[latexpage]

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:

$$A = uv^T,$$

where $v \in \mathb{R}^T$, and $u \in \mathb{R}^m$. 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 $v$ has 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 $u_i/u_j$, 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  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.