Msc Dissertation-2012

i

ACTUARIAL RISK PROCESSES AND RUIN PROBABILITY

ANTONY ODHIAMBO OKUNGU

H00105792

August 2012

Actuarial Mathematics and Statistics

School of Mathematical and Computer Sciences

Dissertation submitted as part of the requirements for the award of

the degree of MSc in Actuarial Science

ii

Abstract

In common day it has become a problem in determining the ruin probability within the classical risk theory

.Previously it has been assumed that premiums rate are independent of the reserves and individual claims also

being independent from each other .As we know claims generated by one policy in an insurance portfolio may

induce claims from other policies within the same portfolio.

In this effect the paper investigates the effects of premium and claim dependence on both the probability of ruin

and time to ruin. We use different formulas in calculating ruin probability and various approximations in a

compound Poisson model.

Acknowledgement

It is a pleasure to thank the many people who made this thesis possible.

It is difficult to overstate my gratitude to my Masters supervisor, Dr.Seva Shneer. With his enthusiasm, his

inspiration, and his great efforts to explain things clearly and simply, he helped to make mathematics fun for

me. Throughout my dissertation-writing period, he provided encouragement, sound advice, good teaching,

good company, and lots of good ideas. I would have been lost without him.

I am indebted to my many student colleagues for providing a stimulating and fun environment in which to

learn and I am grateful to the secretaries and librarians in the math departments of Heriot Watt University, for

helping the departments to run smoothly and for assisting me in many different ways grow.

Most importantly, none of this would have been possible without the love and patience of my family. My

immediate family, to whom this dissertation is dedicated to, has been a constant source of love, concern,

support and strength all these years. I would like to express my heart-felt gratitude to my family

Last but not least Special thanks goes to God for taking me this far.

iii

Table of content

1. Introduction…………………………………………………………………………………………………………1

2. Classical Risk Model……………………………………………………………………………………………….2

2.1. Ruin Theory and Ruin Probability…………………………………………………………...........................2

2.2. Denotation and Assumptions……………………………………………………………………………………3

3. Compound Poisson Model…………………………………………………………………………………………4

3.1. Calculations of Ruin Probability…………………………………………………………………………………6

3.1.1. Pollaczek-Khinchin Formula…...……………………………………………………………………………..7

3.1.2. Differential Equation Formula………………………………………………………………………………..7

3.1.3. Laplace Transform Formula………………………………………………………………………………….10

4. Adjustcent Coefficient…………………………………………………………………………………………….13

4.1. Lundberg Bounds on Ruin Probability…………………………………………………………………………15

4.2. Approximations of the Ruin Probability……………………………………………………………………….15

4.2.1. Crammer-Lundberg Approximation………………………………………………………………………...16

4.2.2. Exponential Approximation…………………………………………………………………………………16

4.2.3. Lundberg Approximation……………………………………………………………………………………16

4.2.4. Beekman-Bowers Approximation…………………………………………………………………………..16

5. Model with reserve-dependent premium rate…………………………………………………………………..17

5.1. Differential Equation Formula…………………………………………………………………………………18

6. Conclusion………………………………………………………………………………………………………..22

7. References……………………………………………………………………………………………………….23

1

1. Introduction

Financial markets in recent times and previous years have had difficulties in being to cover risks they face.

Insurance companies being important f inancial go between. Under a general insurance the insured

party pays an amount of money (the premium) to the insurer at the start of the period of insurance cover. The

insured party will make a claim under the insurance policy each time the insured party has a loss .However it is

always difficult to know the number and amount of claims made. Thereby having to use risk management

to cover risk against a corresponding loss.

An actuarial risk model which is a mathematical description of the behavior of a collection of risks

generated by an insurance portfolio is thereby always necessary to be built. Actuaries derive the loss

distributions that model the number and amount of claims analytically which reduces the sample data to the

analytical expression of the statistical distribution family.

In actuarial risk management it is an important issue to estimate the performance of the portfolio. Ruin

theory is used by actuaries in order to try investigating how practical the insurance process (risk process),

where the analyzed reserve (surplus) of the insurer increases (starting with an initial value) due to premiums

collected and decreases due to insurance claims paid and follow the insurer’s surplus and ruin probability

which can be explained as the probability of insurer’s surplus drops below a specified lower bond such as

minus initial capital. Ruin theory can also serve as a useful tool in to plan for the use of insurer’s funds or

measuring insurance solvency. A typical point in case is a compound Poisson model which is a form of a risk

process assuming that the claims are settled fully as they occur leaving no allowance for interest on the

insurer’s surplus not withstanding is a possibility of insurer expenses.

