Profit test and capital strain for a three-year endowment
A life insurer is profit-testing a three-year without-profit endowment assurance sold to a life aged 50. The sum assured is £100,000, payable at the end of the year of death or on survival to the end of year three. The level annual premium is £33,000, payable in advance. Mortality rates are q50=0.005, q51=0.006 and q52=0.007. The insurer earns 4% per year on assets. Initial expenses are £1,000, and expenses are 10% of each premium. Prudent reserves per policy in force are £30,000 at the end of year one and £65,000 at the end of year two; the terminal reserve is zero. The risk discount rate is 8% per year. The product actuary wants the profit vector, the net present value per policy, and a view on whether the negative early profits are acceptable.
1. Calculate the profit vector at the end of each year per policy in force at the start of the year, and the NPV at 8%.
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Use profit at the end of year t=(opening reserve+premium-expenses)(1+i)-q x sum assured-(1-q) x closing reserve. Opening reserve is zero. Year 1: (0+33,000-4,300) x 1.04 - 0.005 x 100,000 - 0.995 x 30,000 =29,848-500-29,850=-502. Year 2: (30,000+33,000-3,300) x 1.04 - 0.006 x 100,000 - 0.994 x 65,000 =62,088-600-64,610=-3,122. Year 3: (65,000+33,000-3,300) x 1.04 - 0.007 x 100,000 - 0.993 x 0 =98,488-700=97,788. The profit vector is therefore (-502, -3,122, +97,788) per policy in force at the start of each year. NPV at 8% = -502/1.08 - 3,122/1.08^2 + 97,788/1.08^3 = -465 - 2,677 + 77,627 =£74,485.
2. Explain whether the product should be launched if the insurer must hold capital equal to any negative cumulative profit and the 8% risk discount rate does not explicitly include the cost of capital.
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On the central assumptions the NPV is strongly positive, so the product creates value. But the early cashflows are negative: cumulative profit is -£502 at the end of year 1 and -£3,624 at the end of year 2. The insurer must finance that strain with capital, and the 8% risk discount rate may not fully reflect the cost of holding that capital or the risk that the large year-3 profit does not materialise. Launch may still be justified if capital is available and the risks, including mortality, expenses, reserves and lapse behaviour, are understood. The actuary should show the NPV before and after an explicit cost-of-capital charge, and test sensitivity to higher reserves, higher mortality and lower interest rates.
3. State how the result would change if the reserves were £5,000 higher at the end of each of years one and two, all else equal.
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If the closing reserves at the end of years 1 and 2 are each £5,000 higher, year-1 profit falls by 0.995 x 5,000=£4,975. Year-2 profit changes by +5,000 x 1.04 - 0.994 x 5,000 =+5,200-4,970=+£230, because the higher opening reserve earns interest but the higher closing reserve is held for survivors. Year-3 profit rises by +5,000 x 1.04=+£5,200 because the higher opening reserve earns interest and the terminal reserve is unchanged at zero. The new profit vector is approximately (-5,477, -2,892, +102,988). Discounted at 8%, the NPV falls by about £281, so the product remains positive but the early capital strain is larger. More prudent reserves defer profit to later years and reduce NPV when the risk discount rate exceeds the earned rate.