9.1.1 Simple Linear Regression and the Single-Factor Market Model

A regression is a statistical analysis of the relationship that explains the values of a dependent variable as a function of the values of one or more independent variables based on a specified model. The dependent variable is the variable supplied by the researcher that is the focus of the analysis and is determined at least in part by other (independent or explanatory) variables. Independent variables are those explanatory variables that are inputs to the regression and are viewed as causing the observed values of the dependent variable.

In a linear regression, the model that describes the relationship between the dependent variable and the independent variable or variables is linear. A simple linear regression is a linear regression in which the model has only one independent variable. For example, the ex post version of the single-factor market model describes realized excess returns of a security or fund as a linear function of an intercept, the market beta, the market portfolio's realized excess return, and an error term that reflects idiosyncratic risk. An excess return is a total return minus the periodic riskless rate. The single-factor market model based regression equation for asset i, based on a time series of total return data, is as follows:

(9.1)

where R_it is the return of asset i in time period t, R_f is the periodic riskless rate, a_i is the estimated intercept, b_im is the estimated slope coefficient, R_mt is the return of the market portfolio in time period t, and e_it is the residual or estimated error term for asset i at time t.

Equation 9.1 seeks to predict or explain the values of the dependent variable, excess returns E(R_it) − R_f, through movements in the independent variable, the excess return of the market portfolio (R_mt − R_f); b_im is the estimated slope coefficient of the regression and is an estimate of the beta for asset i. The slope coefficient is a measure of the change in a dependent variable with respect to a change in an independent variable. In this example, the slope coefficient estimates the linear sensitivity of the return of asset i to the excess return of the market. The estimate of the intercept of the regression is a_i. The intercept is the value of the dependent variable when all independent variables are zero. In the case of Equation 9.1, the intercept can be interpreted as an estimate of the average ex post alpha of asset i. Finally, the residuals of the regression, e_it, reflect the regression's estimate of the idiosyncratic portion of asset i's realized returns above or below its mean idiosyncratic return (i.e., the regression's estimates of the error term).

9.1.2 Ordinary Least Squares Regression

There are unlimited estimated values that can be inserted for the intercept (a_i) and slope coefficient (b_im) in Equation 9.1. Ordinary least squares regression, the most common regression procedure, selects the intercept and slope that minimize the sum of the squared values of the residuals (the values of e_it). In simple linear regression, the process may be envisioned as drawing a regression line through a scatter plot of the dependent variable and independent variable. The vertical distance between the regression line and each observation is the residual. The least squares fitting criterion minimizes the sum of those distances squared. The use of ordinary least squares has several advantages: It is quick and easy, and the slope coefficient that results has an intuitive interpretation.

Least squares regression has been shown to generate unbiased and most likely estimates of the slope coefficient and intercept if the error terms in the model are (1) normally distributed, (2) uncorrelated, and (3) homoskedastic (i.e., having the same finite variance). Violations of these assumptions are discussed in the next three sections. Other criteria for fitting a model to data also exist.

9.1.3 Outliers

Violations of the assumption that the error term in the model is normally distributed often occur when the data are subject to very large outliers, as is often the case in investment returns.

PROBLEM 1: OUTLIERS. Fat tails (leptokurtic distributions) are synonymous with frequent outliers. Alternative investment returns are especially prone to being leptokurtic. Large outliers dominate a regression, potentially causing the estimates of the slope and intercept to be driven too much by the outliers, rather than by the remaining, more representative data. Ordinary least squares regression seeks to minimize the sum of squared residuals, and the squaring of residuals can cause outliers to have disproportionately higher influence than observations closer to the mean.

RESPONSE 1: A critical but often overlooked task in linear regression is visual observation of the residuals of the regression. At least two plots are advisable for important regressions. Residuals should be plotted on the vertical axis against the independent or explanatory variable on the horizontal axis, and time-series residuals should be plotted on the vertical axis against time on the horizontal axis. The analyst should note extreme outliers to determine if the residuals reflect data errors or economic fact. If the extreme residuals are not the result of errors, the analyst should determine if the underlying economic behavior causing the observation warrants the large level of influence that the outlier has on the estimated parameters. If the outlier is caused by an event that can be reasonably expected to not recur, perhaps the outlier should be removed. An example is a fund experiencing a catastrophic event from short option positions that has amended its investment strategy to disallow short option positions. It is important not to remove outliers corresponding to gains or losses that are likely to be repeated.

