Defined Contribution Plan

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

A Defined Contribution Plan (DC plan) is a Retirement Plan in which the employer and member pay a fixed contribution rate $c$ of salary into the member's own account, and the retirement income is whatever the accumulated account can buy — typically a Life Annuity bought at retirement. The contribution is promised; the benefit is not.

$F_{x_0+n} = \sum_{k=0}^{n-1} c\,S_{x_0+k}\,(1+i)^{\,n-k}$

$P = \frac{F_{x_0+n}}{\ddot{a}^{(12)}_{x_0+n}}$

$R = \frac{P}{S_{x_0+n-1}}$

Symbols: $S_y$ is salary in the year of age $y$ to $y+1$ , with each year's contribution paid at its start; $i$ is the fund's return; $\ddot{a}^{(12)}$ the annuity rate at retirement; $R$ the replacement ratio against the final year's salary (Dickson, Hardy & Waters, ch. 10). With monthly contributions a common approximation is mid-year timing, $(1+i)^{n-k-1/2}$ .
Required contribution rate. $F$ is linear in $c$ , so the rate that meets a target ratio $R^{*}$ is $c^{*} = c\,R^{*}/R$ .
The member bears the risks a Defined Benefit Plan leaves with the Plan Sponsor: investment returns before retirement, annuity prices (interest rates and longevity) at retirement, and — if no annuity is bought — outliving the account. The sponsor's cost is fixed at $c$ times payroll.
With salary growth $g$ , the sum is geometric: $F = c\,S_{x_0}(1+i)^n \frac{1-k^n}{1-k}$ , $k = \frac{1+g}{1+i}$ .

Accumulating a DC Account to a Replacement Ratio A member joins at 35 with salary $50{,}000 in the year of age 35 to 36. Salaries rise 3.5% a year; 10% of salary is contributed at the start of each year to age 65; the account earns 6%. At 65 it buys a monthly life annuity at $\ddot{a}^{(12)}_{65} = 13.0870$ . Find the replacement ratio.
Answer With $k = 1.035/1.06$ and $n = 30$ :
 $\begin{align*} F_{65} &= 5{,}000\,(1.06)^{30}\,\frac{1 - k^{30}}{1 - k} \\ &= 5{,}000 \times 124.516 \\ &= 622{,}580 \end{align*}$ 
 $\begin{align*} P &= \frac{622{,}580}{13.0870} \\ &= 47{,}572 \\ S_{64} &= 50{,}000\,(1.035)^{29} \\ &= 135{,}594 \\ R &= \frac{47{,}572}{135{,}594} \\ &= 35.1\% \end{align*}$ 
Thirty years at 10% of pay replaces about a third of final salary.

Required Contribution Rate for a 60% Target For the member above, find the contribution rate that gives a 60% replacement ratio (a) if the account earns 6%, and (b) if it earns 4%, where 4% gives $R = 25.6\%$ at a 10% rate.
Answer
 $\begin{align*} c^{*}_{6\%} &= 0.10 \times \frac{0.60}{0.35084} \\ &= 17.1\% \end{align*}$ 
 $\begin{align*} c^{*}_{4\%} &= 0.10 \times \frac{0.60}{0.25588} \\ &= 23.4\% \end{align*}$ 
Two points of return cost 6.3 points of pay a year — the price of investment risk, paid by the member.

How it appears on the exam

ALTAM asks for the replacement ratio of a DC member from a salary scale, a contribution rate and an annuity factor, and for the required contribution rate to meet a target replacement ratio, deterministically.

ALTAM sets DC beside DB in the same question, so the same member's FAS and DC ratios can be compared.

RET-101 asks for written answers on DC plan success factors, how DC design (default contributions, investment options, decumulation) manages participant risks, and when a sponsor should prefer DC.

Exam traps

Off-by-one timing: contributions at the start of years 0 to $n-1$ earn $(1+i)^{n-k}$ , not $(1+i)^{n-k-1}$ ; mid-year contributions earn a half-year less.

Dividing $P$ by the salary at age 65 or by an average salary: $R$ uses the final year's salary $S_{x_0+n-1}$ .

Pricing the annuity at the fund's accumulation rate when the question gives a separate annuity factor or rate.

Writing that the sponsor bears investment risk "because it pays most of the contributions": in DC, the sponsor's cost is fixed at $c$ times payroll.