Mean-variance Rule Research Articles

Dispersion indices based on Kerridge inaccuracy measure and Kullback-Leibler divergence

Recently, a new dispersion index, as a measures of information, has been introduced and called varentropy. In this article, we introduce new measures of variability based on two measures of uncertainty, namely, the Kerridge inaccuracy measure and the Kullback-Leibler divergence. Their generating functions are considered and their infinite series representations are given. These new measures and associated properties, bounds and illustrative examples are all presented in detail. Finally, an application of Kullback-Leibler divergence and its dispersion index is illustrated by using the mean-variance rule.

The Mean-Variance Rule for Investors with Reverse S-Shaped Utility

Our paper contributes to the literature by developing the theory of the mean-variance (MV) rules for investors with reverse S-shaped utility. To do so, we first introduce the definition of the MV rule for investors with reverse S-shaped utility. We then set up the conjecture on the preference for different prospects by using the new MV rule that they could get a higher expected utility for the preferred asset under some conditions. Thereafter, we look for the conditions that the conjecture could hold and construct a theorem for this purpose by showing that when the negative (positive) parts of the assets follow one (another) type of location-scale family or the linear combination of location-scale families, then the preferences of the assets is the same as those by using an expected utility for the investors with reverse S-shaped utility. We then extend the theory by developing some properties of portfolio diversification by using the new MV rule. The theory developed in our paper enables academics and practitioners to apply the theory developed in this paper to analyze some important empirical issues and draw inferences on the preferences of investors with reverse S-shaped utility.

The Mean-Variance Rule for Investors with Reverse S-Shaped Utility

The Mean-Variance Rule for Investors with Reverse S-Shaped Utility

Mean–variance combining rules that outperform naïve diversification

The mean–variance framework developed by Markowitz (1952). Portfolio selection, The Journal of Finance, 7(1), 77–91 is still the major model used nowadays in asset allocation and active portfolio management. However, the estimated mean–variance rules often fail to deliver superior performance compared with the simple naïve rule (the equally weighted portfolio) due to the problem of estimation errors. In this paper, I propose a portfolio construction method that is effective in dealing with estimation errors in the optimization process. Particularly, I specify the portfolio weights as an optimal combination of the equally weighted portfolio and a sample zero-investment portfolio. I show analytically that the proposed method alleviates the problem of estimation errors and dominates naïve diversification. I suggest two implementable versions of the combining method and show, empirically, their good performances relative to the naïve rule. The newly developed rules work well, particularly, for portfolios with a medium and high number of assets. Moreover, the outperformance persists generally even in the presence of transaction costs. Since the combinations are theory-based, my study may be interpreted as reaffirming the usefulness of the Markowitz portfolio theory in practice.

Should Investors Choose Both the Highest-Return and Small-Variance Assets When the Mean-Variance Rule Says No? A Study of Health Care and T-Bill in the Investment

The high-speed growth of the health care sector has given this sector an increasingly important role in the stock market. This sector however has the highest mean in our study and a low correlation with the business cycle. On the other hand, T-Bill is also an important asset in investment because of its positive return and low variance. In this paper, we examine the conjecture of whether investors should choose both the highest-return and small-variance assets even when the mean-variance rule says “NO”. This conjecture is explored by making a comparison of the performance of portfolios with and without health care and 6-M T-bill in the U.S. market using portfolio optimization, mean-variance and stochastic dominance approaches. Our findings imply that all risk averters prefer to invest in portfolios with both health care and 6-M T-bill, regardless of whether they buy long or sell short in the market. Our findings do not support the existence of any arbitrage opportunity in the markets we studied but do support market efficiency. We also show that risk averters as well as investors with components of both risk aversion and risk seeking will choose both the highest-return and small-variance assets even when the mean-variance rule says NO.

Are the Combinations of Health Care Sector and T-Bill One of the Best Choices for Investment?

