You decide to open an IRS-approved retirement account at your local brokerage firm. Your best estimate is that it will earn 9 percent.
At the end of each year for the next 25 years, you will deposit $4000 per year into the account (25 total deposits).
3 years after the last deposit, you will begin making annual withdrawals.
A) How much money is in the account 1 year before the first withdrawal?
B) If you want to make 30 annual withdrawals, what amount will you be able to withdraw each year?
C) If you want the account to last forever, what amount will you be able to withdraw each year?
The ANSWERS ARE
A) $251,582.84
B) $24,488.16
C) $22,642.46
To solve these questions, we can use the concept of future value and annuity formulas. The future value formula is used to calculate the value of an investment at a specific time in the future, given an interest rate and the number of years. The annuity formula is used to calculate the future value of a series of equal periodic payments.
Here's how you can find the answers to each question:
A) The amount of money in the account 1 year before the first withdrawal can be determined by calculating the future value of 25 annual deposits of $4,000, earning an interest rate of 9 percent. To solve this, you can use the future value of an annuity formula:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = Future Value
P = Payment per period
r = Interest rate per period
n = Number of periods
In this case, the payment per period (P) is $4,000, the interest rate per period (r) is 9% or 0.09, and the number of periods (n) is 25.
Plugging these values into the formula:
FV = $4,000 * [(1 + 0.09)^25 - 1] / 0.09
Using this formula, the value of the account 1 year before the first withdrawal is approximately $251,582.84.
B) To calculate the amount you will be able to withdraw each year for 30 years, you need to use the annuity formula to find the future value of 30 withdrawals.
FV = P * [(1 + r)^n - 1] / r
In this case, the payment per period (P) is the amount you will be able to withdraw each year, the interest rate per period (r) is 9% or 0.09, and the number of periods (n) is 30.
To calculate the withdrawal amount (P), you can rearrange the annuity formula:
P = FV * r / [(1 + r)^n - 1]
Plugging the values into the formula:
P = $251,582.84 * 0.09 / [(1 + 0.09)^30 - 1]
Using this formula, the withdrawal amount each year will be approximately $24,488.16.
C) To calculate the withdrawal amount that will make the account last forever, we need to use the concept of perpetuity. A perpetuity is a series of equal cash flows that continues indefinitely.
The formula for the present value of a perpetuity is:
PV = P / r
Where:
PV = Present Value
P = Payment per period
r = Interest rate per period
In this case, we want to find the payment per period (P) that will allow the account to last forever. The interest rate per period (r) is 9% or 0.09.
Plugging the values into the formula:
P = PV * r
Where PV is the value we found in part A, which is $251,582.84.
P = $251,582.84 * 0.09
Using this formula, the withdrawal amount each year to make the account last forever will be approximately $22,642.46.
So, the answers to the questions are:
A) The amount of money in the account 1 year before the first withdrawal is approximately $251,582.84.
B) If you want to make 30 annual withdrawals, the withdrawal amount each year will be approximately $24,488.16.
C) If you want the account to last forever, the withdrawal amount each year will be approximately $22,642.46.