2

2. Classical Risk Model.

In a given financial year an insurance company might accumulate a surplus from the collected

premiums and might exceed the claims that have been paid over that period. In this case the surplus will

be used by the insurance company to protect the company against future loss such as time periods when

the claims paid exceed the collected premium. Complications can also arise by the nature of the losses

that are covered by general insurance. Not forgetting that there would be a large number of small claims,

we must also note that there would also be extremely large claims that are several greater than the mean

claim amount. Thus it is even more important that the surplus is modeled, predicted and managed

properly to ensure the continual survival and profitability of the company.

The classical risk model is a particularly useful tool in modeling the surplus process. Its utility comes from

its flexibility, in which each component as premiums, claim numbers and claim amounts can be modeled

separately. The entire line of business can be modeled separately or modeled as a whole instead of on a

policy-by- policy basis. Hence the classical risk model has widespread applications in general insurance

decision-making.

2.1 Ruin Theory and Ruin Probability

Ruin theory investigates how practical is the insurance process (risk process), where the analyzed reserve

(surplus) of the insurer increases (starting with an initial value) due to premiums collected and decreases due to

insurance claims paid.

Ruin Probability is the probability that the surplus will sometimes become negative; from the practical point of

view, it is closely related to the solvency of insurer.

3

2.2 Denotation and assumptions

=Initial surplus (reserve) at time

=Total Premium accumulated up to time

=Number of claims in the interval

∑ =Claim severity (amount of claims)

=Risk process.

With assumptions of a classical insurance risk such as If is the probability space

Carrying (i) a point process ,i.e. an integers valued stochastic process with as,

And for each t<∞ and non-decreasing realization s and (ii) an independent sequence }i =1 of

Positive independent and identically distributed (i.i.d) random variables, then the risk process

{ Is given by

∑

In some cases it is often easier to work with the claim surplus process { denoted by

∑

These definitions follow the ones used in Rolski et al. (1999). A slightly less vigorous, but equivalent, set of

definitions can be found in Beard, Pentik¨ainen, and Pesonen (1969) and Dickson and Waters (1992).

4

3. Compound Poisson Model

The compound Poisson model has nice properties such as derivation as a limit of individual

models.[5] This is the main purpose of Cramer-Lundberg model to assume a continuous time risk model

where the aggregate claims in any interval have a compound Poisson distribution. Moreover, the premium

income is continuous over time and that the premium income in any interval is proportional to the interval

length. This leads to the following model for the surplus of insurance.

∑

Is the initial capital, is the premium rate. The number of claims in is a Poisson process with

rate 𝜆.This model is also called the Cramer-Lundberg process or classical risk process.

We denote the distribution function of claim by G, its moments by and its moment generating

function by .Let .We assume that

Otherwise no insurance company would insure such a risk. Note that for

For an insurance company it is important that stays above a certain level .This level is given by legal

restrictions .By adjusting the initial capital it is no loss of generality to assume this level is 0.

Figure 3.1: Ruin Probability

5

The time to ruin is defined as Let

and the ruin probability in infinite time, i.e. the probability that the

capital insurance company ever drops below zero can be written as . In

the risk process relative safety loading has to be positive, otherwise would be less than and the ruin

probability of the insurance become 1 in the future .The ruin probability in finite time is given by

.

We also note that obviously ).However,the finite time ruin probability may be sometimes also

relavant in cases of finite time.We can also see that is decreasing in and increasing in t.Denoting

the claim times by and by convention

Let .If we only consider the process at the claim times we can see that

∑ is a random walk. From theory of random walk we can see that ruin occurs with probability 1, if

hence we will assume that

𝜆 .

We can interpret the condition that the mean income is strictly larger than the mean outflow, which is also

called the net profit condition. If the profit condition is fulfilled then tends to infinity as .Hence

– is as finite. So we can conclude that .

Practically where is related to the planning horizon of the company, may regard as more

interesting than most insurance managers will closely follow the development of the risk business and

increase the premium if the risk business behaves badly. Therefore, in non-life insurance, it may be natural to

regard equal to four or five years reasonable. [9].

