# Arbitrage pricing theory(APT)

The Arbitrage pricing theory introduced by Ross provides another model for explaining the relationship between return and risk. The theory relates the expected return of an assets to the return from the risk-free assets and a series of other common factors that systematically enhance or detract from the expected return.
Assumption of APT
(i)Investors seek return tempered by risk, they are risk-averse and seek to maximize their terminal wealth.
(ii)An investor can borrow and lend at the risk-free rate.
(iii)There are no market frictions such as transaction costs, taxes or restrictions on short-selling.
(iv) Investors agree on the number and identity of the factors that are important systematically in pricing assets.
(v)There are no riskless arbitrage profit opportunities.
The first three assumptions describe investor behavior in general but fail to describe the factors on which decisions are made. The APT does not make any assumption about the distribution of the return from assets. It does not require that investors make decisions on the basis of the mean and variance second assumption that the risk-free rate is the minimum that an investor would accept for making an investment is a logical part of any assets pricing model.
Driving the arbitrage pricing theory
In the derivation of the APT equilibrium model, Ross does not assume risk aversion and in particular, does not assume that investor makes their decision in the mean-variance framework. Instead, he assumes that the securities rate of return is generated by the following process.
$R_{i}=ER_{i}+\beta&space;_{i}(V-EV)+e_{i}$
Here
$R_{i}$=the rate of return on security i(1=1,2,_______n, where we have securities) with mean $ER_{i}$ V=the value of the factor generating the security returns, where  mean is $E_{i}$ $\beta&space;i$= a coefficient measuring the effect of changes in factor on the rate of return $R_{i}$
$e_{i}$=a random deviation
Note that V is a common factor to all securities, this may be the GNP the stock index or any other factor which one perceives to be appropriate for the generation of securities rate of return.
Practical application of APT
1)APT is an interesting alternative to the CAPM many of the same problems that were discovered during the testing and implementation of the CAPM.
2)An initial empirical test by Roll and Ross(RR).They estimated first the factor betas for securities and their the cross-sectional relationship between security betas and an average rate of return.
3)Roll and Ross estimated the factor betas using a statistical technique called factor analysis. The input to factor analysis is the covariance matrix among the return to the securities in the sample factor analysis determined the set of factor betas that best explain the covariance among the securities in the sample factor analysis determines the set of factor betas the best explain the covariance among the securities in the sample.
4)Factor analysis makes the working assumption that the individual factor variables are equal 100 and then finds that set of factor betas for each stock that will make the covariance among the stocks correspond as closely as possible to the sample covariance as computed directly for the returns.
5)The program continues to add additional factor until the probability that the next portfolio explain a significant fraction of the covariance between stocks goes below some predetermined level after estimates of the factor betas are obtained, the next step is to estimate the value of the factor price associate with each factor.