. What annual investment is required at 8% per year compounded annually to accumulate to
$100,000 at the end of 20 years?
To calculate the annual investment required to accumulate $100,000 at the end of 20 years at 8% interest compounded annually, you can use the formula for the future value of an ordinary annuity:
Future Value = Annual Investment × [(1 + Interest Rate)^(Number of Years) - 1] / Interest Rate
In this case, the future value is $100,000, the interest rate is 8%, and the number of years is 20. Plugging these values into the formula, you can solve for the annual investment:
$100,000 = Annual Investment × [(1 + 0.08)^(20) - 1] / 0.08
Simplifying the equation:
$100,000 = Annual Investment × [1.806111 - 1] / 0.08
$100,000 = Annual Investment × 0.806111 / 0.08
$100,000 = Annual Investment × 10.0763889
Dividing both sides by 10.0763889:
Annual Investment = $100,000 / 10.0763889
Annual Investment ≈ $9,912.02
Therefore, an annual investment of approximately $9,912.02 is required at an 8% interest rate compounded annually to accumulate to $100,000 at the end of 20 years.
To find the annual investment required to accumulate a certain amount, we can use the future value of an ordinary annuity formula. The formula is:
FV = PMT * [(1 + r) ^ n - 1] / r
Where:
FV = Future value (target amount)
PMT = Annual investment
r = Interest rate per compounding period
n = Number of compounding periods (years)
In this case, we know the future value (FV) is $100,000, the interest rate (r) is 8% (or 0.08), and the number of years (n) is 20. We need to solve for the annual investment (PMT).
Substituting the given values into the formula, we have:
$100,000 = PMT * [(1 + 0.08) ^ 20 - 1] / 0.08
Now we can solve for PMT:
PMT = $100,000 * 0.08 / [(1.08 ^ 20) - 1]
PMT = $100,000 * 0.08 / (4.6609637 - 1)
By simplifying the equation, we get:
PMT = $100,000 * 0.08 / 3.6609637
PMT ≈ $8,241.76
Therefore, an approximate annual investment of $8,241.76 is required at an 8% interest rate compounded annually to accumulate to $100,000 at the end of 20 years.