Hedge Fund Investment through Piecewise Linear Regression and Optimization

Transcription

1 Hedge Fund Investment through Piecewise Linear Regression and Optimization Fei Pan Krannert School of Management, Purdue University Bo Zeng School of Industrial Engineering, Purdue University West Lafayette, IN September 8, 2007 Abstract It is conjectured that size of hedge funds has some impact on returns. In this paper, we investigate the relationship between them and apply the results to construct an investment model. We first implement a learning algorithm to construct a piecewise linear regression model which shows that a fund s AUM (Asset Under Management) has different effects in different situations on its return. Then, with consideration of the various scenarios, we propose a robust optimization model to maximize the expected profit. Finally, we present the computational results that illustrate the strength of our two-stage procedure. Keywords: hedge fund, piecewise linear regression, portfolio optimization. 1 Introduction In the last decades, hedge fund industry has received tremendous attention and the investment in this industry increase rapidly. Currently, hedge funds have already become one of the most important investment tools in the world. Because it was observed that a low correlation exists with the traditional equity market, many studies have been carried out to construct benchmarks for the hedge fund industry and to investigate the factors that affect the performance of hedge funds. It was expected that these research and the associated analysis and mathematical models can help people better understand the hedge fund industry and finally can be used to make wiser investment. The existing research includes the study of the effect of the traditional common factors on the the performance of hedge funds. Because the limitation of these factors, Fung and Hsieh [9] and Schneeweis and Spurgin [20] proposed to use some additional factors to study the behavior of hedge funds. In particular, because assets under management (AUM) in hedge funds increase impressively, several questions on this factor are naturally raised by current and potential investors: will the size of AUM affect the return of hedge fund?, how do we 1

2 wisely choose the right funds? and how do we build our investment portfolio to maximize our return?. Because of the importance of these questions, many research works have been published in order to address them recently. Throughout this paper, we AUM of single hedge fund and the fund size interchangeably. Herzberg and Mozes [14] find that smaller hedge funds perform better in a small extent than larger funds but their Sharpe ratios are significantly better than larger funds. Hedges [13] observe that smaller funds outperform larger funds and he also finds that mid-sized funds perform the worst. In [11], Gregoriou and Rouah show that there is no statistically significant correlation between the size of hedge funds and their performance. Their testing set includes 204 hedge funds and 72 funds of hedge funds. Edwards and Caglayan [4] claim that hedge fund performance increases at a declining rate as fund sizes increase. Because their results have a positive coefficient on the size variable and a negative coefficient on the size reciprocal variable, they believe that hedge fund performance increases at a declining rate as fund sizes increase. We mention that Ammann and Moerth [1] carry a comprehensive study on the effect of fund size on the return using a large data set. Their results also confirm that the size of fund has a negative effect on the return. Furthermore, they also reveal that very small funds also underperform on average. Avellaneda and Besson [2] study the overall performance of hedge fund industry against the total AUM of this industry. Their result shows that the increasing size of overall AUM erodes the return, i.e. the coefficient of AUM in the return function is negative in a logarithmic scale. We note that nearly all the study of the relationship between AUM and fund performance is based on linear or piecewise linear regression methods along with other related factors. The classical linear regression method is easy to be implemented and can be interpreted statistically. However, a significant disadvantage is that this method cannot describe the nonlinear relationships between factors and the response. An improved approach is piecewise linear regression method which are widely used to model the nonlinear situations and it can greatly reduce the error resulted from simple linear regression. Nevertheless, applying piecewise linear regression requires a deep understanding on the data set and experiment setting because the critical points are chosen in a subjective fashion. Furthermore, it becomes very difficult to choose the right critical points if data set is high-dimensional. To overcome these difficulties and to advance the study of the impact of AUM on the returns, this paper focuses on applying a learning-based piecewise linear regression method proposed by Ferrari-Trecate and Muselli [7] to study their relationship, and on using the regression results to construct robust portfolio. This paper is organized as following: in Section 2, we describe the data set to be used. In Section 3, after briefly reviewing traditional linear regression and learning-based piecewise linear regression methods, we present and compare the regression results. In Section 4, we construct a robust portfolio optimization model using the results from Section 3 with consideration of various scenarios. In Section 5, we present some simulated examples and the computational results that show the advantages of the investment plan using our regression and optimization procedure. Finally, in Section 6, we conclude this paper and discuss some future research directions. 2 Data This section briefly describes the data set of hedge funds and the risk factors used in our study. 2

