Abstract
Though simple and appealing, mean-variance portfolio choice theory does not describe actual diversification choices by investors, especially their propensity to gamble and the solvency constraints they face. Using 8 million trades realized by 90,000 individual investors, we show that diversification choices are in fact strongly driven by the skewness of returns, especially in bull markets, but also by the amount to be invested in risky assets. Increasing this amount by 10 % leads to increase by 3.8 % the number of stocks in investors’ portfolios, controlling for portfolio skewness. An important contribution of this paper is to show that the strength of the relationship between diversification and the skewness of returns is shaped by market forces. A strong negative relationship exists in bull markets but disappears in bear markets, a result not found in the literature. Our results survive several robustness checks, including controlling for individual heterogeneity and time-variability of stock price co-movements.
Similar content being viewed by others
Notes
Harvey and Siddique (1999, 2000) and Chen et al. (2001) show that the average skewness of single stocks is positive in most periods and the market skewness is negative most of the time. More recently, Albuquerque (2012) got the same results except during the second half of 1987 (due to the Black Monday). The skewness of the equally-weighted market portfolio is negative 77 % of the time.
At the same time, diversification does not reduce by much the portfolio variance because systematic risk is the most important component of total risk in such periods.
For international equity returns, Longin and Solnik (2001) showed that the asymmetry of correlations is statistically significant. Campbell et al. (2002) and Ang and Bekaert (2002, 2004) also identified an asymmetric correlation between bull and bear regimes, with higher correlations appearing in the bear regime and lower correlations in the bull regime.
http://www.eurofidai.org. A part of this database has been recently used by Foucault et al. (2011) in their study of retail trading and volatility on the French market and by Baker et al. (2012) to study the contagion of sentiment across countries, including France and the U.S.
The results (not reported here) are almost identical when considering one year of daily returns.
To measure the standardized skewness of portfolio returns, we use the usual estimate with one quarter of daily returns
$$\begin{aligned} \widehat{S_{k}}^{3}=\frac{\frac{1}{n} { \sum \nolimits _{t=1}^{n}} (r_{t}-\overline{r})^{3}}{\widehat{\sigma }^{3}} \end{aligned}$$(5)where \(\overline{r}\) is the average daily return and \(\widehat{\sigma }^{3}\) the cube of the estimated standard deviation of daily returns. One advantage of Eq. (5) is that it is standardized by variance (or standard deviation).The equation offers a way to take into account the mechanical positive link between variance and skewness illustrated in Sect. 3.
We significantly reject the hypothesis of a random effect model with the Hausman test at the highest level of significance in all models.
Three different measures of the reliability of our results are provided: the \(F\)-statistic (testing whether the vector of regression coefficients is the null vector), the \(R^{2}s,\) overall, between and within, and finally the intraclass correlation \(rho\), which is the fraction of the variance that is due to differences across individuals. Moreover, because there are multiple observations for each investor, the standard deviations of estimates are clustered at the investor level.
It should be noted that we obtain the same results (unreported) when we exclude investors who hold only one stock in any sub-period. Because there are many observations that are clustered approximately at 1 for \(D_{1}\), these systematically underdiversified investors do not drive our main conclusions.
We only present our results for years 2000 to 2006. We also estimate the model in which sub-periods are semesters, but due to a high number of instruments used, our Stata program does not run the complete estimation. Results including a constant instead of semester dummies are available upon request.
The presence of a lagged variable with fixed effects produces biased and inconsistent OLS estimates, which occurs because the lagged dependent variable is correlated with the error term although there is no autocorrelation between terms \(\varepsilon _{jt}\).
Due to variable differentiation, only 57,596 (versus 76,825 in Table 8) investors are examined over a maximum length of 6 years. The estimation uses a total of 22 instrumental variables. Due to the huge number of instrumental variables, we do not perform the same analysis over semesters.
