Actuarial Risk Matrices: The Nearest Positive Semidefinite Matrix Problem

The manner in which a group of insurance risks are interrelated is commonly presented via a correlation matrix. Actuarial risk correlation matrices are often constructed using output from disparate modeling sources and can be subjectively adjusted, for example, increasing the estimated correlation between two risk sources to confer reserving prudence. Hence, while individual elements still obey the assumptions of correlation values, the overall matrix is often not mathematically valid (not positive semidefinite). This can prove problematic in using the matrix in statistical models. The first objective of this article is to review existing techniques that address the nearest positive semidefinite matrix problem in a very general setting. The chief approaches studied are Semidefinite Programming (SDP) and the Alternating Projections Method (APM). The second objective is to finesse the original problem specification to consider imposition of a block structure on the initial risk correlation matrix. This commonly employed technique identifies off-diagonal subsets of the matrix where values can or should be set equal to some constant. This may be due to similarity of the underlying risks and/or with the goal of increasing computational efficiency for processes involving large matrices. Implementation of further linear constraints of this nature requires adaptation of the standard SDP and APM algorithms. In addition, a new Shrinking Method is proposed to provide an alternative solution in the context of this increased complexity. “Nearness” is primarily considered in terms of two summary measures for differences between matrices: the Chebychev Norm (maximum element distance) and the Frobenius Norm (sum of squared element distances). Among the existing methods, adapted to function appropriately for actuarial risk matrices, APM is extremely efficient in producing solutions that are optimal in the Frobenius norm. An efficient algorithm that would return a positive semidefinite matrix that is optimal in Chebychev norm is currently unknown. However, APM is used to highlight the existence of matrices close to such an optimum and exploited, via the Shrinking Method, to find high-quality solutions. All methods are shown to work well both on artificial and real actuarial risk matrices provided under collaboration with Tokio Marine Kiln (TMK). Convergence speeds are calculated and compared and sample data and MATLAB code is provided. Ultimately the APM is identified as being superior in Frobenius distance and convergence speed. The Shrinking Method, building on the output of the APM algorithm, is demonstrated to provide excellent results at low computational cost for minimizing Chebychev distance.     

Robust estimation of historical volatility and correlations in risk management

Financial time series have two features which, in many cases, prevent the use of conventional estimators of volatilities and correlations: leptokurtotic distributions and contamination of data with outliers. Other techniques are required to achieve stable and accurate results. In this paper, we review robust estimators for volatilities and correlations and identify those best suited for use in risk management. The selection criteria were that the estimator should be stable to both fractionally small departures for all data points (fat tails), and to fractionally large departures for a small number of data points (outliers). Since risk management typically deals with thousands of time series at once, another major requirement was the independence of the approach of any manual correction or data pre-processing. We recommend using volatility t-estimators, for which we derived the estimation error formula for the case when the exact shape of the data distribution is unknown. A convenient robust estimator for correlations is Kendall's tau, whose drawback is that it does not guarantee the positivity of the correlation matrix. We chose to use geometric optimization that overcomes this problem by finding the closest correlation matrix to a given matrix in terms of the Hadamard norm. We propose the weights for the norm and demonstrate the efficiency of the algorithm on large-scale problems.     

Statistical rehabilitation of improper correlation matrices

The simplest way to describe the dependence for a set of financial assets is their correlation matrix. This correlation matrix can be improper when it is specified element-wise. We describe a new method for obtaining a positive definite correlation matrix starting from an improper one. The expert's opinion and trust in each pairwise correlation is described by a beta distribution. Then, by combining these individual distributions, a joint distribution over the space of positive definite correlation matrices is obtained using Cholesky factorization, and its mode constitutes the new proper correlation matrix. The optimization is complemented by a visual representation of the entries that were most affected by the legalization procedure. We also sketch a Bayesian approach to the same problem.     

Markowitz投资组合模型的最小二乘算法

本文从一个新的视角来研究Markowitz均值—方差模型。通过将Markowitz均值—方差模型表述为约束最小二乘问题,继而使用约束最小二乘问题的算法研究了协方差矩阵正定和半正定时模型的求解问题,我们给出了计算投资组合有效前沿及最小方差组合的新算法。实证分析表明：最小二乘算法在计算稳定和计算速度方面优于传统算法。     

Portfolio diversification and value at risk under thick-tailedness†

This paper focuses on the study of portfolio diversification and value at risk analysis under heavy-tailedness. We use a notion of diversification based on majorization theory that will be explained in the text. The paper shows that the stylized fact that portfolio diversification is preferable is reversed for extremely heavy-tailed risks or returns. However, the stylized facts on diversification are robust to heavy-tailedness of risks or returns as long as their distributions are moderately heavy-tailed. Extensions of the results to the case of dependence, including convolutions of α-symmetric distributions and models with common shocks are provided.     

