Effect of changes in mortality and interest assumptions

Q: What is the general relationship between interest rates ($ i $) and net premiums ($ P $)?

There is an inverse relationship ; as the interest rate increases, the net premium decreases.

Q: How does a change in the interest rate typically affect the policy value (reserve) $ {}_tV $?

Policy values generally have an inverse relationship with interest rates; an increase in $ i $ usually leads to a decrease in $ {}_tV $.

Q: How does an increase in mortality rates ($ q_x $) affect the net premium?

An increase in mortality leads to an increase in the net premium.

Q: Why is the impact of mortality changes on policy values more complex than its impact on premiums?

Because mortality changes affect both the Present Value of Future Benefits (PVFB) and the Present Value of Future Premiums (PVFP) simultaneously.

Exam FAM – Fundamentals of Actuarial Mathematics · SOA (Society of Actuaries) · 6 min read

Welcome to the Changing World of Actuarial Assumptions!

In your journey through Exam FAM, you’ve learned how to calculate premiums and policy values using specific "best guess" assumptions for interest and mortality. But what happens if those guesses are wrong? What if interest rates rise suddenly, or a medical breakthrough makes people live longer?

In this chapter, we explore how changes in these assumptions affect the Premium (P) and the Policy Value (V). Think of this as a "stress test" for your insurance model. Understanding these relationships is vital because, in the real world, assumptions are rarely static!

1. The Impact of Interest Rate Changes

Interest rates are the "time travel" tool for actuaries. They tell us how much a dollar in the future is worth today. When the interest rate $ i $ changes, it ripples through every calculation.

How Interest Affects Premiums

There is an inverse relationship between interest rates and premiums.
• If $ i $ increases: Money grows faster. The insurance company needs less money upfront to pay for future claims. Therefore, the Net Premium decreases.
• If $ i $ decreases: Money grows slower. You need to collect more now to ensure you have enough later. Therefore, the Net Premium increases.

How Interest Affects Policy Values (Reserves)

Policy values generally follow the same inverse rule as premiums.
Key Rule: An increase in the interest rate $ i $ usually leads to a decrease in the policy value $ {}_tV $.

Analogy: Imagine you are saving for a \$1,000 goal in 10 years. If the bank offers a higher interest rate, you don't need to have as much in your account today to reach that goal. The "Policy Value" is like that account balance; higher interest does more of the "heavy lifting," so the required reserve is lower.

Quick Review:
$ i \uparrow $ leads to $ P \downarrow $ and $ V \downarrow $
$ i \downarrow $ leads to $ P \uparrow $ and $ V \uparrow $

2. The Impact of Mortality Changes

Mortality assumptions represent the "risk" side of the equation. If we assume more people will die (higher $ q_x $), the company expects to pay death benefits sooner.

How Mortality Affects Premiums

This one is intuitive:
• If $ q_x $ increases: The risk of paying a claim is higher. Therefore, the Net Premium increases.
• If $ q_x $ decreases: The risk is lower. Therefore, the Net Premium decreases.

How Mortality Affects Policy Values (The Tricky Part!)

Don't worry if this seems tricky at first—it trips up many students! Unlike premiums, a general increase in mortality doesn't *always* increase the policy value. It depends on the pattern of the change.

The policy value is the difference between the Present Value of Future Benefits (PVFB) and the Present Value of Future Premiums (PVFP).
$ {}_tV = PVFB - PVFP $

If mortality increases, both the PVFB (costs) and the PVFP (expected future income) change. The impact on the policy value depends on which one changes more.

The "Lidstone" Logic (Simplified):
If the rate of increase in mortality is steeper than your original model, the reserves usually need to be higher. However, if mortality increases by a constant amount across all ages, the effect on the reserve might be smaller than you expect because the company is also receiving premiums "sooner" (relative to the shortened life expectancy).

Key Takeaway:
1. Higher mortality always increases the Premium.
2. Higher mortality usually increases the Policy Value, but only if the increase in mortality is "weighted" toward later years or represents a specific "steepening" of the mortality curve.