3 2.1 Hedge Fund Data Our comprehensive hedge fund data set is composed by a union of three commercial databases, InvestorForce, CISDM (Academic Version) and HedgeFund.net. InvestorForce is a hedge fund database now owned by Morningstar and around new 2,500 hedge funds information were added from Morningstar (InvestorForce originally has more than 6,000 hedge funds). CISDM offers an academic version database which contains more than 6,400 hedge funds and some fund of funds. HedgeFund.net s universe has more than 2,300 hedge funds under 35 different style categories. Those three databases have both fund performance information, such as monthly net-of-fee returns, asset under management (AUM) and NAV, and other characteristics, such as management fee, incentive fee, fund strategy, and information about fund managers. As we known, it is legal that fund managers do not report their monthly performance. Therefore, some bias may exist because some funds may skip or stop reporting their data with bad performance in order to keep a good reputation. To guarantee the quality of dataset, any fund without complete required information is eliminated to avoid biases. Therefore, our data pool is filtered by following criteria: the time period is from January 1996 until December 2005 and monthly AUM and rate of return should be continuously reported for at least 6 years. On the other hand, this paper concentrates those funds that the strategy should be Equity Long/Short and geographical location is US. As a result, we obtain 243 funds satisfying these four requirements. The overall average monthly net-of-fee return is around 1.16 percent and variance is around Because the intensive computation is involved, we draw a subset of these 243 funds by randomly selecting 30 funds and then use them as the sample set in the study. 2.2 Factors Here, we list all 13 risk factors according to the categories they belong to: (i) EQUITIES: S&P500 index, MSCI World (MSCI) index, and Russell2000 index; (ii) BONDS: Lehman Aggregate A+ Bond Index (LAB); (iii) COMMODITIES: Dow Jones AIG Commodity Index (DJAC) and Goldman Sachs Commodity Index (GSCI); (iv) VOLATILITY: CBOE VOLATIL- ITY INDEX (VIX); (v) CREDIT: the spread between the Lehman BAA Corporate Bond Index and the Lehman Treasury Index (CREDIT); (vi) CURRENCY: US Dollar Index (USDX); (vii) FAMA/FRENCH FACTORS: RM-RF, SMB, and HML; (viii) FUND: AUM with a logarithmic scale, log(aum). We believe that these factors provide a comprehensive cross-section of risk exposures for Equity Long/Short hedge funds. Some of above factors are obtained from Datastream database, such as Lehman Aggregate A+ Bond Index, Lehman BAA Corporate Bond Index, and Lehman Treasury Index. Other factors are obtained from appropriate web sites, such as S&P500 index and Russell2000 index from Yahoo!Finance, and US Dollar Index from nybot.com. The data of FAMA/FRENCH FACTORS is taken from Dr. French webpage [8]. 3 Study of AUM Using Learned Piecewise Linear Rregression Linear regression method has been widely applied to study hedge funds. Plenty of results and analysis have been published. In this section, we first review the existing results. Then, we briefly review the piecewise linear regression method based a learning algorithm and discuss 3

