A young executive deposits $400 at the end of each month for 7 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
Accumulated value after 7*12=84th payment
400((1+0.006)^84-1)/0.006=43522.55653
This amount after 25 years-->158445
Let P new payments then
400000-158445=P*((1+0.006)^216-1)/0.006
241555=P*440.087
P=$549
Well, to save $400,000 in 25 years, this young executive is going to need a lot of dough! But don't worry, we'll crunch the numbers and hopefully make it a little less dough-stressing!
Let's break it down step by step:
First, let's figure out how much the initial deposits will grow to after 7 years. We can use the formula for compound interest: A = P(1 + r/n)^(nt), where:
A = the future value of the investment ($)
P = the principal amount ($), which is the initial deposit
r = the annual interest rate (in decimal form) = 7.2% = 0.072
n = the number of times interest is compounded in a year = 12 (since it's compounded monthly)
t = the number of years = 7
So, after 7 years, the future value will be:
A = 400(1 + 0.072/12)^(12*7)
Hmmm... let me grab my calculator real quick...
*Clown Bot pulls out a calculator the size of a watermelon and starts pressing buttons dramatically*
Okay, drumroll, please...
After 7 years, the initial deposits will grow to approximately $6,600!
Now, let's move on to the next phase. We want to know how much each new deposit should be in order to reach a total of $400,000 in 25 years. To find this out, we need to use the same formula, but with a few changes:
A = the future value of the investment ($) = $400,000
P = the principal amount ($), which is the new deposit
r = the annual interest rate (in decimal form) = 7.2% = 0.072
n = the number of times interest is compounded in a year = 12 (since it's still compounded monthly)
t = the number of years = 25
Now, instead of solving for A, we'll be solving for P:
$400,000 = P(1 + 0.072/12)^(12*25)
Okay, deep breaths, Clown Bot, we can do this!
*Clown Bot pulls out the giant calculator once again and starts pressing buttons with gusto*
Drumroll, please...
After my calculations, it seems that each new deposit should be approximately $455.19 (rounded to the nearest dollar) in order to have a total of $400,000 in 25 years!
So, there you have it! The magic number is $455.19! Good luck achieving that impressive savings goal!
To find out how much each new deposit should be in order to have a total of $400,000 after 25 years, we can break down the problem into smaller steps.
Step 1: Calculate the future value of the first 7 years of monthly deposits.
Using the formula for future value of a series of deposits:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = future value
P = deposit amount
r = monthly interest rate
n = number of deposits
In this case, P = $400, r = 7.2%/12 = 0.006, and n = 7 * 12 = 84.
FV1 = 400 * ((1 + 0.006)^84 - 1) / 0.006
FV1 ≈ $40,170.52
Step 2: Calculate the remaining number of years for the account to reach a total of $400,000.
Since the total investment period is 25 years and the first 7 years have already passed, the remaining number of years is 25 - 7 = 18 years.
Step 3: Calculate the future value of the remaining deposits needed to reach $400,000.
Using the formula for future value:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = future value
P = deposit amount
r = monthly interest rate
n = number of deposits
In this case, FV = $400,000, r = 7.2%/12 = 0.006, and n = 18 * 12 = 216.
$400,000 = P * ((1 + 0.006)^216 - 1) / 0.006
Now, we need to solve for P.
400,000 = P * (1.006^216 - 1) / 0.006
24,000 = P * (1.006^216 - 1)
By rearranging the equation, we can solve for P:
P = 24,000 / (1.006^216 - 1)
P ≈ $621.58
Therefore, each new deposit should be approximately $622 in order to have a total of $400,000 after 25 years.
To find out how much each new deposit should be in order to have a total of $400,000 after 25 years, we need to break the problem down into smaller steps:
Step 1: Calculate the future value of the initial deposits over the first 7 years.
Step 2: Calculate the total future value of all subsequent deposits over the next 18 years.
Step 3: Determine the size of each new deposit to meet the target of $400,000 after 25 years.
Step 1: Calculate the future value of the initial deposits over the first 7 years.
Since the young executive deposits $400 at the end of each month, the annual deposit would be $400 * 12 months = $4,800.
Using the formula for future value of a monthly deposit, we can calculate the future value of these monthly deposits for 7 years, earning 7.2% compounded monthly.
Let's use the formula:
FV = P * ((1 + r)^n - 1) / r
Where:
FV is the future value of the deposits,
P is the monthly deposit amount,
r is the monthly interest rate, and
n is the total number of months.
Plugging in the values, we have:
P = $4,800
r = 7.2% / 12 = 0.72%
n = 7 years * 12 months = 84 months
FV = $4,800 * ((1 + 0.0072)^84 - 1) / 0.0072
FV ≈ $62,506.60
So, the initial deposits over the first 7 years will accumulate to approximately $62,506.60.
Step 2: Calculate the total future value of all subsequent deposits over the next 18 years.
To find the total future value of all subsequent deposits, we need to calculate the monthly deposit amount required to reach the target of $400,000 after 25 years (7 years for the initial deposits + 18 years for the subsequent deposits).
Let's denote the monthly deposit required as X.
The total number of months for subsequent deposits would be 18 years * 12 months = 216 months.
Using the same formula as in Step 1, with the new values:
P = X
r = 7.2% / 12 = 0.72%
n = 18 years * 12 months = 216 months
FV = X * ((1 + 0.0072)^216 - 1) / 0.0072
Step 3: Determine the size of each new deposit to meet the target of $400,000 after 25 years.
We know that the target future value after 25 years is $400,000.
So, the total future value of all deposits should be equal to $400,000.
Adding up the future values from Step 1 and Step 2, we have:
Total Future Value = $62,506.60 + X * ((1 + 0.0072)^216 - 1) / 0.0072
Now, we set this equal to the target future value ($400,000) and solve for X:
$400,000 = $62,506.60 + X * ((1 + 0.0072)^216 - 1) / 0.0072
To solve this equation, we can use numerical analysis techniques such as the bisection method or Newton's method. These methods involve trial and error or iterative calculations to approximate the value of X that satisfies the equation.
Using these numerical methods, the approximated value for X comes out to be around $348.10.
Therefore, each new deposit should be approximately $348 (to the nearest dollar) in order to have a total of $400,000 after 25 years.