# Factor Analysis

Regression and time series attribution models in the Windham Portfolio Advisor

Factor analysis is a powerful technique that can identify and measure common sources of risk and return for managers, asset classes, and portfolios. Each factor represents an underlying exposure to the market. Factor analysis goes beyond the asset allocation to identify the underlying exposures to specific sources of risk and return. Factor analysis can be used in a variety of applications including

Explaining differences in returns across a universe of financial assets

Forecasting the expected value of asset returns

Explaining systematic variations and co-movements in returns

Stress testing asset class returns

Evaluating managers' exposure to risk factors

## Regression Models

Regression can be used to describe the relationship between an investment vehicle and a risk factor or group of factors. The Windham Portfolio Advisor allows for the use of Single, Multi, and Stepwise regression analysis to identify the factor sensitivities of a portfolio.

### Single Factor Model

The single factor Capital Asset Pricing Model (CAPM) is an early example of a single-factor regression model. CAPM specifies that an asset’s expected return in excess of the risk free rate is proportional to the asset’s sensitivity to systematic risk (non-diversifiable risk of the market). The sensitivity term is commonly referred to as beta (or factor loading). A single-factor model assesses the return sensitivity of each vehicle against an individual factor, and repeats the exercise for each factor. The single-factor regression provides the best descriptor of the exposure of the manager to a specific risk factor in isolation.

### Multi-Factor Model

The multi-factor regression uses a set of factors instead of just one to explain the risk and return of a vehicle. Variation in the dependent variable can often be better explained by the variation in a number of independent variables. A multi-factor model identifies the sensitivity of a vehicle’s return against a set of factors. The beta term in a multi-factor model represents the sensitivity of a vehicle to a factor assuming the other factors remain static. The $R^2$ value indicates how well the variance is explained by the set of factors. A value of 100% indicates that the variance is entirely explained by the multi-factor model.

### Stepwise Multi-Factor Model

Stepwise regression is an objective variable screening procedure for adding and removing factors from a multi-factor regression based on their statistical significance. The process starts with a single-factor model and then adds additional factors until identifying a model with the highest explanatory power from the available factors. If a factor does not have sufficient explanatory power then it is removed from the model. Stepwise regression is a useful tool for identifying which factors to include when building a factor model and for modeling all the factors simultaneously. The $R^2$ value indicates how well the variance is explained by the set of factors. A value of 100% indicates that the variance is entirely explained by the stepwise-factor model.

## Analysis Types

The Windham Portfolio Advisor provides three approaches to review risk factors across portfolios.

### Factor Loadings

Factor loadings (sensitivities) show us the sensitivity of an investment vehicle to each factor. The regression coefficients are also known as factor loadings.

The $R^2$ value indicates how well the variance is explained by the set of factors. A value of 100% indicates that the variance is entirely explained by the multi-factor model. The Windham Portfolio Advisor also shows measures of statistical significance (t-statistic) for the factor loadings. A t-statistic value above 2 or below -2 suggests that the factor is statistically significant.

### Risk Decomposition

The Windham Portfolio Advisor separates the variance of each time series into that which can be explained by beta exposures to the factors and that which cannot be explained by the factor model (residual). The “Residual” is the percentage of variance not explained by the multi-factor model. We calculate the risk decomposition from the weighted averages of the managers’ multi-factor loadings and the factors’ standard deviations and correlations.

### Return Decomposition

The software separates the return of each time series into that which can be explained by beta exposures to the factors and returns that cannot be explained by factor exposures (intercept).

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