# 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!

• I think this would be better in the math or stats stackexchange sites. But there may be some people on here who can answer. – Slade Nov 17 '19 at 16:21
• @Slade okay understood. Thank you. – James Debenham Nov 17 '19 at 18:01

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 am having trouble calculating the expected value of $S(t)$. I know that the formula is given by $S(t) = t * \lambda * m_1$ where, in this case, $t=8$, $\lambda = 10$ and $m_1$ is the first moment of the i-th claim amount $X$. How do I calculate $m_1$? – James Debenham Nov 18 '19 at 14:43
• $m_1$ is the expected value of $X_1$, in notation $m_1 = \mathbb{E}[X_1]$. We know that $X_1$ has an exponential distribution so we only need to find out how to compute the expected value of an exponentially distributed random variable. – Cettt Nov 18 '19 at 14:52
• Can't thank you enough for your clear and helpful responses. Say I wanted to apply this question to calculate $ψ(u)$, using Lundberg's inequality, would I be able to go about it using the results I have established from (a) and (b)? – James Debenham Nov 19 '19 at 12:43
• Hi, I am not fluent on ruin theory :). Better ask a new question where you explain what $\psi$ is. – Cettt Nov 19 '19 at 12:52