Details

Term life insurance is the basic insurance product where the insurance event is the death of the insured. For pure endowment, the insurance event is the survival of the term. The endowment is a mixture of the two other types of insurance.

The net reserve of endowment for years, where the premium is paid in years, is:

( is used for the more standard actuarial notation.)

Assuming the initial costs to be of the benefit, the administrative costs of the benefit, and the collection fee of the gross premium (but as follows the result does not depend on ), we can calculate the gross premium as

,

then the gross reserve is

where

is the net reserve in time ,

is the gross reserve in time ,

is the endowment for years,

is the annual premium paid within years.

The calculation of reserves for term insurance and pure endowment is analogous, using for term life insurance and for pure endowment.

The computation is based on the commutation functions:

,

,

where , , and can be derived using the decremental data and :

and ,

and ,

where

.