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In-Depth Information
quake), it does provide a very good fit. Note that the procedure of trimming the top
-
% of the data before calibration is known as “robust estimation” and it leads to
very similar conclusions (see a recent paper by Chernobai et al.,
).
From the probability plots, we can also infer that the waiting time data can be
described by the log-normal and - to some degree - the exponential distribution.
he maximum likelihood estimates of the parameters of these two distributions
are given by μ
=−
.
and σ
=
.
(log-normal) and β
=
.
(exponential).
RiskProcessanditsVisualization
6.4
he risk process (
.
) is the sum of the initial capital and the premium function mi-
nus the aggregate claim process governed by two random phenomena - the severity
and incidence of claims. In many practical situations, it is reasonable to consider the
counting process N
t
(responsible for the incidence of events) to be (i) a renewal pro-
cess,i.e.,acountingprocesswithinterarrivaltimesthatarei.i.d.positiverandomvari-
ables with a mean of
λ, and (ii) independent of the claim severities
X
k
.Insuch
a case, the premium function can be defined in a natural way as c
(
t
)=(
+
θ
)
μλt,
where μ is the expectation of X
k
and θ
is the relative safety loading on the
premium which “guarantees” the survival of the insurance company. he (homoge-
neous) Poisson process (HPP)is a special case of the renewal process with exponen-
tially distributed waiting times.
Two standard ways of presenting sample trajectories of the risk process are dis-
played in Figs.
.
and
.
. Here, we use the Danish fire losses dataset, which can
be modeled by a log-normal claim amount distribution with parameters μ
=
.
and σ
.
(obtained via maximum likelihood) andaHPPcounting processwith
a monthly intensity λ
=
million DKK.InFig.
.
,five real (discontinuous) sample realizations ofthe resulting
risk process are presented, whereas in Fig.
.
the trajectories are shown in a con-
tinuous fashion. he latter way of depicting sample realizations seems to be more
illustrative. Also note that one of the trajectories falls below zero, indicating a sce-
nario leading to bankruptcy of the insurance company.
=
.
. he company's initial capital is assumed to be u
=
Ruin Probability Plots
6.4.1
Whenexamining thenature ofthe riskassociated with abusinessportfolio,itisoten
interesting to assess how the portfolio may be expected to perform over an extended
period of time. his is where the ruin theory (Grandell,
) comes in handy. Ruin
theory is concerned with the excess of the income c
(
t
)
(with respect to a business
portfolio) over the outgoings, or claims paid, S
. his quantity, referred to as the
insurer's surplus, varies over time. Specifically, ruin is said to occur if the insurer's
surplus reaches a specified lower bound, e.g., minus the initial capital. his can be
observed inFig.
.
,wherethe time ofruin isdenoted byastar. One measureof risk
(
t
)
