## Multifactor Models for risk and return

The arbitrage pricing model's failure to identify specifically the factors in the model may be a strength from a statistical standpoint, but it is a clear weakness from an intuitive standpoint. The solution seems simple: Replace the unidentified statistical factors with specified economic factors, and the resultant model should be intuitive while still retaining much of the strength of the arbitrage pricing model. That is precisely what multi-factor models do.

### Deriving a Multi-Factor Model

Multi-factor models generally are not based on extensive economic rationale but are determined by the data. Once the number of factors has been identified in the arbitrage pricing model, the behavior of the factors over time can be extracted from the data. These factor time series can then be compared to the time series of macroeconomic

Unanticipated Inflation: This is the difference between actual inflation and expected inflation.

Arbitrage: An investment opportunity with no risk that earns a return higher than the riskless rate.

variables to see if any of the variables are correlated, over time, with the identified factors.

For instance, a study from the 1980s suggested that the following macroeconomic variables were highly correlated with the factors that come out of factor analysis: industrial production, changes in the premium paid on corporate bonds over the riskless rate, shifts in the term structure, unanticipated inflation, and changes in the real rate of return.iC) These variables can then be correlated with returns to come up with a model of expected returns, with firm-specific betas calculated relative to each variable. The equation for expected returns will take the following form: E(R) = Rf + Pgnp (E(RGNP)-Rf) + Pi (E(Ri)-Rf) ...+ Ps (E(R6)-Rf) where

Pgnp = Beta relative to changes in industrial production

E(Rgnp) = Expected return on a portfolio with a beta of one on the industrial production factor, and zero on all other factors Pi = Beta relative to changes in inflation

E(Ri) = Expected return on a portfolio with a beta of one on the inflation factor, and zero on all other factors The costs of going from the arbitrage pricing model to a macroeconomic multi-factor model can be traced directly to the errors that can be made in identifying the factors. The economic factors in the model can change over time, as will the risk premium associated with each one. For instance, oil price changes were a significant economic factor driving expected returns in the 1970s but are not as significant in other time periods. Using the wrong factor(s) or missing a significant factor in a multi-factor model can lead to inferior estimates of cost of equity.

In summary, multi factor models, like the arbitrage pricing model, assume that market risk can be captured best using multiple macro economic factors and estimating betas relative to each. Unlike the arbitrage pricing model, multi factor models do attempt to identify the macro economic factors that drive market risk.

10 Chen, N., R. Roll and S.A. Ross, 1986, Economic Forces and the Stock Market, Journal of Business, 1986, v59, 383-404.

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