Simulating and calibrating diversification against black swans

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Abstract

An investor concerned with the downside risk of a black swan only needs a small portfolio to reap the benefits from diversification. This matches actual portfolio sizes, but does contrast with received wisdom from mean–variance analysis and intuition regarding fat tailed distributed returns. The concern for downside risk and the fat tail property of the distribution of returns can explain the low portfolio diversification. A simulation and calibration study is used to demonstrate the relevance of the theory and to disentangle the relative importance of the different effects.

Introduction

Early diversification studies, such as that of Evans and Archer (1968) and Elton and Gruber (1977) have focussed on the benefits from portfolio diversification that can be measured by the volatility of portfolio returns. These studies consider how many randomly selected stocks to include in a portfolio before most idiosyncratic risk is eliminated. Fama (1976), for example, identified the portfolio size that generates a 95% reduction in portfolio variance. Evans and Archer (1968) advocated a statistical criterion to determine the optimal portfolio size using the number of stocks at the point where no further significant reduction in the portfolio dispersion can be obtained. More recently, Campbell et al. (2001, p. 25) concluded that in more recent decades: ‘the increase in idiosyncratic volatility over time has increased the number of randomly selected stocks needed to achieve relatively complete portfolio diversification’.1

Although it is clearly worthwhile and interesting to determine how much it would take to eliminate ‘almost all’ unsystematic risk through diversification, it is unsatisfactory in an economic sense when left to itself. An economic based portfolio size not only assesses the benefits in terms of risk reduction, but also takes into account the associated costs of diversification. Diversification should be increased as long as the marginal benefits exceed the marginal costs of adding one extra security. Statman (1987) has contributed to the literature on diversification by proposing a framework in which the costs and benefits can be balanced. However, even if the costs of diversification are adequately assessed, Statman, 1987, Statman, 2004 nevertheless conclude that the level of diversification in investor's equity portfolios presents a puzzle regarding the mean–variance based portfolio analysis. The level of diversification in the average investor's portfolio found empirically, which hovers at around three or four stocks, appears to be far less than the optimal theoretical level. Consider, though, the Financial Times of August 1, 2011 weekly review of fund management, which opened with the headline: ‘New research shows concentrated funds outperform diversified vehicles’. What should be made of this?

In this paper we investigate the benefits of diversification for an investor who has a concern over downside risk and recognizes the fat tail feature in the distribution of asset returns, while retaining concern regarding the costs of diversification. The concern for downside risk is captured by the value-at-risk (VaR), which has become a popular measure of risk in the banking industry. One might think that fatter than normal tails, that give rise to more outliers, would require even more diversification than in the case of the standard assumption of normally distributed returns. We show, though, that at a given level of (downside) risk, the benefits from diversification come in more rapidly in the case of fat tails. The apparent contradiction stems from the fact that in comparing the tails, one is comparing distributions as the risk level changes. But diversification stems from cross-sectional aggregation at a constant risk level. Diversification at a constant risk level reduces the VaR, but this occurs at a rate that is not necessarily equal to the rate if the risk level is reduced. As long as the second moment is bounded, our simulations and derivations show that the Rate of Diversification for fat tailed distributed returns in terms of reducing the VaR can be better than for normally distributed returns. Ibragimov (2009) considers the case of infinite variance, wherein the effects go in the opposite direction (and diversification may not be a good idea at all).

The downside risk concern is modeled within the safety first framework of Roy (1952) through a VaR constraint.2 Arzac and Bawa (1977) provide an equilibrium analysis, like the CAPM, of the safety-first investor who maximizes expected returns subject to a VaR constraint. Downside risk and the closely related safety-first principle are frequently employed in models of finance and risk management, providing alternatives to the traditional expected utility framework. More recently, the concern for downside risk has received renewed interest, see e.g. Gourieroux et al. (2000), Jansen et al. (2000), and Campbell and Kraussl (2007).

To assess the diversification issue, we consider equally weighted randomly composed portfolios. Kan and Zhou (2007) discuss the difficulties in estimating the parameters, such as the mean and variance, required to construct the optimal portfolio weights. Dash and Loggie (2008) suggest that the equally weighted index turns out to be a powerful investment idea after examining the performance of the S&P 500 Equal Weight Index. Equally weighted portfolios circumvent the difficulty of having to estimate optimal portfolio weights. Moreover, this facilitates an easier comparison across different risk measures, as different utility functions imply different optimal portfolio weights.

