Modelling stochastic mortality for dependent lives

Stochastic mortality, i.e. modelling death arrival via a jump process with stochastic intensity, is gaining an increasing reputation as a way to represent mortality risk. This paper is a first attempt to model the mortality risk of couples of individuals, according to the stochastic intensity approach. Dependence between the survival times of the members of a couple is captured by an Archimedean copula.We also provide a methodology for fitting the joint survival function by working separately on the (analytical) marginals and on the (analytical) copula. First, we provide a sample-based calibration for the intensity, using a time-homogeneous, non mean-reverting, affine process: this gives the marginal survival functions. Then we calibrate and select the best fit copula according to the Wang and Wells [Wang, W., Wells, M.T., 2000b. Model selection and semiparametric inference for bivariate failure-time data. J. Amer. Statis. Assoc. 95, 62-72] methodology for censored data. By coupling the calibrated marginals with the best fit copula, we obtain a joint survival function, which incorporates the stochastic nature of mortality improvements.We apply the methodology to a well known insurance data set, using a sample generation. The best fit copula turns out to be one listed in [Nelsen, R.B., 2006. An Introduction to Copulas, Second ed. In: Springer Series], which implies not only positive dependence, but dependence increasing with age.     

An additive stochastic model of mortality rates: An application to longevity risk in reserve evaluation

The paper proposes an additive continuous-time stochastic mortality model which revises that (B&H model) of Ballotta and Haberman (2006). The structure of the B&H model implies that the future hazard rate is proportional to the stochastic component, thus inducing two questionable features. First, in the B&H model, the uncertainty of the future hazard rate will be enlarged as the base hazard rate increases. However, an increase in the base hazard rate may not cause a dramatic increase suggested by the exponential component of B&H (2006). Second, in the B&H model, the uncertainty of the future hazard rate will be larger in the group which is older and will be greatly augmented by the interaction of age and time. But the uncertainty of the future hazard rate may not increase with an increase in age. The problems can be resolved by our additive structure which is the sum of a deterministic estimator and a stochastic component. Since using the additive structure will contribute to the fact that the stochastic component is independent of age and the base hazard rate, in our model the uncertainty of the future hazard rate will not be affected by an increase in age or in the base hazard rate. We further demonstrate an application of our model by calculating reserves of longevity risks for pure endowments and various common annuity products in the UK. We also compare our results with those of the B&H model.     

Explaining young mortality

Stochastic modeling of mortality rates focuses on fitting linear models to logarithmically adjusted mortality data from the middle or late ages. Whilst this modeling enables insurers to project mortality rates and hence price mortality products it does not provide good fit for younger aged mortality. Mortality rates below the early 20’s are important to model as they give an insight into estimates of the cohort effect for more recent years of birth. It is also important given the cumulative nature of life expectancy to be able to forecast mortality improvements at all ages. When we attempt to fit existing models to a wider age range, 5-89, rather than 20-89 or 50-89, their weaknesses are revealed as the results are not satisfactory. The linear innovations in existing models are not flexible enough to capture the non-linear profile of mortality rates that we see at the lower ages. In this paper, we modify an existing 4 factor model of mortality to enable better fitting to a wider age range, and using data from seven developed countries our empirical results show that the proposed model has a better fit to the actual data, is robust, and has good forecasting ability.     

One-year Value-at-Risk for longevity and mortality

Upcoming new regulation on regulatory required solvency capital for insurers will be predominantly based on a one-year Value-at-Risk measure. This measure aims at covering the risk of the variation in the projection year as well as the risk of changes in the best estimate projection for future years. This paper addresses the issue how to determine this Value-at-Risk for longevity and mortality risk. Naturally, this requires stochastic mortality rates. In the past decennium, a vast literature on stochastic mortality models has been developed. However, very few of them are suitable for determining the one-year Value-at-Risk. This requires a model for mortality trends instead of mortality rates. Therefore, we will introduce a stochastic mortality trend model that fits this purpose. The model is transparent, easy to interpret and based on well known concepts in stochastic mortality modeling. Additionally, we introduce an approximation method based on duration and convexity concepts to apply the stochastic mortality rates to specific insurance portfolios.     

