Capital Asset
Pricing and
Arbitrage
Pricing Theory
Bodie, Kane and MarcusEssentials of Investments 9th Global Edition
7
7.1 THE CAPITAL ASSET PRICING MODEL
Assumptions
Markets are competitive, equally profitable
No investor is wealthy enough to individually affect prices
All information publicly available; all securities public
No taxes on returns, no transaction costs
Unlimited borrowing/lending at risk-free rate
Investors are alike except for initial wealth, risk aversion
Investors plan for single-period horizon; they are rational, mean-
variance optimizers
Use same inputs, consider identical portfolio opportunity sets
7.1 THE CAPITAL ASSET PRICING MODEL
Hypothetical Equilibrium
All investors choose to hold market portfolio
Market portfolio is on efficient frontier, optimal risky portfolio
Risk premium on market portfolio is proportional to variance
of market portfolio and investor’s risk aversion
Risk premium on individual assets is proportional to risk
premium on market portfolio, and the beta coefficient of
security on market portfolio
FIGURE 7.1 EFFICIENT FRONTIER AND
CAPITAL MARKET LINE
7.1 THE CAPITAL ASSET PRICING MODEL
Passive Strategy is Efficient
Mutual fund theorem: All investors desire same portfolio of
risky assets, can be satisfied by single mutual fund
composed of that portfolio
If passive strategy is costless and efficient, why follow active
strategy?
If no one does security analysis, what
brings about efficiency of market portfolio?
7.1 THE CAPITAL ASSET PRICING MODEL
Risk Premium of Market Portfolio
Demand drives prices, lowers expected rate of return/risk
premiums
When premiums fall, investors move funds into risk-free
asset
Equilibrium risk premium of market portfolio proportional
to
Risk of market
Risk aversion of average investor
E(rm) - rf = A sM2
EXPECTED RETURN AND RISK ON INDIVIDUAL SECURITIES
The risk premium on individual securities is a
function of the individual security’s
__________________________________________
What type of individual security risk will matter,
systematic or unsystematic risk?
An individual security’s total risk (s2i) can be
partitioned into systematic and unsystematic risk:
s2i = bi
2 sM2 + s2(ei)
M = market portfolio of all risky securities
contribution to the risk of The market portfolio
7-7
7.1 THE CAPITAL ASSET PRICING MODEL
Individual security’s contribution to the risk of the
market portfolio is a function of the __________ of the
stock’s returns with the market portfolio’s returns and
is measured by BETA
With respect to an individual security, systematic
risk can be measured by bi = [COV(ri,rM)] / s2M
covariance
7-8
7.1 THE CAPITAL ASSET PRICING MODEL
EXPECTED RETURN AND RISK ON INDIVIDUAL SECURITIES
7.1 THE CAPITAL ASSET PRICING MODEL
•
7.1 THE CAPITAL ASSET PRICING MODEL
The Security Market Line (SML)
Represents expected return-beta relationship of CAPM
Graphs individual asset risk premiums as function of asset
risk
Alpha
Abnormal rate of return on security in excess of that
predicted by equilibrium model (CAPM)
FIGURE 7.2 THE SML AND A POSITIVE-
ALPHA STOCK
Equation of the SML (CAPM)
E(ri) = rf + bi[E(rM) - rf]
Slope SML =
=
(E(rM) – rf )/ βM
price of risk for market
SAMPLE CALCULATIONS FOR SML
E(rm) - rf = rf =
bx = 1.25
E(rx) =
by = .6
E(ry) =
Equation of the SML: E(ri) = rf + bi[E(rM) - rf]
0.03 + 1.25(.08) = .13 or 13%
0.03 + 0.6(0.08) = 0.078 or 7.8%
If b = 1?
If b = 0?
.08 .03
Return per unit of systematic risk = 8% & the return due to the TVM = 3%
7-12
7.1 THE CAPITAL ASSET PRICING MODEL
E(r)
SML
ß
ßM
1.0
RM=11%
3%
Rx=13%
ßx
1.25
Ry=7.8%
ßy
.6
.08
GRAPH OF SAMPLE CALCULATIONS
If the CAPM is correct, only β
risk matters in determining
the risk premium for a given
slope of the SML.
7-13
7.1 THE CAPITAL ASSET PRICING MODEL
E(rE(r))
15%15%
SMLSML
ßß1.01.0
RRmm=11%=11%
rrff=3%=3%
1.251.25
Disequilibrium Example
Suppose a security with a b of
____ is offering an expected
return of ____
According to the SML, the E(r)
should be _____
1.2515%
13%
Underpriced: It is offering too high of a rate of return for its level of risk
The difference between the return required for the risk level as measured
by the CAPM in this case and the actual return is called the stock’s _____
denoted by __
What is the __ in this case?
E(r) = 0.03 + 1.25(.08) = 13%
Is the security under or overpriced?
= +2% Positive is good, negative is bad
+ gives the buyer a + abnormal return
alpha
13%
7-14
PORTFOLIO BETAS
βP =
If you put half your money in a stock with a beta of ___
and ____ of your money in a stock with a beta of ___and
the rest in T-bills, what is the portfolio beta?
