Question

Lloyd is a divorce attorney who practices law in Florida. He wants to join the American Divorce Lawyers Association (ADLA), a professional organization for divorce attorneys. The membership dues for the ADLA are $550 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 $5,000 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 7.5%. What is the minimum number of years that Lloyd must remain a member of the ADLA so that the lifetime membership is cheaper (on a present value basis) than paying $550 in annual membership dues? (Note: Round your answer up to the nearest year.)

21 years
18 years
14 years
12 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 4.5% annual interest rate, what is its value as of 2018 (392 years later)?

$987,122,447.84
$635,647,030.80
$859,993,041.68
$747,820,036.24

Answer 1

For the lifetime membership to be cheaper than paying $550 in annual membership dues, we need to calculate the present value of the lifetime membership and compare it to the present value of the stream of annual payments.

The present value of the lifetime membership can be calculated using the formula for the present value of a lump sum:

PV = FV / (1 + r)^n

where FV is the future value (which is the cost of the lifetime membership today, $5,000), r is the annual interest rate (7.5%), and n is the number of years for which we want to calculate the present value.

The present value of the stream of annual payments can be calculated using the formula for the present value of an annuity:

PV = A * [(1 - (1 + r)^-n) / r]

where A is the annual payment ($550), r is the annual interest rate (7.5%), and n is the number of years for which we want to calculate the present value.

We need to find the minimum number of years for which the present value of the lifetime membership is less than the present value of the stream of annual payments. We can do this by setting the two present values equal to each other and solving for n:

5000 / (1 + 0.075)^n = 550 * [(1 - (1 + 0.075)^-n) / 0.075]

Solving this equation gives n ≈ 18.7 years. Rounded up to the nearest year, this means that Lloyd must remain a member of the ADLA for at least 19 years for the lifetime membership to be cheaper than paying $550 in annual membership dues.

Therefore, the answer is 19 years.

For the second question, we can use the formula for the future value of a lump sum:

FV = PV * (1 + r)^n

where PV is the present value ($24), r is the annual interest rate (4.5%), and n is the number of years (392).

Substituting the given values, we get:

FV = 24 * (1 + 0.045)^392

FV ≈ $859,993,041.68

Therefore, the value of the $24 purchase price as of 2018 is approximately $859,993,041.68.