Health care sector plays an increasingly important role in the stock market because it is growing nearly in the entire period and has low correlation with the business cycle. On the other hand, T-Bill is also an important asset in investment because it has positive return and small variance. In this paper, we employ the mean-variance (MV) rules and stochastic dominance (SD) approach to investigate the portfolio performance with and without both health care sector and T-Bill in the US over the period from September 1986 to May 2017. The results show that all the portfolios with health care asset and 6-M TB significantly dominate the corresponding portfolios without health care and 6-M TB, regardless whether short sale is allowed. This study also concludes that the combinations of health care sector and 6-M TB not only reduce risk but also gain better return and all risk-averse investors prefer to invest in portfolios with health care sector and 6-M TB, regardless whether they buy long or sell short the market. By employing the mean-variance (MV) rules and stochastic dominance (SD) approach, this paper investigates the portfolio performance with and without both health care sector and T-Bill in the U.S. market over the period from September 1986 to May 2017. The results show that all the portfolios with health care asset and 6-m TB significantly dominate the corresponding portfolios without health care and 6-m TB, regardless whether short sale is allowed. This study also concludes that the combinations of health care sector and 6-m TB not only reduce risk but also gain better return; all risk-averse investors prefer to invest in portfolios with health care sector and 6-m TB, regardless whether they buy long or sell short the market.

Efficient cluster-based portfolio optimization

The sample mean and covariance matrix of historical data provide a disappointing out-of-sample performance in mean-variance portfolio rules. This poor performance is certainly due to the high estimation error incurred in the optimization model. Our purpose in this article is to find a method that enhances the out-of-sample performance of the portfolio weights. Using hierarchical clustering, we propose an alternative cluster-based portfolio to obtain a sequence of cluster assets. On the basis of Gram-Schmidt orthogonalization, the estimation risk of the data set becomes the sum of the estimations of the clusters in the sequence. The performance of our method and its competitors is compared empirically and via some simulations in high dimension.

Could Omega Ratio Perform Better than Sharpe Ratio?

In this paper, we will investigate whether there is any Sharpe ratio rule or Omega ratio rule that can be used to show that one asset outperforms another asset if it has a higher Sharpe ratio and/or Omega ratio. We find that Sharpe ratio rule could not detect preference of both risk averters and risk seekers in some strong dominance cases. We set up the Omega ratio rule and find that the Omega ratio rule is better than the mean-variance rule because the former could the former can detect the first order stochastic dominated asset but the latter cannot. We also show the superiority of the Omega ratio rule over any Sharpe ratio rule by using hedging funds data and discuss the advantage of using the Omega ratio rule to stochastic dominance rule.

The Preferences of Omega Ratio for Risk Averters and Risk Seekers

It is well-known that under some conditions, the mean-variance rule is equivalent to stochastic dominance rule. Some academics hypothesize that there could exist mean-Omega ratio rule that could be equivalent to stochastic dominance rule under certain conditions. To explore this possible, in this paper, we aim to establish the necessary conditions between Omega ratio and stochastic dominance that leads to the preferences of risk averters/seekers. We find that it is possible to establish the necessary conditions between and Omega ratio and the preferences of risk averters/seekers under the condition that the variables being compared belong to the location-scale family or the same linear combination of location-scale families.

Endogenous Bank Networks and Contagion

I develop a model to study two related questions: how bank decisions to form connections depend on fundamentals; and how financial stability depends on bank network structure. In my model, banks are connected through two layers of networks: interbank debts and banks' common investments in non-financial firms. These layers of interconnections are incentivized by diversified investments when banks maximize their expected equity values according to mean-variance rules. I show that the topologies of the two network layers interact with each other. Comparative statics of a small number of banks indicates that, in equilibrium, as banks become less risk averse, they tend to issue more debts and form more links within the banking sector. Furthermore, I conduct numerical computations for bank default probabilities in a circle network and a more connected network. The results demonstrate that increased bank interconnectedness and common asset holdings significantly reduce systemic stability.

