Bernard Wong

On Optimal Periodic Dividend Strategies in the Dual Model with Diffusion

The dual model with diffusion is appropriate for companies with continuous expenses that are offset by stochastic and irregular gains. Examples include research-based or commission-based companies. In this context, Bayraktar et al. (2013a) show that a dividend barrier strategy is optimal when dividend decisions are made continuously. In practice, however, companies that are capable of issuing dividends make dividend decisions on a periodic (rather than continuous) basis.

Modelling Dependence in Insurance Claims Processes with Lévy Copulas

In this paper we investigate the potential of Levy copulas as a tool for modelling dependence between compound Poisson processes and their applications in insurance. We analyse characteristics regarding the dependence in frequency and dependence in severity allowed by various Levy copula models. Through the

Stochastic Loss Reserving with Dependence: A Flexible Multivariate Tweedie Approach

Stochastic loss reserving with dependence has received increased attention in the last decade. A number of parametric multivariate approaches have been developed to capture dependence between lines of business within an insurers portfolio. Motivated by the richness of the Tweedie family of distributions, we propose a multivariate Tweedie approach to capture cell-wise dependence in loss reserving. This approach provides a transparent introduction of dependence through a common shock structure. In addition, it also has a number of ideal properties, including marginal flexibility, transparency, and tractability including moments that can be obtained in closed form. Theoretical results are illustrated using a simulated data set and a real data set from a property-casualty insurer in the US.

A Note on Realistic Dividends in Actuarial Surplus Models

Because of the profitable nature of risk businesses in the long term, de Finetti suggested that surplus models should allow for cash leakages, as otherwise the surplus would unrealistically grow (on average) to infinity. These leakages were interpreted as ‘dividends’. Subsequent literature on actuarial surplus models with dividend distribution has mainly focussed on dividend strategies that either maximise the expected present value of dividends until ruin or lead to a probability of ruin that is less than one (see Albrecher and Thonhauser, Avanzi for reviews). An increasing number of papers are directly interested in modelling dividend policies that are consistent with actual practice in financial markets. In this short note, we review the corporate finance literature with the specific aim of fleshing out properties that dividend strategies should ideally satisfy, if one wants to model behaviour that is consistent with practice.

Capturing Non-Exchangeable Dependence in Multivariate Loss Processes with Nested Archimedean Lévy Copulas

The class of spectrally positive Lévy processes is a frequent choice for modelling loss processes in areas such as insurance or operational risk. Dependence between such processes (for example, between different lines of business) can be modelled with Lévy copulas. This approach is a parsimonious, efficient, and flexible method which provides many of the advantages akin to distributional copulas for random variables. Literature on Lévy copulas seems to have primarily focused on bivariate processes. When multivariate settings are considered, these usually exhibit an exchangeable dependence structure (whereby all subset of the processes have an identical marginal Lévy copula). In reality, losses are not always associated in an identical way, and models allowing for non-exchangeable dependence patterns are needed. In this paper, we present an approach which enables the development of such models. Inspired by ideas and techniques from the distributional copula literature we investigate the procedure of nesting Archimedean Lévy copulas. We provide a detailed analysis of this construction, and derive conditions under which valid multivariate (nested) Lévy copulas are obtained. Our results are discussed and illustrated, notably with an example of model fitting to data.

On Modelling Long Term Stock Returns with Ergodic Diffusion Processes: Arbitrage and Arbitrage-Free Specifications

We investigate the arbitrage-free property of stock price models where the local martingale component is based on an ergodic diffusion with a specified stationary distribution. These models are particularly useful for long horizon asset-liability management as they allow the modelling of long term stock returns with heavy tail ergodic diffusions, with tractable, time homogeneous dynamics, and which moreover admit a complete financial market, leading to unique pricing and hedging strategies. Unfortunately the standard specifications of these models in literature admit arbitrage opportunities. We investigate in detail the features of the existing model specifications which create these arbitrage opportunities and consequently construct a modification that is arbitrage free.