For example, if an analyst regressed the monthly returns of a U.S. financial stock on U.S. stock market returns over a period including 2007 and 2008, the analyst would probably obtain a very high estimate of the stock's beta due especially to the months in which financial stocks experienced tremendously negative returns and in which the overall market experienced negative returns as well. The analyst would detect these outliers with a plot and then need to decide whether the observed correlations were a representative sample on which to forecast future systematic risk (beta) or the outliers generated an estimate of beta that is unduly indicative of behavior under stressed conditions and therefore unrepresentative of anticipated market conditions.

9.1.4 Autocorrelation

The simplest statistical regression procedures assume that the model's error terms are uncorrelated—including through time. Autocorrelation of the error terms is a violation of that assumption.

PROBLEM 2: AUTOCORRELATION. Violations of the assumption that the error term is uncorrelated through time most often occur when returns are autocorrelated. Many alternative investment return series are especially prone to autocorrelation due to smoothed pricing or illiquidity.

RESPONSE 2: The Durbin-Watson statistic, detailed in Chapter 4, is used to test for autocorrelation of residuals. If the Durbin-Watson statistic indicates autocorrelation, there are several well-established statistical procedures for performing adjusted regressions that provide better results. First-order autocorrelation is a common phenomenon in alternative investments and is reasonably easy to address.

For example, if an analyst regresses the percentage changes in a real estate project's value based on monthly appraisals against the overall market return, the residuals of the regression might exhibit autocorrelation based on a Durbin-Watson test. The autocorrelation may indicate that the appraisal valuations were reflecting value changes on a delayed basis. Such a regression should be corrected for autocorrelation in order to provide a more accurate measure of the correlation between true real estate values and the overall market.

9.1.5 Heteroskedasticity

The simplest statistical regression procedures assume that the variance of the model's error terms is homoskedastic.

PROBLEM 3: HETEROSKEDASTICITY: Heteroskedasticity is the opposite of homoskedasticity. In a regression, heteroskedasticity refers to a situation in which the variance of the error term varies. For example, the variance of the error term may be correlated with an independent variable, may vary through time, or may be related to some other variable or dimension. With homoscedasticity, the variance of the error term is constant.

RESPONSE 3: The same plots used for outlier examination should be used to detect heteroskedasticity (i.e., residuals should be plotted against the independent variable and against time). In this visual analysis, the analyst should look for a pattern in the dispersion of the residuals, such as a <, >, < >, or > < pattern. For example, a < pattern would show generally increasing dispersion of the residuals moving from left to right in the diagram. Heteroskedasticity can be formally detected using various tests. The problem with regression results from data exhibiting heteroskedasticity is that the estimated regression parameters are unduly influenced by the data related to the greatest variance in the error term. The most popular correction is weighted least squares, in which a weighting scheme is developed and applied to the data to reduce the importance of the data subject to higher error-term volatility.

For example, an analyst regressed the returns of a corporate bond against a constant maturity Treasury index. A plot of the residuals through time tends to indicate a > pattern, with earlier observations (to the left) having more dispersion than more recent observations (to the right). The heteroskedasticity is attributable to the declining price volatility of the corporate bond as its maturity nears and its duration declines. The earliest observations with the highest dispersion dominate the regression, generating inefficient estimates. A weighted least squares approach should be used to adjust the influence of the observations toward being more equal over time.

In summary, the accuracy of a regression's results may be adversely affected by three primary issues: outliers, autocorrelation, and heteroskedasticity. The statistical approach should be adjusted as necessary to correct for any of these challenges before using the estimated parameters.

9.1.6 Interpreting a Regression's Goodness of Fit

The first major interpretation of a regression's results is evaluating the overall explanatory power of the regression. The explanatory power of the regression is evaluated as its goodness of fit. The goodness of fit of a regression is the extent to which the model appears to explain the variation in the dependent variable. The r-squared value of the regression, which is also called the coefficient of determination, is often used to assess goodness of fit, especially when comparing models. In a simple linear regression, the r-squared is simply the squared value of the estimated correlation coefficient between the dependent variable and the independent variable. Correlation, discussed in Chapter 4, ranges from −1 to +1, with negative values showing an inverse relationship between two variables, and positive values denoting a direct relationship between two variables. Because the r-squared is equal to a correlation coefficient squared, the range of possible values for r-squared is between zero and 1 and is often expressed as a percentage. When building or explaining financial relationships, larger values of r-squared are preferred, everything else being equal, as the independent variable is explaining a greater portion of the variance in the dependent variable.

