How Final-Average Salary Determines a Pension
A final-average salary formula links a member’s pension at retirement to a specified measure of pensionable pay near the end of the member’s career. Salary may change from year to year, so the benefit calculation must first identify which years count and how their pay is averaged. The resulting salary measure is then combined with credited service and the plan’s accrual rate.
The formula calculates a pension benefit amount under the plan rules. By itself, it does not calculate the actuarial present value of that pension. Valuing expected payments would also require assumptions about such matters as interest, survival, payment timing, and the benefit trigger. In the examples below, the target is the annual pension amount at commencement, not its present value.
Set Up the Formula Before Calculating
Let \(S_j\) be the pensionable salary for year \(j\), as defined by the plan. If the plan uses the average of the final \(n\) years, the final-average salary is the arithmetic mean of those \(n\) annual amounts. If the plan instead selects the highest average over eligible consecutive periods, calculate each eligible average and select the largest one.
For a constant accrual rate \(r\) per year of credited service and \(s\) years of credited service, the annual pension \(B\) is
where \(\bar{S}\) is the plan-defined final-average salary. The rate \(r\) is written as a decimal in calculations: for example, \(1.75\%\) is \(0.0175\). If the plan states that the accrual factor is a percentage of final-average salary for each year of service, then \(r s\) is the total percentage applied to that salary measure.
The formula assumes that the same accrual rate applies to every credited year included in \(s\), and that the plan applies that rate to the final-average salary. It also assumes no separate adjustment or limit changes the benefit. If a plan specifies a different accrual structure or a limit, that rule must be reflected rather than silently using this simple formula.
Why the Service Factor Multiplies the Average
Under a constant-rate final-average formula, each credited year adds the same fraction \(r\) of the specified final-average salary to the annual pension. The salary basis is held at \(\bar{S}\) for this calculation. Summing the accrual for each year gives the formula directly.
This also explains an important feature of a final-average formula: the salaries earned in earlier service years are not individually multiplied by that year’s accrual in this calculation. The plan first establishes the final-average salary and then applies the total service factor to it. That is why using a changing salary separately for every year of service would calculate a different kind of benefit.
Worked Calculations
Worked Example: Highest Three-Year Average in the Final Four Years
A hypothetical plan pays an annual pension equal to \(1.75\%\) of the highest average of three consecutive annual pensionable salaries among the member’s final four years, multiplied by credited service. The final four annual salaries, from earliest to latest, are $100,000, $110,000, $105,000, and $120,000. The member has 32 years of credited service. Find the annual pension and the equal monthly amount.
Assumptions and method. The salary amounts are deterministic and already meet the plan’s pensionable-pay definition. There is no salary probability distribution, interest discounting, or mortality assumption in this benefit-amount calculation. The pension is payable in twelve equal monthly amounts, with no other adjustment.
Step 1: Identify the eligible periods. There are two three-year consecutive periods within the final four years:
The first average is \((\$100{,}000+\$110{,}000+\$105{,}000)/3=\$315{,}000/3=\$105{,}000\).
The second average is \((\$110{,}000+\$105{,}000+\$120{,}000)/3=\$335{,}000/3=\$111{,}666.6667\), before rounding.
Step 2: Select the highest eligible average. Since \(\$111{,}666.6667\) exceeds \(\$105{,}000\), use \(\bar{S}=\$111{,}666.6667\).
Step 3: Calculate the service factor. Convert the accrual rate to a decimal and multiply by service: \(0.0175\times32=0.56\).
Step 4: Calculate the annual pension. The annual amount before rounding is \(\$111{,}666.6667\times0.56=\$62{,}533.3333\).
Step 5: Convert to equal monthly payments. Divide by 12: \(\$62{,}533.3333/12=\$5{,}211.1111\). Rounded to the nearest cent, the annual pension is $62,533.33 and the monthly amount is $5,211.11.
Check and common mistake avoided. The selected average is the greater of the two eligible averages, and the service factor is 56%. Multiplying the selected salary by 0.56 reproduces the annual amount. Using $5,250.00 per month would imply an annual pension of $63,000.00, which does not follow from the stated average and accrual factor.
Worked Example: Average of a Fixed Final Four Years
A hypothetical plan provides an annual pension equal to \(1.6\%\) per year of credited service times the average pensionable salary over the member’s final four years. Those salaries, in chronological order, are $78,000, $82,000, $90,000, and $96,000. The member has 27 years of credited service. Find the annual pension.
Assumptions and method. All four stated amounts are annual pensionable salaries and all four are included. The plan does not select a highest period. Salary is treated as deterministic; no probability distribution, interest rate, or mortality assumption is needed because only the pension amount at commencement is requested.
Step 1: Calculate the specified average. The sum is \(\$78{,}000+\$82{,}000+\$90{,}000+\$96{,}000=\$346{,}000\). Therefore, \(\bar{S}=\$346{,}000/4=\$86{,}500\).
