Building the replicator

Building the replicator is a straightforward process. The parameter estimates are used to construct replication products returns ( ):

( 10 )

Subject to 1 =

Using the parameter estimates ( ) for the individual hedge funds in our sample, we can now reformulate the question of whether or not a hedge-fund strategy can be cloned to a question about how much of a hedge fund’s expected return is due to risk premium from identifiable factors. Hence, the regressions are first performed to each

9 The number of six month time periods

10 The hedge funds monthly return is regressed on historical returns for the variables.

individual hedge fund and then the regression parameters are averaged across the whole sample.

The method is the same as proposed by Sharpe (1992) for “style analysis”. However, the motivation is quite different as the intercept is dropped to create the best estimate of a weighted average of the factors that best replicate the fund returns. As we drop the constant term, the least-square algorithm forces the factor means to fit the means of the fund. Further, the betas are constrained to sum to one to yield a portfolio interpretation for the weights. The method used in this thesis differs from Sharpe’s (1992) original method, also in the respect that it allows negative betas. Negative betas enable occasionally short selling each of the instruments to achieve risk exposures usual to hedge funds.

The monthly beta coefficients are multiplied with monthly returns for the variables to achieve monthly replicator returns. Next, the next section introduces the tools for analysis.

5.4. Analysis of replicators

In the last step of the methodology, the HP and All replicators’ performance and replication quality are analyzed in respect to each other, hedge fund indices and other benchmark indices. The replicators are analysed with respect to the replication quality and with respect to the replicator performance.

Analysis of replicator performance 5.4.1.

The performance of replicators is analyzed with the return component, volatility, risk-adjusted return and liquidity for the replicator. These parameters are then compared with benchmark indices. Liquidity issues are discussed in more detail in the next chapter.

Return is analyzed with the annualized average return and the volatility is analyzed with the annualized standard volatility. The risk-adjusted return is measured with the annualized Sharpe ratio, which is a risk-adjusted performance measurement and is widely used as a comparison measurement. The ratio takes into account both the return and the risk. Equation 12 presents the annualized Sharpe ratio.

( 11 ) The measurement compares the annualized abnormal return¹¹ against the annualized standard deviation that is a proxy for risk. The annualized mean return for the replicators is subtracted with the US treasury 1-month interest rate and is thereafter divided by the annualized standard deviation for the replicators.

The first-order autocorrelation is used as proxy for liquidity. The first-order autocorrelation is the correlation between a series current return and the time-series previous month return. Multiple researchers (Lo, 2011; Getmansky, Lo, Makarov (2004) observe that a positive value for is a sign for illiquidity risk. Equation 14 presents first-order autocorrelation.

( 12 )

A positive autocorrelation indicates that returns for a replicator do not vary as much as they should between months. A positive autocorrelation decreases the volatility of the returns and hence increases the Sharpe ration, which decreases the robustness of the results.

Analysis of replicators’ replication quality 5.4.2.

The replication quality is measured with correlation, adjusted and root-mean squared error (RMSE). These parameters are then compared with benchmark indices.

HFRI index is chosen to compare the replicators to a broad sample of hedge funds. The DJ CS managed futures index is chosen to compare the replicators to a set of hedge funds with the same strategy.

The correlation is a natural measurement for estimating replication quality. However, the correlation only relies on second-order co-movement, and is unable to measure first-order movement. Hence the correlation can be high, even though the replicator constantly yields lower returns. Secondly, the correlation coefficient is unable to consider the magnitude of the movement.

11 Return excess to the benchmark index.

, 1

The adjusted measures the explanatory power of the replicator to the benchmark.

The test is performed to analyze how much the replicator follows the benchmark. The replicator is the independent variable and the benchmark is the dependent variable.

Equation 14 presents the OLS regression.

(13 ) The parameter is analyzed with a standard OLS regression. Equation 15 presents the formula for the adjusted

Adjusted = ^{( )( )}

( 14 )

The is the sample R-square, p is number of parameters and N is total sample size.

Hence, the adjusted penalizes the amount of parameters. A high adjusted show that the independent variable explains a large part of the variation in the dependent variable.

The RMSE is a frequently used measure of the difference between values predicted by a model or an estimator and the observed values. Equation 16 presents the formula for RMSE.

RMSE = √ ∑ ( ̂ ) ( 15 )

The measurement observes the residuals and aggregates them into a single measure of predictive power. The RMSE can also be interpreted as the tracking error between the replicator and its benchmark and a low RMSE indicated good replication quality.

6 DATA

In hedge fund studies multiple different data biases exists, which needs to be considerate to confirm robust results. Therefore, this section begins with a discussion of the major data biases in hedge fund research. Thereafter, the section presents the descriptive statistics for the time-series length, individual hedge funds and the variable returns.