Suppose you wish to invest X dollars in a bank account which pays 5% per year. You want to use this account to pay for costs that appear each year, starting with year 15. The amount you have to pay in year 15 is 276.71 and then the payments grow at a rate of 5% per year until year 38. If you wish to have exactly 1/11 of the amount X left in the account at the end of year 38, what should X be ?
To determine the initial investment amount X, we need to calculate the future value of the payments from year 15 to year 38, discounted by a rate of 5% per year.
Step 1: Calculate the future value of the payments from year 15 to year 38.
We can use the formula for the future value of a series of payments:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = future value of the payments
P = payment amount each year (starts with 276.71)
r = interest rate (5%)
n = number of years (38 - 15 = 23 years)
FV = 276.71 * [(1 + 0.05)^23 - 1] / 0.05
Step 2: Calculate the remaining amount at the end of year 38.
We need to have exactly 1/11 of the initial investment X left in the account at the end of year 38.
Remaining amount = X * (1 - 1/11)
Step 3: Set up equation and solve for X.
Since the remaining amount at the end of year 38 needs to be equal to the future value of the payments from year 15 to year 38, we can set up the equation:
Remaining amount = FV
X * (1 - 1/11) = 276.71 * [(1 + 0.05)^23 - 1] / 0.05
Simplify the equation:
10X/11 = 276.71 * [(1.05)^23 - 1] / 0.05
Now solve for X:
X = [276.71 * [(1.05)^23 - 1] / 0.05] * 11/10
X ≈ $1,142.42
Therefore, the initial investment amount X should be approximately $1,142.42 to ensure that there is exactly 1/11 of X left in the account at the end of year 38.
To find the initial investment amount (X), we need to work backward from year 38 to year 15.
We know that the payment amount in year 38 is 1/11 of X. Let's denote this payment amount as P38.
P38 = X/11
We also know that the payments grow at a rate of 5% each year. This means that the payment amount in year 37 (P37) is 5% less than the payment amount in year 38 (P38).
P37 = P38 - 5/100 * P38
P37 = (1 - 5/100) * P38
P37 = 0.95 * P38
We can continue this process backward until we reach year 15. The payment amount in year 15 (P15) is 276.71, which is the given value.
P15 = 276.71
Now, let's calculate the payment amount in year 16 (P16) using the same formula as before. P16 is 5% less than P15.
P16 = P15 - 5/100 * P15
P16 = 0.95 * P15
We can repeat this process until we reach year 38.
P16 = 0.95 * P15
P17 = 0.95 * P16 = 0.95 * 0.95 * P15
P18 = 0.95 * P17 = 0.95 * 0.95 * 0.95 * P15
...
P37 = 0.95^21 * P15
P38 = 0.95 * P37 = 0.95 * 0.95^21 * P15
We know that P38 is 1/11 of X.
X/11 = 0.95 * 0.95^21 * P15
To find X, we can multiply both sides of the equation by 11.
X = 11 * 0.95 * 0.95^21 * P15
Now, substitute the given value of P15, which is 276.71, into the equation.
X = 11 * 0.95 * 0.95^21 * 276.71
Calculating this expression will give you the value of X, which is the initial investment amount needed.