The time value of money concept can be applied in various situations and is a fundamental concept underlying other financial concepts.

Consider the following example of the application of this concept.

Manuel is a divorce attorney who practices law in New York City. He wants to join the American Divorce Lawyers Association $($ADLA$),$ a professional organization for divorce attorneys. The membership dues for the ADLA are $$500$ per year and must be paid at the beginning of each year. For instance, membership dues for the first year are paid today, and dues for the second year are payable one year from today. However, the ADLA also has an option for members to buy a lifetime membership today for $$4,500$ and never have to pay annual membership dues.

Obviously, the lifetime membership isn$$t a good deal if you only remain a member for a couple of years, but if you remain a member for $40$ years, it$$s a great deal. Suppose that the appropriate annual interest rate is $8\mathrm{.}5\%.$

What is the minimum number of years that Manuel must remain a member of the ADLA so that the lifetime membership is cheaper $($on a present value basis$)$ than paying $$500$ in annual membership dues? $($Note: Round your answer up to the nearest year.$)$

$13$ years

$15$ years

$19$ years

$20$ years

In $1626,$ Dutchman Peter Minuit purchased Manhattan Island from a local Native American tribe. Historians estimate that the price he paid for the island was about $$24$ worth of goods, including beads, trinkets, cloth, kettles, and axe heads. Many people find it laughable that Manhattan Island would be sold for $$24,$ but you need to consider the future value $($FV$)$ of that price in more current times.

If the $$24$ purchase price could have been invested at a $5\mathrm{.}75\%$ annual interest rate, what is its value as of $2017(391$ years later$)?$

$$63,570,730,476\mathrm{.}21$

$$74,789,094,677\mathrm{.}89$

$$86,007,458,879\mathrm{.}57$

$$98,721,604,974\mathrm{.}81$