Since we can have a wide variety of investment styles, single-factor models can yield biased estimates of performance.
For a conditional single-factor model, the regression becomes:
Combining equations (4) and (5) gives a conditional single-factor model with time-varying alphas and betas:
Studies examining the structure of problem behavior by gender have found that single-factor models describe a significant portion of the variance for both male and female participants, but the factor loadings for specific problem behaviors appear to differ by gender (Donovan & Jessor, 1985; Donovan et al.
The single-factor model explained 54% of the variance in these measures, a value that is somewhat higher than values found when first-order, single-factor models are tested with general samples of American youths (Donovan & Jessor, 1985; Donovan et al.
Findings indicated that a single-factor model provided a close fit for these data and compared favorably with three competing two-factor models.
Previous research has found evidence to suggest reverse scoring items perform poorly in single-factor models (Woods, 2006).
When seen collectively, the single-factor model demonstrates a poor fitting model to the data.
Barra's sophisticated process of risk decomposition has proven superior to single-factor models
based solely on historical variables such as price.
Single-factor models may be "good enough" for some applications such as managing portfolios of similar-maturity bonds, but they will result in hedging error when applied to complex securities, such as spread derivatives, for example.
Examples of equilibrium-based interest rate models include the Cox, Ingersoll, and Ross (1985), or CIR, model and the widely used Vasicek (1977) model--both usually implemented as single-factor models.
Exhibit 3 illustrates the need for a multi-factor model, as opposed to a single-factor model
, for systematic risk.