Portfolio selection: An extreme value approach

We show theoretically that lower tail dependence (χ), a measure of the probability that a portfolio will suffer large losses given that the market does, contains important information for risk-averse investors. We then estimate χ for a sample of DJIA stocks and show that it differs systematically from other risk measures including variance, semi-variance, skewness, kurtosis, beta, and coskewness. In out-of-sample tests, portfolios constructed to have low values of χ outperform the market index, the mean return of the stocks in our sample, and portfolios with high values of χ. Our results indicate that χ is conceptually important for risk-averse investors, differs substantially from other risk measures, and provides useful information for portfolio selection.     

Asymptotic dynamics and value-at-risk of large diversified portfolios in a jump-diffusion market

This paper studies the modelling of large diversified portfolios in a financial market with jump-diffusion risks. The portfolios considered include three categories: equal money-weighted portfolios, risk-minimizing portfolios and market indices. Reduced-form dynamics driven jointly by one Brownian motion and one Poisson process are derived for the asymptotics of such portfolios. We prove that derivatives written on a portfolio can be priced by treating the asymptotic dynamics as the underlying process if the number of assets in the portfolio is sufficiently large. Analytical and Monte Carlo value-at-risk can be computed for the portfolios based on their asymptotic dynamics.     

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.     

Measuring large comovements in financial markets

A general, copula-based framework for measuring the dependence among financial time series is presented. Particular emphasis is placed on multivariate conditional Spearman's rho (MCS), a new measure of multivariate conditional dependence that describes the association between large or extreme negative returns—so-called tail dependence. We demonstrate that MCS has a number of advantages over conventional measures of tail dependence, both in theory and in practical applications. In the analysis of univariate financial series, data are filtered to remove temporal dependence as a matter of routine. We show that standard filtering procedures may strongly influence the conclusions drawn concerning tail dependence. We give empirical applications to two large data sets of high-frequency asset returns. Our results have immediate implications for portfolio risk management, derivative pricing and portfolio selection. In this context we address portfolio tail diversification and tail hedging. Amongst other aspects, it is shown that the proposed modeling framework improves the estimation of portfolio risk measures such as the value at risk.     

Using Four‐Moment Tail Risk to Examine Financial and Commodity Instrument Diversification

We consider the effect of higher moments on diversification, since most assets possess a potential for tail losses. In particular, we examine higher‐moment Value‐at‐Risk measures for individual instruments and diversified portfolios. We find that a naïve futures portfolio is consistently superior to common stock indexes. As few as ten randomly chosen instruments diversify away 85% of the unsystematic four‐moment tail risk. We also compare the two‐ and four‐moment tail risks for different size portfolios. Finally, the tail risk for naïve portfolios varies much less over time than other portfolios.     

Reduction of estimation risk in optimal portfolio choice using redundant constraints

It is well known that when the moments of the distribution governing returns are estimated from sample data, the out-of-sample performance of the optimal solution of a mean–variance (MV) portfolio problem deteriorates as a consequence of the so-called “estimation risk”. In this document we provide a theoretical analysis of the effects caused by redundant constraints on the out-of-sample performance of optimal MV portfolios. In particular, we show that the out-of-sample performance of the plug-in estimator of the optimal MV portfolio can be improved by adding any set of redundant linear constraints. We also illustrate our findings when risky assets are equally correlated and identically distributed. In this specific case, we report an emerging trade-off between diversification and estimation risk and that the allocation of estimation risk across portfolios forming the optimal solution changes dramatically in terms of number of assets and correlations.     

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.     

