Paul Adams owns a health club in downtown Los Angeles. He charges his customers
ID: 2783157 • Letter: P
Question
Paul Adams owns a health club in downtown Los Angeles. He charges his customers an annual fee of $700 and has an existing customer base of 640. 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 $129,000 a year and are expected to grow at the inflation rate of 3 percent annually. After five years, Paul plans to buy a luxury boat for $540,000, 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 8 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?
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?
Annual withdrawal $
Explanation / Answer
Earnings for Present Year (Year 0):
Membership Fee = 700, Customer Base = 640, Expense = 129000
So, Revenue = (Membership Fee * Customer Base) - Expense = (700 * 640) - 129000 = 448000 - 129000 = 319000
We need to calculate the FV of revenue at 5th year. The interest rate is 8% or 0.08
FV of Revenue at the end of 5th Year = 319000 * (1 + 0.08)5 = 468715.66 [rounded to 2 decimals]
Earnings for Year 1:
Membership Fee = 700 * 1.07 = 749, Customer Base = 640 * 1.03 = 659, Expense = 129000* 1.03 = 132870 [Customer Base is rounded off to last whole number as customers can be fraction]
So, Revenue = (Membership Fee * Customer Base) - Expense = (749 * 659) - 132870 = 493591 - 132870 = 360721
We need to calculate the FV of revenue at 5th year. The interest rate is 8% or 0.08
FV of Revenue at the end of 5th Year = 360721 * (1 + 0.08)4 = 490756.94 [rounded to 2 decimals]
Earnings for Year 2:
Membership Fee = 749 * 1.07 = 801.43, Customer Base = 659 * 1.03 = 678, Expense = 13287* 1.03 = 136856.10 [All calculations are rounded to 2 decimals & Customer Base is rounded off to last whole number as customers can be fraction]
So, Revenue = (Membership Fee * Customer Base) - Expense = (801.43 * 678) - 136856.10 = 543369.54 - 136856.10 = 406513.44
We need to calculate the FV of revenue at 5th year. The interest rate is 8% or 0.08
FV of Revenue at the end of 5th Year = 406513.44 * (1 + 0.08)3 = 512089.86 [rounded to 2 decimals]
Earnings for Year 3:
Membership Fee = 801.43 * 1.07 = 857.53, Customer Base = 678 * 1.03 = 698, Expense = 136856.10* 1.03 = 140961.78 [All calculations are rounded to 2 decimals & Customer Base is rounded off to last whole number as customers can be fraction]
So, Revenue = (Membership Fee * Customer Base) - Expense = (857.53 * 698) - 140961.78 = 598555.94 - 140961.78 = 457594.16
We need to calculate the FV of revenue at 5th year. The interest rate is 8% or 0.08
FV of Revenue at the end of 5th Year = 457594.16 * (1 + 0.08)2 = 533737.82 [rounded to 2 decimals]
Earnings for Year 4:
Membership Fee = 857.53 * 1.07 = 917.56, Customer Base = 698 * 1.03 = 718, Expense = 140961.78* 1.03 = 145190.64 [All calculations are rounded to 2 decimals & Customer Base is rounded off to last whole number as customers can be fraction]
So, Revenue = (Membership Fee * Customer Base) - Expense = (917.56 * 718) - 145190.64 = 658808.08 - 145190.64 = 513617.44
We need to calculate the FV of revenue at 5th year. The interest rate is 8% or 0.08
FV of Revenue at the end of 5th Year = 513617.44 * (1 + 0.08)1 = 554706.84 [rounded to 2 decimals]
Earnings for Year 5:
Membership Fee = 917.56 * 1.07 = 981.79, Customer Base = 718 * 1.03 = 739, Expense = 145190.64* 1.03 = 149546.36 [All calculations are rounded to 2 decimals & Customer Base is rounded off to last whole number as customers can be fraction]
So, Revenue = (Membership Fee * Customer Base) - Expense = (981.79 * 739) - 149546.36 = 725542.81 - 149546.36 = 575996.45
Payment for Boat at the end of Year 5 = 540000
Paul's savings on the day he starts his world tour assuming he has already paid for his boat = Sum of FV of earnings over year 0 to 4 + Earnings of Year 5 - Payment for Boat
= 468715.66 + 490756.94 + 512089.86 + 533737.82 + 554706.84 + 575996.45 - 540000 = 3136003.57 - 540000 = 2596003.57
Annual amount that Paul can pay can be calculated using PV of Annuity formula:
PV = C * ((1-1/(1+r)T)/r) where C = Annual Payment, r = Interest Rate, T = Number of periods
Here, PV = 2596003.57, r =8%, T = 25
So,
2596003.57 = C * ((1-1/(1+8%)25)/8%)
or, C = 2596003.57/((1-1/(1+8%)25)/8%) = 243190.44 [rounded to 2 decimals]
So,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 = 243190.44