Parametric survival models and life tables

Q: What does the survival function \( S_0(x) \) represent?

It is the probability that a newborn lives to at least age \( x \).

Q: What defines the Exponential Model for mortality?

The force of mortality is constant (\( \mu_x = \mu \)), and survival probability is \( {}_t p_x = e^{-\mu t} \).

Q: What is the functional form of the force of mortality in the Gompertz Model?

\( \mu_x = B \cdot c^x \), where death risk increases exponentially with age.

Q: How does the Makeham Model adjust the Gompertz Model?

It adds an age-independent constant \( A \) to the Gompertz force: \( \mu_x = A + B \cdot c^x \).

Q: What characterizes the force of mortality in a Weibull Model?

The force of mortality is a power function of age: \( \mu_x = k \cdot x^n \).

Q: What is the formula for the probability of death within a fraction of a year \( s \) under the Uniform Distribution of Deaths (UDD)?

\( {}_s q_x = s \cdot q_x \) for \( 0

Q: How do you calculate survival over a fraction of a year \( s \) under the Constant Force of Mortality (CF) assumption?

\( {}_s p_x = (p_x)^s \) for \( 0

Exam FAM – Fundamentals of Actuarial Mathematics · SOA (Society of Actuaries) · 7 min read

Welcome to Mortality Models: Parametric survival models and life tables!

Hello there, future actuary! Welcome to one of the most fundamental chapters of your Exam FAM journey. In this section, we are going to learn how to translate the unpredictable nature of human life into math. Why? Because to price an insurance policy or value a pension, we need to know the probability of someone being alive to receive a benefit or dying to trigger a claim.

Think of this chapter as the "rulebook" for how people age and die in our mathematical models. We'll look at Parametric Models (smooth mathematical formulas) and Life Tables (neatly organized lists of data). Don't worry if the formulas look a bit intimidating at first—we'll break them down piece by piece!

1. The Basics: Survival and Death

Before we look at complex models, we need to understand the two main "characters" in our story: the probability of living and the probability of dying.

• \( S_0(x) \) (Survival Function): This is the probability that a newborn lives to at least age \( x \). It starts at 1 (everyone is alive at birth) and eventually goes to 0 (everyone dies eventually).
• \( F_0(x) \) (Distribution Function): This is the probability that a newborn dies before age \( x \). It is simply \( 1 - S_0(x) \).
• \( {}_t p_x \): This is the probability that someone already aged \( x \) lives for another \( t \) years. Mathematically: \( {}_t p_x = \frac{S_0(x+t)}{S_0(x)} \).
• \( {}_t q_x \): This is the probability that someone aged \( x \) dies within the next \( t \) years. Mathematically: \( {}_t q_x = 1 - {}_t p_x \).

Analogy: The Smartphone Battery

Imagine a smartphone battery. \( S_0(x) \) is the probability that the battery is still working after \( x \) hours. \( {}_t p_x \) is like saying, "If my phone is already at 50% after 4 hours, what are the odds it lasts another 2 hours?"

Key Takeaway

Everything in this chapter is a ratio. To find the probability of surviving from age 40 to 50, you are simply looking at how many people (proportionally) make it to 50 out of those who were alive at 40.

2. The Force of Mortality \( \mu_x \)

The Force of Mortality is a very important concept. It represents the instantaneous rate of death at age \( x \). It is not a probability (it can be greater than 1!), but rather the "intensity" of the risk of dying right at that second.

The formula to remember is:
\( \mu_x = -\frac{1}{S_0(x)} \frac{d}{dx} S_0(x) = -\frac{d}{dx} \ln(S_0(x)) \)

Quick Review: If you are given \( \mu_x \) and need to find the survival probability, use this "integral trick":
\( {}_t p_x = e^{-\int_{0}^{t} \mu_{x+s} ds} \)

Common Mistake to Avoid

Students often confuse \( q_x \) and \( \mu_x \). Remember: \( q_x \) is the probability of dying over a whole year, while \( \mu_x \) is the pressure of death at a single moment.

3. Parametric Survival Models

Actuaries use specific mathematical shapes to describe mortality. Here are the four you need to know for Exam FAM:

A. Exponential Model (Constant Force)

In this model, the risk of dying is the same regardless of age. It's like a machine that never wears out; it only breaks by random accidents.
• Formula: \( \mu_x = \mu \) (a constant)
• Survival: \( {}_t p_x = e^{-\mu t} \)
• Memory Aid: "Constant is Boring." The probability of surviving 10 years is the same whether you are 20 or 80.

B. Gompertz Model

This is more realistic for humans. It assumes the risk of death increases exponentially as we age.
• Formula: \( \mu_x = B \cdot c^x \)
• Survival: \( {}_t p_x = \exp\left[ -\frac{B \cdot c^x (c^t - 1)}{\ln c} \right] \)
• Key Point: If you see a formula where the age \( x \) is in the exponent (like \( 0.001 \cdot 1.1^x \)), think Gompertz!

C. Makeham Model

Makeham took the Gompertz model and added a constant (\( A \)) to account for "accidental death" that doesn't depend on age.
• Formula: \( \mu_x = A + B \cdot c^x \)
• Survival: It's just the Gompertz survival multiplied by \( e^{-At} \).
• Did you know? This was one of the first models to accurately describe adult human mortality across many different populations.

