A young executive deposits $300 at the end of each month for 8 years and then increases the deposits. If the account earns 7.2%, compounded monthly, how much (to the nearest dollar) should each new deposit be in order to have a total of $400,000 after 25 years?
The new deposit should be $400 per month in order to have a total of $400,000 after 25 years.
reason
Well, let me crunch some numbers for you! So, the young executive deposits $300 at the end of each month for 8 years, which gives us a total of $28,800. Now, they want to have a total of $400,000 after 25 years.
First, let's calculate the value of the $28,800 deposited over 8 years at an interest rate of 7.2% compounded monthly. Using the compound interest formula, we get:
A = P(1 + r/n)^(nt)
Where:
A = Total amount
P = Principal amount ($28,800)
r = Annual interest rate (7.2%)
n = Number of times interest is compounded per year (12 months)
t = Time in years (8)
Plugging in the values, we have:
A = 28,800(1 + 0.072/12)^(12*8)
A ≈ 40,726.13
So, after 8 years, the initial deposit of $28,800 would grow to approximately $40,726.13.
To achieve the desired total of $400,000 after 25 years, we need to find out how much additional deposit is required over the remaining 17 years.
Now, let's calculate the new deposit amount using the future value of an ordinary annuity formula:
FV = P * ((1 + r/n)^(nt) - 1) / (r/n)
Where:
FV = Future value ($400,000)
P = Monthly deposit
r = Annual interest rate (7.2%)
n = Number of times interest is compounded per year (12 months)
t = Time in years (17)
Plugging in the values, we have:
400,000 = P * ((1 + 0.072/12)^(12*17) - 1) / (0.072/12)
P ≈ 395.74
So, to have a total of $400,000 after 25 years, each new monthly deposit should be approximately $395.74.
However, since we're dealing with real money and not imaginary decimals, the new deposit should be rounded to the nearest dollar. So, each new deposit should be $396.
Now the young executive can start making those increased deposits and clown his way to $400,000!
To find the amount of each new deposit, we need to calculate the future value of the deposits made in the first 8 years and then calculate the additional deposits required to reach $400,000 after 25 years.
First, let's calculate the future value of the deposits made in the first 8 years. We will use the formula for the future value of an ordinary annuity:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future value
P = Monthly deposit
r = Monthly interest rate (7.2% divided by 12)
n = Number of periods (8 years multiplied by 12 months)
FV = 300 * ((1 + 0.072/12)^(8*12) - 1) / (0.072/12)
FV = 300 * ((1.006)^96 - 1) / 0.006
FV ≈ 40693.66
After 8 years, the deposits made will have a future value of approximately $40,693.66.
Next, let's calculate the additional deposits required to reach $400,000 after 25 years. We will use the future value formula again, but this time with the given values:
FV = 400,000 - 40,693.66 (the remaining amount required)
r = Monthly interest rate (7.2% divided by 12)
n = Number of periods (17 years multiplied by 12 months)
400,000 - 40,693.66 = P * ((1 + 0.072/12)^(17*12) - 1) / (0.072/12)
359,306.34 = P * ((1.006)^204 - 1) / 0.006
Now, let's solve for P:
P = 359,306.34 * 0.006 / ((1.006)^204 - 1)
P ≈ $292.04
Each new deposit should be approximately $292.04 in order to have a total of $400,000 after 25 years.
To solve this problem, we can break it down into a series of steps:
Step 1: Calculate the future value of the initial deposits after 8 years.
Step 2: Calculate the future value of the additional deposits over the remaining 17 years.
Step 3: Calculate the future value of the total deposits after 25 years.
Step 4: Determine the amount of each new deposit to reach a total of $400,000 after 25 years.
Let's go through each step in detail:
Step 1: Calculate the future value of the initial deposits after 8 years.
We can use the formula for the future value of a series of monthly deposits with compound interest:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = future value
P = periodic deposit
r = interest rate per period
n = number of periods
In this case, the periodic deposit (P) is $300, the interest rate (r) is 7.2% per year (0.072/12 = 0.006), and the number of periods (n) is 8 years (8 * 12 = 96 months).
Using the formula, we can calculate the future value of the initial deposits:
FV1 = 300 * ((1 + 0.006)^96 - 1) / 0.006
Step 2: Calculate the future value of the additional deposits over the remaining 17 years.
To calculate the future value of the additional deposits, we need to consider the increased monthly deposits and the additional time period.
The increased monthly deposits will start after 8 years, so the number of periods (n2) for the additional deposits will be 17 years (17 * 12 = 204 months).
Let's assume the amount of each new deposit is X.
Using the same formula as in Step 1, we can calculate the future value of the additional deposits:
FV2 = X * ((1 + 0.006)^204 - 1) / 0.006
Step 3: Calculate the future value of the total deposits after 25 years.
Now, we need to calculate the future value of the total deposits after 25 years by adding the future values of the initial deposits (FV1) and the additional deposits (FV2):
Total Future Value (FV_total) = FV1 + FV2
Step 4: Determine the amount of each new deposit to reach a total of $400,000 after 25 years.
We want to find the value of X, the amount of each new deposit required to reach a total of $400,000 after 25 years.
So we set up the equation:
400,000 = FV_total
Rearranging the equation, we get:
400,000 = FV1 + FV2
Substituting the values from Step 1 and Step 2 into the equation, we have:
400,000 = 300 * ((1 + 0.006)^96 - 1) / 0.006 + X * ((1 + 0.006)^204 - 1) / 0.006
Solving this equation will give us the value of X, which is the amount of each new deposit required.
You can plug in the equation into a financial calculator, spreadsheet software (like Excel), or use numerical methods to solve for X.