Actuarial Notes

Normal Cost

Study guides: Exam ALTAM, Exam RET-201

The Normal Cost (C in AMLCR) of a defined benefit pension plan is the contribution that, paid at the start of a year, funds the benefits that year's service adds, so that if every assumption is borne out the assets keep pace with the actuarial accrued liability. It is defined by the one-year funding equation.

{}_0V + C = \text{EPV}_0(\text{exit benefits in year}) + v\,p_x^{(\tau)}\,{}_1V

C^{\,\text{PUC}} = \alpha\,\hat{S}\,v^{\,65-x}\;{}_{65-x}p_x^{(\tau)}\;\ddot{a}_{65}

  • {}_0V is the accrued liability now and {}_1V the liability one year on for a member still in service, with one more year of service and, under TUC, the new salary. p_x^{(\tau)} is the probability of remaining in service and v is at the valuation rate
  • PUC: the second block, for the age-retirement benefit alone with no exit benefits. The year's normal cost is one year's accrual \alpha on the projected final salary \hat{S}, so it equals {}_0V/n
  • TUC: {}_1V uses next year's salary on all n+1 years, so the normal cost also pays for raising the past n years' benefit to the new salary. It therefore climbs steeply with age, while PUC's grows more evenly
  • The sponsor's total contribution is the normal cost plus an amortization of any unfunded liability; members' contributions, if any, are part of C

PUC Versus TUC Normal Cost A plan pays 1.5\% of the salary in the year before retirement per year of service, from age 65, annually in advance. A member aged 50 has 20 years of service and earned \$80{,}000 in the year just completed; salaries rise 3\% a year. Given {}_{15}p_{50}^{(\tau)} = 0.85, \ddot{a}_{65} = 13.0, i = 5\% and no benefit on any exit before 65, find the normal cost for the coming year under PUC and TUC.

Answer EPV of 1 a year from 65: v^{15}(0.85)(13.0) = 5.31524. Projected final salary: 80{,}000(1.03)^{15} = 124{,}637.39.

\begin{align*}
C^{\,\text{PUC}} &= 0.015(124{,}637.39)(5.31524) \\
&= 9{,}937.16
\end{align*}

Under TUC, v\,p_{50}^{(\tau)}\,{}_1V = 21(0.015)(82{,}400)(5.31524), since v\,p_{50}^{(\tau)} \cdot v^{14}\,{}_{14}p_{51}^{(\tau)} = v^{15}\,{}_{15}p_{50}^{(\tau)}:

\begin{align*}
C^{\,\text{TUC}} &= \left[21(82{,}400) - 20(80{,}000)\right](0.015)(5.31524) \\
&= 130{,}400(0.015)(5.31524) \\
&= 10{,}396.61
\end{align*}

Of the TUC cost, 6{,}569.64 is the new year's accrual and 3{,}826.97 lifts the 20 past years to the 3\% higher salary. At age 50 TUC already costs more than PUC.

Funding Equation With a Withdrawal Benefit A member aged 45 has 15 years of service in a plan paying 1.5\% of final-year salary per year of service from 65 (\ddot{a}_{65} = 13.0, i = 5\%). Her salary was \$60{,}000 last year and is \$61{,}800 this year. In the coming year she dies with probability 0.002 (no benefit) or withdraws at year end with probability 0.04, keeping a deferred pension on her 16 years and final salary \$61{,}800. Given {}_{19}p_{46} = 0.90 for a life no longer subject to withdrawal, and that withdrawal does not change mortality, find the TUC normal cost.

Answer A pension of 1 from 65, valued at age 46: u = v^{19}(0.90)(13.0) = 4.63009. The accrued pension at 46 is 16(0.015)(61{,}800) = 14{,}832. Her accrued pension is paid from 65 whether she stays or leaves, so {}_0V needs survival only: {}_{20}p_{45} = 0.998(0.90) = 0.8982.

\begin{align*}
{}_0V &= 15(0.015)(60{,}000)\,v^{20}(0.998)(0.90)(13.0) \\
&= 59{,}410.63
\end{align*}
\begin{align*}
{}_1V &= 14{,}832(4.63009) \\
&= 68{,}673.45
\end{align*}
\begin{align*}
C &= v(0.04)(68{,}673.45) + v(0.958)(68{,}673.45) - 59{,}410.63 \\
&= 2{,}616.13 + 62{,}656.35 - 59{,}410.63 \\
&= 5{,}861.85
\end{align*}

Because a leaver keeps exactly the benefit a stayer has accrued, the withdrawal and in-service terms combine and the result is the no-exit formula [16(61{,}800) - 15(60{,}000)](0.015)\,v^{20}(0.8982)(13.0).

How it appears on the exam

  • ALTAM asks for the normal cost of an active member under PUC and TUC, for the age-retirement benefit and for early-exit benefits, from a salary scale, service table and annuity values; it is almost always paired with the accrued liability in the same question.
  • ALTAM's interpretation parts ask why TUC normal costs rise faster with age, and which method suits a plan with an older membership.
  • RET-201 asks for normal cost under a range of cost methods in an ongoing-plan valuation, and how it combines with amortization of the unfunded liability into the required contribution.

Exam traps

  • Under TUC, taking the normal cost as one year's accrual on the new salary (6{,}569.64 in the first example) and leaving out the catch-up on past service. The funding equation includes it automatically.
  • Discounting {}_1V with interest only. The liability at time 1 is held only for members still in service, so it needs v\,p_x^{(\tau)}, and exit benefits in the year are added separately.
  • Valuing a withdrawal benefit with the in-service probability {}_{65-x}p^{(\tau)}. Once a member leaves, only mortality applies, so the deferred pension is discounted with the mortality-only {}_tp.