6

3.1 Calculation of Ruin Probability in a Compound Poisson Model

Using different formulas namely Pollaczek-Khinchin, Laplace Transform Differential Equation we try to calculate

the Ultimate ruin probability there by giving the following results.

3.1.1 Pollaczek-Khinchin Formula

Pollaczek-Khinchin formula is used to derive explicit closed form solutions for claim amount distribution.

For other distribution that cannot be derived by Pollaczek-Khinchin, Monte Carlo simulation method can be

applied, and can only give explicit results for the exponential claims. Derived by Asmussen, S. (2000) (Pg. 61).

(3.2)

∑

q=

where

∫

,

Note that the formula can be rewritten in a different way in terms of ( depending on your view.

3.1.2 Differential Equation Formula.

Let h be the small. If Ruin does not occur in the interval (0, ^h] then the Compound

Poisson Process starts at time ^h with a new initial capital of . Let

Denote the probability that ruin do not occur (Survival). Using that the interval times are

Exponentially distributed we have the density 𝜆 of the distribution of and P [

.We get noting that =0 for x<0

7

∫ ∫ 𝜆

. (3.3)

When h tends to 0, Rearranging the terms and dividing by by h yields

c

=

-

∫ ∫ 𝜆

DT. (3.4)

Letting h tends to 0 shows that is differentiable from the right and

C 𝜆[ ∫

] (3.5)

Replacing u by u-ch gives

∫ ∫ 𝜆

DT. (3.6)

Concluding that is continuous .Rearranging the terms, dividing by h and letting h yeilds the equation

C 𝜆[ ∫

]. (3.7)

It can be seen that is differentiable at all points u where is continuous .If has a jump then the

derivatives from the left and from the right do not coincide. But we have shown that is absolutely

continuous .Letting know be the derivative from the right we have that is a density of

Getting rid of the derivative in equation (1.4) we find

( ) =

∫

∫

∫ ∫

(3.8)

=∫

∫ ∫

(3.9)

=∫

∫ ∫

(3.10)

=∫

∫ ∫

(3.11)

8

∫

( ) ∫

(3.12)

Note that ∫

= . Letting we can use the bounded convergence theorem interchange limit

and integral and get

C (1- 𝜆 ∫ 𝜆

(3.13)

Where we used that it follows that

=1-

, =

. (3.14)

Replacing by we obtain

c 𝜆 𝜆 ∫

) (

= ( ∫ ( ) ∫

(3.15)

Exponential Claims ruin probability

Let the claims be Exp ( distributed .Then the equation (1.1) can be written as

C 𝜆[ ∫

(3.16)

Differentiating yields

C 𝜆 dy- 𝜆 (3.17)

Giving a solution of

(

) (3.18)

Because 1 as u we get A=1.Because =1-𝜆/ (ac), the solution is

(

) Or 𝝍 (u) =

(

) (3.19)

9

3.1.3 Laplace Transform Formula.

The notation of Laplace Transform of ruin probability is given by s) = ∫

.

And the Laplace Transform of the density of the claim sizes is s) =∫

,

t) = E ( ), denotes the moment generating function of the distribution of the claim amount.

T ,

s) = E ( ), s (3.2.0)

s) = ∫

∫

(3.2.1)

The Laplace transform defined

s) =∫

, 0 (3.2.2)

s) =∫

, (3.2.3)

We denote by uniqueness s) = ∫

and,

s) =∫

= ∫

dx (3.2.4)

=

∫

) (3.2.5)

By integrating

s) =

-

∫

=

(3.2.6)

, M=

{

s) = ∑

s) = ∑

10

=

(3.2.7)

s) = ∫

dx =

Assume µ~Exp ( )

s) = ∫

=

s) =

=

=

(3.2.7)

s) =

=

(3.2.8)

s) =

=

(

)

Since s= ∫

=

=∫

(

), (3.2.9)

4. Adjustcent Coefficient.

It is particularly useful in the determination of both ruin probabilities and bounds of ruin probabilities. It is

defined as the positive number r giving a measure for risk for a surplus process. It has two major factors

considered in the surplus process namely aggregate claim and premium amount.

=0 and = for all (3.3.0)

11

Figure 1.3: The Adjustment Coefficient.

