In
actuarial science and
applied probability, ruin theory (sometimes risk theory or collective risk theory) uses mathematical models to describe an insurer's vulnerability to insolvency/ruin. In such models key quantities of interest are the probability of ruin, distribution of surplus immediately prior to ruin and deficit at time of ruin.
Classical model
The theoretical foundation of ruin theory, known as the Cramér–Lundberg model (or classical compound-Poisson risk model, classical risk process or Poisson risk process) was introduced in 1903 by the Swedish actuary
Filip Lundberg. Lundberg's work was republished in the 1930s by
Harald Cramér
Harald Cramér (; 25 September 1893 – 5 October 1985) was a Swedish mathematician, actuary, and statistician, specializing in mathematical statistics and probabilistic number theory. John Kingman described him as "one of the giants of stati ...
.
The model describes an insurance company who experiences two opposing cash flows: incoming cash premiums and outgoing claims. Premiums arrive a constant rate ''c'' > 0 from customers and claims arrive according to a
Poisson process
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space with the essential feature that the points occur independently of one ...
with intensity ''λ'' and are
independent and identically distributed
In probability theory and statistics, a collection of random variables is independent and identically distributed if each random variable has the same probability distribution as the others and all are mutually independent. This property is usua ...
non-negative random variables
with distribution ''F'' and mean ''μ'' (they form a
compound Poisson process
A compound Poisson process is a continuous-time (random) stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. A compound Poisso ...
). So for an insurer who starts with initial surplus ''x'', the aggregate assets
are given by:
:
The central object of the model is to investigate the probability that the insurer's surplus level eventually falls below zero (making the firm bankrupt). This quantity, called the probability of ultimate ruin, is defined as
:
where the time of ruin is
with the convention that
. This can be computed exactly using the
Pollaczek–Khinchine formula In queueing theory, a discipline within the mathematical theory of probability, the Pollaczek–Khinchine formula states a relationship between the queue length and service time distribution Laplace transforms for an M/G/1 queue (where jobs arri ...
as (the ruin function here is equivalent to the tail function of the stationary distribution of waiting time in an
M/G/1 queue)
:
where
is the transform of the tail distribution of
,
:
and
denotes the
-fold
convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution' ...
.
In the case where the claim sizes are exponentially distributed, this simplifies to
:
Sparre Andersen model
E. Sparre Andersen extended the classical model in 1957 by allowing claim inter-arrival times to have arbitrary distribution functions.
::
where the claim number process
is a
renewal process and
are independent and identically distributed random variables.
The model furthermore assumes that
almost surely and that
and
are independent. The model is also known as the renewal risk model.
Expected discounted penalty function
Michael R. Powers
Michael Roland Powers (; born November 19, 1959) is the Zurich Insurance Group Chair Professor of Finance at Tsinghua University’s Tsinghua University School of Economics and Management, School of Economics and Management (SEM). Since 2014, he ...
and Gerber and Shiu
analyzed the behavior of the insurer's surplus through the expected discounted penalty function, which is commonly referred to as Gerber-Shiu function in the ruin literature and named after actuarial scientists
Elias S.W. Shiu
Elias Sai Wan Shiu is a Professor of Actuarial science, Actuarial Mathematics at the University of Iowa, and an internationally renowned actuarial scientist.
Biography
Shiu graduated from California Institute of Technology in 1975 after complet ...
and
Hans-Ulrich Gerber. It is arguable whether the function should have been called Powers-Gerber-Shiu function due to the contribution of Powers.
In
Powers
Powers may refer to:
Arts and media
* ''Powers'' (comics), a comic book series by Brian Michael Bendis and Michael Avon Oeming
** ''Powers'' (American TV series), a 2015–2016 series based on the comics
* ''Powers'' (British TV series), a 200 ...
' notation, this is defined as
: