MAXWELL INSTITUTE for MATHEMATICAL SCIENCES
Actuarial and Financial Mathematics Seminars
Abstracts
This study is intended to gauge the risk inherent in defined
contribution
(DC) pension plans on an individual and on an aggregate basis, based on
United States data. Our aim is to gain insight into the consequences of
a
DC pension scheme becoming the predominant pillar of retirement income
for
an entire society. It is necessary for the primary source of retirement
income to, by design, provide a sufficient pension that will offer
financial security to the elderly and will facilitate the transition
from
employment to retirement. Due to the uncertainty in its accumulated
wealth, such a requirement could not be fulfilled by a traditional DC
pension plan if the pension delivery date is fixed. Therefore, rather
than focus on the accumulated wealth at a specified retirement age,
this
study investigates the likely retirement age of DC participants if they
hoped to maintain a fixed standard of living once they have retired,
which
will sustain them till death. Based on the simulated output of a DC
flexible age of retirement model, we decide upon the optimal investment
strategies. We then examine the demographic dynamics in an entire
population of DC pension plan participants. The conclusions drawn
demonstrate the significant role the market plays in the effectiveness
of
the DC pension plan scheme's success or failure. There is a high level
of
uncertainty in the age of retirement of each DC participant, regardless
of
his or her investment strategy. Furthermore, there are large retirement
age discrepancies between the DC participants in different cohorts,
despite their identical characteristics. We find that, even when we
allow
for a wide range of investment strategies amongst the members, the
ratio
of retirees to workers varies significantly over time. This suggests
that
countries dominated by DC schemes of this type may, over time, be
exposed
to significant risk in the size of its labour force.
(The talk is based on a paper by Bonnie-Jeanne MacDonald and Andrew
Cairns.)
We consider a variant of an optimisation problem involving sequential
entry and exit decisions that has emerged in the economics literature.
The problem that we solve aims at maximising a performance criterion of
an ergodic, or long-term average nature, which is better suited to
decision making within a sustainable economic environment. Our results
include a complete characterisation of the optimal strategy, which can
take qualitatively different forms depending on the
problem�@~Ys data, as well as explicit expressions
for the maximal value of the associated performance index.
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abstract
In this talk, we consider consumption strategies for stochastic money
accounts. We constructively prove the existence of discrete finite time
consumption processes fulfilling certain restrictions (e.g. the money
account being always positive and exactly zero at the end) and
pre-specified semilinear projection properties. One possible example
are consumption rates that are a martingale under the mentioned
restrictions. However, it is shown that any consumption strategy with
restrictions as above possesses at least one corresponding semilinear
projection property and could therefore be constructed from it. As an
actuarial application we show how the introduced consumption techniques
can be used to develop bonus strategies for with-profits policies.
Some problems of characterizing the no-arbitrage in markets with
frictions will be discussed. In particular we concentrate on the recent
results for markets with 2 assets of Grigoriev and Cetin and Grigoriev.
The talk will begin by reviewing the background to stochastic
mortality modelling and the original version of the Olivier-Smith
model (OS). The OS model is equivalent to a no-arbitrage model
for interest rates. The model takes as input a full term structure
of mortality by cohort and term to maturity. It then imposes
a term-structure of volatility to describe how survival probabilities
evolve over time. It is a one factor model driven by a sequence
of i.i.d. Gamma random variables with consequent restrictions
on volatilities.
Using some rather crude statistical procedures, we will argue
that the one-factor OS model does not fit historical England and
Wales mortality data very well. Instead more than one factor
is required to model dynamics and a greater degree of flexibility
is required to capture the observed volatility in mortality rates
at different ages.
We will discuss progress towards resolving these issues, with
copulas playing a central in a possible solution.
Any portfolio credit risk model that is to be used to calculate a loss
distribution
associated with defaults and changes in rating must address the
challenge of modelling
dependent defaults and dependent rating migrations. Most industry
models (such as KMV,
CreditMetrics, CreditRisk+) incorporate mechanisms for modelling this
dependence,
generally by assuming conditional independence of defaults and
migrations given common
economic factors. However, the calibration of these mechanisms is often
quite ad hoc,
despite the fact that the tail of the portfolio loss distribution is
extremely sensitive
to small changes in the parameters governing dependence.
We consider the problem of making formal statistical inference for such
models based on
historical default and rating migration data. In the solution we
propose, portfolio
credit models are represented as generalized linear mixed models
(GLMMs) and inference
is made using Markov chain Monte Carlo (MCMC) techniques. This general
framework allows
quite complex models where the random effects essentially play the role
of unobserved
latent factors influencing default and migration rates; to capture
economic cycle
effects the latent factors are allowed to have a dynamic time-series
structure. An
empirical study of Standard and Poors data shows strong evidence for
economic cycles and
also reveals pronounced sectoral heterogeneity in default and migration
rates.
