Markowitz efficiency and size effect: evidence from the UK stock market

Academics and practitioners have frequently debated the relationship between market capitalization and expected return. We apply the Markowitz efficient frontier approach to develop a portfolio performance measure that compares the return of a portfolio to its optimal return, using data from the UK stock market over the period 1985–2012. Our results show that there is a negative relationship between portfolio size and portfolio return during the period under study. When comparing actual portfolio return with achievable return for the same level of risk, we find that as the portfolio size expands, underperformance of the portfolio increases, i.e. the larger the portfolio size, the greater the underperformance. This indicates that Markowitz efficiency is difficult to achieve, particularly in large portfolios. Changing model parameters leads to alternative efficient frontiers that impact upon the measurement of performance. However, the use of alternative efficient frontiers does not affect our result of the size effect on the relative performance of portfolios. Our study shows that the size effect is present over the full period. Our findings also suggest that the excess returns found in small portfolios are likely to be associated with higher levels of diversifiable risk in comparison with larger portfolios. Furthermore, in contrast to other studies, we find no evidence to support the size reversal effect in the data.     

INTERNATIONAL DIVERSIFICATION OF EQUITIES AND FIXED-INCOME SECURITIES

Portfolio selection models have been applied principally to common stocks traded in the United States and in foreign stock markets. This study examines the efficient set of portfolios selected from a choice set that includes returns derived from domestic and international corporate bond and government bond indices as well as domestic and international stock indices. To assess the benefits of international multi-asset diversification, the authors examine the following issues: (1) the extent to which international and domestic fixed-income securities are included in efficient portfolios; (2) the effect on efficient set composition of using the Sharpe portfolio selection model as compared to the Markowitz portfolio selection model; (3) the sensitivity of efficient set characteristics produced from a single-index based portfolio selection model to alternative world market indices; and (4) the correspondence between expected and realized portfolio risk and return for the different portfolio selection models.     

A non-probabilistic approach to efficient portfolios

The notion of efficient portfolios is restated employing a novel approach to asset uncertainty, in which an uncertainty set conveys all information on asset returns. Based on the idea that a portfolio is a function mapping the asset returns to the portfolio return, the width of the image of the uncertainty set is used as a risk measure. The statement of the separation theorem is inherited. Furthermore, non-positive Value at Risk with 100% confidence is found in a class of efficient portfolios.     

Downside risk and asset pricing

We analyze if the value-weighted stock market portfolio is stochastic dominance (SD) efficient relative to benchmark portfolios formed on size, value, and momentum. In the process, we also develop several methodological improvements to the existing tests for SD efficiency. Interestingly, the market portfolio seems third-order SD (TSD) efficient relative to all benchmark sets. By contrast, the market portfolio is inefficient if we replace the TSD criterion with the traditional mean–variance criterion. Combined these results suggest that the mean–variance inefficiency of the market portfolio is caused by the omission of return moments other than variance. Especially downside risk seems to be important for explaining the high average returns of small/value/winner stocks.     

THE CROSS-AUTOCORRELATION OF SIZE-BASED PORTFOLIO RETURNS IS NOT AN ARTIFACT OF PORTFOLIO AUTOCORRELATION

Prior studies find evidence of asymmetric size-based portfolio return cross-autocorrelations where lagged large firm returns lead current small firm returns. However, some studies question whether this economic relation is independent of the effect of portfolio return autocorrelation. We formally test for this independence using size-based portfolios of New York Stock Exchange and American Stock Exchange securities and, separately, portfolios of Nasdaq securities. Results from causality regressions indicate that, across all markets, lagged large firm returns predict current small firm returns, even after controlling for autocorrelation in small firm returns. These cross-autocorrelation patterns are stronger for Nasdaq securities.     

