An important practical question when dealing with multi-manager portfolios is the relationship of fund returns to major markets or sectors. Since in most cases the fund is exposed to a number of markets or sectors, it is reasonable to believe that the proper relationships can be found by multi-dimensional regression analysis. Finding such relationships can be done manually, by selecting the factors you consider important, or automatically, by running the step-wise selection method on a universe of pre-selected factors. Cognity is shipped with a large set of factors which are updated on a daily basis. The system includes several template factor models which can either be used directly out of the box or customized to the set of funds you are working with.

Apart from detecting linear dependencies on markets and sectors, you can also introduce a non-linear relationship. For example, the relationship between positive fund returns and positive returns of the market may be different than the relationship between the corresponding negative returns. This can be accounted for in a factor model and you will obtain two different regression coefficients for positive and negative factor returns. This is referred to as up and down market betas.

You can also include squared returns in the regression model as a proxy for a market volatility factor. This will enable you to capture a non-linear relationship between a fund and a factor. You can also use lagged returns of the fund or the market to take into account a possible autocorrelation effect.

Note that non-linear regression can explain non-linearity in the dependence structure but not the fat-tailed behavior of the factors or the residuals. A non-linear model by itself is not sufficient to account for fat tails of the funds because market factors themselves have fat tails. Therefore, it makes sense to combine a fat-tailed model for the return distribution with a non-linear factor model.

Cognity supports three different factor model fitting methods. The standard ordinary least squares method (OLS) is implemented in the system and can be used together with the stepwise selection procedure. The disadvantage of the OLS method is that it is very sensitive to outliers and that it assumes returns are normally distributed. The L1 fitting method is consistent with a fat-tailed assumption for the factor returns. The L1 method is also less prone to the effect of outliers in the dependent variable. The Robust fitting method can also be used together with the stepwise procedure. It is immune against outliers in the dependent as well as explanatory variables.

You can gain valuable insights by comparing the results of different factor models fitting methods. The OLS method is very sensitive to outliers and hence the estimates you get will be a mix of the behavior of the fund under extreme as well as normal market conditions. The Robust method ignores outliers and can help to identify the relationship between the fund and the markets in business-as-usual conditions. If the estimates of the two methods differ significantly then the relationship of the fund to the markets is dynamic and shifts under extreme market conditions.  

Key Points to Remember:

• Multi-dimensional regression is the proper way to model funds for risk management posurposes
• Non-linear factor exposures  may lead to asymmetry in fund returns but may not be the source of fat tails
• Fat-tails in fund returns arise from exposures to fat-tailed factors
• The choice of a fitting method is important and comparing results from different methods can give valuable insights
• The OLS method is very sensitive to outliers
• The L1 method is compatible with the assumption of fat tailed factor returns
• The robust method is immune to outliers and identifies dependencies in normal market conditions.