Emiliano A. Valdez

Annuity Valuation with Dependent Mortality

Annuities are contractual guarantees that promise to provide periodic income over the lifetime(s) of individuals. Standard insurance industry practice assumes independence of lives when valuing annuities where the promise is based on more than one life. This article investigates the use of dependent mortality models to value this type of annuity. We discuss a broad class of parametric models using a bivariate survivorship function called a copula. Using data from a large insurance company, we calculate maximum likelihood estimates to calibrate the model. The estimation results show strong positive dependence between joint lives with real economic significance. Annuity values are reduced by approximately 5 percent when dependent mortality models are used compared to the standard models that assume independence.

Optimal Capital Allocation Principles

This paper develops a unifying framework for allocating the aggregate capital of a financial firm to its business units. The approach relies on an optimisation argument, requiring that the weighted sum of measures for the deviations of the business unit’s losses from their respective allocated capitals be minimised. This enables the association of alternative allocation rules to specific decision criteria and thus provides the risk manager with flexibility to meet specific target objectives. The underlying general framework reproduces many capital allocation methods that have appeared in the literature and allows for several possible extensions. An application to an insurance market with policyholder protection is additionally provided as an illustration.

Hierarchical Insurance Claims Modeling

This work describes statistical modeling of detailed, microlevel automobile insurance records. We consider 1993–2001 data from a major insurance company in Singapore. By detailed microlevel records, we mean experience at the individual vehicle level, including vehicle and driver characteristics, insurance coverage, and claims experience, by year. The claims experience consists of detailed information on the type of insurance claim, such as whether the claim is due to injury to a third party, property damage to a third party, or claims for damage to the insured, as well as the corresponding claim amount. We propose a hierarchical model for three components, corresponding to the frequency, type, and severity of claims. The first model is a negative binomial regression model for assessing claim frequency. The driver’s gender, age, and no claims discount, as well as vehicle age and type, turn out to be important variables for predicting the event of a claim. The second is a multinomial logit model to predict the type of insurance claim, whether it is third-party injury, third-party property damage, insured’s own damage or some combination. Year, vehicle age, and vehicle type turn out to be important predictors for this component. Our third model is for the severity component. Here we use a generalized beta of the second kind of long-tailed distribution for claim amounts and also incorporate predictor variables. Year, vehicle age, and person’s age turn out to be important predictors for this component. Not surprisingly, we show a significant dependence among the different claim types; we use a t-copula to account for this dependence. The three-component model provides justification for assessing the importance of a rating variable. When taken together, the integrated model allows more efficient prediction of automobile claims compared with than traditional methods. Using simulation, we demonstrate this by developing predictive distributions and calculating premiums under alternative coverage limitations.

Tail Conditional Expectations for Exponential Dispersion Models

There is a growing interest in the use of the tail conditional expectation as a measure of risk. For an institution faced with a random loss, the tail conditional expectation represents the conditional average amount of loss that can be incurred in a fixed period, given that the loss exceeds a specified value. This value is typically based on the quantile of the loss distribution, the so-called value-at-risk. The tail conditional expectation can therefore provide a measure of the amount of capital needed due to exposure to loss. This paper examines this risk measure for “exponential dispersion models”, a wide and popular class of distributions to actuaries which, on one hand, generalizes the Normal and shares some of its many important properties, but on the other hand, contains many distributions of nonnegative random variables like the Gamma and the Inverse Gaussian.

Securitization of Longevity Risk in Reverse Mortgages

The reverse mortgage market has been expanding rapidly in developed economies in recent years. The onset of demographic transition places a rapidly rising number of households in an age window in which reverse mortgages have potential appeal. Increasing prices for residential real estate over the last decade have further stimulated interest.Reverse mortgages involve various risks from the provider-s perspective that may hinder the further development of these financial products. This paper addresses one method of transferring and financing the risks associated with these products through the form of securitization. Securitization is becoming a popular and attractive alternative form of risk transfer of insurance liabilities. Here we demonstrate how to construct a securitization structure for reverse mortgages similar to the one applied in traditional insurance products.Specifically, we investigate the merits of developing survivor bonds and survivor swaps for reverse mortgage products. In the case of survivor bonds, for example, we are able to compute premiums, both analytically and numerically through simulations, and to examine how the longevity risk may be transferred to the financial investors. Our numerical calculations provide an indication of the economic benefits derived from developing survivor bonds to securitize the “longevity risk component” of reverse mortgage products. Moreover, some sensitivity analysis of these economic benefits indicates that these survivor bonds provide for a promising tool for investment diversification.

