Two-life functions dependent on a fixed term as well as age

Introduction: Combining Lives and Time

Welcome to this chapter on Two-life functions dependent on a fixed term as well as age. So far in your CM1 journey, you have learned how to value contracts for a single person and how to value contracts that depend on the survival of two people (Joint Life and Last Survivor).

In the real world, many insurance products don't just last "forever" or until death; they often have a fixed term. For example, a couple might want a life insurance policy that lasts only until their 25-year mortgage is paid off. Or, a pension might be guaranteed to pay out for at least 10 years, regardless of when the policyholders pass away. This chapter teaches you how to mathematically combine the mortality of two individuals with a fixed time horizon \(n\).

Did you know? These functions are the "building blocks" for complex products like joint-life mortgages and family income benefits. Mastering these will make the "Pricing and Reserving" section of CM1 much easier!

1. The "Joint Life" Status with a Fixed Term

Recall that a Joint Life Status (written as \(xy\)) exists only as long as both lives are alive. If we add a fixed term \(n\), we create a status that ends when the first of the three events occurs:
1. Life \(x\) dies.
2. Life \(y\) dies.
3. The term \(n\) expires.

We denote this combined status as \(xy:\overline{n|}\). Think of this like a three-way race where the finish line is reached as soon as anyone (including "Father Time" at year \(n\)) crosses it.

Joint Life Temporary Annuity

A temporary annuity pays a benefit as long as the status \(xy:\overline{n|}\) continues. In other words, payments are made while both are alive, but for no longer than \(n\) years.

The Formula:
\(a_{xy:\overline{n|}} = \sum_{t=1}^{n} v^t \cdot {}_t p_{xy}\)
(Assuming payments are in arrears. If in advance, use \(\ddot{a}_{xy:\overline{n|}}\) and start the sum from \(t=0\) to \(n-1\)).

Important Relationship:
Remember that \({}_t p_{xy} = {}_t p_x \cdot {}_t p_y\) (assuming the lives are independent). So, the probability of the status surviving is simply the product of both individuals' survival probabilities.

Joint Life Endowment Assurance

An endowment assurance on two lives pays out at the end of the year of the first death, or at the end of \(n\) years if both survive.

The Logic:
\(A_{xy:\overline{n|}} = A_{xy:\overline{n|}}^1 + A_{xy:\overline{n|}}^{\: \: \: 1}\)
Where:
• \(A_{xy:\overline{n|}}^1\) is the Term Assurance part (pays if the first death occurs within \(n\) years).
• \(A_{xy:\overline{n|}}^{\: \: \: 1}\) is the Pure Endowment part (pays at time \(n\) if both are still alive). This is equal to \(v^n \cdot {}_n p_{xy}\).

Key Takeaway: The joint life status with a term \(n\) behaves exactly like a single life status with term \(n\), but you replace the single survival probability \({}_t p_x\) with the joint survival probability \({}_t p_{xy}\).

2. The "Last Survivor" Status with a Fixed Term

The Last Survivor Status (written as \(\overline{xy}\)) exists as long as at least one of the lives is still alive. When we add a fixed term \(n\), the status \(\overline{xy}:\overline{n|}\) ends when the last survivor dies OR when the term \(n\) expires.

This is commonly seen in Last Survivor Temporary Annuities. These continue paying as long as at least one person is alive, but stop after \(n\) years regardless.

The "Z-Method" (Inclusion-Exclusion)

When dealing with Last Survivor functions and fixed terms, the easiest way to solve problems is to use the inclusion-exclusion principle. For any function (let's call it \(f\)), the relationship is:
\(f_{\overline{xy}:\overline{n|}} = f_{x:\overline{n|}} + f_{y:\overline{n|}} - f_{xy:\overline{n|}}\)

Example: Last Survivor Temporary Annuity
\(\ddot{a}_{\overline{xy}:\overline{n|}} = \ddot{a}_{x:\overline{n|}} + \ddot{a}_{y:\overline{n|}} - \ddot{a}_{xy:\overline{n|}}\)

Why does this work?
Imagine a benefit paid to two people. If we pay one benefit to \(x\) (for \(n\) years) and one to \(y\) (for \(n\) years), we have paid "double" for the time they were both alive. By subtracting the joint life annuity (\(xy:\overline{n|}\)), we remove that double-payment, leaving exactly one payment as long as either is alive.

