Begin with the expression for the general discrete policy and split off the first year
\n\\begin{eqnarray*}
\n_h V &=& b_{h+1} v q_{x+h} – P_h+
\n\\sum_{s=0}^{\\infty} \\left\\{b_{h+s+1} v^{s+2} ~_{s+1|} q_{x+h} – P_{h+s+1} v^{s+1} ~_{s+1} p_{x+h} \\right\\}
\n\\end{eqnarray*}where we have used \\(s=j-1\\). Now, recall the relation \\(~_{s+1} p_{x+h}= p_{x+h} ~_s p_{x+h+1}\\) and
\n\\begin{eqnarray*}
\n~_{s+1|} q_{x+h} &=& ~_{s+1} p_{x+h} q_{x+h+s+1} \\\\
\n&=& p_{x+h} ~_s p_{x+h+1} q_{x+h+s+1} = p_{x+h} ~_{s|} q_{x+h+1}
\n\\end{eqnarray*}
\nWith this, we have
\n\\begin{eqnarray*}
\n_h V &=& b_{h+1} v q_{x+h} – P_h+
\nvp_{x+h} \\sum_{s=0}^{\\infty} \\left\\{b_{h+1+s} v^{s+1} ~_{s|} q_{x+h+1} – P_{h+1+s} v^s ~_s p_{x+h+1} \\right\\} \\\\
\n&=& b_{h+1} v q_{x+h} – P_h+ vp_{x+h} ~_{h+1} V.
\n\\end{eqnarray*}
\nPut another way, we can express this as<\/p>\n
| \\(_h V+P_h\\)<\/font><\/td>\n | \\(= v q_{x+h} b_{h+1}\\)<\/font><\/td>\n | \\(+vp_{x+h} ~_{h+1} V\\)<\/font><\/td>\n<\/tr>\n| \n\t\t\tpolicy value <\/font> | \n\t\t\tplus premium<\/font> \n\t\t\t \n\t\t\t<\/td>\n \n\t\t\tis sufficient to provide a <\/font> | \n\t\t\t death benefit for the<\/font> \n\t\t\tproportion that fails<\/font>\n\t\t\t<\/td>\n \n\t\t\t plus the policy value<\/font> | \n\t\t\tfor the proportion<\/font> \n\t\t\tthat survives<\/font>\n\t\t\t<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n Example<\/strong> Solution.<\/em><\/p>\n Using the recursive reserve formulation, we have |