Identify, interpret, and apply service table and salary scale functions for pension plan valuation.

A pension actuary valuing a 45-year-old plan member needs two grids: one for the probability she stays employed and reaches each future age, and one for what she will earn if she does. Service tables answer the first question, salary scales answer the second.

The service table is a multiple-decrement table for active plan members. $l_x^{(\tau)}$ is the number of actives at exact age $x$. Total decrement $d_x^{(\tau)}$ splits into causes:

$$d_x^{(\tau)} = d_x^{w} + d_x^{d} + d_x^{i} + d_x^{r}, \qquad l_{x+1}^{(\tau)} = l_x^{(\tau)} - d_x^{(\tau)}$$

The four standard decrements:

• Withdrawal ($w$). Voluntary termination before retirement eligibility. Heavy at young ages and short service.
• Death-in-service ($d$). Active-employee mortality. Lower than population mortality due to active-life selection.
• Disability ($i$). Transition to disabled status, removing the member from the active count.
• Retirement ($r$). Early retirement (commonly ages 55 to 64) plus normal retirement at 65 or 67.

KEY: The superscript $(\tau)$ denotes "all causes". $p_x^{(\tau)}$ is the probability of remaining active one year, surviving every decrement.

Common mistakes

• Treating $s_x$ as salary. The scale is an index. Forgetting to divide by $s_x$ inflates projected salary by a factor of the absolute index level, often 100 or more.
• Wrong scale subscripts for FAS. A 5-year FAS for retirement at exact age 65 uses $s_{60}$ through $s_{64}$, not $s_{61}$ through $s_{65}$. The last year of earnings ends at age 65.
• Ignoring withdrawal in survival. Multiplying by mortality-only survival, with no withdrawal decrement, overstates EPV at young ages by 30 to 60 percent.

Bottom line

• Service table $l_x^{(\tau)}$ tracks active lives subject to four decrements: withdrawal $w$, death $d$, disability $i$, retirement $r$.
• Survival from age $x$ to age $y$ as an active member is ${}_{y-x}p_x^{(\tau)} = l_y^{(\tau)} / l_x^{(\tau)}$.
• Salary scale $s_y$ is an index. Project salary by the ratio $s_y / s_x$; absolute levels are arbitrary.
• Final average salary (FAS) averages the last $n$ years of earnings before retirement, indexed by the last $n$ scale values $s_{r-1}, \ldots, s_{r-n}$.

Exam shortcut

Cancel $s_x$ at the end. Sum the $s_{r-k}$ values first, divide by $n \cdot s_x$ once, then multiply by current salary. Fewer divisions, fewer rounding errors. DECISION: Final-salary plan → salary scale required for FAS. Career-average plan → cumulate annual projected salaries; salary scale still projects each year. Memorize the EPV skeleton: probability of reaching retirement, times annual benefit at retirement, times interest discount, times annuity factor. Four pieces.

The full lesson (about 1,834 words, 12 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 5c

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