Calculate replacement ratios for Defined Contribution (DC) and Defined Benefit (DB) plans, including final average salary (FAS), career average earnings (CAE), and career average revalued earnings (CARE) plans.

Two retirees with identical $120,000 final salaries can land on radically different pensions. One pulls 70 percent of pay; the other pulls 35. The replacement ratio is the single number that tells you which plan you joined.

The replacement ratio measures how much pre-retirement income a pension restores.

$$R=\frac{B}{S_{\text{final}}}$$

Some texts use career-average pay or pre-retirement net income in the denominator. Unless the problem says otherwise, use final salary. Typical adequacy targets sit at 70 to 80 percent of final pay when combined with social-insurance pensions.

Contributions accumulate in an individual account. Investment risk sits with the member. At retirement, the fund converts to income through annuity purchase or systematic withdrawal.

Let $c$ be the combined contribution rate, $S_t$ salary in year $t$, $i$ the fund return, $R$ retirement age, and $E$ entry age.

$$F=\sum _{t=E}^{R-1}c\cdot S_t\cdot (1+i{)}^{R-t}$$

$$B_{DC}=\frac{F}{{\ddot{a}}_R},R_{DC}=\frac{F}{{\ddot{a}}_R\cdot S_{R-1}}$$

KEY: DC replacement ratio depends on three uncertain inputs: contribution rate, accumulated investment return, and annuity factor at retirement (which embeds future interest and mortality). None is guaranteed.

Common mistakes

• Off-by-one on FAS years. ${\text{FAS}}_k$ averages the salaries indexed $R-k$ through $R-1$. Including age $R$ (the year of retirement, when there is no full-year salary) inflates FAS by one growth factor.
• Dropping revaluation in CARE. Setting $r=0$ silently reduces CARE to CAE and understates the benefit by 20 to 40 percent over a full career.
• Mismatched annuity factor. ${\ddot{a}}_R$ must use retirement-age mortality and the plan's post-retirement valuation rate. Using a pre-retirement accumulation rate inside ${\ddot{a}}_R$ gives the wrong pension.

Bottom line

• Replacement ratio $R=B/S_{\text{final}}$, where $B$ is first-year retirement income and $S_{\text{final}}$ is the salary just before retirement; it fixes only the FIRST year and erodes with inflation if the pension is not COLA-indexed.
• DC ratio: accumulate contributions at fund rate $i$, divide by annuity factor ${\ddot{a}}_R$, divide by final salary. Market-driven, not guaranteed.
• FAS DB: $B=\alpha \cdot n\cdot {\text{FAS}}_k$, where $\alpha$ is the accrual rate, $n$ is years of service, and ${\text{FAS}}_k$ averages the last $k$ salaries (heavily weighted to late-career pay).
• CAE: $B=\alpha {\sum }_t{S}_t$. No revaluation; inflation erodes early-career credits, giving the lowest ratio at positive growth.

Exam shortcut

For FAS under steady growth $g$, use $R_{FAS}\approx \alpha n/(1+g{)}^{(k-1)/2}$ via the geometric-mean approximation. With $\alpha =2\%$, $n=25$, $g=3\%$, $k=5$: $R\approx 0.50/1.06\approx 47\%$. DECISION: Salary at entry given with flat growth → final salary is $S_0(1+g{)}^{n-1}$, not $S_0(1+g{)}^n$. Off-by-one is the most common slip. For CAE and CARE, factor out $S_E$ and recognize a geometric series in $(1+g)$ for CAE or in $(1+g)(1+r)$ for CARE.

The full lesson (about 5,620 words, 37 min read) adds 4 worked examples, all 11 common mistakes, a self-check, free in the app.

Learning objectives

• 5a

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