Question

1. 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?

2. You are considering buying a car worth $30,000. The dealer, who is anxious to sell the car, offers you an attractive financing package. You have to make a down-payment of $3,500, and pay the rest over 3 years with monthly payments. The dealer will charge you interest at a constant APR of 2%, which is lower than the market interest rate.

   1. (1) Whatisthemonthlypaymenttothedealer?

   2. (2) Thedealeroffersyouasecondoption:youpaycash,butgeta$2,500discount.Shouldyougofor

      the loan or should you pay cash? Assume that the market annual interest rate (APR) is at 5%.

   3. A foundation announces that it will be offering one CUHK (SZ) scholarship every year for an indefinite number of years. The first scholarship is to be offered exactly one year from now. When the scholarship is offered, the student will receive ¥100,000 annually for a period of four years, beginning from the date the scholarship is offered. This student is then expected to repay the principal amount received (¥400,000) in 10 equal annual installments, interest-free, starting two years after the last payment of the scholarship. This implies that the foundation is really giving an interest-free loan under the guise of a scholarship. The current interest is 6% and is expected to remain unchanged.

      1. (1) What is the PV of the first scholarship (the scholarship includes both money given out to and the

         repayments received from the student)?

      2. (2) The foundation invests a lump sum to fund all future scholarships. Determine the size of the

         investment today.

Answer 1

You have asked multiple unrelated questions in the same post. Further, some of your questions have multiple sub parts. I have therefore addressed the first question in entirety. Please post the balance questions, separately.

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?

The PV of a growing annuity is given by, PV =

Hence, PV = 750 / (12% - 5%) x [1 - (1.05 / 1.12)ⁿ] = 10,714.29 x (1 - 0.9375ⁿ)

In order to justify taking out the life membership, the PV of annual fees = PV of life membership = $ 7,500

Hence, (1 - 0.9375ⁿ) = 7500 / 10,714.29 = 0.7

Hence, 0.9375ⁿ = 1 - 0.7 = 0.3

Take log to the base e on both the sides,

n ln (0.9375) = ln (0.3)

Hence, n = ln (0.3) / ln (0.9375) = 18.66 years

Hence, the minimum life expectancy = n = 18.66 years (it may be rounded off to 19 years, if required) from now.