SOA Exam ALTAM Sample Question 23

Assume the following 4-state joint life model of mortality.

Part a (1 point)

Describe the annuity that has actuarial value $\bar{a}_{xy}^{01}$ .

Write down the Kolmogorov forward differential equations, with boundary conditions, for ${}_tp_{xy}^{00}$ and ${}_tp_{xy}^{01}$ .

You are given the Euler-Maclaurin-Woolhouse approximation formula

$\int_0^{\infty} g(t)\,dt \approx \sum_{k=0}^{\infty} g(k) - \frac{1}{2}g(0) + \frac{1}{12}g'(0).$

Use this approximation, with $g(t) = {}_tp_{xy}^{00}\,e^{-\delta t}$ , to prove that

$\bar{a}_{xy}^{00} \approx \ddot{a}_{xy}^{00} - \frac{1}{2} - \frac{1}{12}\left(\mu_x^{01} + \mu_y^{02} + \delta\right).$

Hint: Use the Kolmogorov forward differential equations, and the chain rule.

Z and A are both 70 years old. They purchase a last survivor annuity, with a level benefit payable continuously.

The single net premium is 1,000,000.

You are given the following information:

(i) The mortality of each life follows the Standard Ultimate Life Table.

(ii) Z and A have independent future lifetimes.

(iii) $i = 0.05$

(iv) $\mu_{70} = 0.009881$

(v) $\bar{a}_{70|70} = 2.0317$

Calculate the level annual rate of benefit, using the Euler-Maclaurin-Woolhouse approximation.

(This is Question 6 from the Spring 2021 Written Answer exam.)