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 $750 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 $8,000 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 7.9%. 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.)

A. 18 years

B. 17 years

C. 20 years

D. 15 years

Answer 1

I am going to do the sum in reverse calculation as per the given options
Year	Annual membership fee (A)	Present value factor @ 7.9% per annum (B)	Present Value of Annual membership fee (C = A*B)	Aggregate present value of membership fee paid at the end of each year (D)
1	$750	1.00	750.00	750.00
2	$750	0.93	695.09	1445.09
3	$750	0.86	644.20	2089.28
4	$750	0.80	597.03	2686.32
5	$750	0.74	553.32	3239.63
6	$750	0.68	512.81	3752.44
7	$750	0.63	475.26	4227.70
8	$750	0.59	440.46	4668.17
9	$750	0.54	408.22	5076.38
10	$750	0.50	378.33	5454.71
11	$750	0.47	350.63	5805.34
12	$750	0.43	324.96	6130.30
13	$750	0.40	301.16	6431.46
14	$750	0.37	279.11	6710.58
15	$750	0.34	258.68	6969.25
16	$750	0.32	239.74	7208.99
17	$750	0.30	222.19	7431.18
18	$750	0.27	205.92	7637.10
19	$750	0.25	190.84	7827.94
20	$750	0.24	176.87	8004.81
Present value factor is calculated based on the given information that Annual membership fee is paid at the beginning of every year
Aggregate present value of membership fee paid at the end of 18th year = $7637.10
Aggregate present value of membership fee paid at the end of 17th year = $7431.18
Aggregate present value of membership fee paid at the end of 20th year = $8004.81
Aggregate present value of membership fee paid at the end of 15th year = $6969.25
Since the value of lifetime membership fee if paid today is equal to $8000,
As per above information paying membership fee at the beginning of each year is better option upto
19 years, thereafter from 20th year it is better to take lifetime membership
Therefore answer is 20 years