Suppose you want to have $600,000 for retirement in 35 years. Your account earns 7% interest.
a) How much would you need to deposit in the account each month?
$
b) How much interest will you earn?
$
a) To deposit $600,000 in 35 years, you would need to deposit $1,068.90 each month.
b) You will earn $420,000 in interest over the 35 years.
To calculate the answers to these questions, we can use a financial formula called the future value of an ordinary annuity. The formula for future value is:
FV = P * ((1 + r)^n - 1) / r
Where:
FV is the future value of the investment
P is the regular payment (amount deposited) per period
r is the interest rate per period
n is the number of periods
Now let's calculate the answers step by step:
a) How much would you need to deposit in the account each month?
In this case, the future value (FV) is $600,000, the interest rate (r) is 7% per year, and the number of periods (n) is 35 years. We need to find the regular payment (P).
Substituting the values into the formula:
$600,000 = P * ((1 + 0.07/12)^(35*12) - 1) / (0.07/12)
Simplifying and solving for P will give us the answer:
P = $600,000 / (((1 + 0.07/12)^(35*12) - 1) / (0.07/12))
Using a calculator or spreadsheet, you can compute this and find the monthly deposit needed.
b) How much interest will you earn?
The interest earned will be the difference between the future value and the total amount deposited.
Total amount deposited = Monthly deposit * Number of periods
Interest earned = Future value - Total amount deposited
Using the previously calculated monthly deposit, multiply it by the number of periods (35 years * 12 months/year) to find the total amount deposited. Then subtract the total amount deposited from the future value to get the interest earned.
To calculate the monthly deposit needed for retirement, we can use the future value of an ordinary annuity formula:
Future Value = R × [(1 + i)^n - 1] / i
Where:
Future Value = $600,000
R = Monthly deposit
i = Monthly interest rate (7% divided by 12)
n = Number of months (35 years multiplied by 12 months)
a) Calculating the monthly deposit:
600,000 = R × [(1 + (0.07/12))^(35*12) - 1] / (0.07/12)
Let's calculate:
600,000 = R × [1.00583^(420) - 1] / (0.0058333)
600,000 = R × [1.01.4626668 - 1] / (0.0058333)
600,000 = R × 174.7411842
R = 600,000 / 174.7411842
R ≈ $3,434.14
So, you would need to deposit approximately $3,434.14 in the account each month.
b) To calculate the interest earned, we can subtract the total amount deposited from the total future value:
Total Deposit = R × (n * 12)
Interest Earned = Future Value - Total Deposit
Let's calculate:
Total Deposit = 3,434.14 × (35 * 12)
Total Deposit = 3,434.14 × 420
Total Deposit ≈ $1,442,243.20
Interest Earned = 600,000 - 1,442,243.20
Interest Earned ≈ -$842,243.20
Based on the calculations, you would earn approximately -$842,243.20 in interest over 35 years.