Ruin theory

Overview

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. 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

{\textstyle c>0}

from customers and claims arrive according to a Poisson process

N

t

{\displaystyle N_{t}}

with intensity

λ

{\textstyle \lambda }

and are independent and identically distributed non-negative random variables

ξ

i

{\displaystyle \xi _{i}}

with distribution

F

{\textstyle F}

and mean

μ

{\textstyle \mu }

Sparre Andersen model

E. Sparre Andersen extended the classical model in 1957 by allowing claim inter-arrival times to have arbitrary distribution functions.

X

t

= x + c t −

∑

i = 1

N

t

ξ

i

for

t ≥ 0 ,

{\displaystyle X_{t}=x+ct-\sum _{i=1}^{N_{t}}\xi _{i}\quad {\text{ for }}t\geq 0,}

where the claim number process

(

N

t

)

t ≥ 0

{\displaystyle (N_{t})_{t\geq 0}}

Expected discounted penalty function

Michael R. Powers 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 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' notation, this is defined as

m ( x ) =

E

x

[

e

− δ τ

K

τ

]

{\displaystyle m(x)=\mathbb {E} ^{x}[e^{-\delta \tau }K_{\tau }]}

, where

δ

{\displaystyle \delta }

is the discounting force of interest,

K

τ

{\displaystyle K_{\tau }}

is a general penalty function reflecting the economic costs to the insurer at the time of ruin, and the expectation

E

Recent developments

Compound-Poisson risk model with constant interest Compound-Poisson risk model with stochastic interest Brownian-motion risk model General diffusion-process model Markov-modulated risk model Accident probability factor (APF) calculator – risk analysis model (@SBH)

See also

Financial risk Volterra integral equation § Application: Ruin theory Chance-constrained portfolio selection

Further reading

Gerber, H.U. (1979). An Introduction to Mathematical Risk Theory. Philadelphia: S.S. Heubner Foundation Monograph Series 8. Asmussen S., Albrecher H. (2010). Ruin Probabilities, 2nd Edition. Singapore: World Scientific Publishing Co.