Economic interpretability

In order to label the common factors extracted, the returns of the CFPs could be regressed against a number of representative indices in the spirit of asset-class-factor

12

An example including the manual calculation as well as a macro version is available from the author on request. 12 > 0 = = = (1 + ) = = ...(4.36) =( ) ...(4.37)

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models and additional asset-bases-style (ABS) factors as described in Fung and Hsieh (2003). The model is of the type described in section 4.3.1: asset returns are the returns of the factor portfolios and the factors are observable returns on asset indices, yield spreads or returns on dummy portfolios. The choice of regressors was in accordance with previous research on asset-class-factor models and their application to hedge funds (see section 3.2.2). In particular, a distinction was made between equity indices, yield curve proxies, the Fama-French and momentum factors, cash proxies and the primitive-trend-following factors derived from Fung and Hsieh (2002a). A brief explanation for each regressor can be found in Table 5.1, outlining the composition as well as the data source.

In addition to Sharpe’s asset factors, the factors derived from Fama and French (1992, 1993, 1996) are considered as explanatory factors in pricing models. Empirical evidence shows that both firm size and book-to-market ratio are proxies for exposure to systematic risk in hedge funds not captured by the Sharpe’s original factors. The two additional factors are denoted SMB (small minus big firm size) and HML (high book-to- market ratio minus low ratio):

with as the return of portfolios with low, medium, and high-market-capitalisation stocks and as the return of portfolios sorted according to low, medium, and high book-to- market ratios. Here is the difference in return between an equally weighted long position in the three small firm portfolios and a short position in three big firm portfolios. is defined as the difference in return between an equally weighted long position in high B portfolios and a short position in low B portfolios.

=1₃ + + 1₃ + + ...(4.38)

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A natural extension of the Fama-French model is the momentum factor WML (winners minus losers) introduced by Carhart (1997). A momentum strategy could be described as ‘buying past winners and shorting past losers’ (Jegadeesh & Titman, 1993). Conversely, a contrarian strategy focuses on past underperformers. Carhart (1997: 73) finds no evidence that momentum funds outperform contrarian funds. WML is the return on a zero-investment portfolio that is long past winners and short past losers. Note here that Fama-French factors are thought to proxy for higher systematic moments (compare Chung et al., 2006). One may differentiate between factors describing the exposure to passive indices (location choice) and factors serving as proxies for a particular trading strategy (style choice).

Fung and Hsieh (2001: 317–326) defined the PTFS as a long position in a look-back straddle. The straddle consists of a call and a put with the same exercise price and expiration, designed to capture the difference between the price of the underlying asset upon maturity and the exercise price. The PTFS attempts to capture the largest price movement of the underlying asset during a specified time interval, where the optimal payout of the PMTS is the maximum price less the minimum price of the asset. The payout profile of the PMTS can be simulated by dynamically rolling standard straddles over the life of the look-back straddle. Fung and Hsieh (2002a) showed empirically that trend-following hedge fund returns are strongly correlated with the returns of the PTFS. A complete list of the regressors including sources is depicted in Table 5.1.

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Allowing for lagged factor exposure for up to three lags, the general asset-class factor model can now be written:

The coefficients are estimated using Ordinary-Least-Squares (OLS). A HAC-consistent standard error estimate of the coefficients is given by the diagonal elements of the matrix below (Newey & West, 1987):

and is defined as follows:

where the weighting function is the Bartlett kernel and is the truncation parameter. The truncation lag is = 4( /100) and is the number of regressors (the floor and function maps a real number to the largest previous or the smallest following integer). In order to avoid multicollinearity between the regressors, a stepwise regression algorithm was employed.

The forward stepwise regression algorithm is described in detail in Neter, Kutner, Nachtsheim and Wasserman (1996: 348–352). At every step, a regressor was entered into the model if the increase in explanatory power outweighed the ‘cost’ associated

= + + + + + + + + + + + = + …(4.40 ) = ( ) ( ) …(4.41 ) = …(4.42) + 1 + 1 + = +1 =1

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with the decrease in degrees of freedom. Entering variables were tested for collinearity with existing regressors and would not enter the model if any (adjusted ) from regressing all regressors on all other independent variables exceeded a specified threshold. Conversely, existing regressors were removed if their removal did not cause a significant decrease in explanatory power in the movements of the regressand. The partial F-test allowed for a formal comparison of the reduced model to the full model at every step. In the two variable case where and are indicators of regressors from :

where is mean square due to regression, is the mean square error, is the sum of squares for error, and denotes the degrees of freedom. The coefficient estimates and standard error thereof are denoted as and { } respectively. The equality above holds in the absence of heteroscedasticity and autocorrelation in the error terms. The partial F-statistic can be calculated from ( { }) to account for HAC-adjusted standard errors as in Equation 4.36.

It should be noted that the stepwise regression procedure does not always reliably remove multicollinearity from the regressors. The reasons for this are two-fold: firstly, in a stepwise process variables are either added or removed from the list of regressors. This accounts for co-linearity at the individual series level but ignores any effects from entering variables being jointly co-linearly related to existing regressors or combinations thereof. Secondly, the calculated F-statistics do not take into consideration the iterative nature of the estimation procedure. The p-values associated with the inclusion/exclusion of parameters from the iterative model should be adjusted to account for the sequential

= _{( , )}( | ) ...(4.43)

= ( ) ( , )÷ ( , )

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parameter selection process (compare Brooks, 2003: pp. 104, 105). Whilst these limitations were acknowledged, it was the purpose of the stepwise procedure to reduce the number of regressors to facilitate interpretation of the factor exposures rather than to minimize the impact from multi-collinearity. This approach is deemed appropriate since the results from the procedure were used in the nomenclature of hedge fund portfolios/indices only.