Free SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) Pension Plans and Retirement Benefits Practice Questions

D is correct. Financial planning guidelines commonly target a replacement ratio of approximately 70%–85% because retirees typically face: (1) lower income taxes (reduced income, no payroll taxes); (2) elimination of work-related expenses (commuting, clothing, lunches); and (3) no further need to save for retirement. These factors mean a retiree needs less gross income than during their working years to maintain a similar standard of living.
B is incorrect; 100% replacement is generally not necessary and would over-fund retirement. C reverses the empirical finding — lower-income earners typically need a higher replacement ratio because a greater share of their spending is non-discretionary.
A is incorrect; replacement ratios apply equally to DB plan participants.
E is incorrect; the net replacement ratio explicitly includes Social Security.

E is correct. For a final average pay plan where salaries are expected to grow:

- **Unit Credit (UC):** the accrued benefit uses current salary only (not projected), so the AAL is the present value of a benefit based on today's salary — the lowest of the three.
- **Entry Age Normal (EAN):** costs are spread as a level amount or percentage from entry age to retirement based on the projected final benefit; the AAL at mid-career reflects accumulated level contributions, producing a moderate AAL.
- **Projected Unit Credit (PUC):** the accrued benefit uses projected final salary (the full projected benefit prorated by service to date), so the AAL at mid-career is the present value of a benefit reflecting projected salary growth — the highest of the three.

Therefore: $\text{AAL}_{\text{UC}} \leq \text{AAL}_{\text{EAN}} \leq \text{AAL}_{\text{PUC}}$. A has the order of EAN and UC reversed and places PUC last. B has an incorrect full ordering. C is false; the methods deliberately produce different AALs with the same PVFB by design. D incorrectly places EAN above PUC.

C is correct. Let $S_0$ be the current salary. End-of-year contributions are $c \cdot S_k = c \cdot S_0 (1.03)^k$ for $k = 1, \ldots, 30$. The accumulated fund at retirement is:

$$F = c S_0 \sum_{k=1}^{30} (1.03)^k (1.07)^{30-k} = c S_0 (1.07)^{30} \sum_{k=1}^{30} \left(\frac{1.03}{1.07}\right)^k = c S_0 (1.07)^{30} \cdot s_{\overline{30}|g}$$

Where $g = (1.03/1.07) - 1$ (negative, meaning the salary series grows slower than the investment return). The target fund is $0.70 \times S_{30} \times 13 = 0.70 \times S_0 (1.03)^{30} \times 13$. Dividing both sides by $S_0 (1.03)^{30}$:

$$c \cdot \left(\frac{1.07}{1.03}\right)^{30} \cdot s_{\overline{30}|g} = 0.70 \times 13.$$

Since $(1.07/1.03)^{30} \cdot s_{\overline{30}|g}$ with $g = 1.03/1.07 - 1$ equals $s_{\overline{30}|g'}$ with $g' = 1.07/1.03 - 1$ by the identity of the salary-adjusted annuity:

$$c = \frac{0.70}{13 \cdot s_{\overline{30}|g'}}, \quad g' = (1.07/1.03) - 1 \approx 0.0388.$$

Why each other option is incorrect:
- (A) Using 7% as the accumulation rate ignores that contributions are salary-proportional; when salary grows at 3% and investments grow at 7%, the effective rate of accumulation for the contribution stream is the excess, not the full 7%.
- (C) Placing 13 in the numerator inverts the formula; the annuity factor converts a fund into an annual income stream, so it must divide the target ratio.
- (D) The product $13 \times 30 = 390$ replaces a compound-interest formula with an arithmetic one, drastically overstating $c$ by ignoring 30 years of compounding.
- (E) Projecting the numerator by $(1.03)^{30}$ without adjusting the denominator creates an inconsistency; the salary growth terms cancel when the derivation is done correctly.