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Week 4
Capital Asset Pricing and
Arbitrage Pricing Theory
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Capital Asset Pricing Model (CAPM)
 CAPM: A model that relates the required rate of
return for a security to its risk measured by beta.
 CAPM predicts the relationship between the risk
and equilibrium expected returns on risky assets.
 CAPM is derived using principles of
diversification with simplified assumptions.
 CAPM is first proposed by Sharpe (1964), and
Markowitz, Sharpe, Lintner and Mossin are
researchers credited with its development.
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CAPM: Assumptions
 Individual investors cannot affect prices by their
individual traders
There are many investors, each with an endowment of
wealth that is small compared with total endowment of all
investors.
 Single-period investment horizon
All investors plan for one identical holding period.
 Investors from portfolios from a universe of publicly
traded financial assets, and have access to unlimited
risk-free borrowing or lending.
 Investors pay neither taxes on returns nor transaction
costs on trades in securities.
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CAPM: Assumptions (continued)
 All investors attempt to construct efficient frontier
portfolio.
Investors are rational mean-variance optimizers
 Information is costless and available to all investors
 Homogeneous expectations
All investors analyze securities in the same way and share
same economic view of the world.
They all end with identical estimates of the probability
distribution of future cash flows from investing in the
available securities.
All investors use the same expected return, standard
deviations, and correlations to generate efficient frontier and
the unique optimal risky portfolio.
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Resulting Equilibrium Conditions
 All investors will choose to hold the same optimal
risky portfolio – Market portfolio.
Market portfolio contains all securities in the market and the
proportion of each security is its market value as the
percentage of total market value.
 The market portfolio will be on the efficient frontier,
and it will be the optimal risky portfolio.
The CML, the line from risk-free rate through the market
portfolio, is the best attainable CAL.
All investors will hold the market portfolio as their optimal
risk portfolio, differing only in the amount invested in it
compared to investment in risk-free asset.
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Resulting Equilibrium Conditions (cont.)
 The risk premium on the market portfolio will be
proportional to the variance of the market portfolio
and investor’s typical degree of risk aversion.
E(rM) – rf = A*sM2
 The risk premium on individual securities will be
proportional to the risk premium on market
portfolio and to the beta coefficient of the security
on the market portfolio.
Rate of return on the market portfolio is the single factor of
the security market.
The beta measures the extent to which returns on the stock
respond to the returns of the market portfolio.
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CAPM: Implications
 Why all investors would hold the market portfolio.
The assumed necessary conformity of investors make all
investors hold the market portfolio.
Each investors uses the market portfolio for the optimal risky
portfolio, the CAL in this case is called CML.
The price adjustment process guarantees that all stocks will be
included in the optimal portfolio.
Suppose Stock A is not included in optimal portfolio.
The demand is zero for Stock A, so Stock A will get cheaper,
and will look relatively more attractive.
Ultimately, Stock A’s price will reach a price at which it is
desirable to be included in the optimal portfolio.
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Capital Market Line
E(r)
CML
M (Market
Portfolio)
E(rM)
rf
sm
s
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CAPM: Implications
 The Passive Strategy is Efficient
The market portfolio proportions are optimal, and in the
simple world of CAPM, all investors use precious
resources in security analysis.
A passive investor who simply holds market portfolio
benefits from the efficiency of that portfolio (CML).
Mutual Fund Theorem: All investors desire the same
portfolio of risky assets and can be satisfied by a single
mutual fund composed of that portfolio.
Logical inconsistency of the CAPM: If a passive strategy is
costless and efficient, why would anyone follow an active
strategy? But if no one does any security analysis, what
brings about the efficiency of market portfolio.
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The Risk Premium on Market Portfolio
 In equilibrium, the risk premium on the market
portfolio [E(rM) – rf], must be just high enough to
induce investors to hold the available supply of
stocks.
If the risk premium is too high compared to the average
degree of risk aversion, there will be excess demand for
securities, and prices will rise.
If it is too low, investors will not hold enough stock to
absorb the supply, and prices will fall.
 In equilibrium risk premium of the market portfolio is
proportional to both the risk of the market, as
measured by the variance of its return, and to the
degree of risk aversion of average investor.
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Expected Return on Individual Securities
 As unsystematic risk can be diversified away,
investors do not require a risk premium for bearing
unsystematic risk.
 Investors only need to be compensated for bearing
systematic risk, which cannot be diversified.
 The contribution of a single security to the risk of a
large diversified portfolio depends only on the
systematic risk of the security measured by its beta.
 Individual security’s risk premium is proportional to
its beta, that is, a function of the contribution of
individual security to the risk of market portfolio.
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Expected Return on Individual Securities
 The ratio of risk premium to beta should be the same
for all securities and portfolios.
[E(rM) – rf] / bM = [E(ri) – rf] / bi, where bM = 1.0
 The CAPM’s expected return-beta relationship
E(ri) = rf + bi [E(rM) – rf], bi = [COV(ri,rM)] / sM2
 Even if one does not hold the market portfolio, a
well-diversified portfolio will be so highly correlated with the market that a stock’s beta relative to
the market still will be a useful risk measure.
