Welcome to the World of Two! Joint-Life Modelling

In your earlier studies, you focused on the life of a single person, usually called \((x)\). But in the real world, many financial decisions involve more than one person—think of a married couple buying a house or a company providing pensions to a retiree and their spouse.

In this chapter, we transition from individual life models to joint-life models. We will explore how to calculate probabilities when two lives are involved. It might seem like double the work, but once you see the patterns, it’s just like building with LEGO blocks!

1. The Two Ways to Look at Joint Lives

When we look at two people, say \((x)\) and \((y)\), we usually care about one of two "statuses":

  • Joint-Life Status \((xy)\): This status exists only as long as both people are alive. It "fails" the moment the first person dies. Analogy: A tennis doubles match—if one partner has to leave, the team status ends.
  • Last-Survivor Status \((\overline{xy})\): This status exists as long as at least one person is alive. It "fails" only when the second (last) person dies. Analogy: A light circuit with two bulbs in parallel—the room only goes dark when BOTH bulbs burn out.

Quick Review: The Math Behind the Status

Let \(T_x\) be the time until \((x)\) dies and \(T_y\) be the time until \((y)\) dies.
1. The time until the joint-life status fails is \(T_{xy} = \min(T_x, T_y)\).
2. The time until the last-survivor status fails is \(T_{\overline{xy}} = \max(T_x, T_y)\).

Key Relationship to Remember:

\(T_x + T_y = T_{xy} + T_{\overline{xy}}\)
This identity is a lifesaver! It basically says that the sum of the individual lifetimes is equal to the time until the first death plus the time until the second death.

2. The Independent Life Model

The simplest way to model two lives is to assume that the death of \((x)\) has absolutely no impact on the death of \((y)\). This is called the Independence Assumption.

Survival Probabilities

If they are independent, the probability that both survive \(t\) years is just the product of their individual survival probabilities:

\[{}_t p_{xy} = {}_t p_x \cdot {}_t p_y\]

For the Last-Survivor status, the probability that at least one survives is:

\[{}_t p_{\overline{xy}} = {}_t p_x + {}_t p_y - {}_t p_{xy}\]

Don't worry if this seems tricky! Just remember the Inclusion-Exclusion Principle from basic probability: \(P(A \cup B) = P(A) + P(B) - P(A \cap B)\). Here, "A or B surviving" is the last-survivor status, and "A and B surviving" is the joint-life status.

The Joint Force of Mortality

If lives are independent, the force of mortality for the joint-life status is simply the sum of the individual forces:

\[\mu_{xy}(t) = \mu_x(t) + \mu_y(t)\]

Quick Review: This means the "hazard" or "risk" of the joint-life status ending is the combined risk of either person dying.

Summary: In the independent model, we just multiply survival probabilities to get the joint-life probability.

3. The Multiple State Model Approach

In Exam ALTAM, we often move beyond independence. We use a Multiple State Model to describe the various "states" a pair of lives can be in. This is much more realistic because it allows the death of one person to affect the mortality of the other (the "broken heart" effect).

The Four States

Imagine a box with four rooms:

  • State 0: Both \((x)\) and \((y)\) are alive.
  • State 1: \((x)\) is dead, but \((y)\) is still alive.
  • State 2: \((y)\) is dead, but \((x)\) is still alive.
  • State 3: Both are dead (The "absorbing" state).

The Transitions (Forces of Mortality)

We use \(\mu_{x+t:y+t}^{jk}\) to represent the force of transition from State \(j\) to State \(k\) at time \(t\).

  • \(\mu_{x+t:y+t}^{01}\): The force of \((x)\) dying first.
  • \(\mu_{x+t:y+t}^{02}\): The force of \((y)\) dying first.
  • \(\mu_{x+t:y+t}^{13}\): The force of \((y)\) dying, given \((x)\) is already dead.
  • \(\mu_{x+t:y+t}^{23}\): The force of \((x)\) dying, given \((y)\) is already dead.

Did you know? If the lives are independent, then \(\mu^{01} = \mu^{23} = \mu_x\) and \(\mu^{02} = \mu^{13} = \mu_y\). If they are dependent, these values will be different!

4. Calculating State Probabilities

To find the probability of being in a specific state at time \(t\), we use the following logic:

Staying in State 0 (Both Alive)

To stay in State 0, no one can die. The total force of leaving State 0 is \(\mu^{01} + \mu^{02}\).

\[{}_t p_{xy}^{00} = \exp \left( -\int_0^t (\mu_{x+s:y+s}^{01} + \mu_{x+s:y+s}^{02}) ds \right)\]

Moving to State 1 (\((x)\) dies, then \((y)\) survives)

This is a "path" calculation. To end up in State 1 at time \(t\):

  1. Both must survive until some time \(s\) (where \(s < t\)).
  2. Exactly at time \(s\), \((x)\) dies (transition \(0 \to 1\)).
  3. From time \(s\) to time \(t\), \((y)\) must stay alive in State 1.

\[{}_t p_{xy}^{01} = \int_0^t {}_s p_{xy}^{00} \cdot \mu_{x+s:y+s}^{01} \cdot {}_{t-s} p_{y+s}^{11} ds\]

Memory Trick: Think of this as "Survival until the event \(\times\) the event happens \(\times\) survival after the event."

5. Common Pitfalls to Avoid

1. Confusion between \(T_{xy}\) and \(T_{\overline{xy}}\): Always double-check if the problem asks for "first death" (Joint-life) or "last death" (Last-survivor).
2. Forgetting the state definitions: In Multiple State Models, \({}_t p_{xy}^{01}\) is the probability that \((y)\) is the only survivor at time \(t\). It is not the same as the probability that \((x)\) dies before time \(t\).
3. Integration limits: When calculating probabilities like \({}_t p_{xy}^{01}\), the integral always goes from \(0\) to \(t\). Ensure your force of mortality functions are correctly substituted.

6. Summary Quick-Review Box

Key Formulas:
- Joint-Life Survival: \({}_t p_{xy} = P(T_{xy} > t)\)
- Last-Survivor Survival: \({}_t p_{\overline{xy}} = {}_t p_x + {}_t p_y - {}_t p_{xy}\)
- Independent Force: \(\mu_{xy} = \mu_x + \mu_y\)
- MSM State 0: \({}_t p_{xy}^{00} = e^{-\int (\text{sum of forces leaving state 0})}\)

Key Takeaway: The Multiple State Model is just a map. Follow the arrows (transitions) and multiply the probabilities along the path!

Don't worry if the integral formulas look scary at first. In most exam questions, the forces of mortality (\(\mu\)) are constants, which makes the integration much simpler—usually just resulting in an exponential function!