Capital Asset Pricing and Arbitrage Pricing Theory

• The capital asset pricing model (CAPM) is a centerpiece of modern financial

economics.

• It was introduced by William F. Sharpe, who was awarded the 1990 Nobel

Prize in Economics.

• The CAPM provides a precise prediction of the relationship we should

observe between the risk of an asset and its expected return.

• This relationship serves two vital functions:

→ It provides a benchmark rate of return for evaluating possible investments.

→ The model helps us make an educated guess as to the expected return on

assets that have not yet been traded in the marketplace.

T.M Sibisi – 3 March 2014

The Capital Asset Pricing Model

The model: Assumptions and Implications

• The conditions that lead to the CAPM ensure competitive security markets

and investors who choose from identical efficient portfolios using the mean-

variance criterion:

1. Markets for securities are perfectly competitive and equally profitable to all

investors:

a. Investors are price-takers not price-makers.

b. All information relevant to security analysis is publicly available at no

cost.

c. All securities are publicly owned and traded, and investors may trade all

of them. Thus, all risky assets are in the investment universe.

d. No taxes on investment returns.

e. No transaction costs.

f. Lending and borrowing at the risk-free rate are unlimited.

T.M Sibisi – 3 Mar 2014


2. Investors are alike in every way except for initial wealth and risk aversion,

hence they all choose investment portfolios in the same manner:

a. Investors plan for the same (single-period) horizon.

b. Investors are rational, mean-variance optimisers.

c. Investors use the same inputs and consider identical portfolio

opportunity sets (homogenous expectations).

• Obviously, these assumptions ignore many-world complexities…

• However, they lead to powerful insights into the nature of equilibrium in

security markets…


• Given these assumptions, these are the implications:

I. All investors will hold the market portfolio 𝑀, which includes all assets of

the security universe.

II. The market portfolio will be on the efficient frontier. Moreover, it will be the

optimal risky portfolio, the tangency point of the capital allocation line to the

efficient frontier (this gives us the CML).

III. The risk premium on the market portfolio will be proportional to the variance

of the market portfolio and investors’ typical degree of risk aversion. That is,

𝐸 𝑟𝑀 − 𝑟𝑓 = 𝐴𝜎𝑀2

IV. 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.

Example

• Suppose the risk-free rate is 5%, the average investor has a risk-aversion

coefficient of 𝐴 = 2, and the standard deviation of the market portfolio is

20%.

• It follows that the estimated equilibrium value of the market risk premium is

2 0.22 = 0.08

• So the expected rate of return on the market must be

𝐸 𝑟𝑀 = 𝑟𝑓 + 𝐸𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚

= 0.05 + 0.08 = 0.13 = 13%

• If the typical risk-aversion coefficient is 𝐴 = 3, what would the required

expected rate of return on the market portfolio be?


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.

• A passive investor who takes a free ride by simply investing in the market

portfolio benefits from the efficiency of that portfolio. In fact, an active investor

who chooses any other portfolio will end on a CAL that is inferior to the CML

used by passive investors.

• This result is also known as the mutual fund theorem because it implies that

only mutual fund of risky assets – the market index fund – is sufficient to satisfy

the investment demands of all investors (separation property).

• If all investors choose to hold the market-index fund, then we can separate

portfolio selection into two components:

1. A technical side, where an efficient mutual fund is created by professional

management.

2. A personal side, where investors choose the allocation of the complete portfolio

between the mutual fund and a risk-free asset according to their risk aversion.

The Passive Strategy is Efficient

• The logical inconsistency of the CAPM is this:

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

the market portfolio?

Expected Returns on Individual Securities

• The CAPM is built on the insight that portfolio risk is what matters to

investors, and portfolio risk is what governs the risk premiums they demand.

• We know that nonsystematic risk can be reduced to a negligible amount

through diversification; therefore, investors do not need to be compensated

for bearing firm-specific risk. However, they do require a risk premium to

compensate for systematic risk…

• Beta (𝛽 ) is the coefficient by which we measure an individual asset’s

sensitivity to market fluctuations. It is therefore a measure of systematic

risk.



