Under new solvency regulations, general insurance companies need to calculate a risk margin to cover possible shortfalls in their liability runoff. A popular approach for the calculation of the risk margin is the so-called cost-of-capital approach. A comprehensive cost-of-capital approach involves the consideration of multiperiod risk measures. Because multiperiod risk measures are rather complex mathematical objects, various proxies are used to estimate this risk margin. Of course, the use of proxies and the study of their quality raises many questions, see IAA position paper [8]. In the present paper we provide a first discourse on multiperiod solvency considerations for a general insurance liability runoff. Within a chain ladder framework, we derive analytic formulas for the risk margin which allow to compare the comprehensive approach to the different proxies used in practice. Moreover, a case study investigates and answers questions raised in [8].

In the aftermath of the 2007-2008 financial crisis, there has been criticism of mathematics and the mathematical models used by the finance industry. We answer these criticisms through a discussion of some of the actuarial models used in the pricing of credit derivatives. As an example, we focus in particular on the Gaussian copula model and its drawbacks. To put this discussion into its proper context, we give a synopsis of the financial crisis and a brief introduction to some of the common credit derivatives and highlight the difficulties in valuing some of them.

We also take a closer look at the risk management issues in part of the insurance industry that came to light during the financial crisis. As a backdrop to this, we recount the events that took place at American International Group during the financial crisis. Finally, through our paper we hope to bring to the attention of a broad actuarial readership some “lessons (to be) learned” or “events not to be forgotten”.

Regulation of insurers' use of genetic information means actuaries are interested in age-at-onset of genetic disorders. Arjas & Haara (1984) suggested marked point processes (MPPs) as useful models for life history data with complex covariates. Age-at-onset distributions (or equivalently, hazard rates) in respect of inherited disorders are often estimated from pedigrees, which are life histories with unusually complex covariates, as well as strong dependencies induced by shared genes. Since Elston (1973) parametric models have often been used, conditioning the likelihood on known genotypes. However, a genotype identii ed by a presymptomatic genetic test is a form of internal covariate (Kalbfleisch & Prentice, 2002). We propose a very general MPP model of a pedigree, including presymptomatic genetic testing, (‘the full model’) and show under what circumstances the partial model leading to Elston's likelihood is valid. In practice, pedigrees are often ascertained retrospectively. Many such events can be modelled by augmenting the natural filtration of the MPP. We show that, except in simple special cases, the partial model is no longer valid, and the resulting likelihoods appear to be intractable. In particular, ascertainment interacts even with independent censoring so that likelihoods no longer factorize. For one simple special case — studies of sibships — we generalise a classical result to age-at-onset data. We conclude that the study of genetic conditions with variable age at onset gains insights from the underlying principles of survival analysis in their modern form, but that great care is needed in translating epidemiological studies into actuarial models.

We tackle problems that appear in the practical application of the Mack method for the estimation of reserving risk and the bootstrapping of ultimate reserve distributions. More specifically, we design a filter for outliers and large jumps, and present a robust version of Mack's variance estimator. A combination of these guarantees a reasonable Mack and bootstrap error even for deficient data. Furthermore, a method is derived that allows us to remove the influence of fluctuations in earning patterns from the reserve risk estimate. It is thereby shown that the relation between underwriting and accident year based loss development patterns is given by a convolution. A numerically stable inversion thereof is obtained by means of a Tikhonov regularization. The reliability of the presented methods is verified with several loss triangles.

In this paper we consider a reinsurance syndicate, assuming that Pareto optimal allocations exist. Under a continuity assumption on preferences, we show that a competitive equilibrium exists and is unique. Our conditions allow for risks that are not bounded, and we show that the most standard models satisfy our set of sufficient conditions, which are thus not restrictive. Our approach is to transform the analysis from an infinite dimensional to a finite dimensional setting.

In recent years, market-consistent valuation approaches have gained an increasing importance for insurance companies. This has triggered an increasing interest among practitioners and academics, and a number of specific studies on such valuation approaches have been published.

In this paper, we present a generic model for the valuation of life insurance contracts and embedded options. Furthermore, we describe various numerical valuation approaches within our generic setup. We particularly focus on contracts containing early exercise features since these present (numerically) challenging valuation problems.

Based on an example of participating life insurance contracts, we illustrate the different approaches and compare their efficiency in a simple and a generalized Black-Scholes setup, respectively. Moreover, we study the impact of the considered early exercise feature on our example contract and analyze the influence of model risk by additionally introducing an exponential Lévy model.

