Robin Cunningham, PhD, FSA
Created 8/6/2018
Chapter 9 is a nice relief after the toils of Chapter 8. In this chapter, we are introduced to actuarial functions and benefits that depend on two lives. Good examples are an annuity that makes level payments until both partners die or a life-insurance policy that pays out on the first death of a couple. Here are the highlights:
Notation - for first-to-die and last-to-die of a couple.
Formulas - Just three or four to know
Independent lives - the most common assumption by far, but not terribly realistic for married partners.
Dependent lives - a bit of challenging material, your primary focus should probably be on the Common Shock model.
Survivorship Probabilities
\({}_tp_{xy}\) - The probability that both \(x\) and \(y\) are alive \(t\) years from now. This is called a joint life probability. As you will recall from a Probability course, for independent lives \[{}_tp_{xy} = {}_tp_{x}\cdot{}_tp_{y}\]
\({}_tp_\overline{xy}\) - The probability that at least one of \(x\) and \(y\) is alive in \(t\) years. This is a last survivor probability. Again, for independent lives \[{}_tp_\overline{xy} = 1 - {}_tq_{x}\cdot{}_tq_{y}\]
\({}_tq_{\overset{1}{x}y}\) - The probability that \(x\) dies in the next \(t\) years and dies before \(y\).
\({}_tq_{\overset{2}{x}y}\) - The probability that \(x\) dies in the next \(t\) years and dies after \(y\).
For most of these, you can intuit their meaning: \(\overline{a}_{xy}\) - The EPV of an annuity that pays 1 per year continuously until the first death of \(x\) and \(y\).
\(\overset{..}{a}\overline{xy}\) - An annuity-due of 1 per year that pays until the last death of \(x\) and \(y\)
\(A_{xy}\) - The EPV of and insurance that pays 1 at the end of the year of the first death between \(x\) and \(y\).
\(\overline{a}_{x|y}\) - Not obvious! This is an annuity that begins payment upon the death of \(x\) and ceases upon the death of \(y\). So no payments are made if \(y\) dies first.
\(\overline{A}_{\overset{1}{x}y}\) - The EPV of an insurance of 1 payable upon the death of \(x\) if \(x\) dies before \(y\).
The first two are based on one key idea: \[T_{xy} + T_{\overline{xy}} = T_x + T_y\] This is true because one of \(x\) and \(y\) will die first. If \(x\) dies first, then \(T_x = T_{xy}\) and \(T_y = T_{\overline{xy}}\). Things switch if \(y\) dies first. The reason this is important is it means that \[A_{xy} + A_{\overline{xy}} = A_x + A_y\] and \[a_{xy} + a_{\overline{xy}} = a_x + a_y\] These incredibly important relations naturally hold for continuous insurance and annuities as well. Also, you should reason logically that \[\overline{a}_{x|y} = \overline{a}_y - \overline{a}_{xy}.\] It’s important that this formula makes logical sense and is not memorized. Finally, it should be no surprise that \[\overline{a}_{xy} = \frac{1 - \overline{A}_{xy}}{\delta} \ \ \ \ \ and \ \ \ \ \ \overline{a}_{\overline{xy}} = \frac{1 - \overline{A}_{\overline{xy}}}{\delta}.\]
I think this is the most manageable chapter since Chapter 5. Keep in mind the couple of formulas from the previous slides, understand the notation and it should not be too bad. Here are some examples to focus on:
Example 9.1 - Easy but informative
Example 9.2 - Excellent Exam-like application
Example 9.3 - Just to be read and understood. Don’t derive unless you are into that sort of thing.
Example 9.5 - Practice with awkward Chapter 8 notation.
Example 9.7 - A pedantic point, but you might as well think about the difference.
Example 9.9 - Practice with Makeham’s Law. I suspect that the exam-writers are morel likely to reference Makeham’s Law (or Gompertz) in a multiple-life or multiple-state setting because tables can be less useful for some types of questions in this setting.
Example 9.10 - This is kind of a must along with 9.11 as both involve common shock.
Example 9.11 - Common Shock
Exercises 9.1, 9.2, 9.3, 9.4, 9.5, 9.9, 9.10, 9.11, 9.15, 9.16abc
Especially with the Makeham’s Law questions, numerical integration is required in some cases and spreadsheet summation in others. Two of these are 9.3 and 9.4, which are in the lectures for the chapter.
There are lectures showing solutions for 9.3, 9.4adf, 9.11.