An insurance company takes on contracts continuously throughout the year yet has a single valuation date (e.g., 1 July 20xx). Thus, even for traditional policies with annual cash flows, one needs the policy value when the duration includes a fraction of a year. Of course, for a fully continuous policy value, no special adjustments need be made.
Let (k) denote the integer duration time and (s) denote the fractional time, so that (0< s< 1) and the duration time is (k+s). For a fully discrete policy, as we have seen, the policy value can be expressed recursively as begin{eqnarray*} _k V&=& v q_{x+k} b_{k+1} - P_k + v p_{x+k} ~_{k+1} V. end{eqnarray*} Now, for (0< s< 1), we define the policy value at fractional duration to be begin{eqnarray*} _{k+s} V &=& v^{1-s} ~_{1-s} q_{x+k+s} ~b_{k+1} + v^{1-s} ~_{1-s} p_{x+k+s} ~_{k+1} V . end{eqnarray*} To evaluate this, under UDD we have begin{eqnarray*} ~_{1-s} q_{x+k+s} &=& frac{(1-s) q_{x+k}}{1- s times q_{x+k}} end{eqnarray*} and begin{eqnarray*} ~_{1-s} p_{x+k+s} &=& frac{p_{x+k}}{1- s times q_{x+k}} . end{eqnarray*} Thus, begin{eqnarray*} _{k+s} V &=& frac{v^{1-s}}{1- s times q_{x+k}} left( (1-s) left{q_{x+k} b_{k+1}right} + p_{x+k} ~_{k+1} V right) \ &=& frac{v^{1-s}}{1- s times q_{x+k}} left( (1-s)left{ (P_k +V_k)(1+i) - p_{x+k} ~_{k+1} V right} + p_{x+k} ~_{k+1} V right) \ &=& frac{v^{1-s}}{1- s times q_{x+k}} left( (1-s) (1+i)(P_k +~_k V) +s times p_{x+k} ~_{k+1} V right) \ &approx & (1-s) (P_k +~_k V) +s times ~_{k+1} V. end{eqnarray*} This approximation assumes a small mortality rate so that (q_{x+k} approx 0) and small interest rate so that (v^{-s} approx 1) and (v^{1-s} approx 1.) It is common to write this approximation as
| (~_{k+s} V =) | ((1-s) ~_k V +s ~_{k+1} V) | +((1-s) P_k) |
| reserve | interpolate terminal reserves |
unearned premiums |
where (_k V, ~_{k+1} V) are known as “terminal reserves” and ((1-s) P_k) is that portion of the annual premium that has been collected but “not earned’ by the valuation date.
Example. (Act Mat, p. 219) Consider a 5-year term policy to (50) with (P = 1000 P_{50:overline{5|}}^{~1} = 6.55692), (V_2 = 1.64), and (V_3 = 1.73). Then, we may compute the policy value at duration (k+s=2.25) as
begin{eqnarray*}
~_{2+0.25} V &approx &
(1-0.25) (P_2 + ~_2 V) +0.25 times ~_3 V \
&=& (1-0.25) (6.55692 +1.64) +0.25 times 1.73 = 6.58 .
end{eqnarray*}