R-squared is also interpreted in an absolute sense. For example, a long-only mutual fund may have an r-squared of perhaps 0.90 (i.e., 90%) in a regression of its returns on the returns of a market index. An r-squared such as 0.90 would often be described as meaning that the independent variable (in this case, the returns of the market index) explained 90% of the variation in the dependent variable (in this case, the returns to the mutual fund). This can be interpreted as indicating that 90% of the fund's returns were explained by the systematic risk (i.e., exposure to the market risk represented by the index). The remaining value, 1 − r², is the idiosyncratic risk, or the risk that is not explained by the market index. In this case, the idiosyncratic risk is 10% of the fund's total risk. The fund's idiosyncratic risk might be due to incomplete diversification, such as holding only 25 stocks and being compared to a very well-diversified benchmark index.

9.1.7 Performing a t-Test on Regression Parameters

The second major interpretation of a regression's results is testing the significance of the parameter estimates. In an application of Equation 9.1, the intercept of the regression is usually interpreted as an estimate of the ex ante alpha, or skill of the fund manager (if a fund's return is being analyzed), or the superior risk-adjusted return of a security (if a security's return is being analyzed). The slope coefficient of the regression is usually interpreted as the beta of the asset, a measure of the asset's systematic risk.

The parameter estimates of the regression are typically examined for statistical significance using a t-test. A t-test is a statistical test that rejects or fails to reject a hypothesis by comparing a t-statistic to a critical value.

For each alpha and beta estimate, the t-statistic is formed. The t-statistic of a parameter is formed by taking the estimated absolute value of the parameter and dividing by its standard error. The resulting t-statistic is compared to a critical value. If the t-statistic exceeds the critical value, the parameter estimate is deemed to be significantly different from zero. The critical value of the t-statistic is found from published lists of critical values based on two parameters: (1) the degrees of freedom, and (2) the desired significance level of the test.

9.2.1 Selecting Factors for Multifactor Regression

In alternative investments, it is clear that a wide variety of risk factors explain realized returns. However, it is not clear the extent to which additional risk factors determine expected returns. An alternative investment analyst must be especially careful when selecting risk factors, because the interpretation of a regression intercept as an estimate of alpha may cause the return of any omitted risk factors to be captured in the intercept and be falsely attributed to alpha. For example, if an equity manager makes an investment in a fund that includes commodities, and if an index representing the commodity market factor is not included in the regression, then any returns attributable to the commodity return factor may be counted as alpha. The Fama-French model, discussed in Chapter 6, is a very popular multifactor model in the analysis of equity returns. Equation 9.2 is the empirical model of the Fama-French approach, in which realized returns of an investment are explained not only by estimated exposure to the stock market index but also by estimated exposure to the anomaly factors of value and size:

(9.2)

Equation 9.2 specifies a multiple regression model. By including the returns corresponding to the size factor, R_s − R_b, and the value factor, R_h − R_l, an analyst can expect that the returns of asset i in time period t, R_it, will be more fully explained and that the parameters that estimate the exposure of the asset to each of the factors (the value of each b) will be more accurately estimated. A typical result of adding more true factors to a model is that the r-squared increases and the alpha estimate declines. The r-squared increases, as the additional factors are explaining a greater portion of the variance in the dependent variable. The estimated alpha typically declines, as returns that were previously attributed to the intercept (alpha) are now explained by systematic risk exposure to the anomaly factors of size and value (beta). A major challenge in multiple regression is deciding which independent variables (factors) to include.

9.2.2 Multicollinearity

Section 9.1 detailed three major challenges with simple regression (outliers, autocorrelation, and heteroskedasticity), each of which is also a challenge in multiple regression. In addition, multiple regression adds the challenge of potential multicollinearity. Multicollinearity is when two or more independent variables in a regression model have high correlation to each other. A primary method of detecting multicollinearity is to examine the correlations between the independent variables.

When two independent variables are highly correlated, there are two primary adverse effects to regression results: (1) The estimates of the slope coefficients for each of the correlated independent variables may be highly inaccurate, and (2) the standard errors for the correlated independent variables may be inflated (large). With multicollinearity, even though the r-squared of a regression may be high, it can be difficult to find independent variables with coefficients that have significant t-statistics.