An extended likelihood framework for modelling discretely observed credit rating transitions

The estimation of the parameters of a continuous-time Markov chain from discrete-time observations, also known as the embedding problem for Markov chains, plays in particular an important role for the modeling of credit rating transitions. This missing data problem boils down to a latent variable setting and thus, maximum likelihood estimation is usually conducted using the expectation-maximization (EM) algorithm. We illustrate that the EM algorithm is likely to get stuck in local maxima of the likelihood function in this specific problem setting and adapt a stochastic approximation simulated annealing scheme (SASEM) as well as a genetic algorithm (GA) to combat this issue. Above that, our main contribution is to extend our method GA by a rejection sampling scheme, which allows one to derive stochastic monotone maximum likelihood estimates in order to obtain proper (non-crossing) multi-year probabilities of default. We advocate the use of this procedure as direct constrained optimization (of the likelihood function) will not be numerically stable due to the large number of side conditions. Furthermore, the monotonicity constraint enables one to combine structural knowledge of the ordinality of credit ratings with real-life data into a statistical estimator, which has a stabilizing effect on far off-diagonal generator matrix elements. We illustrate our methods by Standard and Poor’s credit rating data as well as a simulation study and benchmark our novel procedure against an already existing smoothing algorithm.     

Correlation expansions for CDO pricing

This paper develops numerical approximations for pricing collateralized debt obligations (CDOs) and other portfolio credit derivatives in the multifactor Normal Copula model. A key aspect of pricing portfolio credit derivatives is capturing dependence between the defaults of the elements of the portfolio. But, compared with an independent-obligor model, pricing in a model with correlated defaults is more challenging. Our approach strikes a balance by reducing the problem of pricing in a model with correlated defaults to calculations involving only independent defaults. We develop approximations based on power series expansions in a parameter that scales the underlying correlations. These expansions express a CDO tranche price in a multifactor model as a series of prices in independent-obligor models, which are easy to compute. The approach builds on a classical approximation for multivariate Gaussian probabilities; we introduce an alternative representation that greatly reduces the number of terms required to evaluate the coefficients in the expansion. We also apply this method to the underlying problem of computing joint probabilities of multivariate normal random variables for which the correlation matrix has a factor structure.     

基于雅可比矩阵下一种新的证券组合策略

在当前的金融环境下,不同产业之间交互渗透,各公司相互持股,仅依靠单一证券很难将风险分散出去,本文将目光放于组合证券,运用雅可比矩阵判断其相关性,从中选出相关性为0的组合证券,并对它们进行组合,从而最大程度的降低投资风险。     

The cost of capital- the practitioners view

The application of a SWARCH model to stock market returns allows one to endogenously determine the regime dependence of the stock market volatility. Comparison of the results from a sample of daily data from five major stock markets shows that the majority of the markets switch regimes simultaneously. This fact is used to investigate the relation between market volatility and the behaviour of the variance—;covariance matrix. It is found that the international variance—;covariance matrix is not stable and that changes in the matrix are dependent on the volatility regime. A high level of variance causes an increase in the average correlation coefficient. The co-movement of the markets is further described by a steady increase in the covariance over the whole sample period. It can be shown that both the time component and the regime dependence of the average correlation have separate and significant explanatory power.     

Estimation error in the average correlation of security returns and shrinkage estimation of covariance and correlation matrices

The correlation matrix of security returns is an important input component for mean–variance portfolio analysis. This study uses the average of sample correlations to estimate the correlation matrix and derives an expression of its estimation error in terms of sampling variance. This study then considers the impact of such estimation error on shrinkage estimation, where a weighted average is sought between the sample covariance matrix and an average correlation target, and between the sample correlation matrix and the target. An illustrative example using monthly returns of the current Dow Jones stocks is provided.     

Life Insurance Policies with Statistical Heterogeneous Population

In the paper we consider an endowment insurance contract with a twelve months maturation time. Using the majorization order and Schur-convex functions we derive upper and lower bounds of the premium, the death and survival benefits for a hetrogeneous population of insureds. The bounds are obtained for the exponential, Balducci, and linear approximations.     

Pricing exotic options in a regime switching economy: a Fourier transform method

This article considers the valuation of digital, barrier, and lookback options in a Markovian, regime-switching, Black–Scholes model. In Fourier space, integral representations for the option prices are derived via the theory on matrix Wiener–Hopf factorizations. Our main focus is on the 2-state case where the matrix Wiener–Hopf factorization is available analytically. A comparison to several numerical alternatives (analytical approximations, the Brownian bridge algorithm and a finite element scheme) demonstrates that the pricing formulas are easy to implement and lead to accurate price estimates.     

A Simple Algorithm for the Portfolio Selection Problem

The author presents a rapidly convergent algorithm to solve the general portfolio problem of maximizing concave utility functions subject to linear constraints. The algorithm is based on an iterative use of the Markowitz critical line method for solving quadratic programs. A simple example, taken from the theory of state-contingent claims, is worked out in detail. For technical convergence results, the reader is referred to the appropriate mathematical programming literature.     

