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
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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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