3. Suppose a student wants to be a millionaire in 40 years. If she has an account that pays 8% interest compounded monthly, how much must she deposit each month in order to achieve her goal of having $1,000,000? What is the present value of this annuity?
P(1+.08/12)^40 = 1000000
P*21.72 = 1000000
P = $46040.52
Oops. Monthly deposits.
Let r = 1 + .08/12 = 1.00666666
M (r^480 - 1)/(r-1) = 1000000
3641.007M = 1000000
M = $274.65
Anyone see an error here?
Steve, your method is correct,
I let my calculator carry its maximum digits and got
$286.45
Also the present value of $1000000 40 years from now at 8% compounded monthly
PV (1.00666666..)^480 = 1000000
PV = $41197.40
To determine how much the student needs to deposit each month to achieve her goal of $1,000,000 in 40 years, we can use the formula for the future value of an annuity. The formula is given by:
Future Value = Payment x [(1 + Interest Rate)^(Number of Periods) - 1] / Interest Rate
Here, we need to solve for the Payment.
Let's break down the given information:
Interest Rate: 8% (or 0.08 in decimal form)
Number of Periods: 40 years (since the student wants to be a millionaire in 40 years)
Future Value: $1,000,000
Plugging these values into the formula, we have:
$1,000,000 = Payment x [(1 + 0.08/12)^(40*12) - 1] / (0.08/12)
Now, we can solve this equation for Payment. First, simplify the formula:
$1,000,000 = Payment x [(1 + 0.0067)^(480) - 1] / 0.0067
Next, multiply both sides of the equation by 0.0067 to isolate Payment:
$1,000,000 x 0.0067 = Payment x [(1 + 0.0067)^(480) - 1]
$6,700 = Payment x [(1.0067)^(480) - 1]
Now, divide both sides of the equation by [(1.0067)^(480) - 1] to solve for Payment:
Payment = $6,700 / [(1.0067)^(480) - 1]
Using a calculator or spreadsheet software, we can evaluate this expression to find the value of Payment. It will give us the amount that the student needs to deposit each month to achieve her goal of $1,000,000 in 40 years.
To find the present value of this annuity, we can use the present value formula for an annuity. The formula is given by:
Present Value = Payment x [1 - (1 + Interest Rate)^(-Number of Periods)] / Interest Rate
Using the same values for Payment, Interest Rate, and Number of Periods, we can plug them into the formula to find the present value.
Present Value = Payment x [1 - (1 + 0.08/12)^(-40*12)] / (0.08/12)
Evaluate this expression using a calculator or spreadsheet software to find the present value of the annuity.
Please note that the calculated values for Payment and Present Value will depend on the accuracy of the interest rate, the compounding frequency, and the number of periods used.