To calculate the variability of obligations at time 1, we assume that first year expected mortality has held for the insurer and so there are \\(N(1-q_{40})=N^{\\ast}\\) policies outstanding. Let \\(LIAB(\\textbf{i}) = \\sum_{j=1}^{N*} ~_1 L(\\textbf{i})_j \\) be this sum of liabilities. For the second moment, from the “law of total variation” from probability theory, we have
\n\\begin{eqnarray*}
\n\\textrm{Var}(LIAB(\\textbf{i})) &=&
\n\\textrm{E}[\\textrm{Var}(LIAB(\\textbf{i})|\\textbf{i})]+
\n\\textrm{Var}[\\textrm{E}(LIAB(\\textbf{i})|\\textbf{i})] .
\n\\end{eqnarray*}
\nFor the first term, conditional on the interest environment i, the policies are (i.i.d). Thus,
\n\\begin{eqnarray*}
\n\\textrm{Var}(LIAB(\\textbf{i})|\\textbf{i}) &=& N^{\\ast} \\textrm{Var}(~_1 L(\\textbf{i})|\\textbf{i})
\n\\end{eqnarray*}
\nand so
\n\\begin{eqnarray*}
\n\\textrm{E}[\\textrm{Var}(LIAB(\\textbf{i})|\\textbf{i})] &=& N^{\\ast} \\textrm{E}[\\textrm{Var}(~_1 L(\\textbf{i})|\\textbf{i})] \\\\
\n&=& N^{\\ast}\\sum_{j=1}^3 \\textrm{Var}(_1 L(\\textbf{i})|\\textbf{i}) \\times Pr(\\textbf{i}=i_s) \\\\
\n&=& N^{\\ast}\\left\\{(0.25)(0.042) +(0.50)(0.034)+ (0.25)(0.029) \\right\\}\\\\
\n&=& 0.0349 N^{\\ast}
\n\\end{eqnarray*}
\nFor the second term, conditional on the interest environment i, we have
\n\\begin{eqnarray*}
\n\\textrm{E}(LIAB(\\textbf{i})|\\textbf{i}) &=& N^{\\ast} \\textrm{E}(~_1 L(\\textbf{i})|\\textbf{i})
\n\\end{eqnarray*}
\nand so
\n\\begin{eqnarray*}
\n\\textrm{Var}[\\textrm{E}(LIAB(\\textbf{i})|\\textbf{i})] &=& N^{\\ast2} \\textrm{Var}[\\textrm{E}(~_1 L(\\textbf{i})|\\textbf{i})] \\\\
\n&=& N^{\\ast2} \\left\\{(0.25)(0.049-0.014)^2\\right.\\\\
\n& ~~~~~~~~ +& \\left.(0.50)(0.010-0.014)^2 \\right.\\\\
\n& ~~~~~~~~ +& \\left.(0.25)(-0.015-0.014)^2 \\right\\} \\\\
\n&=& 0.0005 N^{\\ast2}
\n\\end{eqnarray*}
\nSummarizing, we have
\n\\begin{eqnarray*}
\n\\textrm{Var}(LIAB(\\textbf{i})) &=& 0.0349 N^{\\ast}+ 0.0005 N^{\\ast2} .
\n\\end{eqnarray*}
\nThus,
\n\\begin{eqnarray*}
\n\\lim_{N\\rightarrow \\infty} \\frac{\\sqrt{\\textrm{Var}(LIAB(\\textbf{i}))}}{N} & = &\\lim_{N\\rightarrow \\infty}
\n\\frac{\\sqrt{0.0349 N^{\\ast}+ 0.0005 N^{\\ast2}}}{N}\\\\
\n& = & \\sqrt{0.0005} q_{40} > 0.
\n\\end{eqnarray*}
\nWe interpret this to mean that this risk is not diversifiable due to a random interest environment that is common to all policies.<\/p>\n