 If the CAPM holds for any individual asset, it must
hold for any combination of assets (portfolios).
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Security Market Line (SML)
 SML: Graphical representation of the expected
return-beta relationship of the CAPM – Slope is the
market risk premium and intercept is risk-free rate.
 SML vs. CML
CML: Risk premiums of efficient portfolio as a function of
portfolio standard deviation.
SML: Risk premium of individual asset as a function of the
contribution of the asset to the portfolio standard
deviation measured by beta.
 Whenever CAPM holds, all securities must lie on
the SML in market equilibrium; Fairly priced assets
plot exactly on the SML.
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Security Market Line
E(r)
Under-priced
SML
a
E(rM)
rf
M
Over-priced
bM = 1.0
b
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Sample Calculations for SML
E(rM) - rf = .08 rf = .03
bx = 1.25
E(rx) = .03 + 1.25(.08) = .13 or 13%
by = .6
E(ry) = .03 + .6(.08) = .078 or 7.8%
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Graph of Sample Calculations
E(r)
SML
E(rx)=13%
E(rM)=11%
E(ry)=7.8%
rf=3%
by =
0.6
bM =
1.0
bx =
1.25
b
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Application of the CAPM
 The CAPM provides a required rate of return on an
investment for a given risk.
 The SML provides a benchmark for evaluation of
investment performance.
Use in the investment management industry.
Suppose the SML is taken as a benchmark to assess the fair
expected return on a risky asset: An analyst calculate own
expected return; If a stock is perceived as a good buy
(underpriced), it will provide a positive alpha (a).
 The CAPM is useful in capital budgeting decisions.
Managers can use the CAPM to obtain the cutoff IRR or
hurdle rate for the project.
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The CAPM and Index Models
 The CAPM has two limitations.
It relies on the theoretical market portfolio.
It deals with expected as opposed to actual returns.
 To implement the CAPM, we cast it in the form of
an index model using actual market index, and use
realized returns.
 True market portfolio cannot be observable and
would not be easily accessed by investors.
 The CAPM’s reliance of the market portfolio should
not faze us if we can verify that predictions of
CAPM are sufficiently accurate when the index
portfolio is substituted for the market.
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The CAPM and Index Models
 Single-index regression equation with realized
excess returns
ri – rf = ai + bi (rM – rf) + ei
 Expressed in terms of expectation;
E(ri) – rf = ai + bi [E(rM)– rf]
 Comparing this relationship with the CAPM reveals
that the CAPM predicts that ai = 0.
Convert the CAPM predictions about unobserved
expectations of security returns relative to an unobserved
market portfolio into a prediction about the intercept in a
regression of observed variables: realized excess returns of
a security relative to those of a specified index.
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The CAPM and Index Models
 If intercepts of regressions of returns on an index
differ significantly from zero, you cannot tell
whether it is because you chose a bad index to proxy
for the market or because the theory is not useful.
 Index models are widely used to operationalize
CAPM (Security Characteristic Line).
In actuality, few instances of persistent, positive alpha values
have been identified, and future alphas are practically
impossible to predict from past values.
 Even if a single-index model representation is
not fully consistent with the CAPM, the concept
of systematic vs. diversifiable risk is still useful.
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Using Text Example Table 5:
Jan.
Feb.
.
.
Dec.
Mean
Std. Dev.
Excess
GM Return
Excess
Market Return
5.41
-3.44
.
.
2.43
7.24
0.93
.
.
3.90
-.60
4.97
1.75
3.32
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Regression Results:
(rGM - rf) = ai + bGM(rm - rf)
Estimated coefficient
Std error of estimate
a
b
-2.590
(1.547)
1.1357
(0.309)
Variance of residuals = 12.601
Std dev of residuals = 3.550
R-square = 0.575
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Predicting Beta
 Estimate betas to forecast the rate of return based on
the past history.
 Betas exhibit a statistical property called “regression
toward the mean.”
High b (b1) securities in one period tend to exhibit a lower
beta in the future, while low b (b1) securities exhibit a
higher b in future periods.
A simple way to account for this tendency of future betas is
to use as your forecast of beta a weighted average of sample
estimate with the value 1.0.
Suppose past data yield a beta estimate of 0.65. Adjusted
beta = 2/3  0.65 + 1/3  1.0 = 0.77.
 Some complicated techniques have been used, but
not very successful to provide accurate beta estimate.
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The CAPM and The Real World
 In only limited ways, portfolio theory and CAPM
have become accepted tools in the practitioner
community.
 Many empirical studies argue that beta does not tell
the whole story of risk.
 Nevertheless, beta is not dead yet.
Other study shows that when we use more inclusive (even
including human capital) proxy than S&P500 for market
portfolio and allows for the fact that beta changes over time,
the performance of beta is considerably improved.