βP = 0.50(1.5) + 0.30(0.9) + 0.20(0) = 1.02
1.5
30% 0.9
Wi βi
• All portfolio beta expected return combinations
should also fall on the SML.
• All (E(ri) – rf) / βi should be the same for all
stocks.
7-15
7.1 THE CAPITAL ASSET PRICING MODEL
Applications of CAPM
Use SML as benchmark for fair return on risky asset
SML provides “hurdle rate” for internal projects
7.2 CAPM AND INDEX MODELS
7.2 CAPM AND INDEX MODELS
•
TABLE 7.1 MONTHLY RETURN
STATISTICS 01/06 - 12/10
Statistic (%) T-Bills S&P 500 Google
Average rate of return 0.184 0.239 1.125
Average excess return - 0.055 0.941
Standard deviation* 0.177 5.11 10.40
Geometric average 0.180 0.107 0.600
Cumulative total 5-year return 11.65 6.60 43.17
Gain Jan 2006-Oct 2007 9.04 27.45 70.42
Gain Nov 2007-May 2009 2.29 -38.87 -40.99
Gain June 2009-Dec 2010 0.10 36.83 42.36
* The rate on T-bills is known in advance, SD does not reflect risk.
FIGURE 7.4 SCATTER DIAGRAM/SCL: GOOGLE VS.
S&P 500, 01/06-12/10
TABLE 7.2 SCL FOR GOOGLE (S&P 500),
01/06-12/10
Linear Regression
Regression Statistics
R 0.5914
R-square 0.3497
Adjusted R-square 0.3385
SE of regression 8.4585
Total number of observations 60
Regression equation: Google (excess return) = 0.8751 + 1.2031 ×S&P 500 (excess return)
ANOVA
df SS MS F p-level
Regression 1 2231.50 2231.50 31.19 0.0000
Residual 58 4149.65 71.55
Total 59 6381.15
Coefficients
Standard Error t-Statistic p-value LCL UCL
Intercept 0.8751 1.0920 0.8013 0.4262 -1.7375 3.4877
S&P 500 1.2031 0.2154 5.5848 0.0000 0.6877 1.7185
t-Statistic (2%) 2.3924
LCL - Lower confidence interval (95%)
UCL - Upper confidence interval (95%)
7.2 CAPM AND INDEX MODELS
• Estimation results
• Security Characteristic Line (SCL)
• Plot of security’s expected excess return over
risk-free rate as function of excess return on
market
• Required rate = Risk-free rate + β x Expected excess return
of index
7.3 CAPM AND THE REAL WORLD
CAPM is false based on validity of its assumptions
Useful predictor of expected returns
Untestable as a theory
Principles still valid
Investors should diversify
Systematic risk is the risk that matters
Well-diversified risky portfolio can be
suitable for wide range of investors
7.4 MULTIFACTOR MODELS AND CAPM
•
FAMA-FRENCH (FF) 3 FACTOR
MODEL
Fama and French noted that stocks of ____________ and
stocks of firms with a _________________ have had
higher stock returns than predicted by single factor
models.
Problem: Empirical model without a theory
Will the variables continue to have predictive power?
high book to market
smaller firms
7-25
FAMA-FRENCH (FF) 3 FACTOR MODEL
FF proposed a 3 factor model of stock returns as follows:
rM – rf = Market index excess return
Ratio of ______________________________________ measured with a variable called ____:
HML: High minus low or difference in returns between firms with a high versus a low book to market ratio.
_______________ measured by the ____ variable
SMB: Small minus big or the difference in returns between small and large firms.
book value of equity to market value of equity
HML
Firm size variable SMB
7-26
7.4 MULTIFACTOR MODELS AND CAPM
•
FAMA-FRENCH (FF) 3 FACTOR MODEL
rGM – rf =αGM + βM(rM – rf ) + βHMLrHML + βSMBrSMB +
eGM
7-28
TABLE 7.3 MONTHLY RATES OF RETURN,
01/06-12/10
Monthly Excess Return % * Total Return
Security Average
Standard
Deviation
Geometric
Average
Cumulative
Return
T-bill 0 0 0.18 11.65
Market index ** 0.26 5.44 0.30 19.51
SMB 0.34 2.46 0.31 20.70
HML 0.01 2.97 -0.03 -2.06
Google 0.94 10.40 0.60 43.17
*Total return for SMB and HML
** Includes all NYSE, NASDAQ, and AMEX
stocks.
7.5 ARBITRAGE PRICING THEORY
Arbitrage
Relative mispricing creates riskless profit
Arbitrage Pricing Theory (APT)
Risk-return relationships from no-arbitrage considerations in large
capital markets
Well-diversified portfolio
Nonsystematic risk is negligible
Arbitrage portfolio
Positive return, zero-net-investment, risk-free portfolio
FIGURE 7.5 SECURITY
CHARACTERISTIC LINES
7.5 ARBITRAGE PRICING THEORY
Multifactor Generalization of APT and CAPM
Factor portfolio
Well-diversified portfolio constructed to
have beta of 1.0 on one factor and beta of
zero on any other factor Two-Factor Model for APT