Kelly wants to retire in $25$ years. At retirement she wants to be able to withdraw $$100,000$ at the end of each year forever $($she plans on establishing a scholarship fund at her local university after her death$).$ Assuming that her investments can earn $8\%$ compounded semi$-$annually prior to her retirement and only $4\%$ compounded annually after her retirement $($retired people and universities are very conservative investors$),$ how much must Rosie invest each year for the next $25$ years? Assume her first deposit will occur in one year.

Answer: Value required at retirement $=$ PV of perpetuity of $$100,000$ at $4\%=$$$2,500,000$

Need an annuity with $25$ payments with future value of $$2\mathrm{.}5$ million. Interest rate $in=25,FV=$$$2\mathrm{.}5$ million, $I=8\mathrm{.}16,$ solve for PMT $=$$$33,406\mathrm{.}02$

Rosie will need to deposit $$33,406\mathrm{.}02$ every year for the next $25$ years to fund her retirement and scholarship.