Sample Selection Bias, Return Moments, and the Performance of Optimal versus Naive Diversification

I examine the sample selection bias in portfolio horse race. Numerous studies propose mean-variance portfolio rules to outperform the naive portfolio rule. However, the outperformance is often justified by a small number of pre-selected datasets. Using a new performance test based on a large number of datasets, I compare thirteen 1/N outperformers with the naive rule. Results show that not only a majority of the thirteen 1/N outperformers on average underperform the rule significantly, but also none of these mean-variance rules significantly outperform the naive benchmark in more than 10% of the datasets. To further understand portfolio performance, this paper explores the theoretical relations between assets' return moments and the performance of optimal versus naive diversification. These relations not only imply strong performance predictability, but also can be exploited to deliver out-of-sample portfolio benefits.

Lotto, How to Win? Skew Timing Strategies

In this paper, we investigate whether the “lotto investor” can benefit from the time-varying skewness of market portfolio and how to capture the gain using skew timing strategies. We find that empirically applying the mean-variance-skewness (M-V-S) rule of Mitton and Vorkink (2007) generates similar performance as that of traditional mean-variance rule, because the optimal weight of M-V-S still mainly depends on the mean and variance unless the forecasted skewness is extremely large. To improve the M-V-S, we suggest combining two constrained versions of M-V-S, namely the mean-variance (M-V) and mean-skewness (M-S). Discarding the variance, the M-S fully considers the usefulness of skewness and the optimal weight solely depends on mean and skewness. However, M-S investor probably suffers huge loss due to forecasting errors. Combining the M-V with M-S should theoretically perform better than individual rules, and hence better than the M-V-S rule. Empirically, we find that the combination rule indeed generates superior performance, in terms of certainty equivalent returns, Sharpe ratios, and skewness of portfolio returns distribution.

The Performance of Asset Allocation Strategies Across Datasets and Over Time

This paper evaluates the ex-ante performance of popular asset allocation strategies including the classical mean variance rule, the widely used naive diversification and several newly sprouted risk-based techniques. In terms of risk-adjusted return, I find that there are no significant differences between these strategies for country- and industry-based portfolios. For the individual stock portfolio and multi-asset portfolios (stocks and bonds combined), the differences between strategy performances are relatively large, and the superior strategy changes over time. With respect to strategies’ risk loadings, the mean variance rule leads to the most fluctuating exposures in size, value and momentum factors for all four datasets, the other allocation methods are less so. This paper indicates that currently-used allocation strategies are all far from ex-post optimal.

Linear Tests for Decreasing Absolute Risk Aversion Stochastic Dominance

We develop and implement linear formulations of convex stochastic dominance relations based on decreasing absolute risk aversion (DARA) for discrete and polyhedral choice sets. Our approach is based on a piecewise-exponential representation of utility and a local linear approximation to the exponentiation of log marginal utility. An empirical application to historical stock market data suggests that a passive stock market portfolio is DARA stochastic dominance inefficient relative to concentrated portfolios of small-cap stocks. The mean-variance rule and Nth-order stochastic dominance rules substantially underestimate the degree of market portfolio inefficiency because they do not penalize the unfavorable skewness of diversified portfolios, in violation of DARA. This paper was accepted by James Smith, decision analysis.

Portfolio selection in a two-regime world

Abstract Standard mean-variance analysis is based on the assumption of normal return distributions. However, a growing body of literature suggests that the market oscillates between two different regimes – one with low volatility and the other with high volatility. In such a case, even if the return distributions are normal in both regimes, the overall distribution is not – it is a mixture of normals. Mean-variance analysis is inappropriate in this framework, and one must either assume a specific utility function or, alternatively, employ the more general and distribution-free Second degree Stochastic Dominance (SSD) criterion. This paper develops the SSD rule for the case of mixed normals: the SSDMN rule. This rule is a generalization the mean-variance rule. The cost of ignoring regimes and assuming normality when the distributions are actually mixed normal can be quite substantial – it is typically equivalent to an annual rate of return of 2–3 percent.

