To develop intuition, it is helpful to begin with a traditional framework. Here, policies have premiums that are payable annually (at the beginning of the year, by convention) and have benefits that are payable at the end of the year of failure.
Special Case: (n)-Year Term Insurance. For this policy, level premiums are payable at the beginning of each year up until failure, or (n) years if sooner. A death benefit of 1 is payable at the end of the year of failure, if within (n) years. When time point (t=h) is an integer, the loss variable is
begin{eqnarray*}
L_h &=&
left{
begin{array}{cc}
v^{K+1-h} – P ddot{a}_{overline{K+1-h}|} & h leq K < n \
0 - P ddot{a}_{overline{n-h}|} & K ge n
end{array}
right .
end{eqnarray*}
This has expectation, or policy value, (_h V = A_{x+h:overline{n-h|}}^{~1} - P_{x:overline{n|}}^{1} ddot{a}_{x+h:overline{n-h|}}).
See the animated display where the value changes by the interest rate
Special Case: (m)-pay, (n)-Year Endowment Insurance. For this policy, level premiums are payable at the beginning of each year up until failure, or (m) years if sooner. A benefit of 1 is payable at the end of the year of failure or (n) years, whichever is sooner.
When the policy is in the premium payment period, time point (t=h < m). The loss variable is
begin{eqnarray*}
L_h &=&
left{
begin{array}{cc}
v^{K+1-h} - P ddot{a}_{overline{K+1-h}|} & 0 leq K < m \
v^{K+1-h} - P ddot{a}_{overline{m-h}|} & m leq K < n \
v^{n-h} - P ddot{a}_{overline{m-h}|} & K ge n
end{array}
right .
end{eqnarray*}
This has expectation, or policy value, (_h V = A_{x+h:overline{n-h|}} - P_{x:overline{n|}} ~ ddot{a}_{x+h:overline{m-h|}}).
When the policy is past the premium payment period, time point (t=h ge m). In this case, the loss variable is
begin{eqnarray*}
L_h &=&
left{
begin{array}{cc}
v^{K+1-h} & h leq K < n \
v^{n-h} & K ge n
end{array}
right .
end{eqnarray*}
This has expectation, or policy value, (_h V = A_{x+h:overline{n-h|}} ).
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). The loss variable is
begin{eqnarray*}
L_h &=&
left{
begin{array}{cc}
0 & 0 leq K < h \
b_{K+1} v^{K+1-h} - sum_{j=h}^K P_j v^{j-h} & K geq h \
end{array}
right .
end{eqnarray*}
Using a summation by parts, this has expectation
begin{eqnarray*}
_h V &=& sum_{j=0}^{infty}
left(b_{h+j+1} v^{j+1} - sum_{k=0}^y P_{h+k} v^k right) ~_{j|} q_{x+h} \
&=& sum_{j=0}^{infty}b_{h+j+1} v^{j+1} ~_{j|} q_{x+h}
- sum_{j=0}^{infty} P_{h+j} v^j ~_j p_{x+h} \
&=& E (PVFB) - E(PVFP) .
end{eqnarray*}
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