Stop-loss Premiums Research Articles

Is employment group insurance financing of expensive gene therapies threatened in the United States?

Despite the potential of gene therapy to transform the lives of patients with rare genetic diseases, serious concern has been raised about the financing of the high up-front costs for such treatments and about the ability of the employer-sponsored insurance system in the United States, particularly in small firms, to pay for discoveries of this type. In this paper, we provide a conceptual framework and empirical evidence to support the proposition that, at present, private group insurance financing of cost-effective gene therapies is not only feasible and competitively necessary in the labor market for employers, regardless of group size, but also that, currently, the number of US workers in small firms who might be stressed by very high-priced claims is a tiny fraction of the group market for genetic treatments. The current system of employer-paid self-insurance supplemented by stop-loss coverage should be able to facilitate the use of new cost-effective gene therapies. Other alternative methods of financing that have been proposed may not be urgently needed. There are, however, some concerns about the long-term resilience of this system if stop-loss premiums continue to have high growth.

Refinements of bounds for tails of compound distributions and ruin probabilities

In this paper we derive lower bounds for right-tails of compound geometric distributions and ruin probabilities in the classical compound Poisson risk model in the heavy-tailed cases using the truncated Lundberg condition, which improve all the corresponding known lower bounds. Some upper bounds are also derived. Examples are given and numerical comparison for ruin probabilities when the adjustment coefficient does not exist are also considered, illustrating the effectiveness of the proposed new bounds. In addition, several bounds for tails of negative binomial distributions are obtained in terms of the tail of compound geometric distributions as well as bounds in the heavy-tailed cases. Also, some bounds for the stop-loss premium associated with compound negative binomial distributions are given. Using Chernoff's upper bounds we derive two-sided bounds for tails of compound Poisson distributions. Finally, two-sided bounds are given for tails of compound logarithmic distributions in the heavy-tailed cases.

Stop-loss protection for a large P2P insurance pool

This paper considers a peer-to-peer (P2P) insurance scheme where the higher layer is transferred to a (re-)insurer and retained losses are distributed among participants according to the conditional mean risk sharing rule proposed by Denuit and Dhaene (2012). The global retention level of the pool of participants grows proportionally with their number. We study the asymptotic behavior of the individual retention levels, as well as individual cash-backs and stop-loss premiums, as the number of participants increases. The probability that the total loss hits the upper layer protected by the stop-loss treaty is also considered. The results depend on the proportional rate of increase of the global retention level with the number of participants, as well as on the existence of the Esscher transform of the losses brought to the pool.

Convex bound approximations for sums of random variables under multivariate log-generalized hyperbolic distribution and asymptotic equivalences

We propose convex bound approximations for the sum of log-multivariate generalized hyperbolic random variables. We derive explicit formulas for the distributions of convex bounds and for the frequently-used risk measures such as Value-at-Risk, Conditional Tail Expectation and stop-loss premium. We present numerical results showing that such approximations are not only accurate but also robust. Moreover, we further prove that there exist asymptotic equivalences between the sum and its convex bounds. To further illustrate the potentials of the convex bound approximations, we provide an application to capital allocation. We show that our formulas can be easily applied to precisely approximate capital allocation rule based on the conditional tail expectation.

Lower and upper bounds for survival functions of the smallest and largest claim amounts in layer coverages

We consider n risks insured by a layer coverage with deductibles and limits given by respectively. We investigate the optimal allocation of insurance layers from the viewpoint of the insurer. We derive lower and upper bounds for the survival function of the smallest and largest claim amounts using the first stochastic dominance order. We find that assigning a small deductible and a large limit to large risks increases (decreases) stop-loss premiums of the largest (smallest) claim amounts.

Poisson-Gamma mixture processes and applications to premium calculation

In the paper, Poisson-Gamma mixture process is first brought forward, which is dynamically expanded from the well-known Poisson-Gamma mixture model. Some properties on Poisson-Gamma mixture process are presented, including the distribution of increment, Markov property, infinitesimal generator, joint density function of jump/waiting times, and the limit distribution of compound Poisson-Gamma mixture process, etc., which provide a thorough grounding in application of Poisson-Gamma mixture process. At last, some premium calculation principles are presented to show the application of Poisson-Gamma mixture process, which include expected value premium, stop-loss premium, mean-variance premium, and exponential premium.

