Chapter 9Chapter 9

Capital Market TheoryCapital Market Theory

• Explain capital market theory and the Capital Asset Pricing Model (CAPM).

• Discuss the importance and composition of the market portfolio.

• Describe two important relationships in CAPM as represented by the capital market line and the security market line.

• Describe how betas are estimated and how beta is used.

• Discuss the Arbitrage Pricing Theory as an alternative to the Capital Asset Pricing Model.

Learning ObjectivesLearning Objectives

Capital Market TheoryCapital Market Theory

Describes the pricing of capital assets in financial markets

• Relates the required rate of return for any security with the market risk for the security as measured by beta

• Focus on the equilibrium relationship between the risk and expected return on risky assets

• Builds on Markowitz portfolio theory• Each investor is assumed to diversify his or her

portfolio according to the Markowitz model, choosing a location on the efficient frontier that matches his/her return-risk preferences

Capital Asset Pricing ModelCapital Asset Pricing Model

• All investors: Use the same

information to generate an efficient frontier (i.e., identical inputs: E(R), σ, ρ)

Have the same one-period time horizon

Can borrow or lend money at the risk-free rate of return

• No transaction costs

• No personal income (i.e., indifferent between dividends and capital gains) taxes

• No inflation

• No single investor can affect the price of a stock (i.e., price-takers)

• Capital markets are in equilibrium

CAPM AssumptionsCAPM Assumptions

CAPM AssumptionsCAPM Assumptions• These assumptions appear unrealistic• The important issue is how well the theory predicts

(describes) reality• The CAPM is robust since most of its assumptions

can be relaxed without significant effects on the model

• Not all of the CAPM assumptions are unrealistic: Some institutional investors are tax-exempt Significant reduction in transaction costs by using

discount brokers and/or internet For the one-period horizon of the model, inflation

may be fully (or mostly) anticipated

• Most important implication of the CAPM All investors hold the same optimal portfolio of risky

assets • As a result of the assumption that all investors have

the same time horizon and homogenous expectations regarding the expected returns and risks for any given risky asset

The optimal portfolio is at the highest point of tangency between RF and the efficient frontier

The portfolio of all risky assets is the optimal risky portfolio

• Called the market portfolio

Market PortfolioMarket Portfolio

• All risky assets must be in portfolio, so it is completely diversified Contains only systematic risk (cannot be eliminated)

• All securities included in proportion to their market value

• Unobservable, but proxied by S&P/TSX Composite Index in Canada (or the S&P 500 in the US)

• In theory, should contain all risky assets worldwide, both financial and real, in their proper proportions

• Is found by determining which efficient portfolio offers the highest risk premium, given the existence of a risk-free asset

Characteristics of the Market Portfolio (M)Characteristics of the Market Portfolio (M)

The Equilibrium Risk-Return trade-offThe Equilibrium Risk-Return trade-off

• The CAPM is an equilibrium model that encompasses two important relationships: The capital market line (CML), specifies the

equilibrium relationship between expected return and total risk for efficient portfolios

The security market line (SML), specifies the equilibrium relationship between expected return and systematic risk

• Line from RF to L is capital market line (CML)

• x = risk premium

= E(RM) - RF

• y = risk = M

• Slope = x/y = market price of risk for efficient portfolios

= [E(RM) - RF]/M

• y-intercept = RF = price of forgone consumption

E(RM)

RF

RiskM

L

M

y

x

Capital Market LineCapital Market Line

Example: Example: Capital Market LineCapital Market Line

• (Pg 247) Assume that the expected return on portfolio M is 13%, with a standard deviation of 25%, and that RF is 7%

• Calculate the slope of the CML

• Slope of the CML is the market price of risk for efficient portfolios, or the equilibrium price of risk in the market

• Relationship between risk and expected return for portfolio P (Equation for CML):

pM

Mp

RF)R(ERF)R(E

p

M

Mp

RF)R(ERF)R(E

Capital Market LineCapital Market Line

Capital Market LineCapital Market Line

• The following should be noted about the CML: Only efficient portfolios consisting of the risk-free

asset and portfolio M lie on the CML It indicates the optimal expected returns associated

with different portfolio risk levels It must always be upwards sloping because the

price of risk must always be positive Although, it must be upward sloping ex ante (before

the fact), it can be, and sometimes is, downward sloping ex post (after the fact). This merely indicates that the returns actually realized differed from those that were expected

• CML Equation only applies to markets in equilibrium and efficient portfolios (i.e., cannot be used to assess the expected return on individual securities or inefficient portfolios)

• The Security Market Line depicts the tradeoff between risk and expected return for individual securities

• Under CAPM, all investors hold the risky portion of their portfolio in the market portfolio How does an individual security contribute to the risk

of the market portfolio?

Security Market LineSecurity Market Line

• The expected return on any risky asset is directly proportional to its covariance with the market portfolio

• Equation for expected return for an individual stock similar to CML Equation

RF)R(ERF

RF)R(ERF)R(E

Mi

M

M,i

M

Mi

RF)R(ERF

RF)R(ERF)R(E

Mi

M

M,i

M

Mi

Security Market LineSecurity Market Line

• Beta = 1.0 implies as risky as market

• Securities A and B are more risky than the market Beta > 1.0

• Security C is less risky than the market Beta < 1.0

AB

C

E(RM)

RF

0 1.0 2.00.5 1.5

SML

BetaM

E(R)

Security Market LineSecurity Market Line

• The SML represents the trade-off between systematic risk (β) and the expected return for all assets, whether individual securities, inefficient portfolios, or efficient portfolios

• Beta measures systematic risk Measures relative risk compared to the market portfolio of

all stocks Volatility of security i is different (higher or lower) than

that of the market if β_i > 1 or β_i < 1 and is equal to that of the market if β_i = 1

