Competing Decrements in a Pension Scheme
A pension scheme for members aged exactly 60 provides a lump sum of ₹5,00,000 on ill-health retirement, payable at the end of the year of retirement, and ₹2,00,000 on death in service, also at the end of the year. There is no benefit on withdrawal. For the next three years the actuary assumes constant independent rates per year: death 0.02, ill-health retirement 0.05, and withdrawal 0.03. Interest is 6% per annum. A junior actuary values the ill-health benefit by applying the independent retirement rate directly to all survivors each year, ignoring death and withdrawal: 0.05 × ₹5,00,000 / 1.06 = ₹23,585 in year 1, then 0.95 × 0.05 × ₹5,00,000 / 1.06² in year 2, and so on. The scheme auditor says this overstates the liability because death and withdrawal remove members before they can retire. The actuary must produce a corrected valuation and explain the treatment of withdrawal. The scheme rules allow a paid-up benefit on withdrawal, but the actuary has no data on its cost.
1. Derive the dependent probability of ill-health retirement in the first year and compute the corrected expected PV for year 1. Why does it differ from the junior actuary’s figure?
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Under the assumption that decrements are uniformly distributed over the year, the dependent ill-health retirement rate is
q_r^dep = q_r [1 − ½(q_d + q_w)] = 0.05 [1 − ½(0.02 + 0.03)] = 0.04875.
The expected PV for year 1 is 500,000 × 0.04875 / 1.06 = ₹22,995. The junior actuary’s ₹23,585 uses 0.05 directly and therefore ignores the probability that a member dies or withdraws before retiring. The difference, ₹590, is the cost of competing decrements in year 1. The correct rate is lower because a member must survive both death and withdrawal to reach retirement; the uniform-distribution assumption allocates half of the other decrements before retirement on average.
2. Extend the calculation over three years, stating assumptions, and compare with the independent-rate approach.
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First compute the dependent death and withdrawal rates for consistency: q_d^dep = 0.02[1 − ½(0.05+0.03)] = 0.0192; q_w^dep = 0.03[1 − ½(0.02+0.05)] = 0.02895. Total dependent decrement rate = 0.0192 + 0.04875 + 0.02895 = 0.09690, so survival p = 0.90310.
Year 1: 500,000 × 0.04875 / 1.06 = ₹22,995. Year 2: 500,000 × (0.90310 × 0.04875) / 1.06² = ₹19,591. Year 3: 500,000 × (0.90310² × 0.04875) / 1.06³ = ₹16,692. Total = ₹59,278.
The junior actuary’s approach gives 500,000 × [0.05/1.06 + 0.95×0.05/1.06² + 0.95²×0.05/1.06³] = ₹63,666. The independent-rate method overstates the liability by about ₹4,388, or 6.9%, because it lets all survivors remain exposed to retirement and ignores death and withdrawal.
3. Advise how withdrawal should be treated given the paid-up benefit and missing data.
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Withdrawal cannot be treated as a zero-benefit decrement if the scheme rules give a paid-up benefit. The correct approach is to keep withdrawal as a decrement in the retirement-benefit valuation and set up a separate reserve for the paid-up benefit. If no data exist, the actuary should not assume zero cost. Use a proxy from a similar scheme, a prudent percentage of the accumulated premium, or a range of scenarios, and disclose the uncertainty. Ignoring withdrawal altogether would overstate the retirement benefit and understate the paid-up liability. If the paid-up benefit is genuinely negligible and the scheme documentation confirms it, the approximation may be acceptable, but it should be documented and reviewed. The auditor’s point is valid: the corrected retirement value is lower, but the total liability may not fall by the full ₹4,388 once the paid-up benefit is included.