Multi Target Risk Measurement & Portfolio Optimization

Multiple investment targets naturally arise in portfolio management when investments are subject to performance benchmarks such as a stock/bond reference portfolio and absolute drawdown limits. Additional layers of investment targets like inflation outperformance or liability coverage ratios further complicate risk management and portfolio optimization. This paper illustrates a comprehensive approach for managing a portfolio against multiple random or deterministic investment targets concurrently in a single period setting. The approach expands from the well-established body of academic and practical research on downside measures, and in particular the mean-below target (MBT) measure, also known as target shortfall (TS), first lower partial moment (LPM1), put premium (PP) risk measure or mean excess loss (MEL) and stop loss premium (SLP) in actuarial sciences. Despite embedding multiple targets the new approach reduces the mathematical complexity to a single dimension allowing to apply well-known results. Even though targets are co-dependent, the multi-target MBT measure allows for explicit decomposition into marginal single target MBT measures. Besides exploring the properties of such risk measure the paper covers all aspects of performance measurement, cost of capital allocation as well as portfolio optimization with multiple targets. Here, the portfolio optimization of the multi target MBT measure remains of linear programming complexity. The resulting comprehensive portfolio management framework is appealing for its simplicity in application, implementation and communication.

Orthogonal polynomial expansions to evaluate stop-loss premiums

A numerical method is proposed to evaluate the survival function of a compound distribution and the stop-loss premiums associated with a non-proportional global reinsurance treaty. The method relies on a representation of the probability density function in terms of Laguerre polynomials and the gamma density. We compare the method against a well established Laplace transform inversion technique at the end of the paper.

Risk aggregation with dependence and overdispersion based on the compound Poisson INAR(1) process

This paper considers an extension of the classical discrete time risk model for which the claim numbers are assumed to be temporal dependence and overdispersion. The risk model proposed is based on the first-order integer-valued autoregressive (INAR(1)) process with discrete compound Poisson distributed innovations. The explicit expression for the moment generating function of the discounted aggregate claim amount is derived. Some numerical examples are provided to illustrate the impacts of dependence and overdispersion on related quantities such as the stop-loss premium, the value at risk and the tail value at risk.

Comparisons of aggregate claim numbers and amounts: a study of heterogeneity

ABSTRACTIn this paper, we investigate how the heterogeneity among occurrence probabilities and claim severities affects the aggregate claim numbers and aggregate claim amount for an insurance portfolio. We show that higher heterogeneity (and dependence) among occurrence probabilities results in both smaller aggregate claim numbers and aggregate claim amount in the sense of the mean residual lifetime order. We also prove that as the heterogeneity among the claims increases, the aggregate claim amount increases in the sense of the usual stochastic order when the vector of occurrence probabilities is left tail weakly stochastic arrangement increasing. These theoretical findings are applied to (i) study ordering properties of convolutions of binomial random variables, (ii) provide upper bounds for the mean residual lifetime functions of the aggregate claim numbers and amount, and (iii) compare stop-loss premiums and risk capital of different insurance portfolios.

Renewal sums under mixtures of exponentials

We start with applying two methods to derive formulas of a mixture of exponential process, i.e., a renewal process whose inter-arrival time follows a mixture of exponentials. Further, stochastic order properties are discussed when comparing this process to a Poisson process with the same expectation of inter-arrival times. Based on these properties, formulas and ordering properties are given for the non-discounted compound process as well as the discounted one. Explicit formulas for the density functions are also provided for both cases. Under the discounted compound case, several new results are derived for heavy-tailed distributions. Finally, the Laguerre series approximation is proposed and tested for various common actuarial indices, e.g., VaR, CTE and stop-loss premium.

Concentration inequalities for separately convex functions

We provide new comparison inequalities for separately convex functions of independent random variables. Our method is based on the decomposition in Doob martingale. However, we only impose that the martingale increments are stochastically bounded. For this purpose, building on the results of Bentkus (Lith. Math. J. 48 (2008) 237–255; Lith. Math. J. 48 (2008) 137–157; Bounds for the stop loss premium for unbounded risks under the variance constraints (2010) Preprint), we establish comparison inequalities for random variables stochastically dominated from below and from above. We illustrate our main results by showing how they can be used to derive deviation or moment inequalities for functions which are both separately convex and separately Lipschitz, for weighted empirical distribution functions, for suprema of randomized empirical processes and for chaos of order two.

