) is a generaltheoryofasset pricingthat holds that theexpected returnof 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 factor-specificbeta coefficient. The model-derived rate of return will then be used to price the asset correctlythe asset price should equal the expected end of period pricediscountedat the rate implied by the model. If the price diverges,arbitrageshould bring it back into line. The theory was proposed by thein 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.

Risky asset returns are said to follow afactor intensity structureif they can be expressed as:

\displaystyle r_j=a_j+\lambda _j1f_1+\lambda _j2f_2+\cdots +\lambda _jnf_n+\epsilon _j

is the risky assets 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:

That is, the expected return of an assetjis alinearfunction of the assets sensitivities to thenfactors.

Note that there are some assumptions and requirements that have to be fulfilled for the latter to be correct: There must beperfect competitionin the market, and the total number of factors may never surpass the total number of assets (in order to avoid the problem ofmatrix singularity).

Arbitrageis the practice of taking positive expected return from overvalued or undervalued securities in the inefficient market without any incremental risk and zero additional investments.

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 flowsdiscountedat 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-specificbeta coefficient.

A correctly priced asset here may be in fact asyntheticasset – aportfolioconsisting 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 identifyingncorrectly 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 islongthe asset andshortthe 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:

The implication is that at the end of the period the

would have appreciated at the rate implied by the APT, whereas the mispriced asset would have appreciated at

than this rate. The arbitrageur could therefore:

The implication is that at the end of the period the

would have appreciated at the rate implied by the APT, whereas the mispriced asset would have appreciated at

than this rate. The arbitrageur could therefore:

2 use the proceeds to buy back the mispriced asset

Relationship with the capital asset pricing model

The APT along with thecapital 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 thesecurities market linerepresents a single-factor 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 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, thecapital asset pricing modelis considered a demand side model. Its results, although similar to those of the APT, arise from a maximization problem of each investors utility function, and from the resulting market equilibrium (investors are considered to be the consumers of the assets).

As with the CAPM, the factor-specific betas are found via alinear regressionof 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 essentiallyempiricalin nature. Severala prioriguidelines as to the characteristics required of potential factors are, however, suggested:

their impact on asset prices manifests in their

influences (these are, clearly, more likely to be macroeconomic rather than firm-specific in nature)

timely and accurate information on these variables is required

the relationship should be theoretically justifiable on economic grounds

Chen,RollandRoss(1986) identified the following macro-economic factors as significant in explaining security returns:

surprises inGNPas indicated by an industrial production index;

surprises in investor confidence due to changes in default premium in corporate bonds;

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 offactor analysis. More direct indices that might be used are:

the difference in long-term and short-term interest rates;

a diversified stock index such as theS&P 500orNYSE Composite;

Fundamental theorem of arbitrage-free pricing

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The Arbitrage Pricing TheoryProf. William N. Goetzmann,Yale School of Management

The Arbitrage Pricing Theory Approach to Strategic Portfolio PlanningPDF), Richard Roll andStephen A. Ross

The APT, Prof. Tyler Shumway,University of Michigan Business School

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.

Capital asset pricing modelalphabetasecurity characteristic line)