Robust and consistent estimation of generators in credit risk

Bond rating Transition Probability Matrices (TPMs) are built over a one-year time-frame and for many practical purposes, like the assessment of risk in portfolios or the computation of banking Capital Requirements (e.g. the new IFRS 9 regulation), one needs to compute the TPM and probabilities of default over a smaller time interval. In the context of continuous time Markov chains (CTMC) several deterministic and statistical algorithms have been proposed to estimate the generator matrix. We focus on the Expectation-Maximization (EM) algorithm by Bladt and Sorensen. [J. R. Stat. Soc. Ser. B (Stat. Method.), 2005, 67, 395–410] for a CTMC with an absorbing state for such estimation. This work’s contribution is threefold. Firstly, we provide directly computable closed form expressions for quantities appearing in the EM algorithm and associated information matrix, allowing to easy approximation of confidence intervals. Previously, these quantities had to be estimated numerically and considerable computational speedups have been gained. Secondly, we prove convergence to a single set of parameters under very weak conditions (for the TPM problem). Finally, we provide a numerical benchmark of our results against other known algorithms, in particular, on several problems related to credit risk. The EM algorithm we propose, padded with the new formulas (and error criteria), outperforms other known algorithms in several metrics, in particular, with much less overestimation of probabilities of default in higher ratings than other statistical algorithms.     

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.     

Random matrix application to correlations amongst the volatility of assets

In this paper, we apply tools from random matrix theory (RMT) to estimates of correlations across the volatility of various assets in the S&P 500. The volatility inputs are estimated by modelling price fluctuations as a GARCH(1,1) process. The corresponding volatility correlation matrix is then constructed. It is found that the distribution of a significant number of eigenvalues of the volatility correlation matrix matches with the analytical result from RMT. Furthermore, the empirical estimates of short- and long-range correlations amongst eigenvalues, which are within RMT bounds, match with the analytical results for the Gaussian Orthogonal ensemble of RMT. To understand the information content of the largest eigenvectors, we estimate the contribution of the Global Industry Classification Standard industry groups to each eigenvector. In comparison with eigenvectors of correlation matrix for price fluctuations, only few of the largest eigenvectors of the volatility correlation matrix are dominated by a single industry group. We also study correlations between ‘volatility returns’ and log-volatility to find similar results.     

财政教育支出的农村减贫效应研究——基于网络空间结构的分析

运用修正的引力模型测算出我国财政教育支出的空间关联关系,以及财政教育支出对农村减贫的空间关联影响效应。实证结果表明：我国财政教育支出在空间关联上呈现出复杂、多线程的网络结构;区域间财政教育支出差异和人力资本差异与农村贫困联动矩阵呈现显著负相关,区域间技术发展水平差异和经济发展水平差异与农村贫困联动矩阵呈现显著正相关。加强我国农村贫困的治理,需要考虑财政教育支出的空间关联作用,实施定向的、梯度推进的区域协同策略。     

金融服务乡村振兴的障碍与破解机制

2019年底突发新冠肺炎疫情。在持续性疫情的影响下，我国各产业都受到了不同程度的影响。为分析各产业经济受疫情影响程度，本文构建基于协方差矩阵的精准选择模型，分析高频次特性股票数据与疫情发展数据的相关性，该模型解决经济变量存在内在关联的问题以及数据不稳定和不一致的特性带来的统计误差。根据模型检验得出服务业、房地产业和农业受疫情影响相对较大的结论，在此基础上本文进一步剖析相关产业所受疫情影响的深层次原因，并从宏观和中观维度提出有效应对的政策建议。     

Analysis of Dynamic Correlation of Japanese Stock Returns with Network Clustering

In this paper, the dynamic correlation of Japanese stock returns is estimated by using the dynamic conditional correlation (DCC–GARCH) model to study their correlation dynamics empirically. It is difficult to fit the model to the whole stock market jointly at the same time; therefore, a network-based clustering is applied for the dimensionality reduction of the sample data. Two types correlation structures are estimated: homogeneous groups of stocks in a balanced size are created by clustering to observe within-group correlation, while a single portfolio that comprises group portfolio returns is also created to observe between-group correlation. The estimation result reveals dynamic changes in correlation intensity represented by the largest eigenvalue of the estimated correlation matrix. A higher level of correlation intensity and volatility are observed during the crisis periods, namely after both the Lehman collapse and the Great East Japan Earthquake, for the between- and within-group correlations. It is also confirmed that the pattern of correlation change is significantly different between the groups. The proposed method is useful for monitoring dynamic correlation of asset returns efficiently in a large scale of portfolio.     

Continuous Maturity Diversification of Default-Free Bond Portfolios and a Generalization of Efficient Diversification

This paper presents a method for solving the mean-variance portfolio selection problem that is applicable to the case where the number of securities is nondenumerably infinite. Necessary conditions for the existence of an optimal portfolio density are obtained and an expression for the efficient frontier is derived. The conditions for the existence of an optimal portfolio of continuously maturing bonds when their covariance matrix is singular are used to derive an arbitrage-free bond pricing equation. A method for estimating the covariance matrix and the associated efficient frontier is presented.