Commutation Functions Used in Insurance

Commutation functions are used in Insurance to determine the value of benefit of premiums paid at the happening of an event that is insured under the policy contract.

Here are some guidelines for the commutation functions that can be used by a life insurance company;

Please note that 'i' is the rate of interest used

Present Value of Survivors Benefit  at age x = 1/(1+i) * Number of survivors at age x

                                                                                                                   2
Present Value of Benefit Estimated as Payable at age (x+1)={ 1/(1+i) * (Number of survivors at age x * probability of death of a life aged x in a year.}


Present Value of Benefits After Age x = Present Value of Survivors benefit at age x + Present Value of benefit at age (x+1) +Present Value of benefit at age (x+2) + Present Value of benefit at age (x+3) + and so on

Present Value of Benefits Estimated as  Payable After Age x = Present Value of Benefit estimated as payable at age (x+1) + Present Value of Benefit estimated as payable at age (x+2) + Present Value of Benefit estimated as payable at age (x+3) + and so on

From here we can infer that:

1) The present value of benefit of premium payable on death after age x =
Present Value of Benefits Estimated as  Payable After Age x / Present Value of Survivors Benefit  at age x

2) The present value of benefit of premium payable on death after any age x before  age (x+n) or on age (x+n) if alive ={ (Present Value of benefits estimated as payable after age x - Present Value of benefits estimated as payable at age (x + n)+ Present Value of Survivor benefit at age x + n )} / Present Value of Survivors Benefit  at age x

3) The present value of benefit of premium payable on death at any age after x but before (x +n) = Present Value of benefits estimated as payable after age x - Present Value of benefits estimated as payable at age (x+n)/Present Value of Survivors Benefit  at age x

4) The present value of benefit of premium payable on survival at age (x+n) = Present Value of Survivors Benefit  at age x+n / Present Value of Survivors Benefit  at age x

5) The present value of premium payable every year to a life aged x as long as the life is alive ( the first payment of premium is made at the commencement of the contract) =
Present Value of Benefits After Age x/ Present Value of Survivors Benefit  at age x

6) The present value of premium payable every year to a life aged x as long as the life is alive before age (x +n)  ( the first payment of premium is made at the commencement of the contract) = {Present Value of Benefits After Age x - Present Value of Benefits at age (x+n)}/ Present Value of Survivors Benefit  at age x

7) The present value of premium payable every year to a life aged x as long as the life is alive ( the first payment of premium is made at the end of the year of commencement of the contract) =  Present Value of Benefits after age x+1/Present Value of Survivors Benefit  at age x

8) The present value of premium payable every year to a life aged x as long as the life is alive before age (x+n) ( the first payment of premium is made at the end of the year of commencement of the contract)= { Present Value of Benefits after age x+1 - Present Value of benefits at age (x+n+1)}/Present Value of Survivors Benefit  at age x