Chap007 (1)

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Capital Asset

Pricing and

Arbitrage

Pricing Theory

Bodie, Kane and MarcusEssentials of Investments 9th Global Edition

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

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

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?

Risk Premium of Market Portfolio

Demand drives prices, lowers expected rate of return/risk

premiums

When premiums fall, investors move funds into risk-free

Equilibrium risk premium of market portfolio proportional

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

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

EXPECTED RETURN AND RISK ON INDIVIDUAL SECURITIES

The Security Market Line (SML)

Represents expected return-beta relationship of CAPM

Graphs individual asset risk premiums as function of asset

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%

RM=11%

Rx=13%

Ry=7.8%

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.

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%

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

PORTFOLIO BETAS

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

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.

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

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)

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

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

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

Firm size variable SMB

7.4 MULTIFACTOR MODELS AND CAPM

FAMA-FRENCH (FF) 3 FACTOR MODEL

rGM – rf =αGM + βM(rM – rf ) + βHMLrHML + βSMBrSMB +

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

Chap007 (1)

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