Arbitrage Pricing Theory and Factor Analysis
Overview
Arbitrage pricing theory
is a
multi factor model
seeks to explain equity prices in a manner simlar to the
Capital Asset Pricing Model,
which is subsumed under the APT.
APT relies on an assumption of efficient markets, but can be used as a framework for trading, even when this assumption
is relaxed.
Multiple Factors
The Arbitrage Pricing Theory assumes that the asset returns can be written as a multifactor model, that
is the return on asset i is given by.
{% r_i = a_i + \sum b_{ij}f_j + \epsilon_i %}
wher {% a_i %} is some constant and {% \epsilon_i %} is an error term. Under these conditions, the APT states
that there exist numbers {% \lambda_0, \lambda_1 ... \lambda_n %} such that
{% \mathbb{E}[r_i] = \lambda _0 + \sum b_{ij} \lambda_j %}
That is to say, that the constant in each return equation is the same as all the others, and the error
term has an
expected value of zero.
Hand Waving Argument
The arbitrage pricing theory is proven through an arbitrage argument. If the returns can be stated as the
linear equaiton above, then you can construct a porfolio where the factor exposures have been hedged out.
(this can only be done if there are more assets than factors). The epsilons are assumeed to be uncorrelated, and
assuming that the number of assets is large, will also be effectively hedged out.
The resulting portfolio is then a more or less risk free portfolio, and must therefore return the risk free rate.
(see
Luenberger for proof)
Models
