Combining equilibrium,resampling, and analyst’s views in portfolio optimization

This paper proposes the use of a portfolio optimization methodology which combines features of equilibrium models and investor’s views as in Black and Litterman (1992), and also deals with estimation risk as in Michaud (1998). In this way, our combined methodology is able to meet the needs of practitioners for stable and diversified portfolio allocations, while it is theoretically grounded on an equilibrium framework. We empirically test the methodology using a comprehensive sample of developed countries fixed income and equity indices, as well as sub-samples stratified by geographical region, time period, asset class and risk level. In general, our proposed combined methodology generates very competitive portfolios when compared to other methodologies, considering three evaluation dimensions: financial efficiency, diversification, and allocation stability. By generating financially efficient, stable, and diversified portfolio allocations, our methodology is suitable for long-term investors such as Central Banks and Sovereign Wealth Funds.     

Optimal commodity asset allocation with a coherent market risk modeling

This paper fills a fundamental gap in commodity price risk management and optimal portfolio selection literatures by contributing a thorough reflection on trading risk modeling with a dynamic asset allocation process and under the supposition of illiquid and adverse market settings. This paper analyzes, from a portfolio managers' perspective, the performance of liquidity adjusted risk modeling in obtaining efficient and coherent investable commodity portfolios under normal and adverse market conditions. As such, the author argues that liquidity risk associated with the uncertainty of liquidating multiple commodity assets over given holding periods is a key factor in formalizing and measuring overall trading risk and is thus an important component to model, particularly in the wake of the repercussions of the recent 2008 financial crisis. To this end, this article proposes a practical technique for the quantification of liquidity trading risk for large portfolios that consist of multiple commodity assets and whereby the holding periods are adjusted according to the specific needs of each trading portfolio. Specifically, the paper proposes a robust technique to commodity optimal portfolio selection, in a liquidity-adjusted value-at-risk (L-VaR) framework, and particularly from the perspective of large portfolios that have both long and short positions or portfolios that consist of merely pure long trading positions. Moreover, in this paper, the author develops a portfolio selection model and an optimization-algorithm which allocates commodity assets by minimizing the L-VaR subject to applying credible operational and financial constraints based on fundamental asset management considerations. The empirical optimization results indicate that this alternate L-VaR technique can be regarded as a robust portfolio management tool and can have many uses and applications in real-world asset management practices and predominantly for fund managers with large commodity portfolios.     

A conditional asset-pricing model with the optimal orthogonal portfolio

This paper employs a conditional asset-pricing model based on the optimal orthogonal portfolio approach to construct a factor portfolio that embodies all the latent factors important for pricing a given set of test assets. The advantage of this portfolio to the anomaly related mimicking portfolios is its ability to separate out the components of average return that are not related to the return covariation. The performance of this portfolio is evaluated against several conventional factors, using both cross-sectional and time-series regression approaches, as well as the Hansen and Jagannathan (1997) distance measure. Its strong out-of-sample results indicate that our suggested methodology may have important applications in risk management, portfolio selection and performance evaluation.     

Diversification of infrastructure projects for emergent and unknown non‐systematic risks

Allocating resources to competing large‐scale infrastructure projects involves multiple objectives. Traditional decision‐aiding methodologies focus on the trade‐offs among performance and resource objectives. Existing methodologies may fail to account for unknown and emergent risks that are typical of large‐scale infrastructure investment allocation problems. In modern portfolio theory, it is well known that a diversified portfolio can be very effective to reduce non‐systematic risks. The approach of diversification is equally important in choosing robust portfolios of infrastructure projects that may be subject to emergent and unknown risks. In this paper, we demonstrate a methodology to analyze and compare the diversification of portfolios of large‐scale infrastructure projects. We classify and explore several metrics of diversification and integrate them with risk and other performance objectives in a multiobjective approach. We test the new metrics and the methodology in a case study of hundreds of millions of dollars of infrastructure investments. The results suggest that the solutions that consider diversification are more robust to emergent risks, thus, identifying an opportunity to incorporate diversification‐based optimization methodologies to support a variety of problems involving large‐scale infrastructure investments.     

