Survival and death probabilities, deferred and select forms

Welcome to Survival and Death Probabilities!

At the heart of actuarial science is the ability to model the "uncertain future." For life insurance, that uncertainty usually revolves around one big question: How long will a person live?

In this chapter, we transition from the "Theory of Interest" (how money grows) to "Life Contingencies" (how life and death affect cashflows). We will learn the mathematical shorthand actuaries use to describe the probability of survival and death. Don't worry if the notation looks like a secret code at first—once you understand the "story" behind each symbol, it becomes much easier!

1. Survival Probabilities: \( {}_np_x \)

The most fundamental building block is the probability that a life currently aged \( x \) will survive for another \( n \) years. In actuarial notation, we write this as \( {}_np_x \).

Breaking down the symbol:
- The \( x \) (at the bottom right) is the current age of the person.
- The \( n \) (at the bottom left) is the duration we are looking at.

The Formula:
Using the life table function \( l_x \) (which represents the number of people alive at age \( x \)), the probability is:
\( {}_np_x = \frac{l_{x+n}}{l_x} \)

Analogy: Imagine a marathon starts with 1,000 people (\( l_x \)). If 800 people are still running at the 10-mile mark (\( l_{x+n} \)), the probability of a starter reaching that mark is \( \frac{800}{1,000} = 0.8 \).

Key Takeaway:

If the duration \( n \) is 1 year, we often drop the \( n \) and just write \( p_x \).

2. Death Probabilities: \( {}_nq_x \)

The "flip side" of survival is the probability of dying. \( {}_nq_x \) is the probability that a life aged \( x \) will die within the next \( n \) years.

The Formulas:
Since you either survive or you don't, the two probabilities must add up to 1:
\( {}_nq_x = 1 - {}_np_x \)

Alternatively, using the life table:
\( {}_nq_x = \frac{l_x - l_{x+n}}{l_x} \)

Common Mistake Alert: Students often confuse "dying at age x" with "dying within the next year." In actuarial math, \( q_x \) always refers to the probability of someone who has already reached age \( x \) dying before they reach age \( x+1 \).

3. Deferred Death Probabilities: \( {}_{n|m}q_x \)

This is where things get interesting! Sometimes we want to know the probability that someone survives for a while, but then dies during a specific later period. This is called a deferred probability.

The Symbol: \( {}_{n|m}q_x \)
This represents the probability that a life aged \( x \):
1. Survives for \( n \) years, and then
2. Dies within the following \( m \) years.

The Logic:
To calculate this, we look at the people alive at age \( x+n \) and see how many of them die by age \( x+n+m \), but we divide by the people alive at age \( x \) (our starting point).

The Formula:
\( {}_{n|m}q_x = \frac{l_{x+n} - l_{x+n+m}}{l_x} \)

You can also think of it this way:
\( {}_{n|m}q_x = {}_np_x - {}_{n+m}p_x \)

Quick Review: The "Special Case" \( {}_n|q_x \)

If you see \( {}_n|q_x \) (with only one number after the bar), it implies the death happens in the single year following the deferment period. So, \( m = 1 \).
\( {}_n|q_x = \frac{l_{x+n} - l_{x+n+1}}{l_x} = \frac{d_{x+n}}{l_x} \)

4. Select and Ultimate Mortality

Did you know? A 60-year-old who just passed a medical exam to buy life insurance is statistically likely to live longer than a 60-year-old who has been in the insurance pool for 10 years. This is because the new policyholder has been "selected" for their good health.

Select Period

To account for this, actuaries use select mortality. We use square brackets \( [x] \) to denote the age at which the person was "selected" (e.g., the age they bought the policy).

Notation:
- \( l_{[x]} \): Number of people alive at the age of selection \( x \).
- \( l_{[x]+r} \): Number of people alive \( r \) years after being selected at age \( x \).
- \( q_{[x]+r} \): Probability of death for someone selected at age \( x \), now aged \( x+r \).

Ultimate Period

After a certain number of years (the select period), the health advantage of the medical exam wears off. At this point, the mortality rate only depends on the current age, not the age at selection. We call this ultimate mortality.

Example: If the select period is 2 years:
- Year 0: \( q_{[x]} \) (Highly select)
- Year 1: \( q_{[x]+1} \) (Still somewhat select)
- Year 2+: \( q_{x+2} \) (Ultimate—the brackets disappear!)

Key Takeaway:

When solving exam questions, always check the table header. If it has square brackets, you are dealing with select mortality, and you must track the duration since selection carefully.

5. Summary of Key Formulas

Use this table as a quick reference for your studies:

Concept	Symbol	Life Table Expression
Survival for \( n \) years	\( {}_np_x \)	\( \frac{l_{x+n}}{l_x} \)
Death within \( n \) years	\( {}_nq_x \)	\( \frac{l_x - l_{x+n}}{l_x} \)
Death in 1 year	\( q_x \)	\( \frac{d_x}{l_x} \)
Deferred Death	\( {}_{n|m}q_x \)	\( \frac{l_{x+n} - l_{x+n+m}}{l_x} \)
Select Survival	\( {}_np_{[x]} \)	\( \frac{l_{[x]+n}}{l_{[x]}} \)

Practical Study Tip: "Draw the Timeline"

If you find \( {}_{n|m}q_x \) confusing, draw a horizontal line representing time.
1. Mark age \( x \) (today).
2. Mark age \( x+n \) (end of deferment).
3. Mark age \( x+n+m \) (end of observation).
The probability is simply the "gap" between the last two points, divided by the starting population at the first point.

In the next chapter, we will use these probabilities to calculate the Expected Present Value (EPV) of actual insurance contracts!