A new framework for asset price dynamics is introduced
where the concept of noisy information about future cashflows
is used to derive the corresponding asset price processes. In
particular, we price equity by modelling the dividend process first
for a single-dividend paying asset, and then also for the multidividend
case. The share price is given by the sum of the discounted conditional
expectations of the dividends, where the conditional
expectation is taken with respect to the noisy partial information
associated with each impending cash flow. Dividend
growth is taken into account by introducing additional structure
on the dividend process. Various growth models can be considered,
depending on the context. The prices of options on dividend
paying assets are derived and, remarkably, the form of the price
process of a European-style call option is of Black-Scholes type.
For gamma-distributed dividend payments a closed-form expression
for the share-price process is obtained, and a semi-analytical
formula is computed for a European call option.The framework has
yet another interesting twist: It generates a natural family of
stochastic volatility models without the need for specifying on an ad
hoc basis the stochastic dynamics of the volatility. (Work carried out
in collaboration with D. C. Brody, Imperial College London, and L. P.
Hughston, King's College London.)
We consider the situation where a decision maker is able, at a cost, to
initiate and then abandon a running payoff. The objective of the
decision
maker is to maximise the expected discounted cashflow of the system
over an
infinite time horizon. The underlying stochastic process driving the
system is
modelled by a general one-dimensional positive It\^o diffusion and the
initialisation and abandonment costs, the discounting factor, and
the running payoffs can all be functions of the diffusion. A set of
sufficient
conditions on the problem data are identified that admit an explicit
analytic
solution to the problem. This problem has a number of applications in
finance
and economics.
In the course of solving this problem we address the fundamental real
options
question, that of when to capitalise an asset, in a general setting.
This is
related to the pricing of perpetual American options. We also establish
a
range of results that can provide useful tools for developing the
solution to
other stochastic control problems.
abstract
We study a general perturbed risk process with cumulative claims
modelled by a subordinator with finite expectation, and the
perturbation being a spectrally negative Levy process with zero
expectation. We derive a Pollaczek-Hinchin type formula for the
survival probability of that risk process, and give an
interpretation of the formula based on the decomposition of the
dual risk process at modified ladder epochs.
One aim of this talk is to give an overview of term structure models
(HJM and LIBOR) driven by time-inhomogeneous Levy processes. Next, we
present a relationship between caps and floors with the same time to
maturity and "moneyness", in term structure models driven by
time-inhomogeneous Levy processes. (Based on joint work with Ernst
Eberlein and Wolfgang Kluge.)
preprint
We compare two different models for assets and
liabilities for an insurance company that can be considered in
the standard approach to solvency assessment and in particular, in
determining the required target capital. The
first model is suggested by a joint working party by members in CEA,
Comit\'e Europ\'een des Assurances, and is
based on the duration concept and the second one is based on ideas
from Arbitrage Pricing Theory (APT).
We address the issue of finding a strategy to sustain structural
profitability of producing a commodity e.g.
a power plant producing non-storable energy like electricity. The power
plant will either continue to produce
electricity until its profitability reaches a critical low level at
which
the production is suspended and starts
again when it is profitable to produce it, depending on its market
price.
But, if the structural non-profitability
remains for a while, the power plant will face the risk to default and
be
definitely closed.
We suggest a general probabilistic set up to model profitability as a
function of the market price of the commodity,
and find the related optimal strategy to sustain it, under the
constraint
that the power plant faces the risk of
defaulting when being non-profitable under a fixed finite time
interval.
Consider the problem of maximizing utility of terminal wealth for a
financial agent who will receive an unbounded random bequest. We assume
a utility function which supports both positive and negative wealth. We
prove the existence of an optimal trading strategy within a class of
permissible strategies - those strategies whose wealth process is a
supermartingale under all pricing measures with finite relative
entropy.
We also investigate the concept of a marginal utility-based price
(MUBP), and show that a price process is a MUBP if and only if it is a
local martingale under the optimal measure for the utility maximizing
investor.
We consider risk-neutral returns and give an explicit and novel
formula that relates tail asymptotics asymptotics of the implied
volatility
smile. The theory of regular variation provides the (ideal)
mathematical
framework to formulate and prove such results. The practical value of
our
formulae comes from the vast literature on tail asymptotics and our
conditions
are often seen to be true by simple inspection of known results. In
cases
with known moment generating function (but unknown tail asymptotics) we
can play some Tauberian tricks and still apply our formula. This is
joint
work with Shalom Benaim (Cambridge).
This seminar discusses different approaches to long term
demographic forecasting. Demographic forecasts are immensely important
to
society. Yet existing methods are simplistic and often suffer from
shortcomings.
One established method is the ``Lee-Carter" method. The seminar
considers this method, and points out its shortcomings. The method is
cast into ``state-space" form and is
used as a springboard to more sophisticated approaches based on modern
time series methods. These approaches include building in smoothness
via regression, ``functional" methods, double spline methods, methods
based on the Wang transform and ``joint" mortality modelling across
countries and different groups. An attempt is made to evaluate the
different approaches, and arrive at some consensus regarding the
advantages and disadvantages of different methods. Approaches are
illustrated using different mortality data sets.