There are several corrections for multicollinearity. In the case of returns as independent variables, one potential method for correction is to form return spreads between the correlated independent variables. For example, consider a multiple regression equation, with a U.S. stock index and a non-U.S. stock index both serving as independent variables. Because the contemporaneous returns of U.S. and non-U.S. stocks tend to have high correlation with each other, this multiple regression model probably has multicollinearity. The estimated slope coefficients for each of the highly correlated factors would be unreliable and are likely to be statistically insignificant. To avoid this issue, the analyst might start with a U.S. equity index return series as one independent variable and then add the difference (spread) between the returns of the U.S. stock index and the non-U.S. stocks as a second independent variable. This transformation serves to reduce the correlation between the independent variables, now making it possible to better separate the effects of each market segment independently.

9.2.3 Selecting the Number of Factors and Overfitting

Once the list of potential return factors is determined, the next challenge is to determine which of the independent variables should be included and retained in the regression equation. Especially when multicollinearity is a potential issue, rather than running a kitchen sink regression that includes all potential variables, a stepwise regression technique is more appropriate. Stepwise regression is an iterative technique in which variables are added or deleted from the regression equation based on their statistical significance. At each step, the variables with the greatest t-statistics are added to or retained in the model, and variables with insignificant t-statistics are deleted from the model.

Although stepwise regression can be an extremely fast way to consider many independent variables and reduce the number of variables ultimately included, the analyst should be cognizant of the temptation for data dredging. Searching across large data sets with numerous potential independent variables can locate statistically significant relationships over the time period of the regression, but these results may fail to predict or explain the dependent variable using data from outside the sample. Analysts must also be careful to not include too many variables in the regression (i.e., to not overfit the model). Overfitted models explain the past well (i.e., the model explains the data used to fit the model), but they do not predict future relationships well. Ideally, the analyst's knowledge should be used to limit the variables under consideration to those that make economic sense.

9.3.1 The Dummy Variable Approach to Dynamic Risk Exposures

The effectiveness of market-timing strategies can be analyzed by a comparison of their average risk exposures to up markets and their average risk exposures to down markets. Equation 9.3 models different responses of the returns of a fund to up markets and down markets:

(9.3)

The dummy variable, D₁, is set equal to 1 when excess returns on the market index, R_mt − R_f, are positive and set equal to zero when the excess returns are zero or negative. The down market beta, b_i,d, is the responsiveness of the fund's return to the market return when the market return is less than the riskless rate (i.e., when the market's excess return is negative, or down). The coefficient b_i,diff is the difference between the sensitivities or betas of the fund's return to up and down markets. The up market beta, b_i,u, is the responsiveness of the fund's return to the market return when the excess market return is positive, and is estimated as the sum of b_i,d and b_i,diff.

Inspection of Equation 9.3 indicates that in down markets, the coefficient of the market's excess returns is simply b_i,d (the down beta), whereas in up markets, the coefficient is b_i,d + b_i,diff, which is the model's estimate of the up market beta.

Suppose, for example, that b_i,d = 0.5 and b_i,diff = 0.7. When the market index is earning a positive excess return, D₁ = 1, and the total beta exposure of the fund is 1.2, which is the sum of the down beta coefficient, 0.5, and the dummy beta coefficient during up markets, 0.7. When the market index is generating a negative excess return, D₁ = 0, and the total beta exposure of the fund is 0.5. When b_i,diff is greater than zero, the manager is demonstrating a valuable market-timing skill by increasing exposure to market risk during times of positive excess returns and reducing market exposure during times of negative excess returns.

Mathematically equivalent models to Equation 9.3 can be formed through algebra. For example, a model can be derived with explicit up and down betas. An advantage to the model in Equation 9.3 is that it can automatically be used to test for a difference between the up and down betas by testing whether b_i,diff statistically differs from zero. Note that b_i,diff is the key measure of market-timing skill in this model.

9.3.2 The Separate Regression Approach to Dynamic Risk Exposures

A similar approach to the dummy variable approach is to perform separate regressions based on subsamples. If the regression is being performed on a time series, then the analyst simply breaks the data set into two or more subsamples based on a specified condition, especially an independent variable such as a market factor. For example, one subsample could include a rising market, and the other subsample, a declining market. The subsamples could be based on dividing the observations into contiguous time periods or could divide the observations based on the specified condition (e.g., all observations with positive excess returns for a market factor, and all observations with negative excess returns for a market factor).