Value at risk methodology of international index portfolio under soft conditions (fuzzy-stochastic approach)

The approach to modelling uncertainty of the international index portfolio by the value at risk (VAR) methodology under soft conditions by fuzzy-stochastic methodology is described in the paper. The generalised term uncertainty is understood to have two aspects: risk modelled by probability (stochastic methodology) and vagueness sometimes called impreciseness, ambiguity, softness is modelled by fuzzy methodology. Thus, hybrid model is called fuzzy-stochastic model. Input data for a stochastic model are unique distribution functions and crisp (real) data. Input data for fuzzy model are fuzzy numbers and crisp (real) data. Input data for hybrid model are fuzzy probability distribution functions, unique distribution functions, and crisp (real) data. Softly defined VAR model is constructed as hybrid model because it is supposed that the input data are difficult to determine as crisp numbers or as some unique distribution functions. Risk is modelled by stochastic methodology on the VAR basis and vagueness is modelled through the fuzzy numbers. The analytical delta normal VAR methodology for international index portfolio under soft conditions is described including illustrative example. It is shown, that methodology described could be considered to be generalised sensitivity analysis.     

How to build a factor portfolio: Does the allocation strategy matter?

Factor-based allocation embraces the idea of factors, as opposed to asset classes, as the ultimate building blocks of investment portfolios. We examine whether there is a superior way of combining factors in a portfolio and provide a comparison of factor-based allocation strategies within a multiple testing framework. Factor-based allocation is profitable beyond exploiting genuine risk premia, even when applying multiple testing corrections. Investment portfolios can be efficiently diversified using factor-based allocation strategies, as demonstrated by robust economic performance over various economic scenarios. The naïve equally weighted factor portfolio, albeit simple and cost-efficient, cannot be outperformed by more sophisticated allocation strategies.     

Capital market equilibrium with heterogeneous investors

As a two-parameter model that satisfies stochastic dominance, the mean-extended Gini model is used to build efficient portfolios. The model quantifies risk aversion heterogeneity in capital markets. In a simple Edgeworth box framework, we show how capital market equilibrium is achieved for risky assets. This approach provides a richer basis for analysing the pricing of risky assets under heterogeneous preferences. Our main results are: (1) identical investors, who use the same statistic to represent risk, hold identical portfolios of risky assets equal to the market portfolio; and (2) heterogeneous investors as expressed by the variance or the extended Gini hold different risky assets in portfolios, and therefore no one holds the market portfolio.     

International portfolio selection and efficiency analysis

In the risk-return tradeoff, the traditional mean-variance analysis has been widely used for studies of international portfolio efficiency and diversification. Without prior knowledge about either the parametric structure of assets' return distributions or the form of investors' preference functions, the variance may no longer serve as a suitable risk proxy. This article examines international portfolio efficiency and diversification effects through mean-variance and various distribution-free (or less restrictive) risk-return measures. We show empirically that the mean-variance model is appropriate for large or well-diversified portfolios, but may provide biased results for single assets and less diversified portfolios. While stochastic dominance stands as theoretically the most appropriate method of international portfolio selection and efficiency analysis, the lack of optimal search algorithms reduces its practical usefulness. Very little gain is obtained by using the Gini-mean-difference risk measure as compared to the semivariance measure. The semivariance measure is a powerful and convenient discriminator of risky prospects, while stochastic dominance can serve as a benchmark to justify portfolio efficiency.     

Basket CDS pricing with interacting intensities

We propose a factor contagion model for correlated defaults. The model covers the heterogeneous conditionally independent portfolio and the infectious default portfolio as special cases. The model assumes that the hazard rate processes are driven by external common factors as well as defaults of other names in the portfolio. The total hazard construction method is used to derive the joint distribution of default times. The basket CDS rates can be computed analytically for homogeneous contagion portfolios and recursively for general factor contagion portfolios. We extend the results to include the interacting counterparty risk and the stochastic intensity process. The authors thank two anonymous referees for several suggestions which have helped to improve the earlier versions. The authors thank Sheng Miao for help in implementation with C++, Huiqi Pan for help in implementation with Fortran, and Xiaozhou Cao for help in implementation with MAPLE. Harry Zheng thanks the London Mathematical Society for its collaborative grant support (Grant 4544 and Grant 4707).     

