Review of Asset Shares and Emerging Profits
Recall our recursive asset share formula
begin{eqnarray*}
(~_k AS + G_k – e_k)(1+i_k) &= &
q_{[x]+k}^{(d)} left(b_{k+1} + E_{k+1}right) \
&~~~~+& q_{[x]+k}^{(w)} ~_{k+1} CV + p_{[x]+k}^{(tau)} ~_{k+1} AS
end{eqnarray*}
Asset shares are calculated during the year (using experience or another basis). Then, the profit during the year (at time (k+1) ) is
begin{eqnarray*}
Pr_{k+1} &= & left( _k AS + G_k -e_kright) (1+ i_k) –
q_{[x]+k}^{(d)} left(b_{k+1} + E_{k+1}right) \
&~~~~-& q_{[x]+k}^{(w)} ~_{k+1} CV – p_{[x]+k}^{(tau)} ~_{k+1} AS .
end{eqnarray*}
Define (Pi_{k+1} = ~_k p_{[x]}^{(tau)} Pr_{k+1}) to be profits discounted for survivorship.
To summarize profits, we might use the net present value
begin{eqnarray*}
NPV = sum_k left(frac{1}{1+r} right)^k Pi_k
end{eqnarray*}
or another summary measure.