基于机器学习算法的个体索赔准备金评估模型

非寿险赔款准备金对保险公司的风险管理和财务决策具有重要影响。传统的准备金评估方法通常基于汇总的流量三角形数据进行建模,没有充分利用个体索赔案件的信息,且存在参数过度化、难以处理大额赔款和负增量赔款等问题。本文基于每份保单的个体索赔信息,使用随机森林和XGBoost等机器学习算法对案件的赔付状态、赔付金额分别建立了预测模型,改进了传统准备金评估模型的预测效果。实证研究结果表明,影响赔付状态的因素主要是结案状态、报案延迟等跟案件相关的信息,而影响赔付金额的因素则主要是历史赔付金额等反映出险事故严重程度的信息。本文最后还给出了RBNS准备金的预测分布,其结果更加接近准备金的真实值且方差更小,表明在非寿险RBNS准备金评估中,基于机器学习算法的个体索赔准备金评估模型优于传统的准备金评估模型。     

A Bayesian Generalized Linear Model for the Bornhuetter-Ferguson Method of Claims Reserving

Abstract

This paper shows how Bayesian models within the framework of generalized linear models can be applied to claims reserving. The author demonstrates that this approach is closely related to the Bornhuetter-Ferguson technique. Benktander (1976) and Mack (2000) previously studied the Bornhuetter-Ferguson technique and advocated using credibility models. The present paper uses a Bayesian parametric model within the framework of generalized linear models.     

Reversible Jump Markov Chain Monte Carlo Method for Parameter Reduction in Claims Reserving

Abstract

We present an application of the reversible jump Markov chain Monte Carlo (RJMCMC) method to the important problem of setting claims reserves in general insurance business for the outstanding loss liabilities. A measure of the uncertainty in these claims reserves estimates is also needed for solvency purposes. The RJMCMC method described in this paper represents an improvement over the manual processes often employed in practice. In particular, our RJMCMC method describes parameter reduction and tail factor estimation in the claims reserving process, and, moreover, it provides the full predictive distribution of the outstanding loss liabilities.     

Issues in Claims Reserving and Credibility: A Semiparametric Approach With Mixed Models

Using the statistical methodology of semi-parametric regression and its connection with mixed models, this article revisits smoothing models for loss reserving and credibility. Apart from the flexibility inherent to all semiparametric methods, advantages of the semiparametric approach developed here are threefold. First, a Bayesian implementation of these smoothing models is relatively straightforward and allows simulation from the full predictive distribution of quantities of interest. Second, because the constructed models have an interpretation as (generalized) linear mixed models ((G)LMMs), standard statistical theory and software for (G)LMMs can be used. Third, more complicated data sets, dealing, for example, with quarterly development in a reserving context, heavy tails, semi-continuous data, or extensive longitudinal data, can be modeled within this framework.     

Accounting Year Effects Modeling in the Stochastic Chain Ladder Reserving Method

Abstract

In almost all stochastic claims reserving models one assumes that accident years are independent. In practice this assumption is violated most of the time. Typical examples are claims inflation and accounting year effects that influence all accident years simultaneously. We study a Bayesian chain ladder model that allows for accounting (calendar) year effects modeling. A case study of a general liability dataset shows that such accounting year effects contribute substantially to the prediction uncertainty and therefore need a careful treatment within a risk management and solvency framework.     

Pricing Contingent Claims on a Risky Asset and Stochastic Interest Rates: A New Discrete Time Approach

This paper presents a new discrete time approach to pricing contingent claims on a risky asset and stochastic interest rates. The term structure of interest rates is modeled so that arbitrage-free bond prices depend on an observable initial forward rate curve rather than an exogenously specified market price of risk. A restricted binomial process is employed to model both interest rates and an asset price. As a result, a complete market valuation formula obtains. By choosing the parameters of the discrete joint distribution such that, in the limit, the discrete model converges to the continuous one, a model is obtained that requires the estimation of only three parameters. The approach is parsimonious with respect to alternative models in the literature and can be used to price contingent claims on any two state variables. The procedure is used to numerically analyze the effects of the volatility of interest rates on the determination of mortgage contract rates.     

