Ruin probabilities in multivariate risk models with periodic common shock

We propose a multidimensional risk model where the common shock affecting all classes of insurance business is arriving according to a non-homogeneous periodic Poisson process. In this multivariate setting, we derive upper bounds of Lundberg-type for the probability that ruin occurs in all classes simultaneously using the martingale approach via piecewise deterministic Markov processes theory. These results are numerically illustrated in a bivariate risk model, where the beta-shape periodic claim intensity function is considered. Under the assumption of dependent heavy-tailed claims, asymptotic bounds for the finite-time ruin probabilities associated to three types of ruin in this multivariate framework are investigated.     

MDP algorithms for portfolio optimization problems in pure jump markets

We consider the problem of maximizing the expected utility of the terminal wealth of a portfolio in a continuous-time pure jump market with general utility function. This leads to an optimal control problem for piecewise deterministic Markov processes. Using an embedding procedure we solve the problem by looking at a discrete-time contracting Markov decision process. Our aim is to show that this point of view has a number of advantages, in particular as far as computational aspects are concerned. We characterize the value function as the unique fixed point of the dynamic programming operator and prove the existence of optimal portfolios. Moreover, we show that value iteration as well as Howard’s policy improvement algorithm works. Finally, we give error bounds when the utility function is approximated and when we discretize the state space. A numerical example is presented and our approach is compared to the approximating Markov chain method.      

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.     

General convex order on risk aggregation

Using a general notion of convex order, we derive general lower bounds for risk measures of aggregated positions under dependence uncertainty, and this in arbitrary dimensions and for heterogeneous models. We also prove sharpness of the bounds obtained when each marginal distribution has a decreasing density. The main result answers a long-standing open question and yields an insight in optimal dependence structures. A numerical algorithm provides bounds for quantities of interest in risk management. Furthermore, our numerical results suggest that the bounds obtained in this paper are generally sharp for a broader class of models.     

Martingales in life insurance

Abstract

This paper presents a general probabilistic model, including stochastic discounting, for life insurance contracts, either a single policy or a portfolio of policies. In § 4 we define prospective reserves and annual losses in terms of our model and we show that these are generalisations of the corresponding concepts in conventional life insurance mathematics. Our main results are in § 5 where we use the martingale property of the loss process to derive upper bounds for the probability of ruin for the portfolio. These results are illustrated by two numerical examples in § 6.     

Non-exponential bounds for stop-loss premiums and ruin probabilities

We present an inventory of non-exponential bounds for ruin probabilities and stop-loss premiums in the general Sparre-Andersen model (renewal model) of risk theory. Various additional bounds are given if one assumes that the ladder height distribution F associated with the risk process belongs to a certain class of distributions, in particular if it is concave or it exhibits a (positive or negative) aging property. In most cases, these bounds are shown to improve existing ones in the literature and/or possess the correct asymptotic behaviour when the distribution F is subexponential. Since in the classical (compound Poisson) risk model the ladder height distribution is always concave, all the bounds given in the paper are also valid for this model. Finally, in many cases the results of the paper are also valid for any compound geometric distribution.     

Term structure modelling of defaultable bonds

In this paper we present a model of the development of the term structure of defaultable interest rates that is based on a multiple-defaults model. Instead of modelling a cash payoff in default we assume that defaulted debt is restructured and continues to be traded.The term structure of defaultable bond prices is represented in terms of defaultable forward rates similar to the Heath-Jarrow-Morton (HJM) (Heath et al., 1992) approach, and conditions are given under which the dynamics of these rates are arbitrage-free. These conditions are a drift restriction that is closely related to the HJM drift restriction for risk-free bonds, and the restriction that the defaultable short rate must always be not below the risk-free short rate. In its most general version the model is set in a marked point process framework, to allow for jumps in the defaultable rates at times of default.Financial Assistance by Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 303, at the University of Bonn and the DAAD is gratefully acknowledged.I thank Pierre Mella-Barral, David Lando and David Webb for helpful conversations, and the participants of the FMG Conference on Defaultable Bonds (March 1997) in London and the QMF 97 conference in Cairns for helpful comments. All errors are of course my own.     

Option pricing and Esscher transform under regime switching

Summary We consider the option pricing problem when the risky underlying assets are driven by Markov-modulated Geometric Brownian Motion (GBM). That is, the market parameters, for instance, the market interest rate, the appreciation rate and the volatility of the underlying risky asset, depend on unobservable states of the economy which are modelled by a continuous-time Hidden Markov process. The market described by the Markov-modulated GBM model is incomplete in general and, hence, the martingale measure is not unique. We adopt a regime switching random Esscher transform to determine an equivalent martingale pricing measure. As in Miyahara [33], we can justify our pricing result by the minimal entropy martingale measure (MEMM).We would like to thank the referees for many helpful and insightful comments and suggestions.Correspondence to: R. J. Elliott     

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.     

Issues in decentralizing the provision of education

This paper analyzes some of the problems that arise in decentralizing education to the private sector. We concentrate on the difficulties that result from the heterogeneity of students and competition among schools in a location setting. We analyze two main issues, the resources expended by schools and the mix of students in schools, and report on results for two others, the location of schools and their number. For each of these, we investigate the extent to which decentralizing the provision of schooling results in an efficient allocation of resources, and consider the use of vouchers to improve the situation. Our analysis draws on elements of three distinct methodologies: the theory of clubs, location theory, and the theory of monopolistic competition. We find that private schooling will typically be inefficient, but that inefficiency may sometimes be corrected by appropriately designed vouchers.We would like to thank Olivier Debande, Jean-François Wen, two anonymous referees, and seminar participants at the 1st meeting of the Canadian Public Economics Study Group and the 51 st congress of the International Institute of Public Finance for helpful comments. Stefan Buergi and Luc Savard provided useful research assistance. We are also grateful to the SSHRCC and the FCAR for financial support.     

