Bounds for Ruin Probabilities in the Presence of Large Claims and their Comparison

Upper and lower bounds of ruin probabilities for the S. Andersen model with large claims are proposed. The bounds are stated in terms of the corresponding ladder height distribution and have a reasonable accuracy, which is illustrated by numerical examples. Comparison with other known bounds is given.     

Two-Sided Bounds for Ruin Probabilities when the Adjustment Coefficient does not Exist

In this paper, we derive two-sided bounds for the ruin probability in the compound Poisson risk model when the adjustment coefficient of the individual claim size distribution does not exist. These bounds also apply directly to the tails of compound geometric distributions. The upper bound is tighter than that of Dickson (1994). The corresponding lower bound, which holds under the same conditions, is tighter than that of De Vylder and Goovaerts (1984). Even when the adjustment coefficient exists, the upper bound is, in some cases, tighter than Lundberg's bound. These bounds are applicable for any positive distribution function with a finite mean. Examples are given and numerical comparisons with asymptotic formulae for the ruin probability are also considered.     

Bounds for Actuarial Present Values Under the Fractional Independence Assumption

We consider the fractional independence (FI) survival model, studied by Willmot (1997), for which the curtate future lifetime and the fractional part of it satisfy the statistical independence assumption, called the fractional independence assumption.

The ordering of risks of the FI survival model is analyzed, and its consequences for the evaluation of actuarial present values in life insurance is discussed. Our main fractional reduction (FR) theorem states that two FI future lifetime random variables with identical distributed curtate future lifetime are stochastically ordered (stop-loss ordered) if, and only if, their fractional parts are stochastically ordered (stop-loss ordered).

The well-known properties of these stochastic orders allow to find lower and upper bounds for different types of actuarial present values, for example when the random payoff functions of the considered continuous life insurances are convex (concave), or decreasing (increasing), or convex not decreasing (concave not increasing) in the future lifetime as argument. These bounds are obtained under the assumption that some information concerning the moments of the fractional part is given. A distinction is made according to whether the fractional remaining lifetime has a fixed mean or a fixed mean and variance. In the former case, simple unique optimal bounds are obtained in case of a convex (concave) present value function.

The obtained results are illustrated at the most important life insurance quantities in a continuous random environment, which include bounds for net single premiums, net level annual premiums and prospective net reserves.     

Application of Risk Theory to Interpretation of Stochastic Cash-Flow-Testing Results

Abstract

In this paper we consider the Sparre Andersen insurance risk model. Three cases are discussed: the ordinary renewal risk process, stationary renewal risk process, and s-delayed renewal risk process. In the first part of the paper we study the joint distribution of surplus immediately before and at ruin under the renewal insurance risk model. By constructing an exponential martingale, we obtain Lundberg-type upper bounds for the joint distribution. Consequently we obtain bounds for the distribution of the deficit at ruin and ruin probability. In the second part of the paper, we consider the special case of phase-type claims and rederive the closed-form expression for the distribution of the severity of ruin, obtained by Drekic et al. (2003, 2004). Finally, we present some numerical results to illustrate the tightness of the bounds obtained in this paper.     

Flexible Spending Accounts as Insurance

We model flexible spending accounts (FSAs) as a special type of insurance policy. We prove the following results given losses drawn from a continuous distribution: (1) the optimal election amount, F^*, is increasing in the consumer's level of risk aversion; (2) F^* is increasing in the level of the maximum loss; If utility is decreasing in absolute risk aversion (DARA), then F^* is (3) decreasing in income and (4) increasing in the marginal tax rate.     

Two-Sided Bounds for Tails of Compound Negative Binomial Distributions in the Exponential and Heavy-Tailed Cases

This paper derives two-sided bounds for tails of compound negative binomial distributions, both in the exponential and heavy-tailed cases. Two approaches are employed to derive the two-sided bounds in the case of exponential tails. One is the convolution technique, as in Willmot & Lin (1997). The other is based on an identity of compound negative binomial distributions; they can be represented as a compound Poisson distribution with a compound logarithmic distribution as the underlying claims distribution. This connection between the compound negative binomial, Poisson and logarithmic distributions results in two-sided bounds for the tails of the compound negative binomial distribution, which also generalize and improve a result of Willmot & Lin (1997). For the heavy-tailed case, we use the method developed by Cai & Garrido (1999b). In addition, we give two-sided bounds for stop-loss premiums of compound negative binomial distributions. Furthermore, we derive bounds for the stop-loss premiums of general compound distributions among the classes of HNBUE and HNWUE.     

On The Expected Discounted Penalty function for Lévy Risk Processes

Abstract

Dufresne et al. (1991) introduced a general risk model defined as the limit of compound Poisson processes. Such a model is either a compound Poisson process itself or a process with an infinite number of small jumps. Later, in a series of now classical papers, the joint distribution of the time of ruin, the surplus before ruin, and the deficit at ruin was studied (Gerber and Shiu 1997, 1998a, 1998b; Gerber and Landry 1998). These works use the classical and the perturbed risk models and hint that the results can be extended to gamma and inverse Gaussian risk processes.

