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 8.5%. 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 $500 in annual membership dues? (Note: Round your answer up to the nearest year.)

Within 15 years, lifetime membership is cheaper than annual membership.

Explanation:

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Considering if Lloyd choose to pay $4,500 for lifetime membership fee, then

Future value of Lump sum = PV (1 + i)^n  

Where PV = Present value = $4,500

, i = interest rate = 8.5%

, n = no. of compounding period = 15 years  

So, FV = 4,500 * (1 + .085)^15

= 4,500 * 3.3997428788

= $15,298.84

When Lloyd choose to pay $500 for annual membership fee:

Future Value of annuity due = (1 + r) * P[((1 + r)^n - 1) / r]

where P = Periodic payment = $500

i = interest rate = 8.5%

n = no. of compounding period = 15 years

So, FV of annuity due = (1 + .085) * 500[((1 + .085)^15 - 1) / .085]

= (1.085) * 500[(3.3997428788 - 1) / .085]

= (1.085) * 500 * 28.2322691624

= $15,316.01

So, within 15 years lifetime membership is cheaper than annual membership.

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