What Has Been Earned by a Given Date?
A pension formula may depend on both service and pensionable pay. When a question asks for the pension accrued at a specified date, the key task is to apply the plan formula to the service and pay basis attributable to that date—not automatically to the member’s eventual service and eventual salary. The distinction matters because a projection to retirement can include both additional years of service and salary increases that have not occurred yet.
The amount calculated is usually an annual pension rate payable at a stated retirement age, subject to the plan’s conditions. It is not necessarily a payment available immediately at the valuation date. Vesting, commencement age, early- or late-retirement adjustments, and the form of payment may affect the member’s rights or the eventual payments, but they are separate from calculating the accrued amount unless the question explicitly includes them.
There is no single pay convention that applies to every plan. A career-average formula may accumulate a pension slice from each year’s earnings. A final-average formula may apply an accrual rate to total credited service and an average of specified years of pay. For a final-average plan, a question must make clear whether the accrued amount uses pay measured at the valuation date or pay projected to retirement. Do not silently substitute one basis for the other.
Unless stated otherwise in the examples below, the plan is hypothetical, service and salary data are deterministic, and the calculated benefit is a nominal annual pension payable from normal retirement age. No interest discounting or mortality decrement is applied: the task is to calculate a benefit amount, not its present value. The stated formula has no service cap or other adjustment.
Calculating the Accrued Amount
For a simple final-average formula, let \(s_t\) be credited service through date \(t\), \(r\) the annual accrual rate per service year, and \(P_t\) the plan-defined pay measure at that date. If the plan says accrued benefits use the pay measure at \(t\), then:
Here \(B_t\) is an annual pension amount. The formula does not itself specify whether \(P_t\) is current salary, an average over recent years, or another plan-defined measure; that must come from the plan terms or problem statement. A career-average formula has a different structure. If year \(j\) earns a pension slice equal to rate \(r\) times pensionable earnings \(E_j\), and there is no revaluation of earlier slices, then:
where \(n_t\) is the number of credited earning years through \(t\). If the plan revalues earlier slices, the specified revaluation must be applied; it cannot be assumed to be absent or present without support in the question.
Worked Example: Career-Average Pension Earned to Date
A hypothetical career-average plan credits a pension equal to 1.5% of each year’s pensionable earnings. At the end of five completed years, a member’s earnings were $52,000, $54,000, $56,000, $58,000, and $60,000. The member expects to complete seven more years, but no future earnings are to be included in the accrued amount. There is no revaluation of earlier slices. Calculate the accrued annual pension. Assume the data are deterministic; interest discounting and mortality are not relevant to this benefit-amount calculation.
Method. Apply the annual accrual rate to the earnings for completed years only. First total those earnings:
The pension accrued from those years is:
The accrued annual pension is $4,200.00. The seven expected future years contribute nothing to this date-specific calculation. As a check, the five annual pension slices are $780, $810, $840, $870, and $900; their sum is also $4,200. The common mistake avoided here is adding projected future earnings to an accrued benefit.
Separating Service Growth from Salary Growth
A final-average formula can make the projected retirement benefit larger for two distinct reasons: more service is completed, and the pay measure rises. A useful way to see the distinction is to start with \(B_0=rs_0P_0\), the accrued amount at the valuation date, and compare it with a projected amount \(B_1=rs_1P_1\). Add service first at the current pay measure, then apply the pay change to all projected service:
The first term is the pension from future service valued at the original pay measure. The second is the change from applying the pay increase to the total projected service. This identity follows by expanding the right-hand side: \((s_1-s_0)P_0+s_1P_1-s_1P_0=s_1P_1-s_0P_0\). It is an exact arithmetic decomposition, not a claim that every plan legally increases accrued benefits with future salary.
The order of the decomposition is a choice. One can instead value future service at the projected pay measure and apply the pay increase only to the service already completed. The components differ, but their sum must still equal \(B_1-B_0\). State which decomposition is being used when interpreting the components.
Worked Example: Final-Average Accrual and a Retirement Projection
A hypothetical plan provides an annual pension of 1.6% of credited service times the average of the three most recent annual salaries. At a valuation date, a member has nine years of service and the three salaries used by the plan are $62,000, $64,000, and $66,000. A projection assumes four additional years of service and a final three-year average salary of $72,000. For the accrued amount, use the salary average measured at the valuation date. Assume no caps, revaluation, discounting, or mortality adjustment; all figures are deterministic.
Accrued entitlement at the valuation date. The current three-year average is:
Using nine completed service years:
The accrued annual pension is $9,216.00.
