Mortality projection

Welcome to Mortality Projection!

Hi there! Welcome to one of the most fascinating (and important) chapters in your CS2 journey. So far in "Survival Models," we’ve looked at how to calculate the probability of someone dying at a specific age. But here is the catch: mortality isn't static.

If you look at records from 100 years ago, a 60-year-old had a much higher chance of dying than a 60-year-old today. Why? Better medicine, cleaner water, and fewer people smoking! In this chapter, we learn how to "look into the future" to predict how these death rates will continue to change. This is vital for pension funds and insurance companies to ensure they have enough money to pay out for longer-than-expected lives.

1. Why do we project mortality?

We project mortality because of Longevity Risk. This is the risk that people live longer than we expected. If you are an actuary for a pension fund and your members live 5 years longer than you planned, you have to pay them 5 extra years of pension—that’s expensive!

Key Concept: Mortality Improvement
Over time, the force of mortality \(\mu_x\) generally decreases. This means \(q_x\) (the probability of dying) is a function of both age (x) and time (t). We write this as \(q(x, t)\).

2. The Three Approaches to Projection

Don't worry if these sound technical; they are just three different ways of looking at the same problem. Think of them as the "Three E's":

• Expectation (Expert Opinion): We ask doctors, scientists, and demographers what they think will happen. Example: "We expect a cure for certain cancers in 20 years."
• Explanation (Causal): We look at why people die (smoking, heart disease) and model those specific risks. Example: "If smoking rates drop by 10%, mortality should drop by X%."
• Extrapolation (Trend): We look at the past 50 years of data and assume the pattern will continue. This is the most common method in CS2.

Quick Review: The 3 E's

Expectation = Expert brains.
Explanation = Root causes.
Extrapolation = History repeating itself.

3. Mortality Reduction Factors

A simple way to project is to use a Reduction Factor. This is a number (usually less than 1) that we multiply by today's death rate to get tomorrow's death rate.

The formula is: \(q(x, t) = q(x, 0) \times RF(x, t)\)

Where:
\(q(x, 0)\) is the mortality rate now.
\(RF(x, t)\) is the reduction factor for age \(x\) at time \(t\).
\(q(x, t)\) is the projected mortality rate in the future.

Analogy: If a store has a 10% discount every year, the price next year is 0.90 times this year's price. Reduction factors work the same way for death rates!

4. The Lee-Carter Model

This is the "celebrity" model of mortality projection. It’s the one you are most likely to be tested on. It uses extrapolation to project central death rates \(m_{x,t}\).

The Formula:
\(\ln(m_{x,t}) = a_x + b_x \kappa_t + \epsilon_{x,t}\)

Let's break this down into "human English":
• \(a_x\): The "Average" mortality at age \(x\). This describes the general shape of mortality across ages.
• \(\kappa_t\): The "Time Index." This tells us the general trend of mortality over time. Usually, this goes down as years pass.
• \(b_x\): The "Sensitivity." It tells us how much age \(x\) reacts to the general time trend. Some ages see faster improvements than others.
• \(\epsilon_{x,t}\): The "Error Term." The little random bits the model can't explain.

Common Mistake: Students often confuse \(b_x\) and \(\kappa_t\). Just remember: k is for Kalendar (time) and b is for Behavior of a specific age.

Constraints in Lee-Carter

Because the model has many moving parts, we have to "fix" some values to make it solvable. The standard IFoA constraints are:
1. The sum of all \(b_x\) values must equal 1: \(\sum b_x = 1\)
2. The sum of all \(\kappa_t\) values must equal 0: \(\sum \kappa_t = 0\)

5. Age-Period-Cohort (APC) Models

Sometimes, looking at age and year isn't enough. We also need to look at when a person was born. This is the Cohort Effect.

Example: People born just after WWII (the Baby Boomers) might have different health habits than people born during the Great Depression. This "generational" effect travels with them as they age.

The APC Model Formula:
\(\ln(m_{x,t}) = a_x + \kappa_t + \gamma_{t-x}\)

Where \(\gamma_{t-x}\) is the cohort effect (since \(Year - Age = Birth Year\)).

6. The CBD Model (Cairns-Blake-Dowd)

The Lee-Carter model is great, but it can struggle with older ages. The CBD model was designed specifically for older populations (pensioners).

Instead of modeling \(m_{x,t}\), it models the logit of the mortality rate \(q_{x,t}\).
The logit function is: \(logit(p) = \ln(\frac{p}{1-p})\).

The Formula:
\(logit(q_{x,t}) = \kappa^{(1)}_t + \kappa^{(2)}_t (x - \bar{x})\)

In this model:
• \(\kappa^{(1)}_t\): Shifts the whole mortality line up or down (parallel shift).
• \(\kappa^{(2)}_t\): Changes the slope of the line (rotational shift).
• \(\bar{x}\): Is the average age in the data set.

Key Takeaway: CBD assumes that at any time \(t\), the logit of mortality is a straight line relative to age.

7. Summary and Key Limitations

Projecting mortality is not an exact science. Here are the things that can go wrong (and make for great exam points):

1. Model Risk: You picked Lee-Carter, but the CBD model was actually better suited for the data.
2. Parameter Risk: Your formula is right, but your estimates for \(a_x\) or \(b_x\) are slightly off.
3. Random Variation: Even if the model is perfect, actual deaths will fluctuate year to year.
4. Catastrophes: Models based on historical trends cannot predict sudden events like pandemics (e.g., COVID-19) or sudden medical breakthroughs.

Chapter Summary Checklist:

• Can you explain why we project mortality? (Longevity Risk!)
• Do you know the 3 E's (Expectation, Explanation, Extrapolation)?
• Can you write out the Lee-Carter formula from memory?
• Do you understand that CBD is for older ages and uses a straight-line logit approach?
• Do you remember the two main constraints for Lee-Carter (\(\sum b_x=1, \sum \kappa_t=0\))?

Don't worry if the formulas look scary! Most exam questions ask you to explain what the parameters mean or perform a simple calculation using a given formula. Practice identifying which part of the formula represents "age," "time," and "generation," and you'll be well on your way to success!