Graphics Reference
In-Depth Information
Introduction
6.1
his chapter concerns risk processes, which may be the most suitable for computer
visualization of all insurance objects. At the same time, risk processes are basic in-
struments for any non-life actuary - they are needed to calculate the amount of loss
that an insurance company mayincur. heyalso appear naturally in rating-triggered
step-up bonds, where the interest rate is bound to random changes in company rat-
ings, and catastrophe bonds, where the size of the coupon payment depends on the
severity of catastrophic events.
Atypicalmodelofinsurancerisk,theso-calledcollectiveriskmodel,hastwomain
components: one characterizing the frequency (or incidence) of events and another
describing the severity (or size or amount) of the gain or loss resulting from the oc-
currence of an event (Klugman et al.,
; Panjer and Willmot,
; Teugels and
Sundt,
).Incidence and severity are both generally assumed tobestochastic and
independent of each other. Together they form the backbone of a realistic risk pro-
cess. Consequently, both must be calibrated to the available historical data. All three
visualization techniques discussed in Sect.
.
:
mean excess function
limited expected value function
probability plot
arerelatively simple,but at the same time they providevaluable assistance during the
estimation process.
Once the stochastic models governing the incidence and severity of claims have
been identified, they can be combined into the so-called aggregate claim process,
N
t
k
=
S
t
=
X
k
,
(
.
)
where the claim severities are described by the random sequence
X
k
with a finite
mean, and the number of claims in the interval
(
, t
]
is modeled by a counting pro-
cess N
t
, oten called the claim arrival process.
he risk process
R
t
t
describing the capital of an insurance company is then
defined as
R
t
=
u
+
c
(
t
)−
S
t
,
(
.
)
where the non-negative constant u stands for the initial capital of the company and
c
is the premium the company receives for sold insurance policies.
he simplicity of the risk process (
.
) is only illusionary. In most cases no analyt-
ical conclusions regarding the time evolution of the process can be drawn. However,
it is this evolution that is important to practitioners, who have to calculate function-
als of the risk process, like the expected time to ruin and the ruin probability. his
calls for e
cient numerical simulation schemes (Burnecki et al.,
) and powerful
inference tools. In Sect.
.
we will present four such tools:
(
t
)
