Welcome to the Life Table!
In your CM1 journey so far, you have looked at interest rates and certain cashflows. Now, we enter the heart of actuarial work: Life Contingencies. In this chapter, we learn how to use a Life Table to model the uncertainty of human life. Don't worry if the notation looks like a secret code at first—once you understand the "story" the table is telling, the math becomes much more intuitive.
Think of a life table as a "biographical map" of a large group of people. We follow them from birth until the very last person passes away, recording exactly how many are left at each birthday. This allows us to calculate the probability of someone surviving to collect a pension or dying during the term of a life insurance policy.
1. The Fundamental Building Blocks: \(l(x)\) and \(d(x)\)
To build our model, we start with two basic functions that form the spine of any life table.
The "Survivors" function: \(l(x)\)
The symbol \(l(x)\) represents the expected number of people who are still alive at exact age \(x\).
- We usually start with a large arbitrary number at age 0 (like 100,000), called the radix.
- As age \(x\) increases, \(l(x)\) must always stay the same or decrease. You can't have more people alive at age 50 than were alive at age 40!
The "Deaths" function: \(d(x)\)
The symbol \(d(x)\) represents the number of people expected to die between age \(x\) and age \(x+1\).
- It is the "gap" between two consecutive years of survivors.
- The Formula: \(d(x) = l(x) - l(x+1)\)
Quick Analogy: Imagine 100 runners start a marathon. \(l(x)\) is the number of runners still on the track at mile \(x\). \(d(x)\) is the number of runners who dropped out between mile \(x\) and mile \(x+1\).
Key Takeaway: \(l(x)\) is a "snapshot" at a specific moment in time (a birthday), while \(d(x)\) covers the entire year between two birthdays.
2. Calculating Probabilities Using the Life Table
Once we have a table of \(l(x)\) values, we can calculate various survival and death probabilities. In the CM1 syllabus, these are expressed as ratios of life table functions.
Survival Probabilities
\(n_p_x\) is the probability that a person aged \(x\) survives for at least \(n\) more years.
- The Formula: \(n_p_x = \frac{l(x+n)}{l(x)}\)
- Example: If \(x=40\) and \(n=10\), we are looking for the proportion of people alive at 40 who make it to 50.
Death Probabilities
\(n_q_x\) is the probability that a person aged \(x\) dies within the next \(n\) years.
- The Formula: \(n_q_x = \frac{l(x) - l(x+n)}{l(x)}\) or simply \(1 - n_p_x\).
Deferred Death Probabilities
\(n|m_q_x\) (often written as \(n|q_x\) if \(m=1\)) is the probability that someone aged \(x\) survives for \(n\) years, but then dies within the following \(m\) years.
- The Formula: \(n|m_q_x = \frac{l(x+n) - l(x+n+m)}{l(x)}\)
- Thinking Process: To calculate this, you find the number of deaths occurring in that specific "future window" and divide by the number of people alive today at age \(x\).
Common Mistake to Avoid: When calculating \(n|m_q_x\), students often accidentally divide by \(l(x+n)\). Remember, the subscript \(x\) at the end of the symbol tells you who you are talking about now. Always divide by the number of people alive at the current age (\(l(x)\)).
3. Select Mortality: The "Selection" Effect
This is one of the most important concepts in actuarial modeling. Up until now, we assumed that everyone aged \(x\) has the same chance of dying. In the real world, this isn't true!
The Concept: Imagine two 50-year-olds. Person A just walked off the street. Person B just passed a rigorous medical exam to buy a life insurance policy. Who is more likely to die this year? Usually, Person B (the "selected" life) is healthier.
Select Period: The "health boost" from passing a medical exam doesn't last forever. After a few years (the select period), the mortality of the "selected" group starts to look just like the mortality of the general population.
Select Notation
We use square brackets \([x]\) to show the age at which the person was selected (e.g., when they bought the policy).
- \(l[x]+r\): The number of survivors at age \(x+r\), given they were selected at age \(x\).
- \(d[x]+r\): The number of deaths between age \(x+r\) and \(x+r+1\), given they were selected at age \(x\).
Did you know? Most life tables in the IFoA exams have a select period of 2 years. This means:
- At year 0: Use \(l[x]\)
- At year 1: Use \(l[x]+1\)
- At year 2+: The brackets disappear, and you just use the standard "ultimate" value \(l(x+2)\).
4. Working with Select Tables
When you look at a Select Life Table, it often looks like a grid. To find the right value, follow these steps:
Step 1: Identify the Age at Selection \([x]\). This is your starting row.
Step 2: Identify the Duration \(r\). This is the number of years since they joined the policy.
Step 3: Check if \(r\) is greater than the select period. If the select period is 2 years and you need the value for duration 5, the "selection" has worn off. You move to the Ultimate column for age \(x+5\).
Key Relationship:
\(p[x] = \frac{l[x]+1}{l[x]}\) (Probability a life selected at age \(x\) survives 1 year).
\(p[x]+1 = \frac{l(x+2)}{l[x]+1}\) (Probability a life selected at age \(x\) survives the 2nd year, reaching the ultimate table).
Quick Summary Table:
- Standard (Ultimate): \(l(x)\) — Used for general populations or after the select period ends.
- Select: \(l[x]\) — Used for people who just joined/passed a medical.
- Duration: \(l[x]+r\) — Used for people \(r\) years into their policy.
5. Chapter Summary and Tips
The "Logic Flow": If you are ever stuck on a complex probability question (like a deferred select probability), always go back to the basic definition:
"Probability = (Number of people who do what I'm looking for) / (Number of people alive at the start)"
Don't Panic about the Math: In the CM1 exam (especially Paper B in Excel), you will often be given these values or have to look them up. The key is knowing which column to look at.
- Use Select columns if the question mentions "just issued," "new policyholder," or "purchased."
- Use Ultimate columns if the question mentions "long-term policyholder" or if the duration exceeds the select period.
Key Takeaway: The life table is just a tool to turn "Ages" into "Probabilities." Master the notation \(l[x]+r\), and you have mastered the foundation of life insurance pricing!
Next Chapter Preview: We will take these probabilities and combine them with the "Time Value of Money" (interest rates) to calculate the actual price of insurance contracts!