Mathematics of Actuarial Finance
Org: Tom Salisbury (York; Fields)
- ERHAN BAYRAKTAR, University of Michigan, Department of Mathematics, 530 Church
Street, Ann Arbor, MI 48109-1043
Minimizing the Probability of Lifetime Ruin under Borrowing
We determine the optimal investment strategy of an individual who
targets a given rate of consumption and who seeks to minimize the
probability of going bankrupt before she dies, also known as
lifetime ruin. We impose two types of borrowing constraints:
First, we do not allow the individual to borrow money to invest in the
risky asset nor to sell the risky asset short. However, the latter is
not a real restriction because in the unconstrained case, the
individual does not sell the risky asset short. Second, we allow the
individual to borrow money but only at a rate that is higher than the
rate earned on the riskless asset.
We consider two forms of the consumption function:
The first is arguably more realistic, but the second is closely
connected with Merton's model of optimal consumption and investment
under power utility. We demonstrate that connection in this paper, as
well as include numerical examples to illustrate our results.
(1) The individual consumes at a constant (real) dollar rate,
(2) the individual consumes a constant proportion of her
This is a joint work with Virginia R. Young.
- PHELIM BOYLE, School of Accountancy, University of Waterloo
Modeling long term embedded options: Actuarial finance in
Contemporary life insurance products often include embedded options.
These options are markedly different from standard financial options
in that they are long term and their value depends on financial
variables and policyholder behaviour. They are difficult to price and
their risk management is challenging. Not surprisingly, consumers find
it hard to evaluate their worth. In this introductory talk we will
discuss the modeling issues involved in the pricing, valuation and
risk management of these options. Their valuation requires not only
an analysis of the relevant financial variables but also an analysis
of how the actuarial assumptions (such as lapse behaviour) affect the
risk. Each of these presents a challenge, but the problem is
exacerbated by an interaction of these factors. Policyholder
behaviour is influenced to some extent by economic conditions but we
do not have good theories on exactly how policyholders will behave and
there is a paucity of published experience data.
- JOSÉ GARRIDO, Department of Mathematics and Statistics, Concordia
Properties of distortion risk measures
The current actuarial-financial literature does not reach a consensus
on which risk measures should be used in practice. Our objective is
to give at least a partial solution to this problem. In this paper we
study properties that a risk measure must satisfy in order to avoid
some of the "inconsistencies" observed with popular measures like
We review the reasons why certain risk measures, like Conditional
Value at Risk (CVaR) can, in some cases, lead to erroneous decisions.
Some properties are proposed so that risk measures can avoid such
inconsistencies. This leads to the definition of two new families of
risk measures: complete measures and adapted
In particular, we study the set of risk measures that are based on
distortion functions and characterize the completeness and adaptive
properties of these, in terms of the derivative of the distortion
function that defines the risk measure.
This is joint work with Alejandro Balbás and Silvia Mayoral (Madrid).
- SEBASTIAN JAIMUNGAL, University of Toronto
Catastrophe Options with Stochastic Interest Rates
This talk will focus on the pricing and hedging of catastrophe put
options when interest rates are stochastic and losses are generated by
a compound Poisson process. The asset price process is modeled
through a jump-diffusion process that is correlated to the loss
process. We obtain explicit formulæ for the price of the option,
and the Greek hedging parameters Delta, Gamma and Rho. Furthermore,
numerical experiments are carried out to illustrate the effect that
stochastic interest rates and the variance of the loss process have on
option prices. Finally, we explore some simulation results to study
the effectiveness of a Delta-Gamma-Rho hedging scheme.
This is joint work with Tao Wang.
- ALEXANDER MELNIKOV, University of Alberta
Quantile hedging for Actuarial Risk Management
The main goals of the talk are: to investigate how quantile hedging
developed in mathematical finance can be applied to equity-linked life
insurance; to perform actuarial analysis to illustrate risk management
implications for insurance companies; and to examine how the choice of
a particular survival model affects assessment of mortality risk. The
approach will be illustrated by presenting numerical examples based on
appropriate financial and mortality data.
- KRISTIN MOORE, University of Michigan, Department of Mathematics, Ann Arbor,
Optimal and Nearly-Optimal Strategies for Minimizing the
Probability of Ruin in Retirement
The increasing risk of poverty in retirement has been well-documented;
this phenomenon is driven by demographic trends, changes in
employer-sponsored pension plans, and inadequate private retirement
savings. We study the optimal investment strategy for a retiree whose
objective is to minimize the probability of lifetime ruin, namely the
probability that a fixed consumption strategy will lead to zero wealth
while the individual is still alive. We derive a variational
inequality that governs the ruin probability and the optimal strategy,
and we demonstrate that the problem can be recast as a related optimal
stopping problem which yields a free-boundary problem that is more
tractable. In the special case of exponential future lifetime, one
can solve the free-boundary problem explicitly and recover a concise
expression for the optimal asset allocation. For more general
mortality, we numerically calculate the ruin probability and optimal
strategy and examine how they change as we vary the mortality
assumption and parameters of the financial model. In addition, we
consider suboptimal strategies that are easier to implement and
examine the impact on the ruin probabilities.
This is joint work with Virginia Young.