This paper deals with numerical computation of the optimal form of reinsurance from the ceding company point of view, when the cedent seeks to maximize the adjustment coefficient of the retained risk and the reinsurance loading is an increasing function of the variance.

We compare the optimal treaty with the best stop loss policy. The optimal arrangement can provide a significant improvement in the adjustment coefficient when compared to the best stop loss treaty. Further, it is substantially more robust with respect to choice of the retention level than stop-loss treaties.

This study incorporates the survival analysis of unemployment duration into the insurance pricing framework to measure the fairly-priced premium rate for Taiwan's unemployment insurance (UI) program. Our results suggest that the fair premiums range from 0.2041% to 0.2436% under the 1999-2002 scheme and from 0.1388% to 0.1521% under the 2003-2009 scheme for various possible levels of average unemployment duration in Taiwan, and they are all lower than the current UI premium rate, 1%. This result explains in part why there is a persistent surplus in the UI program. The sensitivity analysis results indicate that the fair premium rate decreases with the hazard rate of exiting from unemployment and increases with the probability of entering into unemployment. The effect of the entering probability is found to be larger than that of the exiting probability. We also provide a wide range of systematic risk coefficient (β) values generated from three alternative methods to measure its impact on fair premium rates and find that the effect of β on premium rates is stronger under the 1999-2002 scheme than that of the 2003-2009 scheme.

When analyzing catastrophic risk, traditional measures for evaluating risk, such as the probable maximum loss (PML), value at risk (VaR), tail-VaR, and others, can become practically impossible to obtain analytically in certain types of insurance, such as earthquake, and certain types of reinsurance arrangements, specially non-proportional with reinstatements. Given the available information, it can be very difficult for an insurer to measure its risk exposure. The transfer of risk in this type of insurance is usually done through reinsurance schemes combining diverse types of contracts that can greatly reduce the extreme tail of the cedant's loss distribution. This effect can be assessed mathematically. The PML is dei ned in terms of a very extreme quantile. Also, under standard operating conditions, insurers use several “layers” of non proportional reinsurance that may or may not be combined with some type of proportional reinsurance. The resulting reinsurance structures will then be very complicated to analyze and to evaluate their mitigation or transfer effects analytically, so it may be necessary to use alternative approaches, such as Monte Carlo simulation methods. This is what we do in this paper in order to measure the effect of a complex reinsurance treaty on the risk profile of an insurance company. We compute the pure risk premium, PML as well as a host of results: impact on the insured portfolio, risk transfer effect of reinsurance programs, proportion of times reinsurance is exhausted, percentage of years it was necessary to use the contractual reinstatements, etc. Since the estimators of quantiles are known to be biased, we explore the alternative of using an Extreme Value approach to complement the analysis.

In this paper, we consider various specifications of the general discrete-time risk model in which a serial dependence structure is introduced between the claim numbers for each period. We consider risk models based on compound distributions assuming several examples of discrete variate time series as specific temporal dependence structures: Poisson MA(1) process, Poisson AR(1) process, Markov Bernoulli process and Markov regime-switching process. In these models, we derive expressions for a function that allow us to find the Lundberg coefficient. Specific cases for which an explicit expression can be found for the Lundberg coefficient are also presented. Numerical examples are provided to illustrate different topics discussed in the paper.

This paper aims at providing a mathematical foundation for the terms of the well spread supervisory rule “initial market value of assets must be at least equal to provision plus solvency capital”.

It starts with a risk-adjusted assessment — given by a set of test probabilities — of the future cash-flows coming from a company business plan and attempts to define terms of a supervisory accounting mode.

First, inspired by the idea of “representation” of obligations by “equivalent” assets, we define the supervisory provision (or “liability”) attached to existing obligations. This provision is market consistent according to the mathematical definition by Cheridito, Filipovic and Kupper and satisfies a property of equilibrium between supervision's wish for stress testing and management's possibility for appropriate choice of assets.

The comparison between the initial market price of assets and the supervisory provision defines “solvability” of existing obligations.

In a second step the paper defines a required solvency capital as related to the level of discrepancy between assets and obligations of a company. Solvency of a business plan is defined by requiring as initial market value an additional amount over the one needed for solvability: this is the required solvency capital. A business plan with zero required solvency capital is said to have an optimal replicating asset portfolio.

It is shown that — under a natural additional condition, that of a market prudent set of test probabilities — solvability of an obligation allows for solvency of a related business plan, by choice of the asset portfolio.

The paper emphasizes the distinction between supervisory and market oriented accounting hinted to in the CEIOPS CP 20 consultative paper.

