Deriving Pension Cash-Flow Functions

From Plan Rules to Cash-Flow Functions

A pension provision is not a single formula until its conditions are made explicit. The amount may depend on age, credited service and pensionable salary, while the right to receive it depends on events such as retirement or death and on the member’s status. A cash-flow function puts these ingredients together: it specifies the benefit amount and the conditions under which that amount is payable.

This tutorial develops piecewise functions for hypothetical pension plans. The goal is to represent the plan rules faithfully, not to value the resulting payments. In particular, a benefit amount and its payment conditions must both be stated: a calculated pension is not due if the member has not met the plan’s eligibility conditions.

Definition: Let $r$ be the member’s age at retirement, $s$ the credited service used by the plan, and $S$ the pensionable salary specified by the plan. A pension cash-flow function gives the benefit amount or payment rate as a function of these inputs and the member’s status. A piecewise function lists separate expressions for the distinct conditions under which the plan pays different benefits.

For clarity, throughout the examples, salary and service caps are applied to the stated inputs, and each plan defines its own eligibility tests. When a pension is expressed as an annual amount, that amount is an annual payment rate; the plan’s payment dates must still be specified separately to identify individual cash flows.

Building a Piecewise Pension Formula

A useful order is to translate each rule separately: identify eligibility, apply service and salary limits, calculate the unreduced benefit, and then apply any adjustment such as an early-retirement reduction. Finally, attach the benefit to the status or event that makes it payable.

State the inputs and benefit basis.
Define the age, credited service, salary measure, and the status or event that the plan uses. Do not substitute current salary for a plan-defined final average salary.

Apply caps and floors.
If only the first 30 years of service count, use $\min(s,30)$. If pensionable salary is capped at $C$, use $\min(S,C)$.

Apply eligibility and adjustments.
State the age and service tests explicitly. Apply a reduction only as the plan defines it, including whether it is based on completed years or is prorated.

Attach the payment to the correct status.
Distinguish a member’s retirement pension from a survivor benefit or a pre-retirement death benefit. State when each stream is due and when it stops.

For example, suppose a plan grants an annual pension equal to $k$ times capped service times capped salary, with a reduction of $q$ for each complete year of retirement before age 65. If the member retires at age $r$, the number of complete years early under this rule is $\lfloor\max(65-r,0)\rfloor$. The reduction factor is therefore $1-q\lfloor\max(65-r,0)\rfloor$. This formula represents a complete-year rule; it does not prorate a partial year.

Why the complete-year formula works: The plan applies one reduction of $q$ for each whole year by which retirement precedes age 65. The nonnegative age difference is $\max(65-r,0)$; taking its floor counts only whole years. Multiplying the unreduced pension by one minus $q$ times that count applies exactly one reduction for each counted year. If the plan instead specifies a prorated reduction, the floor must be omitted.

With an eligibility condition of age at least 55 and service at least 10 years, the complete expression is

B(r,s,S)= \begin{cases} k\min(s,30)\min(S,C)\left[1-q\left\lfloor\max(65-r,0)\right\rfloor\right], & r\geq 55,\ s\geq 10,\\ 0,&\text{otherwise.} \end{cases}

Here $C$ is the pensionable-salary cap. The zero branch means no pension is payable under this particular retirement provision when its stated eligibility test is not met; it does not rule out other benefits a plan might provide.

Worked Example: Applying Caps and a Complete-Year Reduction

A hypothetical plan pays an annual retirement pension of 1.6% of capped service times capped pensionable salary. Service is capped at 30 years, pensionable salary at $90,000, and eligibility requires retirement at age 55 or later with at least 10 years of service. The pension is reduced by 4% for each complete year before age 65. A member retires at age 62 with 24 years of service and pensionable salary of $72,000. Find the annual pension.

Method. Apply the eligibility test, then the caps, then the reduction. The common mistake to avoid is applying the reduction to a fractional age difference when the plan specifies complete years only.

