General Discrete Policy. For a general discrete policy, we have (known) premiums (P_h) payable at time (h) and benefits payable at time (b_h). As we have seen, the policy value can be expressed recursively as
begin{eqnarray*}
_h V &=& v q_{x+h} b_{h+1} – P_h + v p_{x+h} ~_{h+1} V.
end{eqnarray*}
Now, multiply each side by (v^h ~_h p_x) to get
begin{eqnarray*}
v^h ~_h p_x ~_h V – v^{h+1} ~_h p_x ~p_{x+h} ~_{h+1} V&=& v^{h+1} ~_h p_x q_{x+h} b_{h+1} – P_h v^h ~_h p_x
end{eqnarray*}
Note the relations (~_{h+1} p_x=~_h p_x ~ p_{x+h}) and ( ~_{h|} q_x=~_h p_x ~ q_{x+h} ). Sum each side of the equation over (h=0, ldots, k-1). On the left-hand side, we have
begin{eqnarray*}
sum_{h=0}^{k-1} left{v^h ~_h p_x ~_h V – v^{h+1} ~_{h+1} p_x ~_{h+1} V right} &=&
v^0 ~_0 p_x ~_0 V – v^k ~_k p_x ~_k V = – ~_k E_x ~_k V ,
end{eqnarray*}
recalling the relation (~_k E_x = v^k ~_k p_x) and assuming that (_0 V=0). On the right-hand side, we have
begin{eqnarray*}
sum_{h=0}^{k-1} left{v^{h+1} ~_{h|} q_x b_{h+1} – P_h v^h ~_h p_x right} .
end{eqnarray*}
Thus, we may write
begin{eqnarray*}
V_k &=& frac{sum_{h=0}^{k-1} P_h v^h ~_h p_x }{ ~_k E_x}-
frac{sum_{h=0}^{k-1} v^{h+1} ~_{h|} q_x b_{h+1} }{ ~_k E_x} \
&=& textrm{Accumulated Value of Premium} – textrm{Accumulated Cost of Insurance} .
end{eqnarray*}
Example
For a fully discrete policy for (x), the first year benefit is 10,000 and the first year premium payable at the beginning of the year is 500. Calculate the policy value at duration 1, assuming (i=5%) and (q_x = 0.03).
Solution
From the retrospective formula, we have
begin{eqnarray*}
_1 V &=& frac{sum_{h=0}^0 P_h v^h ~_h p_x }{ ~_1 E_x}-
frac{sum_{h=0}^0 v^{h+1} ~_{h|} q_x b_{h+1} }{ ~_1 E_x} =
frac{P_0}{ ~_1 E_x}-
frac{ v q_x b_1 }{ ~_1 E_x} \
&=& frac{500}{ v p_x}-
frac{ v q_x (10000)}{ v p_x} = frac{500-10000frac{1}{1.05}(0.03)}{frac{1}{1.05} (1-0.03)} = 231.96 .
end{eqnarray*}