An investment fund with initial value F$0=1$ reported effective annual yields over the past

$5$ years as $9\%,14\%,10\%$ for years $12,3,45$ respectively. During this $5-$year period, the

fund received new contributions of $(0\mathrm{.}4,0\mathrm{.}2,0\mathrm{.}5)$ at times $(1,2,4)($in years from the start

of the $5-$year period$)$ and paid out withdrawals of $(0\mathrm{.}3,0\mathrm{.}2)$ at times $(2,4)$ respectively.

$($a$)$ Calculate the fund values just before contributions and withdrawals.

$($b$)$ Find the following rates of return achieved by the fund over the $5-$year period:

$$ the linked internal rate of return $($LIRR$),$

$$ the money weighted rate of return $($MWRR$),$ and

$$ the time weighted rate of return $($TWRR$).$