Realistic Portfolio Allocation Decision-Making For The Small U.S. Retail Investor

This study introduces new domestic mixed-asset and international equity securities that allow for exact portfolio replication even by small U.S. retail investors. Using these new series, various return characteristics are examined. Finally, three sets of mean-variance analyses are conducted: a domestic equity sector-only portfolio, a domestic mixed-asset portfolio, and an international mixed-asset portfolio. Real estate warrants inclusion to varying degrees in all three portfolios. International equity inclusion was also demonstrated.     

OPTIMAL PORTFOLIOS: MARKOWITZ FULL COVARIANCE VERSUS SIMPLE SELECTION RULES

Two major problems faced by portfolio managers are estimating the risk and return characteristics of individual securities and combining individual security risk and return estimates into optimal portfolios. The second problem is investigated in this paper by using the simple ranking criteria suggested by Elton, Gruber, and Padberg (EGP). The empirical results indicate that the EGP procedure is effective in estimating Markowitz efficient portfolios and can be an effective screening procedure for large numbers of securities.     

The econometrics of efficient portfolios

It is well known that the standard mean variance approach can be inappropriate when return distributions feature skewness, fat tails or multimodes. This is typically the situation for portfolios including derivatives. In this case, it can be necessary to come back to the basic expected utility approach. In this paper, an efficient portfolio maximizes the expected utility of future wealth. This paper presents an analysis of the efficiency frontier, formed by a set of efficient portfolios corresponding to a parameterized class of utility functions. First, we discuss the estimation of an efficient portfolio and introduce several tests of the efficiency hypothesis, depending on what is known about the utility function and the budget level. Next we analyse the shape of the frontier and develop a procedure for testing the separability of the efficiency frontier into K independent funds. The inference is semi-nonparametric because the return distribution is left unspecified. We illustrate our approach by an application to portfolios including derivatives.     

Simple Construction of the Efficient Frontier

We provide simple methods of constructing known results. At the core of our methods is the identification of a simple concise basis that spans the Capital Market Line (CML). We show that a portfolio whose risky assets weights are the product of the inverse variance‐covariance matrix of (nonredundant) security rates of return times the vector of the excess expected rates of return over the risk‐free rate is a CML portfolio. This portfolio and the risk‐free security span the CML. In addition, with this basis, there is immediate construction of the efficient frontier of risky assets (the 'hyperbola'), 'tangency' portfolios, 'reflection' portfolios, and a CAPM relationship. Our method is quick and simple. It is easy to derive, teach, implement, interpret, and remember.     

Linear programing models for portfolio optimization using a benchmark

We consider the problem of constructing a perturbed portfolio by utilizing a benchmark portfolio. We propose two computationally efficient portfolio optimization models, the mean-absolute deviation risk and the Dantzig-type, which can be solved using linear programing. These portfolio models push the existing benchmark toward the efficient frontier through sparse and stable asset selection. We implement these models on two benchmarks, a market index and the equally-weighted portfolio. We carry out an extensive out-of-sample analysis with 11 empirical datasets and simulated data. The proposed portfolios outperform the benchmark portfolio in various performance measures, including the mean return and Sharpe ratio.     

Risks and rewards for momentum and reversal portfolios

Rational asset pricing implies a positive relation between the expected risk-adjusted return and the volatility of a factor-mimicking portfolio. The relation for the momentum portfolio is weak after its return is adjusted for the risks associated with the market return, the size factor, and the book-to-market factor. However, the relation is significantly positive and captures most of the average return on the momentum portfolio after the return is adjusted for the market return and the risk associated with the short-term reversal portfolio return. The result supports the hypothesis that there is a common factor underlying both momentum and short-term reversal. The dynamics of the factor loadings and the correlation structure of the underlying factors have important implications for the risk prices associated with the factor-mimicking portfolios and the risk–return trade-off for momentum and reversal portfolios.     

