NCD system-Markov chain

[guest009]

Hello,
Can someone help me solve this question?
especially calculating (a) and (c) -(the transition matrix)


NCD system:
Probability of making one claim in a given year is 0.1
Level 0- 0% discount
Level1 - 20% discount
Level 2 - 50% discount
The rules for moving between these levels are:

Following a year with one claim, move back two levels,or move to 0 from level 1 or remain at 0

If the insurer didn't have an accident last year but had one the year before, he moves back one level ore remains at 0

If the insurer didn't have an accident the last two years he moves to the next higher level, or remains at 2

Xn - denotes the level of the policyholder in year n

a)calculate:
P(X3=1|X1=1,X2=0)
P(X3=1|X1=0,X2=0)

b)explain why Xn isn't a Markov chain

Expanding the transition matrix:
01 paying full premium and had an accident last year
02 paying full premium and last accident was two years ago


c)Write down the transition matrix of this chain. Is it irreducible?

d)Find the average number of years the insurer moves to level 2, known that now he is paying full premium after last year he had an accident.

e)The cost of a full premium is 5000.Find the average premium the insurer is paying for one year.

f)Knowing the insurer is at level 2, find how the number of years until he first reaches level 0 is distributed.

 

[jensen]

Hello

Have you got your transition matrix out?

 

[guest009]

I think it should be the following matrix, but I have no idea if this is the correct answer.Hope someone can help me figure it out.

01 02 1 2
01 0.1 0.9 0 0
02 0.1 0 0.9 0
1 0 0.1 0 0.9
2 0 0 0.1 0.9

 

[jensen]

Assuming your matrix is right, you should be able to get part (a) by multiplying your matrix three times then read off the appropriate entry base on your state definitions.

Is this a past year question?

 

[didster]

Assuming your matrix is right, you should be able to get part (a) by multiplying your matrix three times then read off the appropriate entry base on your state definitions.

That's not quite right. It's asking the probability of going to level 1 from level 0 knowing what level you were at the time before, ie only one time period under consideration, so multiplying a matrix 3 times isn't really right. You could generate an appropriate matrix to read off but I imagine that this part is meant to be done from basics (see below) as you create a transition matrix lower down.

Following a year with one claim, move back two levels,or move to 0 from level 1 or remain at 0

If the insurer didn't have an accident last year but had one the year before, he moves back one level ore remains at 0

If the insurer didn't have an accident the last two years he moves to the next higher level, or remains at 2

Xn - denotes the level of the policyholder in year n

a)calculate:
P(X3=1|X1=1,X2=0)
P(X3=1|X1=0,X2=0)

You sure this description is accurate? It's not very logical, ie if you had an accident last year, you'll be at level 0 this year and no need to say you move back another level this year (no accident this year but one the year before). It could just be a trick to test understanding but maybe not.

x1=1,x2=0 so you had an accident in year 1, so even if you don't have an accident in year 2 you remain at 0 because you had one the year before. So P(X3=1|X1=1,X2=0)=0 (assuming your description is accurate)

Second one doesn't have enough info (Again I question your description's accuracy)

 

[jensen]

oops! Need to brush up my CT4