帕累托索赔分布中风险参数的经验贝叶斯估计

本文建立了贝叶斯模型,讨论了帕累托索赔额分布中参数的估计问题,得到了风险参数的极大似然估计、贝叶斯估计和信度估计,并证明了这些估计的强相合性.在均方误差的意义下比较了这些估计的好坏,并通过数值模拟对均方误差进行了验证,结果表明,贝叶斯估计比其他估计具有较小的均方误差.最后,给出了结构参数的估计并证明了经验贝叶斯估计和经验贝叶斯信度估计的渐近最优性.     

Computing efficient frontiers using estimated parameters

The mean-variance model for portfolio selection requires estimates of many parameters. This paper investigates the effect of errors in parameter estimates on the results of mean-variance analysis. Using a small amount of historical data to estimate parameters exposes the model to estimation errors. However, using a long time horizon to estimate parametes increasers the possibility of nonstationarity in the parameters. This paper investigates the tradeoff between estimation error and stationarity. A simulation study shows that the effects of estimation error can be surprisingly large. The magnitude of the errors increase with the number of securities in the analysis. Due to the error maximization property of mean-variance analysis, estimates of portfolio performance are optimistically biased predictors of actual portfolio performance. It is important for users of mean-variance analysis to recognize and correct for this phenomenon in order to develop more realistic expectations of the future performance of a portfolio. This paper suggests a method for adjusting for the bias. A statistical test is proposed to check for nonstationarity in historical data.     

Optimal Portfolio Diversification: Empirical Bayes versus Classical Approach

This study extends the research on international portfolio diversification with estimation risk. The model suggested by Jorion is used to derive the predictive distribution of future returns. The estimator is then compared with the classical one for both with and without short sales. It is found that the improvement for various optimal portfolios is greater for the case of short sales than the case of no short sales.     

VaR测度下的风险对冲策略研究

本文分别在正态分布和任意分布设定下讨论最小在险价值(VaR)的风险对冲问题。在正态分布设定下,本文深入讨论最小方差对冲比率和最小VaR对冲比率的性质,并得出最小VaR对冲策略下组合收益率的均值和方差大于最小方差策略下组合收益率的均值和方差。在任意分布设定下,本文构建一种新的VaR对冲模型,该模型引入非参数核估计方法对VaR进行估计,然后基于VaR核估计量建立风险对冲问题,实现风险估计与风险对冲同步进行。实证结果非常稳健地表明,不做任何分布假设下的核估计法得到的风险对冲效果优于最小方差对冲策略和正态分布设定下的最小VaR对冲策略。     

Multivariate approximations to portfolio return distribution

This article proposes a three-step procedure to estimate portfolio return distributions under the multivariate Gram–Charlier (MGC) distribution. The method combines quasi maximum likelihood (QML) estimation for conditional means and variances and the method of moments (MM) estimation for the rest of the density parameters, including the correlation coefficients. The procedure involves consistent estimates even under density misspecification and solves the so-called ‘curse of dimensionality’ of multivariate modelling. Furthermore, the use of a MGC distribution represents a flexible and general approximation to the true distribution of portfolio returns and accounts for all its empirical regularities. An application of such procedure is performed for a portfolio composed of three European indices as an illustration. The MM estimation of the MGC (MGC-MM) is compared with the traditional maximum likelihood of both the MGC and multivariate Student’s t (benchmark) densities. A simulation on Value-at-Risk (VaR) performance for an equally weighted portfolio at 1 and 5 % confidence indicates that the MGC-MM method provides reasonable approximations to the true empirical VaR. Therefore, the procedure seems to be a useful tool for risk managers and practitioners.     

A general approach to Bayesian portfolio optimization

We develop a general approach to portfolio optimization taking account of estimation risk and stylized facts of empirical finance. This is done within a Bayesian framework. The approximation of the posterior distribution of the unknown model parameters is based on a parallel tempering algorithm. The portfolio optimization is done using the first two moments of the predictive discrete asset return distribution. For illustration purposes we apply our method to empirical stock market data where daily asset log-returns are assumed to follow an orthogonal MGARCH process with t-distributed perturbations. Our results are compared with other portfolios suggested by popular optimization strategies.     