For compound Poisson claim number process, this is equivalent to the positive root of 𝜆 (r) =𝜆 +cr, where

(r) =∫

DF(x) is the moment generating function of F, evaluated at r.

It can be shown [15] that the coefficient exists when there exists such that (s) < whenever

s< and (s) = .The adjuscent coefficient is the rudimentary measure of risk in the collective, and

will be used to determine the bounds of the ruin probability.

The adjustcent coefficient can be used to calculate the probability of ruin if r exists then

( ) =

(3.3.1)

12

Because of the extreme difficulty in finding the denominator in (1.3.5), the Laplace

Transform is often the preferred way of calculating the probability of ruin.

A closed form solution for (u) can be found if the in (1.1) are i.i.d exponential random variables, with

parameter .In this case,

( =

(

) (3.3.2)

Gerber et al. (1987) used a different transform to obtain an analytical solution to 𝝍(u) for combination of

exponential distributions. That is

∑ (3.3.3)

Where ∑ .Which is very useful as Feldman and Whitt (19980 have shown that heavy-tail distributions

whose density functions are completely inflected, i.e. (x) for all x>0 and n=1, 2, can be

approximated arbitrarily closely by a hyper exponential distribution. Where complete monotonic distributions

include the Pareto, Weibull and exponential mixtures of inverse Gaussian (Abbate and Whitt 1999), which are

constantly used in practice to model claim amounts.

4.1 Lundberg Bounds on Ruin Probabilities.

It is always difficult to obtain a closed-form solution of probability of ruin hence we construct bounds

(especially upper bounds ) of the this 𝝍(u).Most common bound is attributed to Lundberg and it showed that if

the adjustcent coefficient exists , then

( )≤ .

Asmussen and Nielsen [15] proved with a similar result when the premium rate is right-continuous function of

the reserve, i.e. ∏ = ( ) by replacing the adjuscent coefficient r with an adjuscent coefficient r (u).

Giving an explicit formula of the ruin probability for Exponential claims with the parameter shown as follows

( =

exp (-

) (3.3.4)

13

4.2 Approximations of the Ruin Probability.

As we have shown analytically that there exist results of ruin probability in infinite time in case of an

Exponential distribution, we should be clear to say that Laplace technique does not always work in all claim

distributions therefore we use some approximations.

4.2.1 Crammer-Lundberg Approximation.

Cramer-Lundberg’s asymptotic ruin formula for (u) for large u is given by

(u)=C (3.3.5)

Where C= (R) –

It gives almost accurate results for the light-tailed distributions.

4.2.2 Exponential Approximation.

De Vylder [7] derived this approximation which requires the first three moments to be finite.

(u)=exp {-1-

√

}

(3.3.6)

4.2.3 Lundberg Approximation.

Grandell [8] derived this approximation requires the first three moments to be infinite.

(u)= {1+ (

)

} exp (

) (3.3.7

14

4.2.4 Beekman-Bowers Approximation.

It uses the following representation of ruin probability:

It uses the idea of replacing the conditional probability With a Gamma distribution

fuction by fitting first two moments by Grandel [8] leading to

(u)=

{1-G (u), (3.3.8)

Where the parameters of G are given by

-1) (3.3.9)

2

- ) (3.4.0)

5. Model with reserve – dependent premium rates.

Let } be a risk process with a reserve-dependant premium rate delayed claims and initial capital consider a

compound Poisson model as mentioned earlier in Chapter 3.We have

∑ (3.4.1)

Letting as long it’s below and when above where in this case represents the Capital

base/Investment and both occurrence is dependent on the changes in

Shows cases where dividends were paid equal to

For (u) < 1 we need to assume that both are greater than 𝜆 ,consequently it would mean that

𝜆

15

K P2

P1

U P1

Figure 5.1: Reserve-dependent

5.1 Differential Equation Formula.

K

U

U-

16

Figure 5.2:D.E with 0 ≤ u < k

The Survival probability

0 ≤ u < k`

( ) =

∫

(3.4.2)

U~Exp (

+

∫

) (3.4.3)

(

)

∫

(3.4.4)

(

)

∫

Where x=u-y (3.4.5)

=(

)

having boundary conditions

(3.4.6)

(3.4.7)

= (

)

and from the principle in equation (3.4.6) (3.4.7)

We get,

- (

)

)

Solution

And

(3.4.8)

