For a set of assets with returns , factor loadings , and factors , a general factor model that is used in APT is:where follows a multivariate normal distribution. In general, it is useful to assume that the factors are distributed as:where is the expected risk premium vector and is the factor covariance matrix. Assuming that the noise terms for the returns and factors are uncorrelated, the mean and covariance for the returns are respectively:It is generally assumed that we know the factors in a model, which allows least squares to be utilized. However, an alternative to this is to assume that the factors are latent variables and employ factor analysis - akin to the form used in psychometrics - to extract them. Assumptions of APT Model The APT model for asset valuation is founded on the following assumptions: Investors are risk-averse in nature and possess the same expectations Efficient markets with limited opportunity for arbitrage Perfect capital markets Infinite number of assets Risk factors are indicative of systematic risks that cannot be diversified away and thus impact all financial assets, to some degree. Thus, these factors must be: Non-specific to any individual firm or industry Compensated by the market via a risk premium A random variable Arbitrage Arbitrage is the practice whereby investors take advantage of slight variations in asset valuation from its fair price, to generate a profit. It is the realisation of a positive expected return from overvalued or undervalued securities in the inefficient market without any incremental risk and zero additional investments. Mechanics In the APT context, arbitrage consists of trading in two assets – with at least one being mispriced. The arbitrageur sells the asset which is relatively too expensive and uses the proceeds to buy one which is relatively too cheap. Under the APT, an asset is mispriced if its current price diverges from the price predicted by the model. The asset price today should equal the sum of all future cash flows discounted at the APT rate, where the expected return of the asset is a linear function of various factors, and sensitivity to changes in each factor is represented by a factor-specific beta coefficient. A correctly priced asset here may be in fact a synthetic asset - a portfolio consisting of other correctly priced assets. This portfolio has the same exposure to each of the macroeconomic factors as the mispriced asset. The