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Diversification, gambling and market forces

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.

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Notes

  1. 1.

    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.

  2. 2.

    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.

  3. 3.

    Calvet et al. (2007) obtained the same results for Sweden, except that Swedish investors seem to have slightly more diversified portfolios than U.S. investors. Concerning the performance of individual investors, see for example Barber and Odean (2000), Shu et al. (2004), Entrop et al. (2014).

  4. 4.

    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.

  5. 5.

    We use the same presentation as that of Table 2 of Mitton and Vorkink (2007). The complete statistics for all months of the period are available upon request.

  6. 6.

    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.

  7. 7.

    See Lewis (1999) and Karolyi and Stulz (2003) for a literature review on this topic.

  8. 8.

    The results (not reported here) are almost identical when considering one year of daily returns.

  9. 9.

    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.

  10. 10.

    We significantly reject the hypothesis of a random effect model with the Hausman test at the highest level of significance in all models.

  11. 11.

    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.

  12. 12.

    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.

  13. 13.

    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.

  14. 14.

    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 \(\varepsilon _{jt}\).

  15. 15.

    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.

  16. 16.

    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 \(k.\)

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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.

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Correspondence to Patrick Roger.

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 \(1\le k\le n\),

$$\begin{aligned} E(X_{k})=m_{k}=\frac{1}{n}\end{aligned}$$
(9)
$$\begin{aligned} V(X_{k})=\frac{n-1}{n^{2}}\end{aligned}$$
(10)
$$\begin{aligned} E\left[ (X_{k}-m_{k})^{3}\right]=\frac{(n-1)(n-2)}{n^{3}}\end{aligned}$$
(11)
$$\begin{aligned} cov(X_{k},X_{k^{*}})=-\frac{1}{n^{2}}\text { if }k\ne k^{*} \end{aligned}$$
(12)

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

$$\begin{aligned} E\left[ (X_{i}-\mu _{i})^{3}\right]&=E(X_{i}^{3})-3\mu _{i} E(X_{i}^{2})+3\mu _{i}^{2}E(X_{i})-\mu _{i}^{3}\\ &=\frac{1}{n}-3\frac{1}{n^{2}}+3\frac{1}{n^{3}}-\frac{1}{n^{3}} =\frac{1}{n}-\frac{3}{n^{2}}+\frac{2}{n^{3}} \\ &=\frac{(n-1)(n-2)}{n^{3}} \end{aligned}$$
(13)

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

$$\begin{aligned} \forall k,\mu _{k}&=\frac{1}{n}\\ \sigma _{k}^{2}&=\frac{1}{n}\left( \frac{1}{k}- \frac{1}{n}\right) \end{aligned}$$
(14)

Proof

Proposition 1 allows to write the covariance matrix of the \(n\) AD securities payoffs as

$$\begin{aligned} {\mathbf {V}}_{n}=\frac{1}{n}{\mathbf {I}}_{n}-\frac{1}{n^{2}} {\mathbf {1}}_{(n,n)} \end{aligned}$$
(15)

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

$$\begin{aligned} \sigma _{k}^{2}=\frac{1}{k^{2}}{\mathbf {1}}_{(k)}^{\prime } {\mathbf {V}}_{k}{\mathbf {1}}_{(k)} \end{aligned}$$
(16)

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

$$\begin{aligned} \sigma _{k}^{2}&=\frac{1}{k^{2}}{\mathbf {1}}_{(k)}^{\prime } {\mathbf {V}}_{k}{\mathbf {1}}_{(k)}=\frac{1}{k^{2}}{\mathbf {1}}_{(k)}^{\prime } \left( \frac{1}{n}{\mathbf {I}}_{k}-\frac{1}{n^{2}}{\mathbf {1}}_{(k,k)} \right) {\mathbf {1}}_{(k)} \\ &=\frac{1}{k^{2}n}{\mathbf {1}}_{(k)}^{\prime }{\mathbf {I}}_{k} {\mathbf {1}}_{(k)}-\frac{1}{k^{2}n^{2}}{\mathbf {1}}_{(k)}^{\prime } {\mathbf {1}}_{(k,k)}{\mathbf {1}}_{(k)} \\ &=\frac{1}{kn}-\frac{1}{n^{2}}=\frac{1}{n}\left( \frac{1}{k}-\frac{1}{n}\right) \end{aligned}$$
(17)

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:

$$\begin{aligned} s_{k}^{3}=E\left( \left( \frac{1}{k} {\sum \limits _{j=1}^{k}} X_{j}-\mu _{k}\right) ^{3}\right) =\frac{1}{k^{3}}E\left( \left( {\sum \limits _{j=1}^{k}} X_{j}-\frac{k}{n}\right) ^{3}\right) \end{aligned}$$
(18)

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:

$$\begin{aligned} s_{k}^{3}=\frac{1}{n^{3}}\left[ \left( \frac{n}{k}-1\right) \left( \frac{n}{k}-2\right) \right] \end{aligned}$$
(19)

Proof

$$\begin{aligned} s_{k}^{3}&=\frac{1}{k^{3}}E\left[ (Y_{k}^{3}- \left( \frac{k}{n}\right) ^{3}-3\left( \frac{k}{n}\right) Y_{k}^{2}+3\left( \frac{k}{n}\right) ^{2}Y_{k}\right] \\ &=\frac{1}{k^{3}}\left[ \frac{k}{n}-\left( \frac{k}{n} \right) ^{3}-3\left( \frac{k}{n}\right) ^{2}\,+\,3 \left( \frac{k}{n}\right) ^{3}\right] \end{aligned}$$
(20)

Rearranging terms leads to

$$\begin{aligned} s_{k}^{3}&=\frac{1}{n^{3}}\left[ \left( \frac{n}{k}\right) ^{2} -3\left( \frac{n}{k}\right) +2\right] \\ &=\frac{1}{n^{3}}\left[ \left( \frac{n}{k}-1\right) \left( \frac{n}{k}\,-2\right) \right] \end{aligned}$$
(21)

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:

$$\begin{aligned} s_{k}^{3}=\frac{1}{n}\left[ \left( D_{1}-\frac{1}{n}\right) \left( D_{1}-\frac{2}{n}\right) \right] \end{aligned}$$
(22)

We can also establish a very simple relationship between \(s_{k}^{3}\) and \(\sigma _{k}^{2}\) using Eqs. (14) and (19).

$$\begin{aligned} s_{k}^{3}=\sigma _{k}^{2}\left( D_{1}-\frac{2}{n}\right) \end{aligned}$$
(23)

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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

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Keywords

  • Individual investors
  • Return skewness
  • Diversification
  • Gambling

JEL Classification

  • G02
  • G11