Policy Valuation
It is common to assume that benefits are payable at the moment of failure and that premiums are payable at the beginning of the (m)thly period (e.g., (m =1, 2, mbox{or } 4)).
Traditional actuarial symbols are based on discrete annual cash flows because mortality rates are available no more frequent than annually.
The “UDD” assumption means uniform distribution of deaths within a year. This is not the same as the DeMoivre assumption which is uniform over the lifetime of an individual.
Recall earlier relations. For a whole life policy with a benefit payable at the end of the (m)thly period, the expected present value is
begin{eqnarray*}
A_x^{(m)} = frac{i}{i^{(m)}} A_x,
end{eqnarray*}
where (i^{(m)}) is the (m)thly nominal interest rate determined by (1+i= left(1+frac{i^{(m)}}{m}right)^m). Note that this is an (exact) relationship under the UDD assumption. As (m rightarrow infty), this yields
begin{eqnarray*}
bar{A}_x = frac{i}{delta} A_x .
end{eqnarray*}
Further,
begin{eqnarray*}
ddot{a}_x^{(m)} = alpha(m) ddot{a}_x – beta(m),
end{eqnarray*}
where
begin{eqnarray*}
alpha(m) = frac{id}{i^{(m)} d^{(m)}} textrm{ and } beta(m) = frac{i-i^{(m)}}{i^{(m)} d^{(m)}},
end{eqnarray*}
and (d^{(m)}) is the (m)thly nominal discount rate determined by (1-d= left(1-frac{d^{(m)}}{m}right)^m).
As (m rightarrow infty), this yields
begin{eqnarray*}
bar{a}_x^{(m)} = alpha(infty) ddot{a}_x – beta(infty),
end{eqnarray*}
where
begin{eqnarray*}
alpha(infty) = frac{id}{delta^2} textrm{ and } beta(infty) = frac{i-delta}{delta^2} .
end{eqnarray*}
For approximations, we often use
begin{eqnarray*}
alpha(m) approx 1 textrm{ and } beta(m) approx frac{m-1}{2m} .
end{eqnarray*}
With these assumptions and approximations, one can readily handle common traditional policies.