The popularity of hedge funds, as both an investment vehicle and an object of academic interest, has created interest in the question: ‘How many hedge funds (commodity trading advisors (CTAs), managed futures, etc.) make a diversified portfolio?’. Despite well-documented differences in hedge fund return distributions (most notably non-normality and non-linearity with respect to underlying risk factors) and hedge fund investment costs, virtually all studies heavily borrowed the methodologies designed for individual stock portfolios and applied them to hedge funds. In short, this amounts to a two-step procedure:

Henker and Martin (1998), Amin and Kat (2002) and Lhabitant and Learned (2002) are examples of this approach. The number of hedge funds they deem optimal ranges between 5 and 25. Despite the arbitrariness of the above approach, Brown et al (2011) claim that fund of funds exhibit excess diversification.

We see several shortcomings in the above papers. First, no attempt is made specifying the frictional costs of adding another fund into a portfolio. In the absence of these costs, it is always optimal to naively diversify across all possible investments. Samuelson (1967) made this point early on by stating that investors should diversify as much as possible, aware of the tradeoff between diversification and its costs. Frictional costs arise from fixed monitoring costs per additional funds, as well as the loss of bargaining power for fee rebates when diversifying among too many funds. Second, assets under management do not enter the decision-making problem, even though fixed costs can be spread more easily across a large pool of assets. Clearly it makes a very practical difference whether a decision-maker with 10 million USD or 100 million USD asks for the optimal number of assets to invest in. Third, the reduction in volatility is most valuable for investors with high risk aversion, while investors with low risk aversion will be less willing to incur frictional diversification costs for a reduction in volatility they value only very little. Finally, but most importantly, volatility for an investment will not differ if we reshuffle returns across different states of the world. However, investors have a preference for investments that pay well in bad states (where wealth is down) of the world. Such an investment might be more valuable or offer more protection than an asset that offers a higher SHARPE-ratio or lower volatility. Diversifications studies on hedge funds remain silent on this topic. This is most relevant for CTAs that, due to their trend-following trading style and money management techniques, offer portfolio insurance properties. For a risk-averse investor, it will now matter most how well his portfolio of CTAs performs in those months where he values insurance most highly. Consequently, the normative advice of these papers is limited at best.

Another more recent motivation of our work is the flood of papers motivated by Demiguel et al (2009). The authors show that equal weighting (1/n) is preferable to mean variance optimization if the SHARPE ratio differences between assets are small (adjusted for sample size). This situation is likely to be given for CTAs that are notoriously known for both large return dispersion and little to no persistence as documented in Bhardwaj et al (2008). None of the 1/n papers discusses frictional diversification costs and consequently the optimal number of assets is imposed rather than derived.

The next section first reviews the existing methodology used in diversification studies. We then extend the traditional mean variance framework to account for frictional costs of diversification, differences in assets under management and risk aversion to arrive at a closed form solution for the optimal number of assets. This method works well under the assumptions of normality and for investors who show no interest in the conditional nature o...