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.<\/p>\n
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\n\\begin{eqnarray*}\n_k V&=& v q_{x+k} b_{k+1} - P_k + v p_{x+k} ~_{k+1} V.\n\\end{eqnarray*}\nNow, for \\(0< s< 1\\), we define the policy value at fractional duration to be\n\\begin{eqnarray*}\n_{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 .\n\\end{eqnarray*}\nTo evaluate this, under UDD we have\n\\begin{eqnarray*}\n~_{1-s} q_{x+k+s} &=& \\frac{(1-s) q_{x+k}}{1- s \\times q_{x+k}}\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n~_{1-s} p_{x+k+s} &=& \\frac{p_{x+k}}{1- s \\times q_{x+k}} .\n\\end{eqnarray*}\nThus,\n\\begin{eqnarray*}\n_{k+s} V &=& \\frac{v^{1-s}}{1- s \\times q_{x+k}}\n\\left( (1-s) \\left\\{q_{x+k} b_{k+1}\\right\\} + p_{x+k} ~_{k+1} V \\right) \\\\\n&=& \\frac{v^{1-s}}{1- s \\times q_{x+k}}\n\\left( (1-s)\\left\\{ (P_k +V_k)(1+i) - p_{x+k} ~_{k+1} V\n\\right\\} + p_{x+k} ~_{k+1} V \\right) \\\\\n&=& \\frac{v^{1-s}}{1- s \\times q_{x+k}}\n\\left( (1-s) (1+i)(P_k +~_k V) +s \\times p_{x+k} ~_{k+1} V \\right) \\\\\n&\\approx &\n(1-s) (P_k +~_k V) +s \\times ~_{k+1} V.\n\\end{eqnarray*}\nThis 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.\\)\n\nIt is common to write this approximation as\n\n\n
| \\(~_{k+s} V =\\)<\/td>\n | \\((1-s) ~_k V +s ~_{k+1} V\\)<\/td>\n | +\\((1-s) P_k\\)<\/td>\n<\/tr>\n |
| reserve<\/font><\/td>\n | interpolate terminal reserves \n\t\t\t<\/font><\/td>\n | unearned premiums<\/font><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n 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.<\/p>\n Example.<\/strong> (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 |