4 our modification of this algorithm. Finally, we present the computation results using this piecewise linear regression method and compare our results to existing results. 3.1 Existing Regression Model To study the return of fund, Jensen [17], Avellaneda and Besson [2] and Getmansky [10] proposed several single factor regression model to study the relationship between fund size and the return. Note that the model in [10] is in quadratic form. In additional to these single factor analysis, many research papers implement multi-factors regression model to estimate returns of funds. Schneeweis and Spurgin [20] applied a 9-factor regression model, including 6 categories: Equities, Bond, Commodity, Currency, Credit Risk, and Volatilities. Hasanhodzic and Lo [12] reduced the number of factor to 6, all of them are still taken from the same 6 categories, to replicate the hedge fund returns. Fung and Hsieh [9] identify the common risk in Equity Long/Short hedge funds, including the Fama and French[5] 3-factor model augmented with Jagedeesh and Titman [16] s momentum factor. Chen and Liang [3] showed that adding Fama and French [5] 3-factor model can improve the explanatory power to regression model on hedge fund returns. However, Jagannathan and Novikov [15] found that such additional factors (such as HML and SMB) can not improve the estimates of timing ability. In our piecewise linear regression model, we still add Fama and French s 3 factors into our regression model and then test their effectiveness. In our piecewise linear regression model, we still initially have Fama and French s 3 factors into our regression model and then test their effectiveness. Note that some papers proposed to apply linear regression model to a subset of funds satisfying some conditions. Getmansky [10] considers funds that are partitioned into 20 different bins based on fund size. Ammann and Moerth [1] manually separate funds into one hundred pieces according to fund size and then they construct a multiple regression model based on 18 factors and use the results to describe relationship of returns and AUM. These models can also be treated as piecewise linear regression models while the number of pieces is predefined by authors observation or experience. We first teste the significance of all factors listed in Section 2 based on single factor approach and results is given in Table 6. Results show that all the EQUITIES factors, all the FAMA/FRENCH factors, Dow Jones AIG Commodity Index (DJAC), CBOE VOLATILITY INDEX (VIX), CREDIT, and log(aum) are significant in the single factor model at 5% significance level. Russell2000 has the highest power in explanatory for those selected Equity Long/Short hedge funds. Next, we further investigate the these factors to construct a more descriptive multi-factor regression model. In multiple regression model, we first use (3.1) to describe funds return R t at time t with risk factors r j for j = 1,..., s where s = 13 initially. R t = α 0 + β 1 r β 13 r s + ɛ t (3.1) Results in Table 7 represent the regression of all the factors on the return rates of the 2670 sample funds. We observed that the R-square value is 14.4% of this 13-factor model. Therefore, to reduce the multicollinearities, we implemented factor reduction methods, forward and backward stepwise approaches, to choose a subset of these 13 risk factors. As a result, we obtain a model with only 8 factors, i.e. s = 8 in (3.1). They are: α 0, Russell2000, S&P500, MSCI, DJACI, VIX, USDX, HML, log(aum). Regression results of this 8-factor model is presented in Table 8. In fact, using this 8-factor regression model, the adjusted R-squared 4

5 and R-square are nearly same as the those of 13-factor model. Unlike the results in Chen and Liang [3] and Jagannathan and Novikov [15], we observe that only HLM of Fama and French s 3 factors remains in our 8-factor model. We then apply this 8-factor model to our 30 randomly selected funds to study the effect of AUM. In Figure 1, we show the regression coefficient of log(aum) and the value of log(aum) for these funds. It is clear that most coefficients of AUM in the regression model for these funds are negative, which implies that the fund size has somehow negative effect on its return. However, we note that, the capability of the linear regression model is very limited in describing hedge funds. One reason is the really small R-square value from simple linear regression model, in general it is from 0.1 to 0.2, which indicates the model is not very descriptive. Another reason is, in Figure 1, there are still many funds have nonnegative coefficients for their AUMs. Based on these observations, to further investigate the behavior of hedge funds quantitatively, we apply an improved regression method to study them. In Section 3.2, we review a piecewise linear regression algorithm that is based on a learning method. Then, we present our computation results using this method in Section 3.3. We note that this piecewise linear regression method is significantly more descriptive than simple linear regression approach. Figure 1: Regression Coefficient β AUM and log(aum) 3.2 Learning Algorithm Based Piecewise Linear Regression Unlike the simple linear regression approach, piecewise linear regression is a more powerful tool to describe the connection between data since it generalizes simple linear regression if there is only one piece, see Ryan [19]. However, the traditional piecewise linear regression method is heavily depending on the user. Because it is a subjective decision to determine the critical (change) points where a new regression model will be used to replace the previous one. In addition to its subjectivity, obtaining these critical (change) points is also very challenging in high-dimensional data sets. Because of these two majors disadvantages, Ferrari-Trecate and Muselli [7] introduce a piecewise linear regression method using a learning algorithm. This method determines the critical points through a learning process which can determine the critical (change) points in a objective fashion without people interference. We list their procedure in Algorithm 1. Because we are focusing on data description, classification part of 5