For example, Black finds that hedge funds of funds' behavior changed dramatically from the 1990–97 period to the 1999–2004 period, using 1998's experience with Long-Term Capital Management as a dividing line in hedge fund risk exposures.¹ Using the entire time frame of 1990 to 2004 would have not only obscured the change in behavior over this time period but also obscured the degree to which behavior could be well described within each sub-period.

9.3.3 The Quadratic Approach to Dynamic Risk Exposures

Another approach to assessing market-timing skill uses a quadratic curve (i.e., a squared term) rather than a dummy variable or separate regressions. Consider another skilled but imperfect market timer, such as a skilled trend follower. That market timer might tend to perform exceedingly well with large underlying up or down (i.e., large directional) moves in the market, have modest profits in markets with smaller directional moves, and perform with likely losses during directionless markets. Henriksson and Merton, as well as Treynor and Mazuy, discuss models to explain market-timing performance.² One such model is:

(9.4)

Equation 9.4 provides an accurate fit for a U-shaped profit–loss diagram. In the nonlinear model of Equation 9.4, the squared value of the excess return on the market is used to explain the performance of the fund's excess return. A statistically significant and positive beta coefficient on the squared term in Equation 9.4 is an indication that the manager has been able to successfully time the market, earning positive returns in both strongly rising and strongly falling markets. A significant negative value of b_im indicates that the manager has perverse market-timing skill, in which the average market-timing decision is detracting value from the fund. However, a positive estimated beta in Equation 9.4 can also be obtained by purchasing option straddles. The costs of the option straddles, which would be captured by the intercept, may outweigh the benefits. Further insight into the potential for skill would therefore include examination of the estimated intercept.

9.4.1 Conditional Correlation Modeling Approach

A conditional correlation is a correlation between two variables under specified circumstances. For example, an analyst may estimate the correlation coefficient between a hedge fund's returns and the returns of an equity index during only those months in which the stock market rose by 1% or more. The correlation coefficient being estimated would be a conditional correlation coefficient rather than an unconditional correlation coefficient because the behavior being measured is based on or applicable to a limited set of circumstances. Conditional correlation is constant across conditions when the relationship between two variables is completely linear. Conditional correlation estimation and analysis of differences between estimates can be used to understand nonlinear relationships and is similar to the separate regression approach discussed in section 9.3.2. The differences are that (1) the regression-based approach can include multiple factors, and (2) regression coefficients differ from correlation coefficients by a scale factor related to volatility ratios.

Consider the example in Exhibit 9.1. Underlying Exhibit 9.1 are the correlations, standard deviations, and mean returns during two subsamples based on whether each month has rising or falling prices for the S&P 500 Index as proxied by an S&P 500 exchange-traded fund. The “up” subsample includes the months when the S&P 500 Index experienced a nonnegative total return. In that subsample, the S&P 500 rose with a monthly average return of 3.3%. The “down” subsample is the remaining months in which the index fell. During the second subsample, the S&P 500 Index experienced an average monthly return of −3.8%.

Exhibit 9.1 displays the changes to the estimated correlation coefficients, standard deviations, and mean returns between the two subsamples. For example, the last entry indicates that the mean monthly return of the S&P 500 was 7.1% lower in the down sample (falling equity markets) than in the up sample. The other rows indicate the correlation, standard deviation, and mean return changes for indices corresponding to 13 hedge fund strategies. In each case, the subsample differs based on whether the S&P 500 Index was up or down for the associated months. Twelve of the 13 hedge fund strategies had lower average monthly returns in the down sample months for the S&P 500 than in the up months. The only exception was short bias, which, as anticipated, did better in the down months.

Surprisingly, 11 of the 13 hedge fund indices had a higher correlation to equity market returns in down markets than they had in up markets. When the correlation in the down sample is higher than the correlation in the up sample, it is termed negative conditional correlation. The negative conditional correlation in Exhibit 9.1 is undesirable for hedge fund investors, as investors desire lower correlations during times when stock prices are declining to mitigate losses, and higher correlations when stock prices are rising to extend profits. Positive conditional correlation of investment returns to market returns is when the correlation in the up sample is higher than the correlation in the down sample. Investors prefer investment strategies with positive conditional correlation, since the strategies offer higher participation in profits during bull markets and lower participation in losses during bear markets. The only indices exhibiting positive conditional correlation during this period were managed futures and equity market-neutral funds.

