Welcome to the World of Select Life Tables!
Hello! If you have been studying mortality models, you already know about standard life tables. But have you ever wondered if everyone of the same age has the exact same risk of dying? In the real world, someone who just passed a strict medical exam for a new insurance policy is likely "healthier" than someone of the same age who bought their policy 20 years ago.
In this chapter, we explore Select Life Tables. These tables help actuaries account for the fact that newly "selected" individuals often have lower mortality rates. Don't worry if this seems a bit technical at first—we will break it down step-by-step!
1. The Concept of "Selection"
When a person applies for life insurance, they usually undergo underwriting. This might include medical exams, blood tests, and health questionnaires. Because the insurance company only accepts people who pass these tests, those people are "selected" for their good health.
The Selection Effect: For a period of time after joining a plan, a "selected" individual will have a lower probability of death than a random person of the same age from the general population. However, as time passes, this "health boost" wears off, and their mortality eventually matches the general population again.
Analogy: Think of a brand-new car vs. an older car. Even if they are the same model and year, a car that just had a 100-point inspection and a full tune-up (the "selected" car) is less likely to break down next week than a car that hasn't been serviced in years.
Quick Review: Selection happens because of underwriting. It makes mortality rates lower for a specific period of time.
2. Understanding the Notation
Notation is where most students get tripped up in this chapter. Let's look at the symbols used in Select Life Tables:
The Square Brackets \([x]\): The brackets around the age indicate the age at which the person was selected (e.g., when they bought the policy).
The Duration \(+k\): This indicates how many years have passed since the selection occurred.
Key Symbols:
- \( [x] \): Age at selection.
- \( [x] + k \): A person who was selected at age \(x\) and is now age \(x + k\).
- \( q_{[x]+k} \): The probability that a person selected at age \(x\), who is now age \(x+k\), will die within one year.
- \( \ell_{[x]+k} \): The number of lives surviving at age \(x+k\) who were selected at age \(x\).
Common Mistake to Avoid: Confusing \( q_{[50]+2} \) with \( q_{52} \).
\( q_{[50]+2} \) is the mortality rate for a 52-year-old who was just checked by a doctor 2 years ago.
\( q_{52} \) is the mortality rate for a 52-year-old in the general population (or someone whose "selection" has expired).
3. The Select Period and the Ultimate Table
The Select Period (often denoted as \(d\)) is the number of years the selection effect lasts. After \(d\) years, we assume the person's health is no different from anyone else of the same age.
When the selection effect wears off, we move to the Ultimate Table. In math terms, this means:
If \(k \ge d\), then \( q_{[x]+k} = q_{x+k} \).
Memory Aid: Think of the Ultimate table as the "end of the line." It doesn't matter when you started; once you've been in the group long enough, you're just like everyone else.
How to Read a Select Life Table
Tables are often laid out in rows and columns. Usually:
1. The rows represent the age at selection \([x]\).
2. The columns represent the years since selection (0, 1, 2, etc.).
3. The final column is the Ultimate column, where selection no longer matters.
Key Takeaway: Always check the "select period" in the problem. If the select period is 2 years, then after 2 years, you stop using the brackets and use the standard age values.
4. Probabilities and Calculations
The formulas for Select Life Tables are exactly the same as standard life tables; you just have to be careful with the indices.
Survival Probability:
\( p_{[x]+k} = \frac{\ell_{[x]+k+1}}{\ell_{[x]+k}} \)
Death Probability:
\( q_{[x]+k} = 1 - p_{[x]+k} = \frac{\ell_{[x]+k} - \ell_{[x]+k+1}}{\ell_{[x]+k}} \)
Multi-year Survival:
\( {}_n p_{[x]+k} = \frac{\ell_{[x]+k+n}}{\ell_{[x]+k}} \)
Example: If you want to find the probability that someone selected at age 40 survives 5 years, and the select period is 2 years, you would calculate:
\( {}_5 p_{[40]} = \frac{\ell_{[40]+5}}{\ell_{[40]}} \)
Since 5 is greater than the select period of 2, the numerator becomes the ultimate value: \( \frac{\ell_{45}}{\ell_{[40]}} \).
Step-by-Step Calculation:
1. Identify the age at selection \([x]\).
2. Identify the current duration \(k\).
3. Identify the length of the select period \(d\).
4. If \(k + \text{years you are looking forward} > d\), you will eventually "cross over" into the ultimate part of the table.
5. Why do we use Select Life Tables?
Did you know? If insurance companies used aggregate (general) tables to price policies for healthy people, the prices would be too high, and healthy people wouldn't buy insurance! By using Select Life Tables, companies can offer lower, more accurate rates to new policyholders.
Summary of Select vs. Aggregate vs. Ultimate:
- Select: Based on age at entry and time since entry (Healthiest).
- Ultimate: Based only on attained age, after selection wears off (Higher mortality than select).
- Aggregate: A blend of everyone at a certain age, regardless of when they entered (A "middle" average).
6. Common Exam Pitfalls
Keep an eye out for these "traps" often found on Exam FAM:
1. Not checking the select period: Students often stay in the "Select" columns for too many years. Check if the duration has exceeded the select period!
2. Mixing up \(x\) and \(k\): Remember that the number in brackets is the starting age. The number after the plus sign is how many years have passed.
3. Arithmetic errors: Because these tables involve many small decimals, take your time when punching numbers into your calculator.
Key Takeaway for Exam Day: The relationship is almost always \( q_{[x]+k} < q_{x+k} \). If your "selected" person has a higher chance of dying than the general population, double-check your work!
You've got this! Select Life Tables are just a way of being more specific about who we are studying. Once you master the notation \([x]+k\), the rest is just the same life table math you've already practiced.