Question $3(14$ marks$)$

You won a lottery four years ago and the prize was arranged to be paid out on a perpetrity basis, with yearly payments of $$1m$ starting from the day of winning the lottery.

$($a$)$ On the day of winning the lottery $($i$.$e$.$ four years ago$),$ you asked yourself: "How much wealthier have I become $($today$)$ because of this prize?". Calculate the answer to your question if the relevant yearly interest rate is $4\%.$

$(4$ marks$)$

$($b$)$ If the prize could be arranged to be paid out in another way as below:

"A lump$-$sum of $$5m$ on the day of winning the lottery $$x$ every year on a perpetrity basis beginning $3$ years from the day of winning the lottery."

What value of $x$ would make the Present Value of the prize to be $$30m?$ Assume the relevant interest rate is fixed at $4\%$ per year $($as in part $($a$)).$ Support your answer with calculations. $(6$ marks$)$

$($a$)$ Continued with $($a$).$ Four years later $($i$.$e$.$ today$)($immediately before you receive the yearly SIm payment$),$ what would be your answer to the same question as in part $($a$)?$ Support your answer with calculations.

$(4$ marks$)$