Suppose that claims are made to an insurance company according to a Poisson proc

Question

Suppose that claims are made to an insurance company according to a Poisson
process with rate 10 per day.The amount of a claim is a random variable that has
an exponential distribution with mean $1000. The insurance company receives
payments continuously in time at a constant rate of $11,000 per day. Starting
with an initial capital of $25,000, use simulation to estimate the probability
that the firm’s capital is always positive throughout its first 365 days.

Preferably in Python, but help in any programming language accepted.

Explanation / Answer

Since it is not mentioned we cana use programming language other than python. I have solved it using R language.

If you are not familier with R then you can take a look at the code and try to understand the logic, So you will atlease get an idea of pseudo code.

Which can easily be converted to any programming language easily.

Ans:

Since claims are made under poission process so time difference between two claim arrival will follow exponential distribution. (Interarrival time are exponentially distributed)

R Code:

probabilityOfReachingNegative = c();
probabilityOfAlwaysPositive = c();
for (w in 1:15)
{
reachedNegative= c();
for (k in 1:500)
{
total=0;
i=1;
claimInterArrivalTime= c();
capital = 25000;
while(1==1)
{
temoExpoRandom = rexp(1,10);
total= total+temoExpoRandom;
if(total>=365)
{
break;
}
  
claimInterArrivalTime[i] = temoExpoRandom;
  
i = i+1;
}
  
amount=c();
for (i in 1:length(claimInterArrivalTime))
{
amount[i] = rexp(1,1/1000);
}
reachedNegative[k]=0;
for (i in 1:(length(claimInterArrivalTime)))
{
  
capital = capital + 11000*(claimInterArrivalTime[i])-amount[i];
  
if(capital<=0)
{
reachedNegative[k]=1;
break;
}
}
capital = capital + (365-(total-claimInterArrivalTime[length(claimInterArrivalTime)]))*11000;
}
probabilityOfReachingNegative[w] = sum(reachedNegative)/length(reachedNegative);
}

probabilityOfAlwaysPositive = 1- probabilityOfReachingNegative;

Output:

Probability of always positive capital (15 Runs):

0.894, 0.896, 0.900, 0.900, 0.916, 0.912, 0.924, 0.906, 0.898, 0.908, 0.896, 0.904, 0.884, 0.924, 0.892

Avg = 0.9036

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