The determinants of mortality heterogeneity and implications for pricing annuities

Standard annuities are offered at one price to all individuals of the same age and gender. Individual mortality heterogeneity exposes insurers to adverse selection since only relatively healthy lives are expected to purchase annuities. As a result standard annuities are priced assuming above-average longevity, making them expensive for many individuals. In contrast underwritten annuity prices reflect individual risk factors based on underwriting information, as well as age and gender. While underwriting reduces heterogeneity, mortality risk still varies within each risk class due to unobservable individual risk factors, referred to as frailty. This paper quantifies the impact of heterogeneity due to underwriting factors and frailty on annuity values. Heterogeneity is quantified by fitting Generalized Linear Mixed Models to longitudinal data for a large sample of US males. The results show that heterogeneity remains after underwriting and that frailty significantly impacts the fair value of both standard and underwritten annuities. We develop a method to adjust annuity prices to allow for frailty.     

Long-tail longitudinal modeling of insurance company expenses

The insurance industry is known to have high operating expenses in the financial services sector. Insurers, investors and regulators are interested in models to understand the behavior of expenses. However, the current practice ignores skewness, occasional negative values as well as their temporal dependence.Addressing these three features, this paper develops a longitudinal model of insurance company expenses that can be used for prediction, to identify unusual behavior, and to measure firm efficiency. Specifically, we use a three-parameter asymmetric Laplace density for the marginal distribution of insurers’ expenses in each year. Copula functions are employed to accommodate their temporal dependence. As a function of explanatory variables, the location parameter allows us to analyze an insurer’s expenses in light of the firm’s characteristics. Our model can be interpreted as a longitudinal quantile regression.The analysis is performed using property-casualty insurance company data from the National Association of Insurance Commissioners of years 2001-2006. Due to the long-tailed nature of insurers’ expenses, two alternative approaches are proposed to improve the performance of the longitudinal quantile regression model: rescaling and transformation. Predictive densities are derived that allow one to compare the predictions for individual insurers in a hold-out-sample. Both predictive models are shown to be reasonable with the rescaling method outperforming the transformation method. Compared with standard longitudinal models, our model is shown to be superior in identifying insurers’ unusual behavior.     

On stochastic mortality modeling

In the last decennium a vast literature on stochastic mortality models has been developed. All well-known models have nice features but also disadvantages. In this paper a stochastic mortality model is proposed that aims at combining the nice features from the existing models, while eliminating the disadvantages. More specifically, the model fits historical data very well, is applicable to a full age range, captures the cohort effect, has a non-trivial (but not too complex) correlation structure and has no robustness problems, while the structure of the model remains relatively simple. Also, the paper describes how to incorporate parameter uncertainty in the model. Furthermore, a risk neutral version of the model is given, that can be used for pricing.     

On the optimal design of insurance contracts with guarantees

The paper analyzes insurance contracts where the benefits of the insured depend on the performance of an investment strategy and which guarantee a certain interest rate on the contributions made by the insured. The insured has to decide simultaneously on the investment strategy and the guarantee scheme. For a CRRA insured and in a BS economy, the optimal combination is given by a constant mix strategy and the contribution guarantee scheme. In case the insured has a subsistence level, the CPPI strategy turns out to be optimal for arbitrary schemes. We illustrate our results by numerical examples and analyze the utility losses of a CRRA insured due to the use of a suboptimal combination of investment strategy and guarantee scheme.     

Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio

We develop a pricing rule for life insurance under stochastic mortality in an incomplete market by assuming that the insurance company requires compensation for its risk in the form of a pre-specified instantaneous Sharpe ratio. Our valuation formula satisfies a number of desirable properties, many of which it shares with the standard deviation premium principle. The major result of the paper is that the price per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting price as an expectation with respect to an equivalent martingale measure. Via this representation, one can interpret the instantaneous Sharpe ratio as a market price of mortality risk. Another important result is that if the hazard rate is stochastic, then the risk-adjusted premium is greater than the net premium, even as the number of contracts approaches infinity. Thus, the price reflects the fact that systematic mortality risk cannot be eliminated by selling more life insurance policies. We present a numerical example to illustrate our results, along with the corresponding algorithms.     