It should be noted that the results of Exhibit 9.1, like most similar empirical analyses of correlation, are subject to being dominated by the most extreme outcomes. The two particularly bad months for the S&P 500 (August 1998 and October 2008) exert a strong effect on estimated correlations. The managed futures index was up in both of those months, and the equity market-neutral index was near zero in both months, which likely explains why those indices alone had estimated positive conditional correlation. Careful analysis should be used to judge whether predictions of future behavior should be so heavily influenced by the two largest outliers in terms of S&P 500 returns.

Conditional correlation analysis is not limited to separating a sample into only two subsamples or to separating a sample based only on the behavior of the variable with which the correlations are being estimated. For example, an analyst might examine the correlation between hedge fund strategies and equity returns in three market conditions: increasing interest rates, decreasing interest rates, and stable interest rates.

Further, parameters other than correlation, such as volatilities and means, can be analyzed on a conditional basis. Conditional correlation and other conditional analyses can be viewed as the general concept of examining the behavior of estimated parameters relative to one or more identifiable variables. The development and application of advanced methods to model the dynamic behavior of return distributions is an important frontier of alternative investments.

9.4.2 Rolling Window Modeling Approach

Another method to model changing correlation caused by the dynamic exposures of an investment strategy is to use a rolling window analysis. Rolling window analysis is a relatively advanced technique for analyzing statistical behavior over time, using overlapping subsamples that move evenly through time. When analysts use multiple time periods in a regression or correlation analysis, the data set is typically divided into two, three, or four sub-periods of time, with every observation included in only one subsample. A rolling window analysis chooses a time width for the window, such as 36 months, and performs the regression or correlation analysis for each contiguous 36-month period in the data. The sub-periods use overlapping data as the window moves from the first 36 months of data to the last 36 months of data.

For example, using 10 years of data, a rolling window analysis with a window of 36 months would produce 85 unique outputs. The first analysis and output would use the data from months 1 to 36, the second from months 2 to 37, and the final from months 85 to 120.

Let's discuss the 85 subsamples for a rolling window regression analysis of the returns of a fund against the returns of several market indices. The output of the first regression would show the estimated relationship between the dependent and independent variables over the first three-year (36-month) period (months 1 to 36). The second regression would be the same as the first regression except it would delete the first monthly observation and add the observation of the 37th month. As the regression walks forward in time, the sensitivity of fund returns to each market variable can change to reflect the dynamic allocations of the fund manager. Put together, the estimated parameters, perhaps with their confidence intervals, can be graphed through time to illustrate the dynamic nature of the estimates. It should be noted that even though this rolling window approach would generate 85 sets of regression results, the regressions use overlapping data and are therefore not independent statistical tests. With 10 years of data, there are only three statistically independent three-year regressions that can be performed.

Rolling window analysis and other forms of multiperiod analysis using longer-term returns, such as monthly returns, are often appropriate for determining long-term style drifts. Some fund strategies, such as equity market-neutral, managed futures, and global macro, are more likely to alternate signs of exposures too quickly to be well measured with a long-term analysis. Shorter-term changes in exposures are better analyzed with shorter-term return intervals, such as daily return data.

9.5.1 Understanding Style Analysis and Fund Groupings Based on Asset Classes

Multifactor return models often use the returns of underlying asset classes to explain the returns of investments. For example, the returns of convertible arbitrage hedge funds are often explained based on the returns of asset classes such as equities, bonds, and options.

Style analysis is the process of understanding an investment strategy, especially using a statistical approach, based on grouping funds by their investment strategies or styles. The key question in a style analysis is this: Do investment funds of the same stated investment style have returns that can be explained by the same underlying return factors?

The modern approach in performing style analysis on traditional mutual funds was pioneered by Sharpe, who (1) groups mutual funds by their stated investment styles, and (2) analyzes the performance of each group (i.e., style) relative to the performance of various potentially underlying asset classes.³ Sharpe attributes the returns of mutual funds to the returns of indices corresponding to traditional financial security classes related to the most common holdings of the mutual funds. In other words, Sharpe's style analysis regresses mutual fund returns (as the dependent variable) on the returns of various asset classes (as the independent variables). Sharpe selects several distinct bond indices and numerous groupings of stocks based on size, country, and other attributes as the independent variables. His results indicate that up to 90% of each mutual fund's returns are explained by the returns of a few underlying asset classes. The balance of returns may be attributable to manager skill, including security selection and market timing, or luck.