As states are equally likely, there is no reason to consider different prices for AD securities. Investing \(1/k\) in each of the first \(k\) securities then generates a cost independent of \(k.\)
References
Albuquerque R (2012) Skewness in stock returns: reconciling the evidence on firm versus aggregate returns. Rev Financ Stud 25:1630–1673
Ang A, Bekaert G (2002) International asset allocation with regime shifts. Rev Financ Stud 15:1137–1187
Ang A, Bekaert G (2004) How do regimes affect asset allocation? Financ Anal J 60:86–99
Arellano M, Bond S (1991) Some tests of specification for panel data: Monte carlo evidence and an application to employment equations. Rev Econ Stud 58:277–297
Baker M, Wurgler J, Yuan Y (2012) Global, local, and contagious investor sentiment. J Financ Econ 104:272–287
Bali TG, Cakici N, Whitelaw RF (2011) Maxing out: stocks as lotteries and the cross-section of expected returns. J Financ Econ 99:427–446
Barber B, Odean T (2000) Trading is hazardous to your wealth: the common stock investment performance of individual investors. J Finance 55(2):773–806
Barberis N, Huang M (2008) Stocks as lotteries: the implications of probability weighting for security prices. Am Econ Rev 98:2066–2100
Blume ME, Friend I (1975) The asset structure of individual portfolios and some implications for utility functions. In: Rodney L. White Center for Financial Research Working Papers 10–74. Wharton School Rodney L. White Center for Financial Research
Brunnermeier MK, Gollier C, Parker JA (2007) Optimal beliefs, asset prices, and the preference for skewed returns. NBER Working Papers 12940. National Bureau of Economic Research Inc
Brunnermeier MK, Parker JA (2005) Optimal expectations. Am Econ Rev 95:1092–1118
Calvet LE, Campbell JY, Sodini P (2007) Down or out: assessing the welfare costs of household investment mistakes. J Polit Econ 115:707–747
Campbell R, Koedijk K, Kofman P (2002) Increased correlation in bear markets. Open Access publications from Maastricht University urn:nbn:nl:ui:27–19571. Maastricht University
Chamberlain G, Rothschild M (1983) Arbitrage, factor structure, and mean-variance analysis on large asset markets. Econometrica 51:1281–1304
Chen J, Hong H, Stein JC (2001) Forecasting crashes: trading volume, past returns, and conditional skewness in stock prices. J Financ Econ 61:345–381
Conrad J, Dittmar RJ, Ghysels E (2013) Ex ante skewness and expected stock returns. J Finance 68:85–124
Cook PJ, Clotfelter CT (1993) The peculiar scale economies of lotto. Am Econ Rev 83:634–643
Deck C, Schlesinger H (2010) Exploring higher order risk effects. Rev Econ Stud 77:1403–1420
Ebert S, Wiesen D (2011) Testing for prudence and skewness seeking. Manag Sci 57:1334–1349
Eeckhoudt L, Schlesinger H (2006) Putting risk in its proper place. Am Econ Rev 96:280–289
Entrop O, McKenzie M, Wilkens M, Winkler C (2014) The performance of individual investors in structured products. Rev Quant Finance Account. doi:10.1007/s11156-014-0479-8
Forrest D, Simmons R, Chesters N (2002) Buying a dream: alternative models of demand for lotto. Econ Inq 40:485–496
Foucault T, Sraer D, Thesmar DJ (2011) Individual investors and volatility. J Finance 66:1369–1406
Francis JC, Kim D (2013) Modern portfolio theory: foundations, analysis and new developments. Wiley, New York
Garrett TA, Sobel RS (1999) Gamblers favor skewness, not risk: further evidence from united states’ lottery games. Econ Lett 63:85–90
Goetzmann WN, Kumar A (2007) Equity portfolio diversification. Rev Finance 12:433–463
Gollier C (2005) Optimal illusions and decisions under risk. CESifo Working Paper Series 1382. CESifo Group Munich
Harding MC (2008) Explaining the single factor bias of arbitrage pricing models in finite samples. Econ Lett 99:85–88
Harvey CR, Siddique A (1999) Autoregressive conditional skewness. J Financ Quant Anal 34:465–487
Harvey CR, Siddique A (2000) Conditional skewness in asset pricing tests. J Finance 55:1263–1295
Karolyi GA, Stulz RM (2003) Are financial assets priced locally or globally? In: Constantinides G, Harris M, Stulz RM (eds) Handbook of the economics of finance, volume 1 of Handbook of the Economics of Finance. chapter 16. Elsevier, Amsterdam, pp 975–1020
