Arbitrage pricing theory and multi-factor models

Why look beyond the CAPM?

In the capital asset pricing model (CAPM) the only systematic risk (risk that diversification cannot remove) is the market, and beta is the only risk measure that matters for the cost of equity. Research has repeatedly found that market risk is not the only risk equity investors are paid to bear, so researchers and practitioners look for other systematic sources of return.

A model with one risk factor is a single-factor model . The CAPM is the best-known one. The systematic and nonsystematic risk lesson shows why the diversifiable part earns nothing.

What is arbitrage pricing theory?

In arbitrage pricing theory (APT) , a general framework, expected returns depend on several systematic risk factors, each with its own sensitivity (a factor exposure , or beta) and its own expected factor risk premium .

$$E[r_i] = r_f + \beta_{i,1}E(f_1) + \beta_{i,2}E(f_2) + \cdots + \beta_{i,k}E(f_k)$$

Here $\beta_{i,k}$ is asset i’s exposure to factor k and $E(f_k)$ is that factor’s expected risk premium. Three features matter.

• It does not say what the factors are or how many there are. Research has to find them.
• Every factor is a systematic risk. As in the CAPM, the average investor holds a diversified portfolio, so company-specific risk earns nothing.
• The market’s excess return may or may not be one of the factors.

	CAPM	APT
Sources of systematic risk	One: the market	Several, unnamed
Risk measure	One beta	One beta per factor
Is the market a factor?	Yes, the only one	Possibly
Rewards diversifiable risk?	No	No

Where do factors come from?

The sources of factors are three kinds of observed data: company stock market data (prices, returns, market values), macroeconomic data, and company accounting data (for example book values). A multi-factor model is any model that uses more than one factor. It can be theoretical (APT) or defined from the data.

Published research has proposed and tested hundreds of factors. Many are likely products of data mining (searching until something fits) or chance correlations, and most practitioners and academics believe a small number of factors captures the systematic risks that matter. The search is ongoing: factors are added, and some stop working.

What is the four-factor model?

The four-factor model is a widely used multi-factor model with the market plus three factors built from stock market and accounting data: firm size, value and price momentum. Eugene Fama and Kenneth French added size and value to the market in a three-factor model (1993). Mark Carhart added momentum (1997).

$$E(r_i) = r_f + \beta_{i,Market}E(r_m - r_f) + \beta_{i,Size}E(SMB) + \beta_{i,Value}E(HML) + \beta_{i,Momentum}E(UMD)$$

The three non-market factors are returns on zero-investment portfolio s: long one group of stocks, short an equal amount of another, so no net money goes in.

• The size factor, small minus big (SMB) , is long small market-capitalization stocks and short large ones.
• The value factor, high minus low (HML) , is long stocks with high book-to-market ratios and short stocks with low book-to-market ratios. A high book-to-market ratio means a low price-to-book ratio, the mark of a value stock . A low book-to-market ratio means a high price-to-book ratio, the mark of a growth stock .
• The momentum factor, up minus down (UMD) , is long the stocks with the highest recent returns and short those with the lowest.

Four-factor modelmarket plus three long-short portfolios

• Market Memorize excess market return; average beta 1
• Size: small minus big (SMB) Memorize long small, short big; average beta 0
• Value: high minus low (HML) Memorize long high book-to-market, short low; average beta 0
• Momentum: up minus down (UMD) Memorize long recent winners, short losers; average beta 0

How do you read factor exposures?

Compare each beta with the average stock. The average market beta is 1.0. The average size, value and momentum betas are 0, so negative values are common.

• Size beta above 0: behaves like a small company (does well when small caps beat large caps). Below 0: behaves like a large company.
• Value beta above 0: behaves like a value stock. Below 0: like a growth stock.
• Momentum beta above 0: behaves like recent winners. Below 0: does worse when recent winners keep winning.

Three fictional companies, each estimated on 60 monthly returns:

Company	Market	Size	Value	Momentum	Reading
Arden Software	1.30	−0.30	−0.80	0.10	High market risk, large, growth
Brooke Utilities	0.60	−0.20	0.45	−0.05	Low market risk, large, value
Corvel Mining	1.40	0.90	0.60	−0.30	High market risk, small, value, against momentum

Set the betas yourself and watch each factor’s contribution build the cost of equity.

How do you get a cost of equity from a factor model?

Exactly as with the CAPM. Multiply each beta by its factor’s premium, add them up, add the risk-free rate. Premiums are usually historical averages of the factor portfolios’ returns. If they are monthly, multiply by 12 first (annualizing monthly premiums ).

$$E(r_i) = r_f + \sum_{k}\beta_{i,k}\,E(f_k), \qquad E(f_k)_{annual} = 12 \times E(f_k)_{monthly}$$

Why can the CAPM give a different answer?

Two reasons. The CAPM leaves out the other factors’ contributions, and the market beta from a one-factor regression is unlikely to equal the market beta from a multi-factor regression (it can be higher or lower, and the direction is not predictable).

Exam tip Positive exposures to factors with positive premiums raise the cost of equity above the CAPM’s. Negative exposures lower it. Which model gives the higher number depends on the stock.

Estimates also move. Factor betas depend on the horizon and return frequency used, and factor premiums on the historical period chosen. Neither is a guaranteed or permanent number.

Why do multi-factor models matter beyond the cost of equity?

They also shape investment strategy and how performance is judged, both later in the curriculum. factor investing selects stocks by statistical characteristics that matter for risk and return, and factor indexes package such tilts (see types of equity indexes). Analysts should follow new research, both newly found factors and those shown not to explain returns.

Go deeper Why a long-short portfolio is a factor

A zero-investment portfolio isolates one characteristic. By going long small stocks and short large ones in equal amounts, the market’s general movement cancels out, and what is left is the return difference between small and large. A stock’s beta to that return series says how much of the stock’s return moves with the size difference, which is exactly what its size exposure measures.

Go deeper Where the size and value premiums first appear

The size and value premiums were first met in Quantitative Methods, in the interest rates and required returns lesson, as parts of a required return. Here they become two of the factors in a formal model.