There are two main challenges involved in the application of standard asset allocation methods (e.g. efficient frontier analysis) to the design of optimal portfolios that include hedge funds. One challenge is that it is extremely difficult to obtain a forward-looking estimate of a hedge fund’s expected return. Therefore, we will review advanced techniques consistent with the presence of significant parameter uncertainty in the asset allocation process. Another challenge comes from the fact that hedge fund returns are not in general normally distributed, which makes the use of any asset allocation model based on sole estimates of expected return and volatility somewhat problematic. In what follows, we provide a review of optimal asset allocation models that account for more than the first two moments of hedge fund return distributions.

Asset allocation techniques to construct optimal FoHFs

A classic way to analyse and formalise the benefits of investing in hedge funds is to note the improvement in the risk-return trade-off. The first difficulty comes from the sensitivity of optimisation techniques to differences in expected returns, since portfolio optimisers tend to allocate the largest fraction of capital to the asset class for which estimation error in the expected returns is the largest (e.g., Britten-Jones (1999)¹ or Michaud (1998)²). The second one consists of characterising the risk dimension.

Because of the presence of large estimation risk in estimated expected returns, we suggest the use of an improved estimator for the covariance structure of hedge fund returns, focusing on its use for selecting the one portfolio on the efficient frontier for which no information on expected returns is required, the minimum variance portfolio³. Thus, we explain how an efficient allocation can be implemented by an investor who does not feel confident in her ability to generate a reliable forward-looking estimate of hedge fund expected returns.

A problem with the sample covariance matrix of historical returns is that it may have too many parameters compared to the available data. If the number of assets in the portfolio is N, there are indeed N(N-1)/2 different covariance terms to be estimated. The problem is particularly acute in the context of alternative investment strategies, even when a limited set of funds or indices are considered, because data is scarce given that hedge fund returns are only available on a monthly basis. One possible cure to the curse of dimensionality in covariance matrix estimation is to impose some structure on the covariance matrix to reduce the number of parameters to be estimated. Following Amenc and Martellini (2002)⁴, we use an implicit factor model in an attempt to mitigate model risk and impose endogenous structure. The advantage of that option is that it involves low specification error (because of the “let the data talk” type of approach) and low sampling error (because some structure is imposed). Implicit multi-factor forecasts of asset return covariance matrix can be further improved by noise dressing techniques and optimal selection of the relevant number of factors. The authors suggest selecting the number of factors by applying some explicit results from the theory of random matrices (see Marchenko and Pastur (1967)⁵)⁶.

To represent the alternative investment universe, Amenc and Martellini (2002)⁷ choose to use index returns from Credit Swiss First Boston - Tremont (CSFB-Tremont)⁸. Their methodology for testing minimum variance portfolios⁹ is similar to the one used in Chan et al. (1999)¹⁰ and Jagannathan and Ma (2000)¹¹. They find that the ex-post volatility of the minimum variance portfolio generated using implicit factor based estimation techniques is almost 3 times lower than that of a naively diversified equally-weighted portfolio, and almost 7 times lower than that of the value-weighted Global Tremont Index, such differences being both economically and statistically significant (see Table 1). This indicates that optimal variance minimisation can achieve lower portfolio volatility. Differences in mean returns, on the other hand, are not statistically significant (t-stat = .11 and .16, respectively), suggesting that the improvement in terms of risk control does not necessarily come at the cost of lower expected returns.

Table 1: Multi-Style Multi-Class Strategic Allocation: AI Universe

Source: Amenc and Martellini (2002)¹²

Similar results are obtained when traditional and alternative assets are mixed. The ex-post volatility of the minimum variance portfolio generated using implicit factor based estimation techniques is almost 5 times lower than that of a naively diversified equally-weighted portfolio, and almost 9 times lower than that of the S&P 500 (table 2).

Table 2: Multi-Style Multi-Class Strategic Allocation: AI/TI Universe

Source: Amenc and Martellini (2002)¹³

While it addresses the issue of the sensitivity of optimisation techniques to differences in expected returns, the minimum variance approach does not address the problem of the non-normality of hedge funds’ return distribution as the mean variance framework explicitly excludes third and fourth moments of the distribution from the analysis. The following approaches will fill in this gap.