Insurance Guaranty Schemes in a Contingent Claims Setting

The introduction of an insurance guaranty scheme can have significant influence on the pricing and capital structures in a competitive market. This contribution summarizes the major findings of a working paper written by Schmeiser and Wagner (Working Papers on Risk Management and Insurance (IVW-HSG), No. 80, 2010). The effect on competitive equity-premium combinations is studied while considering a framework with policyholders and equity holders where guaranty fund charges are volume-based, as levied in existing schemes. Several settings with regard to the origin of the fund contributions are assessed and the immediate effects on the incentives of the policyholders and equity holders are analyzed through a one-period contingent claim approach. One result is that introducing a guaranty scheme in a market with competitive conditions entails a shift of equity capital towards minimum solvency requirements. Hence adverse incentives may arise with regard to the overall security level of the industry.     

Recursive Moments of Compound Renewal Sums with Discounted Claims

Under regularity conditions, Le´veille´& Garrido [6] gives a derivation of the first two moments (resp. asymptotic) of a Compound Renewal Present Value Risk (CRPVR) process using renewal theory arguments. In this paper, with the same procedure and assuming that all the moments of the claim severity and the claims number process exist, we get recursive formulas for all the moments (resp. asymptotic) of the CRPVR process.     

Index Option Pricing Models with Stochastic Volatility and Stochastic Interest Rates

This paper specifies a multivariate stochasticvolatility (SV) model for the S & P500 index and spot interest rateprocesses. We first estimate the multivariate SV model via theefficient method of moments (EMM) technique based on observations ofunderlying state variables, and then investigate the respective effects of stochastic interest rates, stochastic volatility, and asymmetric S & P500 index returns on option prices. We compute option prices using both reprojected underlying historical volatilities and the implied risk premiumof stochastic volatility to gauge each model's performance through direct comparison with observed market option prices on the index. Our major empirical findings are summarized as follows. First, while allowing for stochastic volatility can reduce the pricing errors and allowing for asymmetric volatility or leverage effect does help to explain the skewness of the volatility smile, allowing for stochastic interest rates has minimal impact on option prices in our case. Second, similar to Melino and Turnbull (1990), our empirical findings strongly suggest the existence of a non-zero risk premium for stochastic volatility of asset returns. Based on the implied volatility risk premium, the SV models can largely reduce the option pricing errors, suggesting the importance of incorporating the information from the options market in pricing options. Finally, both the model diagnostics and option pricing errors in our study suggest that the Gaussian SV model is not sufficientin modeling short-term kurtosis of asset returns, an SV model withfatter-tailed noise or jump component may have better explanatory power.     

Asset Allocation with a High Dimensional Latent Factor Stochastic Volatility Model

We investigate the implications of time-varying expected returnand volatility on asset allocation in a high dimensional setting.We propose a dynamic factor multivariate stochastic volatility(DFMSV) model that allows the first two moments of returns tovary over time for a large number of assets. We then evaluatethe economic significance of the DFMSV model by examining theperformance of various dynamic portfolio strategies chosen bymean-variance investors in a universe of 36 stocks. We findthat the DFMSV dynamic strategies significantly outperform variousbenchmark strategies out of sample. This outperformance is robustto different performance measures, investor’s objectivefunctions, time periods, and assets.     

Variable Annuities with VIX-Linked Fee Structure under a Heston-Type Stochastic Volatility Model

The Chicago Board of Options Exchange (CBOE) advocates linking variable annuity (VA) fees to its trademark VIX index in a recent white paper. It claims that the VIX-linked fee structure has several advantages over the traditional fixed percentage fee structure. However, the evidence presented is largely based on nonparametric extrapolation of historical data on market prices. Our work lays out a theoretical basis with a parametric model to analyze the impact of the VIX-linked fee structure and to verify some claims from the CBOE. In a Heston-type stochastic volatility setting, we jointly model the dynamics of an equity index (underlying the value of VA policyholders’ accounts) and the VIX index. In this framework, we price a guaranteed minimum maturity benefit with VIX-linked fees. Through numerical examples, we show that the VIX-linked fee reduces the sensitivity of the insurer's liability to market volatility when compared to a VA with the traditional fixed fee rate.     

Stochastic Discount Factor Bounds with Conditioning Information

Hansen and Jagannathan (1991) (hereafter HJ) derive restrictionson the volatility of stochastic discount factors that pricea given set of returns. This article studies the sampling propertiesof HJ bounds that use conditioning information. One approachis to multiply the returns by the lagged variables. We alsostudy optimized HJ bounds with conditioning information fromGallant, Hansen, and Tauchen (1990) and based on portfoliosderived in Ferson and Siegel (2001). We document striking finite-samplebiases in the HJ bounds, where the bounds reject asset-pricingmodels too often. We provide a useful bias correction. We alsoevaluate asymptotic standard errors for the bounds from Hansen,Heaton, and Luttmer (1995).     