This paper formulates and studies a general continuous-time
behavioral portfolio selection model under Kahneman and Tversky's
(cumulative) prospect theory, featuring S-shaped utility (value)
functions
and probability distortions. Unlike the conventional
expected utility maximization model, such a behavioral model could be
easily mis-formulated (a.k.a. ill-posed) if its different components
do not coordinate well with each other. Certain classes of an
ill-posed model are identified. A new, systematic approach, which is
fundamentally different from the ones employed in the utility model,
is developed to solve a well-posed model, assuming a complete market
and
general It\^o processes for asset prices.
The optimal terminal wealth positions, derived in fairly explicit
forms,
possess surprisingly simple structure: they resemble the payoff
of a portfolio of two binary options written on the state density
price.
An example with a two-piece CRRA utility is presented to illustrate the
general results obtained, and is solved completely for all
installations
of the parameters. The effect of the behavioral criterion
on the risky allocations is finally discussed.
In this talk, I will present an insurance risk model. The ruin
probability and absolute ruin probability will be considered.
In some special cases, closed form solutions are obtained.
(This talk is based on some joint work with Hans Gerber).
We study an equilibrium model for the pricing of a defaultable
zero coupon bond in the framework of Back (1992). The market consists
of
an informed agent, noise traders and a market maker who sets the price
using the total order. When the insider doesn't trade, the default time
is
modelled as a totally inaccessible stopping time for the market maker
as
in reduced form credit risk models. We find the equilibrium pricing
rule
for the market maker and show that in the equilibrium the total order
behaves like a Bessel bridge from insider's viewpoint but the insider's
trade cannot be detected by the market. We also prove that in the
equilibrium the default time becomes predictable for the market maker.
The talk is based on results obtained jointly with Elisa Alos, UPF
Barcelona.
We show that the Heston volatility or equivalently the
Cox-Ingersoll-Ross
processs is Malliavin differentiable and give an explicit expression
for the
derivative. This result assures the applicability of Malliavin calculus
in the
framework of the Heston stochastic volatility model and the
Cox-Ingersoll-Ross
model for interest rates. Furthermore we derive conditions on the
parameters
which guarantee the existence of the second Malliavin derivative of the
Heston
volatility. We apply this result in order to obtain an extension of the
classical Hull and White formula to the Heston model with correlation
and
derive an approximate option pricing formula.
The talk explores computation of so-called good-deal price
bounds and their relationship to certain martingale measures and
certain
hedging problems. We provide a unifying framework for the HARA class of
expected utility preferences which includes among others quadratic,
logarithmic and exponential utility functions. We further explore links
with indifference pricing, q-optimal measures and f-divergences.
We investigate the solution od an ordinary differential equation and
discuss its relevance to optimal stopping and singular control
problems.
The ODE
\begin{equation}
\frac{1}{2}\sigma^2(x)w''(x)+ b(x)w'(x) - r(x)w(x) = 0
\end{equation}
plays a fundamental role in the solution of stopping problems and
singular control problems in stochastic control. We shall discuss
some of the properties of its solution and the associated
non-homogeneous equation. We shall use these properties to address
stopping problems and entry and exit problems.
An integral form of the expected utility from terminal wealth is used
as a guide in proposing a new risk-return objective functional. In such
a criterion, the risk-aversion is achieved via a wealth dependent
quadratic penalty of the fractions of invested wealth. In addition to
retaining the properties of the expected utility from terminal wealth,
this approach also offers the investor the means for selecting the
measure of return, the measure of risk, and richer risk-aversion
preferences. The solution to the corresponding portfolio control
problem is carried out via the dynamic programming, and the explicit
closed form solution is presented for an important example.
An implicit approach to the problem of transaction cost is to constrain
the trading strategy to be differentiable and thus of finite variation.
By also using criteria that penalize the rate of change of the trading
strategy, a significant reduction of the eventual transaction cost is
achieved. Simulation results for the pseudo-log-optimal portfolio
illustrate this method.
Capital allocation, in the context of credit portfolio risk, is
often understood as determination of the value-at-risk (VaR) of
the loss distribution and a risk-sensitive break-down of VaR to
the parts of the portfolio. When the loss distribution is inferred
from a Monte-Carlo simulation sample, the break-down of VaR
requires to estimate expectations of loan losses conditional on
portfolio-wide losses. We discuss the question how kernel estimation
methods have
to be adapted for this purpose.
In recent years there was an increasing interest in
better models for dependent risks, and in studying the effect
of dependence on the riskiness of portfolios. In this talk it will
be demonstrated, how copulas and stochastic orderings can
be used in this context. Moreover, these static results will be
extended to a dynamic stochastic process context, showing
some recent findings on dependence properties of Levy copulas.
In a Black Scholes market - consisting of one stock whose prices evolve
like a geometric Brownian motion and one risk free asset - an investor
wants to maximize the asymptotic growth rate of his wealth (portfolio
value). Without transaction costs the optimal policy would be given by
the constant Merton fraction which is the fraction of the wealth to be
invested in the stock. Facing transaction costs it is no longer
adequate
to keep the risky fraction constant.