Adjusted Value-at-Risk approach

Most hedge fund managers follow dynamic investment strategies, which distinguishes them from the buy-and-hold type strategies often practised in traditional investment management. Moreover, the use of static or dynamic positions in derivatives and optional instruments reinforces the non-linear and dynamic character of alternative strategies. However, it is well-known that risk measures such as the beta or the Sharpe ratio do not allow the dynamic and non-linear dimensions of hedge fund risks to be accounted for (see for example Leland (1999)¹⁴ or Lo (2001)¹⁵). On top of that, investors generally display a non-trivial preference for the third and fourth order moments of return distribution (skewness and kurtosis), as is evidenced by the development of measures of extreme risk such as the Value-at-Risk (VaR).

With that in mind, we suggest to use a pragmatic application of the VaR calculation in a fat tail distribution environment, along with its integration into a Mean-VaR optimisation process. The Mean-VaR optimisation method, such as introduced by Favre and Galeano (2002)¹⁶ first consists of calculating a VaR using a normal distribution formula and then a Cornish-Fisher expansion to take the skewness and kurtosis into account.

Within the Gaussian framework, the VaR can be calculated explicitly by using the following formula:

                        P (dW £ -VaR) = 1 - a                      VaR = n s W dt^0.5

The analytical side of this normal VaR formula was then adjusted using the Cornish-Fisher extension (1937) as follows:

The adjusted VaR is therefore equal to:       VaR = W (m - z s)

It should be noted that if the distribution is normal, S and K (represents the excess kurtosis in the formula) are equal to zero and consequently, z=Zc, and we come back to the Gaussian VaR.

Interestingly, we find that efficient frontiers obtained using a Gaussian parametric VaR without a Cornish-Fisher correction for a 99% threshold are very close to those obtained with a VaR adjusted according to the Cornish-Fisher extension, but at a 97.5% threshold. We can therefore consider that investors who only take first and second order moments into account greatly underestimate (a factor of 2.5) the extreme risk to which they are exposed (see Amenc et al. 2003).

Graph 1. Comparison of mean/VaR optimisations in the case of "distressed securities" type strategies (HFR) for the period (from February 1990 through March 2002)

Source: Amenc et al. (2003a)

Note that a Minimum adjusted Value-at-Risk approach can easily be derived from the aforementioned optimisation technique. We only have to mix the implicit multi factor approach of the Minimum VaR approach with the Cornish Fisher expansion of the adjusted Value-at-Risk approach. Such an approach would thus mitigate the problem of the non-normality of hedge funds’ return distributions and the sensitivity of optimisation techniques to differences in expected returns, at the same time

A similar approach consists of replacing the Cornish Fisher VaR with the Conditional VaR that is the expected shortfall (see Agarwal and Naik (2003), or Morton et al. (2003)). Since both approaches take into account the impact of extreme losses, they tend to give comparable results.

Gain - Loss approaches

Since the definition of risk is subject to controversy many different approaches corresponding to the risk profile of different investors have been implemented. As a matter of fact, besides volatility or VaR, a wide range of basic downside risk indicators have traditionally been applied for asset allocation purposes. To take into account the asymmetry of the return distribution of hedge funds, indicators such as minimum return or maximum drawdown measures have been applied to the construction of hedge fund portfolios. In the same vein, risk adjusted measures such as the Sortino ratio have been used in portfolio optimisation programs to emphasise the importance of downside events. Nevertheless, such indicators tend to underestimate the aversion of investors to extreme losses. New measures such as the Omega ratio (see Keating and Shadwick (2002)¹⁷) have therefore been implemented in the context of portfolio construction to make up for this weakness. Such gain-loss oriented tools are more appropriate to FoHF construction since they may take into account the skewness and kurtosis effects just as the VaR Cornish Fisher expansion does, but with a more intuitive formulation. They thus enable both risk minimisation and risk return efficiency.

