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QuestionYou want \$600,000 in 25 years with 8% interest. a) Monthly deposit needed? b) Total interest earned?

Studdy Solution

STEP 1

Assumptions1. The desired retirement amount is $600,000. The interest rate is8%
3. The time for saving is25 years4. The interest is compounded annually5. The deposit is made at the end of each month

STEP 2

First, we need to find the monthly deposit amount. We can do this using the formula for the future value of an ordinary annuity, which is given byV=PM×((1+r)n−1r)V = PM \times \left( \frac{(1 + r)^n -1}{r} \right)where- VV is the future value of the annuity- MTMT is the monthly deposit- rr is the monthly interest rate- nn is the number of months

STEP 3

We need to find the monthly interest rate and the number of months. The monthly interest rate is the annual interest rate divided by12, and the number of months is the number of years times12.
r=8%12=0.00667r = \frac{8\%}{12} =0.00667n=25×12=300n =25 \times12 =300

STEP 4

Now, we can rearrange the formula for the future value of an annuity to solve for the monthly deposit.
MT=FV×r(1+r)n−1MT = FV \times \frac{r}{(1 + r)^n -1}

STEP 5

Plug in the values for the future value, monthly interest rate, and number of months to calculate the monthly deposit.
MT=$600,000×0.00667(1+0.00667)300−1MT = \$600,000 \times \frac{0.00667}{(1 +0.00667)^{300} -1}

STEP 6

Calculate the monthly deposit amount.
MT=$600,000×0.00667(1+0.00667)300−1=$1,036.55MT = \$600,000 \times \frac{0.00667}{(1 +0.00667)^{300} -1} = \$1,036.55You would need to deposit approximately $1,036.55 each month.

STEP 7

Now, we need to find the total interest earned. We can do this by subtracting the total amount deposited from the future value of the annuity.
Interest=FV−(MT×n)Interest = FV - (MT \times n)

STEP 8

Plug in the values for the future value, monthly deposit, and number of months to calculate the interest.
Interest=$600,000−($1,036.55×300)Interest = \$600,000 - (\$1,036.55 \times300)

STEP 9

Calculate the interest amount.
Interest=$600,000−($,036.55×300)=$189,035Interest = \$600,000 - (\$,036.55 \times300) = \$189,035You will earn approximately $189,035 in interest.

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