Question

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 7.4%. 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 $600 in annual membership dues? (Note: Round your answer up to the nearest year.)

Answer 1

Answer:

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 the annual membership dues is 23 years.

Explanation:

The question is incomplete.

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 $600 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 $6,500 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 7.4%. 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 $600 in annual membership dues?

We have to equal the price of the lifetime membership with the present value of the annual membership. The anual membership present value is equal to a annuityof n years.

$6,500=600*\sum_0^n\frac{1}{(1+0.074)^{n}} =600(\frac{1-(1.074)^{-n}}{0.074})\\\\\\\frac{6500}{600}*0.074= 1-(1.074)^{-n}\\\\1.074^{-n}=1-0.8017=0.1983\\\\-n*ln(1.074)=ln(0.1983)\\\\n=-ln(0.1983)/ln(1.074)=-(-1.6180)/0.0714=22.66\approx23$