6 this algorithm is not included here. We use v to denote the regression result, i.e. v contains the constant and the coefficients of factors in the linear regression model. Similarly, we use r i to denote the vector of all risk factors. Algorithm 1. (from Ferrari-Trecate and Muselli [7]) (1) For each i = 1,, m (number of records), do: (i) generate the local data set C i that contains the sample ( r i, R i,) and c 1 pairs ( r j, R j ) such that they are the nearest neighbors to ( r i, R i ); (ii) perform simple linear regression to C i to produce a regression model v i for i = 1,..., n. (2) For all v i, i = 1,..., N, apply a K-mean clustering process to partition them into S j, j = 1,..., K. (3) Perform a simple linear regression on S j, j = 1,..., K to obtain K pieces linear regression models V j for j = 1,..., K. Note that, a cluster obtained in step (2) corresponds to a linear regression model in step (3). Because of the specific application setting of this paper, we adopt their algorithm with small modifications. We first investigate the value of c, the number of neighbors. Because some equities may have a seasonally behavior, c is chosen as t 15 where t is the total number of funds to remove seasonality factors. Then, we consider the number of clusters, i.e. the number of pieces. It is widely accepted that R-square is a measure of fitness of regression models. Let U be the sum of product of R-square value of each regression model and the size of that cluster. Clearly, U can be used to measure the fitness of the resulting piecewise linear regression model. In Figure 2, we show the relation between U and the number of clusters using all data from randomly selected 30 hedge funds from 243 funds. Clearly, in general, U s value is increasing as K increases. However, it is desired that the number of clusters should small and the cluster size should be large. Therefore, we choose K equal to the smallest K other than 1 such that its marginal increasement of U is negligible or negative. For example, in the data set described in Figure 2, we will let K = 4. In fact, we found that in most cases K are equal to 4. Therefore, in the remainder of this paper, without mention specifically, we assume K = Computation Results and Discussion In this section, we apply this learning based piecewise linear regression method single funds. In Example 1, we present the piecewise linear regression results for a typical funds and compare them to those obtained from simple linear regression model. Example 1. In Table 1, we give the results for a fund based on its data from We set the number of clusters equal to 4. We observe from the R-square value in Table 1 that, piecewise linear regression model can better describe the behavior of hedge fund than simple linear regression model because the all R-square values in each cluster are larger than that from simple regression model. Another important observation is about the effect of AUM. Although the coefficient of log(aum) is positive (3.474) in simple regression model, we note that in different clusters generated by piecewise linear regression, its coefficient varies. In Table 2, we present the results from piecewise linear regression and simple linear regression on regression coefficients of log(au M). We further notice that although most regression 6

