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?
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It would take 4.73 years more than 4 1/2 years to accumulate $250,000
To calculate the number of years it will take for John to reach his goal, we can use the future value of an annuity formula.
The future value of an annuity formula is given by:
FV = P * [(1 + r)^n - 1] / r
Where:
FV = future value
P = annual contribution
r = annual interest rate
n = number of years
In this case, we can substitute the given values into the formula:
$250,000 = $5,000 * [(1 + 0.12)^n - 1] / 0.12
Simplifying the equation, we have:
50 = [(1.12)^n - 1]
Next, we need to solve for n.
Using logarithms, we can solve for n:
log[(1.12)^n - 1] = log(50)
n * log(1.12) = log(50) + log(1 + 1/(1.12)^n)
Using logarithmic properties, this equation can be rewritten as:
n = [log(50) + log(1 + 1/(1.12)^n)] / log(1.12)
Using a calculator to evaluate the right side of the equation, we find:
n ≈ 10.21 years
Therefore, it will take approximately 10.21 years for John to reach his goal of $250,000 in the brokerage account.
To calculate the number of years it will take for John to reach his goal of $250,000, we can use the future value of an ordinary annuity formula. This formula calculates the future value of a series of equal periodic payments. In this case, John will be making annual contributions of $5,000 to his brokerage account.
The formula to calculate the future value of an ordinary annuity is:
FV = P * [(1 + r)^n - 1] / r
Where:
- FV is the future value
- P is the annuity payment
- r is the interest rate per period
- n is the number of periods
In this case:
- FV = $250,000 (John's goal)
- P = $5,000 (annual contribution)
- r = 12% (interest rate per period, which we need to convert to a decimal by dividing by 100)
- n is the unknown, the number of years we're trying to find
Substituting the given information into the formula, we have:
$250,000 = $5,000 * [(1 + 0.12)^n - 1] / 0.12
Now, we can rearrange the formula to solve for n:
[(1 + 0.12)^n - 1] / 0.12 = $250,000 / $5,000
Simplifying further:
[(1.12)^n - 1] / 0.12 = 50
To isolate the exponent, we can multiply both sides by 0.12:
(1.12)^n - 1 = 6
Now, we need to solve for n using logarithms. Taking the natural logarithm (ln) of both sides of the equation:
ln((1.12)^n - 1) = ln(6)
Next, we can use logarithmic properties to bring the exponent down:
n * ln(1.12) = ln(6) + ln(1)
Since ln(1) is zero, the equation simplifies further:
n * ln(1.12) = ln(6)
Finally, we can solve for n by dividing both sides of the equation by ln(1.12):
n = ln(6) / ln(1.12)
Calculating the values using a calculator or software, we find:
n ≈ 19.72
Therefore, it will take approximately 19.72 years for John to reach his goal of $250,000 in the brokerage account, assuming an expected annual return of 12% and annual contributions of $5,000.