3. Understanding "Delta" and Sensitivity

In the context of Exam FAM, you might be asked to compare two different mortality tables or two different interest rates to see which produces a higher reserve.

Did you know?
Actuaries call this Sensitivity Analysis. Before a company launches a new product, they test "What if interest rates drop by 1%?" to make sure they won't go bankrupt. It's like checking the weather forecast before a long hike!

Step-by-Step: Comparing Two Assumptions

When comparing the effect of change, follow these steps:
1. Identify the Change: Is interest going up or down? Is mortality becoming "heavier" (higher $ q $) or "lighter" (lower $ q $)?
2. Calculate the New Premium: Use the new $ i $ or $ q_x $ to find the revised premium.
3. Apply to the Reserve Formula: Use the Prospective Formula at time $ t $:
$ {}_tV = (Benefits \times \text{New } A) - (Premium \times \text{New } \ddot{a}) $

4. Common Pitfalls to Avoid

Pitfall 1: Assuming $ V $ and $ P $ always move together.
While they often do, they aren't perfectly joined at the hip. An increase in mortality *always* raises the premium, but if that mortality increase happens mostly at very young ages, the reserve for an older person might actually decrease because the "riskiest" years are already behind them.

Pitfall 2: Forgetting the "Time Value" of Premiums.
When mortality increases, we expect to collect fewer premiums (because people die sooner). This loss of future income is a major reason why reserves often have to increase when mortality assumptions rise.

Pitfall 3: Calculation Errors with $ v $.
Remember that $ v = \frac{1}{1+i} $. If $ i $ increases, $ v $ decreases. Many students accidentally increase $ v $ when they see $ i $ increasing. Be careful!

5. Summary and Key Takeaways

• Interest Rate ($ i $): Think of it as the "Discounting Strength." Higher $ i $ means stronger discounting, which leads to lower Premiums and lower Policy Values.

• Mortality ($ q_x $): Think of it as "Claim Pressure." Higher mortality means more claims, which leads to higher Premiums. The effect on Policy Values is usually an increase, provided the increase in risk is significant in the future years of the policy.

• Memory Trick:
- Interest is Inverse (Interest up = Values down).
- Mortality is More (Mortality up = Premium up).

Quick Review Box:
- Higher Interest: $ P \downarrow $, $ V \downarrow $
- Lower Interest: $ P \uparrow $, $ V \uparrow $
- Higher Mortality: $ P \uparrow $, $ V $ usually $ \uparrow $
- Lower Mortality: $ P \downarrow $, $ V $ usually $ \downarrow $

Keep practicing! You are mastering the core mechanics of how insurance products breathe and change with the economy and human health. You've got this!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the general relationship between interest rates ($ i $) and net premiums ($ P $)?

There is an inverse relationship; as the interest rate increases, the net premium decreases.

How does a change in the interest rate typically affect the policy value (reserve) $ {}_tV $?

Policy values generally have an inverse relationship with interest rates; an increase in $ i $ usually leads to a decrease in $ {}_tV $.

How does an increase in mortality rates ($ q_x $) affect the net premium?

An increase in mortality leads to an increase in the net premium.

Why is the impact of mortality changes on policy values more complex than its impact on premiums?

Because mortality changes affect both the Present Value of Future Benefits (PVFB) and the Present Value of Future Premiums (PVFP) simultaneously.

Under what condition does an increase in mortality generally lead to an increase in policy values?

Policy values usually increase if the mortality increase is weighted toward later years or represents a steepening of the mortality curve.

What is the prospective formula used to calculate a revised policy value during sensitivity analysis?

$ {}_tV = (Benefits \times \text{New } A) - (Premium \times \text{New } \ddot{a}) $

What is a common pitfall regarding the relationship between the interest rate $ i $ and the discount factor $ v $?

Forgetting that they are inversely related: $ v = \frac{1}{1+i} $, meaning if $ i $ increases, $ v $ must decrease.

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