SOA Exam ALTAM Sample Question 6

Study guide: Exam ALTAM

You are analyzing waiting times and expenses associated with optometric procedures. The analysis is based on the multiple state model illustrated below.

An individual who is diagnosed with a specific eye condition moves into State 1, where they are Monitored. When the condition progresses, they move onto a Wait List for surgery (State 2). Surgery (State 3) is performed once their name reaches the top of the wait list. The time spent on the wait list is random, depending on the progression of the disease and the availability of surgical resources.

You are given the following information:

(i) $\mu_{x+t}^{01} = a + be^{c(x+t)}$ where $a = 0.004$ , $b = 0.015$ , and $c = 0.005$ ;

(ii) $\mu_{x+t}^{12} = 0.2; \quad \mu_{x+t}^{23} = 0.4.$

(iii) $\delta = 0.05$

Part a (2 points)

Calculate the probability that a Healthy life age 60 will be diagnosed with the eye condition before age 65.

Part b (1 point)

Show that ${}_tp_x^{12} = \left(e^{-0.2t} - e^{-0.4t}\right)$ .

Part c (1 point)

Calculate the probability that a life currently being Monitored (State 1) will have Surgery (State 3) within 5 years.

Part d (2 points)

(i) Show that $\bar{a}_{x:\overline{5}|}^{11} = 2.85$ to the nearest 0.01. You should calculate the value to the nearest 0.001.

(ii) Show that $\bar{a}_{x:\overline{5}|}^{12} = 0.87$ to the nearest 0.01. You should calculate the value to the nearest 0.001.

(iii) Show algebraically that $\bar{a}_{x:\overline{5}|}^{11} + \bar{a}_{x:\overline{5}|}^{12} + \bar{a}_{x:\overline{5}|}^{13} = \bar{a}_{\overline{5}|}$

Part e (3 points)

The health authority is considering two alternatives for improving care for those diagnosed with the eye condition. One criterion for deciding which course of action to choose is based on a quality of life (QL) index, as follows:

In State 1, $QL = 0.5$ per year.
In State 2, $QL = 0.2$ per year
In State 3, $QL = 1.0$ per year.

The authority will choose one of the following options:

Option A: Open a new surgical center. This would increase $\mu_x^{23}$ to 1.0 without changing any other transition intensities.

Option B: Improve care for those in States 1 and 2. This would improve the QL index to 0.55 in State 1, and to 0.3 in State 2.

Calculate whether the health authority should select Option A or Option B, based on the increase in the expected present value of the 5-year QL of lives currently in State 1, using a force of interest of $\delta = 0.05$ .

(LTAM, Fall 2021 Form B, Question 3)

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