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?
What is the monthly deposit required to accumulate to a fund of $1,000,000 over a period of 40 years with deposits starting at the end of the first month and bearing an interest rate of 8% compounded monthly?
S(n) = $1,000,000]
i = .08/12 = .00666...
n = 40(12) = 48
R = S(n)(i)/[(1+i)^n - 1]
1,000,000(.00666...)/[1.00666...)^48-1]
$17,746.25
As for the present value,
P = R[1 - (1+i)^-n]/i
P=17,746.25[1-(1.00666...)^-40]/.00666...
P = $726,920.
To find out how much the student must deposit each month in order to achieve $1,000,000 in 40 years, we can use the formula for the future value of an annuity.
The formula for the future value of an ordinary annuity is:
FV = P * ((1 + r/n)^(nt) - 1) / (r/n)
Where:
FV = Future value
P = Monthly deposit
r = Annual interest rate (as a decimal)
n = Number of compounding periods per year
t = Number of years
In this case, the annual interest rate is 8%, which is equivalent to 0.08 as a decimal. The interest is compounded monthly, so there are 12 compounding periods per year and the number of years is 40.
Using the formula, we can plug in the values:
$1,000,000 = P * ((1 + 0.08/12)^(12*40) - 1) / (0.08/12)
Now we can solve for P:
$1,000,000 = P * (1.0066667^(480) - 1) / 0.0066667
To solve for P, we can rearrange the equation:
P = $1,000,000 * 0.0066667 / (1.0066667^(480) - 1)
Calculating this, we find that the monthly deposit required to achieve $1,000,000 in 40 years is approximately $619.36.
To find the present value of this annuity, we can use the present value of an ordinary annuity formula:
PV = P * (1 - (1 + r/n)^(-nt)) / (r/n)
By plugging the values into the formula, we can calculate the present value:
PV = $619.36 * (1 - (1 + 0.08/12)^(-12*40)) / (0.08/12)
Calculating this, we find that the present value of this annuity is approximately $98,982.26.