Arbitrage pricing theory
In finance, arbitrage pricing theory (APT) is a multi-factor model for asset pricing which relates various macro-economic (systematic) risk variables to the pricing of financial assets. Proposed by economist Stephen Ross in 1976, it is widely believed to be an improved alternative to its predecessor, the Capital Asset Pricing Model (CAPM). APT is founded upon the law of one price, which suggests that within an equilibrium market, rational investors will implement arbitrage such that the equilibrium price is eventually realised. As such, APT argues that when opportunities for arbitrage are exhausted in a given period , then the expected return of an asset is a linear function of various factors or theoretical market indices, where sensitivities of each factor is represented by a factor-specific beta coefficient or factor loading. Consequently, it provides traders with an indication of ‘true’ asset value and enables exploitation of market discrepancies via arbitrage. The linear factor model structure of the APT is used as the basis for evaluating asset allocation, the performance of managed funds as well as the calculation of cost of capital. Model. APT is a single-period static model, which helps investors understand the trade-off between  risk and return. The average investor aims to optimise the returns for any given level or risk and as such, expects a positive return for bearing greater risk. As per the APT model, risky asset returns are said to follow a "factor intensity structure" if they can be expressed as: Idiosyncratic shocks are assumed to be uncorrelated across assets and uncorrelated with the factors. The APT model states that if asset returns follow a factor structure then the following relation exists between expected returns and the factor sensitivities: That is, the expected return of an asset "j" is a linear  function of the asset's sensitivities to the "n" factors. Note that there are some assumptions and requirements that have to be fulfilled for the latter to be correct: There must be perfect competition in the market, and the total number of factors may never surpass the total number of assets (in order to avoid the problem of matrix singularity). General Model. For a set of assets with returns formula_12, factor loadings formula_13, and factors formula_14, a general factor model that is used in APT is:formula_15where formula_16 follows a multivariate normal distribution. In general, it is useful to assume  that the factors are distributed as:formula_17where formula_18 is the expected risk premium vector and formula_19 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:formula_20It 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: 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 arbitrageur creates the  portfolio by identifying n correctly priced assets (one per risk-factor, plus one) and then weighting the assets such that portfolio beta per factor is the same as for the mispriced asset. When the investor is long the asset and short the portfolio (or vice versa) he has created a position which has a positive expected return (the difference between asset return and portfolio return) and which has a net zero exposure to any macroeconomic factor and is therefore risk free (other than for firm specific risk). The arbitrageur is thus in a position to make a risk-free profit : Difference between the capital asset pricing model. The APT along with the capital asset pricing model (CAPM) is one of two influential theories on asset pricing. The APT differs from the CAPM in that it is less restrictive in its assumptions, making it more flexible for use in a wider range of application. Thus, it possesses greator explanatory power (as opposed to statistical) for expected asset returns. It assumes that each investor will hold a unique portfolio with its own particular array of betas, as opposed to the identical "market portfolio". In some ways, the CAPM can be considered a  special case of the APT in that the securities market line represents a single-factor model of the asset price, where beta is exposed to changes in value of the market. Fundamentally, the CAPM is derived on the premise that all factors in the economy can be reconciled into one factor represented by a market portfolio, thus implying they all have equivalent weight on the asset’s return. In contrast, the APT model suggests that each stock reacts uniquely to various macroeconomic factors and thus the impact of each must be accounted for separately. A disadvantage of APT is that  the selection and the number of factors to use in the model is ambiguous. Most academics use three to five factors to model returns, but the factors selected have not been empirically robust. In many instances the CAPM, as a model to estimate expected returns, has empirically outperformed the more advanced APT. Additionally, the APT can be seen as a "supply-side" model, since its beta coefficients reflect the sensitivity of the underlying asset to economic factors. Thus, factor shocks would cause structural changes in assets' expected returns, or in the case of stocks, in firms' profitabilities. On the other  side, the capital asset pricing model is considered a "demand side" model. Its results, although similar to those of the APT, arise from a maximization problem of each investor's utility function, and from the resulting market equilibrium (investors are considered to be the "consumers" of the assets). Implementation. As with the CAPM, the factor-specific betas are found via a linear regression of historical security returns on the factor in question. Unlike the CAPM, the APT, however, does not itself reveal the identity of its priced factors - the number and nature of these factors is likely to change over  time and between economies. As a result, this issue is essentially empirical in nature. Several "a priori" guidelines as to the characteristics required of potential factors are, however, suggested: Chen, Roll and Ross identified the following macro-economic factors as significant in explaining security returns: As a practical matter, indices or spot or futures market prices may be used in place of macro-economic factors, which are reported at low frequency (e.g. monthly) and often with significant estimation errors. Market indices are sometimes derived by means of factor analysis. More direct "indices" that might be used are: Arbitrage pricing theory