Question $2(13$ marks$)$

Godfrey has just recovered from a serious illness and is planning to start his early retirement at the age of $55($i$.$e$.,$ start the retirement right after his ${55}^{th}$ birthday$).$

According to Godfrey's calculations, he would need an average of $$40,000$ per month for the living expenses during the retirement years. The monthly withdrawal will be made at the beginning of each $($retirement$)$ month. Godfrey would also like to leave his teenage daughter $$20m$ which she would be able to inherit in $10$ years after his retirement.

The retirement plan Godfrey purchases will provide a fixed interest return of $12\%,$ compounded monthly.

a$)$ How much money $($the "target amount"$)$ will Godfrey need to have in his retirement savings account at his ${55}^{th}$ birthday so that he meet all expected expenses if he expects a $15-$year retirement life?

$(5$ marks$)$

To meet the "target amount", the retirement plan provides Godfrey with $2$ choices$/$plans:

Plan $(1)$: make a single deposit of $$4,086,356\mathrm{.}21$ into the retirement plan account $($that earns a fixed interest return of $12\%,$ compounded monthly, as stated above$)$; or

Plan $(2)$: make $72$ month$-$end deposits of $$50,000$ each into the retirement plan account $($the last of which will be made exactly at his ${55}^{th}$ birthday$)$ PLUS one single payment at his ${55}^{th}$ birthday.

b$)$ If Godfrey chooses Plan $(1),$ when at the latest $($in terms of age$)$ will be required to make the lump$-$sum deposit in order to have enough money to meet the retirement expenses? $(4$ marks$)$

c$)$ If Godfrey chooses Plan $(2),$ what is the size of the single payment that he will be required to make at his ${55}^{th}$ birthday in order to have enough money to meet the retirement expenses? $(4$ marks$)$