Question

The annual membership fee at your health club is $750 a year and is expected to increase at 5% per year. A life membership is $7,500 and the discount rate is 12%. In order to justify taking out the life membership, what would be your minimum life expectancy?

Answer 1

We assume that First annual membership fee will be paid in one year from now i.e at t₁ and will be equal to $750

Second annual membership fee will be paid in two years from now i.e. at t₂ and will be equal to $750(1+5%) = 750 x 1.05 = $787.50

Since it is given that annual membership fee increase annually by 5%, therefore they form a growing ordinary annuity such that, First Annual membership Fee = A = $750, Expected increase in annual membership fee = g = 5% per year

Life membership Fee = $7500, Discount rate = r = 12%

Minimum life expectancy will be equal to number of years for which the present value of growing annuity will be equal life membership fee.

Let = Minimum life expectancy = n years

We know that

Present value of Growing Annuity = Life membership Fee = [A/(r-g)] [1 - {(1+g)/(1+r)}ⁿ]

7500 = [750/(12%-5%)] [1 - {(1+5%) / (1+12%)}ⁿ]

7500 = [750/7%] [1 - {(1.05) / (1.12)}ⁿ]

(7500 x 7%) / 750 = 1 - {(1.05) / (1.12)}ⁿ

0.70 = 1 - {(1.05) / (1.12)}ⁿ

{(1.05) / (1.12)}ⁿ = 1 - 0.70

{0.9375}ⁿ = 0.30

Taking natural log both the sides

Ln {0.9375}ⁿ = Ln 0.30

n x Ln 0.9375 = Ln 0.30                                                         [ Using log identity Ln a^b = b x Ln a ]

n x -0.064538 = -1.203972

n = -1.203972 / -0.064538 = 18.6552 years = 19 years (rounded to next complete year because after 18th year or t₁₈, membership fee will be paid in 19th year or t₁₉)

Hence Minimum life expectancy = 19 years