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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 $U(t)$.

ANSWER:

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

(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.

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    $\begingroup$ 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". $\endgroup$ – Alex C Nov 22 '19 at 4:58

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