Powerful Linear Tests for DARA Stochastic Dominance

We develop and implement linear formulations of convex stochastic dominance relations based on decreasing absolute risk aversion for discrete and polyhedral choice sets. Our approach is based on a piecewise-exponential representation of utility and a local linear approximation to the exponentiation of log marginal utility. An empirical application to historical stock market data suggests that a passive stock market portfolio is DSD inefficient relative to concentrated portfolios of small-cap stocks. The mean-variance rule and N-th order stochastic dominance rules substantially underestimate the degree of market portfolio inefficiency, because they do not penalize the unfavorable skewness of diversified portfolios, in violation of DARA.

General linear formulations of stochastic dominance criteria

We develop and implement linear formulations of general Nth order stochastic dominance criteria for discrete probability distributions. Our approach is based on a piece-wise polynomial representation of utility and its derivatives and can be implemented by solving a relatively small system of linear inequalities. This approach allows for comparing a given prospect with a discrete set of alternative prospects as well as for comparison with a polyhedral set of linear combinations of prospects. We also derive a linear dual formulation in terms of lower partial moments and co-lower partial moments. An empirical application to historical stock market data suggests that the passive stock market portfolio is highly inefficient relative to actively managed portfolios for all investment horizons and for nearly all investors. The results also illustrate that the mean–variance rule and second-order stochastic dominance rule may not detect market portfolio inefficiency because of non-trivial violations of non-satiation and prudence.

Big House or Small House, Which One Should We Buy? Evidence from Hong Kong

Big house or small house, which one should we buy? To answer this question, we employed both mean-variance approach and stochastic dominance approach that utilizes the entire yield distribution to rank the performance of the Hong Kong housing market. While mean-variance rules cannot show a clear preference between different classes (sizes) of housing, we find, however, that smaller class housing has first-order stochastic dominance over larger class housing. That means not only risk-averse investors but also any investor with increasing utility function can maximize their expected utilities and expected wealth. We conclude that the smallest class housing provides a better return for both risk-averse and other types of investors and a non-satiation investor will increase his wealth as well as his utility if his investment is shifted to the smallest class housing. This suggests even though there may not be arbitrage opportunity, Hong Kong housing market is not efficient. The answer is simple, in Hong Kong, we should buy small house.

Active Asset Allocation Among a Large Set of Stocks: How Effective is the Parametric Rule?

In this study we measure the value of active money management. We explore this issue by comprehensively examining the parametric rule proposed by Brandt, Santa-Clara and Valkanov (2009) (the BSV rule) out-of-sample for cross-sectional portfolio choice among a large number of assets and comparing this rule to the mean-variance (MV) rule and the naive 1/N rule recently advocated by DeMiguel, Garlappi and Uppal (2009). We find that the BSV rule outperforms both the MV and 1/N rules and the outperformance is robust to investment horizons and stock market states. The BSV rule is effective for investors with different preferences or investment opportunities either. The effectiveness of the BSV rule is robust to data screening criteria, estimation periods, portfolio performance evaluation models, the business cycle, and stock market states. Our results suggest that the BSV rule is useful.

General Linear Formulations of Stochastic Dominance Criteria: With an Analysis of Stock Market Portfolio Efficiency

We develop and implement linear formulations of general N-th order Stochastic Dominance criteria for discrete probability distributions. Our approach is based on a piece-wise polynomial representation of utility and its derivatives and can be implemented by solving a relatively small system of linear inequalities. This approach allows for comparing a given prospect with a discrete set of alternative prospects as well as for comparison with a polyhedral set of linear combinations of prospects. We also derive a linear dual formulation in terms of lower partial moments and co-lower partial moments. An empirical application to historical stock market data suggests that the passive stock market portfolio is highly inefficient relative to actively managed portfolios for all investment horizons and for nearly all investors. The results also illustrate that the mean-variance rule and second-order stochastic dominance rule may not detect market portfolio inefficiency because of non-trivial violations of non-satiation and prudence.