SOA Exam ASTAM Fall 2023, Question 3

Study guide: Exam ASTAM

An actuary is analyzing a sample of 1000 claims, as part of a review of pricing assumptions. Claims are assumed to be independent and identically distributed. Let $x_i$ denote the amount of the $i^{th}$ claim in the sample.

You are given that the sample mean is $\bar{x} = 1015.92$ .

Part a (3 points)

The actuary first fits an exponential distribution to the claims data using maximum likelihood estimation. You are given that the maximum likelihood estimate of the exponential distribution parameter $\theta$ is $\hat{\theta} = \bar{x}$ .

(i) Show that the maximum value of the log-likelihood function under the exponential distribution is -7925 to the nearest 5. You should calculate the value to the nearest 0.1.

(ii) The actuary creates the following plot of the difference between the empirical distribution function and the fitted exponential distribution function for each data point, where the observations are sorted from low to high.

You are given that the critical value for the Kolmogorov-Smirnov test at the 5% significance level is $\dfrac{1.36}{\sqrt{n}}$ .

Conduct a Kolmogorov-Smirnov goodness of fit test at the 5% significance level. You should state clearly your conclusion.

Part b (4 points)

The actuary next fits a gamma distribution to the same claims data. You are given that:

The maximum likelihood estimate of the $\alpha$ parameter is $\hat{\alpha} = 0.92463$ .
$\sum_{i=1}^{1000}\ln x_i = 6293.4$ .

(i) Show that the maximum likelihood estimate of the gamma distribution parameter $\theta$ is $\hat{\theta} = 1100$ to the nearest 1 (as printed; the solution gives 1098.73). You should calculate the value to the nearest 0.01.

(ii) Show that the maximum value of the log-likelihood function under the gamma distribution is -7920 to the nearest 5. You should calculate the value to the nearest 0.1. (Hint: You can use the GAMMA function in Excel.)

(iii) The actuary creates the following plot of the difference between the empirical distribution function and the fitted gamma distribution function for each data point, where the observations are sorted from low to high.

Conduct a Kolmogorov-Smirnov goodness of fit test at the 5% significance level. You should state clearly your conclusion.

Part c (3 points)

State with reasons which of the two distributions is preferred based on each of the following criteria:

(i) The Akaike Information Criterion (AIC).
(ii) The Schwartz Bayes Criterion (SBC/BIC).
(iii) The likelihood ratio test at the 5% significance level.

Part d (1 point)

The actuary is concerned about the risk of claims over 6000.

(i) Calculate the probability that a randomly selected claim exceeds 6000 using the fitted exponential distribution.
(ii) Calculate the probability that a randomly selected claim exceeds 6000 using the fitted gamma distribution.

Part e (1 point)

For the task of determining margins for solvency in reserve calculations, it is important to adequately reflect the tail risk of losses. State with reasons which of the two distributions you would recommend for this task.

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