D. Weibull Model

In this model, the force of mortality is a power function of age.
• Formula: \( \mu_x = k \cdot x^n \)
• Survival: \( S_0(x) = e^{-\frac{k}{n+1} x^{n+1}} \)
• Identifying it: Look for age \( x \) raised to a fixed power (like \( x^2 \) or \( \sqrt{x} \)).

Key Takeaway Summary

• Exponential: \( \mu_x \) is flat.
• Gompertz: \( \mu_x \) grows like a compound interest curve.
• Makeham: Gompertz + a flat "accident" rate.
• Weibull: \( \mu_x \) grows like a polynomial (e.g., \( x^2 \)).

4. The Life Table

If parametric models are the "formulas," Life Tables are the "spreadsheets." They give us a snapshot of a large group of people (called a cohort) as they age.

Standard Notation:
• \( l_x \): The number of people alive at age \( x \). We usually start with a large number like 100,000 (called the radix).
• \( d_x \): The number of people who die between age \( x \) and \( x+1 \). Formula: \( d_x = l_x - l_{x+1} \).
• \( p_x \): The probability an \( x \)-year-old lives to \( x+1 \). Formula: \( p_x = \frac{l_{x+1}}{l_x} \).
• \( q_x \): The probability an \( x \)-year-old dies before \( x+1 \). Formula: \( q_x = \frac{d_x}{l_x} \).

Step-by-Step: Moving through the Table

1. Start with \( l_x \).
2. Multiply by \( q_x \) to find how many people die (\( d_x \)).
3. Subtract \( d_x \) from \( l_x \) to get the next year's survivors (\( l_{x+1} \)).
4. Repeat!

Key Takeaway

\( l_x \) is just a scaled version of the survival function. Specifically, \( S_0(x) = \frac{l_x}{l_0} \). If you can do basic fractions, you can master Life Tables!

5. Fractional Ages (Interpolation)

Life tables usually only give us data for whole years (age 40, 41, 42). But what if we need to know the probability of someone dying at age 40.5? We have to "fill in the blanks." There are two main methods for Exam FAM:

A. Uniform Distribution of Deaths (UDD)

We assume deaths occur evenly throughout the year. This is like a straight line connecting \( l_x \) and \( l_{x+1} \).
• Formula: \( {}_s q_x = s \cdot q_x \) (for \( 0 < s < 1 \))
• Analogy: If 100 people die in a year, UDD assumes exactly 50 die in the first 6 months.

B. Constant Force of Mortality (CF)

We assume the force of mortality (\( \mu \)) is constant between birthdays, even if it changes from one year to the next. This creates a slight curve.
• Formula: \( {}_s p_x = (p_x)^s \)
• Key Difference: CF results in slightly higher survival probabilities compared to UDD within the year.

Quick Review Box

If age is fractional (e.g., 0.5):
• UDD: Use linear math (multiplication/addition).
• Constant Force: Use exponential math (powers/exponents).

Final Tips for Success

• Don't Panic: Many exam questions look complex but just require you to find a ratio of two numbers from a table or a formula.
• Practice the Notation: Make sure you can instantly recognize the difference between \( {}_n q_x \) (dying within \( n \) years) and \( {}_m|{}_n q_x \) (surviving \( m \) years then dying in the following \( n \) years).
• Check Your Work: Probabilities must always be between 0 and 1. If you get 1.5, something went wrong!

You've got this! Keep practicing those \( l_x \) calculations, and the patterns will start to feel like second nature.

Quick check

Can you answer these now?

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

What does the survival function \( S_0(x) \) represent?

It is the probability that a newborn lives to at least age \( x \).

How is the conditional survival probability \( {}_t p_x \) calculated?

\( {}_t p_x = \frac{S_0(x+t)}{S_0(x)} \)

What is the mathematical definition of the Force of Mortality \( \mu_x \)?

\( \mu_x = -\frac{1}{S_0(x)} \frac{d}{dx} S_0(x) = -\frac{d}{dx} \ln(S_0(x)) \)

How do you determine survival probability \( {}_t p_x \) if you are given the force of mortality \( \mu \)?

\( {}_t p_x = e^{-\int_{0}^{t} \mu_{x+s} ds} \)

What defines the Exponential Model for mortality?

The force of mortality is constant (\( \mu_x = \mu \)), and survival probability is \( {}_t p_x = e^{-\mu t} \).

What is the functional form of the force of mortality in the Gompertz Model?

\( \mu_x = B \cdot c^x \), where death risk increases exponentially with age.

How does the Makeham Model adjust the Gompertz Model?

It adds an age-independent constant \( A \) to the Gompertz force: \( \mu_x = A + B \cdot c^x \).

What characterizes the force of mortality in a Weibull Model?

The force of mortality is a power function of age: \( \mu_x = k \cdot x^n \).

In a life table, how do you calculate the number of deaths \( d_x \) between ages \( x \) and \( x+1 \)?

\( d_x = l_x - l_{x+1} \)

What is the relationship between the life table value \( l_x \) and the survival function \( S_0(x) \)?

\( S_0(x) = \frac{l_x}{l_0} \)

What is the formula for the probability of death within a fraction of a year \( s \) under the Uniform Distribution of Deaths (UDD)?

\( {}_s q_x = s \cdot q_x \) for \( 0 < s < 1 \).

How do you calculate survival over a fraction of a year \( s \) under the Constant Force of Mortality (CF) assumption?

\( {}_s p_x = (p_x)^s \) for \( 0 < s < 1 \).

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