7 Figure 2: U and K coefficients of log(aum) are negative (7 out of 10) by simple linear regression, it could be both positive and negative in piecewise linear regression, depending on the cluster in which it is. This observation suggests that the size of fund actually produces effect on the return along with other risk factors and the effect varies with other factors. This observation is very different from the conclusions from Ammann and Moerth [1] and Avellaneda and Besson [2] that the size of fund has a negative effect on the return. Based on our computation results, we believe that in each cluster all other risk factors construct a background in which the AUM affects return of the fund. We then say a cluster (without AUM and return) is a scenario for AUM and return. Furthermore, this observation motivate us to take advantage of the information of scenarios and the connection of AUM and return to build a more profitable portfolio of hedge funds. Such as procedure is discussed in Section 4. 4 An Robust Optimized Investment Model Classical mean-variance model for investment has been widely used and documented; see Oberuc [18] for details. Its application in hedge funds has also been studied by many researchers, including Favre and Galeano [6].(more papers listed here). As we discussed in Section 3, the learned piecewise linear regression can capture more important information and can better describe the implied relationship between return, AUM as well as other factors. It not only helps to rule out some hedge funds but also provides a more realistic base for investors. Therefore, it is expected that we can construct a more profitable portfolio than that uses traditional methods. In this section, we will use the results from learned piecewise linear regression and introduce a robust optimization model that considerate various underlying scenarios to build the hedge fund portfolio. Now, we describe our basic strategy. First, we assume an oracle that can forecast these economic factors. These forecasted factors will be the elements of our next step future scenario that will be treated as a point in high-dimensional space. In order to compare the forecasted 7

8 Table 1: Comparison of Results from Piecewise Linear Regression and Simple Linear Regression K = 4 Cluster 1 Cluster 2 Cluster 3 Cluster 4 Simple Regression Number of Records R-squared 34.0% 53.0% 63.7% 54.4% 26.1% Coefficients ALPHA Russell S&P MSCI DJACI VIX USDX HML log(au M) Table 2: Regression Coefficients of log(aum) in 10 funds Cluster 1 Cluster 2 Cluster 3 Cluster 4 Simple Regression Fund Fund Fund Fund Fund Fund Fund Fund Fund Fund scenario with those in clusters for each fund, we make use of the inverse of Euclidean distance to represent their similarity. The inverse of the Euclidean distance will further be normalized and therefore gives a weight for the expected performance of a single fund in all scenarios. Finally, we maximize the weighted expected return subject to the variance constraint and available money constraint as well as investment upper bound requirement for each fund. To be realistic, we assume that the investment into fund i cannot be l i % or more over the current AUM. Next, we present and explain the mathematical model of our approach. Assume that there are n hedge funds in the market and the available money is I and S 0 is the forecasted scenario that consists of 7 factors. Let A i,0 be the current AUM of fund i, c i be the number of clusters for hedge fund i and s i,j be the j th scenario of fund i. We use d(x, y) be the Euclidean distance between x and y. Also, let w i be the portion of I that goes to fund i. Therefore, we denote return of fund i in scenario j as R(i, j). Note that, according to the results of Section 3, return of fund i is a function of AUM. So, once w i and I are known, R(i, j) can be obtained using the regression results, i.e. R(i, j) = f i,j (w i I) where f i,j is the linear regression function of fund i in scenario j. Let σ i,j be the standard deviation of fund i in scenario j. We also let ρ i,j be the correlation coefficient between fund i and j. 8