Actuarial Applications of a Hierarchical Insurance Claims Model

This paper demonstrates actuarial applications of modern statistical methods that are applied to detailed, micro-level automobile insurance records. We consider 1993-2001 data consisting of policy and claims files from a major Singaporean insurance company. A hierarchical statistical model, developed in prior work (Frees and Valdez (2008)), is fit using the micro-level data. This model allows us to study the accident frequency, loss type and severity jointly and to incorporate individual characteristics such as age, gender and driving history that explain heterogeneity among policyholders. Based on this hierarchical model, one can analyze the risk profile of either a single policy (micro-level) or a portfolio of business (macro-level). This paper investigates three types of actuarial applications. First, we demonstrate the calculation of the predictive mean of losses for individual risk rating. This allows the actuary to differentiate prices based on policyholder characteristics. The nonlinear effects of coverage modifications such as deductibles, policy limits and coinsurance are quantified. Moreover, our flexible structure allows us to “unbundle” contracts and price more primitive elements of the contract, such as coverage type. The second application concerns the predictive distribution of a portfolio of business. We demonstrate the calculation of various risk measures, including value at risk and conditional tail expectation, that are useful in determining economic capital for insurance companies. Third, we examine the effects of several reinsurance treaties. Specifically, we show the predictive loss distributions for both the insurer and reinsurer under quota share and excess-of-loss reinsurance agreements. In addition, we present an example of portfolio reinsurance, in which the combined effect of reinsurance agreements on the risk characteristics of ceding and reinsuring company are described.

CAPM and Option Pricing with Elliptically Contoured Distributions

This article offers an alternative proof of the capital asset pricing model (CAPM) when asset returns follow a multivariate elliptical distribution. Empirical studies continue to demonstrate the inappropriateness of the normality assumption for modeling asset returns. The class of elliptically contoured distributions, which includes the more familiar Normal distribution, provides flexibility in modeling the thickness of tails associated with the possibility that asset returns take extreme values with nonnegligible probabilities. As summarized in this article, this class preserves several properties of the Normal distribution. Within this framework, we prove a new version of Steins lemma for this class of distributions and use this result to derive the CAPM when returns are elliptical. Furthermore, using the probability distortion function approach based on the dual utility theory of choice under uncertainty, we also derive an explicit form solution to call option prices when the underlying is log-elliptically distributed. The Black-Scholes call option price is a special case of this general result when the underlying is log-normally distributed. Copyright (c) The Journal of Risk and Insurance, 2008.

Multivariate Negative Binomial Models for Insurance Claim Counts

It is no longer uncommon these days to find the need in actuarial practice to model claim counts from multiple types of coverage, such as the ratemaking process for bundled insurance contracts. Since different types of claims are conceivably correlated with each other, the multivariate count regression models that emphasize the dependency among claim types are more helpful for inference and prediction purposes. Motivated by the characteristics of an insurance dataset, we investigate alternative approaches to constructing multivariate count models based on the negative binomial distribution. A classical approach to induce correlation is to employ common shock variables. However, this formulation relies on the NB-I distribution which is restrictive for dispersion modeling. To address these issues, we consider two different methods of modeling multivariate claim counts using copulas. The first one works with the discrete count data directly using a mixture of max-id copulas that allows for flexible pair-wise association as well as tail and global dependence. The second one employs elliptical copulas to join continuitized data while preserving the dependence structure of the original counts. The empirical analysis examines a portfolio of auto insurance policies from a Singapore insurer where claim frequency of three types of claims (third party property damage, own damage, and third party bodily injury) are considered. The results demonstrate the superiority of the copula-based approaches over the common shock model. Finally, we implemented the various models in loss predictive applications.

A multilevel analysis of intercompany claim counts

It is common for professional associations and regulators to combine the claims experience of several insurers into a database known as an “intercompany†experience data set. In this paper, we analyze data on claim counts provided by the General Insurance Association of Singapore, an organization consisting of most of the general insurers in Singapore. Our data comes from the financial records of automobile insurance policies followed over a period of nine years. Because the source contains a pooled experience of several insurers, we are able to study company effects on claim behavior, an area that has not been systematically addressed in either the insurance or the actuarial literatures. We analyze this intercompany experience using multilevel models. The multilevel nature of the data is due to: a vehicle is observed over a period of years and is insured by an insurance company under a “fleet†policy. Fleet policies are umbrella-type policies issued to customers whose insurance covers more than a single vehicle. We investigate vehicle, fleet and company effects using various count distribution models (Poisson, negative binomial, zero-inflated and hurdle Poisson). The performance of these various models is compared; we demonstrate how our model can be used to update a priori premiums to a posteriori premiums, a common practice of experience-rated premium calculations. Through this formal model structure, we provide insights into effects that company-specific practice has on claims experience, even after controlling for vehicle and fleet effects.

Testing Adverse Selection With Two‐Dimensional Information: Evidence From the Singapore Auto Insurance Market

This article examines adverse selection in insurance markets with two‐dimensional information: policyholders’ riskiness and degree of risk aversion. We build a theoretical model to make equilibrium predictions on competitive insurance screening. We study several variations on the pattern of information asymmetry. The outcomes range from full separation to partial separation, and complete pooling of risk types. Next, we propose a copula approach to jointly examine policyholders’ coverage choice and accident occurrence in the Singapore automobile insurance market. Furthermore, we invoke the theory to identify subgroups of policyholders for whom one may expect the risk–coverage correlation and adverse selection to arise.