In this paper we work out this extension to a generalized risk model driven by a nondecreasing Lévy process. Unlike the classical case that models the individual claim size distribution and obtains from it the aggregate claims distribution, here the aggregate claims distribution is known in closed form. It is simply the one-dimensional distribution of a subordinator. Embedded in this wide family of risk models we find the gamma, inverse Gaussian, and generalized inverse Gaussian processes. Expressions for the Gerber-Shiu function are given in some of these special cases, and numerical illustrations are provided.     

Insurance contracts portfolios with heterogenous parametric life distributions

In this paper we consider two portfolios: one of m endowment insurance contracts and one of m whole life insurance contracts. We introduce the majorization order, Schur functions, and parametric families of distribution functions. We assume that the owners of the portfolios are exposed to different members of a known parametric family of distributions and study the effect of this stochastic heterogeneity on the premiums and death benefits of the insurance contracts. We show that the premiums paid in both contracts are Schur concave and that the death benefit awarded in the whole life contract is Schur convex. We provide upper and lower bounds for the premiums and for the death benefit, and compute the bounds for four parametric families of distribution functions used frequently in the Actuarial Sciences.     

Potential measures and expected present value of operating costs until ruin in renewal risk models with general interclaim times

In this paper, a Sparre Andersen risk process with arbitrary interclaim time distribution is considered. We analyze various ruin-related quantities in relation to the expected present value of total operating costs until ruin, which was first proposed by Cai et al. [(2009a). On the expectation of total discounted operating costs up to default and its applications. Advances in Applied Probability 41(2), 495–522] in the piecewise-deterministic compound Poisson risk model. The analysis in this paper is applicable to a wide range of quantities including (i) the insurer's expected total discounted utility until ruin; and (ii) the expected discounted aggregate claim amounts until ruin. On one hand, when claims belong to the class of combinations of exponentials, explicit results are obtained using the ruin theoretic approach of conditioning on the first drop via discounted densities (e.g. Willmot [(2007). On the discounted penalty function in the renewal risk model with general interclaim times. Insurance: Mathematics and Economics 41(1), 17–31]). On the other hand, without any distributional assumption on the claims, we also show that the expected present value of total operating costs until ruin can be expressed in terms of some potential measures, which are common tools in the literature of Lévy processes (e.g. Kyprianou [(2014). Fluctuations of L'evy processes with applications: introductory lectures, 2nd ed. Berlin Heidelberg: Springer-Verlag]). These potential measures are identified in terms of the discounted distributions of ascending and descending ladder heights. We shall demonstrate how the formulas resulting from the two seemingly different methods can be reconciled. The cases of (i) stationary renewal risk model and (ii) surplus-dependent premium are briefly discussed as well. Some interesting invariance properties in the former model are shown to hold true, extending a well-known ruin probability result in the literature. Numerical illustrations concerning the expected total discounted utility until ruin are also provided.     

Randomized observation periods for the compound Poisson risk model: the discounted penalty function

In the framework of collective risk theory, we consider a compound Poisson risk model for the surplus process where the process (and hence ruin) can only be observed at random observation times. For Erlang(n) distributed inter-observation times, explicit expressions for the discounted penalty function at ruin are derived. The resulting model contains both the usual continuous-time and the discrete-time risk model as limiting cases, and can be used as an effective approximation scheme for the latter. Numerical examples are given that illustrate the effect of random observation times on various ruin-related quantities.     

Bounds on the estimation error in the chain ladder method

Buchwalder et al. (2006) have illustrated that there are different approaches for the derivation of an estimate for the parameter estimation error in the distribution-free chain ladder reserving method. In this paper, we demonstrate that these approaches provide estimates that are close to each other for typical parameters. This is carried out by proving upper and lower bounds.     

Option Pricing Bounds in Discrete Time

Upper and lower bounds are derived for call options traded at discrete intervals. These bounds are independent of assumptions on the stock price distribution other than a restriction satisfied by the stock being “non-negative beta.” The development of the bounds relies on the single-price law and arbitrage arguments. Both single-period and multiperiod results are produced, and put option bounds follow by extension. The bounds exist as equilibrium values given a consensus on stock price distribution; they are also valid for empirical studies, being adjustable for dividends and commissions.     

Studies in risk theory with numerical illustrations concerning distribution functions and stop loss premiums. Part I

Abstract

The aim of this paper is to analyse two functions that are of general interest in the collective risk theory, namely F, the distribution function of the total amount of claims, and II, the Stop Loss premium. Section 2 presents certain basic formulae. Sections 17-18 present five claim distributions. Corresponding to these claim distributions, the functions F and II were calculated under various assumptions as to the distribution of the number of claims. These calculations were performed on an electronic computer and the numerical method used for this purpose is presented in sections 9, 19 and 20 under the name of the C-method which method has the advantage of furnishing upper and lower limits of the quantities under estimation. The means of these limits, in the following regarded as the “exact” results, are given in Tables 4-20. Sections 11-16 present certain approximation methods. The N-method of section 11 is an Edgeworth expansion, while the G-method given in section 12 is an approximation by a Pearson type III curve. The methods presented in sections 13-16, and denoted AI-A4, are all applications and modifications of the Esscher method. These approximation methods have been applied for the calculation of F and II in the cases mentioned above in which “exact” results were obtained. The results are given in Tables 4-20. The object of this investigation was to obtain information as to the precision of the approximation methods in question, and to compare their relative merits. These results arc discussed in sections 21-24.     