Some new results on aggregate claim amounts from two heterogeneous Marshall–Olkin extended exponential portfolios

ABSTRACTIn this work, we discuss some stochastic comparisons of two aggregate claim amounts. Applications of our results to the value-at-risk and tail-value-at-risk are also mentioned. It is also shown that the aggregate claim amounts of risks exhibiting a weak form of dependence known as positive cumulative dependence (negatively associated) is larger (smaller) in convex order than the corresponding aggregate claim amounts under the theoretical independence assumption. The obtained results show that the correlations between individual risks increase stop-loss premiums corresponding to aggregate claim amounts. The results established here complete and extend the results of Barmalzan, Payandeh Najafabadi, and Balakrishnan (2017, Theory of Probability and its Applications, to appear).

Stop-Loss Reinsurance Premium for Environmental Impairment Liability Insurance

Stop-Loss Reinsurance Premium for Environmental Impairment Liability Insurance

The multivariate Variance Gamma model: basket option pricing and calibration

In this paper, we propose a methodology for pricing basket options in the multivariate Variance Gamma model introduced in Luciano and Schoutens [Quant. Finance 6(5), 385–402]. The stock prices composing the basket are modelled by time-changed geometric Brownian motions with a common Gamma subordinator. Using the additivity property of comonotonic stop-loss premiums together with Gauss-Laguerre polynomials, we express the basket option price as a linear combination of Black & Scholes prices. Furthermore, our new basket option pricing formula enables us to calibrate the multivariate VG model in a fast way. As an illustration, we show that even in the constrained situation where the pairwise correlations between the Brownian motions are assumed to be equal, the multivariate VG model can closely match the observed Dow Jones index options.

Cramér–Von Mises distance estimation for some positive infinitely divisible parametric families with actuarial applications

Cramér–Von Mises (CVM) inference techniques are developed for some positive flexible infinitely divisible parametric families generalizing the compound Poisson family. These larger families appear to be useful for parametric inference for positive data. The methods are based on inverting the characteristic functions. They are numerically implementable whenever the characteristic function has a closed form. In general, likelihood methods based on density functions are more difficult to implement. CVM methods also lead to model testing, with test statistics asymptotically following a chi-square distribution. The methods are for continuous models, but they can also handle models with a discontinuity point at the origin such as the case of compound Poisson models. Simulation studies seem to suggest that CVM estimators are more efficient than moment estimators for the common range of the compound Poisson gamma family. Actuarial applications include estimation of the stop loss premium, and estimation of the present value of cash flows when interest rates are assumed to be driven by a corresponding Lévy process.

The Multivariate Variance Gamma Model: Basket Option Pricing and Calibration

A basket option is an option whose underlying is a portfolio of individual stock prices. Due to the unknown dependence structure between stocks, basket option pricing relies in general on approximations or numerical methods like Monte Carlo simulation. We propose a methodology for pricing basket options in a multivariate Variance Gamma model. The stock prices composing the basket are then modeled by time changed geometric Brownian motions with a common Gamma subordinator. Using the additivity property of comonotonic stop-loss premiums together with Gauss-Laguerre polynomials, we derive a closed-form expression for the basket option price as a linear combination of Black & Scholes prices. This technique manages to approximate the real basket option price in an accurate way. Furthermore, our new basket option pricing formula enables us to calibrate the multivariate VG model in a fast way provided option quotes on the components and the basket itself are available. As an illustration, we show that the multivariate VG model can closely match the observed Dow Jones index options.

Tail asymptotic expansions for L-statistics

We derive higher-order expansions of L-statistics of independent risks X 1, …,X n under conditions on the underlying distribution function F. The new results are applied to derive the asymptotic expansions of ratios of two kinds of risk measures, stop-loss premium and excess return on capital, respectively. Several examples and a Monte Carlo simulation study show the efficiency of our novel asymptotic expansions.

On mixing, compounding, and tail properties of a class of claim number distributions

The mathematical structure underlying a class of discrete claim count distributions is examined in detail. In particular, the mixed Poisson nature of the class is shown to hold fairly generally. Using some ideas involving complete monotonicity, a discussion is provided on the structure of other class members which are well suited for use in aggregate claims analysis. The ideas are then extended to the analysis of the corresponding discrete tail probabilities, which arise in a variety of contexts including the analysis of the stop-loss premium.

On the Integrated Tail of the Deficit in the Renewal Risk Model

Let G(x, y) be the distribution of the deficit at the time of ruin in the renewal risk model. In this paper, we derive a geometric convolution representation for a function related to the integrated tail of the deficit. This integrated tail is a generalization of the stop loss-premium of the ruin probability, and the proposed convolution is a generalization of the equilibrium distribution of a compound geometric distribution (probability of non-ruin).