Reward–risk portfolio selection and stochastic dominance

The portfolio selection problem is traditionally modelled by two different approaches. The first one is based on an axiomatic model of risk-averse preferences, where decision makers are assumed to possess a utility function and the portfolio choice consists in maximizing the expected utility over the set of feasible portfolios. The second approach, first proposed by Markowitz is very intuitive and reduces the portfolio choice to a set of two criteria, reward and risk, with possible tradeoff analysis. Usually the reward–risk model is not consistent with the first approach, even when the decision is independent from the specific form of the risk-averse expected utility function, i.e. when one investment dominates another one by second-order stochastic dominance. In this paper we generalize the reward–risk model for portfolio selection. We define reward measures and risk measures by giving a set of properties these measures should satisfy. One of these properties will be the consistency with second-order stochastic dominance, to obtain a link with the expected utility portfolio selection. We characterize reward and risk measures and we discuss the implication for portfolio selection.     

Learning risk preferences from investment portfolios using inverse optimization

The level of risk an investor can endure, known as risk-preference, is a subjective choice that is tightly related to psychology and behavioral science in decision making. This paper presents a novel approach of measuring risk preference from existing portfolios using inverse optimization on mean–variance portfolio allocation framework. Our approach allows the learner to continuously estimate real-time risk preferences using concurrent observed portfolios and market price data. We demonstrate our methods on robotic investment portfolios and real market data that consists of 20 years of asset pricing and 10 years of mutual fund portfolio holdings. Moreover, the quantified risk preference parameters are validated with two well-known risk measurements currently applied in the field. The proposed methods could lead to practical and fruitful innovations in automated/personalized portfolio management, such as Robo-advising, to augment financial advisors’ decision intelligence in a long-term investment horizon.     

Optimal asset allocation for a large number of investment opportunities

This paper introduces a stock‐picking algorithm that can be used to perform an optimal asset allocation for a large number of investment opportunities. The allocation scheme is based upon the idea of causal risk. Instead of referring to the volatility of the assets time series, the stock‐picking algorithm determines the risk exposure of the portfolio by concerning the non‐forecastability of the assets. The underlying expected return forecasts are based on time‐delay recurrent error correction neural networks, which utilize the last model error as an auxiliary input to evaluate their own misspecification. We demonstrate the profitability of our stock‐picking approach by constructing portfolios from 68 different assets of the German stock market. It turns out that our approach is superior to a preset benchmark portfolio. Copyright © 2005 John Wiley & Sons, Ltd.     

Risk parity portfolios with risk factors

Portfolio construction and risk budgeting are the focus of many studies by academics and practitioners. In particular, diversification has spawned much interest and has been defined very differently. In this paper, we analyse a method to achieve portfolio diversification based on the decomposition of the portfolio’s risk into risk factor contributions. First, we expose the relationship between risk factor and asset contributions. Secondly, we formulate the diversification problem in terms of risk factors as an optimization program. Finally, we illustrate our methodology with a real example of building a strategic asset allocation based on economic factors for a pension fund facing liability constraints.     

Constructing inverse factor volatility portfolios: A risk-based asset allocation for factor investing

In this study, we investigate risk-based asset allocation approaches for factor investing strategies by constructing a multifactor portfolio based on the inverse weighting method. We propose the inverse factor volatility (IFV) strategy, which is the simplified variant of a factor risk parity, assuming constant factor correlation. In IFV portfolio construction, the portfolio's weights are determined by using scaled inverse factor volatility treated as a proxy for a targeted exposure in the optimization. Based on daily stock and index returns on global markets from 2002 to the end of 2017, we implemented the empirical analysis of IFV portfolios among three stock markets: Japan, Euro, and the US. The results obtained reveal that the IFV portfolios significantly outperformed market capitalization weighted portfolios by successfully acquiring factor risk premiums.     

