Kam C. Yuen

Ruin probabilities for time-correlated claims in the compound binomial model

Abstract In this paper we consider the ruin probability for a risk process with time-correlated claims in the compound binomial model. It is assumed that every main claim will produce a by-claim but the occurrence of the by-claim may be delayed. Recursive formulas for the finite time ruin probabilities are obtained and explicit expressions for ultimate ruin probabilities are given in two special cases.

Sums of Pairwise Quasi-Asymptotically Independent Random Variables with Consistent Variation

This article investigates the tail asymptotic behavior of the sum of pairwise quasi-asymptotically independent random variables with consistently varying tails. We prove that the tail probability of the sum is asymptotically equal to the sum of individual tail probabilities. This matches a feature of subexponential distributions. This result is then extended to weighted sums and random sums.

On a correlated aggregate claims model with Poisson and Erlang risk processes

Abstract In this paper we consider a risk model with two dependent classes of insurance business. In this model the two claim number processes are correlated. Claim occurrences of both classes relate to Poisson and Erlang processes. We derive explicit expressions for the ultimate survival probabilities under the assumed model when the claim sizes are exponentially distributed. We also examine the asymptotic property of the ruin probability for this special risk process with general claim size distributions.

On a Mixture GARCH Time-Series Model

Recently, there has been a lot of interest in modelling real data with a heavy-tailed distribution. A popular candidate is the so-called generalized autoregressive conditional heteroscedastic (GARCH) model. Unfortunately, the tails of GARCH models are not thick enough in some applications. In this paper, we propose a mixture generalized autoregressive conditional heteroscedastic (MGARCH) model. The stationarity conditions and the tail behaviour of the MGARCH model are studied. It is shown that MGARCH models have tails thicker than those of the associated GARCH models. Therefore, the MGARCH models are more capable of capturing the heavy-tailed features in real data. Some real examples illustrate the results.

Optimal dynamic reinsurance with dependent risks: variance premium principle

In this paper, we consider the optimal proportional reinsurance strategy in a risk model with two dependent classes of insurance business, where the two claim number processes are correlated through a common shock component. Under the criterion of maximizing the expected exponential utility with the variance premium principle, we adopt a nonstandard approach to examining the existence and uniqueness of the optimal reinsurance strategy. Using the technique of stochastic control theory, closed-form expressions for the optimal strategy and the value function are derived for the compound Poisson risk model as well as for the Brownian motion risk model. From the numerical examples, we see that the optimal results for the compound Poisson risk model are very different from those for the diffusion model. The former depends not only on the safety loading, time, and the interest rate, but also on the claim size distributions and the claim number processes, while the latter depends only on the safety loading, time, and the interest rate.

A discrete-time risk model with interaction between classes of business

In this paper, a discrete-time risk model with dependent classes of business called the interaction (IR) model is proposed. The model assumes that the number of claims in one class is governed not only by its underlying risk but also by the risks in other classes. For a family of claim-number distributions, the IR model is examined. Numerical studies are carried out to compare the finite-time ruin probabilities of the model to those of other correlated aggregate claims models in the literature. For the infinite-time ruin probabilities, comparisons between these models in terms of their adjustment coefficients are also made.

Estimation in the Constant Elasticity of Variance Model

The constant elasticity of variance (CEV) diffusion process can be used to model heteroscedasticity in returns of common stocks. In this diffusion process, the volatility is a function of the stock price and involves two parameters. Similar to the Black-Scholes analysis, the equilibrium price of a call option can be obtained for the CEV model. The purpose of this paper is to propose a new estimation procedure for the CEV model. A merit of our method is that no constraints are imposed on the elasticity parameter of the model. In addition, frequent adjustments of the parameter estimates are not required. Simulation studies indicate that the proposed method is suitable for practical use. As an illustration, real examples on the Hong Kong stock option market are carried out. Various aspects of the method are also discussed.

The Maximum of Randomly Weighted Sums with Long Tails in Insurance and Finance

In risk theory we often encounter stochastic models containing randomly weighted sums. In these sums, each primary real-valued random variable, interpreted as the net loss during a reference period, is associated with a nonnegative random weight, interpreted as the corresponding stochastic discount factor to the origin. Therefore, a weighted sum of m terms, denoted as , represents the stochastic present value of aggregate net losses during the first m periods. Suppose that the primary random variables are independent of each other with long-tailed distributions and are independent of the random weights. We show conditions on the random weights under which the tail probability of —the maximum of the first n weighted sums—is asymptotically equivalent to that of —the last weighted sum.

On optimality of the barrier strategy for a general Lévy risk process

We consider the optimal dividend problem for the insurance risk process in a general Levy process setting. The objective is to find a strategy which maximizes the expected total discounted dividends until the time of ruin. We give sufficient conditions under which the optimal strategy is of barrier type. In particular, we show that if the Levy density is a completely monotone function, then the optimal dividend strategy is a barrier strategy. This approach was inspired by the work of Avram et al. [F. Avram, Z. Palmowski, M.R. Pistorius, On the optimal dividend problem for a spectrally negative Levy process, The Annals of Applied Probability 17 (2007) 156-180], Loeffen [R. Loeffen, On optimality of the barrier strategy in De Finettis dividend problem for spectrally negative Levy processes, The Annals of Applied Probability 18 (2008) 1669-1680] and Kyprianou et al. [A.E. Kyprianou, V. Rivero, R. Song, Convexity and smoothness of scale functions with applications to De Finettis control problem, Journal of Theoretical Probability 23 (2010) 547-564] in which the same problem was considered under the spectrally negative Levy processes setting.

Exact joint laws associated with spectrally negative Lévy processes and applications to insurance risk theory

We consider the spectrally negative Lévy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the minimum, the maximum, and the duration of negative values. We apply our results to insurance risk theory to find an explicit expression for the generalized expected discounted penalty function in terms of scale functions. Furthermore, a new expression for the generalized Dickson’s formula is provided.