Review of Asset Shares and Emerging Profits<\/p>\n
Recall our recursive asset share formula
\n\\begin{eqnarray*}
\n(~_k AS + G_k – e_k)(1+i_k) &= &
\nq_{[x]+k}^{(d)} \\left(b_{k+1} + E_{k+1}\\right) \\\\
\n&~~~~+& q_{[x]+k}^{(w)} ~_{k+1} CV + p_{[x]+k}^{(\\tau)} ~_{k+1} AS
\n\\end{eqnarray*}<\/p>\n
Asset shares are calculated during the year (using experience or another basis). Then, the profit during the year (at time \\(k+1\\) ) is
\n\\begin{eqnarray*}
\nPr_{k+1} &= & \\left( _k AS + G_k -e_k\\right) (1+ i_k) –
\nq_{[x]+k}^{(d)} \\left(b_{k+1} + E_{k+1}\\right) \\\\
\n&~~~~-& q_{[x]+k}^{(w)} ~_{k+1} CV – p_{[x]+k}^{(\\tau)} ~_{k+1} AS .
\n\\end{eqnarray*}<\/p>\n
Define \\(\\Pi_{k+1} = ~_k p_{[x]}^{(\\tau)} Pr_{k+1}\\) to be profits discounted for survivorship.<\/p>\n
To summarize profits, we might use the net present value
\n\\begin{eqnarray*}
\nNPV = \\sum_k \\left(\\frac{1}{1+r} \\right)^k \\Pi_k
\n\\end{eqnarray*}
\nor another summary measure.<\/p>\n