SOA Exam ALTAM Fall 2025, Question 4

Study guide: Exam ALTAM

An insurer is analyzing its mortality experience, using the Alive-Dead model, assuming the force of mortality $\mu_x$ is constant within the age range $[x, x+1)$ . There are $n$ independent lives in the group.

Let $t_j$ denote the waiting time in $[x, x+1)$ for the $j$ -th life in the group, and let $\delta_j = 1$ if the $j$ -th life died in $[x, x+1)$ , $\delta_j = 0$ otherwise.

You are given that the contribution of the $j$ -th life to the likelihood function for $\mu_x$ may be written as $f_j\left(t_j, \delta_j\right) = e^{-t_j\mu_x}\left(\mu_x\right)^{\delta_j}$ .

Part a (5 points)

(i) Briefly describe the circumstances under which $t_j < 1$ and $\delta_j = 0$ for the $j$ -th life.

(ii) Write down the equation for log-likelihood function for $\mu_x$ in terms of $t_j$ and $\delta_j$ , $j = 1, 2, \cdots, n$ .

(iii) Show that the maximum likelihood estimator for $\mu_x$ is $\hat{\mu}_x = \dfrac{d_x}{E_x^c}$ , where $E_x^c$ and $d_x$ are functions you should specify.

(iv) Show that $\text{Var}\left[\hat{\mu}_x\right] \approx \dfrac{d_x}{\left(E_x^c\right)^2}$ .

Part b (1 point)

The following data have been collected for age 65 for two different policy lines. Assume all lives are independent.

$x$	Line A: $E_x^c$	Line A: $d_x$	Line B: $E_x^c$	Line B: $d_x$
65	150,150	1,550	18,070	132

(i) Calculate $\hat{\mu}_{65}$ for each line.

(ii) Estimate the standard deviation of $\hat{\mu}_{65}$ for each line.

Part c (2 points)

Show that the age 65 forces of mortality for the two lines are significantly different from each other, at a 95% confidence level.

Part d (2 points)

Write down three possible reasons why the mortality experience of the two lines might be significantly different from each other.

Answer this question and see the worked solution

SOA Exam ALTAM Fall 2025, Question 4

Part a (5 points)

Part b (1 point)

Part c (2 points)

Part d (2 points)

Concepts tested

More questions from this set