A substitute and concurrent theory to the Capital Asset Pricing Model (CAPM) is one that incorporates multiple factors in explaining the movement of asset prices. The arbitrage pricing model (APT) on the other hand approaches pricing from a different aspect. It is rarely successful to analyse portfolio risks by assessing the weighted sum of its components. Equity portfolios are far more diverse and enormously large for separate component assessment, and the correlation existing between the elements would make a calculation as such untrue. Rather, the portfolio’s risk should be viewed as a single product’s innate risk. The APT represents portfolio risk by a factor model that is linear, where returns are a sum of risk factor returns. Factors may range from macroeconomic to fundamental market indices weighted by sensitivities to changes in each factor. These sensitivities are called factor-specific beta coefficients or more commonly, factor loadings. In addition, the firm-specific or idiosyncratic return is added as a noise factor. This last part, as is the case with all econometric models, is indispensable in explaining whatever the original factors failed to include. In contrast with the CAPM, this is not an equilibrium model; it is not concerned with the efficient portfolio of the investor. Rather, the APT model calculates asset pricing using the different factors and assumes that in the case market pricing deviates from the price suggested by the model, arbitrageurs will make use of the imbalance and veer pricing back to equilibrium levels. At its simplest form, the arbitrage pricing model can have one factor only, the market portfolio factor. This form will give similar results to the Capital Asset Pricing Model (CAPM).

Stephen Ross, who initiated Arbitrage Pricing Theory (APT) in 1976, explained that an asset’s price today should equal the sum of discounted future cash flows, where the expected return of the asset is a linear function of the various factors. It is based on the tenet that in a well functioning security market no arbitrage opportunities should exist.

The core idea of the APT is that only a small number of systematic influences affect the long term average returns of securities. The first ingredient of Ross’s APT is a factor model. Multi-factor models allow an asset to have not just one, but many measures of systematic risk. Each measure captures the sensitivity of the asset to the corresponding pervasive factor. If the factor model holds exactly and assets do not have specific risk, then the law of one price implies that the expected return of any asset is just a linear function of the other assets’ expected return. If this were not the case, arbitrageurs would be able to create a long-short trading strategy that would have no initial cost, but would give positive profits for sure.

According to the above explanation, risky asset return will satisfy the following equation:

E(r

_{j}) = r_{f}+ b_{j1}RP_{1}+ b_{j2}RP_{2}+ b_{j3}RP_{3}+ b_{j4}RP_{4}+ … + b_{jn}RP_{n}

Where E(rj) is the expected return of the asset, RPn the is risk premium of the factor, rf is the risk-free rate and bn is the sensitivity of the asset to factor *n*, also known as factor loading.

Major assumptions of Arbitrage Pricing Theory (APT) are (1) returns can be described by a factor model, (2) there are no arbitrage opportunities, (3) there are a large number of securities so it is possible to form portfolios that diversify the firm-specific risk of individual stocks and (4) the financial markets are friction-less.

Factors may be economic factors (such as interest rates, inflation, GDP) financial factors (market indices, yield curves, exchange rates) fundamentals (like price/earnings ratios, dividend yields), or statistical (e.g. principal component analysis, factor analysis.) The factor model’s beta coefficients i.e. sensitivities may be estimated using cross-sectional regression or time series techniques.

## Relationship between Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT)

The two models approach asset pricing from different aspects. The Arbitrage Pricing Theory (APT) is less restrictive in its assumptions than the Capital Asset Pricing Model (CAPM). It is a rather explanatory model as opposed to statistical. It assumes investors will each hold a portfolio unique to their risk receptiveness with a unique beta, as opposed to the identical market portfolio presumed by the CAPM.

Moreover, the Arbitrage Pricing Theory (APT) presumes an infinite number of investments, which in turn lead to the disappearance of firm-specific risk. It can be viewed as a supply-side model, as its beta coefficients reflect sensitivity of the underlying asset to the different factors. In this sense, factor changes will cause sizable shifts in the asset’s expected returns. On the other hand, the CAPM is a demand-side model. Its results arise from the investors’ utility function maximization problem, and from the resultant market equilibrium. As investors can be considered to be consumers of the asset, the demand approach is reasonable.