### Diversification And Correlation Ambiguity

University of California, San Diego (UCSD) – Rady School of Management

Shanghai University of Finance and Economics, School of Finance

November 18, 2015

**Abstract: **

One of the most important insights of modern finance is diversification in the sense that the optimal portfolio usually contains all available risky assets. In this paper, we show that correlation ambiguity can generate anti-diversification in the sense that the optimal portfolio has exactly one risky assets even if there are N>1 available risky assets. More generally, correlation ambiguity leads to under-diversification in that the optimal portfolio contains only part of available risky assets. With 100 stocks randomly-selected from S&P 500, on average only around 20 stocks will be held in the optimal portfolio when the sets of ambiguous correlations are given by 95% confidence intervals. Our results suggest that aversion to correlation ambiguity may provide an explanation for under-diversification documented in empirical studies.

### Diversification And Correlation Ambiguity – Introduction

WE STUDY PORTFOLIO CHOICE of an agent who is averse to ambiguity in correlations. While previous studies focus on aversion to expected return ambiguity, we study the effect of aversion to correlation ambiguity on portfolio choice. We find that the aversion to correlation ambiguity may lead to anti-diversification, that is, there is only one single risky asset in the optimal portfolio. In general, correlation ambiguity generates under-diversification, in the sense that the optimal portfolio contains only part of available risky assets.

One of the most important insights of modern finance theory is diversification in that the optimal portfolio should contain all available risky assets. This is true for expected utility theories including the Markowitz’s static portfolio choice theory and the Merton’s dynamic portfolio choice theory.

In this paper, we show that anti-diversification happens with correlation ambiguity. If correlations are ambiguous enough, the agent holds only one risky asset. Intuitively, when correlations are totally ambiguous, an optimal portfolio for an agent who is aversion to this ambiguity should be insensitive to correlations. Only portfolios that consist of one asset are insensitive to correlations, the optimal strategy for the agent with ambiguity aversion is to hold the asset which has the greatest Sharpe ratio. The sufficient and necessary condition for occurrence of anti-diversification is characterized in this paper.

More generally, when correlations are not completely ambiguous, that is, when correlations can take values in strict subsets of [-1; 1], we have under-diversification in the sense that the optimal portfolio does not contain all risky assets. The number of risky assets in the optimal portfolio can be substantially smaller than the total number of the available risky assets. For example, given 100 randomly-selected US stocks with ambiguous sets being 95% confidence intervals of correlation estimations, the optimal portfolio has about 20 stocks. As degree of correlation ambiguity increases, stocks with lower Sharpe ratio will tend to be left out of the optimal portfolio until the one with the greatest Sharpe ratio remains. In contrast, without ambiguity aversion all 100 stocks are held under the mean-variance framework. Goldman (1979) coined the term Anti-diversification for holding one risky asset. He shows that, for buy-and-hold strategies, the limit of infinite time horizon leads to anti-diversification.

Under-diversification is documented in many empirical studies. For example, Campbell (2006) suggests that financial portfolios of households contain only a few of risky assets. Goetzmann and Kumar (2008) report that the majority of individual investors hold a single digit number of assets in a sample data during 1991-1996. Among many other empirical findings about under-diversification from various data sets, we refer to Mitton and Vorkink (2007), Calvet, et al. (2008), and Ivkovic, et al. (2008). Our result suggests that correlation ambiguity may be an explanation of these findings.

There are other explanations of under-diversification. Brennan (1975) finds that the optimal number of risky assets in a portfolio is small where there is fixed transacting cost. Liu (2014) proposes a model in which under-diversification may be caused by solvency requirements in the presence of committed consumption. Roche et al. (2013) suggest that financial constraints can lead to under-diversification. Boyle, et al. (2012) can produce under-diversification with ambiguous expected returns.

Jagannathan and Ma (2003) point out that covariance (and correlations) are imprecisely estimated especially when the number of assets is big. Note that the number of correlations increases in N(N -1)=2, thus correlations are more difficult to estimate for a big number of assets. We refer to Engle and Sheppard (2001) and Engle (2002) for estimating a big number of correlations. Besides, empirical studies, for example, Longin and Solnik (2001) and Cappiello et al. (2006) 1, show that correlations are dynamic, hence even harder to estimate than constant correlations in a model.

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