Actuarial Notes

SOA Exam ALTAM Spring 2026, Question 3

Study guide: Exam ALTAM

You are using the Standard Sickness-Death multiple state model to price a whole life insurance policy issued to (50), with a waiver of premiums on sickness.

You are given the following policy information and assumptions:

(i) The death benefit of 100,000 is payable immediately on death.

(ii) Premiums are payable at the start of each of the first 20 years, if the policyholder is Healthy.

(iii) Premiums are waived if the policyholder is Sick at the premium payment date.

(iv) Commissions are 45% of premium in the first year and 8% of premiums paid thereafter. Commission is not paid on waived premiums.

(v) All policyholders are Healthy at the issue date.

(vi) Transitions follow the Standard Sickness-Death model given in the Excel worksheet.

(vii) i = 0.05

(viii) The Two-Term Woolhouse's Formula is used to calculate annual payment annuity values.

Part a (3 points)

(i) Show that {}_{20}p_{50}^{01} = 0.21 to the nearest 0.01. You should calculate the value to the nearest 0.0001.

(ii) You are given that {}_{20}p_{50}^{00} = 0.49695. Show that \ddot{a}_{50:\overline{20}|}^{00} = 11.22 to the nearest 0.01. You should calculate the value to the nearest 0.0001.

Part b (2.5 points)

Let L denote the gross loss-at-issue random variable, and let G denote the gross annual premium. You are given that E[L] = -100.

Show that G = 3{,}340 to the nearest 10. You should calculate the value to the nearest 0.1.

Part c (3 points)

You are given the following additional information:

  • {}_tV^{(j)} denotes the gross premium policy value at time t for a policy in State j at time t.
  • For integer t, {}_tV^{(j)} denotes the value immediately before the time t premium is due to be paid, and {}_{t^+}V^{(j)} denotes the value immediately after the time t premium is due to be paid.
  • \ddot{a}_{60:\overline{10}|}^{00} = 6.9073 \qquad {}_{10}V^{(1)} = 49{,}334
  • \mu_{60}^{01} = 0.014086 \qquad \mu_{60}^{02} = 0.014518 \qquad \mu_{60}^{10} = 0.003443 \qquad \mu_{60}^{12} = 0.02325 (as printed)

(i) Show that {}_{10}V^{(0)} = 25{,}000 to the nearest 100. You should calculate the value to the nearest 1.

(ii) Show that {}_{10^+}V^{(0)} = 28{,}100 to the nearest 100. You should calculate the value to the nearest 1.

(iii) Write down the value of {}_{10^+}V^{(1)}.

Part d (2.5 points)

(i) Show that \displaystyle\lim_{t \to 10^+} \frac{d}{dt}\,{}_tV^{(0)} = 28 to the nearest 1. You should calculate the value to the nearest 0.001.

(ii) Calculate {}_{10.2}V^{(0)} using Euler's method with a step size of h = 0.2.

Answer this question and see the worked solution

Concepts tested

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