Single factor model of financial price data
Consider a data matrix which contains the log-returns of assets over time periods (say, days).
A single-factor model for this data is one based on the assumption that the matrix is a dyad:
where , and . In practice, no component of and is zero (if that is not the case, then a whole row or column of is zero and can be ignored in the analysis).
According to the single factor model, the entire market behaves as follows. At any time , the log-return of asset is of the form .
The vectors and have the following interpretation.
- For any asset, the rate of change in log-returns between two-time instants is given by the ratio , independent of the asset. Hence, gives the time profile for all the assets: every asset shows the same time profile, up to a scaling given by .
- Likewise, for any time , the ratio between the log-returns of two assets and at time is given by , independent of . Hence gives the asset profile for all the time periods. Each time shows the same asset profile, up to a scaling given by .
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.