SOA Exam ASTAM Spring 2025, Question 4

Study guide: Exam ASTAM

An actuary is analyzing the reported losses from an auto insurance with an ordinary deductible of $d$ . Let $X$ denote the ground-up loss random variable, which is assumed to have a Pareto distribution with known $\theta$ and unknown $\alpha$ .

A random sample of 20 reported ground-up losses was collected. Note that losses below the deductible are not reported to the insurer, so only losses above the deductible were sampled.

Part a (3 points)

(i) Determine the log-likelihood function for the sample in terms of $\theta, \alpha, d,$ and the sample data, $x_1, x_2, \cdots, x_{20}$ .

(ii) You are given that $\theta = 5{,}000$ , $d = 1{,}000$ , and that $\displaystyle\sum_{i=1}^{20} \ln(\theta + x_i) = 181.66$ . Show that the maximum likelihood estimate of $\alpha$ , denoted $\hat{\alpha}$ , is 2.6 to the nearest 0.1. You should calculate the value to the nearest 0.001.

Part b (1 point)

Estimate a 95% confidence interval for $\alpha$ using Fisher's information.

Part c (2 points)

Let

$Y = \begin{cases} 0 & \text{if } X \le d \\ X - d & \text{if } X > d \end{cases}$

Calculate the maximum likelihood estimate of $E[Y]$ .

Part d (2 points)

For $X$ with a Pareto distribution with $\theta = 5000$ and $\alpha = \hat{\alpha}$ , show that the loss elimination ratio for an ordinary deductible of $d = 1000$ is 0.25 to the nearest 0.05. You should calculate the value to the nearest 0.001.

Part e (1 point)

The actuary's colleague notes that an ordinary deductible of $d = 1000$ produces a similar reduction in expected payment as does a 75% coinsurance.

Explain why an insurer may prefer to impose an ordinary deductible of 1000 over 75% coinsurance.

Answer this question and see the worked solution