Non-Gaussian diversification: When size matters François Desmoulins-Lebeault a,⇑ , Cécile Kharoubi-Rakotomalala b a Grenoble École de Management, 12, rue Pierre Sémard, 38000 Grenoble, France b ESCP Europe, 79, avenue de la République, 75011 Paris, France article info Article history: Received 16 July 2011 Accepted 3 March 2012 Available online 13 March 2012 JEL classification: G11 G15 G52 Keywords: Diversification benefits Asymmetric diversification Non-Gaussian distributions abstract Classical portfolio theory informs investors that they should have a large number of assets in their port- folios in order to diversify risk. We show that the non-Gaussian features of stock return distribution may not allow for this risk protection in times of crisis. Moreover, we demonstrate empirically that, if inves- tors are risk-averse and consider higher order moments, they have numerous incentives not to diversify their portfolios fully. This is caused by the evolution of both large losses and asymmetry of returns when the numbers of assets in a portfolio change. Ó 2012 Elsevier B.V. All rights reserved. 1. Introduction When a financial crisis occurs, many theoretically well-diversi- fied portfolios react as if diversification was no longer effective. The assets composing these portfolios evolve all at the same time and in the same direction, which implies that the effects of diversifica- tion, quite necessary when markets are hectic, become weaker. Diversification is one of the key features of finance and portfolio analysis since Sharpe (1963) showed that risk can be split into a diversifiable and a systematic part. He also demonstrated that including a growing number of assets in a portfolio eliminates the former, a finding Evans and Archer (1968) empirically confirmed later. Since then, the consensus is that a portfolio of 10–20 assets is sufficient to achieve most of the diversification benefits. Statman (1987) indicates that these numbers have been underestimated and that the optimal size is actually between 30 and 40 assets. Nevertheless, a number of studies highlight that household investors usually hold equity portfolios containing a very limited number of different assets. Kelly (1995) establishes that the med- ian stockholder in the US in 1983 only held one stock. More re- cently, Bertaut (1998), Bertaut and Starr-McCluer (2000) and Goetzman and Kumar (2001) show that even if there has been some improvement over recent years, investors still hold poorly diversified equity portfolios consisting of about five stocks. It appears that educated investors diversify more than others do, but they still hold a surprising amount of idiosyncratic risk. Institutional investors do not exhibit these reduced size portfolios for two main reasons. First, they tend to follow models or to be constrained by risk management models; second, extreme events do not loom as large when they concern your clients’ wealth rather than your own. However, the assumption that underlies the common vision of diversification is that variance is a correct measure of risk, which is the case when either the returns of all assets follow a joint Gaussian distribution or the investors possess utility functions ignoring higher order moments. Fama (1965) and Richardson and Smith (1993) show that the normality hypothesis can be rejected, so consequently we can no longer use variance alone to measure risk and correlation ceases to be an appropriate measure of depen- dence, hence the size/diversification relation does not necessarily hold. New researches, such as that undertaken by You and Daigler (2010a), investigate the impact of these non-Gaussian features on diversification. French and Poterba (1991) illustrate that it is investors’ choices and not institutional constraints that lead to under-diversified portfolios, while Polkovnichenko (2005) argues that rank prefer- ences may contribute to explaining this under-diversification, although most authors seem to point in the direction of poor edu- cation or irrational behavior. However, the non-Gaussian properties of the distribution of re- turns on financial assets may also contribute in part to explaining 0378-4266/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jbankfin.2012.03.006 ⇑ Corresponding author. Tel.: +33 4 76 70 60 77; fax: +33 4 76 60 64 58. E-mail addresses: francois.desmoulins.lebeault@grenoble-em.com (F. Desmoulins- Lebeault), ckharoubi@escpeurope.eu (C. Kharoubi-Rakotomalala). Journal of Banking & Finance 36 (2012) 1987–1996 Contents lists available at SciVerse ScienceDirect Journal of Banking & Finance journal homepage: www.elsevier.com/locate/jbf