Effect of changes in mortality and interest assumptions

Welcome to the "What If?" Chapter!

In the world of actuarial science, we rarely know the future with 100% certainty. We make educated guesses called assumptions about two main things: how much interest we will earn (interest rates) and when people might pass away (mortality rates). But what happens if our guesses are slightly off? What if interest rates suddenly drop? What if medical breakthroughs mean people live much longer?

In this chapter, we explore how the Actuarial Present Value (APV) of insurance and annuities shifts when these assumptions change. This is a favorite topic for exam writers because it tests your intuitive understanding of how these variables interact. Don't worry if it seems a bit abstract at first—we’ll break it down using simple logic and real-world analogies!

1. Changing the Interest Rate Assumption

The interest rate (usually denoted as \(i\)) is the engine that drives the time value of money. When we calculate the present value of a future benefit, we "discount" it back to today.

The Inverse Relationship

The most important rule to remember is: Interest rates and Present Values move in opposite directions.

If the interest rate \(i\) increases:

1. The discount factor \(v = \frac{1}{1+i}\) decreases.
2. Future payments are discounted more heavily.
3. The Actuarial Present Value (APV) of both insurances and annuities decreases.

Analogy: The Delivery Fee
Think of the discount rate like a "delivery fee" for bringing money from the future to the present. If the fee (interest rate) goes up, you get to keep less of that money when it arrives today. Therefore, the value today is lower.

Quick Review:
- \(i \uparrow \implies \text{APV} \downarrow\)
- \(i \downarrow \implies \text{APV} \uparrow\)

2. Changing the Mortality Assumption

Mortality assumptions are based on the probability of death (\(q_x\)). If we assume people are "more likely to die" (higher mortality), it changes our math significantly depending on whether we are looking at an insurance policy or an annuity.

A. Impact on Life Insurance (\(A_x\))

Life insurance pays out when the insured person dies. If we assume higher mortality rates:

1. The benefit is likely to be paid sooner.
2. There is less time for the money to be discounted (meaning the "delivery fee" we talked about earlier is smaller).
3. The APV of Life Insurance increases.

B. Impact on Life Annuities (\(\ddot{a}_x\))

A life annuity pays out as long as the person is alive. If we assume higher mortality rates:

1. The person is expected to die sooner.
2. The company expects to make fewer payments.
3. The APV of the Annuity decreases.

Did you know?
This is why "longevity risk" (people living longer than expected) is a major concern for pension funds. If mortality rates decrease, the APV of their future annuity payments shoots up, which can be very expensive!

3. Comparing Different Insurance Products

Not all products react the same way to mortality changes. Let's look at a few common ones:

Whole Life vs. Term Insurance

If mortality increases at all ages, the APV of Whole Life Insurance (\(A_x\)) definitely increases because the payout is guaranteed eventually—it's just a matter of "when."

However, for Term Insurance (\(A_{x:\bar{n}|}^1\)), a change in mortality is more complex. If mortality increases, the person is more likely to die during the term (increasing the APV), but they are also more likely to die earlier in the term (increasing the APV even more). Generally, higher mortality leads to higher Term Insurance APVs.

Endowment Insurance (\(A_{x:\bar{n}|}\))

This is the "trick question" of the exam! Remember that an Endowment Insurance is a combination of Term Insurance and a Pure Endowment (\(A_{x:\bar{n}|} = A_{x:\bar{n}|}^1 + A_{x:\bar{n}|} \cdot {}_nE_x\)).

1. Higher mortality increases the value of the Term Insurance part (payout on death).
2. Higher mortality decreases the value of the Pure Endowment part (payout for surviving).
3. The total effect depends on which part is larger! Usually, for a typical endowment, the pure endowment effect is very strong, so higher mortality often decreases the total APV.

4. Mathematical Summaries and Mnemonics

To keep these straight in your head during the exam, use this "Cheat Sheet" logic:

The "Speed" Rule

Higher Mortality (\(q_x \uparrow\)) makes everything happen faster.
- If it's a "Death Benefit" (Insurance) \(\implies\) Faster Payout \(\implies\) Higher Value.
- If it's a "Living Benefit" (Annuity) \(\implies\) Shorter Payout \(\implies\) Lower Value.

Directional Summary Table

Assumption Change | Insurance APV | Annuity APV
Interest Rate \(i \uparrow\) | Decreases (\(\downarrow\)) | Decreases (\(\downarrow\))
Mortality Rate \(q_x \uparrow\) | Increases (\(\uparrow\)) | Decreases (\(\downarrow\))

5. Common Pitfalls to Avoid

Mistake 1: Forgetting the interest effect on annuities.
Some students think that because annuities involve many payments, an interest rate change doesn't matter as much. Actually, because annuities often last for decades, they are extremely sensitive to interest rate changes!

Mistake 2: Assuming all mortality changes are equal.
On the exam, they might say "Mortality increases only after age 70." If you are calculating the APV for a 20-year-old for a 10-year term policy, this change has zero effect on the APV because the change happens outside the policy term.

Mistake 3: Confusing \(q_x\) and \(p_x\).
Always read carefully. An increase in mortality means \(q_x\) is higher. An increase in survival probability means \(p_x\) is higher (which is the same as mortality decreasing).

Section Key Takeaways

1. Interest Rates: There is an inverse relationship with all APVs. Higher \(i\) means lower APV.
2. Mortality (Insurance): Higher mortality means earlier payouts, which increases APV.
3. Mortality (Annuities): Higher mortality means fewer payouts, which decreases APV.
4. Endowment Check: Remember that an endowment insurance is a tug-of-war between a death benefit and a survival benefit. Check which one dominates!

Don't worry if this seems tricky at first! The best way to master this is to ask yourself: "Does this change make the money leave the company's pocket faster or slower?" If it leaves faster or more often, the APV goes up. If it leaves slower or less often, the APV goes down.