“Generalized Pareto Fit to the Society of Actuaries’ Large Claims Database,” Ana C. Cebrián,Michel Denuit,and Philippe Lambert,July 2003

Abstract

We consider the issue of modeling the latent or hidden exposure occurring through either incomplete data or an unobserved underlying risk factor. We use the celebrated expectationmaximization (EM) algorithm as a convenient tool in detecting latent (unobserved) risks in finite mixture models of claim severity and in problems where data imputation is needed. We provide examples of applicability of the methodology based on real-life auto injury claim data and compare, when possible, the accuracy of our methods with that of standard techniques. Sample data and an EM algorithm program are included to allow readers to experiment with the EM methodology themselves.     

Credit cycles and macro fundamentals

We use an intensity-based framework to study the relation between macroeconomic fundamentals and cycles in defaults and rating activity. Using Standard and Poor's U.S. corporate rating transition and default data over the period 1980–2005, we directly estimate the default and rating cycle from micro data. We relate this cycle to the business cycle, bank lending conditions, and financial market variables. In line with earlier studies, the macro variables appear to explain part of the default cycle. However, we strongly reject the correct dynamic specification of these models. The problem is solved by adding an unobserved dynamic component to the model, which can be interpreted as an omitted systematic credit risk factor. By accounting for this latent factor, many of the observed macro variables loose their significance. There are a few exceptions, but the economic impact of the observed macro variables for credit risk remains low. We also show that systematic credit risk factors differ over transition types, with risk factors for downgrades being noticeably different from those for upgrades. We conclude that portfolio credit risk models based only on observable systematic risk factors omit one of the strongest determinants of credit risk at the portfolio level. This has obvious consequences for current modeling and risk management practices.     

On the Probability of (Non-) Ruin in Infinite Time

In the context of the classical Poisson ruin model Gerber (1988a,b) and Shiu (1987, 1989) have obtained two formulae for the ruin and non ruin probabilities in infinite time. Here these two formulae are generalized to the case of an arbitrary premium process when all claims are integer-valued, as in Picard & Lefèvre (1997). Moreover, this generalization throws a new light on the two known formulae and it then leads very simply to a third new formula.     

A multivariate Markov chain stock model

ABSTRACT

We propose a dividend stock valuation model where multiple dividend growth series and their dependencies are modelled using a multivariate Markov chain. Our model advances existing Markov chain stock models. First, we determine assumptions that guarantee the finiteness of the price and risk as well as the fulfilment of transversality conditions. Then, we compute the first- and second-order price-dividend ratios by solving corresponding linear systems of equations and show that a different price-dividend ratio is attached to each combination of states of the dividend growth process of each stock. Subsequently, we provide a formula for the computation of the variances and covariances between stocks in a portfolio. Finally, we apply the theoretical model to the dividend series of three US stocks and perform comparisons with existing models. The results could also be applied for actuarial purposes as a general stochastic investment model and for calculating the initial endowment to fund a portfolio of dependent perpetuities.     

Law of large numbers and large deviations for dependent risks

We analyse the mathematical structure of models for large risk portfolios, especially for credit risk models. These risk portfolios are modelled using a multivariate mixture model for the dependence structure between the risks. The dependence structures are characterized by latent variables Θ, which play the role of systematic risks. We show that, depending on the choice of the distribution of Θ, there are different asymptotic behaviours for the aggregated risk portfolio, namely law of large numbers/central limit theorem behaviour and large deviation behaviour.     

Equity and Credibility

We build on previous work concerned with measuring equity and consider the problem of using observed claim data or other information to calculate premiums which maximize equity. When these optimal premiums are used, we show that gathering more information or refining the risk classification always increases equity. We study the case for which the premium is constrained to be an affine function of the claim data and obtain results analogous to classical credibility theory, including the inhomogeneous and homogeneous cases of the Bu¨hlmann-Straub model. We derive formulas for the credibility weights in certain cases.     

The Moments of the Time of Ruin in Markovian Risk Models

Abstract

We present an approach based on matrix-analytic methods to find moments of the time of ruin in Markovian risk models. The approach is applicable when claims occur according to a Markovian arrival process (MAP) and claim sizes are phase distributed with parameters that depend on the state of the MAP. The method involves the construction of a sample-path-equivalent Markov-modulated fluid flow for the risk model. We develop an algorithm for moments of the time of ruin and prove the algorithm is convergent. Examples show that the proposed approach is computationally stable.     

Bivariate Survival Models for Coupled Lives

This article presents alternative models for modeling the dependence of the time of deaths of coupled lives and applies these to a data set from a life annuity portfolio. The data indicates a very high positive correlation in the time of deaths between coupled lives. The analysis concludes that a mixed frailty copula model with Gompertz marginals fits the data very well. Another model that fits the data well is based on a generalized Frank copula.     

