Multi Index Risk
Overview
Multi Index risk models seek to measure the risk associated with a stock by modeling its returns as being
driven by multiple factors. This forms the underlying premise of the
Arbirage Pricing Theory (APT).
Multi Factor Model
{% r = \alpha + \sum_i \beta_i \times f_i + e %}
r is the rate of return of a given asset and {% f_i %} is a factor, usually itself a return.
Often times the factors are chosen to be the returns on various industry indexes or style indexes
(such as value or growth)
In this equation, it is usually assumed that the error terms, e are uncorrelated with each other.
Factors
Calculating Portfolio Risk
To calculate portfolio variance, we need to calculate the
asset covariance matrix. In the multi index model we
have
{% \Sigma = B \Sigma_I B^T + S %}
where {% \Sigma %} is the asset covariance,
{% \Sigma_I %} is the covariance matrix of Index returns,
{% B %} is a matrix of
asset betas, and {% S %} is the diagonal matrix of idiosyncratic
risks.
{% \Sigma_I %} is a measured covariance matrix, using standard
techniques.
{% B %} is a matrix of betas, that is
{% B = [\textbf{b}_1, \textbf{b}_2, ..., \textbf{b}_k] %}
The vector {% \textbf{b}_k %} is each stock beta (exposure) to the
kth index, or factor.
