$($a$)$ An investment is worth $\backslash (200\backslash )$ after $6$ months under the annual interest rate $\backslash (4\backslash \%\backslash )$ compounded quarterly. What was the initial investment amount? $(4$ marks$)$

$($b$)$ An investment of $\backslash (800\backslash )$ is made into an account paying $\backslash (6\backslash \%\backslash )$ monthly interest. How long will it take for its value to grow to at least $\backslash (900\backslash )?(4$ marks$)$

$($c$)$ A hedge fund claims that they can guarantee $\backslash (1\backslash \%\backslash )$ monthly rate of return. Assuming compounding effect, what will be the equivalent annual rate of return? $(5$ marks$)$

$($d$)($i$)$ How is logarithmic return on an investment calculated?

$($ii$)$ Suppose that the logarithmic return over $3$ months on investment subject to continuous compounding is $\backslash (3\backslash \%\backslash ).$ Find the interest rate. $(6$ marks$)$

$($e$)$ A second$-$hand car worth $\backslash (1500\backslash )$ is to be paid off in $2$ equal payments at the end of each year. If the loan interest rate is $\backslash (5\backslash \%\backslash )$ compounded annually, calculate the value of the annual payments? $(6$ marks$)$