Suppose you are 30 years old and would like to retire at age 60. ​Furthermore, you would like to have a retirement fund from which you can draw an income of ​$75, 000 per year-forever! How much would you need to deposit each month to do​ this? Assume a constant APR of 7​% and that the compounding and payment periods are the same.

To draw ​$75,000 per​ year, there must be ​$____ in your savings account when you retire.

Did you ever figure it out? Because I’m stuck on this question too

To calculate the amount you would need to deposit each month to retire with $75,000 per year, we first need to determine the total amount you would need to have in your savings account when you retire.

Since you want to have a constant annual income of $75,000, we can use the formula for the present value of a perpetuity:

PV = PMT / r

Where PV is the present value, PMT is the amount of annual payment, and r is the interest rate.

Plugging in the values, we have:

PV = $75,000 / 0.07

PV = $1,071,428.57

Therefore, you would need to have $1,071,428.57 in your savings account when you retire in order to generate an annual income of $75,000.

Now, to determine how much you need to deposit each month, we can use the formula for the future value of an ordinary annuity:

FV = P * [(1 + r)^n - 1] / r

Where FV is the future value, P is the monthly deposit, r is the interest rate, and n is the number of periods (in this case, the number of months until you retire).

Rearranging the formula to solve for P:

P = FV * [r / (1 + r)^n - 1]

Plugging in the values, we have:

FV = $1,071,428.57 (the present value we calculated earlier)
r = 0.07 (7% APR converted to decimal)
n = (60 - 30) * 12 (years until retirement multiplied by 12 months/year)

P = $1,071,428.57 * [0.07 / (1 + 0.07)^(30 * 12) - 1]

P = $923.02 (rounded to the nearest cent)

Therefore, you would need to deposit approximately $923.02 each month to retire with $75,000 per year, assuming a constant APR of 7%.

To calculate the amount of money you would need in your savings account to retire, we can use the concept of present value. The present value is the current value of a future sum of money, taking into account interest and the time value of money.

In this case, you would like to withdraw $75,000 per year from your retirement fund, indefinitely. If we assume an annual interest rate of 7% and a constant payment of $75,000 per year, we can use the present value of an annuity formula.

The formula for the present value of an annuity is:

PV = PMT * [1 - (1 + r)^(-n)] / r

Where:
PV = Present Value (the amount you need in your savings account at retirement)
PMT = Payment per period ($75,000 per year)
r = Annual interest rate (7% or 0.07)
n = Total number of periods (retirement age - current age = 60 - 30 = 30 years)

Plugging in the values into the formula, we get:

PV = $75,000 * [1 - (1 + 0.07)^(-30)] / 0.07

Now, we can calculate this value to find out how much you would need in your savings account at retirement.