我国交强险索赔频率影响因素分析 ——基于GAM和广东、河南、湖北、山东的经验数据

为分析我国交强险各因素对索赔频率的影响,以2016年广东、河南、湖北、山东四省的保单数据为样本,采用广义可加模型（GAM）对其保单中的驾驶员年龄、汽车车龄和汽车重量进行非参数分析,并对公路里程数等变量进行参数分析。结果表明：索赔频率有明显的地区差异,公路里程数对索赔频率有正向的影响,其中除汽车车型对索赔频率没有影响外,其余变量均有显著影响。     

Unobserved systematic risk factor and default prediction

We conduct a thorough analysis on the role played by the unobserved systematic risk factor in default prediction. We find that this latent factor outweighs the observed systematic risk factors and can substantially improve the in-sample predictive accuracy at the firm, rating group, and aggregate levels. Thus it might be helpful to include the unobserved systematic risk factor when simulating portfolio credit losses. However, we also find that this factor only marginally improves out-of-sample model performance. Therefore, although the models we investigated all show reasonably good ability to rank order firms by default risk, accurate prediction of default rate remains challenging even when the unobserved systematic risk factor is considered.     

Risk parity portfolio optimization under a Markov regime-switching framework

We formulate and solve a risk parity optimization problem under a Markov regime-switching framework to improve parameter estimation and to systematically mitigate the sensitivity of optimal portfolios to estimation error. A regime-switching factor model of returns is introduced to account for the abrupt changes in the behaviour of economic time series associated with financial cycles. This model incorporates market dynamics in an effort to improve parameter estimation. We proceed to use this model for risk parity optimization and also consider the construction of a robust version of the risk parity optimization by introducing uncertainty structures to the estimated market parameters. We test our model by constructing a regime-switching risk parity portfolio based on the Fama–French three-factor model. The out-of-sample computational results show that a regime-switching risk parity portfolio can consistently outperform its nominal counterpart, maintaining a similar ex post level of risk while delivering higher-than-nominal returns over a long-term investment horizon. Moreover, we present a dynamic portfolio rebalancing policy that further magnifies the benefits of a regime-switching portfolio.     

Comparative issues between linear and non-linear risk measures for non-convex portfolio optimization: evidence from the S&P 500

Abstract

This article focuses on inferring critical comparative conclusions as far as the application of both linear and non-linear risk measures in non-convex portfolio optimization problems. We seek to co-assess a set of sophisticated real-world non-convex investment policy limitations, such as cardinality constraints, buy-in thresholds, transaction costs, particular normative rules, etc. within the frame of four popular portfolio selection cases: (a) the mean-variance model, (b) the mean-semi variance model, (c) the mean-MAD (mean-absolute deviation) model and (d) the mean-semi MAD model. In such circumstances, the portfolio selection process reflects to a mixed-integer bi-objective (or in general multiobjective) mathematical programme. We precisely develop all corresponding modelling procedures and then solve the underlying problem by use of a novel generalized algorithm, which was exclusively introduced to cope with the above-mentioned singularities. The validity of the attempt is verified through empirical testing on the S&P 500 universe of securities. The technical conclusions obtained not only confirm certain findings of the particular limited existing theory but also shed light on computational issues and running times. Moreover, the results derived are characterized as encouraging enough, since a sufficient number of efficient or Pareto optimal portfolios produced by the models appear to possess superior out-of-sample returns with respect to the benchmark.     

Longitudinal modeling of insurance claim counts using jitters

Modeling insurance claim counts is a critical component in the ratemaking process for property and casualty insurance. This article explores the usefulness of copulas to model the number of insurance claims for an individual policyholder within a longitudinal context. To address the limitations of copulas commonly attributed to multivariate discrete data, we adopt a ‘jittering’ method to the claim counts which has the effect of continuitizing the data. Elliptical copulas are proposed to accommodate the intertemporal nature of the ‘jittered’ claim counts and the unobservable subject-specific heterogeneity on the frequency of claims. Observable subject-specific effects are accounted in the model by using available covariate information through a regression model. The predictive distribution together with the corresponding credibility of claim frequency can be derived from the model for ratemaking and risk classification purposes. For empirical illustration, we analyze an unbalanced longitudinal dataset of claim counts observed from a portfolio of automobile insurance policies of a general insurer in Singapore. We further establish the validity of the calibrated copula model, and demonstrate that the copula with ‘jittering’ method outperforms standard count regression models.     

On the price of risk in a mean-risk optimization model

We investigate a mean-risk model for portfolio optimization where the risk quantifier is selected as a semi-deviation or as a standard deviation of the portfolio return. We analyse the existence of solutions to the problem under general assumptions. When the short positions are not constrained, we establish a lower bound on the cost of risk associated with optimizing the mean–standard deviation model and show that optimal solutions do not exist for any positive price of risk which is smaller than that bound. If the investment allocations are constrained, then we obtain a lower bound on the price of risk in terms of the shadow prices of said constraints and the data of the problem. A Value-at-Risk constraint in the model implies an upper bound on the price of risk for all feasible portfolios. Furthermore, we provide conditions under which using this upper bound as the cost of risk parameter in the model provides a non-dominated optimal portfolio with respect to the second-order stochastic dominance. Additionally, we study the relationship between minimizing the mean–standard deviation objective and maximizing the coefficient of variation and show that both problems are equivalent when the upper bound is used as the cost of risk. Additional relations between the Value-at-Risk constraint and the coefficient of variation are discussed as well. We illustrate the results numerically.     

