For example, we can consider two insurance companies, A and B. A is a well known insurance company with a big capital and is dealing with a risk with a low variance. We will assume that the global risk of all its customers follow a chi-square distribution with one degree of freedom. Besides, we assume that the initial capital of A is 3 and the premium charged to its customers is 1.05. The reason why it is charged 1.05 is a bit intricate and is not of great importance for this post, I may explain it later, in another post. The idea is that A charges a premium slightly over the expectation of a chi-square with one degree of freedom.
The second firm B is in a very different situation. This is a very new insurance company with a very low capital (let's say 0.1). Its customers have claims which follow a chi square distribution with two degrees of freedom, which means that the insured product is more risky and the variance of the risk is higher too. B is likely to charge its customers with a higher premium than A. Indeed, B has a low capital, the expectation of the claim as well as the variance are higher. So we consider a premium of 3.
To sum up:
| Capital | Premium | Claims | |
| Firm A | k1 = 3 | premium1 = 1.05 | x1 ~ χ(1) |
| Firm B | k2 = 0.1 | premium2 = 3 | x2 ~ χ(2) |
Finally we do a last assumption which is the independence of the claims. This assumption which seems relevant is not such a common place in real insurance problem and is a real source of interesting issues.
Every day, the firms A and B receive their premiums and pay back the claims to their customers.
The probability of ruin is, in risk management, and in the current project of risk management rules for insurances (solvency 2), the measure of interest. For a certain number of days, the company A is in a ruin situation if there has been at least one day where the total amount of claims has been greater than the sum of the initial capital and the total amount of premiums. To estimate such a probability we can easily (at least in this case) simulate the model. The probability of ruin is the probability to be in such a situation. For example, in the next plot, we can see that in a particular simulation for the firm A, A experienced a ruin as soon as the day number 7.
To estimate the probability of ruin for a given number of days, we simulate by Monte Carlo method this situation. For 100 days with a sample of 100,000 simulations, the estimation of the probability of ruin of the company A is 0.66, for the company B, it is 0.41. What is interesting is that if the two companies were merging, which means if they were sharing their claims, theirs premiums and their initial capital, the estimated probability of ruin of the merge company is 0.23. We can see, that merging the company reduce the probability of ruin.
A limit could be pointed out. The problem in merging two companies is that if the merged company experiences a ruin, it is a very serious problem. Indeed, it may be considered as better to have one company ruined often, and very rarely both of them ruined, than a situation where the merged company is sometimes ruined... In our particular case, and this is why I choose two different profile of firms, this is not a relevant problem. The estimated probability of simultaneous ruin of both A and B is 0.26 which is slightly greater than the estimator of the ruin probability of the merged company.
The last plot is an example of a situation where the merged company
would not have been ruined while both A and B would have been ruined
within 100 days.
 |
| The red line is the company B while the black line is A and the purple line is the merged company. Both A and B have been ruined, while the merged company has not been ruined. |
The code (R):
premium1 = 1.05
premium2 = 3horizon = 100
length = 100000
# sum1 is the number of time company 1 is ruined
sum1 = 0
# sum2 is the number of time company 2 is ruined
sum2 = 0
# sum3 is the number of time companies 1 and 2 are ruined
sum3 = 0
# sum is the number of time the merge company is ruined
sum = 0
for(j in 1:length){
k1 = rep(0, horizon+1)
k2 = rep(0, horizon+1)
k = rep(0, horizon+1)
k1[1] = 3
k2[1] = 0.1
k[1] = k1[1]+k2[1]
x1 = rchisq(horizon, 1)
x2 = rchisq(horizon, 2)
ruinX1 = FALSE
ruinX2 = FALSE
ruinX = FALSE
for(i in 1:horizon){
k1[i+1] = premium1 + k1[i] - x1[i]
k2[i+1] = premium2 + k2[i] - x2[i]
k[i+1] = premium1 + premium2 + k[i] - x1[i] - x2[i]
if(k1[i+1]<0){
ruinX1 = TRUE
}
if(k2[i+1]<0){
ruinX2 = TRUE
}
if(k[i+1]<0){
ruinX = TRUE
}
}
if(ruinX1 & ruinX2){
sum3 = sum3 + 1
sum1 = sum1 + 1
sum2 = sum2 + 1
}
else if(ruinX1){
sum1 = sum1 + 1
}
else if(ruinX2){
sum2 = sum2 + 1
}
if(ruinX){
sum = sum + 1
}
}
# prob of ruin of A
prob1 = sum1 / length
# prob1
# prob of ruin of B
prob2 = sum2 / length
# prob2
# prob of ruin of the merged firm
prob = sum / length
# prob
# prob of ruin of A AND B
prob3 = sum3 / length
# prob3
plot(k1, type = 'l', xlab = 'days', ylab = 'available capital', ylim =c(min(k2, k, k1),max(k2, k, k1)))
lines(k2, col = 'red')
lines(k, col = 'purple')
abline(0, 0)
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