Arbitrage Pricing Theory

Arbitrage Pricing Theory (APT) this theory is based on the idea that in competitive markets, arbitrage will ensure that riskless assets provide the same expected return created in 1976 by Stephen Ross, this theory predicts a relationship between the returns of aportfolio and the returns of a single asset through a linear combination of many independent macro-economic variables it isoften viewed asan alternative to the capital asset pricing model (CAPM),since the APThasmore flexible assumption requirements. Whereas the CAPM formula requires the market's expected return, APT uses the risky asset's expected return and the risk premium ofa number ofmacro-economic factors. like the CAPM, APT is an equilibrium model as to how security prices are determined

APT and CAPM APT allows the individual investor more freedom to develop a model that explains the expected return for a particular asset

The expected return on an asset is a function of many factors as well as the sensitivity of the stock to these factors" while in the CAPM theory, the expected return on a stock can be described by the movement of that stock relative to the rest of the stock market

From a practical standpoint, CAPM remains the dominant pricing model used today. When compared to the Arbitrage Pricing Theory, the Capital Asset Pricing Model is both elegant and relatively simple to calculate.

where:

E(rj) = the asset's expected rate of return

rf = the risk-free rate

bj = the sensitivity of the asset's return to the particular factor

RP = the risk premium associated with the particular factorThe APT Formula:E(rj) = rf + bj1RP1 + bj2RP2 + bj3RP3 + bj4RP4 + ... + bjnRPn

APT Factor Models Approach Two-factor Model (actual return on a security)where: a= the return when all factors have zero values fn = the value (uncertain) of factor n bnj = the reaction coefficient depicting the change in the security's return to a one-unit change in the factor ej = the error termRj = a + b1jF1 + b2jF2 + ej

Leeny Kelly Company's stock is related to the following factors with respect to actual return:

Suppose that the a term for the stock is 14 percent and that for the period the unanticipated change in factor 1 is 5 percent, factor 2 is 2 percent, and factor 3 is 10 percent. If the error term is zero, what would be the stock's actual return for the period?Rj = a + .8(F1)+ 1.2(F2) + .3(F3) + ej Rj= a + .8(F1)+ 1.2(F2) + .3(F3) + ej= .14 + .8(.05) + 1.2(.02) + .3(.10) + 0= 3.24%

APT Factor Models Approach Two-factor Model (expected return on a security)where: 0 = corresponds to the return on a risk-free asset 1 = the expected excess return (above the risk-free rate) when: b1j = 1 and b2j = 0()j = 0 + 1b1j + 2b2j

Suppose Torquay Resorts Limited's stock is related to two factors where the reaction coefficients, b1j and b2j are 1.4 and .8, respectively. If the risk-free rate is 8 percent, and 1 is 6 percent and 2 is 2 percent, the stock's expected return is:()j = 0 + 1b1j + 2b2j ()j = .08 + .06(1.4) - .02(.8) = 14.8%The first factor reflects risk aversion and must be compensated for with a higher expected return, whereas the second is a thing of value to investors and lowers the return they expect. Thus, the 's represent market prices associated with factor risks.

APT Factor Models Approach More than two factors Model (expected return)

()j = A + 1 b1j + 2b2j + + nbnjwhere: 1 = represents the expected return in excess of the risk-free rate when the reaction coefficient for the first factor, b1j = is 1.0

Suppose returns required in the market by investors are a function of two factors according to the following equation, where the risk-free rate is 7 percent. Quigley Manufacturing Company and Zolotny Basic Products Corporation both have the same reaction coefficients to the factors, such that b1j = 1.3 and b2j = .9()j= A + 1 b1j + 2b2j + + nbnj()j= .07 + .04(b1j) - .01(b2j)= .07 + .04(1.3) - .01(9)= 11.3%

Roll-Ross Five Factorschanges in expected inflation

unanticipated changes in inflation

Unanticipated changes in industrial production

unanticipated changes in the yield differential between low and high-grade bonds (the default-risk premium)

unanticipated changes in the yield differential between long-term and short-term bonds (the term structure of interest rates)

Roll-Ross may be expressed as:

()j= 0 + 1(b1jEA inflation) + 2(b2jUA inflation)+ 3 (b3jUA industrial production)+ 4 (b4jUA bond risk premium)+ 5 (b5jUA long minus short rate)

where: EA = is an expected changeUA = represents an unanticipated change

()j= 0 + 1(b1jEA inflation) + 2(b2jUA inflation)+ 3 (b3jUA industrial production)+ 4 (b4jUA bond risk premium)+ 5 (b5jUA long minus short rate)()j= .00412 - .00013(b1jEA inflation) - .00063(b2jUA inflation)+ .01359(b3jUA industrial production)+ .00721(b4jUA bond risk premium)- .00521(b5jUA long minus short rate)Suppose the b's for CRR Corporation are:b1= 1.8b2 = 2.4b3 = .9b4= .5b5= 1.1

Under these conditions, the expected return for the stock is()crr= .00412 - .00013(1.8) - .00063(2.4) + .01359(.9)+ .00721(.5) - .00521(1.1)= 1.25%

**The APT and CAPM are the two most influential theories on stock and asset pricing today. The APT model is different from the CAPM in that it is far less restrictive in its assumptions.the APT makes a lot of sense because it removes the CAPM restrictions and basically states**The error term is security specific or unsystematic**In contrast to the actual return Parameters represent risk premiums for the types of risk associated with particular factorsThe parameters can be positive or negative

**where the number of factors is n and each A represents a market price of risk and the b's for all other factors are zeroThe equation tells us that a security's expected return is the risk-free rate, plus risk premiums for each of the n factors.**Roll and Ross believe the truth lies in five specific factors.They suggest that different securities have different sensitivities to these systematic factors and that the major sources of security portfolio risk are captured in them***