Calculate the required contribution rate to meet a target replacement ratio for a DC plan participant, using a deterministic approach.

A 35-year-old wants 70% of her final salary replaced for life starting at 65. Markets, salary growth, and annuity rates are all fixed assumptions. The question is one number: what fraction of pay must flow into the account each year?

A defined contribution plan promises a contribution stream, not a benefit. The participant bears investment and longevity risk. To translate a benefit target into a contribution rate, you fix every economic input and solve the balance equation. The standard deterministic assumptions are:

• Annual salary growth $s$, compounded annually.
• Pre-retirement investment return $i$, credited annually to the account.
• Post-retirement valuation rate $i'$ (may equal $i$ or be lower).
• Annuity factor at retirement $\ddot{a}_r$, reflecting mortality and $i'$.
• Years to retirement $n = r - x$, where $x$ is current age and $r$ is retirement age.
• Target replacement ratio $R$, applied to final salary.

KEY: The contribution rate $c$ is a fraction of current salary paid each year. The same $c$ is applied to the rising salary, so contributions grow with pay.

Common mistakes

• Using $\ddot{s}_{\overline{n}|i}$ instead of the salary-weighted factor. Plugging straight annuity-due accumulation forgets that contributions grow at rate $s$. Always either build the salary-weighted sum or compute $\ddot{s}$ at the real rate $j$.
• Picking the wrong final-salary index. If the problem says "final salary," confirm whether it means $S_0(1+s)^{n-1}$, $S_0(1+s)^{n}$, or a final-three-year average. The trap value differs by a factor of $(1+s)$, commonly 3 to 5% in error.
• Mixing pre- and post-retirement rates. The annuity factor $\ddot{a}_r$ must use the post-retirement rate $i'$. Using $i$ (pre-retirement) inflates the annuity factor and understates $c$.

Bottom line

• Replacement ratio $R = \text{first-year retirement income} / \text{final salary}$, usually targeted net of Social Security.
• Three blocks: project final salary, accumulate contributions to retirement, convert the fund to an annuity; equate to target income and solve for rate $c$.
• Closed-form rate: $c = R \cdot \ddot{a}_r / S(n, s, i)$; $\ddot{a}_r$ uses post-retirement assumptions (not pre-retirement returns) and $S_0$ cancels.
• Salary-weighted factor: $S(n, s, i) = \sum_{k=0}^{n-1} (1+s)^{k} (1+i)^{n-1-k} = \frac{(1+i)^n - (1+s)^n}{i - s}$, or $n(1+i)^{n-1}$ when $i = s$.

Exam shortcut

Convert to the real rate first: $j = (1+i)/(1+s) - 1$. Then $c = R \ddot{a}_r / \ddot{s}_{\overline{n}|j}$. One line, one division, done. DECISION: Problem gives separate $i$ and $s$ → real-rate shortcut. Problem gives only one rate → set $s = 0$ and use $\ddot{s}_{\overline{n}|i}$ directly. For a sanity check, the contribution rate for a 50% net target, 30-year horizon, and 3% real return is roughly 9 to 12% of pay.

The full lesson (about 5,278 words, 35 min read) adds 5 worked examples, all 15 common mistakes, a self-check, free in the app.

Learning objectives

• 5b

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