Timing and diversification: A state-dependent asset allocation approach

Abstract

The influence of changing economic environment leads the distribution of stock market returns to be time-varying. A conditionally optimal investment hence requires a dynamic adjustment of asset allocation. In this context, this paper examines the improvement in portfolio performance by simulating portfolio strategies that are conditioned on the Markov regime switching behaviour of stock market returns. Including a memory effect eliminates the empirical shortcoming of discrete state models, namely that they produce a standard and an extreme state in stock returns. So far, this has prevented the regimes from being used as a valuable conditioning variable. Based on a discrete state indicator variable, is presented evidence of considerable performance improvement relative to the static model due to optimal shifting between aggressive and well diversified portfolio structures.     

Ruin Probabilities And Capital Requirement for Open Automobile Portfolios With a Bonus-Malus System Based on Claim Counts

For a large motor insurance portfolio, on an open environment, we study the impact of experience rating in finite and continuous time ruin probabilities. We consider a model for calculating ruin probabilities applicable to large portfolios with a Markovian Bonus-Malus System (BMS), based on claim counts, for an automobile portfolio using the classical risk framework model. New challenges are brought when an open portfolio scenario is introduced. When compared with a classical BMS approach ruin probabilities may change significantly. By using a BMS of a Portuguese insurer, we illustrate and discuss the impact of the proposed formulation on the initial surplus required to target a given ruin probability. Under an open portfolio setup, we show that we may have a significant impact on capital requirements when compared with the classical BMS, by having a significant reduction on the initial surplus needed to maintain a fixed level of the ruin probability.     

Bayesian inference for generalized linear mixed models of portfolio credit risk

The aims of this paper are threefold. First, we highlight the usefulness of generalized linear mixed models (GLMMs) in the modelling of portfolio credit default risk. The GLMM-setting allows for a flexible specification of the systematic portfolio risk in terms of observed fixed effects and unobserved random effects, in order to explain the phenomena of default dependence and time-inhomogeneity in historical default data. Second, we show that computational Bayesian techniques such as the Gibbs sampler can be successfully applied to fit models with serially correlated random effects, which are special instances of state space models. Third, we provide an empirical study using Standard and Poor's data on U.S. firms. A model incorporating rating category and sector effects, and a macroeconomic proxy variable for state-of-the-economy suggests the presence of a residual, cyclical, latent component in the systematic risk.     

Systematic Risk and Empirical Research

We show here that risky asset returns generating processes stated in terms of factors which include both accounting and non-accounting based measures of risk (e.g. book to market ratios) imply, under fairly standard regularity conditions, that the Sharpe-Lintner-Black asset pricing model beta is a 'sufficient' statistic in the sense that it captures all important attributes of the returns generating process in a single number. We then derive the parametric relationship between betas based on inefficient index portfolios and betas based on the market or tangency portfolio. We demonstrate that the relationship between risky asset expected returns and betas computed on the basis of inefficient index portfolios is both consistent with the predictions of the Capital Asset Pricing Model and the multi-factor asset pricing models of Fama and French (1992, 1993, 1995 and 1996). The 'trick' is to realise that inefficient index portfolios are composed of the market portfolio and a collection of inefficient but self financing 'kernel' or 'arbitrage' portfolios. It then follows that there is a perfect linear cross sectional relationship between risky asset expected returns, betas based on inefficient index portfolios and the arbitrage portfolios. Hence, if we happen to stumble across variables that span the same subspace as the vectors representing the arbitrage portfolios, it is easy to create the illusion that risky asset expected returns depend on variables other than 'beta'.     

Life-cycle portfolio choice with liquid and illiquid financial assets

Traditionally, quantitative models that have studied households׳ portfolio choices have focused exclusively on the different risk properties of alternative financial assets. We introduce differences in liquidity across assets in the standard life-cycle model of portfolio choice. More precisely, in our model, stocks are subject to transaction costs, as considered in recent macroliterature. We show that when these costs are calibrated to match the observed infrequency of households׳ trading, the model is able to generate patterns of portfolio stock allocation over age and wealth that are constant or moderately increasing, thus more in line with the existing empirical evidence.     

Financial and Demographic Risks of a Portfolio of Life Insurance Policies with Stochastic Interest Rates

Abstract

We refer to a recent paper by G. Parker (1997) in which the risk of a portfolio of life insurance policies (namely the risk related to the entire contractual life) is studied by separating the demographic component from the financial component. In our paper, after making a brief summary of Parker’s model, we propose two additional contributions: 1. We first give the problem a different formalization, thus allowing a portfolio risk analysis by management periods and a study of the risk due to the interactions among years;

2. We elaborate on a powerful and flexible algorithm for calculating the probability distribution of the sum of random variables that proves useful to solve not only the problems discussed in this paper concerning the risk analysis but also various other problems.

In the paper, we also show, for both contributions, some applications made under the same financial and demographic assumptions taken by Parker; we also compare our results with Parker’s results.