## THE MULTIFACTOR MODEL OF RISK & RETURN #

The equation for a multifactor model for stock i can be expressed as follows:

** Ri =E(Ri) + ****βi1F1 + ****βi2F2 + . . .+ ****βik FK + ei**

Where:

Ri = return on stock i

E(R_{i}) = expected return for stock i

β_{ij} = j th factor beta for stock i

F_{j} = deviation of macroeconomic factory from its expected value

e_{i} = firm-specific return for stock i.

The factor beta, βij , equals the sensitivity of the stock return to a 1-unit change in the factor.

The firm-specific return, ep is the portion of the stock’s return that is unexplained by macro factors.

A single-factor security market line (SML) is analogous to the capital asset pricing model (CAPM). In the single-factor SML, systematic risk is measured as the exposure of the asset to a well-diversified market index portfolio.

A multifactor model can be used to hedge away multiple factor risks. To do so, the investor can create factor portfolios, which are well-diversified portfolios with beta equal to one for a single risk factor, and betas equal to zero on the remaining risk factors.

**THE LAW OF ONE PRICE & ARBITRAGE OPPORTUNITY**

According to the Law of One Price, identical assets selling in different locations should be priced identically in the different locations. The action of buying an asset in the cheaper market and simultaneously selling that asset in the more expensive market is called arbitrage. The actions of arbitrageurs cause prices to rise in the cheaper market and fall in the expensive market. The simultaneous trades will continue until the asset trades at one price in both markets, at which point the arbitrage opportunity will be fully exploited.

## WELL DIVERSIFIED PORTFOLIOS #

- The part of an individual security’s risk that is uncorrelated with the volatility of the market portfolio is that security’s non systematic risk .
- The part of an individual security’s risk that arises because of the positive covariance of that security’s returns with overall market returns is called its systematic risk.
- Portfolio risk reduction through diversification comes from reducing non systematic risk.
- The standardized measure of systematic risk is beta.
- Because non systematic (diversifiable) risk can be avoided (without cost) by efficient diversification, there is no added expected return for bearing non systematic risk.

## HEDGING EXPOSURES TO MULTIPLE FACTORS #

- A multifactor model can be used to hedge away multiple factor risks.
- To do so, the investor can create factor portfolios, which are well-diversified portfolios with beta equal to one for a single risk factor, and betas equal to zero on the remaining risk factors.
- Factor portfolios can be used to hedge multiple risk factors by combining the original portfolio with offsetting positions in the factor portfolios.

## THE ARBITRAGE PRICING THEORY #

The arbitrage pricing theory (APT) describes expected returns as a linear function of exposures to common (i.e., macroeconomic) risk factors. The APT model can be written as:

**E(Ri) = RF + ****βi1 [E(R1)- RF] + ****βi2 [E(R2)- RF] + …+ ****βik[E(Rk)- RF] **

The assumptions underlying the APT model are as follows:

- Returns follow a k-factor process:
- Ri = E(Ri) + βi1 F1+ βi2 F2+ …+ βikFK + ei
- Well-diversified portfolios can be formed.
- No arbitrage opportunities exist.

The CAPM can be considered a special restrictive case of the APT in which there is only one risk factor

THE FAMA FRENCH THREE – FACTOR MODEL

In addition to the market return factor (RM – RF), the Fama-French three-factor model specifies the following two factors: SMB (small minus big) is the firm size factor equal to the difference in returns between portfolios of small and big firms (RS- RB).

HML (high minus low) is the book-to-market (i.e., book value per share divided by stock price) factor equal to the difference in returns between portfolios of high and low book-to-market firms (RH — RL).

The equation for the Fama-French three-factor model is:

Ri – RF = αi + βi,M(RM – RF) + βi,_{SMB *}SMB + βi,_{HML}HML + ei