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.

The primary objective of the paper is to explore using reinsurance as a risk management tool for an insurance company. We consider an insurance company whose surplus can be modeled by a Brownian motion with drift and that the surplus can be invested in a risky or riskless asset. Under the above Black-Scholes type framework and using the objective of minimizing the ruin probability of the insurer, we formally establish that the excess-of-loss reinsurance treaty is optimal among the class of plausible reinsurance treaties. We also obtain the optimal level of retention as well as provide an explicit expression of the minimal probability of ruin.

In this paper, we discuss how to define conservative biometric bases in life insurance. The first approach is based on cumulative hazard (or survival probabilities), the second one on the hazard itself, and the third one on the rate of increase of the hazard. The second case has been studied in the literature and the sum-at-risk plays a central role in defining safe-side requirements. The two other cases appear to be new and concepts related to sum-at-risk are defined.

Some extensions to the delayed renewal risk models are considered. In particular, the independence assumption between the interclaim time and the subsequent claim size is relaxed, and the classical Gerber-Shiu penalty function is generalized by incorporating more variables. As a result, general structures regarding various joint densities of ruin related quantities as well as their probabilistic interpretations are provided. The numerical example in case of time-dependent claim sizes is provided, and also the usual delayed model with time-independent claim sizes is discussed including a special case with exponential claim sizes. Furthermore, asymptotic formulas for the associated compound geometric tail for the present model are derived using two alternative methods.

We first present a simple matrix-based recursive formula for calculating the distribution function of compound phase-type random variables. Then we extend the results to the case when the number of claims follows a bivariate matrix negative binomial (BMNB) distribution detailed herein. Further, extending the results in Hipp (2006), we provide speedy recursive formulas for both the univariate and the bivariate models when the claim sizes follow discrete phase-type distributions. Numerical examples are provided.

In this paper, we reexamine the two optimal reinsurance problems studied in Cai et al. (2008), in which the objectives are to find the optimal reinsurance contracts that minimize the value-at-risk (VaR) and the conditional tail expectation (CTE) of the total risk exposure under the expectation premium principle. We provide a simpler and more transparent approach to solve these problems by using intuitive geometric arguments. The usefulness of this approach is further demonstrated by solving the VaR-minimization problem when the expectation premium principle is replaced by Wang's premium principle.

We consider the unit-linked endowment with guarantee and periodic premiums, where at each premium payment date the insurance company invests a certain fraction of the premium into a risky reference portfolio. In the dual random environment of stochastic interest rates with deterministic volatilities and mortality risk, and for a fixed guarantee, simple analytical lower and upper bounds for the fair periodic premium are explicitly derived. We also consider contracts with guaranteed minimum benefits that vary over time and we obtain tight lower and upper bounds for both fair periodic premiums and guaranteed minimum benefits that increase over time. The numerical illustrations of our results reveal that the analytical bounds are very tight. Moreover, the simple, fast and very reliable analytical numerical calculations with controlled accuracy avoid time consuming Monte Carlo calculations and are almost always preferred by practitioners. Some analytical closed-form solutions for one- and two-year maturity dates are also stated.

In this article we propose a bootstrap test for the probability of ruin in the compound Poisson risk process. We adopt the P-value approach, which leads to a more complete assessment of the underlying risk than the probability of ruin alone. We provide second-order accurate P-values for this testing problem and consider both parametric and nonparametric estimators of the individual claim amount distribution. Simulation studies show that the suggested bootstrap P-values are very accurate and outperform their analogues based on the asymptotic normal approximation.

Sensitivity analysis, or so-called ‘stress-testing’, has long been part of the actuarial contribution to pricing, reserving and management of capital levels in both life and non-life assurance. Recent developments in the area of derivatives pricing have seen the application of adjoint methods to the calculation of option price sensitivities including the well-known ‘Greeks’ or partial derivatives of option prices with respect to model parameters. These methods have been the foundation for efficient and simple calculations of a vast number of sensitivities to model parameters in financial mathematics. This methodology has yet to be applied to actuarial problems in insurance or in pensions. In this paper we consider a model for a defined benefit pension scheme and use adjoint methods to illustrate the sensitivity of fund valuation results to key inputs such as mortality rates, interest rates and levels of salary rate inflation. The method of adjoints is illustrated in the paper and numerical results are presented. Efficient calculation of the sensitivity of key valuation results to model inputs is useful information for practising actuaries as it provides guidance as to the relative ultimate importance of various judgments made in the formation of a liability valuation basis.