math question for class

Question

Suppose you want to have $400,000 for retirement in 25 years. Your account earns 9% interest. How much would you need to deposit in the account each month?

Answer 1

Assuming you're starting out your retirement account with no money invested, let's start by listing out the five key variables that we need to solve annuities:

N (number of periods) = 25 years * 12 months per year = 300 months

I/YR (interest per period) = 9% annual interest / 12 months per year = 0.75% monthly interest

PV (present value) = $0 since we don't have any money in the account at the start

PMT = ? (we're solving for this)

and FV (future value) = $400,000 since you want that amount by the end of 25 years

The first method to find PMT then is to use a financial calculator. Just plug in the values above to the corresponding keys, then click the PMT key to find your monthly payment of $356.79 (the calculator will show the payment as a negative, but note that the negative is only there to tell you that the payment is leaving your wallet, not that you have a negative payment).

You can also solve this question using the PMT function on Excel. Click on any cell in the sheet and enter =PMT(0.75%, 300, 0, 400000) to get your monthly payment of $356.79. Excel also displays this payment as negative for the same reason a financial calculator does, to tell you that the money is leaving your wallet but not that the payment itself is negative.

Answer: $356.79 per month

Answer 2

To calculate how much you would need to deposit in the account each month to reach your retirement goal of $400,000 in 25 years with a 9% annual interest rate, you can use the formula for the future value of an ordinary annuity:

FV = PMT * [(1 + r)^nt - 1] / r

Where:

FV = future value of the annuity

PMT = monthly deposit

r = monthly interest rate (annual interest rate divided by 12)

n = number of times the interest is compounded per year (12 for monthly compounding)

t = number of years

First, we need to find the monthly interest rate by dividing the annual interest rate by 12:

r = 9% / 12 = 0.0075

Next, we can rearrange the formula to solve for the monthly deposit (PMT):

PMT = FV / [((1 + r)^nt - 1) / r]

Now, we can plug in the values:

PMT = $400,000 / [((1 + 0.0075)^(12 * 25) - 1) / 0.0075]

PMT = $400,000 / [(1.0075^300 - 1) / 0.0075]

After calculating the expression in the brackets:

PMT = $400,000 / 1,121.12194

Finally, we find the required monthly deposit:

PMT ≈ $356.79

Therefore, you would need to deposit approximately $356.79 per month to reach your retirement goal.