Actuarial Notes

SOA Exam ALTAM Sample Question 18

Study guide: Exam ALTAM

The mortality of a couple, (x) and (y), is modeled using the Markov multiple-state model described in the following diagram:

You are given:

  • \mu_{x+t:y+t}^{01} = A + Bc^{x+t} \quad\mu_{x+t:y+t}^{02} = A + Bc^{y+t} \quad\mu_{y+t}^{13} = D + Bc^{y+t} \quad\mu_{x+t}^{23} = E + Bc^{x+t}
  • A = 0.0001 \quad B = 10^{-5} \quad c = 1.12 \quad D = 0.00015 \quad E = 0.0002

Part a (1 point)

State with reasons whether (x) and (y) have independent future lifetimes under this model.

Part b (3 points)

(i) Write down the Kolmogorov forward differential equation for {}_tp_{xy}^{00}, and give the associated boundary condition.

(ii) Starting from the equation in (i), prove that

{}_tp_{xy}^{00} = \exp\left\{-\int_0^t \left(\mu_{x+r:y+r}^{01} + \mu_{x+r:y+r}^{02}\right)dr\right\}.

Part c (3 points)

A couple, who are ages x = 50 and y = 55, purchases a special single premium, deferred joint and last survivor annuity. The annuity will pay 50,000 per year while both are alive, and will pay 30,000 per year while only one is alive. Payments are continuous, and begin 10 years after the annuity purchase.

If neither life survives the deferred period, then a sum of 3 times the single premium is paid immediately on the second death.

There are no other benefits.

You are given:

(i) {}_{10}p_{50:55}^{00} = 0.86041 \quad{}_{10}p_{50:55}^{01} = 0.04835 \quad{}_{10}p_{50:55}^{02} = 0.08628

(ii) \bar{A}_{50:55:\overline{10}|}^{03} = 0.003421

(iii) \bar{a}_{60:65}^{00} = 8.8219 \quad\bar{a}_{60:65}^{01} = 1.3768 \quad\bar{a}_{60:65}^{02} = 3.0175

(iv) \bar{a}_{65}^{11} = 10.1948 \quad\bar{a}_{60}^{22} = 11.8302

(v) i = 0.05

(i) Show that the expected present value of the future benefits at time 10, if both lives survive to time 10, is 573,000 to the nearest 1000. You should calculate the value to the nearest 100.

(ii) Calculate the single net premium for the policy.

Part d (4 points)

(i) Determine {}_{10}V^{(0)}, {}_{10}V^{(1)} and {}_{10}V^{(2)} for the policy.

(ii) Write down Thiele's differential equation for the policy value at t, t \ge 10, assuming both lives survive to t.

(iii) Using Euler's forward method, with a step size of h = 0.5, calculate the policy value required at 10.5 assuming both lives are alive at that time.

(MLC, Fall 2015, Question 2)

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