Chapter17 - Ruin Theory

Question 1

Describe N(t) and state its distribution.

Answer 1

N(t) is the number of claims arising within a time period of [0,t] for all t>0.

If N(1) ~ Poi(lambda) then N(t)~Poi(lambda*t)
(This is called the Poisson Process)

Question 2

State the conditions for the claim number process.

Answer 2

1. N(0) = 0
2. N(t) must be integer value for all t>0
3. N(s)<N(t) for s<t - claims are non-decreasing over time
   4.When s<t, N(t)-N(s) represent the number of claims over the period (s,t]

Question 3

What type of Process is the Poisson Process?

Answer 3

A counting process

Question 4

State the conditions for a claim number process to be a Poisson process

Answer 4

1. N(0) = 0 and N(s)<=N(t) when s<t
2. Maximum of one claim in a very short time interval h
3. The number of claims in a time interval of length h does not depend on when that time interval starts
   4.For s<t , the number of claims in the time interval (s,t] is independent of the number of claims up to time s

Question 5

What is the distribution of T1, the time until the first claim?

Answer 5

T1~exp(lambda)

P(T1<= t) = 1-exp(-lamba*t)

Question 6

What is meant by the compound Poisson process?

Answer 6

The Poisson process for the number of claims will be combined with a claim amount distribution to give a compound Poisson process for the aggregate claims.

Question 7

State the 3 assumptions for the aggregate claims process S(t)
to be a compound Poisson process.

Answer 7

1. The random variables Xi are independently and identically distributed
2. The random variables Xi are independent of N(t) for all t>0
3. The stochastic process N(t) is a Poisson process whose parameter is lambda

Question 8

What is the kth moment about zero of Xi’s

Answer 8

m_k = E[x^k]

Question 9

Define the moment generating function Mx(r)

Answer 9

Mx(r)=E[exp(-r*x))

Where Mx(r) is dependent on the distribution of x

Question 10

What is the mean and variance of S(t)

Answer 10

If S(t)=x1+x2+…+xN then (N, number of claims and N~Poi(lambda)
E[s(t)] = lamdaE[x] = lambdam1
Var[s(t)]=lamdaE[x^2]=lambdam2

Question 12

Define and state Lundbergs inequality

Answer 12

Lundberg’s inequality tells us that we can find an upper bound for the probability of Ultimate ruin (Y(U))

Y(U) <=exp(-R*U)
R - the adjustment coefficient
U initial surplus

Question 13

What does large values of R, the Adjustment Coefficient, imply.

Answer 13

R measures risk - it in a inverse measure of risk. So a large R implies the probability of ultimate ruin, Y(U), reduces.

Question 14

What is the equation to calculate R?

Answer 14

lambda*Mx(r) = lambda + CR

Where C is the the premium income per unit of time

Question 15

What is the upper bound for R?

Answer 15

lambda + CR = lambdaMx(R)
lambda + CR = lambdaE[exp(Rx)]
lambda + CR = lambdaE[1+ Rx + (Rx)^2/2
lambda + CR = lambda(E(1) + RE(x) + 1/2R^2E(x^2)

lambda + CR = lambda(1+ Rm1 + 1/2R^2m2)
R< 2(c-lambam1)/lambdam2

If c = (1+o)lamda*m1 where o is the insurers loading factor, then

R<(2om1)/m2

Question 16

When can you define a low bound for R?

Answer 16

When the distribution of X, the individual claim amounts, has an upper limit, say M.

Question 17

What is the lower bound for R?

Answer 17

M is the upper bound to the distribution of X. e.g. X~uni(0,100), M=100

R>1/Mlog(c/lamda*m1)

If c=(1+o)lamda*m1 then

R>1/Mlog(1+o)

Question 18

Define proportional reinsurance.

Answer 18

alpha is the retained proportion that the insurer will pay of every claims.

Y=alphaX
Z=(1-alpha)X

Question 19

What is the mean and variance of the claim amount paid by the direct insurer and reinsurer under proportional reinsurance?

Answer 19

Y = amount paid by direct insurer
Z = amount paid by reinsurer

E[Y] = E[alphaX] = alphaE[X}
E[Z] = E[(1-alpha)X] = (1-alpha)^2E[X]

Var[Y] = Var[alphax] = alpha^2Var[X]
Var[Z] = Var[(1-alpha)X] = (1-alpha)^2Var[X]