Mrs. Kay who is 24 years old plans to retire at the age of 60. Mrs. Kay would like to be able to withdraw $120,000 per year from her retirement account for 40 years after retirement beginning a year after her retirement. She already has $15,000 in her retirement investment account that earns 7% per year. After retirement she plans to take less risk so she will move her investment into an account that earns 4.5% per year. How much does she need to invest each year in order to reach her goal?
can you explain how you got the answer.
To calculate how much Mrs. Kay needs to invest each year to reach her goal, we will use the future value of annuity formula to determine the amount she needs to have saved by the time she retires, and then use the present value of annuity formula to determine how much she needs to invest each year to reach that goal. Here are the steps:
1. Determine how many years Mrs. Kay will be saving before retirement:
She plans to retire at age 60 and is currently 24 years old, so she will be saving for 60 - 24 = 36 years.
2. Determine the future value of the withdrawals she plans to make during her retirement:
Mrs. Kay wants to withdraw $120,000 per year for 40 years, beginning one year after she retires. Since the account will be earning 4.5% interest per year, we can use the future value of annuity formula:
FV = P * [(1 + r)^n - 1] / r
where FV = future value of withdrawals, P = annual withdrawal amount ($120,000), r = annual interest rate (4.5% = 0.045), and n = number of years she plans to make withdrawals (40).
FV = $120,000 * [(1 + 0.045)^40 - 1] / 0.045
FV ≈ $8,676,696
So, Mrs. Kay needs to have $8,676,696 saved by the time she retires to be able to make her desired withdrawals.
3. Determine how much she needs to invest each year to achieve that future value:
Mrs. Kay will be investing in an account that earns 7% interest per year and she plans to invest the same amount each year. Using the present value of annuity formula:
PV = FV / [(1 + r)^n - 1] / r
where PV = present value (the amount she needs to invest each year), FV = future value of the account ($8,676,696), r = annual interest rate (7% = 0.07), and n = number of years she will be saving (36).
PV = $8,676,696 / [(1 + 0.07)^36 - 1] / 0.07
PV ≈ $6,095
So, Mrs. Kay needs to invest approximately $6,095 each year in order to reach her goal of having $8,676,696 saved by the time she retires.
To find out how much Mrs. Kay needs to invest each year in order to reach her retirement goal, we can use the concept of present value of an annuity.
First, let's calculate the total amount Mrs. Kay needs to accumulate by the time she retires.
Mrs. Kay plans to withdraw $120,000 per year for 40 years after retirement. So, the total amount she needs for those 40 years can be calculated as follows:
Total amount needed = Annual withdrawal amount * Number of years
= $120,000 * 40
= $4,800,000
Now, let's calculate the future value of her current investment. Mrs. Kay already has $15,000 in her retirement investment account that earns 7% interest per year. This can be calculated using the future value of a single sum formula:
Future value = Present value * (1 + interest rate)^number of periods
= $15,000 * (1 + 0.07)^40
= $15,000 * (1.07)^40
≈ $173,981.14 (approx.)
So, the total amount already accumulated by Mrs. Kay is approximately $173,981.14.
To determine how much she needs to invest each year to reach her retirement goal, we can subtract the already accumulated amount from the total amount needed:
Amount needed to invest each year = Total amount needed - Amount already accumulated
= $4,800,000 - $173,981.14
= $4,626,018.86
Now, we need to calculate the annual investment amount using the present value of an annuity formula:
Annual investment amount = Amount needed to invest each year / Present value factor
The present value factor can be calculated using the formula:
Present value factor = (1 - (1 + interest rate)^(-number of periods)) / interest rate
For Mrs. Kay, the interest rate is 4.5% (0.045) because she plans to move her investment to an account that earns 4.5% per year.
Using this information, we can calculate the present value factor:
Present value factor = (1 - (1 + 0.045)^(-40)) / 0.045
≈ 21.52664912 (approx.)
Finally, we can calculate the annual investment amount:
Annual investment amount = $4,626,018.86 / 21.52664912
≈ $214,763.65 (approx.)
Therefore, Mrs. Kay needs to invest approximately $214,763.65 each year to reach her retirement goal of being able to withdraw $120,000 per year for 40 years.
To find out how much Mrs. Kay needs to invest each year in order to reach her retirement goal, we can break down the problem into smaller steps.
Step 1: Calculate the total amount needed for retirement:
Mrs. Kay plans to withdraw $120,000 per year for 40 years after retirement. So, the total amount needed for retirement is calculated as:
Total amount needed = Annual withdrawal amount * Number of years.
Total amount needed = $120,000 * 40 = $4,800,000.
Step 2: Calculate the amount Mrs. Kay already has:
Mrs. Kay already has $15,000 in her retirement investment account.
Step 3: Find the difference between the total amount needed and the amount already saved:
Difference = Total amount needed - Amount already saved.
Difference = $4,800,000 - $15,000 = $4,785,000.
Step 4: Calculate the annual contribution needed:
Mrs. Kay plans to retire in 36 years (from her current age of 24 until age 60). To accumulate the difference amount in this period, we can use the formula for calculating the future value of an annuity:
Future Value = Annual contribution * [((1 + interest rate) ^ number of years) - 1] / interest rate.
Here, the future value is the difference amount calculated in Step 3, and the interest rate is the rate at which Mrs. Kay's retirement investment account earns.
Plugging the values into the formula, the equation becomes:
$4,785,000 = Annual contribution * [((1 + 0.045) ^ 36) - 1] / 0.045.
Let's solve for the annual contribution:
Multiply both sides of the equation by 0.045:
0.045 * $4,785,000 = Annual contribution * [((1 + 0.045) ^ 36) - 1].
$215,325 = Annual contribution * [1.045 ^ 36 - 1].
Divide both sides of the equation by [1.045 ^ 36 - 1]:
Annual contribution = $215,325 / [1.045 ^ 36 - 1].
Using a calculator, we can find that [1.045 ^ 36 - 1] ≈ 20.4915.
Therefore, the annual contribution needed is:
Annual contribution ≈ $215,325 / 20.4915 ≈ $10,528.
So, Mrs. Kay needs to invest approximately $10,528 each year to reach her retirement goal of being able to withdraw $120,000 per year for 40 years after retirement.