Welcome to the World of Pension Math!
Hello there! If you've made it to the ALTAM exam, you already know that actuarial science is more than just life and death—it’s also about planning for a comfortable future. In this chapter, we are diving into Service Tables and Salary Scale Functions. These are the tools we use to predict how many people will stay at a company and how much money they will earn before they retire.
Don't worry if this seems a bit dense at first. Think of this chapter as building a "simulated office." We want to know: Who stays? Who goes? And how big will their paychecks get? Let’s break it down step-by-step!
1. The Service Table (Multiple Decrement Model)
In basic FAM-L, you dealt with mortality tables where the only way "out" was death. In a pension plan, things are more realistic (and a bit more complicated). A Service Table tracks a group of employees and accounts for all the different ways they might leave the plan.
What is a Service Table?
A service table is essentially a Multiple Decrement Table. We track a cohort of employees starting at an age (like 25 or 30) until everyone has left the service. There are four main ways employees leave:
1. Death (the classic decrement)
2. Withdrawal (quitting or being laid off)
3. Disability (leaving due to health reasons)
4. Retirement (the goal!)
Key Notation to Remember
You’ll see these symbols a lot, so let's get comfortable with them:
\( l_x^{(\tau)} \): The number of people still "in service" at age \( x \). The \(\tau\) stands for "total."
\( d_x^{(j)} \): The number of people who leave between age \( x \) and \( x+1 \) for a specific reason \( j \). For example, \( d_x^{(w)} \) might be those who withdraw (quit).
\( q_x^{(j)} \): The probability that an employee age \( x \) will leave for reason \( j \) within one year.
The Golden Rule of Service Tables:
\( l_{x+1}^{(\tau)} = l_x^{(\tau)} - d_x^{(1)} - d_x^{(2)} - ... - d_x^{(n)} \)
Essentially, the number of people next year equals the number of people this year minus everyone who left for any reason.
Analogy: The Leaky Bucket
Imagine a bucket of water (the employees). There aren't just one, but four different holes in the bottom of the bucket. One hole is labeled "Retirement," another "Quitting," etc. The Service Table simply measures how much water is left in the bucket after it leaks through all those holes over time.
Quick Review: Key Points
• \( l_x^{(\tau)} \) is the surviving population.
• \( d_x^{(j)} = l_x^{(\tau)} \times q_x^{(j)} \).
• The total decrement rate is the sum of the individual rates: \( q_x^{(\tau)} = \sum q_x^{(j)} \).
2. Salary Scale Functions
Now that we know who is working, we need to know how much they are earning. Most people don't earn the same salary for 40 years; they get raises! We use a Salary Scale Function, denoted as \( s_x \), to model this growth.
Understanding \( s_x \)
The value \( s_x \) is not a dollar amount. It is a relative index. It tells us the ratio of salary at one age compared to another.
If you know someone’s salary at age \( x \) (let's call it \( Sal_x \)), you can predict their salary at age \( y \) using this formula:
\( Sal_y = Sal_x \times \frac{s_y}{s_x} \)
Why do salaries increase?
Actuaries usually consider three factors in the salary scale:
1. Inflation: The general rise in prices.
2. Productivity: General economic growth.
3. Merit/Promotions: The specific employee getting better at their job as they get older.
Did you know? In many exam problems, you might be given a constant annual growth rate \( g \). In that case, the salary scale looks like a geometric progression: \( s_x = (1+g)^x \).
Common Mistake: The "Middle of the Year" Trap
Pay close attention to when the salary is paid.
• If the salary is for the year following age \( x \), we usually use \( s_x \).
• Sometimes problems talk about the rate of salary at a exact moment in time. Always read carefully to see if you need to average two years or use a mid-year value!
Key Takeaway
The Salary Scale \( s_x \) is a multiplier. To find a future salary, you divide by where you are and multiply by where you're going.
3. Calculating Average Salaries
Most pension plans don't base your benefit on just your very last paycheck. Instead, they use an average of your highest or final years of earnings. This is called the Final Average Salary (FAS).
Final Average Salary (3-year example)
If a plan uses a 3-year FAS, and an employee retires at age \( R \), the FAS is the average of the salaries earned in the three years before retirement:
\( FAS = \frac{1}{3} [Sal_{R-1} + Sal_{R-2} + Sal_{R-3}] \)
Using our salary scale \( s_x \), if we know the salary at current age \( x \), we can write this as:
\( FAS = \frac{Sal_x}{3} \times [ \frac{s_{R-1}}{s_x} + \frac{s_{R-2}}{s_x} + \frac{s_{R-3}}{s_x} ] \)
Step-by-Step: Projecting FAS
1. Identify the current age \( x \) and the current salary \( Sal_x \).
2. Identify the retirement age \( R \).
3. List the years needed (e.g., for a 5-year average, you need salaries at ages \( R-1, R-2, R-3, R-4, R-5 \)).
4. Project each salary using the ratio \( \frac{s_{target}}{s_x} \).
5. Average them by adding them up and dividing by the number of years.
4. Bringing it All Together: Valuation
Why do we use both the service table and the salary scale? Because to calculate the cost of a pension today, we need to know:
1. The Probability that the person survives in service to receive the benefit (from the Service Table).
2. The Amount of the benefit, which is usually a percentage of their salary (from the Salary Scale).
3. The Time Value of Money (discounting the future cost to the present).
Example Concept:
The Actuarial Present Value (APV) of a future retirement benefit often looks something like this (simplified):
\( APV = (\text{Probability of staying until retirement}) \times (\text{Discount factor}) \times (\text{Projected Benefit based on Salary}) \)
Memory Aid: The "PAS" Method
When solving these problems, remember PAS:
P - Probability: Use the service table \( \frac{l_R^{(\tau)}}{l_x^{(\tau)}} \).
A - Amount: Use the salary scale and the benefit formula.
S - Sum/Standardize: Discount the value back to today using interest rates.
5. Summary and Tips for Success
• Stay Organized: Service table problems often involve big tables. Keep your columns straight!
• Watch the Ages: Does the person retire at age 65? That means their last year of work (and last salary) was from age 64 to 65.
• Total vs. Individual: Remember that \( q_x^{(\tau)} \) is the probability of leaving for any reason. Don't mix it up with the probability of leaving for a specific reason \( q_x^{(j)} \).
• Practice the FAS: FAS calculations are almost guaranteed to show up. Practice 3-year and 5-year averages until they feel like second nature.
Don't worry if the math feels heavy. Just remember: we are just predicting a career path—how long it lasts and how much it pays. You've got this!