John Roberts has $42,180.53 in a brokerage account, and he plans to contribute an additional
$5,000 to the account at the end of every year. The brokerage account has an expected annual
return of 12 percent. If John’s goal is to accumulate $250,000 in the account, how many years
will it take for John to reach his goal?
I do not need anyone to do the wok I just need to know how to solve the problem. Thanks
To solve this problem, we can use the concept of future value of an annuity.
Here are the steps to calculate the number of years it will take for John to reach his goal:
1. Determine the future value of the regular contributions:
- John plans to contribute $5,000 at the end of every year.
- The expected annual return on the account is 12 percent.
- We can calculate the future value at the end of each year using the formula for future value of an annuity:
FV = P * [(1 + r)^n - 1] / r
Where:
FV is the future value of the annuity,
P is the regular contribution per period ($5,000 in this case),
r is the annual interest rate (12% = 0.12),
n is the number of years.
- Calculate the future value (FV1) at the end of the last contribution year (n) using the given values.
2. Determine the future value of the existing balance at the end of n years:
- The initial balance is $42,180.53.
- Calculate the future value of the existing balance (FV2) using the formula for compound interest:
FV = PV * (1 + r)^n
Where:
FV is the future value,
PV is the present value (the existing balance),
r is the annual interest rate (12% = 0.12),
n is the number of years.
3. Determine the total future value (FV) at the end of n years:
- Total future value = FV1 + FV2
4. Set up the equation:
- We want to find the number of years (n) required to reach a future value of $250,000.
- Set the total future value equal to $250,000 and solve for n.
Total Future Value = $250,000 = FV1 + FV2
5. Solve the equation for n:
- Simplify the equation and solve for n using algebraic steps.
Once you have the equation solved, you will find the number of years it will take for John to reach his goal of accumulating $250,000 in his brokerage account.
Using a spreadsheet method, in which $5000 and the accrued 12% interest of the balance on deposit a year earlier are added each year, and doing the problem "by hand", I get the time required to be almost exactly 11 years.
Continue this process 11 times:
(end of) initial deposit intrst total
Year # balance
1 42181 5000 5062 52243
2 52243 5000 6269 62512
etc