Monthly Premium Conversion on an Endowment Assurance
A 35-year-old male is buying a 25-year endowment assurance with a sum assured of ₹20,00,000. The office quotes an annual premium of ₹54,000, calculated on a gross premium basis at 5% interest using a standard mortality table. Expenses are 6% of every premium, plus renewal expenses of ₹1,200 at the start of each policy year after the first; the annual premium already includes these. The administration system can only collect monthly premiums in advance and cannot levy a separate policy fee. The operations team proposes to set the monthly premium at ₹54,000 ÷ 12 = ₹4,500, arguing that this is fair because the same total is collected each year. The pricing actuary knows that the annual annuity-due factor is 14.20 and the corresponding 12thly annuity-due factor (payments of 1/12 each month) is 13.67; the sum assured and survival benefits are unchanged by payment frequency. The product is sold in a competitive market, and a 4% lapse rate applies in the first year. The actuary must advise the monthly premium and document the basis for the pricing committee.
1. Calculate the monthly premium consistent with the annual premium under the equivalence principle, showing how the annuity factors and expense treatment are used. State any assumption about the fixed renewal expense.
Read the worked answer
Let P_a = ₹54,000 be the annual premium and P_m the monthly premium. The gross premium equation is
P_a × ä × (1 − 0.06) = B + E_fixed,
where B is the PV of benefits and E_fixed is the PV of the fixed renewal expenses. For monthly premiums in advance, using the 12thly annuity-due factor for payments of 1/12 each month:
12 P_m × ä^(12) × (1 − 0.06) = B + E_fixed.
Equating the right-hand sides gives P_m = P_a × ä / (12 × ä^(12)) = 54,000 × 14.20 / (12 × 13.67) = ₹4,674.47 per month.
The 6% commission cancels because it is proportional to premium. The fixed renewal expense also cancels because it is incurred annually at the start of each policy year and its PV does not depend on whether the premium is paid annually or monthly. If the system spreads the fixed expense into the monthly premium, the same ratio still applies, but the cash-flow timing should be checked.
2. Explain why dividing the annual premium by 12 is not actuarially neutral and quantify the shortfall.
Read the worked answer
Dividing ₹54,000 by 12 gives ₹4,500 per month. That is not actuarially neutral because monthly premiums are paid later within each year than a single annual premium in advance. The insurer loses interest on the deferred premiums and also carries the risk that a policy lapses before the later monthly payments are collected. The correct monthly premium is ₹4,674.47, so the proposal undercharges by ₹174.47 per month, or ₹2,093.64 per year (3.88% of the correct annual equivalent). Over the 25-year term the shortfall has a significant present value and would reduce the profit margin. Rounding down to a market-friendly figure such as ₹4,650 would require an explicit profit test and a decision on whether to recover the difference through lower expenses or accept a lower margin.
3. Should the first-year lapse rate or competitive pressure change the recommended premium? How should the decision be documented?
Read the worked answer
The first-year lapse rate matters because acquisition costs are incurred at outset, while monthly premiums arrive over time. A 4% first-year lapse rate increases the risk that the insurer does not recover initial expenses, and monthly collection makes the early cash flow worse than under annual premiums. The conversion above assumes the same lapse experience; if monthly premiums change lapse behaviour, the pricing basis must be re-run. Competitive pressure is not a reason to ignore the arithmetic: the insurer can round to ₹4,675, reduce commission, or adjust the product design. The pricing committee should document the annuity factors, expense assumptions, lapse assumption, rounding rule, and a sensitivity showing the profit impact of charging ₹4,500 instead of ₹4,674.