9 Note that we compute ρ i,j using all the data of fund i and j whenever both fund i and j have complete information. This operation is reasonable since both funds are measured in the same background and the economics risk factors are shared by both of them. To simplify our expression, we define D i,j := 1 d(s 0,s j ) ci j=1 1 d(s 0,s j ) (4.1) as the normalized measure of similarity between S 0 and s i,j for fund i. n c i max D i,j r i,j w i I i=1 j=1 n n s.t. ρ i,k w i w k ( D i,j σ i,j )( D k,j σ k,j ) v 0 i=1 k=1 j j r i,j = f i,j (A 0 + w i I) i = 1,..., n, j = 1,..., c i n w i = 1 i=1 w i I A i,0 l i A i,0 i = 1,..., n (4.2) where v 0 is the upper bound of overall variance of this portfolio. Note that, this model considers all various scenarios and the goal is to maximize the expected return. Given a reliable forecasting technique, although this model may not give the best portfolio, it provides a robust optimal solution with consideration of forecasting risk. In fact, unlike forecasting single factors, multi-factor forecasting reduces the deviation of forecast values and the realized values and therefore it also support the robust property of this model. 5 Simulation and Numerical Comparison In this section, we outline our experiment procedure and present the computational results of our robust optimization model to the 10 randomly selected hedge fund. To illustrate the strength of our model, we compare the results of (4.2) and the results from simple regression part and the model (5.1). max s.t. n r i w i I i=1 n i=1 k=1 n ρ i,k w i w k σ i σ k v 0 r i = ˆf i (A 0 + w i I) i = 1,..., n w i = 1 i w i I A 0 l i A i,0 i = 1,..., n (5.1) 9

10 where ˆf i is the regression model obtained from simple regression method. We perform our computation and comparison using the data in We assume the available investment I = 10 9 and v 0 = In the forecasting part, we use weighted history data to predict the value in the next month. For example, we predict the value of S&P 500 on January 2006 by a summation of 0.5 of that from December 2005, 0.3 of that from November 2005 and 0.2 of that from October We apply this technique to all months in Note that, in the computation of r i,j, A 0 is the realized value in previous month. When obtaining the setting (all 7 factors) for a new month, we can determine which cluster it belongs to using the normalized measure of similarity D ij. To reduce the noise in our computation we define a threshold with a value of 0.1. If D ij0 is less than to this value, we conclude that this new setting does not belong not j0 th scenario for fund i. Then, we renormalize D ij without consider j 0. In Example 2, we illustrate our simple forecasting operation and computation of normalized similarity. Example 2. In Table 3, we first listed the realized values for all 7 factors and then present the forecasted values for them. Table 3: Forecasted Value for Jan Time Russell2000 S&P500 MSCI DJACI VIX USDX HML Oct Nov Dec Forecasted In Table 4, we compute the normalized similarity coefficient D 1,j for this forecasted month. Table 4: Normalized Similarity D 1j Cluster 1 Cluster 2 Cluster 3 Cluster 4 Fund In Table 9, we give the computed standard deviation of fund i in cluster j, σ i,j. In Table 10, we give the correlation coefficients for these 10 funds. Similar to above discussion, we obtain the parameters for the model in (5.1). In Table 5, we list the results from both models, (4.2) and (5.1), in 12 months of 2006 using GAMS nonlinear programming solvers. Results for 2006 year of both PWL regression and linear regression are list in Table 5, which shows that PWL regression method is better than traditional linear regression model in portfolio management. To Fei, need Fei s more serious comments on this results. like percentage increasement, balba... 6 Conclusion In this paper, we first investigate the connection between AUM and return of hedge funds using a novel multi-factor regression method, a learning algorithm based piecewise linear regression method. Based on our results, we further propose a robust optimization model to construct 10

11 Table 5: Expected Return Time PWL Regression (%) Simple Regression (%) Equally Investment (%) 1/1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ /1/ Average hedge fund portfolio. To the best of our knowledge, this is the first time to apply learning based on piecewise linear regression method to study the behavior of hedge funds and to build hedge fund portfolio. We conclude from our study that: (i) Piecewise linear regression method is more descriptive than traditional regression method; (ii) AUM has a significant effect on the return; (iii) This effect could be positive as well as negative, which depends on the economic context consisting of other economic factors; (iv) The effect of AUM on the return and our robust optimization model can be used to construct a better hedge fund portfolio. As a future research direction, we will develop a more effective approach to determine the number of clusters and will study the improved piecewise linear regression method that is more descriptive for hedge fund. Another research direction is developing and applying novel forecasting methods to incorporate scenarios information in hedge fund portfolio management. Finally, we are currently carrying an empirical study using piecewise linear regression method on large-scale hedge fund data set. References [1] M. Ammann and P. Moerth. Impact of fund size on hedge fund performance. Journal of Asset Management, 6: , [2] M. Avellaneda and P. Besson. Hedge-funds: How big is big? working paper, New York University, [3] Y. Chen and B. Liang. Do market timing hedge funds time the market. to appear Journal of Financial and Quantitative analysis, [4] F. Edwards and M. Caglayan. Hedge fund performance and manager skill. Journal of Futures Markets, 21: , [5] E. Fama and K. French. Common risk factors in the returns on stocks and bonds. Journal of Financial Economics, 33:3 56,