Estimating a covariance matrix for market risk management and the case of credit default swaps

We analyze covariance matrix estimation from the perspective of market risk management, where the goal is to obtain accurate estimates of portfolio risk across essentially all portfolios—even those with small standard deviations. We propose a simple but effective visualisation tool to assess bias across a wide range of portfolios. We employ a portfolio perspective to determine covariance matrix loss functions particularly suitable for market risk management. Proper regularisation of the covariance matrix estimate significantly improves performance. These methods are applied to credit default swaps, for which covariance matrices are used to set portfolio margin requirements for central clearing. Among the methods we test, the graphical lasso estimator performs particularly well. The graphical lasso and a hierarchical clustering estimator also yield economically meaningful representations of market structure through a graphical model and a hierarchy, respectively.     

Non-Gaussian diversification: When size matters

Classical portfolio theory informs investors that they should have a large number of assets in their portfolios in order to diversify risk. We show that the non-Gaussian features of stock return distribution may not allow for this risk protection in times of crisis. Moreover, we demonstrate empirically that, if investors are risk-averse and consider higher order moments, they have numerous incentives not to diversify their portfolios fully. This is caused by the evolution of both large losses and asymmetry of returns when the numbers of assets in a portfolio change.     

Would emerging market pension funds benefit from international diversification: investigating wealth accumulations for pension participants

In recent years, investment portfolio selection is growing in importance for many emerging market pension funds, as pension reforms replace traditional pay-as-you-go systems with advanced funding systems. Various investment regulations are applied to the funded pensions, particularly in the form of portfolio limits for equities and international assets. With a bootstrap simulation approach, this paper attempts to quantify the impacts on retirement benefits of restricting international assets from the investment portfolios of emerging market pension funds. We find that, on average, over half of the pension portfolios of emerging market countries should be in international assets in order to maximize the expected utility of moderate and conservative pension fund participants. More generally, international assets can play a significant role in the investment portfolios for workers with risk aversion varying from aggressive to conservative. With few exceptions, the entire probability distribution of wealth accumulations at retirement could be shifted higher with the inclusion of international assets.     

Where is the distribution tail threshold? A tale on tail and copulas in financial risk measurement

Estimating the market risk is conditioned by the fat tail of the distribution of returns. But the tail index depends on the threshold of this distribution fat tail. We propose a methodology based on the decomposition of the series into positive outliers, Gaussian central part and negative outliers and uses the latter to estimate this cutoff point. Additionally, from this decomposition, we estimate extreme dependence correlation matrix which is used in the measurement of portfolio risk. For a sample consisting of six assets (Bitcoin, Gold, Brent, Standard&Poor-500, Nasdaq and Real Estate index), we find that our methodology presents better results, in terms of normality and volatility of the tail index, than the Kolmogorov–Smirnov distance, and its unnecessary capital consumption is lower. Also, in the measurement of the risk of a portfolio, the results of our proposal improve those of a t-Student copula and allow us to estimate the extreme dependence and the corresponding indexes avoiding the implicit restrictions of the elliptic and Archimedean copulas.     

The dynamics of squared returns under contemporaneous aggregation of GARCH models

The paper investigates the properties of a portfolio composed of a large number of assets driven by a strong multivariate GARCH(1,1) process with heterogeneous parameters. The aggregate return is shown to be a weak GARCH process with a (possibly large) number of lags, which reflects the moments of the distribution of the individual persistence parameters. The paper describes a consistent estimator of the aggregate return dynamics, based on nonlinear least squares. The proposed aggregation-corrected estimator (ACE) performs very well and outperforms some competing estimators in forecasting the daily variance of U.S. stocks portfolios at different horizons.     

Climate change financial risks: Implications for asset pricing and interest rates

In addition to tail macroeconomic events (e.g. wars, financial crises and pandemics), climate change poses a threat to financial stability — with extreme climatic events increasing in frequency and intensity and policy risks putting pressure on asset valuations. We study the effect of a changing climate on asset prices and interest rates through the lens of a dynamic CAPM with rare disasters, time-varying risk and recursive preferences. In our model, a changing climate makes tail events more frequent and less predictable, increasing the premium of climate risk; interestingly, this change may not be fully reflected in the overall market risk premium that includes both components of risk: macroeconomic and environmental. Our results also support the hypothesis of a declining real rate of interest as the planet warms, while the increasing risk of climate policy reduces the participation of brown assets in the market portfolio.     