- MANUEL MORALES, Department of Mathematics and Statistics, York University
On the discounted penalty function for the generalized
inverse Gaussian process
We will review, from a historical point of view, the use of Lévy
processes in ruin theory. We focus on the decomposition for the ruin
probability and we argue how its convolution structure is inherited from
the Lévy family of processes. We will discuss the notion of discounted
penalty function in the framework of Lévy risk processes. The problem of
finding expressions for this function in a risk model driven by a Lévy
process will be addressed. The particular example using a generalized
inverse Gaussian process will be discussed. In this case, integral
expressions for the discounted penalty function are available. Actual
computation of ruin probabilities, distribution of the time of ruin and
joint distribution of the process prior and at the moment of ruin, are
carried out for a this example. Finally, forms for the discounted penalty
function in more general Lévy risk models will be presented.
This is joint work with Jose Garrido (Concordia).
- DAVID PROMISLOW, Dept. of Mathematics & Statistics, York University,
Pension fund switching
A recent trend in pensions is to allow employees a choice between
defined benefit and defined contribution plans. In some cases there
are possibilities to switch from one to the other. A particular
example is the State of Florida, which in 2002 implemented a scheme
whereby employees could choose a self managed DC plan, in place of the
usual DB plan, but at any time before maturity they could elect to
switch back to the DB plan, upon payment of the accrued benefit
obligation. We analyze the effect of this provision on the various
This is joint work with Moshe Milevsky.
- KEN SENG TAN, University of Waterloo, Dept. of Statistics and Actuarial
CTE and Capital Allocation under the Skew Elliptical
In recent years, there has been a growing interest among actuaries and
finance experts on adopting the Conditional Tail Expectation (CTE) as a
"coherent" risk measure for risk management. More recently CTE has
been proposed in the context of capital allocation in which CTE
provides a convenient way of determining the capital requirement for
individual lines of business among correlated business units.
Analytic results are derived in Panjer (2002) in the case of
multivariate Normal risks. Landsman and Valdez (2003) extend the
results of Panjer (2002) to elliptical distributions. In this paper,
we further generalize these results by considering a relatively new
class of distributions known as the skew elliptical distributions.
These distributions have the desirable properties that they need not
be symmetric and there is an additional parameter which regulates the
This is joint work with Jun Cai.
- STEVEN VANDUFFEL, Katholieke Universiteit Leuven, and Fortis Central Risk
Closed-Form Approximations for Constant Continuous Annuities
For a series of cash flows, the stochastically discounted or
compounded value is often a key quantity of interest in finance and
actuarial science. Unfortunately, even for the most realistic rate of
return models, it may be too difficult to obtain analytic expressions
for the risk measures involving this discounted sum. Some recent
research has demonstrated that in the case where the return process
follows a Brownian motion, the so-called comonotonic approximations
usually provide excellent and robust estimates of risk measures
associated with discounted sums of cash flows involving log-normal
We will derive analytic approximations for risk measures in case one
considers the continuous counterpart of a discounted sum of log-normal
returns. Although one may consider the discrete sums as providing a
more realistic situation than their continuous counterpart,
considering the continuous setting leads to more tractable explicit
formulas and may therefore provide further insight necessary to expand
the theory and to exploit new ideas for later developments. Moreover,
the closed-form approximations we derive in this continuous set-up can
then be compared more effectively with some exact results, thereby
facilitating a discussion about the accuracy of the approximations.
Indeed, in the discrete setting, one must always compare
approximations with results from simulation procedures, which always
gives room for debate.
Our numerical comparisons reveal that the comonotonic "maximal
variance" lower bound approximation provides an excellent fit for
several risk measures associated with integrals involving log-normal
returns. Similar results to those we derive for continuous annuities
can also be obtained in the case of continuously compounding which
therefore opens a roadmap for deriving closed-form approximations for
the prices of Asian options. Future research will also focus on
optimal portfolio slection problems.
This is joint work with Jan Dhaene and Emil Valdez.
- JIAN WANG, School of Computer Science, University of Waterloo
Hedging with a Correlated Asset: An Insurance Approach
Hedging a contingent claim with an asset which is not perfectly
correlated with the underlying asset results in an imperfect
hedge. One example arises in the case of segregated funds. These are
guarantees provided by insurance companies on mutual fund investments.
In many cases, the underlying asset is a mutual fund managed by the
insurer providing the guarantee. As the insurance company cannot take
a short position in its own fund, the guarantee would typically be
hedged using index futures, which will lead to an imperfect hedge. We
price the residual risk from hedging with a correlated asset using an
actuarial standard deviation principle in infinitesimal time, which
leads to a nonlinear partial differential equation. A fully implicit,
monotone discretization method is developed for solving the pricing
PDE. This method is shown to converge to the viscosity solution,
provided certain grid conditions are satisfied. An algorithm is
devised to ensure that these conditions hold. Monte Carlo simulations
are used to illustrate features of the profit and loss distribution
from hedging a contingent claim with an imperfectly correlated asset.
This is joint work with P. A. Forsyth, K. R. Vetzal, and
H. A. Windcliff.
- VIRGINIA YOUNG, Department of Mathematics, University of Michigan
Correspondence between Lifetime Minimum Wealth and Utility of
We establish when the two problems of minimizing a decreasing function
of lifetime minimum wealth and of maximizing utility of lifetime
consumption result in the same optimal investment strategy. To this
end, we equate the two investment strategies and show that if the
individual consumes at the same rate in both problems-the
consumption rate is a control in the problem of maximizing
utility-then the investment strategies are equal only when the
consumption function is linear in wealth. It then follows that the
corresponding investment strategy is also linear in wealth and the
implied utility function exhibits hyperbolic absolute risk aversion.
This is joint work with Erhan Bayraktar.