Projected retirement benefit. Projected service is \(9+4=13\) years, and the projected salary average is $72,000:
Thus the projected benefit exceeds the accrued amount by \(14{,}976-9{,}216=\$5{,}760\). To reconcile the difference using the service-first decomposition, future service at the current average pay adds:
The salary increase from $64,000 to $72,000 then affects all 13 projected service years:
The components sum to \(4{,}096+1{,}664=\$5{,}760\), matching the direct difference. The common mistake avoided is to apply the projected salary only to future service, or to use projected service and salary while labeling the result as the amount accrued at the valuation date.
Partial Service and the Pay Basis
When service is stated in years and months, use the plan’s proration rule. If each completed month earns one-twelfth of a service year, for example, eight months is \(8/12\) of a year. Do not round the service period to a whole year unless the plan explicitly requires that treatment. Similarly, an average-pay calculation must use the particular periods specified; a recent salary figure is not automatically the same as a multi-year average.
Worked Example: Accrued Pension with Partial-Year Service
A hypothetical final-average plan accrues 1.8% per service year. It credits service proportionally by completed month. At the valuation date, a member has six years and eight months of service. The plan’s pay measure is the average of the two most recent completed annual salaries, which are $66,000 and $70,000. A projection assumes two further years of service and a final average salary of $74,000. Calculate the accrued amount and the projected amount. Assume deterministic salaries, no service cap, no pay revaluation beyond the stated projection, and no interest or mortality adjustment.
Convert service to years without rounding:
The pay measure at the valuation date is:
Therefore the accrued annual pension is:
Projected service is \(20/3+2=26/3\) years. Applying the projected average salary gives:
The increase is \(\$11{,}544-\$8{,}160=\$3{,}384\). Using the service-first decomposition, two future years at the current pay measure add:
The $6,000 rise in the average pay measure then adds, across \(26/3\) projected service years:
The check is \(2{,}448+936=\$3{,}384\), equal to the direct difference. Rounding the service to seven years would change both the accrued amount and the projection, so the plan’s monthly proration rule must be followed.
Common Errors and a Reliable Calculation Process
The most important exam distinction is between the benefit earned by the stated date and a projection that assumes the member remains employed. For a final-average formula, a future salary increase may affect a projected benefit on past service, but it belongs in the accrued calculation only if the question’s accrued-benefit basis or plan terms call for that salary measure. The calculation should not decide that issue by implication.
- Using future service in an accrued amount. Count only service credited through the stated valuation date.
- Applying a projected salary without being asked to project. Identify whether the requested pay measure is measured now or at a future date.
- Dropping salary increases on past service in a projection. If the formula uses projected final-average pay for all service, the projected pay measure applies to all projected service, not just future service.
- Rounding service too early. Preserve the specified month or fractional-year precision through the calculation.
- Calling an annual pension a present value. A benefit amount and the present value of that benefit are different quantities; discounting and survival assumptions are needed only for a valuation task.
Determine whether the question asks for an accrued annual pension at a given date or a projected retirement benefit.
Identify the accrual rate, credited service, pay measure, and any proration or revaluation rule.
Use only the service and salary basis appropriate to the requested date; keep projected assumptions separate.
If a future benefit is also required, calculate it independently and check that its difference from the accrued amount reflects the stated future service and pay assumptions.
Accrued entitlement is a building block for pension analysis because it identifies what the plan formula attributes to service already completed. Keeping it distinct from projected benefits prevents future assumptions from being mistaken for an entitlement already earned. This calculation framework is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.
Check Your Understanding
For each hypothetical plan, calculate the requested amount from the stated formula and assumptions. No discounting or mortality adjustment is required.
- A career-average plan credits 2% of each year’s pensionable earnings, with no revaluation. Earnings for three completed years are $45,000, $48,000, and $51,000; two further years are projected but are not part of the accrued amount. Find the accrued annual pension. (Answer: $2,880.00)
- A final-average plan accrues 1.4% per service year using the average of the two most recent annual salaries. At the valuation date, the member has 12 years of service and salaries of $80,000 and $84,000 in those periods. Find the accrued annual pension using the pay average at that date. (Answer: $13,776.00)
- A plan accrues 1.5% per service year using a specified pay average. At the valuation date the member has 7.5 years of service and the pay average is $60,000. A projection assumes two more years of service and a future pay average of $66,000. Find both the accrued and projected annual pensions. (Answer: Accrued $6,750.00; projected $9,405.00)
- A final-average plan accrues 1.75% per year of service. At the valuation date, service is 10 years and the pay average is $90,000. A retirement projection assumes 13 years of total service and a pay average of $96,000. Find the accrued amount, projected amount, and increase. (Answer: Accrued $15,750.00; projected $21,840.00; increase $6,090.00)