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 $650 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 $7,000 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 5.9%. 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 $650 in annual membership dues? (Note: Round your answer up to the nearest year.)

Answer 1

16 years

Step-by-step explanation:

we can use the present value of an annuity due formula:

PV of annuity due = payment + {payment x [1 - (1 + r)ⁿ⁻¹]/r}

• present value = 7,000
• payment = 650
• r = 0.059

7,000 = 650 + {650 x [1 - (1 + 0.059)ⁿ⁻¹]/0.059}

6,350 = 650 x [1 - (1 + 0.059)ⁿ⁻¹]/0.059

6,350 / 650 = 9.769230769 = [1 - (1 + 0.059)ⁿ⁻¹]/0.059

9.769230769 x 0.059 = [1 - (1 + 0.059)ⁿ⁻¹]

0.576384615 = 1 - (1 + 0.059)ⁿ⁻¹

(1 + 0.059)ⁿ⁻¹ = 0.423615384

n - 1 = log 0.423615384 / log 1.059 = 0.373028275 / 0.02489596 = 14.98336571

n = 14.98336571 + 1 = 15.98336571

if an attorney remains am ember for at least 16 years (16 ≥ 15.98336571), then the lifetime membership fee is better