#### Subject 8. Arbitrage pricing theory and the factor model

An arbitrage arises if an investor can construct a zero investment portfolio with a sure profit. Since no investment is required, an investor can create large positions to secure large levels of profit. In efficient markets, profitable arbitrage opportunities will quickly disappear.

The arbitrage pricing theory (APT) describes the expected return on an asset (or portfolio) as a linear function of the risk of the asset with respect to a set of factors. It makes the following assumptions:

• A factor model describes asset returns. The APT model, however, does not tell us which factors are relevant.
• The returns of a security are based on the systematic risk exposure of the security. This means that asset-specific risk can be eliminated. APT applies to well-diversified portfolios.
• No arbitrage opportunities exist among well-diversified portfolios. APT does not rely on the existence of a market portfolio. It is based purely on no-arbitrage conditions.

Consider a world where investors are broadly diversified, but there may be multiple sources of risk in the economy. Instead of everyone caring solely about the market portfolio, investors actually care about lots of things, including shifts in stock index levels, interest rates, inflation, changes in GDP or other broad macro-economic factors that are difficult to purge from your portfolio through diversification.

The multifactor APT assumes that the stochastic process generating asset returns can be represented as K factor model of the form

E(RP) = RF + λ1 βP,1 + ... + λk βP,k

where
RF = risk free rate.
λj = the risk premium for factor j. Each λ stands for the expected risk premium associated with each risk factor. Each λj equals the risk premium for a portfolio (a pure factor portfolio) with factor sensitivity of 1 to factor j and a sensitivity of 0 to all other factors.
βP,j = the sensitivity of the portfolio to factor j. Note that the APT does not require that one of the risk factors is the market portfolio.
K = the number of factors.

The method of exploiting arbitrage opportunities in the APT framework is detailed in the example 12 of the reading.

APT versus Multifactor Models

• The APT is a cross-sectional equilibrium model. It explains the variation across assets' expected returns. The multifactor model is a time-series regression. It explains the variation over time in returns for one asset.
• The intercept term in a macroeconomic factor model is the investment's expected return. The intercept in the APT is the risk-free rate.

#### Practice Question 1

Which pricing model provides no guidance concerning the determination of the risk premium on factor portfolios?

I. The multifactor APT.
II. The CAPM.

#### Practice Question 2

The following factors might affect stock returns:

I. interest rate fluctuations.
III. inflation rates.

Correct Answer: I, II and III

#### Practice Question 3

Consider a well-diversified portfolio, A, in a two-factor economy. The risk-free rate is 6%, the risk premium on the first factor portfolio is 4% and the risk premium on the second factor portfolio is 3%. If portfolio A has a beta of 1.2 on the first factor and .8 on the second factor, what is its expected return?

A. 8.0%
B. 10.3%
C. 13.2%

#### Practice Question 4

Regarding factor portfolios:

I. A factor portfolio is a well-diversified portfolio that has a beta coefficient equal to one for a specific factor and zero for all other factors.
II. Since there are many assets, such portfolios can be constructed for each factor.
III. If a portfolio manager wants to bet on a source of risk, he or she may use factor portfolios.

Correct Answer: I, II and III

#### Practice Question 5

The arbitrage pricing theory assumes:

I. capital markets are perfectly competitive.
II. investors always prefer more wealth to less wealth with certainty.
III. The utility function is quadratic in nature.
IV. The stochastic process generating asset returns can be represented as a K factor model.
V. Security returns are normally distributed.

A. I, II and IV.
B. I, II, III, IV and V.
C. II, III, IV and V.

#### Practice Question 6

Which statement(s) is/are true about APT?

I. It uses the development of a security's price over time to calculate the expected return.
II. It uses risk premiums based on stock specific variables.
III. It uses key macroeconomic factors relevant to determine a security's returns.
IV. Investors are risk-averse.

A. I and III
B. III only
C. III and IV

#### Practice Question 7

The major assumptions that are not required in the arbitrage pricing theory include: