Variance of the CTE Estimator

Abstract

The Conditional Tail Expectation (CTE), also called Expected Shortfall or Tail-VaR, is a robust, convenient, practical, and coherent measure for quantifying financial risk exposure. The CTE is quickly becoming the preferred measure for statutory balance sheet valuation whenever real-world stochastic methods are used to set liability provisions. We look at some statistical properties of the methods that are commonly used to estimate the CTE and develop a simple formula for the variance of the CTE estimator that is valid in the large sample limit. We also show that the formula works well for finite sample sizes. Formula results are compared with sample values from realworld Monte Carlo simulations for some common loss distributions, including equity-linked annuities with investment guarantees, whole life insurance and operational risks. We develop the CTE variance formula in the general case using a system of biased weights and explore importance sampling, a form of variance reduction, as a way to improve the quality of the estimators for a given sample size. The paper closes with a discussion of practical applications.     

On the Class of Erlang Mixtures with Risk Theoretic Applications

Abstract

A wide variety of distributions are shown to be of mixed-Erlang type. Useful computational formulas result for many quantities of interest in a risk-theoretic context when the claim size distribution is an Erlang mixture. In particular, the aggregate claims distribution and related quantities such as stop-loss moments are discussed, as well as ruin-theoretic quantities including infinitetime ruin probabilities and the distribution of the deficit at ruin. A very useful application of the results is the computation of finite-time ruin probabilities, with numerical examples given. Finally, extensions of the results to more general gamma mixtures are briefly examined.     

Social Transfers And Income Inequality In Old Age

Abstract

This paper examines variation in old-age income inequality between industrialized nations with modern welfare systems. The analysis of income inequality across countries with different retirement income systems provides a perspective on public pension policy choices and designs and their distributional implications. Because of the progressive nature of public pension programs, we hypothesize that there is an inverse relationship between the quality of public pension benefits and old-age income inequality—that is, countries with comprehensive, universal, and generous public pension systems will exhibit more equal distributions of income in old age.

Luxembourg Income Study data indeed show that cross-national variation in old-age income inequality is partly explained by differences in the percentage of seniors’ total income derived from public pension transfers. Sweden, for example, has the highest level of government transfers and the lowest level of old-age income inequality, while Israel and the United States have the lowest levels of dependency on government transfers and the highest levels of income inequality. A notable exception is Canada, where public transfers represent only a moderate portion of elderly income, yet old-age income inequality is relatively low. These findings suggest that quality of public pension benefits does indeed play a role in explaining differences in old-age income inequality between industrialized nations, yet these variations are also likely influenced by other factors.     

Compound Poisson Model with Covariates

Abstract

Pet insurance in North America continues to be a growing industry. Unlike in Europe, where some countries have as much as 50% of the pet population insured, very few pets in North America are insured. Pricing practices in the past have relied on market share objectives more so than on actual experience. Pricing still continues to be performed on this basis with little consideration for actuarial principles and techniques. Developments of mortality and morbidity models to be used in the pricing model and new product development are essential for pet insurance. This paper examines insurance claims as experienced in the Canadian market. The time-to-event data are investigated using the Cox’s proportional hazards model. The claim number follows a nonhomogenous Poisson process with covariates. The claim size random variable is assumed to follow a lognormal distribution. These two models work well for aggregate claims with covariates. The first three central moments of the aggregate claims for one insured animal, as well as for a block of insured animals, are derived. We illustrate the models using data collected over an eight-year period.     

An Extreme Value Analysis Of Advanced Age Mortality Data

Abstract

Extreme value theory describes the behavior of random variables at extremely high or low levels. The application of extreme value theory to statistics allows us to fit models to data from the upper tail of a distribution. This paper presents a statistical analysis of advanced age mortality data, using extreme value models to quantify the upper tail of the distribution of human life spans.

Our analysis focuses on mortality data from two sources. Statistics Canada publishes the annual number of deaths in Canada, broken down by angender and age. We use the deaths data from 1949 to 1997 in our analysis. The Japanese Ministry of Health, Labor, and Welfare also publishes detailed annual mortality data, including the 10 oldest reported ages at death in each year. We analyze the Japanese data over the period from 1980 to 2000.

Using the r-largest and peaks-over-threshold approaches to extreme value modeling, we fit generalized extreme value and generalized Pareto distributions to the life span data. Changes in distribution by birth cohort or over time are modeled through the use of covariates. We then evaluate the appropriateness of the fitted models and discuss reasons for their shortcomings. Finally, we use our findings to address the existence of a finite upper bound on the life span distribution and the behavior of the force of mortality at advanced ages.     

Analysis of a Generalized Penalty Function in a Semi-Markovian Risk Model

Abstract

In this paper an extension of the semi-Markovian risk model studied by Albrecher and Boxma (2005) is considered by allowing for general interclaim times. In such a model, we follow the ideas of Cheung et al. (2010b) and consider a generalization of the Gerber-Shiu function by incorporating two more random variables in the traditional penalty function, namely, the minimum surplus level before ruin and the surplus level immediately after the second last claim prior to ruin. It is shown that the generalized Gerber-Shiu function satisfies a matrix defective renewal equation. Detailed examples are also considered when either the interclaim times or the claim sizes are exponentially distributed. Finally, we also consider the case where the claim arrival process follows a Markovian arrival process. Probabilistic arguments are used to derive the discounted joint distribution of four random variables of interest in this risk model by capitalizing on an existing connection with a particular fluid flow process.     

Multivariate Models of Equity Returns for Investment Guarantees Valuation

Abstract

In this paper we investigate the valuation of investment guarantees in a multivariate (discrete-time) framework. We present how to build multivariate models in general, and we survey the most important multivariate GARCH models. A direct multivariate application of regime-switching models is also discussed, as is the estimation of these models using maximum likelihood and their comparison in a multivariate setting. The computation of the CTE provision is further presented. We have estimated the models with a multivariate dataset (Canada, United States, United Kingdom, and Japan), and we compared the quality of their fit using multiple criteria and tests. We observe that multivariate GARCH models provide a better overall fit than regime-switching models. However, regime-switching models appropriately represent the fat tails of the returns distribution, which is where most GARCH models fail. This leads to significant differences in the value of the CTE provisions, and, in general, provisions computed with regime-switching models are higher. Thus, the results from this multivariate analysis are in line with what was obtained in the literature of univariate models.     

On the Joint Distributions of the Time to Ruin,the Surplus Prior to Ruin,and the Deficit at Ruin in the Classical Risk Model

Abstract

The seminal paper by Gerber and Shiu (1998) unified and extended the study of the event of ruin and related quantities, including the time at which the event of ruin occurs, the deficit at the time of ruin, and the surplus immediately prior to ruin. The first two of these quantities are fundamentally important for risk management techniques that utilize the ideas of Value-at-Risk and Tail Value-at-Risk. As is well known, calculation of these and related quantities requires knowledge of the associated probability distributions. In this paper we derive an explicit expression for the joint (defective) distribution of the time to ruin, the surplus immediately prior to ruin, and the deficit at ruin in the classical compound Poisson risk model. As a by-product, we obtain expressions for the three bivariate distributions generated by the time to ruin, the surplus prior to ruin, and the deficit at ruin. Finally, we consider mixed Erlang claim sizes and show how the joint (defective) distribution of the time to ruin, the surplus prior to ruin, and the deficit at ruin can be calculated.