Paul Adams owns a health club in downtown Los Angeles. He charges his customers
ID: 2700314 • Letter: P
Question
Paul Adams owns a health club in downtown Los Angeles. He charges his customers an annual fee of $550 and has an existing customer base of 700. Paul plans to raise the annual fee by 6.2 percent every year and expects the club membership to grow at a constant rate of 2.3 percent for the next five years. The overall expenses of running the health club are $73,400 a year and are expected to grow at the inflation rate of 2.4 percent annually. After five years, Paul plans to buy a luxury boat for $481,000, close the health club, and travel the world in his boat for the rest of his life.
What is the annual amount that Paul can spend while on his world tour if he will have no money left in the bank when he dies? Assume Paul has a remaining life of 17 years and earns 8.6 percent on his savings. (Do not include the dollar sign ($). Round your answer to 2 decimal places. (e.g., 32.16))
Paul Adams owns a health club in downtown Los Angeles. He charges his customers an annual fee of $550 and has an existing customer base of 700. Paul plans to raise the annual fee by 6.2 percent every year and expects the club membership to grow at a constant rate of 2.3 percent for the next five years. The overall expenses of running the health club are $73,400 a year and are expected to grow at the inflation rate of 2.4 percent annually. After five years, Paul plans to buy a luxury boat for $481,000, close the health club, and travel the world in his boat for the rest of his life.
Explanation / Answer
As Paul has remaining 25 yrs to live, 5 yrs are with the club & 20 yrs on boat.
1. Cash flows, no of customers, inc in fees, inc in exenses are shown in Table above.
2. Yearly CFs are invested at 9% pa. So CF of Y1 = 175000 will be invested for 5 yrs at 9%. So FV = PV*(1+i)^n = 175000*(1+9%)^5 = $269259.19.
CF of Y2 will be invested for 4 yrs & so on.
3. Sum of FV of these CFs is $2,181,159.58. Less cost of Boat & Paul has a Lump sum remaining as $1,681,159.58.
Now Paul is looking at finding the amount which he can spend each year from this lump sumfor next 20 yrs at end of which he will have zero money.
So this is like an annuity whose PVA ie present value of annuity is $1,681,159.58, duration n = 20yrs & Int rate i=9%. We need to find PMT which is teh amount that Paul will spend each year.
Present value of annuity PVA = PMT(PVIFAi,n).
So PVA = PMT*(1/i - 1/(i(1+i)^n))
ie PVA = PMT*(1/9% - 1/(9%*(1+9%)^20)) = PMT*(11.11-1.98) = 9.13*PMT
So PMT = PVA/9.13 = $1,681,159.58/9.13 = $184,135.77
So Paul can spend $184,135.77 every year for 20 yrs after buying the boat.
Note: It is not clear when the 25 yr period is starting. If it is starting after buying the boat, then pl see below calculation
PVA = PMT*(1/9% - 1/(9%*(1+9%)^25)) = PMT*(11.11-1.29) = 9.82*PMT
So PMT = PVA/9.82 = $1,681,159.58/9.82 = $171,197.51
So Paul can spend $171,197.51 every year for 25 yrs after buying the boat.