To compare the incremental benefits and costs from diversification for different risk measures, define the ‘Rate of Diversification’ as the derivative of the benefits with regard to the number of assets. For independent risk drivers that are either normally or fat tailed distributed, it is relatively straightforward to obtain the rates of diversification. In an equilibrium setting, when stock returns are dependent, there can be multiple rates stemming from different contributing factors. Since it can not be determined a priori which source dominates with a limited number of assets, we employ a numerical calibration and simulation study. These calibrations and simulations complement the empirical work in our companion paper, see Hyung and de Vries (2010).

We find that, in comparison to a mean–variance investor, the concern for extreme3 downside risk in combination with the fat tail feature produces more focused portfolios. At first, this may seem counter-intuitive. However, as we show, the normal distribution is conducive to high diversification, as this reduces the power by which the (exponential type) tail of the loss distribution declines. Per contrast, diversification only affects the scale of fat tailed distributions, which is a more limited effect.

In Section 2 we briefly review the analysis of Statman (2004) and Hyung and de Vries (2010). Section 3 presents analytical expressions for the rates of the diversification benefits. Following this, we conduct a calibration and simulation study in 4 Numerical calibration study, 5 Simulation study, respectively. Conclusions are provided in the final section.

Section snippets

Risk measures and diversification cost benefit analysis

The costs of diversification are the transaction costs in buying and selling, holding costs and monitoring costs of assets. Statman (2004) defines the concept of ‘additional net cost’ as the net cost of increasing diversification from any n-stock portfolio to a fully diversified portfolio. In accordance with the work of Statman, we assume constant net additional costs. The optimal portfolio size is the point at which the marginal cost of adding one extra security equals the benefit of the

Normal versus fat-tailed distribution

We compare the benefits from diversification for different types of investors who alternatively employ the VaR and standard deviation as measures of risk. The two measures imply quite different levels of diversification depending on whether the returns follow a normal distribution or a distribution that exhibits fat tails.

Numerical calibration study

In the previous section, we compared the incremental benefits from diversification for different risk measures and under different assumptions regarding the distribution of returns. In the case of factor models, when returns are cross-sectionally dependent, there is more than one rate driving the diversification benefits. In those cases it is not only the power in the exponent (which determines the rate) that is important, but the scaling constants are important as well for the determination of

Simulation study

In this section we consider the case of heterogeneous idiosyncratic risk and varying correlation. The effects of heterogeneity and varying correlation are most easily studied by simulation. The design of the simulations is as follows. In this simulation study we use the same parameter values as in the calibration exercise of Section 4, except for the scale parameters, c.f. (18). The returns ri derive from a factor model (14) in which the innovations are i.i.d. with a three parameter version of

Conclusion

The diversification puzzle is that actual portfolios of individuals contain a much smaller amount of different stocks than mean–variance theory would prescribe. The calibration and simulation studies demonstrate that the concern over downside risk at a sufficiently low probability level combined with the fat tail phenomenon can be used to replicate the low diversification phenomenon. Our companion paper, Hyung and de Vries (2010), furnishes empirical evidence for this finding. The fat tail

Acknowledgments

Hyung gratefully acknowledges support by National Research Foundation of Korea—Grant funded by the Korean Government (KRF-2009-327-B00300). Both authors are very grateful for the excellent comments by the referees on an earlier draft.

References (21)

  • E.R. Arzac et al.

    Portfolio choice and equilibrium in capital markets with safety-first investors

    Journal of Financial Economics

    (1977)
  • D. Jansen et al.

    Portfolio selection with limited down-side risk

    Journal of Empirical Finance

    (2000)
  • G.Y.N. Tang

    How efficient is naive portfolio diversification? An educational note

    Omega

    (2004)
  • H. Benjelloun

    Evans and Archer—forty years later

    Investment Management and Financial Innovations

    (2010)
  • J. Campbell et al.

    Revisiting the home bias puzzle: downside equity risk

    Journal of International Money and Finance

    (2007)
  • J. Campbell et al.

    Have individual stocks become more volatile? An empirical exploration of idiosyncratic risk

    Journal of Finance

    (2001)
  • S. Dash et al.

    Equal Weight Indexing: Five Years Later

    (2008)
  • V. DeMiguel et al.

    Optimal versus naive diversification: how inefficient is the 1/N portfolio strategy?

    The Review of Financial Studies

    (2009)
  • D.L. Domian et al.

    Diversification in portfolio of individual stocks: 100 stocks are not enough

    The Financial Review

    (2007)
  • E.J. Elton et al.

    Risk reduction and portfolio size: an analytical solution

    Journal of Business

    (1977)
There are more references available in the full text version of this article.

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