Art as an investment: risk, return and portfolio diversification in major painting markets

The present paper examines risk, return and the prospects for portfolio diversification among major painting and financial markets over the period 1976–2001. The art markets examined are Contemporary Masters, French Impressionists, Modern European, 19th Century European, Old Masters, Surrealists, 20th Century English and Modern US paintings. The financial markets comprise US Treasury bills, corporate and government bonds and small and large company stocks. In common with the published literature in this area, the present study finds that the returns on paintings are much lower and the risks much higher than conventional investment markets. Moreover, while low correlations of returns suggest that opportunities for portfolio diversification in art works alone and in conjunction with equity markets exist, the construction of Markowitz mean‐variance efficient portfolios indicates that no diversification gains are provided by art in financial asset portfolios. However, diversification benefits in portfolios comprised solely of art works are possible, with Contemporary Masters, 19th Century European, Old Masters and 20th Century English paintings dominating the efficient frontier during the period in question.     

Potential performance and tests of portfolio efficiency

The potential performance of an asset set may be obtained by choosing the portfolio proportions to maximize the Sharpe (1966) performance measure. If a portfolio has a Sharpe measure equivalent to the potential performance of the underlying set of assets, then it is efficient. Multivariate statistical procedures for comparing potential performance and testing portfolio efficiency are developed and then evaluated using simulations. Two likelihood ratio statistics are then used to compare stock and bond indices against sets of 20 and 40 portfolios. The procedures are also compared to the Gibbons (1982) methodology for testing financial models.     

Is Bitcoin a better portfolio diversifier than gold? A copula and sectoral analysis for China

This paper aims to compare Bitcoin with gold in the diversification of Chinese portfolios using daily data over the 2010–2020 period. We propose a new development of copula-based joint distribution function of returns to simulate the Value-at-Risk and expected shortfall of portfolios including Bitcoin (or gold) and those without it. The stochastic dominance method is also used to compare the return distributions of the three types of portfolios. Empirical results show that gold is a better portfolio diversifier than Bitcoin as it helps better reduce the risk of portfolios. On the other hand, Bitcoin better increases the return but also increases the risk. The stochastic dominance results further show that portfolios diversified by gold dominate those diversified by Bitcoin. Based on these findings, we conclude that in China, gold is a better portfolio diversifier than Bitcoin for risk-averse investors. However, for risk-seeking investors, Bitcoin can be a better choice. This result is found to be robust to the time, frequency and currency effects.     

Household portfolio optimization with XTFs? An empirical study using the SHS-base

XTFs are plain-vanilla Exchange Traded Funds (ETFs) which replicate a broad, internationally diversified market index. We question, if XTFs can optimize the performance of households’ portfolios when taking multiple relevant asset classes into account, not only stocks. As opposed to most existing studies, we apply representative household portfolio data to estimate households’ portfolios. Households’ portfolios in our sample show similar compositions and can be grouped into one of three stylized portfolio compositions which exhibit asset class concentrations on cash/savings, mutual funds and individual stocks. For each stylized portfolio, we first investigate if an easily investable 60/40 stock/bond XTF portfolio which is risk-adjusted (including (de-)leverage costs) to the risk of the stylized portfolios, achieves higher returns than the stylized portfolios. This is the case for all stylized portfolios, even those with concentrations on cash/savings or mutual funds. Second, we examine risk/return-changes when replacing the entire risky assets of the stylized portfolios with the 60/40 stock/bond XTF portfolio including transaction costs. This leads to return enhancements in all stylized portfolios and particularly in the portfolio with high stock concentrations to risk reductions. Overall, we find that XTFs are generally suitable to optimize the performance of households’ portfolios under consideration of multiple relevant asset classes.     

REGULATORY INFLUENCES ON PORTFOLIO PERFORMANCE: SHORT SELLING AND REGULATION T

Several models are developed to examine the portfolio effect of short selling. Three things are demonstrated in this study. First, that for many assets, short selling is a useful strategy for reducing risk when constructing mean-variance efficient portfolios. Second, Regulation T can be used in combination with short selling to further improve expected portfolio performance. Third, the performance of the suggested models is superior to previously suggested allocation models. Ex ante and ex post tests are conducted to arrive at the above conclusions.     