We consider a combination of fixed (proportional to wealth) and
proportional
costs which punish the trading frequency as well as the magnitude of
the
transactions. Then an optimal trading strategy will consists of a
sequence
of stopping times and the optimal transactions at those times. So we
have to
deal with impulse control strategies which can be described as
solutions of
quasi-variational inequalities.
Motivated by various structural results we first look at a restricted
class
of trading strategies which can be described by four parameters, two
for
the stopping boundaries and two for the new risk fractions. In this
class the
problem can be simplified by renewal arguments to one period between
two
trading times, where we have to weight the new risky fractions by their
invariant distribution. This yields an explicit functional that has
only to
be maximized in these four parameters. So the computation of the best
strategy
in this class is very simple. Then we use the corresponding
quasi-variational
inequalities to prove that an optimal solution exists and that it can
be found
in this class. Solutions for short selling and borrowing can be given.
The surplus or book value of an insurance
company is a random process. This can be described by
the Cramer-Lundberg model or by a diffusion process.
According to De Finetti it is at some point optimal to
pay dividends to the shareholders. However, doing so
often imply that the book value at some point will be
non-positive with probability one. In my talk, I will
focus on the case where the risk process is modelled
by a Brownina motion, and discuss how issuance of
equity or bail-out loans from the benefactors influence
the optimal dividend strategy and the value of the
insurance company. I will discuss various cases of
costs associated with issuance of equity, and indicate
possible extensions to the case of risk processes
modelled by general diffusions or Levy processes.
Due to the new regulatory guidelines known as Basel II for banking and
Solvency 2 for insurance, the financial industry is looking for
qualitative approaches to and quantitative models for operational risk.
This talk gives an overview of the Basel II requirements for
quantitative modeling of operational risk and discusses several
possible approaches. Special focus is laid on the advanced measurement
approach and the
calculation of the operational-risk capital charge. We also raise
several issues concerning diversification effects and overall
quantitative risk management consequences of extremely heavy-tailed
data.
Any portfolio credit risk model that is to be used to calculate a loss
distribution associated with defaults and changes in rating must
address
the challenge of modeling dependent defaults and dependent rating
migrations. Most industry models (such as KMV, CreditMetrics,
CreditRisk+) incorporate mechanisms for modeling this dependence,
generally by assuming conditional independence of defaults and
migrations
given common economic factors. However, the calibration of these
mechanisms is often quite ad hoc, despite the fact that the tail of the
portfolio loss distribution is extremely sensitive to small changes in
the parameters governing dependence.
We consider the problem of making formal statistical inference for such
models based on historical default and rating migration data. In the
solution we propose, portfolio credit models are represented as
generalized linear mixed models (GLMMs) and inference is made using
Markov chain Monte Carlo (MCMC) techniques. This general framework
allows
quite complex models where the random effects essentially play the role
of unobserved latent factors influencing default and migration rates;
to
capture economic cycle effects the latent factors are allowed to have a
dynamic time-series structure. An empirical study of Standard and Poors
data shows strong evidence for economic cycles and also reveals
pronounced sectoral heterogeneity in default and migration
rates.
Over the past two decades, catastrophe risk modelling has expanded to
cover a wide range of natural and man-made hazards. Underlying all
catastrophe risk modelling is the realization that historical
experience, however extensive, provides only a limited window on
future disaster outcomes. In respect of influenza pandemic, 1918 is
not the worst case scenario. Concerned over the number of human H5N1
cases, life and health insurers worldwide have sought a quantitative
risk assessment of their pandemic loss potential. RMS has
developed a catastrophe pandemic risk model that is constructed using
a large stochastic set of pandemic scenarios. This model will be
described and compared with some actuarial approaches.
We consider the problem of determining optimal retention
levels for insurers willing to mitigate their risk exposure by
purchasing proportional reinsurance. We revisit De Finetti's classical
results in continuous-time and allow retention levels to change
dynamically in response to claims experience
and market performance. We also take up some ideas from dynamic
reinsurance markets to intertwine De Finetti's work and Markowitz's
mean-variance portfolio theory.
Many actuaries and financial economists use Monte Carlo simulation
methods,
for which of course they require a random number generator. Many would
use
what is supplied within the computer system available to them. But
there is
a lot to random number generation. It is of interest in itself to see
how
it may be done, and if you understand the principles, you may be able
to use
your own system, which you can control much better than whatever is
provided. I shall share some of my experiences with you.
