现代投资组合理论的基础与前沿分析

马柯维茨的投资组合理论奠定了现代金融理论的基石。自提出以来,就受到关注,国内外学者对其进行了大量的研究。本文在对该理论进行介绍的基础上,介绍了国内学者的研究成果。     

Risk-Adjusted Economic Value Analysis

Abstract

The ability of commonly used profitability measures to reflect risk exposure appropriately is evaluated and found lacking. As an alternative, a modern portfolio theory approach, based on utility theory, is recommended. Generalized formulas for calculating risk-adjusted economic values by deriving risk adjustments from certainty equivalents are developed by using the Markowitz expected utility maxim. Practical applications are described. Where appropriate, simplifying assumptions are shown to result in closed-form solutions, thereby reducing the need for extensive, stochastic cashflow simulations. The resulting formulas can be used to measure financial performance on a risk-adjusted basis consistently across different lines of business or to evaluate risk exposures in strategic alternatives.     

Markowitz meets Talmud: A combination of sophisticated and naive diversification strategies

The modern portfolio theory pioneered by Markowitz (1952) is widely used in practice and extensively taught to MBAs. However, the estimated Markowitz portfolio rule and most of its extensions not only underperform the naive 1/N rule (that invests equally across N assets) in simulations, but also lose money on a risk-adjusted basis in many real data sets. In this paper, we propose an optimal combination of the naive 1/N rule with one of the four sophisticated strategies—the Markowitz rule, the Jorion (1986) rule, the MacKinlay and Pástor (2000) rule, and the Kan and Zhou (2007) rule—as a way to improve performance. We find that the combined rules not only have a significant impact in improving the sophisticated strategies, but also outperform the 1/N rule in most scenarios. Since the combinations are theory-based, our study may be interpreted as reaffirming the usefulness of the Markowitz theory in practice.     

Portfolio Selection and Optimization with Higher Moments: Evidence from the Indian Stock Market

The Markowitz portfolio optimization model, popularly known as the Mean-Variance model, assumes that stockreturns follow normal distribution. But when stock returns do not follow normal distribution, this model wouldbe inadequate as it would prescribe sub-optimal portfolios. Stock market literature often deliberates that stock returns are non-normal. In such context the Markowitz model would not be sufficient to estimate the portfolio risks. The purpose of this paper is to expand the original Markowitz portfolio theory (mean-variance) via adding the higher order moments like skewness (third moment about the mean) and kurtosis (fourth moment about the mean) in the return characteristics. The research paper investigates the impact of including higher moments using multi-objective programming model for portfolio stock selection and optimization. The empirical results indicate that the inclusion of higher moments had a considerable impact in estimating the returns behavior of portfolios. The portfolios optimized using all the four moments, generated higher returns for the given level of risk in comparison to the returns of the Markowitz model during the study period 2000–2011. The results of this study would be immensely useful to fund managers, portfolio managers and investors as it would help them in understanding the Indian stock market behavior better and also in selecting alternative portfolio selection models.     

Portfolio optimization algorithms,simplified criteria,and security selection: A contrast and evaluation

The Markowitz full covariance model provides a general framework for analysis of the porfolio selection problem. Three alternative solution methodologies have been developed to facilitate normative applications, but this article shows that they lead to markedly different stock selection and portfolio weighting decisions. In sample-based applications, incompatibilities arise due to model misspecifications and different distributional assumptions, and from the interactive effects of estimation error, optimization model selection bias, and conflicting distributional assumptions.     

Portfolio selection with higher moments

We propose a method for optimal portfolio selection using a Bayesian decision theoretic framework that addresses two major shortcomings of the traditional Markowitz approach: the ability to handle higher moments and parameter uncertainty. We employ the skew normal distribution which has many attractive features for modeling multivariate returns. Our results suggest that it is important to incorporate higher order moments in portfolio selection. Further, our comparison to other methods where parameter uncertainty is either ignored or accommodated in an ad hoc way, shows that our approach leads to higher expected utility than competing methods, such as the resampling methods that are common in the practice of finance.     

Portfolio selection subject to experts' judgments

Since Markowitz [Markowitz, H. M. (1952). Portfolio selection. The Journal of Finance, 7, 77-91.], mean-variance theory has assumed that risky-asset returns to be random variables. The theory deals with this uncertainty by further assuming that investors hold homogeneous beliefs regarding the probability distribution governing return uncertainty. While the theory deals with return uncertainty, it fails to address measurement imprecision. In his original work, Markowitz recognized the need to combine randomness with heterogeneous expert judgment resulting in such imprecision. The main objective contributions of the paper are (i) to explore the implications of fuzzy return indeterminacy on mean-variance optimal portfolio choice, (ii) to use bid-ask spread as a proxy measure of the indeterminacy or “fuzzy” nature of random returns, and (iii) to introduce a brief, self-contained glimpse of empirical representations to practitioners unfamiliar with the fuzzy modeling field. Exposition, such as this one, is expected to open new collaborations between other branches of fuzzy mathematics and asset-pricing theories.     

What really happens if the positive definiteness requirement on the covariance matrix of returns is relaxed in efficient portfolio selection?