On optimal investment in a reinsurance context with a point process market model

We study an insurance model where the risk can be controlled by reinsurance and investment in the financial market. We consider a finite planning horizon where the timing of the events, namely the arrivals of a claim and the change of the price of the underlying asset(s), corresponds to a Poisson point process. The objective is the maximization of the expected total utility and this leads to a nonstandard stochastic control problem with a possibly unbounded number of discrete random time points over the given finite planning horizon. Exploiting the contraction property of an appropriate dynamic programming operator, we obtain a value-iteration type algorithm to compute the optimal value and strategy and derive its speed of convergence. Following Schäl (2004) we consider also the specific case of exponential utility functions whereby negative values of the risk process are penalized, thus combining features of ruin minimization and utility maximization. For this case we are able to derive an explicit solution. Results of numerical computations are also reported.     

Indifference pricing of pure endowments and life annuities under stochastic hazard and interest rates

We study indifference pricing of mortality contingent claims in a fully stochastic model. We assume both stochastic interest rates and stochastic hazard rates governing the population mortality. In this setting we compute the indifference price charged by an insurer that uses exponential utility and sells k contingent claims to k independent but homogeneous individuals. Throughout we focus on the examples of pure endowments and temporary life annuities. We begin with a continuous-time model where we derive the linear pdes satisfied by the indifference prices and carry out extensive comparative statics. In particular, we show that the price-per-risk grows as more contracts are sold. We then also provide a more flexible discrete-time analog that permits general hazard rate dynamics. In the latter case we construct a simulation-based algorithm for pricing general mortality-contingent claims and illustrate with a numerical example.     

Longevity risk in portfolios of pension annuities

We analyze the importance of longevity risk for the solvency of portfolios of pension annuities. We distinguish two types of mortality risk. Micro-longevity risk quantifies the risk related to uncertainty in the time of death if survival probabilities are known with certainty, while macro-longevity risk is due to uncertain future survival probabilities. We use a generalized two-factor Lee-Carter mortality model to produce forecasts of future mortality rates, and to assess the relative importance of micro- and macro-longevity risk for funding ratio uncertainty. The results show that if financial market risk is fully hedged so that uncertainty in future lifetime is the only source of uncertainty, pension funds are exposed to a substantial amount of risk. Systematic and non-systematic deviations from expected survival imply that, depending on the size of the portfolio, buffers that reduce the probability of underfunding to 2.5% at a 5-year horizon have to be of the order of magnitude of 7% to 8% of the initial value of the liabilities.     

On the group of modular units and the ideal class group

J.-M. Kim, S. Bae and I.-S. Lee showed that there exists an isomorphism between the p-primary part of the ideal class group and p-primary part of the unit group modulo cyclotomic unit group in Q⁺(ζpn) for all sufficiently large n under some conditions. In the present paper, we shall give an analogue of their result for modular units.     

Ruin probability in the presence of risky investments

We consider an insurance company in the case when the premium rate is a bounded non-negative random function _ct$c_t$ and the capital of the insurance company is invested in a risky asset whose price follows a geometric Brownian motion with mean return a   and volatility σ>0$\sigma >0$. If β?2a/σ²-1>0$\beta ?2a/{\sigma }^2-1>0$ we find exact the asymptotic upper and lower bounds for the ruin probability Ψ(u)$\Psi (u)$ as the initial endowment u   tends to infinity, i.e. we show that _C*u^-β?Ψ(u)?C^*u^-β$C_{*}{u}^{-\beta }?\Psi (u)?C^{*}{u}^{-\beta }$ for sufficiently large u  . Moreover if _ct=c^*e^γt$c_t=c^{*}{\mathrm{e}}^{\gamma t}$ with γ?0$\gamma ?0$ we find the exact asymptotics of the ruin probability, namely Ψ(u)∼u^-β$\Psi (u)\sim u^{-\beta }$. If β?0$\beta ?0$, we show that Ψ(u)=1$\Psi (u)=1$ for any u?0$u?0$.     

