Mortality is not completely predictable. How important is that to the replication argument? Does approximate matching of cash flows help the insurer?
Diversifiable risks in a portfolio.
Consider (N) risks in a portfolio, (X_1, ldots, X_N). If the variance of the average becomes small sufficiently fast (as the portfolio size grows), then the risks are said to be diversifiable. Here is one condition to ensure that the variability of the average becomes small:
begin{eqnarray*}
lim_{Nrightarrow infty} frac{sqrt{textrm{Var}left( sum_{i=1}^N X_i right)}}{N} = lim_{Nrightarrow infty} sqrt{textrm{Var}left( bar{X} right)} = 0.
end{eqnarray*}
Non-diversifiable risks – Risks that do not meet the condition to be diversifiable.
To assess whether a portfolio is diversifiable, it is often helpful to decompose the variance into components. To illustrate, we show how to decompose the variance of the loss (L) into two components based on an interest environment (textbf{i}). The first component, (textrm{E}[textrm{Var}(L|textbf{i})]), is the risk due to uncertainty over future lifetimes; the second, (textrm{Var}(textrm{E}[L|textbf{i}])), is due to uncertainty over interest rates.
begin{eqnarray*}
textrm{Var}(L) = textrm{E}[textrm{Var}(L|textbf{i})] + textrm{Var}[textrm{E}(L|textbf{i})]
end{eqnarray*}