Welcome to Joint Life Premiums!
Hello there! If you’ve made it this far in your ALTAM journey, you already know how to calculate premiums for a single person. In this chapter, we are simply taking those same concepts and applying them to couples (or any group of two lives). Whether it’s a husband and wife buying a "first-to-die" policy to pay off a mortgage, or a couple buying "last-survivor" insurance for estate planning, the math follows the same logic you’ve already mastered. Don't worry if it looks intimidating—we’ll break it down step-by-step!
1. The Foundation: The Equivalence Principle
Before we dive into the formulas, let’s remember the Golden Rule of Actuarial Science: The Equivalence Principle. To find the premium, we set the value of what comes in equal to the value of what goes out.
Expected Present Value (EPV) of Premiums = EPV of Benefits
In the context of joint lives, we just need to be very careful about when the premiums stop and when the benefit is paid. Is it when the first person dies, or when the last person dies?
Quick Review: The Two Main Statuses
- Joint Life Status (\(xy\)): This status "fails" (ends) as soon as the first person dies. Think of it like a light circuit that breaks if either bulb burns out.
- Last Survivor Status (\(\overline{xy}\)): This status "fails" only when the very last person dies. Think of this like a backup generator—it keeps running until everyone is gone.
2. Premiums for Joint Life Insurance (\(P_{xy}\))
A Joint Life Insurance policy pays out immediately when the first death occurs. Because the payout happens sooner (on average) than a single-life policy, these premiums are usually higher than those for a single person of the same age.
The Formula
For a fully discrete whole life insurance on \((x)\) and \((y)\) payable at the end of the year of the first death, the annual premium \(P_{xy}\) is:
\( P_{xy} = \frac{A_{xy}}{\ddot{a}_{xy}} \)
Breakdown:
- \(A_{xy}\): The EPV of the insurance benefit (paid when the first death occurs).
- \(\ddot{a}_{xy}\): The EPV of a life annuity-due (premiums are paid only while both are alive).
Analogy: Imagine two roommates sharing a subscription. The deal is: "We pay every year as long as we both live here. The moment one of us moves out, the contract ends and the company pays a moving bonus."
Key Takeaway:
For Joint Life (\(xy\)), everything—the benefit and the premium payments—is tied to the first death.
3. Premiums for Last Survivor Insurance (\(P_{\overline{xy}}\))
A Last Survivor Insurance policy (often called "Second-to-Die" insurance) pays out only when both people have passed away. These are very common in estate planning to help heirs pay taxes.
The Formula
The annual premium \(P_{\overline{xy}}\) is:
\( P_{\overline{xy}} = \frac{A_{\overline{xy}}}{\ddot{a}_{\overline{xy}}} \)
Wait! A Common Pitfall:
Students often ask: "Do premiums continue after the first death?"
Yes! In the standard \(P_{\overline{xy}}\) model, premiums are paid as long as at least one person is alive. This is why we divide by \(\ddot{a}_{\overline{xy}}\).
Did you know?
\(P_{\overline{xy}}\) is significantly cheaper than \(P_{xy}\). Why? Because the insurance company gets to hold onto the money longer (until the second death) and they likely collect more premium payments over time!
Quick Review Box:
Joint Life (\(xy\)): Premium stops at 1st death. Benefit paid at 1st death.
Last Survivor (\(\overline{xy}\)): Premium stops at 2nd death. Benefit paid at 2nd death.
4. Working with the Premium Relations
On the ALTAM exam, you might not be given \(A_{\overline{xy}}\) or \(\ddot{a}_{\overline{xy}}\) directly. You might have to use the Addition Rule (also known as the Inclusion-Exclusion Principle).
Remember this essential relationship:
\( \text{Status (x)} + \text{Status (y)} = \text{Status (xy)} + \text{Status (\)\overline{xy}\))} \)
This means:
\( \ddot{a}_{\overline{xy}} = \ddot{a}_x + \ddot{a}_y - \ddot{a}_{xy} \)
\( A_{\overline{xy}} = A_x + A_y - A_{xy} \)
Step-by-Step Process for Solving Premium Problems:
1. Identify if the policy is Joint Life or Last Survivor.
2. Calculate the individual components (\(A_x, A_y, A_{xy}\) and \(\ddot{a}_x, \ddot{a}_y, \ddot{a}_{xy}\)).
3. Use the Addition Rule to find the "Last Survivor" versions if needed.
4. Divide the Benefit EPV by the Annuity EPV.
5. Special Case: Premiums Payable for a Limited Time
Sometimes, a couple might want to pay premiums only while both are alive, even though the insurance is Last Survivor. This is a very common exam trick!
The Scenario: Insurance pays at the second death, but premiums stop at the first death.
The Formula: \( P = \frac{A_{\overline{xy}}}{\ddot{a}_{xy}} \)
Notice the denominator! Since premiums stop when the first person dies, we use the joint-life annuity \(\ddot{a}_{xy}\). This premium will be higher than the standard \(P_{\overline{xy}}\) because the payment period is shorter.
6. Common Mistakes to Avoid
- Confusing the Denominator: Always ask yourself: "When do the payments stop?" If they stop when either person dies, use \(\ddot{a}_{xy}\). If they stop only when both are dead, use \(\ddot{a}_{\overline{xy}}\).
- Forgetting the Interest Rate: Ensure your \(A\) and \(\ddot{a}\) values are calculated using the same interest rate \(i\).
- Mixing up Continuous and Discrete: Watch out for \(P_{xy}\) (discrete, end of year) vs \(\bar{P}_{xy}\) (continuous). If the exam says "payable at the moment of death," you need \(\bar{A}_{xy}\).
Summary Checklist
- Joint Life (\(xy\)): Pay at 1st death, premiums until 1st death. \( P_{xy} = A_{xy} / \ddot{a}_{xy} \).
- Last Survivor (\(\overline{xy}\)): Pay at 2nd death, premiums until 2nd death. \( P_{\overline{xy}} = A_{\overline{xy}} / \ddot{a}_{\overline{xy}} \).
- Inclusion-Exclusion: Use it to find \(\overline{xy}\) values from single-life and joint-life values.
- Logic Check: \( P_{xy} \) should be the most expensive, followed by single-life premiums, and \( P_{\overline{xy}} \) should be the cheapest (assuming ages are similar).
Keep practicing! The more you see these symbols, the more they will feel like a second language. You've got this!