Fung and Hsieh use data on hedge funds to apply a style analysis approach analogous to that conducted by Sharpe but to alternative investments.⁴ They focus on using indices of traditional asset returns to explain the returns of hedge funds. Contrary to Sharpe's results for mutual funds, Fung and Hsieh find that the amount of variation of hedge fund returns that is explained by financial asset class returns is low: R-squared measures are less than 25% for almost half of the hedge funds studied.

In summary, traditional mutual fund returns are well explained by the returns of the asset classes that the funds hold, but the same is not true for hedge funds. Empirical evidence indicates that the returns on most hedge funds are not well explained by passive return indices of their underlying assets. For example, managed futures funds hold positions in commodity futures, but an analyst should not expect that a particular managed futures fund will have returns that are highly correlated to commodity price indices. Managed futures funds have actively traded long and short positions, and their returns depend more on the extent to which managers can time changes in commodity prices.

9.5.2 Understanding Funds Based on Strategies

Another interesting question is whether funds with the same stated investment strategy or style have similar returns or returns that respond to similar risk factors. For example, in traditional investments, the returns of an equity fund are compared with the returns of other equity funds to detect the extent to which the equity funds respond to the same underlying risk factors, such as Fama-French factors. The analy- sis is often taken to a finer level of detail so that a U.S. large-cap growth fund is compared with other such funds. Grouping funds by strategies or styles and analyzing the returns of funds with the returns of other funds of similar style is commonly performed in both traditional and alternative investments.

The stated strategy or style of a traditional mutual fund is usually quite clear from examining its publicly available listings of assets and from the fact that most traditional mutual funds maintain relatively stable portfolios. However, hedge fund portfolios are often opaque, can be very diverse, and can have changing portfolios and risk exposures. Further, a hedge fund may not identify itself as following a particular style that can be used to associate that hedge fund with other hedge funds. Even if a group of hedge funds can be identified with the same style (e.g., equity market timing), the funds within that style group may have very different trading strategies and very different returns. Finally, a hedge fund's strategy or style may change or drift through time.

Fung and Hsieh use data on hedge funds and a principal components analysis to find return commonalities among hedge fund returns. Principal components analysis is a statistical technique that groups the observations in a large data set into smaller sets of similar types based on commonalities in the data. Thus, principal components analysis identifies subgroups of observations that tend to behave similarly. Fung and Hsieh find that the returns of many hedge funds can be moderately explained by viewing most of the funds as behaving as if they belong in one of five groups or trading styles, which they labeled as (1) systems/opportunistic, (2) global macro, (3) value, (4) systems/trend following, and (5) distressed. They estimate that these five hedge fund styles explain about 45% of the cross-sectional variation in hedge fund returns. Their work suggests that cross-sectional hedge fund returns are better explained by their trading styles than by their correlations with traditional asset classes. Therefore, the returns of a global macro fund tend to be explained better by the fund's tendency to behave like other global macro funds than by its mixture of underlying traditional asset classes.

In summary, a hedge fund's return is explained better by its trading style than by the returns of hedge funds with the same stated style or the returns of the asset classes that it trades. For example, an equity market-neutral fund is unlikely to have returns highly correlated with all other equity market-neutral funds, because the returns of these funds are driven by distinct idiosyncratic risks. Also, an equity market-neutral fund is unlikely to have returns highly correlated with underlying equity market indices, because the fund strives to hedge its returns against equity market fluctuations. Rather, an equity market-neutral fund with trades based on a trading style, such as trend following, is more likely to have returns correlated with other funds that use trend following, whether they are equity market-neutral funds or not.

9.5.3 Understanding Funds Based on Marketwide Factors

The pioneering work of Fama and French, discussed in Chapter 6, indicates that individual equity returns can be explained by identifiable marketwide factors, such as size. The key to this type of analysis is the reliance on an arbitrage-free model of returns that applies to all assets and all funds in the market. Researchers develop relevant factors by (1) developing a concept of how the returns experienced by underlying securities in the market might vary based on a particular variable (e.g., size), (2) dividing the sample into two subgroups based on that variable (e.g., a large-cap group and a small-cap group), (3) estimating the return spread from being long one of the groups and short the other group, and (4) empirically examining whether returns from the entire sample of securities are consistently explained by the return spread.