Approximation methods for piecewise deterministic Markov processes and their costs

In this paper, we analyse piecewise deterministic Markov processes (PDMPs), as introduced in Davis (1984). Many models in insurance mathematics can be formulated in terms of the general concept of PDMPs. There one is interested in computing certain quantities of interest such as the probability of ruin or the value of an insurance company. Instead of explicitly solving the related integro-(partial) differential equation (an approach which can only be used in few special cases), we adapt the problem in a manner that allows us to apply deterministic numerical integration algorithms such as quasi-Monte Carlo rules; this is in contrast to applying random integration algorithms such as Monte Carlo. To this end, we reformulate a general cost functional as a fixed point of a particular integral operator, which allows for iterative approximation of the functional. Furthermore, we introduce a smoothing technique which is applied to the integrands involved, in order to use error bounds for deterministic cubature rules. We prove a convergence result for our PDMPs approximation, which is of independent interest as it justifies phase-type approximations on the process level. We illustrate the smoothing technique for a risk-theoretic example, and compare deterministic and Monte Carlo integration.     

Behavioral Biases and Investment

We investigate the way investors react to prior gains/losses. We directly examine investor reactions to different definitions of gains and losses (i.e., overall wealth, paper gains and losses, and realized capital gains and losses) and investigate how gains and losses in one category of wealth (e.g., real estate) affect holdings in other categories (e.g., financial assets). We show that investors change their holdings of risky assets as a function of both financial and real estate gains. Prior gains increase risk-taking, while prior losses reduce it. To interpret our results, we consider and compare three alternative hypotheses of investor behavior: prospect theory, house money effect and standard utility theory with decreasing risk aversion. Our evidence fails to support loss aversion, pointing in the direction of the house money effect or standard utility theory. Investors consider wealth in its entirety, and risk-taking in financial markets is affected by gains/losses in overall wealth, financial wealth, and real estate wealth. We appreciate the helpful comments of: O. Bondarenko, F. De Jong, B. Dumas, H. Hau, P. Hillion, R. Jaganathan, M. Lettau, P.Maenhout, M. Huang, S. Mullanaithen, T. Odean, J. Peress, R. Shiller, P. Sodini, M. Suominen, A. Subrahmanyan, B. Swaminathan, R. Thaler, L. Tepla, P. Veronesi, M. Weber and the participants of the Summer Financial Markets Symposium at Gerzensee and the NBER Behavioral Finance Meeting, Fall 2002. We are grateful to Sven-Ivan Sundqvist for numerous helpful discussions and for providing us with the data. Financial support from Inquiry Europe is acknowledged. Andrei Simonov also acknowledges financial support from the Jan Wallander and Tom Hedelius Foundation. Any remaining errors are our own.     

Bounding Bermudan swaptions in a swap-rate market model

Abstract

We develop a new method for finding upper bounds for Bermudan swaptions in a swap-rate market model. By comparing with lower bounds found by exercise boundary parametrization, we find that the bounds are well within bid-offer spread. As an application, we study the dependence of Bermudan swaption prices on the number of instantaneous factors used in the model. We also establish an equivalence with LIBOR market models and show that virtually identical lower bounds for Bermudan swaptions are obtained.     

What is going on in the oil market?

Conclusion In this article, we propose a consistent view on the recent oil-price history based on fundamental data and economic theory. We sum up: After the turn of the century three major stylized shocks have hit market. First, the demand curve has shifted fight outwards, mainly driven, as extensively reported in the media, by sustained growth in China and other Asian Countries. Second, supply disruptions in countries with low extraction costs (Iraq and Venezuela) have shifted the supply curve to the left. Third, we show that speculators adjust their inventories in order to take advantage of predictable price fluctuations and play themselves a major role in the price formation. Optimal storage theory implies that aggregate inventories are negatively related to the oil price and positively to the volatility of supply and demand shocks. We provide evidence that the political events in the last years have increased volatility and induced the inventory curve to shift right outwards. We analyze in a graphical framework the interaction of all these shocks and conclude that speculators have caused the oil price to overshoot in the short run its long-run fundamental value. However, this is not at all attributable to market failure or the harmfulness of speculators. In fact, the opposite is true. Speculators have in general a dampening effect on the oil price. The record oil price in the very recent history is partly a consequence of speculators maintaining or building-up inventories to cope with the supply and demand shocks to come. Hence, high prices represent a short-term toll for future price stability. It follows from our analysis that the oil price is expected to fall towards its long-term mean, provided that no further shocks hit the economy and, critically, the oil supply. As we saw, this prediction is consistent with the observed prices in the futures markets. Also in terms of future price volatility, the outlook is rather upbeat. The increased inventory levels held by speculators will cushion the spot market against fluctuations in natural supply and demand and limit the degree to which the currently high underlying volatility will translate into higher price volatility. We would like to thank Manuel Ammann, Bernd Brommundt, Stephan Kessler, Ralf Seiz, Michael Verhofen, Hemrich von Wyss, and two anonymous referees for helpful comments We are particularly indebted to Sergej Peisotchenko at United Energy System (UES) of Russia and Jan Gjerde at Shell for clearing us up on technical aspects of oil production