Multi Index Risk

Overview


Multi Index risk models seek to measure the risk associated with an asset by modeling its returns as being driven by multiple factors. This forms the underlying premise of the Arbirage Pricing Theory (APT). The multi-index model is an extension of the single index model.

Index models are examples of Latent Variable and Factor Models.

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.

Linear Algebra Formulation


Multi Index models can be stated using the linear algebra and matrices, which makes the equatoins more succint and easier to implement.
{% \vec{r}_t = \vec{\alpha} + \textbf{B} \vec{f}_t + \vec{\epsilon}_t %}
Here we have

  • {% \vec{r}_t %} is an {% n %} dimensional vector of asset (arithmetic) excess (return minus the risk free rate) returns
  • {% \vec{\alpha} %} is a {% n %} dimensional vector of alphas
  • {% \vec{f}_t %} is a {% m %} dimensional (less than {% n %}) vector of factor returns
  • {% \textbf{B} %} is an {% n %} by {% m %} matrix of portfolio weights of each factor
The asset returns, the factor returns and the epsilons are indexed by time, {% t %}.



There are a typical set of assumptions that are applied to this model.

  • {% \vec{\epsilon}_t %} is independent of {% \vec{f}_t %}
  • {% \mathbb{E}[\vec{\epsilon}_t] = 0 %}

Topics


  • Choosing Factors
  • Estimating Factor Coefficients
  • Standard Form
  • Factor Transformations

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.