Common Shock Models for Claim Arrays

The paper is concerned with multiple claim arrays. We construct a broad and flexible family of models, where dependency is induced by common shock components. Models incorporate dependencies between observations both within arrays and between arrays. Arrays are of general shape (possibly with holes), but include the usual cases of claim triangles and trapezia that appear in the literature. General forms of dependency are considered, with cell-, row-, column-, diagonal-wise, and other, forms of dependency as special cases. In recognition of the extensive use by practitioners of large correlation matrices for the estimation of diversification benefits in capital modelling, substantial effort is applied to practical interpretation of such matrices generated by the models constructed here. Indeed, the literature does not document any methodology by which practitioners, who often parametrise those correlations by means of informed guesswork, may do so in a disciplined and parsimonious manner. In fact, this motivated the work presented in this paper.Reasonably realistic examples are examined, in which an expression is obtained for the general entry in the correlation matrix in terms of a limited set of parameters, each of which has a straightforward intuitive meaning to the practitioner. This will maximise chance of obtaining a reliable matrix. This construction is illustrated by a numerical example. Finally, the generated correlation matrix is then combined with heuristic estimates of tail dependency to arrive at a t-copula which might be used to construct capital margins dealing with the extreme right tail.

On the Interface between Optimal Periodic and Continuous Dividend Strategies in the Presence of Transaction Costs

In the classical optimal dividends problem, dividend decisions are allowed to be made at any point in time - according to a continuous strategy. Depending on the surplus process that is considered and whether dividend payouts are bounded or not, optimal strategies are generally of a band, barrier, or threshold type. In reality, while surpluses change continuously, dividends are generally paid on a periodic basis. Because of this, the actuarial literature has recently considered strategies where dividends are only allowed to be distributed at (random) discrete times - according to a periodic strategy.In this paper, we focus on the Brownian risk model. In this context, the optimal continuous and periodic strategies have previously been shown (independently of one another) to be of barrier type. We analyse the interface between continuous and periodic strategies when transaction costs are introduced. In some cases, a hybrid strategy proves optimal. In such a strategy, decisions are allowed to be made either at any time (continuously), or periodically at a lower cost. We show under which combination of parameters a pure continuous, pure periodic or hybrid (including both continuous and periodic dividend payments) barrier strategy is optimal. Results are illustrated.

Arbitrage and approximate arbitrage: the fundamental theorem of asset pricing

We consider an incomplete market model where asset prices are modelled by Ito processes, and derive the first fundamental theorem of asset pricing using standard stochastic calculus techniques. This contrasts with the sophisticated functional analytic theorems required in the comprehensive works of F. Delbaen and W. Schachermayer (1993) No Arbitrage and the Fundamental Theorem of Asset Pricing, pp. 37–38; Math. Finance 4 (1994), pp. 343–348; Math. Ann. 300 (1994), pp. 464–520; Ann. Appl. Probab. 5 (1995), pp. 926–645 and Proc. Sympos. Appl. Math. 57 (1999), pp. 49–58, and the comparative lack of transparency of the associated technical conditions. An additional benefit is that a clear relationship between no arbitrage and the existence of equivalent local martingale measures is also presented.

On the Distribution of the Excedents of Funds with Assets and Liabilities in Presence of Solvency and Recovery Requirements

In this paper, we consider a profitable, risky setting with two separate, correlated asset and liability processes (first introduced by Gerber and Shiu, 2003). The company that is considered is allowed to distribute excess profits (traditionally referred to as dividends in the literature), but is regulated and is subject to particular regulatory (solvency) constraints. Because of the bivariate nature of the surplus formulation, such distributions of excess profits can take two alternative forms. These can originate from a reduction of assets (and hence a payment to owners), but also from an increase of liabilities (when these represent the wealth of owners, such as in pension funds). The latter is particularly relevant if distributions of assets do not make sense because of the context, such as in regulated pension funds where assets are locked until retirement. In this paper, we extend the model of Gerber and Shiu (2003) and consider recovery requirements for the distribution of excess funds. Such recovery requirements are an extension of the plain vanilla solvency constraints considered in Paulsen (2003), and require funds to reach a higher level of funding than the solvency level (if and after it is triggered) before excess funds can be distributed again. We obtain closed form expressions for the expected present value of distributions (asset decrements or liability increments) when a distribution barrier is used. The optimal barrier level, as well as its existence and uniqueness are discussed.