The Diversification Benefits of Hedge Funds and Funds of Hedge Funds
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The diversification benefits of hedge funds and funds of hedge funds Maher Kooli School of Business and Management, University of Quebec in Montreal (UQAM), CP 6192, Succursale Centre-Ville, Montreal Quebec, Canada, H3C 4R2. Tel: þ 1 514 987 3000, ext. 2082; Fax: þ 1 514 987 0422; E-mail: [email protected] Received (in revised form): 7th September, 2006 Maher Kooli is a professor of finance at the School of Business and Management, University of Quebec in Montreal (UQAM). He holds a PhD in finance from Laval University (Quebec) and was a postdoctoral researcher in finance at the Center of Interuniversity Research and Analysis on Organisations. He also worked as a research advisor for la Caisse de Depot et Placement de Quebec (CDP Capital). His current research interests include alternative investments, initial public offerings and mergers and acquisitions. Practical applications The hedge funds (HF) and the funds of HF universe is evolving rapidly. To explain the growing attraction of these alternative investments, investors often rely on the existence of free lunch. The paper examines the diversification benefits of HF and funds of HF. We rely on mean–variance spanning tests to analyse this issue. It is found that including HF or fund of HF portfolios to a set of benchmark portfolios (US stocks only) provides an extra return for a unit increase in standard deviation. However, this conclusion is less evident when we consider an internationally diversified portfolio as a benchmark. Our message to portfolio managers is that the value added by HF is uncertain and depends upon the opportunity set that is considered. Abstract funds of HF, as an asset class, does bring diversification We examine whether investors can improve their benefits for mean–variance investors, even when we investment opportunity set through the addition of a enlarge our investment opportunity set by including hedge fund (HF) or fund of HF portfolio to different sets fixed income, international assets and commodities. of benchmark portfolios. Using data from 1994 to Derivatives Use, Trading & Regulation (2007) 12, 2004, we find that HF, as an asset class, improve the 290–300. doi:10.1057/palgrave.dutr.1850053 mean–variance frontier of sets of benchmark portfolios sorted by firm size and book-to-market ratio. However, Keywords: hedge funds; funds of funds; mean- we find that the improvement comes mainly from a variance spanning test leftward shift of the global minimum-variance portfolio rather than the tangency portfolio. Furthermore, Derivatives Use, INTRODUCTION Trading & Regulation, investors who already hold a diversified portfolio do not Vol. 12 No. 4, 2007, pp. 290–300 improve their investment opportunity set by adding HF Hedge fund (HF) investment has increased r 2007 Palgrave Macmillan Ltd portfolios. On the other hand, we find that investing in dramatically in the last decade. For instance, Van 1357-0927 $30.00 290 Derivatives Use, Trading & Regulation Volume 12 Number 4 2007 www.palgrave-journals.com/dutr Hedge Fund Advisors International, Inc. reports The rest of the paper is organised as follows: a total of 8,100 global HF managing close $1.2 The next section sets out the methods used to trillion in capital at the end of year 2005. To evaluate mean–variance spanning; The explain the growing attraction of these funds, subsequent section describes the data for HF, investors often rely on the existence of free FOF and benchmark portfolios; The lunch. Indeed, HF table higher average returns penultimate section discusses the empirical than standard market indexes and higher Sharpe results and the final section concludes the paper. ratio. More important, they tend to have lower correlations with most passive investments. Low TESTING FOR DIVERSIFICATION correlation means that adding HF to a portfolio BENEFITS can actually reduce risk and still provide the benefits of higher average returns. However, HF (or FOF) can play an important role as a these conclusions depend on the investor’s ability diversifier due to its low or negative correlation to achieve the performance of passive to other asset classes. This highlights the investments. For those who do not meet the important role HF (or FOF) may play in the definition of accredited investor, investing in asset allocation mix as it protects the portfolio Funds of funds (FOF) is presently the only way from undesirable moves in the equity market. to gain access to absolute return strategies. We rely on mean–variance spanning tests to Indeed, FOF let investors with as little as evaluate the economic diversification gains of $10,000 to