Actuarial Notes

SOA Exam ALTAM Sample Question 1

Study guide: Exam ALTAM

A long-term care provider offers three care levels. Transitions are modeled as a Markov multiple-state model. Transitions and states are shown in the following diagram:

Part a (1 point)

Write down Kolmogorov's forward differential equations with the associated boundary conditions for this model for:

(i) {}_tp_x^{10}

(ii) {}_tp_x^{11}

Part b (3 points)

Estimate {}_1p_{80}^{10} using Euler's forward method, a step size of h = 1/3, and the transition intensities and probabilities in the table below:

t {}_tp_{80}^{11} \mu_{80+t}^{01} \mu_{80+t}^{03} \mu_{80+t}^{10} \mu_{80+t}^{12} \mu_{80+t}^{13} \mu_{80+t}^{23}
0 1.00000 0.10000 0.02981 0.08000 0.15000 0.05962 0.11924
1/3 0.90346 0.10000 0.03082 0.08000 0.15000 0.06164 0.12328
2/3 0.81652 0.10000 0.03186 0.08000 0.15000 0.06373 0.12746
1 -- 0.10000 0.03294 0.08000 0.15000 0.06589 0.13178

Part c (4 points)

You are given the following annuity values at 5%:

x \bar{a}_x^{00} \bar{a}_x^{01} \bar{a}_x^{02} \bar{a}_x^{11} \bar{a}_x^{10} \bar{a}_x^{12} \bar{a}_x^{22}
80 5.5793 1.3813 0.6109 3.0936 1.6719 1.7206 4.4712
85 4.8066 1.0396 0.3403 2.6723 0.8834 1.0883 3.2367

You are also given:

  • {}_5p_{80}^{00} = 0.53880 \qquad {}_5p_{80}^{01} = 0.17327 \qquad {}_5p_{80}^{02} = 0.06956
  • i = 0.05
  • Residents pay a service fee of 8000 per year continuously while in Level 0 or Level 1 Care.
  • Level 2 Care costs are paid at a continuous rate of 30,000 per year for lives age 80 to 85, and 40,000 per year for lives older than 85.

A person, who is age 80, is currently in Level 0 Care.

(i) Calculate the expected present value of the person's future Level 0 and Level 1 service fees.

(ii) Calculate the expected present value of the person's future Level 2 Care costs.

(MLC, Fall 2014, Question 4.)

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