### Factor Models

 Factor Models

In general, factor models are used to predict random variables Y, with the help of explanatory variables, X. The basic idea behind these models is the relation between the dependent and the independent variables. The independent variables constitute the factors that determine the dependent variables.

The explanatory variables must be carefully selected, as they are to be the factors that influence the dependent variables. A linear factor model can be formulized as follows:
Ri = βi + βi1F1 + βi2F2 + ... + βikFk
where Ri is the return of fund i, and F1, F2, ..., Fk are the k factors that are claimed to influence the fund’s return. We assume that there are n funds, i = 1, 2, ..., n.

The beta coefficients βi1, βi2, ..., βik reflect the sensitivities of the fund to specified factors. These coefficients designate the change in the return on the fund per unit of change in the specified factor. The error term εi capture all randomness in the relationship.

A popular factor model known as the CAPM has only one factor, k = 1. Models with a unique factor are called single-factor models, whereas models with more than one factor are called multifactor models.

For hedge funds and managed futures, certain multifactor models are available in explaining managed futures and hedge fund returns. The factors used are justified on the distinctiveness of hedge fund manager trading styles.

The single-factor model assumes that the factors are linearly related to fund returns, but nonlinearity of factor models is also possible. The linear multifactor model given above does not have the time dimension and is therefore static, but dynamic factor analysis is possible when the time dimension with subscript t is introduced.

The following technical assumptions must be satisfied to make use of estimation by the Ordinary Least Squares (OLS) method and statistical inference in factor models:
• The expected value of the error term must be zero, E(εi) = 0, i = 1, 2, ..., n.
• Factors and error terms should be uncorrelated, Cov(Fj, εi) = 0, j = 1, 2, ..., k.
• Error terms should not be autocorrelated, Cov(εi, εj) = 0, i ≠ j.
• All error terms must have the same variance, E(<ε2) = σ2.

Some additional assumptions of time series analysis such as stationarity of each series must be imposed for dynamic factor analysis.

Factor models are introduced in the literature to facilitate the interpretation of a voluminous data set to reveal factors determining fund returns. Multifactor models can be categorized into broad classes of macroeconomic (macroeconomic indicators like interest rate series are used as factors), fundamental (factors concerning securities or firms, like firm size or dividend yield are used), and statistical models. Factor models are helpful in making decisions on asset valuation and are extensively referred in portfolio theory.

The researcher must determine the appropriate factors in the analysis to produce a meaningful relationship. The coefficient of determination, R2, can be used as a benchmark criteria to assess the goodness of fit. There are several serious attempts in literature to work out the main factors that explain the hedge fund returns.

Agarwal and Naik use the factor model approach to figure out that hedge fund returns are attributable to risk factors consisting of indices representing equities (Russell 3000 Index, lagged Russell 3000 Index, MSCI World Excluding the USA Index, and MSCI Emerging Markets Index), bonds (Salomon Brothers Government and Corporate Bond Index, Salomon Brothers World Government Bond Index, and Lehman High Yield Index), Federal Reserve Bank competitiveness-weighted dollar index, and the Goldman Sachs commodity index as well as the three zero-investment strategies representing Fama-French’s “size” factor (small-minus-big or SMB), “book-to-market” factor (high-minus-low or HML), Carhart’s “momentum” factor (winners minus losers), and the change in the default-spread (the difference between the yield on the BAA-rated corporate bonds and the 10-year Treasury bonds) to capture credit risk.

In a similar study, Fung and Hsieh explain the HFR fund of funds index with two equity risk factors (S&P 500, SC-LC), “... two interest rate risk factors (the change in the yield of the 10 year treasury, and the change in the credit spread), and three trend-following factors (the portfolio returns of options on currencies, commodities, and long-term bonds).”

In a similar attempt, Schneeweis and Spurgin explain the hedge fund performance index with the independent variables of nominal and absolute values of the SP500, GSCI, SBBI, and USDX, the intramonth standard deviation of the SP500, GSCI, bond, and USDX, and the nominal value of the MLM index. Meredith and Figueiredo present a more detailed study of factor models to explain the returns for every strategy.

The factors they use are small cap stock minus large cap stocks, value stocks minus growth stocks, winners minus losers, GSCI, Russell 3000 (with up to four lags), Citigroup high yield composite, MSCI emerging markets, Fed dollar weighted index, MLCBI, reserve moving average, and traded implied volatility (change in VIX).