Defined Benefit Plan

Study guides: Exam FAM, Exam ALTAM, Exam RET-101

A Defined Benefit Plan (DB plan) is a Retirement Plan that promises each member a pension set by a formula — usually an accrual rate $\alpha$ times years of service $n$ times a salary measure — and leaves the Plan Sponsor to contribute whatever it takes to fund that promise. The salary measure names the three classic designs: final average salary (FAS), career average earnings (CAE) and career average revalued earnings (CARE).

$B_{\text{FAS}} = n\,\alpha\,S_{\text{Fin}}$

$B_{\text{CAE}} = \alpha \sum_{k=0}^{n-1} S_{x_0+k}$

$B_{\text{CARE}} = \alpha \sum_{k=0}^{n-1} S_{x_0+k}\,\frac{I_{\text{ret}}}{I_{x_0+k}}$

Symbols (Dickson, Hardy & Waters, ch. 10): $\alpha$ is the accrual rate (typically 1%–2%), $S_y$ the salary earned in the year of age $y$ to $y+1$ , $x_0$ the entry age, $S_{\text{Fin}}$ the average salary over the last few years before retirement, $I$ a revaluation index (prices or wages).
Salary scale. Future salaries are projected with $s_y$ : $S_{y} = S_x\,s_y/s_x$ . FAS benefits depend on it; CAE benefits do not, which is why a CAE plan needs a higher $\alpha$ for the same replacement ratio.
Risks — investment, longevity and, for FAS, salary inflation — are the sponsor's; the member's pension does not move with the fund. The cost is measured by Valuation: the Actuarial Accrued Liability for service to date and the Normal Cost for the year's accrual.
The benefit is paid as a Life Annuity, so its value is $B$ times an annuity factor discounted for interest and survival to retirement. Compare Defined Contribution Plan.

FAS versus CAE Replacement Ratio with a Salary Scale A member joins at 35 and retires at exact age 65 with 30 years' service. Salary in the year of age 35 to 36 is $60{,}000 and $s_y = 1.03^{\,y}$ . The accrual rate is 1.5%. Find the replacement ratio (a) under a FAS plan using the average of the last three years' salaries, and (b) under a CAE plan.
Answer $S_y = 60{,}000\,(1.03)^{y-35}$ , so $S_{62} = 133{,}277$ , $S_{63} = 137{,}276$ , $S_{64} = 141{,}394$ .
 $\begin{align*} S_{\text{Fin}} &= \tfrac{1}{3}\,(133{,}277 + 137{,}276 + 141{,}394) \\ &= 137{,}316 \\ B_{\text{FAS}} &= 30 \times 0.015 \times 137{,}316 \\ &= 61{,}792 \\ R_{\text{FAS}} &= \frac{61{,}792}{141{,}394} \\ &= 43.7\% \end{align*}$ 
Total career salary is $\sum_{y=35}^{64} S_y = 2{,}854{,}525$ .
 $\begin{align*} B_{\text{CAE}} &= 0.015 \times 2{,}854{,}525 \\ &= 42{,}818 \\ R_{\text{CAE}} &= \frac{42{,}818}{141{,}394} \\ &= 30.3\% \end{align*}$ 
Unrevalued early salaries drag the CAE pension down; matching 43.7% would need $\alpha \approx 2.16\%$ .

Value of the Accrued Benefit: TUC versus PUC An active member is exactly 50 with 15 years' service. Salary in the year of age 49 to 50 was $80{,}000; $s_y = 1.03^{\,y}$ . The plan pays $1.5\%$ of the final year's salary per year of service from 65, monthly for life. Retirement at 65 is the only exit apart from death, ${}_{15}E_{50} = 0.46151$ and $\ddot{a}^{(12)}_{65} = 13.0870$ (SUSM, 5%). Find the actuarial liability for the accrued benefit under the traditional unit credit (current salary) and projected unit credit (projected salary) methods.
Answer
 $\begin{align*} B_{\text{TUC}} &= 15 \times 0.015 \times 80{,}000 \\ &= 18{,}000 \\ S_{64} &= 80{,}000 \times \frac{s_{64}}{s_{49}} \\ &= 80{,}000 \times 1.03^{15} \\ &= 124{,}637 \\ B_{\text{PUC}} &= 15 \times 0.015 \times 124{,}637 \\ &= 28{,}043 \end{align*}$ 
 $\begin{align*} AL_{\text{TUC}} &= 18{,}000 \times 0.46151 \times 13.0870 \\ &= 108{,}716 \\ AL_{\text{PUC}} &= 28{,}043 \times 0.46151 \times 13.0870 \\ &= 169{,}376 \end{align*}$ 
PUC holds 56% more today because it prices the salary growth the sponsor's promise is already exposed to.

How it appears on the exam

ALTAM asks for replacement ratios under FAS, CAE and CARE plans with a salary scale, and for accrued benefits, including benefits on early exit.

ALTAM then values them: the accrued liability and normal cost under PUC and TUC, often paired with a service table for withdrawal and early retirement.

RET-101 asks for written comparisons of traditional DB plans with DC and hybrid designs — cost volatility, who bears longevity and investment risk, and how the plan suits the sponsor's goals.

Exam traps

Projecting with the wrong salary-scale ages: salary in year $y$ to $y+1$ is $S_x\,s_y/s_x$ ; projecting the final salary as $S_x\,s_{65}/s_x$ instead of $s_{64}/s_x$ overstates it by a year of growth.

A salary rate at exact age $x$ is centred on $x - \tfrac{1}{2}$ : divide by $s_{x-1/2}$ (often $\tfrac{1}{2}(s_{x-1}+s_x)$ ), not by $s_x$ .

Averaging the wrong salaries: FAS over the last three years before 65 is ages 62, 63, 64 — not 63, 64, 65.

Using total career salary in the FAS formula, or $n\,\alpha\,S_{\text{Fin}}$ for CAE: CAE already sums over service, so multiplying by $n$ again counts it twice.