Actuarial Science: Life Contigencie I. Do not copy others or else THUMB DOWN $!$

An individual decides to start saving for retirement. They want to purchase an annuity of $$4,000$ per month starting at age $65.$ Assume mortality follows the Standard Ultimate Life Table and $$ i $=5\%,$ and apply the Woolhouse formula in $3$ terms as required.

a$)$ Assume they purchase this policy at age $45.$ What is the monthly premium payable from age $45$ to retirement as calculated by the equivalence principle?

b$)$ Assume instead that they purchase this policy at age $25.$ What is the monthly premium payable from age $25$ to retirement as calculated by the equivalence principle?

c$)$ Explain in words why the magnitude of change in premiums is greater than the magnitude of change in the number of premiums, and what the implications of this would be on an educated investor. $($Woolhouse formula in $3$ terms is provided, with m $=12$ in this problem$)$