Kelly M (1995) All their eggs in one basket: portfolio diversification of us households. J Econ Behav Organ 27:87–96
Kim WC, Fabozzi FJ, Cheridito P, Fox C (2014) Controlling portfolio skewness and kurtosis without directly optimizing third and fourth moments. Econ Lett 122:154–158
Kimball MS (1990) Precautionary saving in the small and in the large. Econometrica 58:53–73
Kraus A, Litzenberger RH (1976) Skewness preference and the valuation of risk assets. J Finance 31:1085–1100
Kumar A (2007) Do the diversification choices of individual investors influence stock returns? J Financ Mark 10:362–390
Kumar A (2009) Who gambles in the stock market? J Finance 64:1889–1933
Lease RC, Lewellen WG, Schlarbaum GG (1974) The individual investor: attributes and attitudes. J Finance 29:413–433
Lesmond DA, Ogden JP, Trzcinka CA (1999) A new estimate of transaction costs. Rev Financ Stud 12:1113–1141
Lewis KK (1999) Trying to explain home bias in equities and consumption. J Econ Lit 37:571–608
Liu H (2014) Solvency constraint, underdiversification, and idiosyncratic risks. J Financ Quant Anal 49:1–39
Longin F, Solnik B (2001) Correlation structure of international equity markets during extremely volatile periods. J Finance 56:649–676
Menezes C, Geiss C, Tressler J (1980) Increasing downside risk. Am Econ Rev 70:921–932
Mitton T, Vorkink K (2007) Equilibrium underdiversification and the preference for skewness. Rev Financ Stud 20:1255–1288
Nagel S (2005) Short sales, institutional investors and the cross-section of stock returns. J Financ Econ 78:277–309
Odean T (1999) Do investors trade too much? Am Econ Rev 89:1279–1298
Ortobelli S, Biglova A, Huber I, Racheva-Iotova B, Stoyanov S (2005) Portfolio choice with heavy-tailed distributions. J Concr Appl Math 3:353–376
Shefrin H, Statman M (2000) Behavioral portfolio theory. J Financ Quant Anal 35:127–151
Shu PG, Chiu SB, Chen HC, Yeh YH (2004) Does trading improve individual investor performance? Rev Quant Finance Account 22:199–217
Stoyanov S, Rachev S, Racheva-Iotova B, Fabozzi F (2011) Fat-tailed models for risk estimation. J Portf Manag 37:107–117
Tarazona-Gomez M (2004) Are individuals prudent? An experimental approach using lottery choices. Technical Report, Copenhagen Business School
Tversky A, Kahneman D (1992) Advances in prospect theory: cumulative representation of uncertainty. J Risk Uncertain 5:297–323
Walker I, Young J (2001) An economist’s guide to lottery design. Econ J 111:700–722
Xu J (2007) Price convexity and skewness. J Finance 62:2521–2552
Acknowledgments
We thank an anonymous referee for suggesting improvements in preceding versions of the paper. We also thank Laurent Deville, Gunter Franke, Burton Hollifield, Jens Jackwerth, Gregory Nini, Charles Noussair, Winfried Pohlmeier, Mark Seasholes, Pierre Six, Marc Willinger, the participants of the DMM meeting (2011, Montpellier), the Konztanz-Strasbourg Workshop (2011, Königsfeld), the Behavioral Insurance Meeting (2011, München), the French Finance Association Meeting (2014) for comments and suggestions. The financial supports of OEE (Observatoire de l’Epargne Européenne) and CCR Asset Management are gratefully acknowledged. We thank Tristan Roger for valuable research assistance and computer programming.
Author information
Authors and Affiliations
Corresponding author
Appendix: Moments of equally-weighted portfolios of Arrow–debreu securities
Appendix: Moments of equally-weighted portfolios of Arrow–debreu securities
Let \(\Omega =\left\{ \omega _{1},\ldots ,\omega _{n}\right\}\) denote a finite state-space with \(n\) equally-likely states of nature and assume that all Arrow–Debreu securities, denoted \(X_{1},\ldots, X_{n}\), are traded. \(X_{i}\) pays 1 in state \(\omega _{i}\) and 0 elsewhere. \((p_{1},\ldots,p_{n})\) stands for a sequence of equally-weighted portfolios containing respectively \(1,2,\ldots, n,\) AD securities. Without loss of generality, we assume that \(p_{k}\) contains \(1/k\) units of each of the first \(k\) securities.Footnote 16
Before analyzing portfolios, we briefly recall the elementary properties of the moments of AD securities.
