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. Furthermore, the newer APT model is more dynamic being utilised in more theoretical application than the preceding CAPM model. A 1986 article written by Gregory Connor and Robert Korajczyk, utilised the APT framework and applied it to portfolio performance measurement suggesting that the Jensen coefficient is an acceptable measurement of portfolio performance.

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:

:<math>r_j = a_j + \beta_{j1}f_1 + \beta_{j2}f_2 + \cdots + \beta_{jn}f_n + \epsilon_j</math>

:where

:*<math>a_j</math> is a constant for asset <math>j</math>.

:*<math>f_i</math> is a systematic factor for 1≤i≤n.

:*<math>\beta_{ji}</math> is the sensitivity of the <math>j</math>th asset to factor <math>f_i</math>, 1≤i≤n, also called factor loading.

:* and <math>\epsilon_j</math> 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 model states that if asset returns follow a factor structure then the following relation exists between expected returns and the factor sensitivities:

:<math>\mathbb{E}\left(r_j\right) = r_f + \beta_{j1}RP_1 + \beta_{j2}RP_2 + \cdots + \beta_{jn}RP_n</math>

:where

:* <math>RP_i</math> is the risk premium of factor <math>f_i</math>.

:* <math>r_f</math> is the risk-free 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 <math>r\in\mathbb{R}^{m}</math>, factor loadings <math>\Lambda \in\mathbb{R}^{m\times n}</math>, and factors <math>f\in\mathbb{R}^{n}</math>, a general factor model that is used in APT is:<math display="block">r = r_{f} + \Lambda f + \epsilon, \quad \epsilon \sim \mathcal{N}(0,\Psi)</math>where <math>\epsilon</math> follows a multivariate normal distribution. In general, it is useful to assume that the factors are distributed as:<math display="block">f \sim \mathcal{N}(\mu,\Omega)</math>where <math>\mu</math> is the expected risk premium vector and <math>\Omega</math> 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:<math display="block">\mathbb{E}(r) = r_{f} + \Lambda \mu, \quad \text{Cov}(r) = \Lambda\Omega\Lambda^{T} + \Psi</math>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:

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:

their impact on asset prices manifests in their unexpected movements and they are completely unpredictable to the market at the beginning of each period

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 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:

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

International arbitrage pricing theory

International arbitrage pricing theory (IAPT) is an important extension of the base idea of arbitrage pricing theory which further considers factors such as exchange rate risk. In 1983 Bruno Solnik created an extension of the original arbitrage pricing theory to include risk related to international exchange rates hence making the model applicable international markets with multi-currency transactions. Solnik suggested that there may be several factors common to all international assets, and conversely, there may be other common factors applicable to certain markets based on nationality.

Fama and French originally proposed a three-factor model in 1995 which, consistent with the suggestion from Solnik above suggests that integrated international markets may experience a common set of factors, hence making it possible to price assets in all integrated markets using their model. The Fama and French three factor model attempts to explain stock returns based on market risk, size, and value.

A 2012 paper aimed to empirically investigate Solnik’s IAPT model and the suggestion that base currency fluctuations have a direct and comprehendible effect on the risk premiums of assets. This was tested by generating a returns relation which broke down individual investor returns into currency and non-currency (universal) returns. The paper utilised Fama and French’s three factor model (explained above) to estimate international currency impacts on common factors. It was concluded that the total foreign exchange risk in international markets consisted of the immediate exchange rate risk and the residual market factors. This, along with empirical data tests validates the idea that foreign currency fluctuations have a direct effect on risk premiums and the factor loadings included in the APT model, hence, confirming the validity of the IAPT model.

References

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.