In this thesis the performances of different approximations are compared for a standard actuarial and
financial problem: the estimation of quantiles and conditional
tail expectations of the final value of a series of discrete cash
flows.
To calculate the risk measures such as quantiles and Conditional
Tail Expectations, one needs the distribution function of the
final wealth. The final value of a series of discrete payments in
the considered model is the sum of dependent lognormal random
variables. Unfortunately, its distribution function cannot be
determined analytically. Thus usually one has to use
time-consuming Monte Carlo simulations. Computational time still
remains a serious drawback of Monte Carlo simulations, thus
several analytical techniques for approximating the distribution
function of final wealth are proposed in the frame of this thesis.
These are the widely used moment-matching approximations and
innovative comonotonic approximations.
Moment-matching methods approximate the unknown distribution
function by a given one in such a way that some characteristics
(in the present case the first two moments) coincide. The ideas of
two well-known approximations are described briefly. Analytical
formulas for valuing quantiles and Conditional Tail Expectations
are derived for both approximations.
Recently, a large group of scientists from Catholic University
Leuven in Belgium has derived comonotonic upper and comonotonic
lower bounds for sums of dependent lognormal random variables.
These bounds are bounds in the terms of "convex order". In order
to provide the theoretical background for comonotonic
approximations several fundamental ordering concepts such as
stochastic dominance, stop-loss and convex order and some
important relations between them are introduced. The last two
concepts are closely related. Both stochastic orders express which
of two random variables is the "less dangerous/more attractive"
one.
The central idea of comonotonic upper bound approximation is to
replace the original sum, presenting final wealth, by a new sum,
for which the components have the same marginal distributions as
the components in the original sum, but with "more dangerous/less
attractive" dependence structure. The upper bound, or saying
mathematically, convex largest sum is obtained when the components
of the sum are the components of comonotonic random vector.
Therefore, fundamental concepts of comonotonicity theory which are
important for the derivation of convex bounds are introduced. The
most wide-spread examples of comonotonicity which emerge in
financial context are described.
In addition to the upper bound a lower bound can be derived as
well. This provides one with a measure of the reliability of the
upper bound. The lower bound approach is based on the technique of
conditioning. It is obtained by applying Jensen's inequality for
conditional expectations to the original sum of dependent random
variables. Two slightly different version of conditioning random
variable are considered in the context of this thesis. They give
rise to two different approaches which are referred to as
comonotonic lower bound and comonotonic "maximal variance" lower
bound approaches.
Special attention is given to the class of distortion risk
measures. It is shown that the quantile risk measure as well as
Conditional Tail Expectation (under some additional conditions)
belong to this class. It is proved that both risk measures being
under consideration are additive for a sum of comonotonic random
variables, i.e. quantile and Conditional Tail Expectation for a
comonotonic upper and lower bounds can easily be obtained by
summing the corresponding risk measures of the marginals involved.
A special subclass of distortion risk measures which is referred
to as class of concave distortion risk measures is also under
consideration. It is shown that quantile risk measure is not a
concave distortion risk measure while Conditional Tail Expectation
(under some additional conditions) is a concave distortion risk
measure. A theoretical justification for the fact that "concave"
Conditional Tail Expectation preserves convex order relation
between random variables is given. It is shown that this property
does not necessarily hold for the quantile risk measure, as it is
not a concave risk measure.
Finally, the accuracy and efficiency of two moment-matching,
comonotonic upper bound, comonotonic lower bound and "maximal
variance" lower bound approximations are examined for a wide range
of parameters by comparing with the results obtained by Monte
Carlo simulation. It is justified by numerical results that,
generally, in the current situation lower bound approach
outperforms other methods. Moreover, the preservation of convex
order relation between the convex bounds for the final wealth by
Conditional Tail Expectation is demonstrated by numerical results.
It is justified numerically that this property does not
necessarily hold true for the quantile.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa.de:swb:ch1-200700024 |
| Date | 09 January 2007 |
| Creators | Karniychuk, Maryna |
| Contributors | TU Chemnitz, Fakultät für Mathematik |
| Publisher | Universitätsbibliothek Chemnitz |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:masterThesis |
| Format | application/pdf, text/plain, application/zip |
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