A new method for sampling a Variance Gamma (VG) process, called
Dirichlet bridge sampling, is proposed and is shown to represent a
generalization of the known gamma bridge sampling method. Dirichlet
bridge sampling allows immediate generation of the entire trajectory of
the process over a certain period of time, avoiding sequential sampling
at arbitrary intermediate points in time as, instead, it is the case
with the gamma bridge method. We explore the efficiency of the proposed
simulation methodology and apply some variance reduction techniques
such as stratification. The proposed method is then tested against
approaches already existing in the literature, such as sequential
sampling and bridge sampling of the VG process. In particular, we price
path-dependent options such as Asian options, lookback options and
barrier options. In addition, we use the proposed technique in order to
calculate the market consistent value of some participating life
insurance contracts. This application is particularly important for the
insurance industry, following the introduction of market based
accounting standards and stricter capital requirements in accordance
with the EU Solvency II project.
Capital allocation when aggregate requirements are given by coherent
risk measures has been exhaustively studied. Approaches based on
marginal costs yield allocations that provide no incentives to split
the portfolio, which is consistent with the subadditivity property of
the risk measure. This presentation discusses the capital allocation
problem with convex risk measures, which relax the positive
homogeneity/subadditivity property of coherent ones. In that context,
aggregation penalties are applied and there may be a legitimate case
for splitting a portfolio. It is shown that the convexity property has
very strong implications in a capital allocation context, since it
implies ad infinitum splitting of portfolios, if such splitting can
take place at no additional cost. Finally, in a modest attempt to
inject some realism into the model, constraints are imposed on possible
portfolio splits and appropriate solutions are sought in the theory of
coalitional games.
A price process is scale-invariant if and only if the returns
distribution is independent of the price measurement scale. We show
that most stochastic processes used for pricing options on financial
assets have this property and that many models not previously
recognised as scale-invariant are indeed so. We also prove that price
hedge ratios for a wide class of contingent claims under a wide class
of pricing models are model-free. In particular, previous results on
model-free price hedge ratios of vanilla options based on
scale-invariant models are extended to any contingent claim with
homogeneous pay-off, including complex, path-dependent options.
However, model-free hedge ratios only have the minimum variance
property in scale-invariant stochastic volatility models when
price-volatility correlation is zero. In other stochastic volatility
models and in scale-invariant local volatility models, model-free hedge
ratios are not minimum variance ratios and our empirical results
demonstrate that they are less efficient than minimum variance hedge
ratios.
Global portfolio optimization
models rank among the proudest achievements of modern finance theory,
but for years practitioners have struggled to put them to work. In
1992, Back and Litterman put the problem down to difficulties
portfolio managers have in expanding views about some expected asset
returns into full probabilistic forecasts about all asset returns.
They propose a method to mitigate personal forecasts so that the
ensuing optimal portfolio does not depart too much from a chosen
reference portfolio. But we find that their method lacks a sound
rationale, requires ad hoc inputs and is limited in scope. We
propose a more general method based on a least discrimination
principle. It produces a probabilistic forecast that is true to
personal views but is otherwise as close as possible, in an expected
utility sense, to the forecast implied by the reference portfolio.
The least discrimination method produces optimal portfolios for a
variety of views, including views on volatility and correlation and
produces the corresponding optimal portfolios, including options when
appropriate. It also justifies a simple linear interpolation between
market and personal forecasts should a compromise be reached.
We present two situations where optimal decisions are to be made
in a Multi-state model: a) Decisions on protection against risk in a
life
insurance context, including risk of losing income due to disability or
unemployment; b) Portfolio optimization with credit risky assets where
the
credit risk is modelled by a (conditional) Markov chain. The two
situations seem very different but they turn out to be closely related
by
the mathematics it takes to solve the optimization problems.
There are two important
sources of uncertainty: randomness and fuzziness. Randomness models
the stochastic variability of all possible outcomes of a situation
and fuzziness relates to the unsharp boundaries of the parameters of
the model. In this sense, randomness is largely an instrument of a
normative analysis that focuses on the future, while fuzziness is
more an instrument of a descriptive analysis reflecting the past and
its implications. Clearly, randomness and fuzziness are
complementary, and so a natural question is how fuzzy variables could
interact with random variables. This presentation focuses on one
important dimension of this issue, fuzzy random variables (FRVs). The
goal is to model these FRVs and, in doing so, to illustrate how
naturally compatible and complementary randomness and fuzziness are.
Sources of heterogeneity in rating migration behavior are explored
using a continuous time Markov chain based framework. Working in
continuous time circumvents the embedding problem, mitigates the
censoring effect and facilitates term structure modelling with
arbitrary prediction horizons. Classical estimation provides ample
evidence of heterogeneity. However, adopting a Bayesian estimation
procedure can help mitigate the problems arising from data sparsity
and reduce estimation error. The transition probability matrices
estimated for different issuer profiles can be quite different from
each other. Using the CreditRisk+ framework, and a sample credit
portfolio, it can be shown that ignoring heterogeneity may give
erroneous estimates of VaR and a misleading picture of the risk
capital.
A class of stochastic models for the pricing of inflation-linked assets
is proposed. The nominal and the real pricing kernels, in terms of
which the consumer price index can be expressed, are modelled by
introducing a bivariate utility function depending on (a) aggregate
consumption, and (b) the aggregate real liquidity benefit conferred by
the money supply. Consumption and money supply policies are chosen such
that the expected joint utility obtained over a specified time horizon
is maximised, subject to a budget constraint that takes into account
the ?value? of the liquidity benefit associated with the money supply.
