IV$.$ You want to save $$2,000$ today for retirement in $40$ years. You have to choose between the two plans listed in $($i$)$ and $($ii$).$

$($i$)$ Pay no taxes today, put the money in an interest$-$yielding account, and pay taxes equal to $20\%$ of the total amount withdrawn at retirement. $($In the U$.$S$.,$ such an account is known as a regular individual retirement account, or IRA.$)$

$($ii$)$ Pay taxes equivalent to $15\%$ of the investment amount today, put the remainder in an interest$-$yielding account, and pay no taxes when you withdraw your funds at retirement. $($In the U$.$S$.,$ this is known as a Roth IRA.$)$

a$.$ What is the expected present discounted value of each of these plans if the interest rate is $1\%?10\%?$

b$.$ Which plan would you choose in each case? Explain your logic clearly.