May  9, 2021

IFID/MITACS Conference on
Financial Engineering for Actuarial Mathematics

to be held at
Fields Institute, 222 College Street, Toronto

Sunday November 9 - Monday November 10, 2008

Speaker Abstracts

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Erhan Bayraktar, University of Michigan
Proving the Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control

We consider three closely related problems in optimal control: (1) minimizing the probability of lifetime ruin when the rate of consumption is stochastic and when the individual can invest in a Black-Scholes financial market; (2) minimizing the probability of lifetime ruin when the rate of consumption is constant but the individual can invest in two risky correlated assets; and (3) a controller-and-stopper problem: first, the controller controls the drift and volatility of a process in order to maximize a running reward based on that process; then, the stopper chooses the time to stop the running reward and rewards the controller a final amount at that time. Our primary goal is to show that the minimal probability of ruin for Problem 1, whose stochas- tic representation does not have a classical form as the utility maximization problem does, is the unique classical solution of its Hamilton-Jacobi-Bellman (HJB) equation, which is a non-linear boundary-value problem. It is not clear a priori that the value functions of the first two problems are regular (convex, smooth solutions of the corresponding HJBs), and here we give a novel tech- nique in proving their regularity. To this end, we reduce the dimension of Problem 1 by considering Problem 2. An important step to show that the value functions of Problems 1 and 2 are regular is to construct a regular, convex sequence of functions that uniformly converges to the value function of Problem 2. After an extensive analysis of Problem 3, which has the structure of a classical control problem, we construct this regular, convex sequence by forming a sequence of Legendre transforms of problems of the form (3). That is, Problem 3, which is itself an interesting problem to analyze, has a key role in the analysis of the minimum probability of ruin.

This is a joint work with Virginia R. Young.

Sid Browne, Brevan Howard US Asset Management LP
Active portfolio management: investment goals and portfolio constraints (or how to invest, if you must)

Steven Haberman, Cass Business School
The Lee-Carter Mortality Model for Mortality Dynamics: Recent Devlopments

The Lee Carter methodology has proved to be an elegant and effective method of forecasting demographic variables including mortality rates and it has gained wide acceptance. As this modelling framework has been used and tested for a wide range of populations, it has been found that it does not necessarily capture all of the features of past trends in certain applications. We propose to extend this modelling framework by introducing a new feature - namely, an age-cohort effect in the context of a wider class of generalized non-linear models. This seems to provide a better fit to past trends for certain applications and has an important impact on forecasted mortality rates and derived quantities like expectations of life and annuity values. We consider the effect of generalizing the choice of error distribution in these models. We also consider risk measurement and assess simulation strategies for measuring the risk inherent in predictions of mortality rates, expectations of life and annuity values. Finally, we consider the issue of back-testing and how robust the parameter estimates are to the choice of data. Illustrations of these developments will be provided using general population and annuity purchaser data sets from the UK.

X. Sheldon Lin, University of Toronto
Pricing Perpetual Catastrophe Put Options and Related Issues

The expected discounted penalty function proposed in the seminal paper Gerber and Shiu (1998) has been widely used to analyze the time of ruin, the surplus immediately before ruin and the deficit at ruin of insurance risk models in ruin theory. However, few of its applications can be found beyond except that Gerber and Landry (1998) explored its use for the pricing of perpetual American put options. In this talk, I will discuss the use of the expected discounted penalty function and mathematical tools developed for the function to evaluate perpetual American catastrophe put options. Assuming that catastrophe losses follow a mixture of Erlang distributions, I will show that an analytical (semi-closed) expression for the price of perpetual American catastrophe put options can be obtained. I will then discuss the fitting of mixtures of Erlang distributions to catastrophe loss data and possible uses of the expected discounted penalty function for other types of options.