Where F is the cumulative distribution function, MAR or Minimum Acceptable Return is the loss threshold, and [a,b] the interval on which asset returns are defined.

Note that in practice the Omega based approach, though very simple, tends to give results that converge to those obtained through the Mean/Adjusted VaR or Mean/Expected Shortfall approaches. Nevertheless, when the number of observations is limited, the Omega ratio proves to be particularly inaccurate. This problem, however, is mitigated when one disposes of more than 200 data points (see Favre-Bull and Pache (2003)¹⁸). Due to the scarcity of data one should be cautious when applying the Omega ratio to hedge funds. In this respect, there is no doubt that this compelling allocation technique will gather pace when the alternative industry matures.

Optimal Allocation to Hedge Funds: an Empirical Analysis – Cvitanic et al. (2003)¹⁹

In practice, managers of FoHFs generate estimates for expected hedge fund returns, or abnormal returns, from a mix of quantitative (improved estimators of expected returns) and qualitative analysis (due diligence). In what follows, we describe a methodology that can be used by a sophisticated investor who has access to reliable, albeit imperfect, estimates of hedge fund alphas. We offer an explicit solution for the optimal allocation problem of a non-myopic investor with incomplete information who allocates wealth between a risk-free security, a passive portfolio and a set of hedge funds (see Cvitanic et al. (2003)²⁰). This is based on the theory of stochastic control in a continuous-time setting with Bayesian update (Kalman filter approach).

Uncertainty about risky asset prices in the economy is represented by a standard filtered probability space on which a 2-dimensional Brownian motion W=(W¹,W²) is defined. We assume that the investor can choose among three assets, a risk-free asset and two risky assets. The first of these has a price that we denote by P_t and we interpret it as a traditional long-only portfolio, e.g., the S&P 500. The second security, whose price we denote by A_t, is a hedge fund.

In this setting, we consider a risk averse investor who has access to the three securities described above and who maximizes utility of final wealth, where preferences are assumed to be represented by a power utility with risk-aversion coefficient denoted by 1-a, where a<0 (a=0 corresponds to a logarithmic -myopic- utility).

We assume that the investor observes neither the constant mean returns vector nor the source of noise but observes the price processes. Define the “risk premium” vector process as follows:

Because investors do not have good estimates for expected returns, we assume that the vector of risk-premium has a normal prior distribution, independent of the Brownian motion W:

Where m_P is the mean estimate of the uncertain expected return on the traditional portfolio and m_A the mean estimate of the uncertain expected return on the hedge fund).

As can be seen from the fact that the off-diagonal terms in the covariance matrix of priors on risk-premium vector are zero, we assume that the priors are independent (see Cvitanic et al. (2003b)²¹ for the general case of correlated priors). In this setup, Cvitanic et al. (2003a)²² show that the optimal holdings in the traditional portfolio and the hedge fund can be expressed in the following form:

where T is the investor’s time-horizon, and where a = m_A – r - b(m_P-r), is the expected value of the abnormal return alpha of the hedge fund for the investor with incomplete information, i.e., the best estimate that an investor has about the hedge fund abnormal return.

As expected, an increase in the expected alpha leads the investor to hold more of the active portfolio, everything else being equal. On the other hand, an increase in the uncertainty around alpha leads the investor to hold less (or short less) of the active portfolio, everything else being equal. An increase in the time-horizon also leads the investor to hold less (or short less) of the active portfolio. On the other hand, when there is no uncertainty around alpha, the solution is time-horizon independent. Finally, an increase in the specific risk of the active portfolio leads the investor to hold less (or short less) of it, everything else being equal.

An important question that investors in hedge funds often ask is where they should take the money they are planning to allocate to the hedge fund from. In the context of the above-presented model, we can give a quantitative answer to that question. The changes in holdings due to the introduction of the active portfolio are:

andare respectively the optimal holdings in the traditional portfolio and risk-free asset in the absence of the hedge fund.