The Markowitz critical line method for mean–variance portfolio construction has remained highly influential today, since its introduction to the finance world six decades ago. The Markowitz algorithm is so versatile and computationally efficient that it can accommodate any number of linear constraints in addition to full allocations of investment funds and disallowance of short sales. For the Markowitz algorithm to work, the covariance matrix of returns, which is positive semi-definite, need not be positive definite. As a positive semi-definite matrix may not be invertible, it is intriguing that the Markowitz algorithm always works, although matrix inversion is required in each step of the iterative procedure involved. By examining some relevant algebraic features in the Markowitz algorithm, this paper is able to identify and explain intuitively the consequences of relaxing the positive definiteness requirement, as well as drawing some implications from the perspective of portfolio diversification. For the examination, the sample covariance matrix is based on insufficient return observations and is thus positive semi-definite but not positive definite. The results of the examination can facilitate a better understanding of the inner workings of the highly sophisticated Markowitz approach by the many investors who use it as a tool to assist portfolio decisions and by the many students who are introduced pedagogically to its special cases.     

¿Han sido el IBEX35 y el IPC definiciones financieramente eficientes del portafolio de mercado?

The present paper questions the financial efficiency of the most used market portfolio proxies in Spain and Mexico (IBEX35 and IPC) in order to determine if these can be considered a proper market portfolio proxy. The paper questions if they can be used as “neutrals”, according to Black & Litterman (1992) proposals in portfolio management. For this purpose, two discrete event simulations that use the Markowtiz-Tobin-Sharpe-Linter model (Markowitz, 1987, p.5) are performed with monthly data of the stock members of these indices in a February 2001 to December 2010 time window. The results are compared by using the Sharpe ratio (Sharpe, 1966) and show that the equilibrium assumptions in the market do not hold, leading to conclude that these market portfolio proxies are inefficient.     

On the robustness of risk-based asset allocations

Since the subprime crisis, portfolios based on risk diversification are of great interest to both academic researchers and market practitioners. They have also been employed by several asset management firms and their performance appears promising. Since they do not rely on estimates of expected returns, they are assumed to be robust. The same argument holds for minimum variance and equally weighted portfolios. In this paper, we consider a Monte Carlo simulation, as well as an empirical global portfolio dataset, to study the effect of estimation errors on the outcomes of two recently proposed asset allocations, the equally weighted risk contribution (ERC) and the principal component analysis (PCA) portfolio. The ERC portfolio is more robust to changes in the input parameters and has a smaller estimation error than the Markowitz approaches, whereas the PCA portfolio is even more unstable than the classical approaches. In the worst-case scenario, neither approach delivers what it promises. However, in every case the resulting return?Crisk relationship is dominated by the Markowitz approaches.     

Dynamic Trading with Predictable Returns and Transaction Costs

We derive a closed‐form optimal dynamic portfolio policy when trading is costly and security returns are predictable by signals with different mean‐reversion speeds. The optimal strategy is characterized by two principles: (1) aim in front of the target, and (2) trade partially toward the current aim. Specifically, the optimal updated portfolio is a linear combination of the existing portfolio and an “aim portfolio,” which is a weighted average of the current Markowitz portfolio (the moving target) and the expected Markowitz portfolios on all future dates (where the target is moving). Intuitively, predictors with slower mean‐reversion (alpha decay) get more weight in the aim portfolio. We implement the optimal strategy for commodity futures and find superior net returns relative to more naive benchmarks.     

Can utility optimization explain the demand for structured investment products?

In this paper, we first show that for classical rational investors with correct beliefs and constant absolute or constant relative risk aversion, the utility gains from structured products over and above a portfolio consisting of the risk-free asset and the market portfolio are typically much smaller than their fees. This result holds irrespectively of whether the investors can continuously trade the risk-free asset and the market portfolio at no costs or whether they can just buy the assets and hold them to maturity of the structured product. However, when considering behavioural utility functions, such as prospect theory, or investors with incorrect beliefs (arising from probability weighting or probability misestimation), the utility gain can be sizable.     

Diversification in financial and crypto markets

This article investigates the conditional value at risk (CVaR) of two portfolio optimiza- tion approaches containing assets from the financial and crypto markets. We first catch the conditional interdependence structure among each variable through the vine-copula-GARCH model before merging it with the Mean-CVaR model. We then optimize each portfolio and find out the optimal allocation while evaluating the precise risk. The results indicate that the D-Vine copula is more suitable for both portfolios and that, when different conditional stock indices information are being taken into consideration, the crypto-market components can act as a weak hedge/safe haven against financial market indices. Furthermore, as CVaR is found to outperform the mean-variance of Markowitz in both portfolios, both risk measures similarly show that when including cryptocurrencies in a portfolio, the S&P 500 shall not be included. Additionally, the inclusion of Ethereum in a portfolio already containing Bitcoin does not boost the return.     