β_i = 1 means that for every 1% change in the market’s return, on average security i’s returns change 1%

β_M = 1 and β_RF = 0

Security Market LineSecurity Market Line

Security Market LineSecurity Market Line

• β is useful for comparing the relative systematic risk of different stocks and, in practice, is used by investors to judge a stock’s riskiness

• βs may vary widely across companies in different industries and within a give industry (e.g., β for Barrick Gold Corp. is 0.63 and for Eldorado Gold Corp. is 1.2)

• All securities should lie on the SML The expected return on the security should be only

that return needed to compensate for systematic risk

ErUnderpriced SML: Er = rf + (Erm – rf)

Overpriced

rf

Underpriced expected return > required return according to CAPM lie “above” SML

Overpriced expected return < required return according to CAPM lie “below” SML

Correctly priced expected return = required return according to CAPM lie along SML

SML and Asset ValuesSML and Asset Values

β

• Required rate of return on an asset (ki) is composed of risk-free rate (RF) risk premium (i [ E(RM) - RF ])

• Market risk premium adjusted for specific security

ki = RF +i [ E(RM) - RF ] The greater the systematic risk, the greater

the required return

CAPM’s Expected Return-Beta CAPM’s Expected Return-Beta RelationshipRelationship

Example: Example: Expected Return-Beta Expected Return-Beta RelationshipRelationship

• (Pg 252) If the β estimate for security i is 1.04, the RF is 0.0241, and the expected return on the market is estimated to be 0.10.

• Calculate the required return on security i

• To use the SML, an investor needs estimates of the return on the risk-free asset, the expected return on the market index, and the β for an individual security

• Treasury Bill rate used to estimate RF• Expected return for the market index in not

observable Estimated using past market returns and taking an

expected value• Estimating individual security betas is difficult

β is only company-specific factor in CAPM Requires asset-specific forecast

Estimating the SMLEstimating the SML

• Market model Relates the return on each stock to the return on the

market, assuming a linear relationship with an intercept and slope

Rit =i +i RMt + eit

It is identical to the single-index model except that it does not make the assumption that the error terms of the different securities are uncorrelated

The Market Model produces an estimate of return for any stock

Estimating BetaEstimating Beta

Estimating BetaEstimating Beta

• Characteristic line A regression equation used to estimate β by

regressing stock returns on market returns Line fitted to total returns for a security relative to

total returns for the market index (Fig 9.6 pg 255) β is the slope of the characteristic line

• Betas change with a company’s situation (e.g., earnings & cash flows) Not stationary over time

• Estimating a future beta May differ from the historical beta

• RMt represents the total of all marketable assets in the economy Approximated with a stock market index, which, in

turn, approximates return on all common stocks

How Accurate Are Beta Estimates?How Accurate Are Beta Estimates?

• No one correct number of observations and time periods for calculating beta. As a result, estimates of β will vary

• The regression estimates of the true and from the characteristic line are subject to estimation error

• Portfolio betas more reliable than individual security betas

• βs for large portfolios are stable (show less change from period to period) because of the averaging effect (i.e., errors involved in estimating βs tend to cancel out)

How Accurate Are Beta Estimates?How Accurate Are Beta Estimates?

Test of CAPMTest of CAPM

• Previous empirical results indicate that:

Ri =a1 +i a2

The SML appears to be linear (i.e., the trade-ff between expected return and risk is an upward sloping straight line

The intercept term, a1, is generally found to be higher than RF

The slope of the CAPM, a2, is generally found to be less steep than predicted by the theory (i.e., overpredicts returns for low-β stocks and underpredicts returns for high-β stock)

Test of CAPMTest of CAPM

• Fama and French (1992) Indicate that the CAPM’s sole risk factor, market risk

β, posses no explanatory power in discriminating among the cross-sectional returns of US stocks

Common market equity (ME) and the ratio of book equity to market equity (BE/ME) combine to explain the cross-section of expected returns

Three-factor model (overall market factors, factors related to firm size, & BE/ME ratio)

• Empirical SML is “flatter” than predicted SML• Fama and French (1992)

Market Size Book-to-market ratio

• Roll’s Critique True market portfolio is unobservable Tests of CAPM are merely tests of the mean-

variance efficiency of the chosen market proxy

Test of CAPMTest of CAPM

• Based on the Law of One Price Two otherwise identical assets cannot sell at

different prices Equilibrium prices adjust to eliminate all

arbitrage opportunities• Unlike CAPM, APT does not assume

single-period investment horizon, absence of personal taxes, riskless borrowing or lending, mean-variance decisions

Arbitrage Pricing TheoryArbitrage Pricing Theory

• APT assumes returns generated by a factor model

• Factor Characteristics Each risk must have a pervasive influence on

stock returns Risk factors must influence expected return and

have nonzero prices Risk factors must be unpredictable to the market

FactorsFactors

• Most important are the deviations of the factors from their expected values

• The expected return-risk relationship for the APT can be described as:

E(Rit) =a0+bi1 (risk premium for factor 1) +bi2 (risk premium for factor 2) +… +bin (risk premium for factor n)

APT ModelAPT Model

• Reduces to CAPM if there is only one factor and that factor is market risk

• Roll and Ross (1980) Factors: Changes in expected inflation Unanticipated changes in inflation Unanticipated changes in industrial production Unanticipated changes in the default risk

premium Unanticipated changes in the term structure of

interest rates

APT ModelAPT Model

• Factors are not well specified ex ante To implement the APT model, the factors that

account for the differences among security returns are required

• CAPM identifies market portfolio as single factor

• Neither CAPM or APT has been proven superior Both rely on unobservable expectations

Problems with APTProblems with APT