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 $i^{\prime }$ (may equal $i$ or be lower).
• Annuity factor at retirement ${\ddot{a}}_r$, reflecting mortality and $i^{\prime }$.
• 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 $i^{\prime }$. 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 7,325 words, 49 min read) adds 6 worked examples, all 19 common mistakes, a self-check, free in the app.

Learning objectives

• 5b

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