# Classical Cramer Lundberg model - Ruin Theory Simple Question

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 $$X_i \sim \text{Exp}( \frac{1}{20} )$$. The time unit is 1 week. Assume that 1 month is 4 weeks.'

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

$$E[S(t)] = \lambda t m_1 = 1600$$, $$Var[S(t)] = \lambda t m_2 = 64,000$$, $$E[U(t)] = u + ct - E[S(t)] = -1448.8$$, $$Var[U(t)] = Var[S(t)] = 64,000$$

(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).

Now, the question I am trying to answer, using the information I've already worked out above is:

• Calculate the ruin probability $$ψ(u)$$, using Lundberg's inequality.

• Now calculate the probability of ruin at t=2 months, assuming that the surplus process at any given time can be approximated with a suitable Normal distribution.

How do these two results compare to eachother?

Does anyone have any ideas? Am I posting in the right stack exchange for Financial Mathematics related questions? Thanks.

• Lundberg's Inequality is $\psi(u)\le \exp(-r^* u)$. You may want to refresh your memory how to calculate $r^*$, the so-called "adjustment coefficient". – Alex C Nov 22 '19 at 4:58