As a result, when the optimal holding in the hedge fund p_A is positive (i.e., when the perceived hedge fund abnormal return a is positive), we have: D p_B £ D p_P Û b £ ½. We find that the introduction of the active fund leads investors to optimally withdraw an amount from the money market account larger than that taken out of the passive fund when the active fund has a beta lower than 1/2. This result suggests that low beta hedge funds may actually serve as natural substitutes for a significant portion of an investor’s risk-free asset holdings, while high beta hedge funds can be regarded as substitutes for a portion of equity holdings.

Note that neither the prior on the expected return of the passive fund asset nor the volatility of that fund have any impact on that decision. It should be noted that the condition b£ 1/2 holds for most non-directional hedge fund strategies. This, on the other hand, would be relatively unusual for traditional long-only active strategies.

               

Footnotes:

¹Britten-Jones, M., 1999, The sampling error in estimates of mean-variance efficient portfolio weights, Journal of Finance, April 1999, Vol.54, Issue 2, p.655-671

²Michaud, R., 1998, Efficient asset management: a practical guide to stock portfolio optimization and asset allocation, Harvard Business School Press, 1998

³Alternatively, one motivation in focusing on the minimum variance portfolio is to note that it is the efficient portfolio obtained under the null hypothesis of no informative content in the cross-section of expected returns.

⁴Amenc, N. and Martellini, L., 2002, Portfolio optimization and hedge fund style allocation decisions, Journal of Alternative Investments, 5, 2, p.7-20

⁵Marchenko, V., and L. Pastur, 1967, Eigenvalue distribution in some ensembles of random matrices, Math. USSR Sbornik 72, p.536-567

⁶Another decision rule would be: keep sufficient factors to explain x% of the covariation in the portfolio.

⁷See Amenc and Martellini (2002) – Opus Cit.4

⁸Note that very similar results were obtained with the HFR and EACM indices

⁹We use the previous 48 months of observations (beginning of 1994 to end of 1998) to estimate the covariance matrix of the returns of the 9 hedge fund sub-indices. We form a portfolio which is set to be held during 6 months and then repeat the same process. So, the minimum variance portfolio has ex-post monthly returns from early 1999 to the end of 2000.

¹⁰Chan, L., Karceski, J. and Lakonishok, J., 1999, On Portfolio Optimization: Forecasting Covariances and Choosing the Risk Model, Review of Financial Studies, 12, p.937-74

¹¹Jagannathan R. and T. Ma, 2000, Covariance Matrix Estimation: Myth and Reality, Working Paper, Northwestern University

¹²See Amenc and Martellini (2002) – Opus Cit.4

¹³See Amenc and Martellini (2002) – Opus Cit.4

¹⁴Leland, H., 1999, Beyond mean-variance: risk and performance measures for portfolios with nonsymmetric distributions, Working Paper, Haas School of Business, U.C. Berkeley

¹⁵Lo, A, 2001 Risk Management for Hedge Funds: Introduction and overview, Financial Analyst Journal, Vol. 57, p.16-33

¹⁶Favre, L., and Galeano J. A., 2002, Mean Modified Value-at-Risk Optimization with Hedge Funds, Journal of Alternative Investments, Fall 2002, Vol.5, N°2, p.21-25

¹⁷Keating, C.and Shadwick, W., 2002, A Universal Performance Measure, Journal of Performance Measurement, Spring 2002, Vol.6, N°3, p.59-84.

¹⁸Favre-Bull, A. and Pache, S., 2003, The Omega Measure: Hedge Fund Portfolio Optimization, MBF’s Master Thesis, University of Lausanne

¹⁹Cvitanic, J., Lazrak, A., Martellini, L. and Zapatero,F., 2003a, Optimal Allocation to Hedge funds: an Empirical Analysis, Quantitative Finance, Vol.3 (2003), p.1-12

²⁰See Cvitanic et al. (2003a) – Opus Cit.19

²¹Cvitanic, J., Lazrak, A., Martellini, L. and Zapatero, F., 2003b, Revisiting Treynor and Black (1973): an Intertemporal Model of Active Management, Working Paper, USC

²²See Cvitanic et al. (2003a) – Opus Cit.19