In mathematical finance, multiple factor models are asset pricing models that can be used to estimate the discount rate for the valuation of financial assets; they may in turn be used to manage portfolio risk.
They are generally extensions of the single-factor capital asset pricing model (CAPM).
The multifactor equity risk model was first developed by Barr Rosenberg and Vinay Marathe.
are risk exposure values calculated from fundamental and technical data,
are factor returns determined by a cross-sectional regression for each time period and
For instance the model might be fit over the 3000 highest capitalization US common stocks.
is then used for Markowitz portfolio construction which involves maximizing the quadratic utility function subject to linear constraints on the vector of asset holdings
Nicolo G. Torre made a number of improvements to this framework which importantly sharpened the risk control achievable by these means.
Torre modified this scheme by introducing an explicit market factor (with unit exposure for each asset.)
To keep the model identified by imposed the condition that the industry factor returns sum to zero in each time period.
Explicitly identifying the market factor then permitted Torre to estimate the variance of this factor using a leveraged GARCH(1,1) model due to Robert Engle and Tim Bollerslev Here and w, a, b1 and b2 are parameters fit from long time series estimations using maximum likelihood methods.
In particular it accounts for the convergence of asset returns and consequent loss of diversification that occurs in portfolios during periods of market turbulence.
In the risk model industry factors carry about half the explanatory power after the market effect is accounted for.
Difficulties can be reduced by introducing a large number of narrowly defined industries, but this approach is in tension with the demands of risk estimation.
Torre resolved this problem by introducing several hundred narrowly defined mini-industries and then applying guided clustering techniques to combine the mini-industries into industry groupings suitable for risk estimation.
In the initial Rosenberg approach factor and specific returns are assumed to be normally distributed.
However experience turns up a number of outlying observations that are both too large and too frequent to be fit by a normal distribution Although introduction of a GARCH market factor partly reduces this difficulty, it does not eliminate it.
In the case of a single factor the mixing model is easily stated.
As such it makes possible construction of portfolios which behave in more expected manners during periods of market turbulence.
The problem of how to construct a multi-asset class risk model then arises.
A first approach was made by Beckers, Rudd and Stefek for the global equity market.
Torre resolved these difficulties by introducing a two-stage factor analysis.
This is particularly relevant for global equity portfolios and for enterprise wide risk management.
Many academics have attempted to construct factor models with a fairly small number of parameters.