Don't worry if this seems tricky at first! Just remember: Last Survivor = Person 1 + Person 2 - Both. This rule applies to assurances, annuities, and probabilities.

3. Functions Dependent on a Fixed Term: Special Cases

The syllabus specifically mentions functions dependent on a fixed term as an extension. Two very common "exam-style" applications are Guaranteed Annuities and Deferred Benefits involving two lives.

Guaranteed Annuities on Two Lives

A common contract is an annuity payable for the greater of a fixed term \(n\) and the lifetime of the survivors. This is often written as \(\ddot{a}_{\overline{xy}:\overline{n|}}\) but in a different context: the status exists for the full \(n\) years for certain, plus any additional time \(x\) or \(y\) survive.

Step-by-Step Logic for \(\ddot{a}_{\overline{xy}}\) with an \(n\)-year guarantee:
1. Value the certain part: \(\ddot{a}_{\overline{n|}}\) (This is a standard financial annuity-certain).
2. Add the "contingent" part: The payments that happen after \(n\) years if someone is still alive.
3. This is expressed as: \(\ddot{a}_{\overline{n|}} + v^n \cdot {}_n p_{\overline{xy}} \cdot \ddot{a}_{\overline{x+n:y+n}}\)
(Or more simply: \(\ddot{a}_{\overline{n|}} + {}_n|\ddot{a}_{\overline{xy}}\)).

Deferred Joint Life Benefits

A deferred annuity \({}_n|\ddot{a}_{xy}\) means: "Wait \(n\) years. If both are alive at that point, start paying a joint-life annuity."

The Formula:
\({}_n|\ddot{a}_{xy} = v^n \cdot {}_n p_{xy} \cdot \ddot{a}_{x+n:y+n}\)

Common Mistake to Avoid:
Be careful with the probability! For a deferred joint life annuity, both must survive the deferred period. For a deferred last survivor annuity, only one needs to survive.

4. Summary of Key Formulae

Here is a Quick Review of the most important relationships in this chapter:

1. Probability Relationship:
\({}_t p_{\overline{xy}:\overline{n|}}\) is 1 if \(t \le n\) and someone is alive. Usually, we look at the survival of the status. The probability that the joint status \(xy:\overline{n|}\) is still active at time \(t\) is:
• \({}_t p_{xy}\) if \(t < n\)
• \(0\) if \(t \ge n\)

2. Annuity Relationship:
\(\ddot{a}_{xy:\overline{n|}} = \ddot{a}_{xy} - {}_n|\ddot{a}_{xy}\)
(A temporary annuity is a whole-life annuity minus the deferred part we don't want).

3. Last Survivor "Z" Rule:
Always use \(\text{Last Survivor} = X + Y - \text{Joint}\) when a fixed term \(n\) is involved to keep your calculations clean.

5. Practical Tips for the Exam

Tip 1: Read the timing carefully!
Does the payment happen at the first death or the second death? Is there a fixed term? Use the notation to guide you. If you see a line over the ages (\(\overline{xy}\)), it's last survivor. If there is a "1" over a status (e.g., \(A_{xy:\overline{n|}}^1\)), it only pays if that specific status "dies" first.

Tip 2: Excel (Paper B) Efficiency
In CM1B, you will often need to build a table of \({}_t p_x\) and \({}_t p_y\). To get the joint life version, just create a column multiplying them together: \({}_t p_{xy} = {}_t p_x \cdot {}_t p_y\). To get the temporary version, simply stop your sum at row \(n\).

Tip 3: The "At Least One" Logic
If a question asks for the probability that "at least one life is alive at the end of \(n\) years," that is simply \({}_n p_{\overline{xy}} = {}_n p_x + {}_n p_y - {}_n p_x \cdot {}_n p_y\). Or, even easier: \(1 - (1 - {}_n p_x)(1 - {}_n p_y)\), which is \(1 - (\text{probability both are dead})\).

Key Takeaway: Adding a fixed term \(n\) to two-life functions doesn't change the underlying logic of Actuarial Mathematics. It simply means our sums and integrals now have an upper limit of \(n\) instead of \(\infty\). Use the inclusion-exclusion principle for last survivors, and always check if the contract is "Joint Life" (ends on first death) or "Last Survivor" (ends on second death).