Michael Ludkovski, University of California, Santa Barbara
Optimal Risk Sharing under Distorted Probabilities

We study optimal risk sharing among $n$ agents endowed with distortion risk measures. Our model includes market frictions that can either represent linear transaction costs or risk premia charged by a clearing house for the agents. Risk sharing under third-party
constraints is also considered. We obtain an explicit formula for Pareto optimal allocations. In particular, we find that a stop-loss or
deductible risk sharing is optimal in the case of two agents and several common distortion functions. This extends recent result of
Jouini et al. (2006) to the problem with unbounded risks and market frictions. This is joint work with Jenny Young (U of Michigan).

Manuel Morales, University of Montreal
On a New Generalization of the Expected Discounted Penalty Function

The Expected Discounted Penalty Function (EDPF) was introduced in a series of now classical papers [Gerber and Shiu (1997), (1998a), (1998b) and Gerber and Landry (1998)]. Motivated by applications in finance and risk management, and inspired by recent developments in fluctuation theory for L\'evy processes, we propose an extended definition of the expected discounted penalty function that takes into account a new ruin-related random variable. In addition to the surplus before ruin and deficit at ruin, we extend the EDPF to include the surplus at the last minimum before ruin. We provide a defective renewal equation for the generalized EDPF in a setting involving a subordinator and a spectrally negative Levy process. Well-known results for the classical EDPF are also revisited by using a fluctuation identity for first-passage times of Levy processes. Potential applications in finance will be briefly discussed.
Authors: Enrico Biffis and Manuel Morales

Ragnar Norberg, London School of Economics
Management of Finacial and Demographic Risk in Life Insurance and Pensions

Traditional paradigms - the principle of equivalence and notions of reserves. Management of financial risk, mortality risk and longevity risk: 1. The with profit scheme - what it is and what it might be. 2. Unit-linked insurance - traditional and conceivable forms. 3. Alternative risk transfer through market operations - securitization of mortality risk, optimal hedging and optimal design of derivatives, and a few words about swaps. General discussion of the role of financial instruments in life insurance and pensions: Can the markets come to our rescue? The fair value fairyland.

Annamaria Olivieri, University of Parma
Developing a Stochastic Mortality Model for Internal Assessments

In this talk, we take the point of view of an insurer dealing with life annuities, which aims at building up a (partial) internal model in order to quantify the impact of mortality risks, namely process and longevity risk, in view of taking appropriate risk management actions. We assume that a life table providing a best-estimate assessment of annuitants’ future mortality is available; conversely, no access to data sets and methodology underlying the construction of the life table is at the insurer’s disposal. In spite of this, a stochastic approach is required.

We focus on the number of deaths in a given cohort, which we model allowing for a random mortality rate. In particular, we extend the traditional Poisson-Gamma or Pólya-Eggenberg scheme, which involves a static distribution, by introducing age- and time-dependent parameters. Further, we define a Bayesian-inferential procedure for updating the parameters to experience in some situations. The
setting we define is practicable, while allowing for process and longevity risk in a rigorous way.

The model is then implemented for capital allocation purposes. We investigate the amount of the required capital for a given life annuity portfolio, based on solvency targets which could be adopted within internal models. The outcomes of such an investigation are compared with the capital required according to some standard rules, in particular those proposed within the Solvency 2 project.

Keywords: Life annuities, Random fluctuations, Systematic deviations, Process
risk, Longevity risk, Solvency, Insurance risk management, Internal models.

Gordon Willmot, University of Waterloo
Generalized penalty functions in dependent Sparre Andersen models

The structure of various Gerber-Shiu functions in Sparre Andersen models allowing for possible dependence between claim sizes and interclaim times is examined. The penalty function is assumed to depend on some or all of the surplus immediately prior to ruin, the deficit at ruin, the minimum surplus before ruin, and the surplus immediately after the second last claim before ruin. Defective joint and marginal distributions involving these quantities are derived. Many of the properties in the Sparre Andersen model without dependence are seen to hold in the present model as well. A discussion of Lundberg's fundamental equation and the generalized adjustment coefficient is given. The usual Sparre Andersen model without dependence is also discussed, and in particular the case with exponential claim sizes is considered. This talk is based on joint work with Eric Cheung, David Landriault, and Jae-Kyung Woo.



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