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
begin{eqnarray*}
textrm{Var}(LIAB(textbf{i})) &=&
textrm{E}[textrm{Var}(LIAB(textbf{i})|textbf{i})]+
textrm{Var}[textrm{E}(LIAB(textbf{i})|textbf{i})] .
end{eqnarray*}
For the first term, conditional on the interest environment i, the policies are (i.i.d). Thus,
begin{eqnarray*}
textrm{Var}(LIAB(textbf{i})|textbf{i}) &=& N^{ast} textrm{Var}(~_1 L(textbf{i})|textbf{i})
end{eqnarray*}
and so
begin{eqnarray*}
textrm{E}[textrm{Var}(LIAB(textbf{i})|textbf{i})] &=& N^{ast} textrm{E}[textrm{Var}(~_1 L(textbf{i})|textbf{i})] \
&=& N^{ast}sum_{j=1}^3 textrm{Var}(_1 L(textbf{i})|textbf{i}) times Pr(textbf{i}=i_s) \
&=& N^{ast}left{(0.25)(0.042) +(0.50)(0.034)+ (0.25)(0.029) right}\
&=& 0.0349 N^{ast}
end{eqnarray*}
For the second term, conditional on the interest environment i, we have
begin{eqnarray*}
textrm{E}(LIAB(textbf{i})|textbf{i}) &=& N^{ast} textrm{E}(~_1 L(textbf{i})|textbf{i})
end{eqnarray*}
and so
begin{eqnarray*}
textrm{Var}[textrm{E}(LIAB(textbf{i})|textbf{i})] &=& N^{ast2} textrm{Var}[textrm{E}(~_1 L(textbf{i})|textbf{i})] \
&=& N^{ast2} left{(0.25)(0.049-0.014)^2right.\
& ~~~~~~~~ +& left.(0.50)(0.010-0.014)^2 right.\
& ~~~~~~~~ +& left.(0.25)(-0.015-0.014)^2 right} \
&=& 0.0005 N^{ast2}
end{eqnarray*}
Summarizing, we have
begin{eqnarray*}
textrm{Var}(LIAB(textbf{i})) &=& 0.0349 N^{ast}+ 0.0005 N^{ast2} .
end{eqnarray*}
Thus,
begin{eqnarray*}
lim_{Nrightarrow infty} frac{sqrt{textrm{Var}(LIAB(textbf{i}))}}{N} & = &lim_{Nrightarrow infty}
frac{sqrt{0.0349 N^{ast}+ 0.0005 N^{ast2}}}{N}\
& = & sqrt{0.0005} q_{40} > 0.
end{eqnarray*}
We interpret this to mean that this risk is not diversifiable due to a random interest environment that is common to all policies.