A General Semi-Markov Model for Coupled Lifetimes

Joint-life annuities with a high last survivor benefit play an important role in the optimal annuity portfolio for a retired couple. The dependence between coupled lifetimes is crucial for valuing joint-life annuities. Existing bivariate modeling of coupled lifetimes is based on outdated data with limited observation periods and does not take into account mortality improvement. In this article, we propose a transparent and dynamic framework for modeling coupled lifetime dependence caused by both marital status and common mortality improvement factors. Dependence due to marital status is captured by a semi-Markov joint life model. Dependence due to common mortality improvement, which represents the correlation between mortality improvement patterns of coupled lives, is incorporated by a two-population mortality improvement model. The proposed model is applied to pricing the longevity risk in last survivor annuities sold in the United States and the United Kingdom.     

Multivariate Tweedie lifetimes: the impact of dependence

Systematic longevity risk is increasingly relevant for public pension schemes and insurance companies that provide life benefits. In view of this, mortality models should incorporate dependence between lives. However, the independent lifetime assumption is still heavily relied upon in the risk management of life insurance and annuity portfolios. This paper applies a multivariate Tweedie distribution to incorporate dependence, which it induces through a common shock component. Model parameter estimation is developed based on the method of moments and generalized to allow for truncated observations. The estimation procedure is explicitly developed for various important distributions belonging to the Tweedie family, and finally assessed using simulation.     

Markovian Approaches to Joint-Life Mortality

Abstract

Many insurance products provide benefits that are contingent on the combined survival status of two lives. To value such benefits accurately, we require a statistical model for the impact of the survivorship of one life on another. In this paper we first set up two models, one Markov and one semi-Markov, to model the dependence between the lifetimes of a husband and wife. From the models we can measure the extent of three types of dependence: (1) the instantaneous dependence due to a catastrophic event that affect both lives, (2) the short-term impact of spousal death, and (3) the long-term association between lifetimes. Then we apply the models to a set of jointlife and last-survivor annuity data from a large Canadian insurance company. Given the fitted models, we study the impact of dependence on annuity values and examine the potential inaccuracy in pricing if we assume lifetimes are independent. Finally, we compare our Markovian models with two copula models considered in previous research on modeling joint-life mortality.     

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.     

Pricing catastrophe risk in life (re)insurance

What is the catastrophe risk a life insurance company faces? What is the correct price of a catastrophe cover? During a review of the current standard model, due to Strickler, we found that this model has some serious shortcomings. We therefore present a new model for the pricing of catastrophe excess of loss cover (Cat XL). The new model for annual claim cost C is based on a compound Poisson process of catastrophe costs. To evaluate the distribution of the cost of each catastrophe, we use the Peaks Over Threshold model for the total number of lost lives in each catastrophe and the beta binomial model for the proportion of these corresponding to customers of the insurance company. To be able to estimate the parameters of the model, international and Swedish data were collected and compiled, listing accidents claiming at least twenty and four lives, respectively. Fitting the new model to data, we find the fit to be good. Finally we give the price of a Cat XL contract and perform a sensitivity analysis of how some of the parameters affect the expected value and standard deviation of the cost and thus the price.     

Household Life Insurance Demand

Abstract

What types of households own life insurance? Who owns term life and who owns whole life insurance? These are questions of great interest to insurers that operate in a highly competitive market. To answer these questions, we jointly examine household demand of two types of insurance, term and whole life, using data from the Survey of Consumer Finances, a probability sample of the U.S. population. We model both the frequency and the severity of demand for insurance, building on the work of Lin and Grace by using explanatory variables that they developed. For the frequency portion, the household decisions about whether to own term and whole life insurance are modeled simultaneously with a bivariate probit regression model. Given ownership of life insurance by a household, the amounts of insurance are analyzed using generalized linear models with a normal copula. The copula permits the bivariate modeling of insurance amounts for households who own both term and whole life insurance, about 20% of our sample. These models allow analysts to predict who owns life insurance and how much they own, an important input to the marketing process.

Moreover, our findings suggest that household demand for term and whole life insurance is jointly determined. After controlling for explanatory variables, there exists a negative relationship for a household’s decision to own both whole and term life insurance (the frequency part) and a positive relationship for the amount of insurance purchased (the severity part). This indicates that the greater the probability of holding one type, the smaller the probability of holding the other type of life insurance. However, higher demand for both types of insurance exists when a household decides to own both. This mixed effect extends prior work that established a negative relationship, suggesting that term life insurance and whole life insurance are substitutes for one another. In contrast, our findings reveal that the ownership decision involves substitution, but, for households owning both types of insurance, amounts are positively related. Therefore, term and whole life insurance are substitutes in the frequency yet complements in the severity.     

The Impact of Disability Insurance on a Portfolio of Life Insurances

The aim of this article is to study the impact of disability insurance on an insurer's risk situation for a portfolio that also consists of annuity and term life contracts. We provide a model framework using discrete time nonhomogeneous bivariate Markov renewal processes and in a simulation study focus on diversification benefits as well as potential natural hedging effects (risk-minimizing or risk-immunizing portfolio compositions) that may arise within the portfolio because of the different types of biometric risks. Our analyses emphasize that disability insurances are a less efficient tool to hedge shocks to mortality and that their high sensitivity toward shocks to disability risks cannot be easily counterbalanced by other life insurance products. However, the addition of disability insurance can still considerably lower the overall company risk.     

