Benefit Reserves
\[_tV+\text{APV at }t\text{ of }\textbf{future}\text{ premiums}=\text{APV at }t\text{ of }\textbf{future}\text{ benefits}\]
Recursion Relation
\[({_tV}+P_t)(1+i)=b_{t+1}q_{x+t}+{_{t+1}V}p_{x+t}\]
Rearrangement yields Fackler’s accumulation formula: \[_{t+1}V=\frac{({_tV}+P_t)(1+i)-b_{t+1}q_{x+t}}{p_{x+t}}\]
In terms of Net Amount at Risk (top-up on reserve to equate benefit), \[({_tV}+P_t)(1+i)={_{t+1}V}+(b_{t+1}-{_{t+1}V})q_{x+t}\]