For any choice of the bivariate utility function, the resulting model
determines a relation between the rate of consumption, the price level,
and the money supply. The model also produces explicit expressions for
the real and nominal pricing kernels, and hence establishes a basis for
the arbitrage-free valuation of inflation-linked securities. In
conclusion I will also make some remarks about the modelling of
interest rates and inflation in an information-based setting. (Work in
collaboration with L. P. Hughston, King's College London).
For most households, choosing whether to rent or buy a home is
a difficult, multifaceted problem. Not only do households have to
grapple with the uncertainties of future movements of rents and house
prices and the substantial cost of changing residence. Housing tenure
decisions are further complicated if households' exposure to labour
income risk varies across occupations, industries and regions. Then,
potential correlations with these background risks may influence the
rent or buy decision. In this study, we present preliminary empirical
evidence, derived from the German Socio-Economic Panel (GSOEP), that
both labour income growth and rent growth varies across industries and
regions. We find that income-rent correlations have a statistically
significant influence on industry-specific average rental shares in
West-German federal states. However, the economic significance of the
relationship between real rent growth and real income growth on the
decision to rent or own is rather low. A one standard deviation of the
income-rent correlation implies an increase in rental shares of about
1.75 percentage points. (Work in collaboration with Martin Wersing and
Axel Werwatz).
We use a unique dataset of bond downgrades from a niche rating company
that has been found to be reacting faster to publicly available
information than its competitors. Using regime-switching models we
propose risk measures to quantify stock return disturbances (distress
costs) associated with the timing of downgrades. These risk measures
are based on the Capital Asset Pricing Model (CAPM) and use the
estimated parameters of the regime-switching models in a method that
resembles a dynamic event study. We observe a noticeable switch from a
low-volatility to a high-volatility regime one day before the day of
downgrades. On average the volatility in stock returns triples around
the time of downgrades and the stock return process remains in the
high-volatility regime for about three days. Using our proposed risk
measure we find that stock returns are associated with distress costs
of about twenty-two*d percent (where “d” is the daily market price of
risk) over a window of ten days before and after downgrades. These
costs can be further separated between bond rating companies that are
designated by the SEC as nationally recognized to rate debt and those
which are not. (Work in collaboration with Shaun Wang, Georgia State
University)
With the aid of Taylor-based approximations, this paper presents
results for pricing insurance contracts by using indifference pricing
under general utility functions. We discuss the connection between the
resulting “theoretical” indifference prices and the pricing
rule-of-thumb that practitioners use: Best Estimate plus a “Market
Value Margin”. Furthermore, we compare our approximations with known
analytical results for exponential and power utility.
Recent work on copula theory has reinvigorated the use of quantile
functions for the Monte Carlo simulation of marginal distributions. The
first half of this talk discusses quantile functions as solutions of
certain non-linear ordinary and partial differential equations. The PDE
representation leads us to a natural generalization to a collection of
multivariate distributions in which quite exotic combinations of
marginal distributions are coupled together in a natural way. (Joint
work with G. Steinbrecher)
Under the moratorium on the use of genetic tests in insurance
underwriting, greater emphasis is placed on family history information
in predicting risk. We conducted literature-based systematic reviews
and meta-analyses on 7 common diseases (colorectal cancer, breast
cancer, lung cancer, prostate cancer, ovarian cancer, stroke and
multiple sclerosis) to estimate pooled relative risks. To allow
individual risk prediction, we used life-table methods based on
population disease and mortality data to convert relative risks to
absolute risks for different patterns of family history. I will present
data from the different diseases with comments on the availability and
reliability of population data, validity of methods and conclusions
that can be drawn from our risk estimation efforts.
Starting from a rather general description of the “disability process”
(that is, the development through time of an individual occurrence of
disability), we show that a reasonable approximation to the related
probabilistic structure leads to the multistate model. In actuarial
practice, the multistate model is used, for example, for pricing and
reserving in Income Protection and Long Term Care business. Conversely,
calculations for other insurance products in the “health” area are
commonly based on simpler (and often less rigorous) methods. We first
show that, as the features of the multistate model allow for several
disability degrees, a rigorous modelling for Personal Accident
Insurance can be obtained; in this context, risk factors (and hence
rating factors) can be represented by an appropriate choice of the
transition intensities. Secondly, as the multistate model provides a
sound framework for interpreting practical calculation methods used in
the health insurance area, we discuss some pricing and reserving
formulae for Personal Accident Insurance and Sickness Insurance.
We start with a brief introduction to portfolio credit risk modelling.
In the main part of the talk a new information-based approach for
modelling the dynamic evolution of a portfolio of credit risky
securities is
proposed. In this context market prices of liquidly traded derivatives
are
given by the solution of a nonlinear filtering problem. This problem is
solved
via the innovations approach to nonlinear filtering. Moreover, we
derive
the ensuing asset price dynamics and compute risk-minimizing hedging
strategies. We conclude with some (preliminary) numerical results.
