Course 3: Nov 2001 Prob 14

[aniasat]

Problem no 14 in Nov 2001 goes as follows:

You are simulating a continuous surplus process, where claims occur according to a Poisson process with frequency 2, and severity is given by a Pareto distribution with parameters a = 2 and q = 1000 . The initial surplus is 2000, and the relative security loading is 0.1. Premium is collected continuously, and the process terminates if surplus is ever negative. You simulate the time between claims using the inverse transform method (where small numbers correspond to small times between claims) using the following values from the uniform distribution on [0,1]: 0.83, 0.54, 0.48, 0.14. You simulate the severities of the claims using the inverse transform method (where small numbers correspond to small claim sizes) using the following values from the uniform
distribution on [0,1]: 0.89, 0.36, 0.70, 0.61.

Calculate the simulated surplus at time 1.

(A) 1109
(B) 1935
(C) 2185
(D) 4200
(E) Surplus becomes negative at some time in [0,1].

The solution uses the inverse transform method by substituting F(x) = u (the random numer generated) and then inverting it.

However when Mahler discusses simulation of Poisson Process he has asked us the use S(x) = u for the Poisson process where S(x) is the survival function.

Does the SOA solution use F(x) = u coz it is written that we are supposed to use Inverse Transform???

Also do the statements like " large random numbers are associated with large losses" have any impact on the solution of a problem.

Thank you very much in advance.

Best of luck!!!

Cheers,
anita.

[Macroman]

Problem no 14 in Nov 2001 goes as follows:

....

Also do the statements like " large random numbers are associated with large losses" have any impact on the solution of a problem.



If the loss amount has an upper bound as in losses are distributed U(0,1000) then it is equally plausible to say "small random numbers are associated with large losses"....ie random numbers on U(0,1)
rand num............ loss
0 .......................1000
0.1....................... 900
0.9 ...................... 100
....

You can use S(x), S(x) = 1 - F(x). The SOA is basically doing this conversion at a later stage with the same results. I think converting to the survival function first generally makes more sense.

[Gandalf]

If the loss amount has an upper bound as in losses are distributed U(0,1000) then it is equally plausible to say "small random numbers are associated with large losses"....ie random numbers on U(0,1)
rand num............ loss
0 .......................1000
0.1....................... 900
0.9 ...................... 100
....

You can use S(x), S(x) = 1 - F(x). The SOA is basically doing this conversion at a later stage with the same results. I think converting to the survival function first generally makes more sense.

Why does it mattter if the loss amount has an upper bound? Couldn't you use S(x) regardless?

[Macroman]

Sure you can use S(x) regardless...

You could also always transform the losses to reverse order losses regardless, because even if the losses themselves are not specified with an upper bound, the random numbers that could be produced by any given workable scheme would have an upper bound. IE usually they will say that random numbers are generated from a U(0,1) distribution, They could specify something else but the random numbers need to have some kind of upper limit and that upper limit needs to translate into some definite loss on the loss distribution.

[Toonces]

Also do the statements like " large random numbers are associated with large losses" have any impact on the solution of a problem.
That statement is exactly what you use to determine whether you use F(x)=u or S(x)=u which is the same thing as F(x)=1-u.

In this example, the first 2 claim size simulations are .89 and .36. Using S(x)=u, you get claim sizes of 60 and 667. Using F(x)=u, you get claim sizes of 2015 and 250. Since Large random numbers are associated with large losses, the second way is what they want, not the first way.

[Macroman]

thanks Toonces, nice concise explanation..

[boognish]

Use exponential to simulate times b/w claims, b/c its a poisson process? Or can you take the product of the random numbers and see if its equal to 1-p(0) for a poisson.

Thanks.

[Macroman]

The solution requires use of the cumulative Poisson distribution. The random number from the arrival time data is translated back to an arrival time using the inverse of the cumulative distribution function.