SOA Exam ALTAM Sample Question 10

Study guide: Exam ALTAM

You are using the following multiple state Markov model to analyze a long term care product which provides coverage until the death of the insured. Assume all relevant transition intensities exist.

You are given the following information:

(i) The insurance pays a benefit of 3000 per year, payable continuously, while the policyholder is in State 2.

(ii) Policyholders may opt to have benefits start immediately on transition into State 2 or, for a lower premium, may select a 6-month waiting period before benefits begin.

(iii) Benefits are valued at a force of interest of $\delta = 0.05$

(iv) Transition intensities for all ages $x \ge 90$ are

$\mu_x^{12} = 0.10 \quad \mu_x^{13} = 0.04 \quad \mu_x^{14} = 0.10$

$\mu_x^{23} = 0.04 \quad \mu_x^{24} = 0.30 \quad \mu_x^{34} = 0.20$

Part a (2 points)

Derive the Kolmogorov forward differential equation for ${}_tp_x^{12}$ .

Part b (1 point)

Show that ${}_tp_{90}^{12} = e^{-0.24t} - e^{-0.34t}$

Part c (2 points)

Calculate the expected present value of the LTC benefit in State 2 for a life currently age 90 and in State 1, assuming benefits begin immediately on transition.

Part d (1 point)

$\bar{a}_{90:\overline{0.5}|}^{\overline{22}} = 0.45$ to the nearest 0.01. Calculate $\bar{a}_{90:\overline{0.5}|}^{\overline{22}}$ to the nearest 0.001.

Part e (2 points)

Calculate the expected present value of the LTC benefit in State 2 for a life currently age 90 and in State 1, assuming a waiting period of 6 months between transition to State 2 and the start of the benefit payments.

Part f (2 points)

Suggest two reasons why an insurer uses waiting periods in long term health products.

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