Longevity risk (the risk that future mortality rates are lower than
anticipated) has, in recent years, become a focus of attention
in the insurance and pensions industry. Alongside this a number of
new models have been developed to describe the stochastic evolution
of mortality rates over time. In this extended seminar, we will review
a number of these models, ranging from the simple Lee-Carter model to
more complex multifactor
models incorporating a cohort effect. As a starting point, one might
use a quantitative criterion such as the BIC to identify which models
are the best. However, this on its own reveals only part of the
picture.
In the talk we will discuss a variety of additional criteria that
can be used to give a much clearer picture of the merits of each model.
Amongst these, criteria that relate to the robustness of a particular
model give a clear indication that one model with a high BIC is
unreliable.
My talk will be about optimal investment in a model of currency trading
with transaction costs. The model is general enough to allow a
discontinuous bid-offer spread. The investor wishes to maximise their
"direct" utility of consumption, which is measured in terms of
consumption assets linked to some (but not necessarily all) of the
traded currencies.
The analysis will center on two conditions under which the
straightforward existence of a dual minimiser leads to the existence of
an optimal terminal wealth. The first condition is a well known growth
condition on the dual function. The second weaker, and more natural
condition is that of "asymptotic satiability" of the value
function.
I will start with a review of results in Dickson (ASTIN Bulletin, 2008)
about the joint density of the time of ruin and deficit at ruin in the
classical risk model. I will then show how these ideas can be applied
to the Erlang(2) risk model. Some interesting contrasts arise between
these two models, particularly in the special case when the initial
surplus
is 0. An exact solution for the joint density will be presented in the
case of Erlang(2) claims, and some computational issues will be
discussed.
Structured products are
popular with retail investors. Many of these products provide a
guaranteed return combined with some participation in the performance
of the equity market. These contracts often have complex
(path-dependent) designs. We explain why expected utility maximizing
investors should prefer (European) contracts. However, if consumers
overweight the probability of getting the maximum possible return they
may prefer the more complex contracts. We explore this explanation and
provide evidence that sellers encourage this type of overweighting by
the projections they select in the prospectus documents.
Ruin probability is considered for a Markov process with one level of
switching between two independent Levy processes one of which is
spectrally negative and another is a compound Poisson process with
drift. There is given a partition of the real line into two sets and,
with each set, there is associated a probability distribution. When a
Markov process takes a value in one of these sets, its next increment
has the distribution associated with this set. Explicit representations
were found for the ruin probability in terms of ladder heights. As a
consequence, results were obtained for a risk process, where the
premium rate and the claim size depend on whether current reserve is
above or below a certain threshold.
We consider the problem of optimal investment under the threat of a
crash of uncertain height. Further, we do not know if and when
the crash happens. For this a new stock price model will be introduced
and the approach of worst-case portfolio optimization will be
developed. The computed optimal portfolio strategies show a much more
realistic behaviour than the ones obtained in the standard Merton
setting. Applications to both portfolio problems in financial and
actuarial models are given.
The talk may be divided into three parts. The first two parts introduce
ideas and methods that appear
in the analysis of rare events for heavy-tailed stochastic processes.
In the last part, these ideas are used in an analysis of a ruin
problem. The outline of the talk is as follows.
Part I - Heavy tails - introduces and explains the concept of
regular variation. This begins with the Pareto distribution and the
aim is to show that it is natural and very useful to look at regular
variation in a general setting, a weak convergence approach.
Part II - Extremes for stochastic processes - focuses on understanding
the extremal behaviour of stochastic integral processes
driven by heavy-tailed noise. This is an illustration of the
ideas discussed in Part I and will be important in the analysis in...
Part III - Ruin probabilities under optimal investments.
Explicit results for the asymptotics of ruin probabilities are found
without strong distributional assumptions for the claim sizes and
the processes representing the investment possibilities.
One aim here is to
illustrate the usefulness of the approach discussed in Parts I and II.
Based on joint work with Mirela Predescu, Greg Gupton, Ahmet Kocagil,
Wei Liu, Alexander Reyngold,
Quantitative Research, Fitch Solutions.
This talk reviews a statistical model that ranks the liquidity of
reference entities in the single-name credit default swap (CDS) market.
A reduced-form approach is adopted: a handful of price- and market
activity-based predictors are selected to signal the liquidity
characteristics of each reference entity. When combined in a regression
model, these predictors provide a basis for ranking reference entities
on their relative liquidity.
The model's main contribution is that it
generates a liquidity score for each reference entity. This provides a
framework for: evaluating overall CDS market liquidity;
assessing the liquidity in each sector;
comparing corporate and sovereign liquidity;
understanding the relationship between liquidity and credit quality.
The data covers well over a 1000 reference entities from different
geographical regions over a near three-year period. The model has thus
been statistically validated in different ways. It is shown to have
high discriminatory power in separating those names which the wider
market perceives to be liquid from those which it does not. The results
are shown to be significant across different geographical regions. The
model also performs well on walk-forward tests, which is especially
reassuring given that the model estimation period spans the recent
credit and banking crisis.
