Pension actuarial mathematics

A defined benefit plan promises a pension set by a formula, for example 1.5% of final pay for each year of service. The employer bears the risk that the money will be enough. The actuary answers two different questions: what the promise is worth today (valuation) and at what pace to contribute to pay for it (funding). The first depends on the assumptions; the second also depends on the cost method.

The basic model follows each active participant through a service table with several decrements: death, disability, withdrawal and retirement. Their probabilities combine into a total probability of remaining in service, and expected cash flows are discounted at the valuation interest rate. On that skeleton sit the salary scale, the benefit formula, vesting, payment forms and ancillary benefits.

Every cost method splits the present value of future benefits between past and future. The part attributed to service already rendered is the actuarial (accrued) liability; the part charged in each future year is the normal cost. The difference between the liability and assets is the unfunded liability, which is amortized over a period.

The balance equation is the same for all: today's liability plus normal cost, accumulated with interest, covers next year's expected liability for those who remain, plus benefits paid during the year. When experience differs from the assumptions, actuarial gains or losses arise, and each method treats them differently.

Methods fall into two families. Accrued benefit methods (unit credit) assign to each year the benefit earned in it, so cost rises with the participant's age. Projected benefit methods (entry age normal, aggregate, frozen initial liability, attained age normal) spread the total cost evenly, which gives steadier contributions and funds earlier. No method changes the ultimate cost of the plan, which depends on benefits paid and investment returns earned; they change the pace and volatility of contributions.

The unfunded liability is amortized over a fixed, open or closed period, in level amounts or as a rising percent of payroll. Assets may be valued at market or with a smoothing method that recognizes investment gains and losses gradually, usually over five years and within a corridor around market value. The choice is documented in the plan's funding policy.

Gains and losses are analyzed by source: investment, mortality, withdrawal, salary, retirement and changes in assumptions or plan provisions. That analysis lets the actuary revise assumptions based on experience and explain to the sponsor why the contribution changed.

The Employee Retirement Income Security Act (ERISA) was signed on September 2, 1974. Title I protects participants' rights (reporting, disclosure, fiduciary duties, participation, vesting and funding) and is administered mainly by the Department of Labor; Title II amended the Internal Revenue Code, applied by the IRS; Title III created the Joint Board for the Enrollment of Actuaries, which enrolls the actuaries who sign plan valuations; and Title IV created the Pension Benefit Guaranty Corporation (PBGC), the federal insurer of defined benefit pensions.

The Pension Protection Act of 2006 rewrote single-employer funding rules in section 430 of the Code. The minimum required contribution is the target normal cost plus amortization of the shortfall relative to the funding target, the present value of benefits accrued at the start of the year, discounted with three corporate bond segment rates (cash flows at 0 to 5 years, 5 to 20 and over 20). Shortfalls were originally amortized over 7 years; the American Rescue Plan Act of 2021 moved this to 15 years, reset prior amortization bases and stabilized rates with a corridor around their 25-year average, which the November 2021 infrastructure law extended.

Section 436 restricts benefits based on the adjusted funding target attainment percentage (AFTAP): below 80%, liability-increasing amendments are barred and accelerated payments are limited, and below 60%, accruals are frozen and accelerated payments are prohibited. Multiemployer plans follow a different regime: a funding standard account (section 431) and a zone system (section 432) with endangered, critical and, since 2014, critical and declining status; ARPA also created a special financial assistance program administered by the PBGC.

Each year the enrolled actuary signs Schedule SB (single employer) or Schedule MB (multiemployer) of Form 5500. For single-employer plans terminating in 2026, the PBGC guarantees at most $7,789.77 a month at age 65 as a straight life annuity, and it has published $8,113.64 for 2027. In professional practice, ASOP No. 4, revised in December 2021, governs how obligations are measured and cost methods are chosen.

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