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Valuation of Liabilities.

25. A valuation of the policies at the end of any given period is called a valuation of the liabilities.

26. To find the value of a policy.

If the rates of sickness and mortality experienced and the rate of interest realised correspond with the rates of sickness, mortality, and interest adopted in computing the premiums, the fund in hand to provide for the claims under the policies is the premiums received at the end of each year less the claims by sickness and death at the end of each year, accumulated at compound interest. Thus the accumulated fund at the end of each year is the accumulation of the premiums received less the claims paid at the early ages to meet the claims which exceed the premiums at the advanced ages. The share of each person living at the end of each year is the value of each policy.

Let _nV_x denote the present value of a policy for £1 on the life of a person of the age x at entry after it has been n years in force, the premium being due a year hence, A_x the value of the sum assured on the life x, and P_x its risk annual premium; then at the end of a year the value of the policy will be equal to the value of the sum assured on the life x+1 less the value of the future premiums, that is

₁V_x=A_x+l-P_x¹a_x

at the end of two years

₂V_x=A_x+2-P_x²a_x

at the end of n years

_nV_x=A_x+n-P_xⁿa_x

Rule. The value of a policy is the value of the sum assured at the present aye less the value of the future premiums, computed by the rates of sickness and mortality adopted, and the rate of interest which trill most likely he realised in the future.^*

If the accumulated fund in hand is equal to the sum of the values of the sums assured less the sum of the values of the future premiums the Society is solvent.