Arbitrage Pricing Theory – Part One

Last time out, I promised we would look at the risk premia associated with various hedge fund strategies.  But first I’d like to finish the discussion of pricing models that we began with our look at the Capital Asset Pricing Model.

Arbitrage Pricing Theory (APT) is a multi-factor model conceived by Stephen Ross in 1976. It is a linear equation in which a series of input variables, such as economic indicators and market indices, are each assigned their own betas to determine the expected return of a target asset. These factor-specific betas (b) fine-tune the sensitivity of the target asset’s rate of return to the particular factor.  Here are the equations:

Suppose that asset returns are driven by a few (i) common systematic factors plus non-systematic noise:

ri= E(ri) +bi1 F1 + bi2 F2 +· · ·+bin Fn + εi for i = 1, 2, . . .n  where:

  • E(ri) is the expected return on asset i
  • F1, . . ., Fn are the latest data on common systematic factors driving all asset returns
  • bin gives how sensitive the return on asset i with respect to news on the n-th factor (factor loading)
  • εi is the idiosyncratic noise component in asset i’s return that is unrelated to other asset returns; it has a mean of zero

APT claims that for an arbitrary asset, its expected return depends only on its factor exposure:

If we define RPi = the risk premium on factor i = the return from factor i above the risk-free rate = (rFn−Rf), then the APT equation is:

E(ri) = Rf +bi1 RP1+. . .+bin RPn where

  • RPn is the premium on factor n
  • bin is asset i’s loading of factor n
  • Rf is the risk-free rate

So you can see, the expected return of an asset i is a linear function of the assets’ betas to the n factors. This equation assumes ideal markets and no surplus of the number of factors above the number of assets.

In Part Two, we’ll explore the assumptions and implications of the APT equation.

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Copyright 2011 Eric Bank, Freelance Writer

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