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?
1000000=P/(1.08)^480
1000000=P/1.105103
P=1,105,103
To find out how much the student must deposit each month, we can use the concept of an ordinary annuity. An ordinary annuity is a series of equal cash flows that occur at the end of each period. In this case, the cash flows are the monthly deposits.
Let's break down the problem into the given information:
- Interest rate: 8% per year
- Compounding period: Monthly
- Time horizon: 40 years
- Future value goal: $1,000,000
To calculate the monthly deposit needed, we can use the present value of an ordinary annuity formula. The formula is:
PMT = (FV * r) / ((1 + r)^n - 1)
Where:
PMT = Monthly deposit
FV = Future value goal
r = Interest rate per compounding period
n = Total number of compounding periods
Substituting the given values into the formula:
PMT = ($1,000,000 * 0.08/12) / ((1 + 0.08/12)^(40*12) - 1)
Simplifying the equation:
PMT = $689.47 (approximately)
Therefore, the student must deposit approximately $689.47 each month to achieve her goal of having $1,000,000 in 40 years.
To calculate the present value of this annuity, we can use the present value of an ordinary annuity formula:
PV = PMT * ((1 - (1 + r)^(-n)) / r)
Substituting the values into this formula, we get:
PV = $689.47 * ((1 - (1 + 0.08/12)^(-40*12)) / (0.08/12))
Simplifying the equation:
PV = $132,282.94 (approximately)
Therefore, the present value of this annuity is approximately $132,282.94.