Question

In: Finance

Paul Adams owns a health club in downtown Los Angeles. He charges his customers an annual...

Paul Adams owns a health club in downtown Los Angeles. He charges his customers an annual fee of $940 and has an existing customer base of 490. Paul plans to raise the annual fee by 7 percent every year and expects the club membership to grow at a constant rate of 3 percent for the next five years. The overall expenses of running the health club are $114,000 a year and are expected to grow at the inflation rate of 4 percent annually. After five years, Paul plans to buy a luxury boat for $400,000, close the health club, and travel the world in his boat for the rest of his life. Assume Paul has a remaining life of 25 years and earns 10 percent on his savings.

How much will Paul have in his savings on the day he starts his world tour assuming he has already paid for his boat? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) Account value at retirement $

What is the annual amount that Paul can spend while on his world tour if he will have no money left in the bank when he dies? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) Annual withdrawal $

Solutions

Expert Solution

First Part:

Assumption: All savings take place at the end of year.

Year, n Linkage 1 2 3 4 5
y-o-y growth in
Annual membership fees 7% 7% 7% 7%
Customer base 3% 3% 3% 3%
Expenses 4% 4% 4% 4%
Annual membership fees A                     940                1,006                 1,076              1,152               1,232
[x] Customer base B                     490                   505                    520                 535                  551
Revenue C = A x B              460,600           507,627            559,456         616,576          679,529
[-] Expenses D              114,000           118,560            123,302         128,234          133,364
Savings E = C - D              346,600           389,067            436,154         488,342          546,165
Interest rate F 10%
FV factor FVF = (1+F)^(5-n)                1.4641             1.3310              1.2100           1.1000            1.0000
FV of savings FV = FVF x E              507,457           517,849            527,746         537,176          546,165
Total FV of Savings sum of all FV          2,636,392.65

  Paul have $ 2,636,392.65 in his savings on the day he starts his world tour assuming he has already paid for his boat.

Second part

Assumption: All withdrawals take place at the end of year.

Let A be the annual amount that Paul can spend while on his world tour if he will have no money left in the bank when he dies.

Hence, PV of annuity A over 25 - 5 = 20 years should be equal to $ 2,636,392.65

Hence, A / i x [1 - (1 + i)-n] = A / 10% x [1 - (1 + 10%)-20] =  8.5136 x A = $ 2,636,392.65

Hence, A =  $ 2,636,392.65 / 8.5136 = $  309,669.69


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