Cohort and value-based multi-country longevity risk management

ABSTRACT

Multi-country risk management of longevity risk provides new opportunities to hedge mortality and interest rate risks in guaranteed lifetime income streams. This requires consideration of both interest rate and mortality risks in multiple countries. For this purpose, we develop value-based longevity indexes for multiple cohorts in two different countries that take into account the major sources of risks impacting life insurance portfolios, mortality and interest rates. To construct the indexes we propose a cohort-based affine model for multi-country mortality and use an arbitrage-free multi-country Nelson–Siegel model for the dynamics of interest rates. Index-based longevity hedging strategies have the advantages of efficiency, liquidity and lower cost but introduce basis risk. Graphical risk metrics are a way to effectively capture the relationship between an insurer's portfolio and hedging strategies. We illustrate the effectiveness of using a value-based index for longevity risk management between two countries using graphical basis risk metrics. To show the impact of both interest rate and mortality risk we use Australia and the UK as domestic and foreign countries, and, to show the impact of mortality only, we use the male populations of the Netherlands and France with common interest rates and basis risk arising only from differences in mortality risks.     

Types of dependence and time-dependent association between two lifetimes in single parameter copula models

Spreeuw, J. Types of dependence and time-dependent association between two lifetimes in single parameter copula models. Scandinavian Actuarial Journal. Most publications on modeling insurance contracts on two lives, assuming dependence of the two lifetimes involved, focus on the time of inception of the contract. The dependence between the lifetimes is usually modeled through a copula and the effect of this dependence on the pricing of a joint life policy is measured. This paper investigates the effect of association at the outset on the mortality in the future. The conditional law of mortality of an individual, given his survival and given the life status of the partner is derived. The conditional joint survival distribution of a couple at any duration, given that the two lives are then alive, is also derived. We analyze how the degree of dependence between the two members of a couple varies throughout the duration of a contract. We have done that for (mainly Archimedean) copula models, with one parameter for the degree of dependence. The conditional distributions hence derived provide the basis for the calculation of prospective provisions.     

A Re-Examination of the Joint Mortality Functions

Abstract

Mortality analysis involving multiple lives is easily one of the more complicated aspects in the theory of life contingencies. In this paper, we re-investigate joint mortality functions and in particular, we examine an assertion that relates the joint-life and last-survivor random variables. This common assertion states that the sum of the lifetimes of the joint-life and the last-survivor statuses is equal to the sum of the lifetimes of the single statuses. However, we show that this assertion is not precisely correct. We therefore offer a modification to the statuses definitions so that this common assertion holds.     

Stochastic Orderings of Convex-Type for Discrete Bivariate Risks

New classes of order relations for discrete bivariate random vectors are introduced that essentially compare the expectations of real functions of convex-type of the random vectors. For the actuarial context, attention is focused on the so-called increasing convex orderings between discrete bivariate risks. First, various characterizations and properties of these orderings are derived. Then, they are used for comparing two similar portfolios with dependent risks and for constructing bounds on several multilife insurance premiums.     

CreditRisk+ Model with Dependent Risk Factors

The CreditRisk⁺ model is widely used in industry for computing the loss of a credit portfolio. The standard CreditRisk⁺ model assumes independence among a set of common risk factors, a simplified assumption that leads to computational ease. In this article, we propose to model the common risk factors by a class of multivariate extreme copulas as a generalization of bivariate Fréchet copulas. Further we present a conditional compound Poisson model to approximate the credit portfolio and provide a cost-efficient recursive algorithm to calculate the loss distribution. The new model is more flexible than the standard model, with computational advantages compared to other dependence models of risk factors.     

Recovery Process Model for Two Companies

The quantification of the recovery rate for the debt of the defaulted small company is one of the most important problems for banks and their supervisors. However, the data of the real recovery rates is seldom available for academic study. Therefore, there have been a few studies for the recovery rate for the debt. The recovery process model for a single company is introduced by Itoh (Asia Pac Financ Mark 2008). In this paper, we extend the model of Itoh (Asia Pac Financ Mark 2008) to two defaulted companies, and we model the recovery processes using an inhomogeneous bivariate compound Poisson process with the delays. In other word, we assume that the recoveries are occurred by the shocks, and that there are individual shocks that affect only one company and common shocks that affect two companies. Moreover, we assume that there are delays between the shock points and the recovery points. Therefore, the recovery points of two companies are correlated, but the recoveries do not occur synchronously almost surely. We derive the correlation of the recovery rates of the debts of two defaulted companies, and the expected value and the standard deviation of the recovery rate for the defaulted debt portfolio. Furthermore, we present two methods based on the Vernic recursion formula and the Monte Calro simulation for calculating the probability distribution function of the recovery process, and illustrate several numerical results.     