Bounds for the sum of dependent risks and worst Value-at-Risk with monotone marginal densities

In quantitative risk management, it is important and challenging to find sharp bounds for the distribution of the sum of dependent risks with given marginal distributions, but an unspecified dependence structure. These bounds are directly related to the problem of obtaining the worst Value-at-Risk of the total risk. Using the idea of complete mixability, we provide a new lower bound for any given marginal distributions and give a necessary and sufficient condition for the sharpness of this new bound. For the sum of dependent risks with an identical distribution, which has either a monotone density or a tail-monotone density, the explicit values of the worst Value-at-Risk and bounds on the distribution of the total risk are obtained. Some examples are given to illustrate the new results.     

Stationarity aspects of the sparre andersen risk process and the corresponding ruin probabilitles

Abstract

The Sparre Andersen risk model assumes that the interclaim times (also the time between the origin and the first claim epoch is considered as an interclaim time) and the amounts of claim are independent random variables such that the interclaim times have the common distribution function K(t), t|>/ 0, K(O)= 0 and the amounts of claim have the common distribution function P(y), - ∞ < y < ∞. Although the Sparre Andersen risk process is not a process with strictly stationary increments in continuous time it is asymptotically so if K(t) is not a lattice distribution. That is an immediate consequence of known properties of renewal processes. Another also immediate consequence of such properties is the fact that if we assume that the time between the origin and the first claim epoch has not K(t) but as its distribution function (k^b1 denotes the mean of K(t)) then the so modified Sparre Andersen process has stationary increments (this works even if K(t) is a lattice distribution).

In the present paper some consequences of the above-mentioned stationarity properties are given for the corresponding ruin probabilities in the case when the gross risk premium is positive.     

MEAN GINI CAPITAL ASSET PRICING MODEL: SOME EMPIRICAL EVIDENCE

The capital asset pricing model of Sharpe (1964) and Lintner (1965) provides a valid approach to portfolio selection if either the distribution of asset returns is jointly normal or the investor's preference function is quadratic. Various authors have questioned the validity of these assumptions, and Roll (1977) raises the question whether the traditional CAPM can be tested. An alternative Capital Asset Pricing Model has been proposed by Shalit and Yitzhaki (1984). In this model the extended mean Gini coefficient is used to measure risk. As little research has been conducted on this model, this paper estimates systematic risk as derived from the extended mean Gini model for a sample of Australian companies and compares the empirical security market line with the predicted extended mean Gini security market line.     

A ruin model with a resampled environment

ABSTRACT

This paper considers a Cramér–Lundberg risk setting, where the components of the underlying model change over time. We allow the more general setting of the cumulative claim process being modeled as a spectrally positive Lévy process. We provide an intuitively appealing mechanism to create such parameter uncertainty: at Poisson epochs, we resample the model components from a finite number of d settings. It results in a setup that is particularly suited to describe situations in which the risk reserve dynamics are affected by external processes. We extend the classical Cramér–Lundberg approximation (asymptotically characterizing the all-time ruin probability in a light-tailed setting) to this more general setup. In addition, for the situation that the driving Lévy processes are sums of Brownian motions and compound Poisson processes, we find an explicit uniform bound on the ruin probability. In passing we propose an importance-sampling algorithm facilitating efficient estimation, and prove it has bounded relative error. In a series of numerical experiments we assess the accuracy of the asymptotics and bounds, and illustrate that neglecting the resampling can lead to substantial underestimation of the risk.     

Portfolio optimization under a generalized hyperbolic skewed t distribution and exponential utility

In this paper, we show that if asset returns follow a generalized hyperbolic skewed t distribution, the investor has an exponential utility function and a riskless asset is available, the optimal portfolio weights can be found either in closed form or using a successive approximation scheme. We also derive lower bounds for the certainty equivalent return generated by the optimal portfolios. Finally, we present a study of the performance of mean–variance analysis and Taylor’s series expected utility expansion (up to the fourth moment) to compute optimal portfolios in this framework.     

On the relationship between classical chain ladder and granular reserving

We connect classical chain ladder to granular reserving. This is done by defining explicitly how the classical run-off triangles are generated from individual iid observations in continuous time. One important result is that the development factors have a one to one correspondence to a histogram estimator of a hazard running in reversed development time. A second result is that chain ladder has a systematic bias if the row effect has not the same distribution when conditioned on any of the aggregated periods. This means that the chain ladder assumptions on one level of aggregation, say yearly, are different from the chain ladder assumptions when aggregated in quarters and the optimal level of aggregation is a classical bias variance trade-off depending on the data-set. We introduce smooth development factors arising from non-parametric hazard kernel smoother improving the estimation significantly.