Net premium and reserve for a three-year endowment assurance
A life insurer issues a three-year endowment assurance to a life aged exactly 40. The sum assured is £100,000, payable at the end of the year of death or on survival to the end of year three. Premiums are level annual premiums payable in advance while the policy is in force. Mortality follows a table with q40=0.0010, q41=0.0015 and q42=0.0020. The insurer uses an effective interest rate of 5% per year and ignores expenses. The actuarial team is asked to calculate the net annual premium and the net premium reserve at the end of the first year, just before the second premium is paid. A separate sensitivity exercise will value the same contract at 4% interest, and the team must explain why the reserve changes.
1. Calculate the net annual premium for the three-year endowment assurance.
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First find the EPV of the benefit. The probabilities are: death in year 1 =0.0010; death in year 2 =0.9990 x 0.0015=0.0014985; death in year 3 =0.9990 x 0.9985 x 0.0020=0.0019950; survival to the end of year 3 =0.9990 x 0.9985 x 0.9980=0.9955065. At 5%, v=0.952381, v^2=0.907029 and v^3=0.863838. EPV benefit=100,000 x [0.0010v + 0.0014985v^2 + (0.0019950+0.9955065)v^3] =100,000 x [0.0009524+0.0013592+0.8616793]=£86,399. The EPV of a premium annuity-due is 1+0.9990v+0.9990 x 0.9985v^2 =1+0.951429+0.904763=2.856192. Therefore the net annual premium is P=86,399/2.856192=£30,250, to the nearest pound.
2. Calculate the net premium reserve at the end of year one, just before the second premium is paid, using the prospective method.
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At the end of year 1, just before the second premium, the life is aged 41 and two years remain. The prospective reserve is the EPV of the future benefit minus the EPV of future premiums. Benefit EPV=100,000 x [0.0015v + 0.9985v^2] =100,000 x [0.0014286+0.9056689]=£90,710. Future premiums are the immediate second premium and the third premium payable one year later if the life survives: EPV=P x [1+0.9985v] =30,250 x [1+0.950952]=30,250 x 1.950952=£59,016. The reserve is 90,710-59,016=£31,694. Equivalently, the insurer needs about £31,700 per policy at that point to cover the remaining guaranteed benefit after allowing for future premiums.
3. Explain why a fall in the interest assumption from 5% to 4% increases the reserve, and state one practical action the insurer might take.
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A lower interest rate increases the present value of the future benefit more than it increases the present value of the future premiums, because the benefit is paid later than the premiums. The premiums are paid in advance, while the sum assured is paid at the end of the year of death or at the end of year three. Therefore the prospective reserve rises. At 4% the discount factors and the annuity factor are larger, but the long-dated survival benefit is affected most. Practically, the insurer may need to hold more capital, revisit premium rates for new business, and use asset-liability matching or hedging to reduce interest-rate risk. It should not simply reduce the reserve, because the reserve is meant to be a prudent estimate of the liability.