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 7.9%. 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 $750 in annual membership dues? (Note: Round your answer up to the nearest year.)

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 $800 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 $8,500 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 7.9%. 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 $800 in annual membership dues?

A. 14 years

B. 13 years

C. 19 years

D. 23 years

19 years (Kindly note the above question is bit different in numbers because I was unable to find the remainder part of your question. But it provides excellent understanding of the question).

Explanation:

Here, we have four options and we will start finding "Advance Annuity" from the least number which is 13% and then we move to a greater number.

Best Choice:

The best choice will be the one which is almost equal to or smaller than the value of the lifetime membership $8,500 and takes least time.

For 13 Years

Annuity = Cashflow * (1 + Annuity Factor at 7.6%)

Annuity at 7.6% for 13 years = (1 -  (1 + 7.6%)^-(13 - 1) / 7.6% = 7.6948

Annuity = $800 * (1 + 7.6948) =  $6,956

Similarly for 14 Years:

Advance Annuity = Cashflow * (1 + Annuity Factor at 7.6%)

Advance Annuity at 7.6% for 14 years = (1 -  (1 + 7.9%)^-(14 - 1) / 7.6% = 8.0807

Advance Annuity = $800 * (1 + 8.0807) = $7265

Similarly for 19 Years:

Advance Annuity = Cashflow * (1 + Annuity Factor at 7.6%)

Advance Annuity at 7.6% for 19 years = (1 -  (1 + 7.9%)^-(19 - 1) / 7.6% = 9.6377

Advance Annuity = $800 * (1 + 9.6377) = $8510.16

As the annuity value slightly crosses $8500, hence it is somewhat between 18 to 19 years. Thatswhy we will not consider the last option with 23 years as the nearest option is 19 years.

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