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).

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

    \[A = uv^T,\]

where v \in \mathbb{R}^T, and u \in \mathbb{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 \leq t \leq T, the log-return of asset i \; (1 \leq i \leq m) is of the form A_{it} = u_iv_t.

The vectors u and v have the following interpretation.

  • For any asset, the rate of change in log-returns between two-time instants t_1 \leq 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_jindependent 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.

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