Single factor model of financial price data

Consider a m times T data matrix which contains the log-returns of m assets over T time periods (say, days).

A single-factor model for this data is one based on the assumption that the matrix is a dyad:

 A = u v^T,

where v in mathbf{R}^T, and u in mathbf{R}^m. In practice, no component of u and v is zero (if that is not the case, then a whole row or column of A is zero, and can be ignored in the analysis).

According to the single factor model, the entire market behaves as follows. At any time t (1 le t le T), the log-return of asset i (1 le i le m) is of the form

 A_{it} = u_i v_t.

The vectors u and v has the following interpretation.

  • For any asset, the rate of change in log-returns between two time instants t_1le t_2 is given by the ratio v_{t_2}/v_{t_1}, independent of the asset. Hence, v gives the time profile for all the assets: every asset shows the same time profile, up to a scaling given by u.

  • Likewise, for any time t, the ratio between the log-returns of two assets i and j at time t is given by u_i/u_j, independent of t. Hence u gives the asset profile for all the time periods. Each time shows the same asset profile, up to a scaling given by v.

While single-factor models may seem crude, they often offer a reasonable amount of information. It turns out that with many financial market data, a good single factor model involves a time profile v equal to the log-returns of the average of all the assets, or some weighted average (such as the SP 500 index). With this model, all assets follow the profile of the entire market.