Compound Poisson Claims and Surplus Moments in Ruin Theory

# Statistics related question about ruin theory


# Statistics related question about ruin theory












I am trying to solve the following problem:

'An insurance company has an initial surplus of 150 and premium loading factor of 15%. Assume that claims arrive according to a compound Poisson process $(S(t))_{t≥0}$ with parameter $λ = 10$ and claim size $X_i ∼ exp( 1/20 )$. The time unit is 1 week. Assume that 1 month is 4 weeks.'

(a) Calculate the average number of claims on any given day, week and month, and the probability that at least one claim occurs within the next 3 days. Calculate also the probability that at least 3 claims occur in the next 3 days.

(b) Let t = 2 months. Calculate the mean and variance of $S(t)$ and of $U(t)$.

(For reference in case of different notation usage, $S(t)$ represents the aggregate claim amount i.e. the claims paid, and $U(t)$ denotes the surplus process. $U(t) = u + ct - S(t)$ where $c$ is the rate of income of premiums per unit time, $t$ is time).

This question popped up as an exercise regarding the topic of ruin theory. I know it is heavily intertwined with statistical theory, but I hope I am posting this question to the relevant page. I'm not too sure how to begin this question so any explanations or pointers would be helpful. Do let me know if anything extra needs clarifying. Thank you!

## Answer by Cettt (score 2, accepted)

https://quant.stackexchange.com/a/49777

this is classical Cramer Lundberg Model in ruin theory. In it the total number of claims is modeled using a compound Poisson process: $$ S(t) = \sum_{k = 1}^{N(t)} X_k, $$ where $X_1 \sim Exp(0.05)$.

And the surplus is given by $$ U(t) = u + c \cdot t - S(t), $$ where $u$ is the initial surplus (in your case $u = 150$) and $c$ is equal to $15\%$.

Part (a) only deals with the total number of claims but not the size. The number of claims is modeled with a Poisson Process with parameter $\lambda = 10$. This means that the total number of claims after $t$ weeks has a Poisson distribution with parameter $t\cdot \lambda$.

All you need to know to solve (a) is therefore characteristics of a Poisson distribution: if $X \sim Pois(\lambda)$ than the expectation of $X$ is equal to $\lambda$. For example the expected number of claims in one week ($t = 1$) is the expected value of a random variable which has a $Pois(10)$ distribution. Therefore this expectation is equal to 10.

For (b) note that you have to first calculate the mean and variance of $S(t)$. You can find the appropriate formula in the wikipedia link about Compound Poisson Processes. The mean and variance of $U(t)$ are then very easy to calculate: $$ \mathbb{E}[U(t)] = 150 + 0.15t - \mathbb{E}[S(t)], \quad Var(U(t)) = Var(S(t)). $$

I hope this is helpful.