Mortality curves and heterogeneity in mortality

Welcome to the World of Mortality Curves and Heterogeneity!

Hello there! If you’ve made it to Exam ALTAM, you already know that people age and eventually pass away. But have you ever wondered why some 90-year-olds are still running marathons while others struggle in their 60s? That "difference" is what we call heterogeneity. In this chapter, we’re going to move beyond simple mortality tables and look at the mathematical "curves" that describe aging and the hidden factors that make each individual unique. Understanding this isn't just cool science—it's vital for pricing pensions and life insurance accurately!

1. Mortality Laws: The Shape of Aging

Before we look at differences between people, we need a "baseline" for how humans age. Actuaries use mathematical formulas called Mortality Laws to describe the force of mortality (\(\mu_x\)).

The Gompertz Law

Benjamin Gompertz noticed in 1825 that the risk of dying increases exponentially as we get older. Imagine a snowball rolling down a hill, getting faster and faster—that’s your risk of death after age 30.
Formula: \(\mu_x = B \cdot c^x\)
Where:
- B is the base level of mortality.
- c is the "growth rate" of mortality (usually around 1.1, meaning risk increases about 10% per year of age).

The Makeham Law

Makeham realized that Gompertz forgot one thing: accidents! Some risks don't care how old you are (like being struck by lightning). He added a constant term, A, to the formula.
Formula: \(\mu_x = A + B \cdot c^x\)
- A: Age-independent risk (accidents).
- B \cdot c^x: Age-dependent risk (biological aging).

Quick Review:
- Gompertz: Only biological aging.
- Makeham: Biological aging + random accidents.

2. Heterogeneity: Why We Aren't All the Same

In basic models, we assume everyone aged 40 has the exact same probability of dying. But in reality, some people have "good" genes or healthy habits, and others don't. This diversity is called heterogeneity.

The Marathon Analogy:
Imagine a huge marathon. At the start, you have elite athletes and casual joggers.
1. As the race goes on, the casual joggers (the "frail" individuals) drop out first.
2. By mile 20, the people left are mostly the elite athletes (the "robust" individuals).
3. Because the "weakest" runners drop out early, the average speed of the remaining pack might actually look faster than you'd expect! This is the core of heterogeneity in mortality.

Key Concept: Selection Effect
As a group ages, the individuals with the highest risk die first. This leaves behind a "survivor group" that is healthier on average than the original group started out. This can make it look like mortality is slowing down at very old ages.

3. Frailty Models

How do we put this into a math equation? We use a Frailty Variable, usually denoted as Z.

The Individual Force of Mortality

Each person has their own frailty value, \(Z\).
\(\mu(x+t | Z=z) = z \cdot \mu^*(x+t)\)
- \(z\): The individual's personal "multiplier." If \(z > 1\), they are more frail (higher risk). If \(z < 1\), they are more robust (lower risk).
- \(\mu^*(x+t)\): The "baseline" force of mortality for a standard person.

Did you know?
We usually assume the average frailty at birth is 1 (\(E[Z] = 1\)). This keeps our baseline (\(\mu^*\)) meaningful as the average starting risk.

The Population Force of Mortality

This is where students often get tripped up! The mortality rate for the whole population, denoted \(\bar{\mu}_{x+t}\), is the average of the individual mortalities of those who are still alive.
Formula: \(\bar{\mu}_{x+t} = E[Z | T > t] \cdot \mu^*(x+t)\)

Common Mistake Alert:
Don't just use the starting average of \(Z\)! As time passes, the high-\(z\) people die. Therefore, \(E[Z | T > t]\) decreases over time. The population risk is always less than or equal to the average risk of the original group.

4. The Gamma-Gompertz Model

In Exam ALTAM, the most common distribution used for frailty \(Z\) is the Gamma distribution. Why? Because the math works out beautifully (it's "mathematically tractable").

If \(Z\) follows a Gamma distribution with mean 1 and variance \(\sigma^2\):
1. The population survival function \(S_p(t)\) becomes a specific power function of the baseline survival.
2. The population force of mortality \(\bar{\mu}_{x+t}\) will eventually level off or increase much slower than the individual \(\mu\). This is often called mortality deceleration.

Memory Trick:
Think of Gamma as the "Great Leveler." It takes an exponentially increasing risk (Gompertz) and levels it out because the frailest people are gone.

5. Summary and Key Takeaways

Don't let the notation scare you! Here is what you absolutely must remember for the exam:

1. Individual vs. Population: Individual mortality is what happens to one person with a fixed \(z\). Population mortality is what we observe in a group, which changes as "frail" members die off.

2. The Selection Effect: At older ages, the population mortality rate is lower than it would be if everyone were identical, because only the "strongest" (lowest \(z\)) have survived.

3. Impact on Pricing: If an actuary ignores heterogeneity, they might overestimate how many people will die at very old ages. For a pension company, this is dangerous because they might not save enough money to pay the survivors!

Quick Review Box:
- \(z \cdot \mu^*\): Individual risk.
- Selection: The process where high-risk people die early.
- Deceleration: The flattening of the mortality curve at extreme old ages (e.g., 100+).
- Gamma: The go-to distribution for modeling frailty \(Z\).

Keep practicing those formulas! You're doing great. Remember, every complex model is just a way of trying to describe the real, messy world of human life. You've got this!