The member meets both eligibility conditions: $62\geq55$ and $24\geq10$. Capped service is $\min(24,30)=24$ years, and capped salary is $\min(72{,}000,90{,}000)=\$72{,}000$.

The unreduced annual pension is

0.016(24)(\$72{,}000)=0.384(\$72{,}000)=\$27{,}648.

The complete years before age 65 are $\lfloor65-62\rfloor=3$, so the reduction is $3(0.04)=0.12$, or 12%. The reduction factor is $1-0.12=0.88$. Thus the annual pension is

\$27{,}648(0.88)=\$24{,}330.24.

Check. The reduction is less than 100%, the amount is below the unreduced pension, and all eligibility tests are met. The annual pension is $24,330.24.

Partial Years, Caps, and Eligibility Branches

A piecewise expression should reflect the exact wording of each plan rule. “Four percent for each complete year early” differs from “four percent per year, prorated.” The first uses a floor; the second uses the unrounded age difference. Likewise, a cap changes the value entering the benefit formula, not the eligibility test, unless the plan expressly says otherwise.

Worked Example: A Partial Year Before Normal Retirement Age

Use the same hypothetical plan as in the preceding example. A member retires at age 64.5, has 18.5 years of credited service, and has pensionable salary of $100,000. The plan still requires age 55 and 10 years of service, caps service at 30 years and salary at $90,000, and reduces the pension by 4% per complete year before age 65. Find the annual pension.

Method. Test eligibility and apply each cap before calculating the reduction. The key point is that $65-64.5=0.5$ is not a complete year.

Eligibility is satisfied because $64.5\geq55$ and $18.5\geq10$. Capped service is $\min(18.5,30)=18.5$. Capped salary is $\min(100{,}000,90{,}000)=\$90{,}000$.

The unreduced pension is

0.016(18.5)(\$90{,}000)=0.296(\$90{,}000)=\$26{,}640.

The number of complete years early is $\lfloor\max(65-64.5,0)\rfloor=\lfloor0.5\rfloor=0$. The reduction factor is $1-0.04(0)=1$, so the annual pension remains $26,640.00.

Check. The salary cap reduces the salary basis from $100,000 to $90,000, while no early-retirement reduction applies under a complete-year rule. Applying a 2% reduction would incorrectly prorate the stated 4% annual reduction.

Separate Benefits by Member Status

A cash-flow function can also distinguish benefits that depend on whether the member is active, retired, or deceased. Each branch needs its own payment condition. For instance, a retirement pension payable while the member is alive must not be added to a survivor pension that begins only after the member’s death, unless the plan explicitly provides an overlapping payment.

The following notation is useful for describing the branches. Let $A$ mean the member is active, $R$ mean the member is retired and alive, and $D$ mean the member has died. The labels describe mutually exclusive statuses at a given time. If a plan provides a benefit only when an eligible spouse survives, that condition must also be stated; it should not be assumed from the member’s death alone.

Worked Example: Retirement and Survivor Cash-Flow Branches

A hypothetical plan pays an annual pension of 1.8% of service, capped at 30 years, times pensionable salary, capped at $100,000. Retirement is permitted from age 60 with at least 20 years of service. While a retired member is alive, the pension is payable to the member. After the member’s death, an eligible surviving spouse receives 50% of that annual pension, beginning with the next scheduled annual payment. If the member dies before retirement, an eligible spouse instead receives a one-time lump sum equal to 1.5 times pensionable salary. Assume salary is not capped for this lump sum. A member retires at age 60 with 22 years of service and pensionable salary of $80,000, then dies at age 68 leaving an eligible spouse. State the benefit amounts and branches.

Method. Calculate the retirement pension first, then use its amount to determine the survivor pension. Keep the post-retirement survivor stream separate from the alternative pre-retirement death branch.

