Actuarial Notes

Pension Plan

Study guides: Exam ALTAM, Exam CP-351, Exam RET-101, Exam RET-201, Exam RET-301

A pension plan is an arrangement through which a plan sponsor, usually an employer, provides retirement income to employees, financed by contributions accumulated in a fund. In a defined benefit (DB) plan the benefit is set by formula from salary and service and the sponsor bears the investment and longevity risk; in a defined contribution (DC) plan the contributions are set and the member's benefit is whatever the account buys.

B = \alpha \, n \, \bar S

R = \frac{\text{pension income in the year after retirement}}{\text{salary in the year before retirement}}

\frac{S_y}{S_x} = \frac{s_y}{s_x}

c\sum_{t=0}^{n-1} S_{x+t}\,(1+i)^{n-t} = \text{account at retirement}

  • In the DB formula \alpha is the accrual rate (e.g. 1.5\%), n the years of service and \bar S the pensionable salary: final average salary (FAS), career average earnings (CAE), or career average revalued earnings (CARE). R is the replacement ratio
  • The salary scale s_y projects salary from age x to age y; in the DC equation c is the contribution rate, paid at the start of each year here, and the account buys an annuity at retirement
  • A DB plan is valued for several purposes with different assumptions: going-concern funding (contributions, via a cost method's normal cost and accrued liability), solvency or hypothetical wind-up (benefits if the plan ended today, at annuity-purchase rates), termination, and accounting
  • The liabilities are long-duration and often inflation-linked, so their value is very sensitive to interest rates and longevity; liability-driven investing matches the asset duration to the liability's (a partial immunization) instead of maximizing return

Replacement Ratio under a Final-Average-Salary Plan A member aged exactly 40, hired at 30, earns \$70{,}000 in the year of age 40 to 41. Salaries rise 3\% a year. The plan pays 1.6\% of the 3-year final average salary per year of service at age 65. Find the annual pension and the replacement ratio.

Answer The final year's salary (age 64 to 65) is 70{,}000(1.03)^{24} = 142{,}296, and the 3-year average is

\begin{align*}
\bar S &= \frac{142{,}296}{3}\left(1 + 1.03^{-1} + 1.03^{-2}\right) \\
       &= 138{,}191 \\
B &= 0.016 \times 35 \times 138{,}191 \\
  &= 77{,}387 \\
R &= \frac{77{,}387}{142{,}296} \\
  &= 0.544
\end{align*}

The pension of \$77{,}387 replaces 54.4\% of final salary; averaging over three years costs about 2.9\% of the benefit.

DC Contribution Rate for a Target Replacement Ratio A member aged 35 earns \$50{,}000 this year; salaries rise 3\% a year. Contributions of c times salary are paid at the start of each year to age 65 and earn 6\%. At 65 the account buys a pension at \$12.0 per \$1 of annual income. Find c for a replacement ratio of 50\%.

Answer Final salary is 50{,}000(1.03)^{29} = 117{,}828, so the target pension is 58{,}914 and the account must reach 12.0 \times 58{,}914 = 706{,}970.

\begin{align*}
\sum_{t=0}^{29} 50{,}000(1.03)^t(1.06)^{30-t} &= 50{,}000(1.06)^{30}\,\frac{1 - (1.03/1.06)^{30}}{1 - 1.03/1.06} \\
                                              &= 5{,}858{,}671 \\
c &= \frac{706{,}970}{5{,}858{,}671} \\
  &= 0.1207
\end{align*}

A contribution rate of about 12.1\% meets the target, but only if every assumption is realized; the member bears any shortfall.

Interest Rate Hedge Ratio of a DB Plan A DB plan has liabilities of \$200 million with modified duration 14. Its \$200 million of assets are \$80 million of bonds with duration 6 and \$120 million of equities (treat as duration 0). Find the interest rate hedge ratio, the surplus change if rates fall 1\%, and the receive-fixed swap notional (duration 18) that raises the hedge ratio to 80\%.

Answer

\begin{align*}
\text{Hedge ratio} &= \frac{80 \times 6}{200 \times 14} \\
                   &= 17.1\% \\
\Delta\text{Surplus} &\approx (480 - 2{,}800)(0.01) \\
                     &= -23.2 \text{ million} \\
\text{Notional} &= \frac{0.8(2{,}800) - 480}{18} \\
                &= 97.8 \text{ million}
\end{align*}

The plan loses \$23.2 million of surplus on a 1\% fall; a \$97.8 million swap overlay cuts that to about \$5.6 million without selling equities.

How it appears on the exam

  • ALTAM (Pension Plans and Retirement Benefits): compute replacement ratios for FAS, CAE and CARE plans, the DC contribution rate for a target ratio, and use service-table and salary-scale functions to value accrued benefits.
  • RET-201 / RET-301 ("differentiate between the various purposes for valuing pension plans"): explain how going-concern, solvency or hypothetical wind-up, termination and accounting valuations differ in purpose, assumptions and method, usually as a written answer for a described plan.
  • CP-351 ("asset allocation strategies appropriate to underlying liability profiles such as pension plans"): critique an allocation using duration, hedge ratio and surplus sensitivity, and propose a liability-driven investment strategy.

Exam traps

  • Wrong salary in the replacement ratio. The denominator is the salary in the year before retirement, not the final average salary and not the salary projected to the retirement age itself.
  • Counting years of salary increase. A salary for the year of age 40 to 41 grows 24 times to reach the year 64 to 65, not 25.
  • Using going-concern assumptions for a wind-up valuation. Solvency and hypothetical wind-up liabilities use settlement (annuity purchase and commuted value) rates and include no future salary increases; a going-concern discount rate understates them.
  • Treating asset share as hedge ratio. A plan 40\% invested in bonds is not 40\% hedged; the hedge ratio compares dollar durations, and short bonds hedge little of a long liability.