Alternative investments: asset allocation in multimanagement

Asset allocations in models can be difficult to apply. Below, we explain optimal asset allocation models that take a broader view of hedge funds return distribution

There are two main challenges in applying standard asset allocation methods, for example efficient frontier analysis, to designing optimal portfolios including hedge funds.

One 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 hedge fund returns generally not being normally distributed, making the use of any asset allocation model based on sole estimates of expected return and volatility somewhat problematic.

Here 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

Minimum VAR approach

A classic way to analyse and formalise benefits of investing in hedge funds is to note 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 expected returns is largest (for example, 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 using 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 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 alternative investment strategies, even when a limited set of funds or indices are considered, because data is scarce as hedge fund returns are only available monthly. One possible cure to the curse of dimensionality in covariance matrix estimation is imposing 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 to mitigate model risk and impose endogenous structure. The advantage of that option is it involves low specification error, because of the 'let the data talk' 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 CSFB-Tremont8. 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 the ex-post volatility of the minimum variance portfolio generated using implicit factor-based estimation techniques is several times lower than that of a naively diversified equally weighted portfolio, and almost seven times lower than that of the value-weighted Global Tremont Index, such differences being both economically and statistically significant (see table 1).

This indicates 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 the improvement in terms of risk control does not necessarily come at cost of lower expected returns.

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 five times lower than that of a naively diversified equally-weighted portfolio, and almost nine times lower than that of the S&P 500 Index (table 2).

While it addresses the issue of 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 fill this gap.

Adjusted VaR approach

Most hedge fund managers follow dynamic investment strategies, distinguishing them from buy-and-hold 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 risk measures such as the beta or Sharpe ratio do not allow hedge fund risks' dynamic and non-linear dimensions to be accounted for (see for example Leland (1999)14 or Lo (2001)15).

Additionally investors generally display a non-trivial preference for third- and fourth-order moments of return distribution (skewness and kurtosis), as evidenced by development of measures of extreme risk such as the value at risk (VaR).

With that in mind, we suggest using 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, 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:

where n = number of standard deviations at (1-a)

s = annual standard deviation

W = current portfolio value

dt = fraction of the year

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

where: Zc = the critical value of the probability (1-a)

S = the skewness

K = the excess kurtosis (i.e. kurtosis minus 3)

So the adjusted VaR is equal to:

VaR = W (m - z s)

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 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).

Note that a minimum adjusted VaR 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 VaR approach. Such an approach would thus mitigate the problem of the non-normality of hedge funds' return distributions and 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 account for 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 hedge funds' return distribution, indicators such as minimum return or maximum drawdown measures have been applied to constructing 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 investors' aversion to extreme losses.

New measures such as the Omega ratio (see Keating and Shadwick (2002)17) have therefore been implemented in 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 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 (Minimum Acceptable Return) is the loss threshold, and [a,b] the interval on which asset returns are defined.

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 particularly inaccurate. This problem, however, is mitigated when one disposes of more than 200 data points (Favre-Bull and Pache (2003)18). Due to data scarcity one should be cautious when applying the Omega ratio to hedge funds. In this respect, there is no doubt 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, FoHF managers 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). Here we describe a methodology that can be used by a sophisticated investor with 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 (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 two-dimensional Brownian motion W=(W1,W2) is defined. We assume the investor can choose among three assets, one risk-free and two risky. The price of the first is Pt and we interpret it as a traditional long-only portfolio, for example the S&P 500. The second security, a hedge fund's, price, is At.

In this setting, we consider a risk averse investor with access to the three securities and who maximises 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 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 the vector of risk-premium has a normal prior distribution, independent of the Brownian motion W:

where mP is the mean estimate of the uncertain expected return on the traditional portfolio and mA the mean estimate of the uncertain expected return on the hedge fund.

As can be seen from the fact the off-diagonal terms in the covariance matrix of priors on risk-premium vector are zero, we assume the priors are independent (Cvitanic et al. (2003b)21 for the general case of correlated priors). In this setup, Cvitanic et al. (2003a)22 show 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 is the expected value of the abnormal return alpha of the hedge fund for the investor with incomplete information, that is, the best estimate an investor has about the hedge fund abnormal return.

As expected, an increase in expected alpha leads the investor to hold more of the active portfolio, everything else being equal. On the other hand, an increase in 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 investors 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 specific risk of the active portfolio leads investors to hold less (or short less) of it, everything else being equal.

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

and are 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 pA is positive (that is, when the perceived hedge fund abnormal return a is positive), we have: D pB £ D pPp € b £ 0.5. We find 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 0.5. This result suggests 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 neither the prior on the passive fund asset's expected return nor that fund's volatility have any impact on that decision. It should be noted the condition b£ 0.5 holds for most non-directional hedge fund strategies. This, on the other hand, would be relatively unusual for traditional long-only active strategies.

Footnotes

1 Britten-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

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

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

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

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

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

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

8 Very similar results were obtained with HFR and EACM indices

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

10 Chan, 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

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

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

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

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

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

16 Favre, 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

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

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

19 Cvitanic, 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

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

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

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