The retirement eligibility conditions are met: age $60\geq60$ and service $22\geq20$. Capped service is $\min(22,30)=22$ years, and capped salary is $\min(80{,}000,100{,}000)=\$80{,}000$. The member’s annual pension is

0.018(22)(\$80{,}000)=0.396(\$80{,}000)=\$31{,}680.

The member receives the $31,680 annual pension while retired and alive. After death at age 68, the eligible spouse receives 50% of that amount:

0.50(\$31{,}680)=\$15{,}840

per year, beginning with the next scheduled annual payment. The pre-retirement death lump sum is not payable in this scenario, because the member died after retirement.

The branches can be written as an annual benefit rate, with the separate lump-sum branch shown alongside it:

\begin{array}{ll} \$31{,}680 & \text{if retired and alive},\\ \$15{,}840 & \text{if the member has died after retirement and an eligible spouse survives},\\ 0 & \text{otherwise under these two annual pension provisions}; \end{array} \qquad \begin{array}{ll} 1.5S & \text{as a lump sum if death occurs before retirement and an eligible spouse survives},\\ 0 & \text{otherwise under this lump-sum provision}. \end{array}

Check. The survivor pension is exactly half the member’s annual pension. Since this member died after retirement, only the post-retirement survivor branch applies; the pre-retirement lump sum is not added.

Common Translation Errors

A correct formula is a compact statement of the plan, so small translation errors can change both eligibility and amount. Check the conditions as carefully as the arithmetic.

Prorating a complete-year reduction. The incorrect calculation at age 64.5 is $4\%(65-64.5)=2\%$. Under a rule applying 4% for each complete year, the correct count is $\lfloor0.5\rfloor=0$, so the reduction is 0%. A prorated reduction is appropriate only if the plan says so.
Applying a cap after the percentage. If the formula caps salary at $90,000, use $\min(S,90{,}000)$ as the salary basis before multiplying by the accrual rate and service. Do not calculate using the uncapped salary and then try to subtract a cap from the pension.
Confusing entitlement and payment. An eligible member may have earned a pension, but the plan may specify that payment begins at retirement or on a later date. State the due condition as well as the amount.
Combining mutually exclusive branches. A pre-retirement death benefit and a post-retirement survivor pension may be alternative benefits. Do not add them unless the plan expressly provides both.

This translation skill is useful before any valuation: the cash-flow function identifies what is payable, while a later valuation step can attach payment dates, probabilities and discounting. Accurate conditions also make it easier to check that each branch corresponds to a stated plan rule. This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each question, identify the applicable branch and calculate the requested benefit. Assume all amounts are hypothetical.

A plan pays 1.6% of capped service times capped salary, with service capped at 30 years and salary capped at $90,000. It requires age 55 and 10 years of service and reduces the pension by 4% per complete year before age 65. A member retires at age 61 with 15 years of service and salary of $80,000. Find the annual pension. (Answer: $16,128.00)
Use the plan in Question 1. A member retires at age 64.2 with 12 years of service and salary of $105,000. Find the annual pension, applying the complete-year rule. (Answer: $17,280.00)
A plan pays 1.5% of service capped at 30 years times salary capped at $90,000. Eligibility requires age 55 and at least 10 years of service; the pension is reduced by 5% per complete year before age 65. A member retires at age 60 with 33 years of service and salary of $95,000. Find the annual pension. (Answer: $30,375.00)
A plan pays a $24,000 annual pension to a retired member while alive and 50% of that amount to an eligible surviving spouse after the member’s death. The member dies after retirement. What annual survivor pension is payable, and does the pre-retirement death branch apply? (Answer: $12,000 per year; no, the member died after retirement.)
A plan requires both age at least 60 and service of at least 20 years for retirement. A member retires at age 62 with 19.5 years of service. The formula would otherwise give $28,000 annually. Under this provision, what annual retirement pension is payable? (Answer: $0; the service eligibility condition is not met.)

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