participate in a diversified mix of HF, HF or FOF portfolios for investors (DeRoon 2 3 while traditional HF require a minimum and Nijman; Kan and Zhou ). These tests investment of $250,000. FOF can offer the most enable us to analyse the effect on the mean– attractive risk-adjusted rates of return with low variance frontier of adding new assets to a set of to zero correlation to most traditional portfolios. benchmark assets. For example, we can say that However, the diversification across manager spanning occurs when the mean–variance styles comes at the cost of a multiplication of the frontier of the set of benchmark portfolios and fees paid by the investor (Brown et al.1). that of the benchmark portfolios plus a HF The purpose of this article is a better portfolio (or a FOF portfolio) coincide. In this understanding of the diversification benefits of case, investors do not benefit from adding a HF HF and funds of HF. We rely on mean–variance (or a FOF) to their current portfolio. spanning tests to analyse this issue. Throughout In finite samples, the question of whether an the paper, we consider that a set of asset observed shift is statistically significant can be returns provides diversification benefits relative tested using regression-based tests of mean– to a set of benchmark returns if adding these variance spanning. returns to the benchmark leads to a significant leftward shift in the mean–standard deviation Asymptotic and finite sample test frontier. We should, however, note that even statistics if the analysis is based on historical returns, the In this section, we briefly describe the statistical benefits we find could have little bearing on tests used to examine whether adding a HF future performance. portfolio can significantly improve the investment Diversification benefits of hedge funds 291 opportunity set relative to a set of benchmark and Lagrange multiplier tests for mean–variance assets. For purposes of convenience, we follow spanning. The exact finite sample distribution of the notation and treatment in Kan and Zhou. the likelihood ratio test under the null, as in We denote by K the set of benchmark Huberman and Kandel, is: portfolios (ie, without HF or FOF portfolios) with 1 T À K À N return R and by N the set of test assets (the HF À 1 1t U 1=2 N ð6Þ or FOF portfolio) with return R .Weestimate 2t F for N ¼ 2 the following model using ordinary least squares: 2N;2ðTÀKÀNÞ R2t ¼ a þ bR1t þ x ; t ¼ 1; 2; ...T; t ð1Þ 1 T À K À 1 ð ¼ þ Þ À 1 R XB E in matrix notation U1=2 2 ð7Þ Following Huberman and Kandel,4 the null F2;2ðTÀKÀ1Þ for N ¼ 1 hypothesis of ‘spanning’ is: where U ¼ |Gˆ|/|Hˆ þ Gˆ| H : a ¼ 0 ; d ¼ 1 À b1 ¼ 0 ; ð2Þ 0 N N K N In general, the test for mean–variance where 0N is defined as the zero vector of N spanning can be divided into two parts: (1) the elements. We then denote by l1 and by l2 the spanning of the global minimum-variance two eigenvalues of the matrix HˆGˆÀ1 (see portfolio and (2) the spanning of the tangency Appendix A for the definition of Hˆ and Gˆ). The portfolio. Therefore, we can re-write the Wald distribution of the asymptotic Wald test statistic test as ÀÁ ! of the null hypothesis is: 2 s^ R1 2 W ¼T À 1 W ¼ Tðl þ l Þw ð3Þ 2 1 2 2N ðÞs^ R !ð8Þ In this case, if we fail to reject the null hypothesis, 1 þ y^ ðRGMV Þ2 þ T R 1 À 1 then the benchmark assets span the mean–variance ^ GMV 2 1 þ yR1ðR Þ frontier of the benchmark plus a HF or FOF 1 portfolio, that is investors are not able to enlarge where R is the return of the benchmark assets their investment opportunity set by adding a HF or plus the HF portfolio; s^ 2 and sˆ 2 are the global R1 R FOF portfolio. On the other hand, if the null minimum variance of the benchmark assets and hypothesis is rejected, adding a HF or FOF portfolio that of the benchmark assets plus a HF or FOF ^ GMV does improve the investment opportunity set. portfolio, respectively; yR1 ðR1 Þ is the slope of The corresponding asymptotic likelihood the asymptote of the mean–variance frontier for ˆ GMV ratio and Lagrange multiplier tests are: the benchmark assets; and yR(R1 ) is the slope X2 of the tangency line of the mean–variance 2 LR ¼ T lnð1 þ liÞw2N ð4Þ frontier for the benchmark portfolios plus a HF i¼1 or FOF portfolio. X2 l LM ¼ T i w2 ð5Þ The first term measures the change of the þ l 2N i¼1 1 i global minimum-variance portfolios due to the As the Wald test is not the uniformly most addition of a HF or FOF portfolio. The second powerful test, we also use the likelihood ratio term measures whether there is an improvement 292 Kooli of the squared tangency slope after adding a HF Mean–variance intersection tests or FOF portfolio to the set of benchmark A mean–variance intersection test is equivalent portfolios.