Bayesian approaches for analyzing earthquake catastrophic risk

Extreme value theory has been widely used in analyzing catastrophic risk. The theory mentioned that the generalized Pareto distribution (GPD) could be used to estimate the limiting distribution of the excess value over a certain threshold; thus the tail behaviors are analyzed. However, the central behavior is important because it may affect the estimation of model parameters in GPD, and the evaluation of catastrophic insurance premiums also depends on the central behavior. This paper proposes four mixture models to model earthquake catastrophic loss and proposes Bayesian approaches to estimate the unknown parameters and the threshold in these mixture models. MCMC methods are used to calculate the Bayesian estimates of model parameters, and deviance information criterion values are obtained for model comparison. The earthquake loss of Yunnan province is analyzed to illustrate the proposed methods. Results show that the estimation of the threshold and the shape and scale of GPD are quite different. Value-at-risk and expected shortfall for the proposed mixture models are calculated under different confidence levels.     

An exact solution to a robust portfolio choice problem with multiple risk measures under ambiguous distribution

This paper proposes a unified framework to solve distributionally robust mean-risk optimization problem that simultaneously uses variance, value-at-risk (VaR) and conditional value-at-risk (CVaR) as a triple-risk measure. It provides investors with more flexibility to find portfolios in the sense that it allows investors to optimize a return-risk profile in the presence of estimation error. We derive a closed-form expression for the optimal portfolio strategy to the robust mean-multiple risk portfolio selection model under distribution and mean return ambiguity (RMP). Specially, the robust mean-variance, robust maximum return, robust minimum VaR and robust minimum CVaR efficient portfolios are all special instances of RMP portfolios. We analytically and numerically show that the resulting portfolio weight converges to the minimum variance portfolio when the level of ambiguity aversion is in a high value. Using numerical experiment with simulated data, we demonstrate that our robust portfolios under ambiguity are more stable over time than the non-robust portfolios.     

Determination of the portfolio selection for a property-liability insurance company

This paper presents an analysis of a portfolio model which can be used to assist a property-liability insurance company in determining the optimal composition of the insurance and investment portfolios. By introducing insurer's threshold risk and relaxing some non-realistic assumptions made in traditional chance constraint insurance and investment portfolio models, we propose a method for an insurer to maximize his return threshold for a given threshold risk level. This proposed model can be used to optimize the composition of underwriting and investment portfolios regarding the insurer's threshold risk level, as well as to generate the efficient frontier by adjusting insurer's threshold risk levels. A numerical example is given based on the industry's aggregated data for a sixteen year period.     

Calculating risk neutral probabilities and optimal portfolio policies in a dynamic investment model with downside risk control

This paper presents a method for solving multiperiod investment models with downside risk control characterized by the portfolio’s worst outcome. The stochastic programming problem is decomposed into two subproblems: a nonlinear optimization model identifying the optimal terminal wealth distribution and a stochastic linear programming model replicating the identified optimal portfolio wealth. The replicating portfolio coincides with the optimal solution to the investor’s problem if the market is frictionless. The multiperiod stochastic linear programming model tests for the absence of arbitrage opportunities and its dual feasible solutions generate all risk neutral probability measures. When there are constraints such as liquidity or position requirements, the method yields approximate portfolio policies by minimizing the initial cost of the replication portfolio. A numerical example illustrates the difference between the replicating result and the optimal unconstrained portfolio.     

Asymptotic behavior of Mean-CVaR portfolio selection model under nonparametric framework

Portfolio selection is an important issue in finance and it involves the balance between risk and return. This paper investigates portfolio selection under Mean-CVa R model in a nonparametric framework with α-mixing data as financial data tends to be dependent. Many works have provided some insight into the performance of portfolio selection from the aspects of data and simulation while in this paper we concentrate on the asymptotic behaviors of the optimal solutions and risk estimation in theory.     

一个组合分布模型及其参数估计

本文根据极值分布理论,提出了一个由原始分布和尾分布组成的组合分布模型,研究了组合分布模型中原始分布和尾分布的确定方法,建立了组合分布模型参数估计的加权最优化模型,实例计算说明,组合分布较好地反映了风险变量极值事件的风险。     

Conditional value-at-risk in portfolio optimization: Coherent but fragile

We evaluate conditional value-at-risk (CVaR) as a risk measure in data-driven portfolio optimization. We show that portfolios obtained by solving mean-CVaR and global minimum CVaR problems are unreliable due to estimation errors of CVaR and/or the mean, which are magnified by optimization. This problem is exacerbated when the tail of the return distribution is made heavier. We conclude that CVaR, a coherent risk measure, is fragile in portfolio optimization due to estimation errors.     

