Actuarial Notes

Retirement Plan

Study guides: Exam FAM, Exam CP-321, Exam RET-101, Exam RET-201, Exam RET-301

A Retirement Plan is an arrangement under which a Plan Sponsor (an employer, a group of employers or a union) and usually the employees set money aside during working years to pay income to members after they retire. It is a Defined Benefit Plan when the benefit is promised and contributions adjust to fund it, a Defined Contribution Plan when the contribution is promised and the benefit is whatever the account buys, and a hybrid (cash balance, target benefit, shared risk) when the two are mixed.

C + I = B + E

F_{t+1} = F_t + C_t + I_t - B_t - E_t

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

  • The funding equation. Over the plan's life, contributions C plus investment income I must equal benefits B plus expenses E; year by year the fund F_t rolls forward the same way. Funding decisions only change when C is paid and how much I does the work, never the ultimate B + E.
  • Replacement ratio R is the yardstick of benefit adequacy (Dickson, Hardy & Waters, ch. 10): a DB formula fixes it in advance, a DC account only projects it. The plan's ratio is read together with government benefits (Social Insurance).
  • Who bears the risk is what separates the types: in DB, investment, longevity and inflation risk sit with the sponsor; in DC they sit with the member. Hybrids share them — a target benefit plan fixes contributions and lets benefits adjust.
  • A Pension Plan is a retirement plan paying a lifetime income; the plan's rules — formula, eligibility, vesting, retirement ages, indexation — are its Plan Design.

Rolling the Pension Fund Forward One Year A DB fund holds $50{,}000{,}000 at the start of the year. The sponsor contributes $4{,}000{,}000 at the start of the year, benefits of $6{,}000{,}000 are paid at mid-year, expenses of $500{,}000 are paid at year-end, and the fund earns 6% effective. Find the investment income and the year-end fund.

Answer

\begin{align*}
I &= 0.06\,(50{,}000{,}000 + 4{,}000{,}000) - 6{,}000{,}000\,(1.06^{0.5} - 1) \\
&= 3{,}240{,}000 - 177{,}378 \\
&= 3{,}062{,}622
\end{align*}
\begin{align*}
F_1 &= 50{,}000{,}000 + 4{,}000{,}000 + 3{,}062{,}622 - 6{,}000{,}000 - 500{,}000 \\
&= 50{,}562{,}622
\end{align*}

Investment income, not contributions, covered half the year's benefits — the I in C + I = B + E doing its share.

Same Money, Different Risk Bearer: DB versus DC $100{,}000 is set aside for a member aged 55 who retires at 65; the pension is bought with \ddot{a}^{(12)}_{65} = 13.0870. Ignore death before 65. (a) In a DC plan, find the annual pension if the account earns 6% a year, and if it earns 3%. (b) In a DB plan the sponsor promised the pension that 6% would buy. If assets earn 3%, how much must the sponsor add at 65?

Answer (a) DC: the member takes whatever the account buys.

\begin{align*}
P_{6\%} &= \frac{100{,}000\,(1.06)^{10}}{13.0870} \\
&= \frac{179{,}085}{13.0870} \\
&= 13{,}684
\end{align*}
\begin{align*}
P_{3\%} &= \frac{100{,}000\,(1.03)^{10}}{13.0870} \\
&= \frac{134{,}392}{13.0870} \\
&= 10{,}269
\end{align*}

(b) DB: the pension stays $13{,}684, so the sponsor makes up the shortfall.

\begin{align*}
\text{Top-up} &= 179{,}085 - 134{,}392 \\
&= 44{,}693
\end{align*}

The same 3% return costs the DC member a quarter of their pension and costs the DB sponsor $44{,}693 — investment risk lands on whoever did not fix their side of the deal.

How it appears on the exam

  • RET-101 asks you to identify and prioritize a sponsor's goals for the retirement plan, assess whether they are feasible, and recommend a plan type — written answers comparing DB, DC and hybrids (target benefit, shared risk, executive plans) on cost, risk and adequacy.
  • RET-101 also asks how plan design manages the risks participants and retirees face (longevity, investment, inflation), including in multi-employer plans.
  • RET-201 and RET-301 set the plan inside its funding, accounting and Canadian regulatory frame: the sponsor's funding options, and the assumptions chosen for a funding valuation.
  • ALTAM computes replacement ratios for DB and DC plans — see the two archetype pages.

Exam traps

  • Saying a DC plan "has no risk": it has all the same risks, they simply sit with the member — the written answer must name who bears each one.
  • Treating a target benefit plan as DB because it states a target benefit: contributions are fixed, so benefits are cut when the fund falls short.
  • Thinking a funding method lowers cost: by C + I = B + E it only moves contributions in time; the ultimate cost changes only through B, E or I.
  • Measuring R against final average salary, or against the salary in the year of retirement: the denominator is the salary in the year before retirement.