There are three distinguishing characteristics to multifactor analysis using marketwide return factors: (1) using tradable factors that are identified as the spread between the returns of two groups of stocks (e.g., the return of small-cap stocks minus the return of large-cap stocks), (2) using empirically identified factors rather than factors identified with theory, and (3) for each asset (or each fund), finding empirically estimated exposures to the factors rather than risk exposures identified through fundamental analysis of the asset or fund.

The factors in a Fama-French style of analysis are referred to as tradable because an investor could receive the returns of each factor by holding long positions in one set of stocks (e.g., small-cap stocks) and short positions in another set of stocks (e.g., large-cap stocks). When the factors are tradable, there are two important economic implications: (1) The intercept of the model in an efficient market must be equal to the riskless rate, and (2) the model itself can be described as an arbitrage-free relationship, because if the model did not have a mean-zero error term or intercept equal to the riskless rate, there would be an arbitrage opportunity. In other words, if some error terms were consistently positive or negative, a market participant could earn superior risk-adjusted returns with long positions in the assets with generally positive error terms, and short positions in the assets with consistently negative error terms. Fama and French identified a market factor, a size factor, and a value factor. Other researchers claim evidence of many other factors.

How can this marketwide factor approach of Fama and French be extended to alternative investments? To what extent can hedge fund returns be well explained by marketwide factors? Fung and Hsieh propose seven observable and tradable factors:

1. The return of the S&P 500 minus the risk-free return
2. Small-cap stock returns minus large-cap stock returns
3. The return of the 10-year Treasury bond minus the risk-free return
4. The return of Baa-rated bonds minus the return of the 10-year Treasury bond
5. The return of a portfolio of call and put options on bonds
6. The return of a portfolio of call and put options on currencies
7. The return of a portfolio of call and put options on commodities⁵

The options portfolios (factors 5, 6, and 7) refer to portfolios of calls and puts that are constructed to mimic the behavior of a series of look-back options. A look-back option has a payoff that is based on the value of the underlying asset over a reference period rather than simply the value of the underlying asset at the option's expiration date. Fung and Hsieh estimate that 90% of the return variation in diversified portfolios of hedge funds can be explained by those seven factors. However, individual hedge fund returns are not so well explained.

In summary, there are three distinguishing characteristics to multifactor analysis using marketwide return factors: tradable factors, empirically identified factors, and empirically estimated exposures. In empirical testing, an individual hedge fund's return is not explained well by marketwide factors. However, diversified portfolios of hedge funds can be well explained by seven factors, which include two equity market factors, two bond market factors, and three look-back option factors related to three different markets.

9.5.4 Understanding Funds Based on Specialized Market Factors

A final and emerging approach to analyzing hedge fund returns with multiple factors is related to hedge fund replication. As introduced in Chapter 2, hedge fund replication is the process of mimicking the performance of a particular hedge fund investment strategy using different assets or a different investment process. For example, a convertible bond arbitrage fund may hold long positions in convertible bonds, hedged with short positions in equities that are selected using a proprietary model and the skilled discretion of the fund's manager. One hedge fund replication strategy might be to try to replicate the convertible bond strategy's returns using different underlying assets, such as a portfolio of equity indices, bond indices, and call options. This strategy is often used to create liquid products that attempt to replicate a strategy that uses illiquid securities by designing a strategy that uses liquid securities. Another fund replication strategy might be to try to replicate the returns of a skill-based proprietary strategy using a naïve and mechanical trading model applied to positions similar to the positions being held by the fund being replicated.

In the context of multifactor return models, hedge fund replication involves identifying specialized market factors and estimating fund exposures to those factors such that a portfolio of other securities can be constructed that generates beta similar to a selected fund.

The difference between this approach and the marketwide factor approach is that here, the factors are selected to be tailored to the specifics of a particular fund rather than gathered as marketwide factors. Factors in a marketwide approach are selected based on how they explain returns of all of the assets in a market. The factors in a specialized market factor approach are specifically identified and selected to represent the returns to a specific fund. The factors may be identified empirically by searching for historical correlations between a fund's returns and potential factors, or they may be identified through an understanding of the fund's strategy.

In summary, the specialized market factor approach to hedge fund replication uses the returns of specially chosen factors to explain the return of each particular fund. This approach assumes that the manager's beta exposure and pursuit of alpha may be predictable enough that the returns of the fund can be closely linked to these specialized market-based factors. For example, the returns to a U.S. merger arbitrage hedge fund may be highly correlated with a factor that contains hedged positions in all announced U.S. mergers.