A dynamic model of active portfolio management with benchmark orientation

This paper studies optimal dynamic portfolios for investors concerned with the performance of their portfolios relative to a benchmark. Assuming that asset returns follow a multi-linear factor model similar to the structure of Ross (1976) [Ross, S., 1976. The arbitrage theory of the capital asset pricing model. Journal of Economic Theory, 13, 342–360] and that portfolio managers adopt a mean tracking error analysis similar to that of Roll (1992) [Roll, R., 1992. A mean/variance analysis of tracking error. Journal of Portfolio Management, 18, 13–22], we develop a dynamic model of active portfolio management maximizing risk adjusted excess return over a selected benchmark. Unlike the case of constant proportional portfolios for standard utility maximization, our optimal portfolio policy is state dependent, being a function of time to investment horizon, the return on the benchmark portfolio, and the return on the investment portfolio. We define a dynamic performance measure which relates portfolio’s return to its risk sensitivity. Abnormal returns at each point in time are quantified as the difference between the realized and the model-fitted returns. Risk sensitivity is estimated through a dynamic matching that minimizes the total fitted error of portfolio returns. For illustration, we analyze eight representative mutual funds in the U.S. market and show how this model can be used in practice.     

Liquidity risk,leverage and long-run IPO returns

We examine the risk-return characteristics of a rolling portfolio investment strategy where more than 6000 Nasdaq initial public offering (IPO) stocks are bought and held for up to 5 years. The average long-run portfolio return is low, but IPO stocks appear as “longshots”, as 5-year buy-and-hold returns of 1000% or more are somewhat more frequent than for non-issuing Nasdaq firms matched on size and book-to-market ratio. The typical IPO firm is of average Nasdaq market capitalization but has relatively low book-to-market ratio. We also show that IPO firms exhibit relatively high stock turnover and low leverage, which may lower systematic risk exposures. To examine this possibility, we launch an easily constructed “low-minus-high” (LMH) stock turnover portfolio as a liquidity risk factor. The LMH factor produces significant betas for broad-based stock portfolios, as well as for our IPO portfolio and a comparison portfolio of seasoned equity offerings. The factor-model estimation also includes standard characteristic-based risk factors, and we explore mimicking portfolios for leverage-related macroeconomic risks. Because they track macroeconomic aggregates, these mimicking portfolios are relatively immune to market sentiment effects. Overall, we cannot reject the hypothesis that the realized return on the IPO portfolio is commensurable with the portfolio's risk exposures, as defined here.     

Using industry momentum to improve portfolio performance

Minimum-variance portfolios, which ignore the mean and focus on the (co)variances of asset returns, outperform mean–variance approaches in out-of-sample tests. Despite these promising results, minimum-variance policies fail to deliver a superior performance compared with the simple 1/N rule. In this paper, we propose a parametric portfolio policy that uses industry return momentum to improve portfolio performance. Our portfolio policies outperform a broad selection of established portfolio strategies in terms of Sharpe ratio and certainty equivalent returns. The proposed policies are particularly suitable for investors because portfolio turnover is only moderately increased compared to standard minimum-variance portfolios.     

FILTER RULE TESTS OF THE ECONOMIC SIGNIFICANCE OF SERIAL DEPENDENCIES IN DAILY STOCK RETURNS

In this paper we examine the ability of filter rules to predict variation in expected daily returns for a sample of 120 Dow Jones and S&P 100 stocks from 1963 through 1989. Equally weighted portfolios of filter-rule-traded stocks consistently outperform a buy-and-hold portfolio of the same stocks before accounting for transaction costs. The difference in returns between filter rule and buy-and-hold portfolios is eliminated by one-way transaction costs of 12 basis points. The economic significance of daily stock return autocorrelations is estimated. A marginal 1 percent increase in a first-order autocorrelation increases filter rule returns by an estimated 3.84 percent.