Diversification versus optimality: is there really a diversification puzzle?

ABSTRACT

In this paper, we provide a general valuation of the diversification attitude of investors. First, we empirically examine the diversification of mean-variance optimal choices in the US stock market during the 11-year period 2003–2013. We then analyze the diversification problem from the perspective of risk-averse investors and risk-seeking investors. Second, we prove that investors’ optimal choices will be similar if their utility functions are not too distant, independent of their tolerance (or aversion) to risk. Finally, we discuss investors’ attitude towards diversification when the choices available to investors depend on several parameters.

KEYWORDS:

• Diversification puzzle
• stochastic dominance
• risk-seeking investors
• risk-averse investors
• utility functions
• compensatory risk premium

JEL CLASSIFICATION:

• C44
• D81
• G11

Acknowledgements

The authors thank the Co-Editor, David A. Peel, and the anonymous referees for substantive comments that helped significantly improve the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ Indeed, Boyle et al. (2011) justify the scarce diversification by introducing the ambiguity as a further decision parameter in the portfolio problem (i.e. the optimal portfolio choices of their model depend on the mean, the variance, and an ambiguity parameter).

² We assume that the random variables $X$ are gross returns $X_{t,i}=\frac{P_{t,i}}{P_{t-1,i}}$ (where $P_{t,i}$ is the price at time $t$ of i–th asset), returns $X_{t,i}=\frac{P_{t,i}-P_{t-1,i}}{P_{t-1,i}}$ or log returns $X_{t,i}=ln\left(\frac{P_{t,i}}{P_{t-1,i}}\right).$ While returns and log returns could be negative, the gross returns are positive random variables, and, thus, if no short sales are allowed, all portfolios of gross returns are positive random variables. All the results we get in the following empirical analysis can be obtained using all the different definitions of returns. However, in Corollary 3 and in the theoretical results of Section V, we explicitly use the gross returns because they are positive random variables.

³ That is, the i–th components of vectors ${\vec{\alpha }}^0,{\vec{\beta }}^0$ are given by ${\alpha }_i^0={\alpha }_{n-i+1:n}$, ${\beta }_i^0={\beta }_{n-i+1:n}$ where ${\alpha }_{k:n},$${\beta }_{k:n}$ indicate the k–th component of the vectors $\vec{\alpha },\vec{\beta }$ sorted in ascending order.

⁴ Observe that we obtain very similar results (not reported here) on portfolio diversification of the optimal mean-variance portfolios using or not using other techniques to reduce the dimensionality (see Ortobelli and Tichý (2015)). Clearly, this further aspect emphasizes the robustness of the proposed empirical analysis.

⁵ Because in the ordering columns of Table 2 each row reports the percentage of times the portfolio is dominated by the adjacent portfolio (with a greater level of the mean or with a smaller level of the mean), then the last row (corresponding to the maximum mean portfolio) does not report any values.

⁶ The assumption that the returns have the same finite variance and the same correlation is consistent with the maximum ambiguity in the variance-covariance estimation (see, among others, Pflug, Pichler, and Wozabal (2012), Boyle et al. (2011)). This assumption implies that the ‘1/N’ portfolio is the GMV portfolio, that is mean-variance efficient, even if it is first-order stochastically dominated according to Ortobelli et al. (2003).

⁷ For concave and convex ordering we use the same notation often used in ordering theory (see, among others, Shaked and Shanthikumar (1993)).

⁸ Generally, we can use the distance: $d(u_1,u_2)=\underset{X}{sup}E(|u_1(X)-u_2(X)|)$

when we want to identify the distance between standardized utility functions $u_i$ applied to a class of choices $X$ for which we can guarantee that $\underset{X}{sup}E(\left|u_1(X)\right|)<\mathrm{\infty }$ for $\mathrm{i}=1,2.$ Thus, when we use this distance considering the class of all admissible portfolios $\vec{\alpha }\in S_n$, we implicitly assume that $\underset{\vec{\alpha }\in S_n}{sup}E(\left|u_i({\vec{\alpha }}^{\mathrm{\prime }}\vec{X})\right|)<\mathrm{\infty }$ for $\mathrm{i}=1,2.$

⁹ We observe that the theoretical results of this section can be easily extended to scale and translation invariant families of random variables. This alternative assumption is useful when we consider random variables which are not positive such as portfolios of log returns (or assuming that short sales are allowed).

¹⁰ See Buhlmann (1970) who first proposed the $u$-mean certainty equivalent (see also Ben–Tal and Teboulle (2007) and the references therein). Moreover, for any standardized utility function $u$ belonging to $U_1^A$ the $u$–mean certainty equivalent $M_u(X)$ can be used as a monotone positive translation equivariant reward measure ${\mu }_X$ to define the parametric family $\sigma {\tau }_k^{+}(\bar{a}).$

¹¹ Observe that we use the concept of translation invariance in the sense of Gaivoronski and Pflug (2004) (i.e. ${\rho }_{X+t}={\rho }_X)$, and we call translation equivariance the property ${\mu }_{X+t}={\mu }_X+t.$ Moreover, note that the $u$–mean certainty equivalent and the optimized certainty equivalent are translation equivariant (see, among others, Ben–Tal and Teboulle (2007)). Moreover, the classic definition of certainty equivalent (i.e. the quantity $X^{\ast }$ solution of the equation $E(u(X))=u(X^{\ast })$) is not a translation equivariant reward measure, except for the linear utility function $u(x)=x$.

¹² See, among others, Buhlmann (1970) and Ingersoll (1987).

¹³ See, among others, Ben–Tal and Teboulle (2007) and Goovaerts, De Vylder, and Haezendonck (1984).

¹⁴ See, among others, Ben–Tal and Teboulle (2007) and the references therein.

Additional information

Funding

This research has been partially supported by grants: from Agencia Nacional de Innovación e Investigación (ANII) in Uruguay; Asia University, Hang Seng Management College, China Medical University Hospital, Lingnan University, the Research Grants Council (RGC) of Hong Kong (project number 12500915), Italian funds MURST 2018; the Czech Science Foundation (GACR) (project number 17–19981S), VSB–TU Ostrava (SGS research project SP2018/34), and Ministry of Science and Technology (MOST), R.O.C.