17

Assuming u~Exp (

For u ≥ k u

U-

K

U-y

Figure 5.2:D.E with u ≥ k

( )=

∫

. (3.4.9)

We denote,

(

)

∫

∫

(3.5.0)

=(

)

having boundary conditions , we know have the

following equation

(3.5.1)

= (

)

and from the principle in equation (3.4.6)

Taking an assumption of from

We get new equation written as

18

( ) =

∫

(3.5.1)

From equation (3.4.6) through differentiation we can find

( ) =

(

)

and = (

)

(3.5.2)

The following boundary condition ∫

now gets a new representation using the same

principle as equation (3.4.6)

∫ (

)

(3.5.3)

Equation (3.4.7) can now be written as

(

)

=

(

)

∫

(

)

We can now say that

(

)

= (

)

(3.5.4)

Solutions,

𝜆

𝜆 [

𝜆

(

)

𝜆

( (

) )]

19

6. Conclusion

Insurance companies have each a contract of insurance that brings a risk which occurs because of random

frequency and positive random amount of claim. One of the main tasks of actuaries is to determine

insurance and risk premiums to prevent from ruin of the insurance. To manage this risk, Ruin

probabilities are important tools which help actuaries to control ruin risk. The probability of ruin is

used for decision taking, for instance the premium calculation and the solvency of a company. For an

actuary it is important to be able to take a good decision in reasonable time.

Even though Classical Risk Model is really popular in actuarial science, it has some limitations; this was

notably commented by Taylor and Buchanan (1988) that while this theory is well developed and well known,

there are a number of respects in which it lacks realism to a point which militates against its practical use

without substantial modification. The main source of the limitations arise from the assumptions of

independency that are placed on the claim frequency and claim size distributions. For example, in home

and contents insurance where storm-water damage is covered, the rate of occurrence would be affected by

climate patterns which shows us the dependency of increments within the claim frequency process {N(t)}.The

claim amount themselves may be dependent .Hence consequently generating claims of similar amounts.

We can obviously see that the assumption of the independent increments would not be enough or sufficient in

modeling a real- world insurance process. Therefore dependence of parameters in the model would have

effects on the distribution of the aggregate claims {S (t)}, and the probability of ruin 𝝍 (u) and the time τ.

20

7. References

[1] Embrechts, P., Kaufmann, R., and Samorodnisky, G. (2002). Ruin theory revisited: stochastic models

for Operational Risk. Risk Management for Central Bank Foreign Reserves, European Central Bank,

Frankfurt, 243-261

[2] Burnecki , K., Härd le , W., and Weron , R . (2004). Simulation of Risk Process

Encyclopedia of Actuarial Science, Wiley, Chichester.,1564-1570

[3] Ammeter, H. (1948). A Generalization of the Collective Theory of Risk in

Regard to Fluctuating Basic Properties. Skand. Aktueriedskr. 31:171-198

[4] Embrechts, P. and Klüppelberg, C. (1993). Some Aspects of Insurance Mathematics. Theory Probb.

Appl. 38: 262-295

[5] Grandell, J. (1991). Aspects of Risk Theory. Springer. New York.

[6] Grandell, J. and Segerdahl, C.-0 (1971). A Comparison of Some Approximations of

Ruin Probabilities, Skand. Aktuarietidskr.144-158.

[7] De Vylder, F.E (1996). Advanced Risk Theory. A self-contained Introduction, Editions de l’Universite

de Bruxelles and Swiss Association of Actuaries.

[8] Grandell, J. (2000). Simple Approximations of Ruin Probabilities. Insurance Math. Economic. 26:157-

173

[9] De Vylder, F.E. (1978) A Practical Solution to the Problem of Ultimate Ruin Probability. Scand.

Actuar. J.: 114-119.

[10] B u r n e c k i , K ., M i s t a , P ., a n d W e r o n A . (2003). A N e w D e V y l d e r T y p e

Approximation of the Ruin probability in Infinite Time, Research Report Hsc/03/05. [11]Asmussen, S.

(2000). Ruin Probabilities, World Scientific, Singapore.

[12] Encyclopedia of Actuarial Science (2006) John Wiley & Sons, Ltd

[13] Kaas, Rob. Modern Actuarial Risk Theory. Hingham, MA, USA: Kluwer Academic

Publishers, 2001. p iii