Arbitrage Pricing Theory (APT)

The APT is the general model that subdivides systematic risk into smaller components, which need to specific in advance. These define arbitrage pricing plan. APT states that the expected risk premium on a stock should depend on the expected risk premium associated with each factor and the stock’s sensitivity to each of the factors (b1, b2, b3, etc.). For any individual stock there are two sources of risk.

•Systematic risk that stems from the pervasive macroeconomic factors which cannot be eliminated by diversification. Any macro-economic factors including market sentiment, which impact upon investors returns may be incorporate into the APP, such as interest rates, inflation, investor risk attitudes, industrial productivity etc.

•risk arising from possible events that are unique to the company.

The return equation for a n-factor APP conforms to the following simple linear relationship for the expected return on the j-th security in portfolio:

rj=a+b1*r1+b2*r2+b3*r3+b4*r4+….+bn*rn+εpt, where

•rj-expected rate of return on security j,

•ri-expected return on factor i, (i=1,2,3,4) or

•a-constant for security j,

•bj- is the sensitivity of the j-th asset to factor, also called factor loading or slope of rj

The expected risk premium on the j-th security is defined as the difference between its expected return rj and the risk free rate of interest rf associated with each factor’s return ri and the security’s sensitivity to each of these factors (bi). The n-factor equation is given by (rj- rf)=b1(r1- rf)+b2(r2- rf)+b3(r3- rf)+b4(r4- rf) +…. +bn*(rn - rf), - is expected risk premium.

The general APT is still a linear model. Theoretically it assumes that unsystematic (unique) risk can be eliminated in a well-diversified portfolio, leaving only the portfolio’s sensitivity to unexpected changes in macro –economic factors. Diversification does eliminate unique risk, and diversified investors can therefore ignore it when deciding whether to buy or sell a stock. The expected risk premium on a stock is affected by factor or macroeconomic risk; it is not affected by unique risk.

1)If you plug in a value of zero for each of the b’s in the formula, the expected risk premium is zero. A diversified portfolio that is constructed to have zero sensitivity to each macroeconomic factor is essentially risk free and therefore must be priced to offer the risk-free rate of interest. If the portfolio offered a higher return, investors could make a risk-free (or “arbitrage”) profit by borrowing to buy the portfolio. If it offered a lower return, you could make an arbitrage profit by running the strategy in reverse; in other words, you would sell the diversified zero-sensitivity portfolio and invest the proceeds in U.S. Treasury bills.

2)A diversified portfolio that is constructed to have exposure to, say, factor 1, will offer a risk premium, which will vary in direct proportion to the portfolio’s sensitivity to that factor. For example, imagine that you construct two portfolios, A and B, which are affected only by factor 1. If portfolio A is twice as sensitive to factor 1 as portfolio B, portfolio A must offer twice the risk premium. Therefore, if you divided your money equally between U.S. Treasury bills and portfolio A, your combined portfolio would have exactly the same sensitivity to factor 1 as portfolio B and would offer the same risk premium. Suppose that the arbitrage pricing formula did not hold. For example, suppose that the combination of Treasury bills and portfolio A offered a higher return. In that case investors could make an arbitrage profit by selling portfolio B and investing the proceeds in the mixture of bills and portfolio A.