Arbitrage Pricing 1077 assets'returns. And thus, portfolios of the first k+ 1 assets are perfect substitutes for all other assets in the market. Since perfect substitutes must be priced equally, there must be restrictions on the individual returns generated by the model. This is the core of the APt: there are only a few systematic components of risk existing in nature. As a consequence, many portfolios are close substitutes and as such, they must have the same value What are the common or systematic factors? This question is equivalent to asking what causes the particular values of covariance terms in the CAPM. If there are only a few systematic components of risk, one would expect these to be related to fundamental economic aggregates, such as GNP, or to interest rates or weather(although no causality is implied by such relations). The factor model formalism suggests that a whole theoretical and empirical structure must be explored to better understand what economic forces actually affect returns systematically. But in testing the APT, it is no more appropriate for us to examine this issue than it would be for tests of the CaPm to examine what, if anything, causes returns to be multivariate normal. In both instances, the return generating process is taken as one of the primitive assumptions of the theory. We do consider the basic underlying causes of the generating process of returns to be a potentially important area of research, but we think it is an area that can be investigated separately from testing asset pricing theories Now let us develop the APT itself from the return generating pr Consider an individual who is currently holding a portfolio and is contemplating n alteration of his portfolio. Any new portfolio will differ from the old portfolio by investment proportions x(i=1 ) which is the dollar amount purchased or sold of asset i as a fraction of total invested wealth. The sum of the x. proportions, since the new portfolio and the old portfolio put the same wealth into the n assets. In other words, additional purchases of assets must be financed by sales of others Portfolios that use no wealth such as x =(xy xn) are called In deciding whether or not to alter his current holdings, an individual will examine all the available arbitrage portfolios. The additional return obtainable from altering the current portfolio by n is given by x=∑x (∑xE)+(∑xb1)61 (∑xbA)+∑x Consider the arbitrage portfolio chosen in the following fashion. First, we will eep each element, x, of order 1/n in size; i.e. we will choose the arbitrage fied. Second that An underscored symbol indicates a vector or matrix