Single factor model of financial price data

L. El Ghaoui

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:

$A = uv^{T}$ ,

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 $t (1 \le t \le T)$ , 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 $t_{1}\le t_{2}$ 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.

Single factor model of financial price data

License

Share This Book