Policy Values, Asset Shares and the Cost of Not Declaring a Bonus
A with-profit fund contains 12,000 in-force 20-year endowment assurances sold to lives then aged 45, each with a sum assured of ₹1,000,000 and annual premiums payable in advance. The premium basis was 6% interest, mortality at 90% of the standard table, expenses of 40% of the first premium plus ₹1,500 per policy initially, 5% of each renewal premium plus ₹300 per policy, and ₹1,000 per claim. The resulting gross premium was ₹36,270. Two years of poor investment returns have constrained asset shares, which now stand at ₹192,000 per policy at duration five. The actuary must value the contracts at duration five on a policy value basis of 4.5% interest and 100% of the standard table, with expenses unchanged. At that basis A(50:15) = 0.615 and ä(50:15) = 11.00. The fund's free assets are ₹400 million. Management wants to know whether it can keep the current terminal bonus scale.
1. Recalculate the gross premium from the stated basis to confirm the ₹36,270 figure, showing the equivalence-principle equation.
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At 6% interest, A(45:20) = 0.3860 and ä(45:20) = 11.72. Let P be the annual premium. Equating present values of premiums and benefits plus expenses at outset:
P × ä = 1,000,000 × A + 1,000 × A + 0.05P × (ä − 1) + 300 × (ä − 1) + 0.40P + 1,500
Terms in order: the sum assured, the claim settlement expense, renewal expenses proportional to premium (14... in fact 19 renewals, PV (ä − 1)), renewal expenses per policy, the first-year expense loading and the per-policy initial cost.
Numerically:
- 1,000,000 × 0.3860 = 386,000
- 1,000 × 0.3860 = 386
- 0.05 × 10.72 × P = 0.536P
- 300 × 10.72 = 3,216
- Initial: 0.40P + 1,500
So 11.72P = 391,102 + 0.936P, giving 10.784P = 391,102 and P = ₹36,266, i.e. ₹36,270 to the nearest ten.
The equation also shows the structure of the answer: the sum assured term (386,000) dominates, the expense terms are about 1.4% of the premium, and the denominator 10.784 rather than 11.72 reflects the fact that expenses absorb roughly 8% of premium income - a useful check that the premium is not being set on a net-premium basis.
2. Calculate the policy value at duration five on the new basis, compare it with the asset share, and state the aggregate strain.
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Prospective reserve immediately after the fifth premium, age 50, 15 years to run:
V = 1,000,000 × A + 1,000 × A + (0.05P + 300) × (ä − 1) − P × ä
= 615,000 + 615 + (0.05 × 36,270 + 300) × 10.00 − 36,270 × 11.00 = 615,000 + 615 + (1,813.50 + 300) × 10.00 − 398,970 = 615,000 + 615 + 21,135 − 398,970 = ₹237,780 per policy
Comparison. The asset share is ₹192,000, so the reserve exceeds the assets backing the contract by ₹45,780 per policy. Across 12,000 policies that is ₹549 million, against free assets of ₹400 million. The fund is short by about ₹149 million against its own valuation basis.
Interpretation. The gap is not primarily a bonus problem - it is a basis-change problem. On the original 6% basis the reserve at duration five would have been of the order of ₹181,000, which the asset share roughly covers. Moving the valuation rate down 1.5% and mortality up by a tenth has raised the reserve by roughly 31%. The reserve is the difference between two large quantities (the claim term of ₹615,615 plus expenses, against the premium term of ₹398,970), so a 10–13% increase in each component produces a 30% increase in the net figure. That amplification is the key insight and is why duration-based intuition applied crudely to the net reserve understates the risk.
3. Advise management on the terminal bonus declaration and on how the position changes if interest rates fall a further 1% before the next valuation.
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Can the terminal bonus scale continue? No. A declaration of terminal bonus is a promise to pay from asset shares plus the inherited estate. Here the assets available (₹192,000 per policy) are below what the valuation basis says must be held (₹237,780). Declaring further bonus would increase the reserve further, deepen the deficit, and constitute a transfer from the estate to policyholders who are already underfunded. The correct sequence is: (i) suspend or scale back the terminal bonus rate, (ii) tell management that the shortfall of roughly ₹149 million must be met from the estate or from new capital, (iii) review whether the 4.5% basis is the regulatory minimum or a best-estimate basis, because that determines whether the shortfall is a published solvency position or an internal strain.
A further 1% fall. With a duration of about 8 on each term, the claim term rises to roughly ₹665,000 and the premium term to about ₹431,000, giving a reserve in the region of ₹256,000 - another ₹18,000 per policy, ₹216 million in aggregate, on a fund that is already short. Convexity means the true figure is higher, not lower.
Levers. Reduce declared bonuses; use reinsurance or an interest rate swap overlaying the long tail; treat lapse experience honestly - a reserve that relies on lapses to release surplus is lapse-supported and collapses exactly when lapses fall; and if the position cannot be funded, present a formal recovery plan rather than a dividend declaration the fund cannot afford.