impact of increase t on probability of ruin

[Molly]

Hi, bit of a general question here but the figures i have used are from sept 2020 question 8 part iii-> part iv

Completely understand the theory of this - when t increases the probability of ruin will also increase.

However i just wanted to test this on excel for increasing values of t - for September 2020 question 8 part iii)
in this example,
E(S)=2500
Var(S)=1654167
U=2000
c=10
no. policies=500

so for t=1, x=2000+500*10*1=7000
using 1-norm.dist(x, mean, sd, true) i get the correct answer for part iii which is 0.0002337

however i then tested calculating the prob of ruin for increased values of t
for example i used t=1.25
now x=2000+500*10*1.25=8250
using 1-norm.dist(x, mean, sd, true) i get 0.0000039

so the prob of ruin i have found for t=1.25 is actually less than what i found for t=1 which isnt right.

Does anyone know what has happened here? am i doing something wrong?

thank you!

 

[John Potter]

This hits upon one of the crucial points of Ruin theory to understand and why it is quite hard for the Examiners to set a meaningful question on the topic for Paper A. Without looking in detail at the precise questions you quote, it looks like you're calculating the probability of ruin AT time t, which is actually a lower bound for probability of ruin BY time t.

Let's say that P[ruin AT time 5] = 0.02
and P[ruin AT time 6] = 0.01

Then we can say that P[ruin BY time 6] >= 0.02

The P[ruin AT time n] could be an increasing/constant/decreasing function of n, depending on how big the premiums versus the claims are. The P[ruin BY time n] must be an increasing function of n

 

[Molly]

Hi John,

Ah thank you so much! That makes much more sense.

So for this example - i find that at time 0, P[ruin AT time 0]=0.6513.
since this example is a decreasing function of N, then i can say that for any time T the probability of ruin BY time T >=0.6513, since no value of t gives a greater probabiltiy than at time 0, and by your response it seems that for P[ruin BY time T] we take the largest value of P[ruin BY time t] such that t<=T?

Thank you!

 

[John Potter]

Molly,

I think you mean "since no value of t gives a greater probability than at time 0, and by your response it seems that for P[ruin BY time T] we take the largest value of P[ruin AT time t] such that t<=T?"

Good idea but, no, this isn't true either. If we find the largest value of P[ruin AT time t] for values of t < T, all we can stay still is that is a lower bound for the P[ruin BY time T].

Using my numbers again...

P[ruin AT time 5] = 0.02
P[ruin AT time 6] = 0.01

And let's also hypothesise that there are no claims except at times 5 and 6 so that ruin is impossible at other times.

Even then, all we can say is that P[ruin BY time 6] >= 0.02. What we really need to know is
P[ruin AT time 6 | not ruined at time 5], then we could calculate an actual answer.

In Paper B, we can simulate claims and know for sure that the only time we can go bust is at the times of the claims. So more meaningful questions can be asked.

John

 

[Molly]

Molly,

I think you mean "since no value of t gives a greater probability than at time 0, and by your response it seems that for P[ruin BY time T] we take the largest value of P[ruin AT time t] such that t<=T?"

Good idea but, no, this isn't true either. If we find the largest value of P[ruin AT time t] for values of t < T, all we can stay still is that is a lower bound for the P[ruin BY time T].

Using my numbers again...

P[ruin AT time 5] = 0.02
P[ruin AT time 6] = 0.01

And let's also hypothesise that there are no claims except at times 5 and 6 so that ruin is impossible at other times.

Even then, all we can say is that P[ruin BY time 6] >= 0.02. What we really need to know is
P[ruin AT time 6 | not ruined at time 5], then we could calculate an actual answer.

In Paper B, we can simulate claims and know for sure that the only time we can go bust is at the times of the claims. So more meaningful questions can be asked.

John

Hi John, thank you very much that makes more sense - i now see what you mean when you say "why it is quite hard for the Examiners to set a meaningful question on the topic for Paper A" - Thank you!