Random matrix models for datasets with fixed time horizons

This paper examines the use of random matrix theory as it has been applied to model large financial datasets, especially for the purpose of estimating the bias inherent in Mean-Variance portfolio allocation when a sample covariance matrix is substituted for the true underlying covariance. Such problems were observed and modeled in the seminal work of Laloux et al. [Noise dressing of financial correlation matrices. Phys. Rev. Lett., 1999, 83, 1467] and rigorously proved by Bai et al. [Enhancement of the applicability of Markowitz's portfolio optimization by utilizing random matrix theory. Math. Finance, 2009, 19, 639–667] under minimal assumptions. If the returns on assets to be held in the portfolio are assumed independent and stationary, then these results are universal in that they do not depend on the precise distribution of returns. This universality has been somewhat misrepresented in the literature, however, as asymptotic results require that an arbitrarily long time horizon be available before such predictions necessarily become accurate. In order to reconcile these models with the highly non-Gaussian returns observed in real financial data, a new ensemble of random rectangular matrices is introduced, modeled on the observations of independent Lévy processes over a fixed time horizon.     

Venture Capital and Other Private Equity: a Survey

We review the theory and evidence on venture capital (VC) and other private equity: why professional private equity exists, what private equity managers do with their portfolio companies, what returns they earn, who earns more and why, what determines the design of contracts signed between (i) private equity managers and their portfolio companies and (ii) private equity managers and their investors (limited partners), and how/whether these contractual designs affect outcomes. Findings highlight the importance of private ownership, and information asymmetry and illiquidity associated with it, as a key explanatory factor of what makes private equity different from other asset classes.     

Optimal Asset Allocation Over the Business Cycle

Utilizing a broadly diversified portfolio of nine equity and debt assets, we show our portfolio's in-sample Markowitz return/risk profile considerably improved by keying asset proportions to cyclical changes in economic activity. For comparative purposes, we use the same assets in a hypothetical buy-and-hold benchmark portfolio. We find the variance/covariance structure of our portfolio to be considerably altered by the phase of the business cycle, with the diversification benefits enjoyed during expansions substantially diluted during recessions. Thus, cyclical reallocation appears to be more important in maintaining Markowitz efficiency during recessions vis-a-vis expansions. In the latter case we find expansion reallocation producing a 3.53% increase in our portfolio's return-to-risk ratio (relative to a buy-and-hold position), while for recessions optimal reallocation leads to a 79.14% increase.     

Some problems connected with the calculation of stop loss premiums for large portfolios

Abstract

Experience will not, as a rule, be sufficient to permit a direct empirical determination of the premium rates of a Stop Loss Cover when the portfolio to be reinsured is a large one, and when the treaty runs on such terms that there is only a small probability of the total amount of claims exceeding a stipulated limit. In such cases we have to fall back upon mathematical models from the theory of probability—especially the collective theory of risk—and upon such assumptions as—insofar as they cannot be empirically grounded —may be considered reasonable with regard to the nature of the problem.     

Islamic and conventional portfolios optimization under investor sentiment states: Bayesian vs Markowitz portfolio analysis

This paper investigates the portfolio optimization under investor’s sentiment states of Hidden Markov model and over a different time horizon during the period 2004–2016. To compare the efficient portfolios of the Islamic and the conventional stock indexes, we have employed two approaches: the Bayesian and Markowitz mean-variance. Our findings reveal that the Bayesian efficient frontier of Islamic and conventional stock portfolios is affected by the investor’s sentiment state and the time horizon. Our findings also indicate that the investor’s sentiment regimes change the Islamic and the conventional optimal diversified portfolios.Moreover, the results show that the potential diversification benefits seem to be more important when using the Bayesian approach than when applying the Markowitz approach. This finding is valid for the bearish, depressed, bullish and calm states in Islamic stock markets. However, the diversification of potential portfolios is significant only for the bullish and the bubble states in the conventional financial markets.The findings of the study provided additional evidence for investors to exploit googling investor sentiment states to evaluate the portfolio performance and make an optimal portfolio allocation.     

A dynamic balance sheet management model for a Canadian chartered bank

This paper presents a linear programming (LP) model based on Markowitz portfolio theory for solving the balance sheet management problem for the domestic assets and liabilities of a Canadian chartered bank. Given the bank's initial position, its economic forecasts, and the constraints under which it operates, the model will determine the current and expected future balance sheet adjustments which will meet the bank's expected profits goal with the minimum possible risk. By parametrically varying the expected profits goal, the model will generate the set of risk-return efficient decisions. Bankers need examine only the set of efficient decisions to choose their optimal solution.     

Mean-variance optimization using forward-looking return estimates

Despite its theoretical appeal, Markowitz mean-variance portfolio optimization is plagued by practical issues. It is especially difficult to obtain reliable estimates of a stock’s expected return. Recent research has therefore focused on minimum volatility portfolio optimization, which implicitly assumes that expected returns for all assets are equal. We argue that investors are better off using the implied cost of capital based on analysts’ earnings forecasts as a forward-looking return estimate. Correcting for predictable analyst forecast errors, we demonstrate that mean-variance optimized portfolios based on these estimates outperform on both an absolute and a risk-adjusted basis the minimum volatility portfolio as well as naive benchmarks, such as the value-weighted and equally-weighted market portfolio. The results continue to hold when extending the sample to international markets, using different methods for estimating the forward-looking return, including transaction costs, and using different optimization constraints.