Step 2: Calculate the service factor. The decimal accrual rate is \(0.016\), so \(0.016\times27=0.432\), or 43.2%.
Step 3: Calculate the annual pension. \(B=\$86{,}500\times0.432=\$37{,}368.00\).
Check and common mistake avoided. The salary average includes exactly the four final years specified, and the service percentage is 43.2%, not 0.432%. Applying 43.2% to $86,500 gives $37,368.00. Selecting a different four-year window or applying the accrual rate to only one year of service would not follow the plan formula.
Worked Example: Comparing Eligible Averages Before Applying Service
A hypothetical plan uses the highest average of two consecutive annual pensionable salaries among a member’s final three years. The accrual rate is \(1.8\%\) per year, and the member has 25 years of credited service. The three salaries, from earliest to latest, are $88,000, $104,000, and $98,000. Find the annual pension and equal monthly amount.
Assumptions and method. The stated salaries are the full pensionable amounts for the three years. The plan’s highest-average rule applies to the two possible consecutive pairs. No probability distribution, interest discounting, or mortality assumption is part of this pension-amount calculation. Payments are divided into twelve equal monthly amounts without any other adjustment.
Step 1: Calculate both eligible averages. The first pair gives \((\$88{,}000+\$104{,}000)/2=\$192{,}000/2=\$96{,}000\). The second pair gives \((\$104{,}000+\$98{,}000)/2=\$202{,}000/2=\$101{,}000\).
Step 2: Select the plan-defined salary measure. The higher average is \(\bar{S}=\$101{,}000\).
Step 3: Calculate the service factor. \(0.018\times25=0.45\), or 45%.
Step 4: Calculate the annual pension. \(B=\$101{,}000\times0.45=\$45{,}450.00\).
Step 5: Calculate the monthly amount. \(\$45{,}450.00/12=\$3{,}787.50\).
Check and common mistake avoided. The selected pair is the second and third salaries, not necessarily the latest salary alone. The monthly amounts total \(12\times\$3{,}787.50=\$45{,}450.00\), agreeing with the annual pension.
Common Errors and Interpretation
Most errors arise before the multiplication: a candidate may choose the wrong years, use the wrong salary definition, or overlook whether the plan requires a fixed final-period average or the highest eligible average. A correct service calculation cannot repair an incorrectly determined salary measure.
- Using only the latest salary. A highest-average rule requires calculation of every eligible consecutive-period average before selection. The latest year may not produce the highest result.
- Choosing nonconsecutive years. If the rule specifies consecutive years, a member cannot assemble the highest individual salaries from separated years unless the plan explicitly permits that selection.
- Confusing the rate with the total service factor. An accrual rate of \(1.75\%\) is \(0.0175\). For 32 years, the factor is \(0.0175\times32=0.56\), not \(0.0175\) and not \(1.75\times32\).
- Mixing annual and monthly amounts. The formula ordinarily gives an annual pension when salary is annual. Divide by 12 only if the question specifies twelve equal monthly payments and no other payment adjustment.
- Treating a benefit amount as a present value. The annual pension formula does not discount future payments. A present-value calculation needs additional timing and contingency assumptions.
Final-average formulas are useful because they make the salary basis explicit. In a pension valuation under decrements, the benefit amount can then be attached to the state or event specified by the plan, while the probability and timing of that payment are handled separately. Keeping those steps distinct helps prevent a salary formula from being confused with a valuation of contingent cash flows.
Check Your Understanding
For each problem, assume the stated salaries are deterministic pensionable annual salaries and that no adjustment applies beyond the stated formula. Unless specified otherwise, find the annual pension amount at commencement, not its present value.
- A hypothetical plan uses the average of the final three annual salaries. They are $84,000, $90,000, and $96,000. The accrual rate is \(1.5\%\) per year and credited service is 30 years. Find the annual pension. (Answer: $40,500.00)
- A plan selects the highest average of two consecutive salaries among the final three: $92,000, $100,000, and $98,000. The accrual rate is \(1.75\%\) per year and service is 24 years. Find the annual pension. (Answer: $41,580.00)
- A plan uses the average of the final four salaries: $70,000, $76,000, $82,000, and $88,000. With 28 years of service and a \(1.8\%\) accrual rate, find the annual pension. (Answer: $39,816.00)
- A member’s plan uses a final-average salary of $110,000 and an accrual rate of \(1.6\%\) per year. The annual pension is $52,800. Assuming the constant-rate formula with no other adjustment, find credited service. (Answer: 30 years)
- A plan selects the highest average of three consecutive salaries among the final five. Salaries are $80,000, $84,000, $90,000, $86,000, and $100,000, in chronological order. The member has 22 years of service, the accrual rate is \(1.75\%\), and the annual pension is paid in twelve equal monthly amounts. Find the monthly amount. (Answer: $2,951.67)