Performance analysis of a collateralized fund obligation (CFO) equity tranche

This article examines the performance of the junior tranche of a collateralized fund obligation (CFO), i.e. the residual claim (equity) on a securitized portfolio of hedge funds. We use a polynomial goal programming model to create optimal portfolios of hedge funds, conditional to investor preferences and diversification constraints (maximum allocation per strategy). For each portfolio, we build CFO structures that have different levels of leverage, and analyze both the stand-alone performance as well as potential diversification benefits (low systematic risk exposures) of investing in the equity tranche of these structures. We find that the unconstrained mean-variance portfolio yields a high performance, but greater exposure to systematic risk. We observe the exact opposite picture in the case of unconstrained optimization, where a skewness bias is added, thus proving the existence of a trade-off between stand-alone performance and low exposure to systematic risk factors. We provide evidence that leveraged exposure to these hedge fund portfolios through the structuring of CFOs creates value for the equity tranche investor, even during the recent financial crisis.     

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.     

On measuring nonlinear risk with scarce observations

We propose a methodology for estimating the risk of portfolios that exhibit nonlinear dependence on the risk driving factors and have scarce observations, which is typical for portfolios of investments in hedge funds. The methodology consists of two steps: first, regressing the portfolio return on nonlinear functions of each single risk driving factor and second, merging together the obtained estimates taking into account the dependence between different factors. Performing the second step leads us to a certain probabilistic problem, for which we propose an analytic and computationally feasible solution for the case where the joint law of the factors is a Gaussian copula. A typical practical application can be to estimate the risk of a hedge fund or a portfolio of hedge funds. As a theoretical consequence of our results, we propose a new definition of the factor risk, i.e., the risk of a portfolio brought by a given factor.     

Fair valuation of cliquet-style return guarantees in (homogeneous and) heterogeneous life insurance portfolios

Participating life insurance contracts allow the policyholder to participate in the annual return of a reference portfolio. Additionally, they are often equipped with an annual (cliquet-style) return guarantee. The current low interest rate environment has again refreshed the discussion on risk management and fair valuation of such embedded options. While this problem is typically discussed from the viewpoint of a single contract or a homogeneous* insurance portfolio, contracts are, in practice, managed within a heterogeneous insurance portfolio. Their valuation must then – unlike the case of asset portfolios – take account of portfolio effects: Their premiums are invested in the same reference portfolio; the contracts interact by a joint reserve, individual surrender options and joint default risk of the policy sponsor. Here, we discuss the impact of portfolio effects on the fair valuation of insurance contracts jointly managed in (homogeneous and) heterogeneous life insurance portfolios. First, in a rather general setting, including stochastic interest rates, we consider the case that otherwise homogeneous contracts interact due to the default risk of the policy sponsor. Second, and more importantly, we then also consider the case when policies are allowed to differ in further aspects like the guaranteed rate or time to maturity. We also provide an extensive numerical example for further analysis.     

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.     

Multi-asset portfolio optimization and out-of-sample performance: an evaluation of Black–Litterman,mean-variance,and naïve diversification approaches

The Black–Litterman model aims to enhance asset allocation decisions by overcoming the problems of mean-variance portfolio optimization. We propose a sample-based version of the Black–Litterman model and implement it on a multi-asset portfolio consisting of global stocks, bonds, and commodity indices, covering the period from January 1993 to December 2011. We test its out-of-sample performance relative to other asset allocation models and find that Black–Litterman optimized portfolios significantly outperform naïve-diversified portfolios (1/N rule and strategic weights), and consistently perform better than mean-variance, Bayes–Stein, and minimum-variance strategies in terms of out-of-sample Sharpe ratios, even after controlling for different levels of risk aversion, investment constraints, and transaction costs. The BL model generates portfolios with lower risk, less extreme asset allocations, and higher diversification across asset classes. Sensitivity analyses indicate that these advantages are due to more stable mixed return estimates that incorporate the reliability of return predictions, smaller estimation errors, and lower turnover.