Chapter 7 investments capital asset pricing model (CAPM)

• The risk premium on the market portfolio will be proportional to the variance of the market portfolio and investors’ typical degree of risk aversion.. rel-• The risk premium on the mar

Chapter 7

Capital Asset Pricing

Model (CAPM)

7.1 The Capital Asset Pricing Model

Capital Asset Pricing Model (CAPM)

• Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development

• Predicts the relationship between the risk and rium expected returns on risky assets

equilib-• Approach the CAPM in a simplified setting and then add complexity to the environment

Simplifying Assumptions

Individuals are alike, with the notable exceptions of initial wealth

and risk aversion

• Individual investors are price takers

: There are many investors, each with an endowment of wealth that is small compared with the total endowment

of all investors

• Single-period investment horizon

• Investments are limited to traded financial assets

• No taxes and no transaction costs

Simplifying Assumptions (cont.)

• Information is costless and available to all investors

• Investors are rational mean-variance optimizers

: All investors attempt to construct efficient frontier lios

portfo-• Homogeneous expectations

: All investors analyze securities in the same way and share the same economic view of the world

Hypothetical Equilibrium

• All investors will hold the same portfolio for risky assets; the “market portfolio” Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value

• The market portfolio will be on the efficient frontier It will be the mal risky portfolio, the tangency point of the capital allocation line

opti-(CAL) to the efficient frontier

• The capital market line (CML), line from the risk-free rate through the market portfolio, is the best attainable capital allocation line

• The risk premium on the market portfolio will be proportional to the

variance of the market portfolio and investors’ typical degree of risk

aversion

E(r M ) – r f = A* ss M 2

• The risk premium on individual assets will be proportional to the risk premium on the market portfolio (M) and to the beta coefficient of

the security on the market portfolio

Hypothetical Equilibrium (cont.)

Why All investors Would Hold the Market P

ortfolio

• With all assumptions, all investors should hold the same optimal risky portfolio

• They all derive identical efficient frontiers and find the same tangency portfolio for the capital allocation line from the risk-free asset to that frontier

• With everyone choosing to hold the same risky portfolio, stocks will be

repre-sented in the aggregate risky portfolio in the same proportion as they are in

each investor’s risky portfolio

• The market portfolio is the aggregate of all individual portfolios

• Each investor uses the market portfolio for the optimal risky portfolio, the CAL in this case is called CML

The Passive Strategy is Efficient

• The CAPM implies that a passive strategy, using the CML as the optimal CAL, is a powerful alternative to an active strategy

• The market portfolio proportions are a result of a profit-oriented

“buy” and “ sell” orders that cease only when there is no more profit to be made

• It implies that only one mutual fund of risky assets, the market portfolio, is sufficient to satisfy the investment demands of all in-

vestors  called a mutual fund theorem

• If a passive strategy is costless and efficient, why would anyone follow an active strategy? But if no one does not any security

analysis, what brings about the efficiency of the market portfolio?

The Risk Premium of the Market Portfolio

• Think about the decision of how much to invest in the market portfolio

M and how much in the risk-free asset

• In equilibrium, the risk premium on the market portfolio must be just

high enough to induce investors to hold the available supply of stocks

• Investors purchase stocks  their demand drives up prices  lower

ex-pected rates of return and risk premium  Given lower risk premium, atively more risk-averse investors will move their funds to risk-free as-set from the risky market portfolio

rel-• The risk premium on the market portfolio will be proportional to both the risk of the market and risk aversion of the investor

E(r M ) – r f = A* ss M 2

Expected Returns on Individual Securities

• The appropriate risk premium will be determined by its contribution

on the risk of investors’ overall portfolios

• The risk premium on individual securities is a function of the vidual security’s contribution to the risk of the market portfolio.

indi-• What type of individual security risk will matter, systematic or

unsystematic risk?

