Begin with the expression for the general discrete policy and split off the first year
begin{eqnarray*}
_h V &=& b_{h+1} v q_{x+h} – P_h+
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}
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
begin{eqnarray*}
~_{s+1|} q_{x+h} &=& ~_{s+1} p_{x+h} q_{x+h+s+1} \
&=& p_{x+h} ~_s p_{x+h+1} q_{x+h+s+1} = p_{x+h} ~_{s|} q_{x+h+1}
end{eqnarray*}
With this, we have
begin{eqnarray*}
_h V &=& b_{h+1} v q_{x+h} – P_h+
vp_{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} \
&=& b_{h+1} v q_{x+h} – P_h+ vp_{x+h} ~_{h+1} V.
end{eqnarray*}
Put another way, we can express this as
| (_h V+P_h) | (= v q_{x+h} b_{h+1}) | (+vp_{x+h} ~_{h+1} V) |
|
policy value plus premium |
is sufficient to provide a death benefit for the proportion that fails |
plus the policy value for the proportion that survives |
Example
Consider a special fully discrete 10- year endowment policy to (30). Level premiums are payable for 10 years. The maturity value is 1 and a level (i=6%) interest is assumed. For simplicity, mortality is given as (~_k p_{30} = 0.98^k). The death benefit is 1 plus the policy value. Calculate (V_3).
Solution.
Using the recursive reserve formulation, we have
begin{eqnarray*}
(1.06)(_h V + P) &=& b_{h+1} q_{x+h} + p_{x+h} ~_{h+1} V \
&=& (1+ ~_{h+1} V) q_{x+h} + p_{x+h} ~_{h+1} V = 0.02 + ~_{h+1} V
end{eqnarray*}
because ( q_{x+h}=1-p_{x+h}=1- frac{~_{h+1} p_x}{~ _h p_x} = 0.02). We re-write this as
begin{eqnarray*}
1.06 ~_h V – ~_{h+1} V &=& q_{x+h} – 1.06 P
end{eqnarray*}
Multiplying each side by (v^{h+1}) yields
begin{eqnarray*}
v^h ~_h V – v^{h+1} ~_{h+1} V &=& v^{h+1} q_{x+h} – v^h P .
end{eqnarray*}
Summing both sides over (h=0, ldots, 9) yields
begin{eqnarray*}
v^0 ~_0 V – v^{10} ~_{10} V &=& sum_{h=0}^9 left( v^{h+1} q_{x+h} – v^h P right)\
&=& (v(0.02) – 1.06 P) ddot{a}_{overline{10|}}
end{eqnarray*}
Because (_0 V=0) and (_{10} V=1), we have
begin{eqnarray*}
P &=& frac{v^{10}}{ddot{a}_{overline{10|}}} +0.02 v = 0.09044.
end{eqnarray*}