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?
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To determine how much the student needs to deposit each month, we can use the future value of an annuity formula. The formula for the future value of an annuity is:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = Future value (in this case, $1,000,000)
P = Monthly deposit
r = Monthly interest rate (8% divided by 12 to get the monthly rate)
n = Number of months (40 years multiplied by 12 months)
Let's calculate it step by step:
Step 1: Calculate the monthly interest rate.
r = 8% / 12 = 0.08 / 12 = 0.0067
Step 2: Calculate the total number of months.
n = 40 years * 12 months/year = 480 months
Step 3: Plug the values into the formula and solve for P.
$1,000,000 = P * [(1 + 0.0067)^480 - 1] / 0.0067
To find P, we can rearrange the equation:
P = $1,000,000 * 0.0067 / [(1 + 0.0067)^480 - 1]
P ≈ $1,422.06
Therefore, the student needs to deposit approximately $1,422.06 each month to achieve her goal of having $1,000,000 in 40 years.
Now let's calculate the present value of this annuity, which represents the lump sum equivalent of all the future deposits.
The formula for the present value of an annuity is:
PV = P * [1 - (1 + r)^(-n)] / r
Let's calculate it step by step:
Step 1: Plug the values into the formula and solve for PV.
PV = $1,422.06 * [1 - (1 + 0.0067)^(-480)] / 0.0067
PV ≈ $197,742.14
Therefore, the present value of this annuity is approximately $197,742.14.