Confused on the method used to solve this problem and could really use some help. The 5% is throwing me off as we don't know 5% on what amount and we assume that amount compounds over time?
You are currently 25 years old and have just spent the past weekend in Vegas. As you are leaving the hotel, you decide to drop one more quarter in to the progressive slot machine. Amazingly enough, you hit the jackpot paying out $3,500,000.
You decide that you want to spend as much as possible but you also want to set yourself up for a comfortable retirement beginning at age 55. You know that you can safely earn a 5% return. Upon retirement, you want to be able to pull from your investment $60,000 for 20 years.
How much of your winnings do you have to invest today to provide for your retirement demands?
To solve this problem, we need to calculate the present value of the retirement withdrawals using the given interest rate and time period. Here's how you can do it step by step:
Step 1: Determine the present value of the retirement withdrawals.
Since the retirement withdrawals are $60,000 per year for 20 years, we can calculate the present value using the formula for the present value of an annuity:
PV = PMT * [(1 - (1 + r)^(-n)) / r]
Where:
PV = Present Value
PMT = Payment per period ($60,000)
r = Interest rate per period (5% or 0.05)
n = Number of periods (20 years)
Plugging in the values:
PV = $60,000 * [(1 - (1 + 0.05)^(-20)) / 0.05]
PV ≈ $746,275.63
So the present value of the retirement withdrawals is approximately $746,275.63.
Step 2: Calculate the amount of winnings you need to invest today to cover the retirement withdrawals.
Since we want to set ourselves up for retirement, we need to invest an amount that will grow to the present value of the retirement withdrawals. This investment will compound at a rate of 5% over the remaining time until retirement (from age 25 to 55, which is 30 years).
To calculate the required investment amount, we need to use the formula for the future value of a lump sum:
FV = PV * (1 + r)^n
Where:
FV = Future Value (the required investment)
PV = Present Value ($746,275.63)
r = Interest rate per period (5% or 0.05)
n = Number of periods (30 years)
Plugging in the values:
FV = $746,275.63 * (1 + 0.05)^30
FV ≈ $3,773,404.99
Therefore, you need to invest approximately $3,773,404.99 of your winnings today to cover the retirement withdrawals.
Note: The assumption made in this calculation is that the investment will grow at a constant rate of 5% per year and no additional contributions or withdrawals will be made. Also, keep in mind that actual investment returns may vary.