: Nonsystematic risk can be diversified away through diversification  Need to be compensated only for bearing systematic risk

Expected Returns on Individual Securities

• An individual security’s total risk (s2

i) can be partitioned into tematic and unsystematic risk:

sys-s2i = sbi2 sM2 + s2(ei)

M = market portfolio of all risky securities

• Individual security’s contribution to the risk of the market portfolio is

a function of the covariance 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 sbi s= [COV(r i ,r M )] / s 2

M

Expected Returns on Individual Securities

• The risk premium of an asset is proportional to its beta

• The ratio of risk premium to beta should be the same for any two securities or portfolios

• If we were to compare the ratio of risk premium to systematic risk for the market portfolio, which has a beta of 1, with the correspond-ing ratio for any stock, its relationship is the following

[E(r M ) – r f ] / 1 = [E(r i ) – r f ] / b i

• CAPM’s expected return-beta relationship:

E(r i ) = r f + b i [E(r M ) – r f ]

Expected Returns on Individual Securities

• CAPM’s expected return-beta relationship:

E(r i ) = r f + b i [E(r M ) – r f ]

 The rate of return on any asset exceeds the risk-free rate by

a risk premium equal to the asset’s systematic risk

measure (Beta) times the risk premium of the market portfolio

 Only systematic risk matters to investors who can diversify

and that systematic risk is measured by the beta of the security

• In reality, even if one does not hold the precise market portfolio, a diversified portfolio will be so highly correlated with the market that a stock’s beta relative to the market still be a useful risk measure

well-The Security Market Line

• Consider the expected return-beta relationship as a reward-risk tion

equa-• The beta of a security is the appropriate measure of its risk in that

beta is proportional to the risk the security contributes to the mal risky portfolio

opti-• The beta of a stock measures the stock’s contribution to the standard deviation of the market portfolio

• Expect the required risk premium to be a function of beta

• The security’s risk premium is directly proportional to both the beta and the risk premium of the market portfolio; bi [E(r M ) – r f ]

Individual Stocks: Security Market Line

Slope SML =

= Equation of the SML (CAPM) E(ri) = rf + bi[E(rM) - rf]

(E(rM) – rf )/ βMprice of risk for market

The Security Market Line

• CML

- Graphs the risk premiums of efficient portfolios as a function of portfolio

standard deviation

- Standard deviation is a valid measure of risk for portfolios that are

candidates for an investor’s complete portfolio

• SML

- Graphs individual asset risk premiums as s function of asset risk

- The relevant measure of risk for an individual asset is not the asset’s standard deviation

- The contribution of the asset to the portfolio standard deviation is

measured by the asset’s beta

- The SML is valid both for portfolios and individual assets

Capital Market Line (CML)

Sample Calculations for SML

E(rm) - rf = 0.08 rf = 0.03

bx = 1.25E(rx) =

by = 6E(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? Also, If b = 0?

SML

ß ß

Ry=7.8%

ß 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.

Whenever the CAPM holds, all securities must lie on the SML

in market equilibrium

E(r) 15%

SML

ß 1.0

According to the SML, the E(r) should be _

Expected rate of return > Required rate of return

 People want to hold a stock

 A stock price will go up.

 Its expected return will go down

 Therefore, the stock is underpriced, and you should buy

it now

E(r) 15%

SML

ß 1.0

According to the SML, the E(r) should be _

1.25

15%

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%

More on alpha and beta

E(rM) = 14% βS = 1.5 rf = 5%

Required return = rf + βS[E(rM) – rf]

= 5 +1.5[14-5] = 18.5%

If you believe the stock will actually provide a retur

n of 17%, what is the implied alpha?

 = 17% - 18.5% = -1.5%

A stock with a negative alpha plots below the SML

& gives the buyer a negative abnormal return.

7.2 The CAPM and Index Models

Security Characteristic Line (SCL)

Excess Returns (i)

.

.

.

.

.

.

. .

Dispersion of the points

around the line

mea-sures .

The statistic is

called se

unsystematic risk

7.3 The CAPM and the Real World

Evaluating the CAPM

• The CAPM is “false” based on the validity of its assumpti on.

• The CAPM could still be a useful predictor of expected returns That is an empirical question

- Huge measurability problems because the market portfolio is unobservable

e.

Evaluating the CAPM

• However, the practicality of the CAPM is testable.

Betas are not as useful at predicting returns as other measurable factors may be

- More advanced versions of the CAPM that do a better job at estimating the market portfolio are useful at predicting stoc

k returns

- Still widely used and well understood

Evaluating the CAPM

• The principles we learn from the CAPM

are still entirely valid.

- Investors should diversify

- Systematic risk is the risk that matters

- A well diversified risky portfolio can be suitable for a wide range of investors

- Differences in risk tolerances can be handled

by changing the asset allocation decisions

in the complete portfolio