Handout on a compound Poisson process and
ruin probability WITH ANSWERS.
For a particular group of clients, a flow of claims arriving at an
insurance company may be represented as a compound Poisson process, that is, we
deal with the classical scheme. Let the amount of a particular claim be equal
to either 2 or 3 or 4 with probabilities 1/4, 1/2, and 1/4, respectively. Let
the mean number of claims the company receives per day is 10. Assume that the
company chooses for its activity a relative loading coefficient \theta =0.1.
1.
Assume that
the third claim has just arrived. What is the expected waiting time for the
fourth claim? 1/10.
What is the
probability that the fourth claim will come in less than three hours? 1-exp{-(3/24)*10}.
2.
What is the probability that during two days
less than 25 claims will come? SUM_{k=0}^24
[(exp{-20}}(20)^k/k!]
3.
What is the
expected value of an individual claim?
3.
4.
What is the
expected value of the aggregate claim the company receives per day, per hour? 30,
30/24. What premium does the company get per day, per hour? 33, 33/24.
5.
Write an
equation for the adjacent coefficient R. (1/4)exp{2R}+(1/2)exp{3R}+(1/4)exp{4R}=1+3.3R
6.
Does this
equation involve \lambda ? NO.
7.
Solve the
equation numerically, that is, find an approximate solution (in any way you
like). I got around .055
8.
Find an
approximate value of the initial capital for which the ruin probability for the
company will be less than 0.03. ln(100/3)/(.055).
About 64.
9.
Think how to
answer all questions above, if a particular claim is, say, uniformly
distributed on [2,4]. (You are not required to calculate everything.
The first four
answers are the same. In Question 5 you should replace
(1/4)exp{2R}+(1/2)exp{3R}+(1/4)exp{4R}
by [exp{4R}-exp{2R}]/2R.
10. Think how to answer all questions above, if the number of
claims during a day is exactly equal to 10, that is, we consider the discrete
time scheme, and day is a unit of time.
The first
four answers are the same. In Question 5 the equation is
(1/4)exp{2R}+(1/2)exp{3R}+(1/4)exp{4R}=exp{3.3R}.