Welcome to Transition Intensities: Bridging Data and Models!
In your CS2 journey so far, you’ve likely worked with survival models where the transition intensity (like the force of mortality, \(\mu_x\)) is already given to you. But where do those numbers come from? In the real world, actuaries don't find \(\mu_x\) in a textbook; they have to estimate it from data.
This chapter is all about how we take raw data—like lists of birthdays and dates of death—and turn them into the transition intensities we use to price pensions and insurance. Don't worry if this seems a bit abstract at first; we're essentially just counting how many events happened and dividing it by how much time people "risked" having those events happen!
1. The Core Principle: The Force of Mortality
Before we dive into the math, let’s look at the basic "recipe" for estimating a transition intensity (let's use \(\mu_x\) as our example):
Estimated Intensity = (Number of Deaths) / (Total Time Spent at Risk)
In actuarial notation, we write this as:
\( \hat{\mu}_x = \frac{d_x}{E_x^c} \)
Where:
- \(d_x\) is the number of deaths observed at age \(x\).
- \(E_x^c\) is the Central Exposure to Risk at age \(x\).
Analogy: The Speed Camera
Imagine you want to know the "intensity" of speeding on a specific road. You count how many cars get caught (the deaths, \(d_x\)) and divide it by the total time all cars spent driving on that road (the exposure, \(E_x^c\)). If 10 cars got caught and total driving time was 100 hours, your "speeding intensity" is 0.1 per hour.
Key Takeaway: To estimate the intensity, we always need two things: a count of events and the total time people were "at risk" of that event occurring.
2. Calculating Exact Exposure
The most precise way to find \(E_x^c\) is to use Exact Data. This means we know the exact date every person entered the observation, the exact date they left (either by dying, the study ending, or withdrawing), and their exact birthday.
How to calculate it step-by-step:
1. For every individual in the study, look at the time they spent between age \(x\) and \(x+1\).
2. If they were under observation for that entire year, they contribute 1 year to \(E_x^c\).
3. If they joined halfway through the year or died 3 months into the year, they only contribute 0.5 or 0.25 years respectively.
4. Add them all up!
Example: If you have 3 people:
- Person A: Observed for the full year at age 60.
- Person B: Observed for 6 months at age 60, then died.
- Person C: Observed for 3 months at age 60, then the study ended.
The Exact Central Exposure \(E_{60}^c = 1 + 0.5 + 0.25 = 1.75\) person-years.
3. The Census Method: Dealing with Messy Data
In reality, insurance companies often don't have a "stopwatch" on every policyholder. Instead, they take "snapshots" (censuses) at specific dates (like January 1st every year).
The Trapezium Approximation
If we only know how many people were alive at age \(x\) on Jan 1, 2022 (\(P_{x,0}\)) and how many were alive at age \(x\) on Jan 1, 2023 (\(P_{x,1}\)), we assume the population changed linearly throughout the year.
The formula for the Central Exposure is:
\( E_x^c \approx \int_0^1 P_{x,t} dt \approx \frac{1}{2} (P_{x,0} + P_{x,1}) \)
Did you know? This is called the Trapezium Rule because if you graph the population over time, you are essentially calculating the area of a trapezoid under the curve!
4. Age Definitions and the "Rate Interval"
This is where students often get tripped up. Not all "Age 60s" are the same! We need to define the Rate Interval—the period of life over which we are counting deaths and exposure.
Common Age Definitions:
- Age Last Birthday (\(x\)): The person is between exact age \(x\) and \(x+1\). (This is how we usually talk about age in daily life).
- Age Nearest Birthday (\(x\)): The person is between exact age \(x - 0.5\) and \(x + 0.5\).
- Age Next Birthday (\(x\)): The person is between exact age \(x-1\) and \(x\).
Quick Tip: Always identify the average age at the start of the interval.
- For "Age Last Birthday \(x\)", the average age at the start is \(x + 0.5\).
- For "Age Nearest Birthday \(x\)", the average age at the start is \(x\).
5. The Poisson Model for Deaths
In CS2, we often assume that the number of deaths (\(D_x\)) follows a Poisson distribution. This is a very common exam topic!
If we assume the force of mortality \(\mu_x\) is constant over the age interval, then:
\( D_x \sim \text{Poisson}(\mu_x \cdot E_x^c) \)
From this, we can derive the Maximum Likelihood Estimator (MLE):
\( \hat{\mu}_x = \frac{d_x}{E_x^c} \)
And the Variance of our estimator:
\( Var(\hat{\mu}_x) = \frac{\mu_x}{E_x^c} \approx \frac{d_x}{(E_x^c)^2} \)
Memory Aid: Think of \(\mu_x \cdot E_x^c\) as the "Expected Number of Deaths." In a Poisson distribution, the Mean = \(\lambda\). So here, \(\lambda = \text{Intensity} \times \text{Exposure}\).
6. Common Mistakes to Avoid
1. Mixing up Central and Initial Exposure: Central Exposure (\(E_x^c\)) is used to estimate intensities (\(\mu_x\)). Initial Exposure (\(E_x\)) is used to estimate probabilities (\(q_x\)). Don't swap them!
2. The "Half-Year" Rule: When using census data, if a census is taken on July 1st (mid-year) and you want the exposure for the calendar year, that single census count \(P_{x, \text{July}}\) is often used as the average population, so \(E_x^c \approx P_{x, \text{July}}\).
3. Misaligning \(d_x\) and \(E_x^c\): Ensure the deaths you are counting occurred in the exact same age interval and time period as the exposure you are calculating.
Summary Review Box
- The Goal: Estimate \(\mu_x\) using \(\frac{d_x}{E_x^c}\).
- Exact Data: Use the actual time spent by each person (the most accurate).
- Census Data: Use snapshots of the population and assume linear change (the most practical).
- Poisson Assumption: \(D_x \sim \text{Poisson}(\mu_x E_x^c)\).
- Watch your ages! Check if the data is "last birthday," "nearest birthday," or "next birthday" to ensure your \(\mu\) is assigned to the correct age.
Don't worry if the census formulas look intimidating! Just remember they are all just different ways of estimating the "average" number of people alive during the year and multiplying by 1 year.