On the Laplace Transform of the Aggregate Discounted Claims with Markovian Arrivals

Abstract

We present an explicit formula for the Laplace transform of the distribution of the aggregate discounted claims when interclaim times follow a Markovian arrival process. In addition, we derive explicit formulas for the first two moments and then show that the higher moments may be obtained by numerically solving a system of ordinary differential equations.     

Minimizing the Probability of Ruin When Claims Follow Brownian Motion with Drift

Abstract

We extend the work of Browne (1995) and Schmidli (2001), in which they minimize the probability of ruin of an insurer facing a claim process modeled by a Brownian motion with drift. We consider two controls to minimize the probability of ruin: (1) investing in a risky asset and (2) purchasing quota-share reinsurance. We obtain an analytic expression for the minimum probability of ruin and the corresponding optimal controls, and we demonstrate our results with numerical examples.     

Vacancies and Rent Dispersion in a Stochastic Search Model with Generalized Tenant Demand

I model imperfect information, derive a downward sloping market demand curve, and explain vacancies in a partial equilibrium model of a rental housing market. Tenants can be completely described by an exogenous demand curve, perhaps arising from differences in income, preferred location, or tastes, and view vacant units based on a stochastic arrival of rental information. Free entry of these landlords induces excess rental housing capacity (equilibrium vacancies). I determine the equilibrium distribution of rents for vacant units, show that this rent distribution may be discontinuous, and explore the equilibrium vacancy rate to changes in exogenous parameters. The resulting characterization of equilibrium distributions of rents may be amenable to econometric modeling exploring the relationship between market rents and vacancies.     

On the Moments of the Time of Ruin with Applications to Phase-Type Claims

Abstract

We describe an approach to the evaluation of the moments of the time of ruin in the classical Poisson risk model. The methodology employed involves the expression of these moments in terms of linear combinations of convolutions involving compound negative binomial distributions. We then adapt the results for use in the practically important case involving phase-type claim size distributions. We present numerical examples to illuminate the influence of claim size variability on the moments of the time of ruin.     

Stochastic Migration Models with Application to Corporate Risk

In this article we explain how to use rating histories providedby the internal scoring systems of banks and rating agenciesin order to predict the future risk of a set of borrowers. Themethod is developed following the steps suggested by the BasleCommittee. To introduce both migration correlation and non-Markovianserial dependence, we consider rating histories with stochastictransition matrices. We develop the methodology to estimateboth the number and dynamics of the factors influencing thetransitions and we explain how to use the model for prediction.As an illustration, the ordered probit model with unobservabledynamic factor is estimated from French data on corporate risk.     

Good-Deal Option Price Bounds for a Non-Traded Event with Stochastic Return: A Note

Cochrane and Sa'a-Requejo (2000, Journal of Political Economy) proposed the good-deal price bounds for the European call option on an event that is not a traded asset, but is correlated with a traded asset that can be used as an approximate hedge. One remarkable feature of their model is that the return on an event process explicitly appears in the option price bounds formula, which offered a contrast with the standard option pricing model. We show that the good-deal option price bounds on a non-traded event are obtained as a closed-form formula, when the return on an event is governed by a mean reverting process.     

A Statistical Method for Monitoring a Change in the Rate of Nonacceptable Inpatient Claims

Abstract

This paper is an extension of earlier work (Rosenberg 1998; Rosenberg, Andrews, and Lenk 1999; Rosenberg and Griffith 2000) that introduced a statistical control model to supplement current efforts inexpensively to help reduce unnecessary expenditures. The application of the study was to predict the rate of nonacceptable inpatient claims (NACs). In that work, a statistical model was proposed to link information obtained through an expensive audit with inexpensive information that is readily available to estimate the probability that a claim is a NAC. The premise was that a statistical system can be developed to supplement the expensive audit for additional control between audits.

Estimates of the NAC rate obtained from the statistical model are used as input in a statistical monitor to assess whether the NAC rate had changed over time. The statistical monitor is the subject of this paper. The idea is that subgroups of claims can be analyzed inexpensively with the statistical monitor to determine whether any current intervention is required prior to the time of the next scheduled audit, or whether adjustments are needed in the determination of claims to be sampled for the audit. In this study, the estimate for the NAC rate at t ₀ is compared against the estimate of the NAC rate at some later time t ₁. A decision rule is proposed to assess whether a change in the NAC rate has occurred for that subgroup. The methodology is also applicable to other health care measurements.