SOA Exam ALTAM Spring 2023, Question 4

A Continuing Care Retirement Community (CCRC) offers new residents Modified Life Care contracts.

The CCRC contains three different types of accommodation: Independent Living Units (ILU), Assisted Living Units (ALU), and Specialized Nursing Facilities (SNF). The following Markov model is used to determine fees.

You are given:

(i) The CCRC incurs the following monthly costs per resident, payable continuously, by type of accommodation.

Accommodation	Monthly cost
ILU	3,000
ALU	7,000
SNF	15,000

(ii) Each resident pays a lump sum fee immediately on entry and pays residence fees continuously while in the CCRC.

(iii) Residence fees are a fixed proportion, $\gamma$ , of the monthly costs.

(iv) The value of $\gamma$ is determined such that the EPV of future costs is equal to the EPV of all fees, including the entry fee.

(v) $i = 0.05$

(vi) The following actuarial functions, evaluated at 5%.

$x$	$\bar{a}_x^{00}$	$\bar{a}_x^{01}$	$\bar{a}_x^{02}$	$\bar{a}_x^{11}$	$\bar{a}_x^{12}$	$\bar{a}_x^{22}$
65	11.4106	1.3570	0.3745	11.8352	0.7979	10.6905
70	9.5210	1.7037	0.4942	10.1754	0.9960	9.1961

An individual enters the CCRC, in an ILU, at age 65. Their entry fee is 150,000.

Part a (1 point)

Show that the expected present value of future costs is 592,200 to the nearest 100. You should calculate the value to the nearest 10.

Part b (2 points)

Show that $\gamma = 0.75$ to the nearest 0.01. You should calculate the value to the nearest 0.001.

Part c (2 points)

Let ${}_tV_x^{(j)}$ denote the EPV at time $t$ of the CCRC's future costs minus future income, for an individual who entered the CCRC at age $x$ , and who is now in state $j$ at age $x+t$ .

Show that ${}_5V_{65}^{(1)}$ is 261,900 to the nearest 100. You should calculate the value to the nearest 10.

Part d (4 points)

You are also given:

${}_5V_{65}^{(2)} = 419{,}280$
$\mu_{70}^{12} = 0.00156$ and $\mu_{70}^{13} = 0.01194$

(i) (1 point) Write down Thiele's differential equation for ${}_tV_{65}^{(1)}$ .

(ii) (1 point) Show that $\dfrac{d}{dt}\left({}_tV_{65}^{(1)}\right)$ evaluated at $t = 5$ is $-5{,}620$ to the nearest 10. You should calculate the value to the nearest 1.

(iii) (2 points) Using Euler's forward method, with a step size of $h = 0.25$ , calculate ${}_{5.25}V_{65}^{(1)}$ .

Part e (1 point)

The CCRC introduces a Full Life Care contract option, under which residence fees are level throughout their residency, regardless of the accommodation type. Entry fees are the same for both contract options.

State with reasons whether an individual would pay a higher residence fee while in ILU under the Full Life Care contract or the Modified Life Care contract.

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