SOA Exam ALTAM Spring 2024, Question 4

Study guide: Exam ALTAM

An insurer is analyzing its long term care insurance data. The objective of the study is to estimate transition intensities for the underlying multiple state model illustrated in the following diagram.

Transition intensities are assumed to be constant over each year of age.

Let $T_x^{(j)}$ denote the total observed waiting time in State $j$ for lives between ages $x$ and $x + 1$ , and let $D_x^{ij}$ denote the total number of transitions from State $i$ to State $j$ by lives between ages $x$ and $x + 1$ .

Part a (2 points)

At the start of the two-year observation period, a person age 70.2 is in State 0. At age 71.1 she transitions to State 1, and at age 71.4 she transitions back to State 0. At age 71.9 she transitions to State 2, where she remains until the end of the observation period.

(i) Determine the person's contribution to $T_{70}^{(j)}$ and $T_{71}^{(j)}$ for $j = 0, 1, 2$ .

(ii) Write down the person's contribution to $D_{70}^{ij}$ and $D_{71}^{ij}$ for $i, j = 0, 1, 2$ , where $i \ne j$ .

Part b (4 points)

For a given age $x$ :

(i) Write down the log-likelihood function for estimating the transition intensities $\mu_x^{ij}$ , in terms of $T_x^{(j)}$ and $D_x^{ij}$ for $i, j = 0, 1, 2, 3$ and $i \ne j$ .

(ii) Derive the formula for the maximum likelihood estimator of $\mu_x^{01}$ .

(iii) State with reasons whether the maximum likelihood estimators, $\hat{\mu}_x^{0j}$ , are dependent or independent for different $j$ .

(iv) Derive a formula for the standard deviation of the maximum likelihood estimator, $\hat{\mu}_x^{01}$ .

Part c (3 points)

You are given the following aggregated information from the study.

$x$	$T_x^{(0)}$	$T_x^{(1)}$	$T_x^{(2)}$	$D_x^{01}$	$D_x^{10}$	$D_x^{02}$	$D_x^{12}$	$D_x^{23}$
70	3214.0	632.4	317.3	150	42	40	15	20
71	3019.7	640.4	350.7	170	40	55	16	30

(i) Show that $\hat{\mu}_{70}^{0\bullet} + \hat{\mu}_{71}^{0\bullet}$ is 0.13 to the nearest 0.01. You should calculate the value to the nearest 0.001.

(ii) Calculate ${}_2\hat{p}_{70}^{\overline{00}}$ , i.e. the maximum likelihood estimate of ${}_2p_{70}^{\overline{00}}$ .

(iii) Calculate an approximate 95% linear confidence interval for $\mu_{70}^{0\bullet} + \mu_{71}^{0\bullet}$ .

(iv) Calculate an approximate 95% confidence interval for ${}_2p_{70}^{\overline{00}}$ .

Answer this question and see the worked solution

SOA Exam ALTAM Spring 2024, Question 4

Part a (2 points)

Part b (4 points)

Part c (3 points)

Concepts tested

More questions from this set