Proposition 1
For any \(1\le k\le n\),
Proof
The first point is obvious since states are equally-likely. \(\sigma _{i}^{2}=E(X_{i}^{2})-E(X_{i})^{2}=\frac{1}{n}- \frac{1}{n^{2}}\) since \(X_{i}^{m}=X_{i}\) for any positive integer \(m\). The third central moment is calculated as follows
Finally, we get \(cov(X_{i},X_{j})=E(X_{i}X_{j})- E(X_{i}) E(X_{j})=-1/n^{2}\) since \(X_{i}X_{j}\equiv 0\) when \(i\ne j\).
Consider now a portfolio \(p_{k}\) invested in the first \(k\) AD securities and denote \(\mu _{k}\) (\(\sigma _{k}^{2})\) the expectation (variance) of payoffs of \(p_{k}\).
Proposition 2
Proof
Proposition 1 allows to write the covariance matrix of the \(n\) AD securities payoffs as
where \({\mathbf {I}}_{n}\) is the \((n,n)\) identity matrix and \({\mathbf {1}}_{(n,n)}\) is a \((n,n)\) matrix containing only ones. As \(p_{k}=\frac{1}{k} { \sum \nolimits _{i=1}^{k}} X_{i}\), we get
where \({\mathbf {1}}_{(k)}\) denotes a column vector of ones with \(k\) components and \({\mathbf {V}}_{k}\) the square matrix of the first \(k\) rows and columns of \({\mathbf {V}}_{n}\).
Equation (15) implies \({\mathbf {V}}_{k}=\frac{1}{n} {\mathbf {I}}_{k}-\frac{1}{n^{2}}{\mathbf {1}}_{(k,k)}.\) We then write
As expected, the variance of the equally-weighted portfolio decreases with the number of AD securities in the portfolio. The case \(k=n\) gives \(\sigma _{n}^{2}=0\) which is consistent withe fact that \(p_{n}\) is a risk-free portfolio paying \(1/n\) in each state.
The inverse of the number of stocks in portfolios is often considered as a measure of diversification (denoted \(D_{1}\) by Mitton and Vorkink (2007)). Proposition 2 shows that the variance of returns increases linearly with \(D_{1}\). When \(k=n\), the portfolio is risk-free and the variance of payoffs is equal to 0.
Denote now \(s_{k}^{3}\) the third central moment of \(p_{k}\) defined by:
Denoting \(Y_{k}={ \sum \nolimits _{j=1}^{k}} X_{j}\) gives \(s_{k}^{3}=\frac{1}{k^{3}}E((Y_{k}-\frac{k}{n})^{3})\). The specificities of AD securities imply that \(E(Y_{k}^{3})=E(Y_{k}^{2})=\frac{k}{n}\). In fact, these relations simply come from the fact that \(X_{j}^{m}X_{j^{*}}^{t}=0\) for any pair \((m,t)\) of strictly positive integers and different indices \(j\) and \(j^{*}\). We now get easily \(s_{k}^{3}\).
Proposition 3
The central third moment of \(p_{k}\) is valued:
Proof
Rearranging terms leads to
We know that \(k<n;\) consequently an equally weighted portfolio has a positive skewness as long as the number of AD securities it contains is lower than \(n/2\). Beyond this threshold, skewness becomes negative. When \(n\) is even, the distribution of returns is symmetric for \(k=n/2\), leading to a zero skewness for the portfolio return. Using the above diversification measure \(D_{1}\), we get that the third order moment increases quadratically in \(D_{1}\). In fact, we have:
We can also establish a very simple relationship between \(s_{k}^{3}\) and \(\sigma _{k}^{2}\) using Eqs. (14) and (19).
Rights and permissions
About this article
Cite this article
Broihanne, MH., Merli, M. & Roger, P. Diversification, gambling and market forces. Rev Quant Finan Acc 47, 129–157 (2016). https://doi.org/10.1007/s11156-015-0497-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11156-015-0497-1