Question

Eleanor is a divorce attorney who practices law in Dallas. She 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 8.5%. What is the minimum number of years that Eleanor 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.)

Answer 1

Answer:

15 years

Explanation:

Future value of Lumpsum = PV (1 + i)^n

PV = Present value = $4,500  

i = interest rate = 8.5%

n = no. of compounding period = 15 years

So, FV = 4,500 * (1 + .085)^15

= 4,500 * 3.3997428788

= $15,298.84

When Eleanor choose to pay $500 for annual membership fee:

Future Value of annuity due = (1 + r) * P[((1 + r)n - 1) / r]

where P = Periodic payment = $500

i = interest rate = 8.5%

n = no. of compuding period = 15 years

So, FV of annuity due = (1 + .085) * 500[((1 + .085)^15 - 1) / .085]

= (1.085) * 500[(3.3997428788 - 1) / .085]

= (1.085) * 500 * 28.2322691624

= $15,316.01

So, within 15 years lifetime membership is cheaper than annual membership.