Welcome to the World of Pension Funding!

Hello there! If you’ve made it to the ALTAM exam, you’ve already conquered some of the toughest mountains in actuarial science. Today, we are going to dive into how companies actually calculate the cost of a pension plan. Think of this as the "accounting" side of life contingencies.

In this chapter, we focus on two heavy hitters: the Traditional Unit Credit (TUC) method and the Projected Unit Credit (PUC) method. These are the tools actuaries use to answer two big questions for an employer:
1. "How much do I owe my employees right now for their past work?" (Actuarial Accrued Liability)
2. "How much will this year’s work cost me?" (Normal Cost)

Don't worry if these terms sound intimidating. We’ll break them down using simple analogies and step-by-step logic.

Did you know? The choice between TUC and PUC can significantly change how a company's financial health looks on paper, even if the total benefit paid to the employee remains the same!

Core Concepts: AL and NC

Before we look at the specific methods, let's define our two main variables. Imagine you are building a Lego castle for a friend. You agree to add one room every year for 30 years.

1. Actuarial Accrued Liability (AL)

The AL is the value today of the "rooms" you have already built. It represents the value of all retirement benefits earned by the employee up to the current date. It is the "debt" the pension plan has to the worker for their past service.

2. Normal Cost (NC)

The NC is the value today of the "room" you are building this year. It is the cost of the benefits allocated to the current year of service. If the employee works this year, the employer "owes" them this extra slice of their future pension.

Quick Review:
- AL: Looking backward (What have we earned so far?)
- NC: Looking at right now (What is being earned this year?)

The Traditional Unit Credit (TUC) Method

The TUC method is the "What you see is what you get" approach. It calculates benefits based on the employee's current salary at the time of the valuation. It does not try to guess what the employee will be making in 20 years.

How to calculate TUC:

1. Calculate the benefit earned to date based on current service and current salary.
2. Discount that benefit back to today using interest (\(v\)) and the probability of surviving in service to retirement (\({}_{r-x}p_x^{(\tau)}\)).
3. Multiply by the value of an annuity at retirement (\(a_{\ddot{r}}^{(12)}\)).

The TUC Formula for AL:
\( AL_x = b \times n \times v^{r-x} \times {}_{r-x}p_x^{(\tau)} \times \ddot{a}_r \)
Where:
\( b \)= Benefit per year of service (based on current salary)
\( n \)= Years of service already completed
\( r \)= Retirement age
\( x \)= Current age

The TUC Formula for NC:
\( NC_x = b \times 1 \times v^{r-x} \times {}_{r-x}p_x^{(\tau)} \times \ddot{a}_r \)
Note: The NC is just the value of one year's "unit" of benefit!

Key Takeaway: TUC is commonly used for "Flat Dollar" plans (e.g., "$50 per month for every year worked") or "Career Average" plans where future salary increases don't retroactively change the value of past years.

The Projected Unit Credit (PUC) Method

The PUC method is more "forward-looking." It assumes that the employee’s salary will grow over time. Even when calculating the value of past service, we use the projected final salary at retirement.

Analogy: Imagine you are promised 1% of your "Final Year Cake" for every year you work. Even if the cake is small today, PUC assumes the cake will be huge when you retire, and it calculates your current slice based on that future huge cake.

How to calculate PUC:

1. Project the employee's salary to retirement using a salary scale.
2. Calculate the total estimated benefit at retirement.
3. AL: Take the total benefit and multiply by \(\frac{\text{Years of Service to Date}}{\text{Total Potential Years of Service}}\).
4. NC: Take the total benefit and multiply by \(\frac{1}{\text{Total Potential Years of Service}}\).

The PUC Formula for AL:
\( AL_x = [\text{Projected Total Benefit}] \times \frac{n}{N} \times v^{r-x} \times {}_{r-x}p_x^{(\tau)} \times \ddot{a}_r \)
Where:
\( N \)= Total years of service the employee will have at retirement.

The PUC Formula for NC:
\( NC_x = [\text{Projected Total Benefit}] \times \frac{1}{N} \times v^{r-x} \times {}_{r-x}p_x^{(\tau)} \times \ddot{a}_r \)

Common Mistake: Students often forget to use the future projected salary for PUC. If the problem mentions a "Salary Scale" or "Final Average Pay," you are almost certainly in PUC territory!

Comparing TUC and PUC

Why do we have both? Because they tell different stories.

1. Salary Increases

In a Final Average Pay plan, if an employee gets a big raise, their TUC AL will jump suddenly because the "current salary" used for all past years just increased. In PUC, we already expected the salary to increase, so the jump is usually smaller (unless the raise was higher than the assumed salary scale).

2. Normal Cost Trends

Under both methods, the Normal Cost generally increases as an employee gets older. Why? Because the retirement date is getting closer, so the discount factor (\(v^{r-x}\)) gets larger (closer to 1), and there is less time for the employee to leave the company or die (\({}_{r-x}p_x^{(\tau)}\) gets larger).

3. Memory Aid: "P" for PUC

Think PUC = Projected. If the benefit depends on what you earn at the end of your career, use Projected Unit Credit.

Key Takeaway: PUC results in a higher AL and NC in the early years of a career compared to TUC because it anticipates those high end-of-career salaries from day one.

Summary and Success Tips

You've just mastered the two primary ways to value a pension! Here is a quick checklist for your practice problems:

  • Identify the Method: Does the problem ask for TUC (Current Salary) or PUC (Projected Salary)?
  • Check the Service: For AL, use years of service already completed (\(n\)). For NC, use one year of service.
  • Don't Forget Decrements: Always multiply by the probability of reaching retirement (\({}_{r-x}p_x^{(\tau)}\)). In ALTAM, this often includes death, disability, and withdrawal.
  • Watch the Annuity: Ensure your \(\ddot{a}_r\) matches the payment frequency (usually monthly, \( \ddot{a}_r^{(12)} \)).

Encouragement: Pension mathematics can feel like a lot of "moving parts," but at its heart, it’s just finding the Present Value of a future promise. Keep practicing the AL and NC formulas until they feel like second nature. You've got this!

Quick Review Box:
TUC: \(AL = (\text{Past Service}) \times (\text{Benefit on Current Salary}) \times PV \text{ Factor} \)
PUC: \(AL = \frac{\text{Past Service}}{\text{Total Service}} \times (\text{Benefit on Final Salary}) \times PV \text{ Factor} \)
NC: The cost of exactly one year of service credit.