Changes in reserves are expenses to a company and using a larger net premium basis can mean large expenses that may penalize smaller, fast-growing companies. Thus, a number of modified premium reserve bases have been introduced over the years, the most commonly used one being the
Full Preliminary Term Approach. Let (P^n) be the valuation net premium that we assume level for now. Further, define, for valuation purposes,
* (_1 P_{[x]}^n = b_1 v q_{[x]} ) a modified first year cost of insurance and
* (P^{FPT}) a modified premium for subsequent years.
The idea is the (P^n – _1 P_{[x]}^n) is the additional first year expense allowance. Thus, we have ( 0 leq _1 P_{[x]}^n leq P^n leq P^{FPT}).
Let (h) be the number of premium-payment periods and (j) ((leq h)) be the number of years that (P^{FPT}>P^n). Then,
begin{eqnarray*}
_1 P_{[x]}^n + P^{FPT} a_{x:overline{j-1|}} = P^n ddot{a}_{x:overline{j|}} .
end{eqnarray*}
This is equivalent to considering splitting the policy into two components, a 1-year term and a separate contract issued to the same life one year later, if the life survives.
With this (V_k^{FPT} = V_k^n, kge j), that is, the full preliminary term reserve equals the net premium reserve when the premiums are the same. For (1 le k < j), we have begin{eqnarray*} V_k^{FPT} = textrm{EPV Future Benefits} - P^{FPT} ddot{a}_{[x]+k:overline{j-k|}} - P^n ~_{j-k|h-j} ddot{a}_{[x]+k} . end{eqnarray*} Thus, for (1 le k < j) we have ( V_k^{FPT} < V_k^n) and begin{eqnarray*} V_k^n - V_k^{FPT}& =& left{textrm{EPV Future Benefits} - P^{FPT} ddot{a}_{[x]+k:overline{k-j|}} - P^n ~_{j-k|h-j} ddot{a}_{[x]+k} right}\ & ~~~ -& left{textrm{EPV Future Benefits} - P^n ddot{a}_{[x]+k:overline{k-j|}} - P^n ~_{j-k|h-j} ddot{a}_{[x]+k} right} \ & = & (P^{FPT}- P^n) ddot{a}_{[x]+k:overline{k-j|}} . end{eqnarray*} We can think of the modified reserve (V_k^{FPT}) as the net premium reserve minus a reserve for the unamortized excess premium. [previous][next]