Negative Reserves on a Decreasing Term Block
A life insurer is valuing a block of 15-year decreasing term assurances issued to 45-year-old male non-smokers. The sum assured is ₹10,00,000 at outset and falls by ₹60,000 at the end of each policy year; premiums are level annual and payable throughout the term. The valuation basis is AM92 Ultimate mortality and 4% interest. At duration 5 (attained age 50), the appointed actuary’s model produces future benefit PV of ₹2,65,000 and future net premium PV of ₹2,95,000, so the net premium reserve is negative ₹30,000. The finance team wants to zeroise all negative reserves to avoid showing an asset for policies that may lapse. A reinsurer separately requires a gross premium reserve; including future expenses of ₹40,000 and future gross premiums of ₹3,30,000 gives a gross reserve of negative ₹25,000. The company’s solvency regime permits negative reserves only where policyholder behaviour and expense assumptions are demonstrably reliable. The block has a 12% annual lapse rate and no paid-up option. The actuary must decide the valuation reserve and explain the profit signature effect.
1. Calculate and interpret the net premium reserve at duration 5. Is zeroising appropriate under the stated solvency regime?
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Net premium reserve = PV future benefits − PV future net premiums = ₹2,65,000 − ₹2,95,000 = −₹30,000. The negative sign means the policy is an asset: future net premiums are expected to exceed future benefits by ₹30,000. This is plausible because the sum assured has fallen to ₹10,00,000 − 5×₹60,000 = ₹7,00,000, while ten years of level premiums remain.
Under the stated regime, holding −₹30,000 is allowed only if policyholder behaviour and expense assumptions are demonstrably reliable. A 12% annual lapse rate with no paid-up option makes future premium income uncertain: if the policy lapses, the insurer loses the future net premiums that support the asset. The prudent statutory reserve is therefore max(−₹30,000, 0) = ₹0, with the −₹30,000 shown only as a best-estimate embedded-value asset, subject to a lapse sensitivity.
2. Explain how the gross premium reserve differs from the net premium reserve, and recommend a treatment for the reinsurance treaty.
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Gross premium reserve = PV benefits + PV future expenses − PV future gross premiums = ₹2,65,000 + ₹40,000 − ₹3,30,000 = −₹25,000. It is also negative, but smaller in magnitude than the net premium reserve. The difference arises because gross premiums include an expense loading: the office expects ₹3,30,000 of gross premiums against ₹2,65,000 of benefits and ₹40,000 of expenses.
For the reinsurer, the treaty definition controls. If the treaty requires a non-negative reserve, hold ₹0 and disclose the −₹25,000 as an off-balance-sheet asset. If negative reserves are permitted, hold −₹25,000 but stress lapses and expense overruns. The profit signature matters: zeroising releases ₹30,000 of profit at duration 5 but creates future strain if lapses occur; holding the negative reserve defers profit and is more conservative only if the future premiums are robust.
3. How should the 12% lapse rate affect the reserve and the zeroisation decision?
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A 12% annual lapse rate should be modelled explicitly in both reserves. For the gross premium reserve, reduce the PV of future gross premiums by the probability that the policy remains in force; if lapses are selective against the insurer, also strengthen mortality. A rough check: ten-year survival at 12% lapse is 0.88^10 ≈ 0.28, so most of the future premium PV may not be collected. That can turn the −₹30,000 net reserve positive. With no paid-up option, a lapse also removes the policy and any future benefits, but the insurer loses the profitable premium stream and may retain acquisition-cost strain.
The zeroisation decision therefore should not be based only on the sign of the reserve. Given high lapse and no paid-up option, zeroising is the more defensible statutory treatment; the negative reserve can be recognised only in a best-estimate model with lapse and anti-selection stresses.