SOA Exam ASTAM Sample Question 13

Study guide: Exam ASTAM

An insurance company offers telematics insurance to its automobile insurance policyholders. For a portfolio with $n$ new policies, a total of $M$ claims is recorded in the first $t_0$ months of cover. You may assume that each policy gives rise to a maximum of one claim and that claims arise independently of the others.

Assume that for each policy, the time (measured in months) until a claim occurs is modeled as a random variable $T$ , with

$\Pr\left[T \le t\right] = F(t) = 1 - e^{-\beta t} \qquad t > 0, \quad \beta > 0$

Part a

Write down the likelihood of observing $M$ claims in $t_0$ months.

Part b

Determine the maximum likelihood estimate of $\beta$ in terms of $M$ , $t_0$ , and $n$ .

Part c

Explain, but do not perform the calculations, how you could construct an approximate confidence interval for $\beta$ .

Part d

Explain carefully how you can use your estimated value of $\beta$ to estimate the average time until a claim occurs, denoted $\mu$ .

Part e

Explain carefully how you can estimate the variance of your estimate for $\mu$ .

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