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 $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 8.9%. 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?
To determine the minimum number of years that Lloyd must remain a member for the lifetime membership to be cheaper than paying annual membership dues, we need to compare the present values of the two options.
Option 1: Paying annual membership dues of $750 per year.
Option 2: Buying a lifetime membership for $7,000.
We can use the formula for calculating the present value of future cash flows:
PV = CF / (1 + r)^n
Where PV is the present value, CF is the cash flow, r is the annual interest rate, and n is the number of years.
For Option 1, the present value can be calculated as follows:
PV1 = $750 / (1 + 0.089)^1 + $750 / (1 + 0.089)^2 + ...
For Option 2, the present value is simply the cost of the lifetime membership:
PV2 = $7,000
To determine the minimum number of years for Option 2 to be cheaper, we need to find the point at which the present value of Option 1 exceeds the present value of Option 2.
Let's calculate the present values for Option 1 for different numbers of years:
PV1 (1 year) = $750 / (1 + 0.089)^1 = $692.71
PV1 (2 years) = $750 / (1 + 0.089)^1 + $750 / (1 + 0.089)^2 = $1,356.50
It's clear that the present value of Option 1 will continue to increase with each additional year. So, we need to find the minimum number of years for the present value of Option 1 to exceed the present value of Option 2.
PV1 (40 years) = $750 / (1 + 0.089)^1 + $750 / (1 + 0.089)^2 + ... + $750 / (1 + 0.089)^40 = $9,171.51
Since the present value of Option 1 after 40 years is $9,171.51, which is greater than the present value of Option 2 ($7,000), Lloyd must remain a member of the ADLA for at least 40 years for the lifetime membership to be cheaper than paying annual membership dues.
To determine the minimum number of years that Lloyd must remain a member of the ADLA for the lifetime membership to be cheaper than paying $750 in annual dues, we need to compare the present value of the two options.
Let's calculate the present value of the annual membership dues option first:
PV_annual_dues = $750 / (1 + 0.089)^1 (assuming the interest rate is compounded annually)
Now let's calculate the present value of the lifetime membership:
PV_lifetime_membership = $7,000 / (1 + 0.089)^40
To find the minimum number of years, we need to equate the two present values and solve for the number of years.
PV_annual_dues = PV_lifetime_membership
$750 / (1 + 0.089)^1 = $7,000 / (1 + 0.089)^40
Simplifying the equation:
$750 * (1 + 0.089)^40 = $7,000
Dividing both sides by $750:
(1 + 0.089)^40 = 7,000 / 750
Taking the 40th root of both sides:
1 + 0.089 = (7,000 / 750)^(1/40)
Approximately:
1.089 = 1.0734
Subtracting 1 from both sides:
0.089 = 0.0734
Hence, the lifetime membership is cheaper (on a present value basis) than paying $750 in annual membership dues when Lloyd remains a member for at least 40 years.