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QuestionTo have \$500,000 in 20 years at 9% interest, how much to deposit monthly and total interest earned?

Studdy Solution

STEP 1

Assumptions1. The desired retirement amount is \$500,000. The interest rate is9%
3. The time for retirement is20 years4. The interest is compounded monthly5. The monthly deposit is constant

STEP 2

First, we need to find the monthly deposit. We can use the formula for the future value of an ordinary annuity, which is given byV=×((1+r)n−1r)V = \times \left( \frac{(1 + r)^n -1}{r} \right)WhereV = future value (the desired retirement amount) = monthly depositr = monthly interest raten = number of months

STEP 3

We need to convert the annual interest rate to a monthly rate and the number of years to months.
r=Annual interest rate12=9%12=0.0075r = \frac{Annual\, interest\, rate}{12} = \frac{9\%}{12} =0.0075n=Years×12=20×12=240n = Years \times12 =20 \times12 =240

STEP 4

Now, we can rearrange the formula to solve for, the monthly deposit.
=V×r(1+r)n−1 = \frac{V \times r}{(1 + r)^n -1}

STEP 5

Plug in the values for FV, r, and n to calculate the monthly deposit.
=$500,000×0.0075(1+0.0075)240−1 = \frac{\$500,000 \times0.0075}{(1 +0.0075)^{240} -1}

STEP 6

Calculate the monthly deposit.
≈$1,028.61 \approx \$1,028.61So, you would need to deposit approximately \$1,028.61 each month.

STEP 7

To find the total interest earned, we first calculate the total amount deposited over the20 years.
Total deposits=Monthly deposit×Number of monthsTotal\, deposits = Monthly\, deposit \times Number\, of\, monthsTotal deposits=$1,028.61×240Total\, deposits = \$1,028.61 \times240

STEP 8

Calculate the total deposits.
Total deposits≈$246,866.40Total\, deposits \approx \$246,866.40

STEP 9

The total interest earned is the difference between the desired retirement amount and the total deposits.
Interest earned=Desired retirement amount−Total depositsInterest\, earned = Desired\, retirement\, amount - Total\, depositsInterest earned=$500,000−$246,866.40Interest\, earned = \$500,000 - \$246,866.40

STEP 10

Calculate the interest earned.
Interest earned≈$253,133.60Interest\, earned \approx \$253,133.60So, you will earn approximately \$253,133.60 in interest.

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