The logic of the model still compelling and more sophisticated pricing models all rely on the key distinction between
systematic vs. diversifiable risk.
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Arbitrage Pricing Theory
 APT is introduced by Ross (1976).
 Like the CAPM, APT predicts the relationship
between the risk and equilibrium expected returns
on risky assets.
 However, the APT relies on no-arbitrage condition
rather than the market portfolio.
 To explain the APT, we begin with the concept of
Arbitrage, which is the exploitation of relative
mispricing among two or more securities to earn
risk-free profits
 A riskless arbitrage opportunity arises if an investor
can construct a zero investment portfolio with a
sure profit.
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Arbitrage Pricing Theory
 Since no investment is required, an investor can
create large positions to secure large levels of profit.
One has to be able to sell short at least one asset and use
the proceeds to purchase one or more assets.
 An obvious case of an arbitrage opportunity arises
in the violation of the law of one price: When an
asset is trading at different prices in two markets,
sell short in the high priced market and buys it in the
low priced market.
 In efficient markets, profitable arbitrage opportunity
will quickly disappear – Program trading and index
arbitrage.
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Arbitrage Example from Text

Current
Stock Price$
A
10
B
10
C
10
D
10
Expected
Return%
25.0
20.0
32.5
22.5

Mean Stan.Dev.
Portfolio P(A,B,C) 25.83
6.40
D
22.25
8.58
Standard
Dev. %
29.58
33.91
48.15
8.58
Correlation
0.94
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Arbitrage Action and Returns
Expected
Return
* P
* D
Standard
Deviation
Short 3 shares of D and Buy 1 of A, B & C to form P
(Arbitrage Portfolio: Zero-investment Portfolio).
You earn a higher rate on the investment than you
pay on the short sale.
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Arbitrage Pricing Theory
 The critical property of an arbitrage portfolio is than
any investor, regardless of risk aversion or wealth,
will want to take an infinite position in it so that
profits will be driven to an infinite level.
 Because those large positions will force some prices
up and/or some down until the opportunity is
vanished, we can derive restrictions on security
prices that satisfy that no arbitrage opportunities are
left in the market place.
 There is an important distinction between arbitrage
and CAPM risk-versus-return dominance arguments
in support of equilibrium price relationships.
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Arbitrage Pricing Theory
 Risk-vs-Return Dominance argument in CAPM
It holds that when an equilibrium relationship is violated,
many investors will make limited portfolio changes,
depending on wealth and risk-aversion.
Aggregation of limited portfolio changes over many
investors will restore the equilibrium price.
 Arbitrage argument in APT
When arbitrage opportunities exist, each investor wants to
take as large a position as possible.
It will not take many investors to restore equilibrium.
Implications derived from no-arbitrage argument is
stronger, because they do not depend on a large, welleducated investors.
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Well-diversified Portfolio and APT
 APT: A theory of risk-return relationship derived
from no arbitrage conditions in large capital market.
 APT posits a single-factor security market.
Ri = ai + biRM + e, where Ri = (ri – rf)
 Suppose we construct a well-diversified portfolio
with a given beta – No firm-specific risk.
Rp = ap + bpRM
If the portfolio beta is zero, Rp = ap, implying a riskless
excess return over risk-free rate.
This implies that ap should be zero, or else an immediate
arbitrage opportunity opens up (borrow at risk free rate
and buy zero-beta portfolio).
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Well-diversified Portfolio and APT
 Portfolio V (bv and av) and Portfolio U (bu and au).
To form zero-beta portfolio (V+U); buy Portfolio V and
sell Portfolio U with proportions of wv = [-bu/(bv-bu)], and
wu = [bv/(bv-bu)]
Riskless portfolio, but non-zero excess return unless av
and au equal zero.
Beta(V+U) = bv[-bu/(bv-bu)] + bu[bv/(bv-bu)] = 0
R(V+U) = av[-bu/(bv-bu)] + au[bv/(bv-bu)]  0
 The alpha of any well-diversified portfolio must be
zero, even if the beta is not zero.
 (rp – rf) = bp(rM– rf); E(rp) = rf + bp[E(rM) – rf]: Same
as CAPM without any assumption about either
investor preferences or access to market portfolio.
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APT and CAPM Compared
 APT applies only to well diversified portfolios and
not necessarily to individual stocks in equilibrium.
 However, APT relationship must almost surely hold
true for individual securities.
If APT relationship is violated by many individual assets, it
would be virtually impossible for all well-diversified
portfolios to satisfy the relationship.
 APT serves many of same functions as the CAPM.
 APT is more general in that it gets to an expected
return and beta relationship without the assumption
of the market portfolio.
 APT can be extended to multifactor models.
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Multifactor Generalization of APT
Use a multifactor version of APT to
accommodate multiple sources of risk.
Generalize the single-factor model to a twofactor model: Ri = ai + bi1RM1 + bi2RM2 + e.
Two-Factor APT
E(rp) =
rf + bp1 [E(rM1) – rf] + bp2 [E(rM2) – rf]
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