Following the rules: Integrating asset allocation and annuitization in retirement portfolios

Financial advisers have developed standardized payout strategies to help Baby Boomers manage their money in their golden years. Prominent among these are phased withdrawal plans offered by mutual funds including the “self-annuitization” or default rules encouraged under US tax law, and fixed payout annuities offered by insurers. Using a utility-based framework, and taking account of stochastic capital markets and uncertain lifetimes, we first evaluate these rules on a stand-alone basis for a wide range of risk aversion. Next, we permit the consumer to integrate these standardized payout strategies at retirement and compare the results. We show that integrated strategies can enhance retirees’ well-being by 25%-50% for low/moderate levels of risk aversion when compared to full annuitization at retirement. Finally, we examine how welfare changes if the consumer is permitted to switch to a fixed annuity at an optimal point after retirement. This affords the retiree the chance to benefit from the equity premium when younger, and exploit the mortality credit in later life. For moderately risk-averse retirees, the optimal switching age lies between 80 and 85.     

Non-archimedean analytic curves in the complements of hypersurface divisors

We study the degeneration dimension of non-archimedean analytic maps into the complement of hypersurface divisors of smooth projective varieties. We also show that there exist no non-archimedean analytic maps into where Di, 1?i?n, are hypersurfaces of degree at least 2 in general position and intersecting transversally. Moreover, we prove that there exist no non-archimedean analytic maps into when D₁, D₂ are generic plane curves with degD₁+degD₂?4.     

Optimal investment strategy to minimize the ruin probability of an insurance company under borrowing constraints

We consider that the surplus of an insurance company follows a Cramér-Lundberg process. The management has the possibility of investing part of the surplus in a risky asset. We consider that the risky asset is a stock whose price process is a geometric Brownian motion. Our aim is to find a dynamic choice of the investment policy which minimizes the ruin probability of the company. We impose that the ratio between the amount invested in the risky asset and the surplus should be smaller than a given positive bound a. For instance the case a=1 means that the management cannot borrow money to buy stocks.[Hipp, C., Plum, M., 2000. Optimal investment for insurers. Insurance: Mathematics and Economics 27, 215-228] and [Schmidli, H., 2002. On minimizing the ruin probability by investment and reinsurance. Ann. Appl. Probab. 12, 890-907] solved this problem without borrowing constraints. They found that the ratio between the amount invested in the risky asset and the surplus goes to infinity as the surplus approaches zero, so the optimal strategies of the constrained and unconstrained problems never coincide.We characterize the optimal value function as the classical solution of the associated Hamilton-Jacobi-Bellman equation. This equation is a second-order non-linear integro-differential equation. We obtain numerical solutions for some claim-size distributions and compare our results with those of the unconstrained case.     

Pension fund investments and the valuation of liabilities under conditional indexation

This paper reviews the investment policy of collective pension plans. We focus on funds with a collective Defined Contribution character. We suggest two reasons to invest in equities: the lack of a well-developed market in index-linked bonds, and deliberate deviations from the Defined Benefit nature of the plan. Furthermore, this paper assesses the value of limited or conditional indexation options found in many plans.     

A utility-based comparison of pension funds and life insurance companies under regulatory constraints

This paper compares two different types of annuity providers, i.e. defined benefit pension funds and life insurance companies. One of the key differences is that the residual risk in pension funds is collectively borne by the beneficiaries and the sponsor’s shareholders while in the case of life insurers it is borne by the external shareholders. First, this paper employs a contingent claim approach to evaluate the risk return tradeoff for annuitants. For that, we take into account the differences in contract specifications and in regulatory regimes. Second, a welfare analysis is conducted to examine whether a consumer with power utility experiences utility gains if she chooses a defined benefit plan or a life annuity contract over a defined contribution plan. We demonstrate that regulation can be designed to support a level playing field amongst different financial institutions.