Credit Risk in Private Debt Portfolios

Default, loss severity, and average loss rates for a large sample of privately placed bonds are presented and compared with loss experience for publicly issued bonds. The chance of very large portfolio losses is estimated and some determinants of such losses are analyzed. Results show ex ante riskier classes of private debt perform better on average than public debt. Both diversification and the riskiness of individual portfolio assets influence the bad tail of the portfolio loss distribution. Private placements are similar to corporate loans in that both are monitored private debt. The results are thus relevant to management and securitization of private debt portfolios generally.     

A model for tax advantages of portfolios with many assets

Taxable portfolios present challenges for optimization models with even a limited number of assets. Holding many assets, however, has a distinct tax advantage over holding few assets. In this paper, we develop a model that takes an extreme view of a portfolio as a continuum of assets to gain the broadest possible advantage from holding many assets. We find the optimal strategy for trading in this portfolio in the absence of transaction costs and develop bounding approximations on the optimal value. We compare the results in a simulation study to a portfolio consisting only of a market index and show that the multi-asset portfolio’s tax advantage can lead either to significant consumption or bequest increases.     

Optimal portfolios with minimum capital requirements

We propose a novel approach to active risk management based on the recent Basel II regulations to obtain optimal portfolios with minimum capital requirements. In order to avoid regulatory penalties due to an excessive number of Value-at-Risk (VaR) violations, capital requirements are minimized subject to a given number of violations over the previous trading year. Capital requirements are based on the recent Basel II amendments to account for the ‘stressed’ VaR, that is, the downside risk of the portfolio under extreme adverse market conditions. An empirical application for two portfolios involving different types of assets and alternative stress scenarios demonstrates that the proposed approach delivers an improved balance between capital requirement levels and the number of VaR exceedances. Furthermore, the risk-adjusted performance of the proposed approach is superior to that of minimum-VaR and minimum-stressed VaR portfolios.     

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.     

Least-squares approach to risk parity in portfolio selection

The risk parity portfolio selection problem aims to find such portfolios for which the contributions of risk from all assets are equally weighted. Portfolios constructed using the risk parity approach are a compromise between two well-known diversification techniques: minimum variance optimization and the equal weighting approach. In this paper, we discuss the problem of finding portfolios that satisfy risk parity over either individual assets or groups of assets. We describe the set of all risk parity solutions by using convex optimization techniques over orthants and we show that this set may contain an exponential number of solutions. We then propose an alternative non-convex least-squares model whose set of optimal solutions includes all risk parity solutions, and propose a modified formulation which aims at selecting the most desirable risk parity solution according to a given criterion. When general bounds are considered, a risk parity solution may not exist. In this case, the non-convex least-squares model seeks a feasible portfolio which is as close to risk parity as possible. Furthermore, we propose an alternating linearization framework to solve this non-convex model. Numerical experiments indicate the effectiveness of our technique in terms of both speed and accuracy.     

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.     

基于余弦相似度(CS)和最小生成树(MST)的基金市场复杂网络研究

以中国基金市场中123家基金公司持有的投资组合为样本,综合运用余弦相似度(CS)和最小生成树(MST)方法,考量基金市场复杂网络。结果显示：各家基金公司持有股票组合的相似程度比持有债券组合的相似程度更高,表明他们持有的债券组合较之股票组合更加多元化,基金公司持有的股票相对集中于市值大、成长性高的公司。同时,全部资产投资组合、股票投资组合和债券投资组合等三类基金MST网络的节点度均服从幂律分布,表明大多数基金公司以少数强影响力基金公司为中心聚集起来,彼此之间具有较强的业务关联。此种网络结构特征可能导致市场风险向基金聚集团体集中,其抵御系统性风险的能力偏弱,也不利于满足投资者的理财多元化需求。