Problem $2.($Continuous compounding, $11)$ Suppose you have $$1$ million and you can invest

it in a bank account that pays interests at an annual percentage rate $($APR$)$ of $5\%.$ You want

to invest your $$1$m for $3$ years.

$($a$).$ Suppose you are allowed to reinvest the interest at the end of every year $($i$.$e$.,$ annual

compounding$).$ What is the balance of your account at the end of year $3?(2)$

$($b$).$ Suppose you are allowed to reinvest the interest at the end of every month $($i$.$e$.,$ monthly

compounding$).$ What is the balance of your account at the end of year $3?(2)$

$($c$).$ Suppose you are allowed to reinvest the interest at the end of every day $($i$.$e$.,$ daily

compounding$).$ What is the balance of your account at the end of year $3?($Assume there are

$365$ days in a year.$)(2)$

$($d$).$ Suppose you are allowed to reinvest the interest at the end of every hour $($i$.$e$.,$ hourly

compounding$).$ What is the balance of your account at the end of year $3?($Assume there are

$365$ days in a year and $24$ hours in a day.$)(2)$

$($e$).$ The formula of continuous compounding is

lim

$!->$# $$1+$

$)!\backslash$times $\%$

$=$ &$\%,$

where $$ denotes the compounding frequency and $$ is the number of years. Use the

formula of continuous compounding to calculate the balance of your account at the end of

year $3.$ Compare the answers in $(1)-(5).$ What do you conclude? $(3)$