Policy values for insurance and annuities on joint lives

Welcome to Policy Values for Joint Lives!

In your previous studies, you’ve mastered how to calculate policy values (reserves) for a single person. But life—and insurance—often happens in pairs! Whether it’s a married couple wanting to ensure the survivor is taken care of, or business partners protecting their company, joint life products are incredibly common.

In this chapter, we will learn how to calculate and manage the money an insurance company must hold (the policy value) when two lives are involved. Don't worry if this seems like a lot to juggle; we’re essentially taking the single-life concepts you already know and applying them to a "team" of lives. Let’s dive in!

1. The Core Concept: What is a Joint Policy Value?

Just like with single lives, the Policy Value at time \(t\) (denoted as \(_tV\)) is the amount of money the insurer needs to have on hand to meet future obligations.

The fundamental formula remains the same:
Policy Value = (Present Value of Future Benefits) – (Present Value of Future Premiums)

The "twist" in this chapter is that the timing of those benefits and premiums depends on two people (\(x\) and \(y\)) instead of one.

Two Main Flavors of Joint Life Status:

• Joint Life Status (\(xy\)): The status fails when the first person dies. Think of this as "First-to-Die" insurance.
• Last Survivor Status (\(\overline{xy}\)): The status fails when the last person dies. Think of this as "Second-to-Die" insurance.

Did you know? Last survivor insurance is very popular for estate planning. Couples often use it to pay estate taxes that are only due after both spouses have passed away.

2. Policy Values for Joint Life Status (\(xy\))

For a policy that pays a benefit immediately upon the first death of \(x\) or \(y\), the policy value \(_tV_{xy}\) is calculated based on the probability that the joint status survives.

The Discrete Case (End-of-year payments):

If we have a Joint Life Whole Life Insurance with annual premiums \(P\), the prospective policy value at time \(t\) (provided both are still alive) is:
\(_tV = A_{x+t:y+t} - P \cdot \ddot{a}_{x+t:y+t}\)

Quick Review: Remember that \(\ddot{a}_{x:y}\) uses the probability \(_{k}p_{xy}\), which is the probability that both are alive in \(k\) years. Since \(_{k}p_{xy} = _{k}p_{x} \cdot _{k}p_{y}\) (assuming independence), the value of the annuity decreases faster than a single-life annuity because it’s "easier" for the status to fail (only one person has to die!).

3. Policy Values for Last Survivor Status (\(\overline{xy}\))

This is where it gets interesting! A last survivor policy stays active as long as at least one person is alive.

The policy value \(_tV_{\overline{xy}}\) depends heavily on who is still alive at time \(t\). We usually look at the value while both are alive.

The Relationship Formula:

You might remember this handy "inclusion-exclusion" style identity from your earlier joint life studies:
\(V_{\overline{xy}} = V_x + V_y - V_{xy}\)

While this is a great theoretical link, for ALTAM, we often use the Multiple State Model approach to find these values more accurately.

Common Mistake to Avoid: Don't assume the premium stays the same after the first death unless the contract says so! Some policies have "premium waiver" features where premiums stop after the first death, while others require the survivor to keep paying.

4. The Multiple State Model Perspective

ALTAM loves the state-transition approach. Think of a joint life policy as a journey through different "states":

• State 0: Both \(x\) and \(y\) are alive.
• State 1: \(x\) is alive, but \(y\) has died.
• State 2: \(y\) is alive, but \(x\) has died.
• State 3: Both have died (The "Dead State").

In this framework, the policy value at time \(t\) depends on which state the policy is in:

1. \(_tV^{(0)}\): The reserve needed while both are alive.
2. \(_tV^{(1)}\): The reserve needed if only \(x\) is alive. (This is usually just a single-life reserve for \(x\)).
3. \(_tV^{(2)}\): The reserve needed if only \(y\) is alive. (This is usually just a single-life reserve for \(y\)).

Analogy: Imagine a two-player video game. \(_tV^{(0)}\) is the "lives" or "health" the team has while both players are in the game. \(_tV^{(1)}\) is what remains when Player 2 gets a "Game Over" but Player 1 is still playing.

5. Recursion and Thiele’s Equation for Joint Lives

Just like with single lives, policy values grow over time. The Recursion Formula is your best friend for discrete-time problems.

The Discrete Recursion (Both Alive at \(t\)):

\((_tV^{(0)} + P_t)(1+i) = q_{x+t:y+t}^{(01)} \cdot \text{Benefit}_{x} + q_{x+t:y+t}^{(02)} \cdot \text{Benefit}_{y} + q_{x+t:y+t}^{(03)} \cdot \text{Benefit}_{both} + p_{x+t:y+t}^{(00)} \cdot _{t+1}V^{(0)} + p_{x+t:y+t}^{(01)} \cdot _{t+1}V^{(1)} + p_{x+t:y+t}^{(02)} \cdot _{t+1}V^{(2)}\)

Wait, that looks scary! Let's break it down:
The left side is the money you have (Old Reserve + Premium) plus interest.
The right side is what you expect to pay out:
- Death benefits if \(x\) or \(y\) dies.
- The new reserve needed for whichever state you end up in next year (\(0, 1\), or \(2\)).

Step-by-Step for Recursion Problems:

1. Identify the states and the possible transitions.
2. Determine the benefits paid upon each transition (e.g., \(S^{(01)}\) might be the payment if \(y\) dies).
3. Use the probabilities (like \(q_{xy}\)) to weight the future values.
4. Work backward if you need to find the premium, or forward if you are finding the reserve.

6. Summary and Key Takeaways

Key Points to Remember:

• First-to-Die (\(xy\)): Ends when the first death occurs. Reservestend to be smaller because the "risk period" is shorter.
• Last-to-Die (\(\overline{xy}\)): Ends when the second death occurs. Reserves are usually higher because the policy stays on the books longer.
• State Transitions: Always identify which state you are in. \(_tV^{(0)}\) is the reserve when both are alive.
• The "Jump" in Reserves: In a last-survivor policy, when the first person dies, the policy value often "jumps" to the single-life policy value of the survivor (\(_tV^{(1)}\) or \(_tV^{(2)}\)).

Key Takeaway: Policy values for joint lives are just a weighted average of future possibilities. If you can track which state the "team" is in and what happens when they move to a new state, you can solve any problem in this chapter!

Keep practicing! Joint lives can be tricky because of the notation, but the logic is exactly the same as the single-life math you've already conquered. You've got this!