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 find 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 membership dues, we need to compare the present value of the two options.
Let's calculate the present value of paying $750 in annual membership dues for a given number of years. We'll use the formula for the present value of an annuity:
PV = C * (1 - (1 + r)^(-n)) / r
Where PV is the present value, C is the annual payment, r is the interest rate, and n is the number of years.
In this case, C = $750, r = 8.9% = 0.089, and we'll let n be the number of years Lloyd remains a member.
Therefore, the present value of paying $750 per year for n years is:
PV_dues = $750 * (1 - (1 + 0.089)^(-n)) / 0.089
Next, we need to calculate the present value of the lifetime membership option. Since the lifetime membership fee is paid upfront, its present value is simply the fee itself.
PV_lifetime = $7,000
To determine the minimum number of years when the lifetime membership is cheaper, we set up the inequality:
PV_lifetime < PV_dues
$7,000 < $750 * (1 - (1 + 0.089)^(-n)) / 0.089
To solve this inequality, we'll use trial and error or an iterative method. However, to save time, we can make an educated guess based on the given scenario.
If Lloyd remains a member for 40 years, the annual dues payments would add up to $750 * 40 = $30,000. In this case, the present value of the dues would be:
PV_dues = $750 * (1 - (1 + 0.089)^(-40)) / 0.089 = $8,689.51
Since $7,000 (present value of the lifetime membership fee) is less than $8,689.51 (present value of dues for 40 years), the lifetime membership is cheaper if Lloyd remains a member for 40 years or more.
Therefore, the minimum number of years Lloyd must remain a member of the ADLA for the lifetime membership to be cheaper is 40 years.
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 annual membership dues, we need to compare the present value of the lifetime membership to the present value of the annual dues.
Let's calculate the present value of the lifetime membership first.
Step 1: Calculate the present value of $750 per year for 40 years.
PV_annual = $750 * (1 - (1 + r)^-40) / r
PV_annual = $750 * (1 - (1 + 0.089)^-40) / 0.089
PV_annual ≈ $13,451.86
Step 2: Calculate the present value of the lifetime membership.
PV_lifetime = $7,000
Now, let's compare the two present values.
If PV_lifetime < PV_annual, then the lifetime membership is cheaper.
PV_lifetime < PV_annual
$7,000 < $13,451.86
The lifetime membership is cheaper than paying annual dues after the minimum number of years required for it to become cheaper. To find this minimum number of years, we can use the formula to calculate the number of years (n) required for the two present values to be equal.
PV_annual = $750 * (1 - (1 + r)^-n) / r
PV_annual = $7,000
$750 * (1 - (1 + 0.089)^-n) / 0.089 = $7,000
Simplifying the equation:
(1 - (1 + 0.089)^-n) / 0.089 = $7,000 / $750
(1 - 1.089^-n) / 0.089 = 9.3333
1 - 1.089^-n = 0.8342
1.089^-n = 0.1658
Taking the natural logarithm (ln) of both sides:
ln(1.089^-n) = ln(0.1658)
-n * ln(1.089) = ln(0.1658)
n = ln(0.1658) / ln(1.089)
n ≈ 17.08
Lloyd must remain a member of the ADLA for at least 18 years (rounded up) to make the lifetime membership cheaper than paying $750 in annual membership dues on a present value basis.