Theoretical and empirical estimates of mean–variance portfolio sensitivity

This paper studies properties of an estimator of mean–variance portfolio weights in a market model with multiple risky assets and a riskless asset. Theoretical formulas for the mean square error are derived in the case when asset excess returns are multivariate normally distributed and serially independent. The sensitivity of the portfolio estimator to errors arising from the estimation of the covariance matrix and the mean vector is quantified. It turns out that the relative contribution of the covariance matrix error depends mainly on the Sharpe ratio of the market portfolio and the sampling frequency of historical data. Theoretical studies are complemented by an investigation of the distribution of portfolio estimator for empirical datasets. An appropriately crafted bootstrapping method is employed to compute the empirical mean square error. Empirical and theoretical estimates are in good agreement, with the empirical values being, in general, higher.     

风险资产市场组合的概率分布和均值估计

探讨CAPM中风险资产市场组合的概率分布和均值估计问题.在股票价格行为模型用维纳过程(又称布朗运动)表述的前提下,证明了CAPM中的市场组合服从加法逻辑正态分布的结论,进而给出了市场组合均值的3种估计.以此为基础进行CAPM的实证检验,才具有理论上的严密性.     

基于半参数多元Copula-GARCH模型的开放式基金投资组合风险分析

利用Copula技术对我国开放式基金市场的投资组合进行了风险分析。为克服传统Copula模型对金融尾部数据刻画能力的不足,建立了半参数的多元Copula-GARCH模型,灵活地对各支基金的边缘分布进行拟合,刻画了开放式基金投资组合的相依结构。并利用基于Copula技术的蒙特卡洛模拟,对投资组合进行了VaR分析,结果证实了所建立模型的可行性和有效性。     

Filter‐based portfolio strategies in an HMM setting with varying correlation parametrizations

We consider portfolio optimization in a regime‐switching market. The assets of the portfolio are modeled through a hidden Markov model (HMM) in discrete time, where drift and volatility of the single assets are allowed to switch between different states. We consider different parametrizations of the involved asset covariances: statewise uncorrelated assets (though linked through the common Markov chain), assets correlated in a state‐independent way, and assets where the correlation varies from state to state. As a benchmark, we also consider a model without regime switches. We utilize a filter‐based expectation‐maximization (EM) algorithm to obtain optimal parameter estimates within this multivariate HMM and present parameter estimators in all three HMM settings. We discuss the impact of these different models on the performance of several portfolio strategies. Our findings show that for simulated returns, our strategies in many settings outperform naïve investment strategies, like the equal weights strategy. Information criteria can be used to detect the best model for estimation as well as for portfolio optimization. A second study using real data confirms these findings.     

Modelling dynamic portfolio risk using risk drivers of elliptical processes

The situation of a limited availability of historical data is frequently encountered in portfolio risk estimation, especially in credit risk estimation. This makes it difficult, for example, to find statistically significant temporal structures in the data on the single asset level. By contrast, there is often a broader availability of cross-sectional data, i.e. a large number of assets in the portfolio. This paper proposes a stochastic dynamic model which takes this situation into account. The modelling framework is based on multivariate elliptical processes which model portfolio risk via sub-portfolio specific volatility indices called portfolio risk drivers. The dynamics of the risk drivers are modelled by multiplicative error models (MEMs)-as introduced by Engle [Engle, R.F., 2002. New frontiers for ARCH models. J. Appl. Econom. 17, 425-446]-or by traditional ARMA models. The model is calibrated to Moody’s KMV Credit Monitor asset returns (also known as firm-value returns) given on a monthly basis for 756 listed European companies at 115 time points from 1996 to 2005. This database is used by financial institutions to assess the credit quality of firms. The proposed risk drivers capture the volatility structure of asset returns in different industry sectors. A characteristic cyclical as well as a seasonal temporal structure of the risk drivers is found across all industry sectors. In addition, each risk driver exhibits idiosyncratic developments. We also identify correlations between the risk drivers and selected macroeconomic variables. These findings may improve the estimation of risk measures such as the (portfolio) Value at Risk. The proposed methods are general and can be applied to any series of multivariate asset or equity returns in finance and insurance.     

A new class of coherent risk measures based on p‐norms and their applications

To exercise better control on the lower tail of the loss distribution and to easily describe the investor's risk attitude, a new class of coherent risk measures is proposed in this paper by taking the minimization of p‐norms of lower losses with respect to some reference point. We demonstrate that the new risk measure has satisfactory mathematical properties such as convexity, continuity with respect to parameters included in its definition, the relations between two new risk measures are also examined. The application of the new risk measures for optimal portfolio selection is illustrated by using trade data from the Chinese stock markets. Empirical results not only support our theoretical conclusions, but also show the practicability of the portfolio selection model with our new risk measures. Copyright © 2006 John Wiley & Sons, Ltd.