Actuarial Notes

SOA Exam ALTAM Fall 2025, Question 3

Study guide: Exam ALTAM

An insurance company uses the following model to price joint life insurance products.

Let \mu_z^s denote the force of mortality for a life age z under the Standard Ultimate Survival Model (SUSM).

The transition intensities of the joint life model are:

\mu_{x:y}^{01} = \mu_y^s - \lambda; \qquad \mu_{x:y}^{02} = \mu_x^s - \lambda; \qquad \mu_{x:y}^{04} = \lambda

\mu_{x:y}^{13} = \mu_x^s; \qquad \mu_{x:y}^{23} = \mu_y^s

Part a (4 points)

(i) Show that the individual force of mortality for (x) is the same whether (y) is alive or dead, and hence that the marginal distribution of T_x is the same as the distribution under the SUSM.

(ii) Your colleague states that because the individual force of mortality for (x) is the same whether (y) is alive or dead, and similarly for (y), then T_x and T_y must be independent. Explain in words why he is wrong.

(iii) Another colleague states that because \mu_x^{02} < \mu_x^{13} this model incorporates a "broken heart" effect. State with reasons whether she is correct.

Part b (2 points)

Let {}_tp_z^s denote the t-year survival probability for a life age z under the SUSM.

Show that {}_tp_{x:y}^{00} = {}_tp_x^s\;{}_tp_y^s\;e^{\lambda t}.

Part c (4 points)

A couple, x = 50 and y = 55, purchases a special whole life reversionary annuity. The annuity is valued using the model above.

You are given the following information:

  1. An annuity of 50,000 per year is payable continuously, starting immediately on the first death, and continuing until the second death.
  2. Premiums are paid continuously at a rate of P per year while both (x) and (y) are alive.
  3. If both lives die simultaneously, a death benefit of the total premium paid is paid immediately on death. That is, the death benefit payable at time t if both lives die simultaneously at that time is tP.
  4. You should use Woolhouse's two term formula to evaluate continuous annuity functions.
  5. i = 0.05
  6. \lambda = 0.0005
  7. \bar{a}_{50:55}^{00} = 14.7443

Let \left(\bar{I}\bar{a}\right)_{x:y}^{00} denote the EPV of a continuously increasing joint life annuity, with a rate of payment of t per year at time t, payable continuously while (x) and (y) both survive.

(i) Show that the EPV of the common shock death benefit is \lambda P\left(\bar{I}\bar{a}\right)_{50:55}^{00}.

(ii) Show that the EPV of the annuity benefit is 129,800 to the nearest 100. You should calculate the value to the nearest 0.1.

(iii) You are given that \left(\bar{I}\bar{a}\right)_{50:55}^{00} = 167.13. Calculate the annual net premium for the policy.

Part d (1 point)

Write down Thiele's differential equation for the net premium policy value at time t > 0, assuming both lives are alive.

Answer this question and see the worked solution

Concepts tested

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