• The CAPM’s expected return-beta relationship is as follows:

𝐸 𝑟𝑖 − 𝑟𝑓 = 𝛽𝑖 𝐸 𝑟𝑀 − 𝑟𝑓

→ 𝐸 𝑟𝑖 = 𝑟𝑓 +𝛽𝑖 𝐸 𝑟𝑀 − 𝑟𝑓

• The mean-beta relationship of the CAPM implies, for example, that a

security with high variance but a low beta of 0.5 will carry only a third of the

risk premium of a security with a low variance but high beta of 1.5.

• Bottom line… Only systematic risk matters to investors who can diversify!

T. M Sibisi – 3 March 2014

Example…

• Suppose the risk premium of the market portfolio is 9% and we estimate the

beta of Dell as 𝛽𝐷 = 1.3.

• What is the risk premium predicted on the stock?

• If the T-bill rate was 5%, what is the expected return on the Dell stock?

• If the estimate of the beta of Dell were only 1.2, what would be the required

risk premium of Dell be?



• In reality, investment managers generally construct risky portfolios that differ

from the market index. However, this does not necessarily invalidate the

CAPM.

• Recall that all well-diversified portfolios shed almost all firm-specific risk and

are subject only to systematic risk. Even if one does not hold the precise

market portfolio, a well-diversified portfolio will be so highly correlated to the

market that a stock’s beta relative to the market will still be a useful

measure.

• If the mean-beta relationship holds for any individual asset, it must hold for

any combination of assets. The beta of a portfolio is simply the weighted

average of the betas of the stocks in the portfolio, using the portfolio weights

as the respective proportions.


Example…

• If the market risk premium is 7%, consider the following portfolio:

• What is the beta of the portfolio and risk premium of the portfolio?


Asset Beta Portfolio weight Risk Premium

Microsoft 1.2 0.5 9.0%

American Electric power 0.8 0.3 6.0

Gold 0.0 0.2 0.0

The Security Market Line

• The expected return-beta relationship is a reward-risk equation.

• Beta is the proportional variance that an individual stock contributes to the

optimal portfolio (Market portfolio).

• Let 𝑅𝑖 = 𝑟𝑖 − 𝑟𝑓 and 𝑅𝑀= 𝑟𝑀 − 𝑟𝑓, econometrically, CAPM can be specified

as:

𝑅𝑖 = 𝛼 + 𝛽𝑖 𝑅𝑀 + 𝑒𝑖 𝑒𝑖~𝑁(0,1)

Taking the expectation of this equation yields the CAPM as you know it…

𝐸 𝑅𝑖 = 𝛼 + 𝛽𝑖 𝑅𝑀

If markets are in equilibrium, then 𝛼 = 0.



• The variance is given by

𝑉𝑎𝑟 𝑅𝑖 = 𝑉𝑎𝑟 𝛼 + 𝛽𝑖 𝑅𝑀 + 𝑒𝑖

𝜎𝑖2 = 𝛽𝑖

2𝜎𝑀2 + 𝜎2 𝑒𝑖

= 𝑆𝑦𝑡𝑒𝑚𝑎𝑡𝑖𝑐 𝑅𝑖𝑠𝑘 + 𝐹𝑖𝑟𝑚 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑟𝑖𝑠𝑘

Because only systematic risk is priced and firm-specific risk can be diversified,

we then have it that,

𝜎𝑖2 = 𝛽𝑖

2𝜎𝑀2

→ 𝛽𝑖2 =

𝜎𝑖2

𝜎𝑀2

which shows that beta is the proportion of variance.T. M Sibisi – 3 March 2014


• Because the beta of a stock measures the stock’s contribution to the

standard deviation of the market portfolio, we expect the risk premium to be

a function of beta.

• The security market line is the graphical representation of the expected

return-beta relationship of the CAPM.




𝛽

𝐸 𝑟𝑃

1.0 1.2

15.6

14

SML

17Stock

M

𝛼

6

Capital Asset Pricing and Arbitrage Pricing Theory

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