Arbitrage pricing theory
In finance, arbitrage pricing theory (APT) is a general theory of asset pricing that holds that the expected return of a financial asset can be modeled as a linear function of various factors or theoretical market indices, where sensitivity to changes in each factor is represented by a factorspecific beta coefficient. The modelderived rate of return will then be used to price the asset correctly—the asset price should equal the expected end of period price discounted at the rate implied by the model. If the price diverges, arbitrage should bring it back into line. The theory was proposed by the economist Stephen Ross in 1976. The linear factor model structure of the APT is used as the basis for many of the commercial risk systems employed by asset managers.
Model
Risky asset returns are said to follow a factor intensity structure if they can be expressed as:
 where
 is a constant for asset
 is a systematic factor
 is the sensitivity of the th asset to factor , also called factor loading,
 and is the risky asset's idiosyncratic random shock with mean zero.
Idiosyncratic shocks are assumed to be uncorrelated across assets and uncorrelated with the factors.
The APT states that if asset returns follow a factor structure then the following relation exists between expected returns and the factor sensitivities:
 where
 is the risk premium of the factor,
 is the riskfree rate,
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 , 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.
Arbitrage
Arbitrage is the practice of taking 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 factorspecific 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 riskfactor, 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 netzero 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 riskfree profit:
Where today's price is too low:

Where today's price is too high:

Relationship with 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. It allows for an explanatory (as opposed to statistical) model of 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 singlefactor model of the asset price, where beta is exposed to changes in value of the market.
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.[1]
Additionally, the APT can be seen as a "supplyside" 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 factorspecific 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:
 their impact on asset prices manifests in their unexpected movements
 they should represent undiversifiable influences (these are, clearly, more likely to be macroeconomic rather than firmspecific in nature)
 timely and accurate information on these variables is required
 the relationship should be theoretically justifiable on economic grounds
Chen, Roll and Ross (1986) identified the following macroeconomic factors as significant in explaining security returns:
 surprises in inflation;
 surprises in GNP as indicated by an industrial production index;
 surprises in investor confidence due to changes in default premium in corporate bonds;
 surprise shifts in the yield curve.
As a practical matter, indices or spot or futures market prices may be used in place of macroeconomic 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:
 shortterm interest rates;
 the difference in longterm and shortterm interest rates;
 a diversified stock index such as the S&P 500 or NYSE Composite;
 oil prices
 gold or other precious metal prices
 Currency exchange rates
See also
 Beta coefficient
 Capital asset pricing model
 Carhart fourfactor model
 Cost of capital
 Earnings response coefficient
 Efficientmarket hypothesis
 Fundamental theorem of arbitragefree pricing
 Investment theory
 Roll's critique
 Rational pricing
 Modern portfolio theory
 Postmodern portfolio theory
 Value investing
 Fama–French threefactor model
 Risk factor (finance)
References
 French, Jordan (1 March 2017). "Macroeconomic Forces and Arbitrage Pricing Theory". Journal of Comparative Asian Development. 16 (1): 1–20. doi:10.1080/15339114.2017.1297245.
 Burmeister, Edwin; Wall, Kent D. (1986). "The arbitrage pricing theory and macroeconomic factor measures". Financial Review. 21 (1): 1–20. doi:10.1111/j.15406288.1986.tb01103.x.
 Chen, N. F.; Ingersoll, E. (1983). "Exact Pricing in Linear Factor Models with Finitely Many Assets: A Note". Journal of Finance. 38 (3): 985–988. doi:10.2307/2328092. JSTOR 2328092.
 Roll, Richard; Ross, Stephen (1980). "An empirical investigation of the arbitrage pricing theory". Journal of Finance. 35 (5): 1073–1103. doi:10.2307/2327087. JSTOR 2327087.
 Ross, Stephen (1976). "The arbitrage theory of capital asset pricing". Journal of Economic Theory. 13 (3): 341–360. doi:10.1016/00220531(76)900466.
 Chen, NaiFu; Roll, Richard; Ross, Stephen (1986). "Economic Forces and the Stock Market" (PDF). Journal of Business. 59 (3): 383–403. doi:10.1086/296344. Archived from the original (PDF) on 20090320. Retrieved 20081201.
External links
 The Arbitrage Pricing Theory Prof. William N. Goetzmann, Yale School of Management
 The Arbitrage Pricing Theory Approach to Strategic Portfolio Planning (PDF), Richard Roll and Stephen A. Ross
 The APT, Prof. Tyler Shumway, University of Michigan Business School
 The arbitrage pricing theory Investment Analysts Society of South Africa
 References on the Arbitrage Pricing Theory, Prof. Robert A. Korajczyk, Kellogg School of Management
 Chapter 12: Arbitrage Pricing Theory (APT), Prof. Jiang Wang, Massachusetts Institute of Technology.