As of June 2008, the single-name CDS market
had notional outstanding of over USD 33 trillion with total (netted)
market valuation of approximately USD 2 trillion, an increase of over
65% over the previous six months. Given the large volume and increased
market valuation, and the recent liquidity and counterparty-related
concerns in various markets, this OTC market (along with other credit
derivatives) is under consideration by legislative bodies and
regulators for closer scrutiny and regulation going forward. The
results of this research should partially address some of the concerns
raised. It also provides a concrete framework for managing liquidity
risk in this market.
This talk aims to show how ideas of correlation can change the
way Retail Banks model Retail Credit Risk in practice, and how
correlation can help them understand cyclic Credit Risk phenomena that
otherwise require ad hoc solutions. This is illustrated by analysis of
long-term default time-series at portfolio level, and at obligor level.
Unlike Wholesale Credit Risk, where data is thin, Retail Credit Risk
can
take advantage of huge datasets to apply data-hungry General Linear
Mixed Modelling techniques. Never-the-less, Retail Operations remain
strongly influenced by its zero-correlation traditions, and this talk
will examine the cultural and technical issues that are expected to
arise as Retail catches up with Wholesale in its adoption of
correlation.
Geroscience is the scientific interface between ageing and age-related
disease. As a modern discipline, it is one of a number of 21st century
innovations that will drive change in mortality risk in coming decades.
A longevity risk modelling initiative will be described that models
mortality risk at an individual level, and recognizes the intrinsic
randomness in the process of medical discovery in simulating future
trajectories of mortality improvement.
In this paper the performance of locally risk-minimizing hedge
strategies for European options in stochastic volatility models is
studied from an experimental as well as from an empirical perspective.
These hedge strategies are derived for a large class of diffusion-type
stochastic volatility models, and they are as easy to implement as
usual delta hedges. Our simulation results on model risk show that the
locally risk-minimizing hedges are robust with respect to uncertainty
and even misconceptions about the underlying data generating process.
The empirical study indicates that locally risk-minimizing hedge
strategies consistently produce lower standard deviations of
profit-and-loss-ratios than delta hedges (over different time periods
as well as in different markets). The more skewed the market and the
more out-of-the-money the option, the higher the benefit.
The theory of risk measurement has been extensively developed over the
past ten years or so, but there has been comparatively little effort
devoted to using this theory to inform portfolio choice. One theme of
this paper is to study how an investor in a conventional log-Brownian
market would invest to optimize expected utility of terminal wealth,
when subjected to a bound on his risk, as measured by a coherent
law-invariant risk measure. Results of Kusuoka lead to remarkably
complete expressions for the solution to this problem.
The second theme of the paper is to discuss how one would actually
manage (not just measure) risk. We study a principal/agent problem,
where the principal is required to satisfy some risk constraint. The
principal proposes a compensation package to the agent, who then
optimises selfishly ignoring the risk constraint. The principal can
pick a compensation package that induces the agent to select the
principal's optimal choice. We consider two possibilities: firstly,
that the principal chooses a contract which is cheapest subject to
satisfying the agent's participation constraint; and secondly, a robust
contract which perfectly aligns the objectives of principal and agent.
The two typically differ little in price, though their form can look
surprisingly different.
In recent literature, different methods have been proposed on how to
define and model stochastic mortality. In most of these approaches, the
so-called spot force of mortality is modeled as a stochastic process.
In contrast to such spot force models, forward force mortality models
infer dynamics on the entire age/term-structure of mortality.
This paper considers forward models defined based on best-estimate
forecasts of survival probabilities as can be found in so-called
best-estimate generation life tables. We show that the forward approach
bears profound advantages in view of actuarial applications and provide
a detailed analysis of forward mortality models driven by
finite-dimensional Brownian motion. In particular, we address the
relationship to other modeling approaches, the consistency problem of
parametric forward models, and the existence of finite dimensional
realizations for Gaussian forward models.
All results are illustrated based on a simple example with an affine
specification.
The talk is based on joint work with Daniel Bauer (Georgia State
University) and Fred Espen Benth (University of Oslo).
Under new solvency regulations general insurance companies need to
calculate a risk margin for the risks that go beyond
the best estimate liabilities. One approach currently used
is the so-called cost-of-capital approach where the companies
calculate the necessary risk bearing capital and then build
reserves for the price of this risk bearing capital.
Since a general insurance liability runoff takes several years
this involves multiperiod risk measures. Because multiperiod
risk measures are often difficult to handle, companies calculate
the (univariate) risk measure for the next accounting year and then
use a proxy for the remaining accounting years that is based on
that univariate risk measure.
We derive a rigorous multiperiod risk measure approach for
a specific chain-ladder claims reserving model, where one
is still able to calculate or approximate the margin analytically.
Using these analytical formulas we then compare our results to
the proxies used in practice.