A young executive deposits $400 at the end of each month for 7 years and then increases the deposits. If the account earns 7.2%, compounded monthly, how much (to the nearest dollar) should each new deposit be in order to have a total of $400,000 after 25 years
Suppose a state lottery prize of $2 million is to be paid in 20 payments of $100,000 each at the end of each of the next 20 years. If money is worth 9%, compounded annually, what is the present value of the prize?
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To find out how much each new deposit should be in order to have a total of $400,000 after 25 years, we can break down the problem into smaller steps.
Step 1: Calculate the future value of the monthly deposits made during the first 7 years:
We know that the executive deposits $400 at the end of each month for 7 years. We can use the formula for the future value of an ordinary annuity to calculate the total future value of these deposits.
Future Value = Payment x (((1 + Interest Rate)^(Number of Payments) - 1) / Interest Rate)
In this case, the payment is $400, the interest rate is 7.2% (or 0.072 when expressed as a decimal), and the number of payments is 7 years x 12 months/year = 84 months.
Future Value of monthly deposits = $400 x (((1 + 0.072)^(84) - 1) / 0.072)
After calculating this, we get the value of the monthly deposits after 7 years.
Step 2: Calculate the total future value needed after 25 years:
We want to have a total of $400,000 after 25 years. Since we already calculated the future value of the monthly deposits made during the first 7 years, we need to calculate the future value needed for the remaining 18 years.
To find the future value needed after 25 years, we can subtract the previously calculated future value of the monthly deposits from $400,000.
Remaining future value = $400,000 - Future Value of monthly deposits
Step 3: Calculate the new monthly deposits needed for the remaining 18 years:
We now need to find out how much the executive should deposit each month during the remaining 18 years in order to reach the remaining future value.
Using the same formula for the future value of an ordinary annuity, we can calculate the new monthly deposits:
New Deposits = Remaining future value / (((1 + Interest Rate)^(Number of Payments) - 1) / Interest Rate)
In this case, the remaining future value is the previously calculated remaining future value, the interest rate is the same 7.2% (or 0.072 when expressed as a decimal), and the number of payments is 18 years x 12 months/year = 216 months.
After calculating this, we get the value of the new monthly deposits needed for the remaining 18 years.
By following these steps, we can find the amount of each new deposit to have a total of $400,000 after 25 years.
Split the problem in two parts, 25 years and 18 years.
The interest rate is known for the first part (7% p.a. or per month?) and compounded monthly. So the future value after 25 years is determined, say A.
Assuming A<400,000=target, subtract
remaining future value B=400,000-A and calculate deposit required for 18 remaining years at the given interest rate.
For the second part, the future value after (a further) 18 years is known,