## The Arbitrage Pricing Model

The restrictive assumptions in the capital asset pricing model and its dependence upon the market portfolio have for long been viewed with skepticism by both academics and practitioners. In the late seventies, an alternative and more general model for measuring risk called the arbitrage pricing model was developed.9

### 1. Assumptions

The arbitrage pricing model is built on the simple premise that two investments with the same exposure to risk should be priced to earn the same expected returns. An alternate way of saying this is that if two portfolios have the same exposure to risk but offer different expected returns, investors can buy the portfolio that has the higher expected returns and sell the one with lower expected returns, until the expected returns converge.

Like the capital asset pricing model, the arbitrage pricing model begins by breaking risk down into two components. The first is firm specific and covers information that affects primarily the firm. The second is the market risk that affects all investment; this would include unanticipated changes in a number of economic variables, including gross national product, inflation, and interest rates. Incorporating this into the return model above

R = E(R) + m + 8 where m is the market-wide component of unanticipated risk and 8 is the firm-specific component.

Arbitrage: An investment that requires no investment, involves no risk but still delivers a sure profit.

9 Ross, Stephen A., 1976, The Arbitrage Theory Of Capital Asset Pricing, Journal of Economic Theory, v13(3), 341-360.

### 2. The Sources of Market-Wide Risk

While both the capital asset pricing model and the arbitrage pricing model make a distinction between firm-specific and market-wide risk, they part ways when it comes to measuring the market risk. The CAPM assumes that all of the market risk is captured in the market portfolio, whereas the arbitrage pricing model allows for multiple sources of market-wide risk, and measures the sensitivity of investments to each source with what a factor betas. In general, the market component of unanticipated returns can be decomposed into economic factors:

where

Pj = Sensitivity of investment to unanticipated changes in factor j Fj = Unanticipated changes in factor j

### 3. The Effects of Diversification

The benefits of diversification have been discussed extensively in our treatment of the capital asset pricing model. The primary point of that discussion was that diversification of investments into portfolios eliminate firm-specific risk. The arbitrage pricing model makes the same point and concludes that the return on a portfolio will not have a firm-specific component of unanticipated returns. The return on a portfolio can then be written as the sum of two weighted averages -that of the anticipated returns in the portfolio and that of the factor betas:

Rp = (wiRi+w2R2+...+wnRn)+ (wiPi,i+w2pi,2+...+wnpi,n) Fi + (wiP2,1+w2p2,2+...+wnp2,n) F2

where, wj = Portfolio weight on asset j Rj = Expected return on asset j Pi,j= Beta on factor i for asset j Note that the firm specific component of returns (8) in the individual firm equation disappears in the portfolio as a result of diversification.

### 4. Expected Returns and Betas

The fact that the beta of a portfolio is the weighted average of the betas of the assets in the portfolio, in conjunction with the absence of arbitrage, leads to the conclusion that expected returns should be linearly related to betas. To see why, assume that there is only one factor and that there are three portfolios. Portfolio A has a beta of 2.0, and an expected return on 20%; portfolio B has a beta of 1.0 and an expected return of 12%; and portfolio C has a beta of 1.5, and an expected return on 14%. Note that the investor can put half of his wealth in portfolio A and half in portfolio B and end up with a portfolio with a beta of 1.5 and an expected return of 16%. Consequently no investor will choose to hold portfolio C until the prices of assets in that portfolio drop and the expected return increases to 16%. Alternatively, an investor can buy the combination of portfolio A and B, with an expected return of 16%, and sell portfolio C with an expected return of 15%, and pure profit of 1% without taking any risk and investing any money. To prevent this "arbitrage" from occurring, the expected returns on every portfolio should be a linear function of the beta to prevent this f. This argument can be extended to multiple factors, with the same results. Therefore, the expected return on an asset can be written as E(R) = Rf + P1 [E(R1)-Rf] + P2 [E(R2)-Rf] ...+ Pn [E(Rn)-Rf]

where

Rf = Expected return on a zero-beta portfolio

E(Rj) = Expected return on a portfolio with a factor beta of 1 for factor j, and zero for all other factors.

The terms in the brackets can be considered to be risk premiums for each of the factors in the model.

Note that the capital asset pricing model can be considered to be a special case of the arbitrage pricing model, where there is only one economic factor driving market-wide returns and the market portfolio is the factor.

5. The APM in Practice

The arbitrage pricing model requires estimates of each of the factor betas and factor risk premiums in addtion to the riskless rate. In practice, these are usually estimated using historical data on stocks and a statistical technique called factor analysis. Intuitively, a factor analysis examines the historical data looking for common patterns that affect broad groups of stocks (rather than just one sector or a few stocks). It provides two output measures:

1. It specifies the number of common factors that affected the historical data that it worked on.

2. It measures the beta of each investment relative to each of the common factors, and provides an estimate of the actual risk premium earned by each factor. The factor analysis does not, however, identify the factors in economic terms.

In summary, in the arbitrage pricing model the market or non-diversifiable risk in an investment is measured relative to multiple unspecified macro economic factors, with the sensitivity of the investment relative to each factor being measured by a factor beta. The number of factors, the factor betas and factor risk premiums can all be estimated using a factor analysis.

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