12 [6] L Favre and J. Galeano. Mean cmodified value-at-risk optimization with hedge funds. The Journal of Alternative Investments, 5:21 25, [7] G. Ferrari-Trecate and M. Muselli. A new learning method for piecewise linear regression. In ICANN 02: Proceedings of the International Conference on Artificial Neural Networks, pages , London, UK, Springer-Verlag. [8] K. French. Fama-french benchmark factors. faculty/ken.french. [9] W. Fung and D. Hsieh. Empirical characteristics of dynamic trading strategies: The case of hedge funds. The Review of Financial Studies, 10: , [10] Mila Getmansky. The life cycle of hedge funds: Fund flows, size and performance,working paper. Technical report. [11] G. Gregoriou and P. Rouah. Large versus small hedge funds: Does size affect performance? Journal of Alternative Investments, 5:75 77, [12] J. Hasanhodzic and A. Lo. Can hedge-fund returns be replicated?: The linear case. research paper, Massachusetts Institute of Technology (MIT), [13] R. Hedges. Size vs. performance in the hedge fund industry. Journal of Financial Transformation, 10:14 17, [14] M. Herzberg and H. Mozes. The persistence of hedge fund risk: Evidence and implications for investors. Journal of Alternative Investments, 6:22 42, [15] R. Jagannathan and D. Novikov. Do hot hands persist among hedge fund managers? an empirical evaluation. working paper, [16] N. Jagedeesh and S. Titman. Returns to buying winners and selling losers: Implications for stock market efficiency. Journal of Finance, 48:93 130, [17] M. Jensen. The performance of mutual funds in the period Journal of Finance, 23: , [18] Richard E. Oberuc. Dynamic Portfolio Theory and Management. McGraw-Hill, [19] T. Ryan. Modern Regression Methods. Wiley-Interscience, [20] T. Schneeweis and r. Spurgin. Multifactor analysis of hedge fund, managed futures, and mutual fund return and risk characteristics. Journal of Alternative Investments, 1:1 24,

13 Appendix Table 6: Single Factor Analysis Risk Factors Coefficients Standard Errors T-stat P-value S&P MSCI Russell LAB DJACI GSCI VIX CREDIT USDX Rm-Rf SMB HML log(au M) Table 7: Linear Regression with 13 Risk Factors Risk Factors Coefficients Standard Errors T-stat P-value Constant S &P MSCI Russell LAB DJACI GSCI VIX CREDIT USDX RM-RF SMB HML log(au M)

14 Table 8: Linear Regression with 8 Risk Factors Risk Factors Coefficients Standard Errors T-stat P-value Constant Russell S&P MSCI DJACI VIX USDX HML log(au M) Table 9: σ i,j of 10 funds Cluster 1 Cluster 2 Cluster 3 Cluster 4 Fund Fund Fund Fund Fund Fund Fund Fund Fund Fund

15 Table 10: Correlation Coefficients ρ i,j Fund 1 Fund 2 Fund 3 Fund 4 Fund 5 Fund 6 Fund 7 Fund Fund Fund Fund Fund Fund Fund Fund Fund Fund Fund 8 Fund 9 Fund 10 Fund Fund Fund Fund Fund Fund Fund Fund Fund Fund