“The Future of Risk Classification in the Age of Predictive DNA-Based Testing”, Donald C. Chambers,January 1999

Abstract

The correlation among multiple lines of business plays an important role in quantifying the uncertainty of loss reserves for insurance portfolios. To accommodate correlation, most multivariate loss-reserving methods focus on the pairwise association between corresponding cells in multiple run-off triangles. However, such practice usually relies on the independence assumption across accident years and ignores the calendar year effects that could affect all open claims simultaneously and induce dependencies among loss triangles. To address this issue, we study a Bayesian log-normal model in the prediction of outstanding claims for dependent lines of business. In addition to the pairwise correlation, our method allows for an explicit examination of the correlation due to common calendar year effects. Further, different specifications of the calendar year trend are considered to reflect valuation actuaries’ prior knowledge of claim development. In a case study, we analyze an insurance portfolio of personal and commercial auto lines from a major U.S. property-casualty insurer. It is shown that the incorporation of calendar year effects improves model fit significantly, though it contributes substantively to the predictive variability. The availability of the realizations of predicted claims permits us to perform a retrospective test, which suggests that extra prediction uncertainty is indispensable in modern risk management practices.     

The Systemic Risk of Cross-Border Banking: Evidence from the Sudden Stop and Interbank Stress Contagion in East Asia

This article investigates the systemic risk of cross-border banking in East Asia. Using the recursive bivariate probit model, we jointly test the probability of the sudden stop in international lending and its simultaneous effect on the host countries’ interbank markets. The empirical results suggest that the risk of a sudden stop is associated with global liquidity shock; host country productivity shock; and the common lender contagion effect. This facilitates the transmission of interbank stress from advanced economies to emerging markets. However, the tension is mitigated by the “flight-home effect” caused by domestic investors’ repatriation. The sudden stop is more likely to occur in countries with lower financial openness but higher financial risk. Lending flows to the banking sectors are more sensitive to shocks than the flows to the non-bank private sectors.     

Long-Term Care Insurance and Life Insurance Demand

This article investigates the interaction between life insurance and long-term care insurance markets on the demand side. In the model utility depends on both consumption and bequest, and utility from consumption is contingent on the state of health. While the demand for life insurance increases both with decreasing income and with a rising degree of altruism, the influences of these two parameters on the demand for long-term care insurance are ambiguous. If the utility shock arising from disability declines, both insurance demands will rise.     

Ruin probabilities in multivariate risk models with periodic common shock

We propose a multidimensional risk model where the common shock affecting all classes of insurance business is arriving according to a non-homogeneous periodic Poisson process. In this multivariate setting, we derive upper bounds of Lundberg-type for the probability that ruin occurs in all classes simultaneously using the martingale approach via piecewise deterministic Markov processes theory. These results are numerically illustrated in a bivariate risk model, where the beta-shape periodic claim intensity function is considered. Under the assumption of dependent heavy-tailed claims, asymptotic bounds for the finite-time ruin probabilities associated to three types of ruin in this multivariate framework are investigated.     

Pricing Participating Inflation Retirement Funds Through Option Modeling and Copulas

Abstract

Pension plans and life insurances offering minimum performance guarantees are very common worldwide. In the Brazilian market, the customers of a common type of defined contribution plan have the right to receive, over their savings, the positive difference between the return of a specified investment fund, usually a fixed income fund, and the minimum guaranteed rate, commonly defined as the composition of a fixed interest rate and a floating inflation rate. This instrument can be characterized as an option to exchange one asset, the minimum guaranteed rate, for another, the return of the specified investment fund. In this paper we provide a closed formula to evaluate this liability that depends on two stochastic rates assuming bivariate normality. We also explore the use of copulas for the modeling of the dependence structure and price the options using Monte Carlo simulation to compare the effects of the copula specification in their values. An application with real data is provided. The model makes use of a one-factor Vasicek framework for the term structures of interest rate and inflation rate.     

Multivariate dependence of implied volatilities from equity options as measure of systemic risk

This paper presents a methodology to examine the multivariate tail dependence of the implied volatility of equity options as an early warning indicator of systemic risk within the financial sector. Using non-parametric methods of estimating changes in the dependence structure in response to common shocks affecting individual risk profiles, possible linkages during periods of stress are quantifiable while recognizing that large shocks are transmitted across financial markets differently than small shocks. Before and during the initial phase of the financial crisis, we find that systemic risk increased globally as early as February 2007 — months before the unraveling of the U.S. subprime mortgage crisis and long before the collapse of Lehman Brothers. The average (multivariate) dependence among a global sample of banks and insurance companies increased by almost 30% while joint tail risk declined by about the same order of magnitude, indicating that co-movements of large changes in equity volatility were more likely to occur and responses to extreme shocks became more differentiated as distress escalated. The key policy consideration flowing from our analysis is that complementary measures of joint tail risk at high data frequency are essential to the robust measurement of systemic risk, which could enhance market-based early warning mechanisms as part of macroprudential surveillance.     

About a duration index for life insurance contracts

Abstract

In this paper, after having defined the duration for a “generic” life insurance contract, we bring out some of its properties. We also prove that, in some cases, duration is a natural extension of well-known duration indices.