0? Is this still consistent with
CAPM
95% confidence interval (-1.5690 to 3.3470)
Beta = 1.2384
Is beta statistically different from 0?
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CAPM is a benchmark about the fair (required) expected
return on a risk asset. Investors calculate the return they
actually earn based on their input and compare with the return
they get from the CAPM
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Compare the performance of the mutual fund: we use alpha or
risk-adjusted return rather than regular return
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Compute the cost of equity for capital budgeting
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Typical Tests
◦ Time-series test (Black-Jensen-Scholes)
Rit  R ft   i   i ( RMt  R ft )   it
Test to see if  i  0.
◦ Cross-sectional tests (Fama-MacBeth)
~
~
Rit   'i   i RMt  ~it
R i   0   1 ˆi   2CHARi   i , ( i residual)
Test to see if  0  0,  1  RM  RFR ,  2  0.
◦ Most tests are done in portfolios
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High beta portfolios do not necessarily generate
high returns
◦ Controversial results: low betas have positive alpha, high
betas have negative alpha.
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Size and book-to-market value ratio seem to have
explanatory power for returns
◦ Fama and French
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Momentum in returns
◦ Relative strength
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Is the CAPM wrong?
◦ Problems with the proxy for market portfolio
◦ Possible missing risk factors -> Multi-factor
models
◦ Relaxing assumptions
Important intuitions from the CAPM
◦ Diversification
◦ Only covariance with systematic risks matters
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CAPM is powerful at the conceptual level. It is a useful way to
think about risk and return
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Empirical data does not support CAPM fully but it is simple,
logical, easy to use, so use CAPM with caution
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Zero-Beta Model
◦ Helps to explain positive alphas on low beta
stocks and negative alphas on high beta
stocks
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Consideration of labor income and nontraded assets
Merton’s Multiperiod Model and hedge
portfolios
◦ Incorporation of the effects of changes in the
real rate of interest and inflation
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CAPM is limited (true market portfolio is
unobservable), nice idea though!
Other factors also matter, e.g., Fama-French
book-to-market and size factors
Arbitrage Pricing Theory (APT): no free lunch
(for diversified portfolio)!
Chapter 9: Asset Pricing Theories
FIN 2802, Spring 09 - Tang
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CAPM is a single factor model. The market risk premium is
the only factor
In CAPM, all the news, uncertainties affect the market, then
the market affect the stock individually
In APT, there are n factors that can influence stock return so
there will be n-sources of risk or n-channels of uncertainties
Empirical evidence support APT (more than 1 factor affect
stock return), but unable to identify these factors.
So if the purpose is to get cost of capital only, then APT is
appropriate
If we want to know sources of risk then APT is not useful
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Use of more than a single factor
Requires formation of factor portfolios
What factors?
◦ Factors that are important to
performance of the general economy
◦ Fama-French Three Factor Model
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Work of Chen, Roll, and Ross
◦ Chose a set of factors based on the ability
of the factors to paint a broad picture of
the macro-economy
IP = % change in industrial production
EI = % change in expected inflation
UI = % change in unanticipated inflation
CG = excess return of long-term corporate
bond over long-term government bond
 GB = excess return of long-term government
bond over T-bills
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Fama and French propose three factors:
◦ The excess market return, rM-rRF.
◦ the return on, S, a portfolio of small firms (where size is
based on the market value of equity) minus the return on B,
a portfolio of big firms. This return is called rSMB, for S
minus B.
◦ the return on, H, a portfolio of firms with high book-tomarket ratios (using market equity and book equity) minus
the return on L, a portfolio of firms with low book-to-market
ratios. This return is called rHML, for H minus L.
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The factors chosen are variables that on past
evidence seem to predict average returns well
and may capture the risk premiums
rit  i  iM RMt  iSMB SMBt  iHML HMLt  eit
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Where:
◦ SMB = Small Minus Big, i.e., the return of a portfolio of small
stocks in excess of the return on a portfolio of large stocks
◦ HML = High Minus Low, i.e., the return of a portfolio of stocks
with a high book to-market ratio in excess of the return on a
portfolio of stocks with a low book-to-market ratio
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A multi-index CAPM will inherit its risk
factors from sources of risk that a broad
group of investors deem important
enough to hedge
The APT is largely silent on where to look
for priced sources of risk