Assume you are planning to invest $5,000 each year for six years and will earn 10 percent per year. Determine the future value of this annuity if your first $5,000 is invested at the end of the first year.
To determine the future value of this annuity, we can use the formula for the future value of an ordinary annuity:
FV = P * ((1 + r)^n - 1) / r
Where:
FV is the future value
P is the annual investment amount ($5,000)
r is the annual interest rate (10% or 0.10)
n is the number of years (6)
In this case, the first $5,000 is invested at the end of the first year, so we need to calculate the future value starting from the second year. Let's calculate the future value using the formula:
FV = $5,000 * ((1 + 0.10)^5 - 1) / 0.10
FV = $5,000 * ((1.10^5) - 1) / 0.10
FV = $5,000 * (1.61 - 1) / 0.10
FV = $5,000 * 0.61 / 0.10
FV = $5,000 * 6.1
FV = $30,500
Therefore, the future value of this annuity, if your first $5,000 is invested at the end of the first year, is $30,500.
To determine the future value of an annuity, we need to calculate the future value of each individual investment and then sum them up.
Since you are investing $5,000 each year for a total of six years, we need to calculate the future value of $5,000 invested at the end of each year for six years.
To calculate the future value of each investment, we can use the formula for compound interest:
Future Value = Present Value * (1 + Interest Rate)^Number of Periods
In this case, the Present Value (PV) is $5,000, the Interest Rate (r) is 10% (or 0.10), and the Number of Periods (n) is six years.
For the first investment at the end of the first year, the future value would be:
Future Value1 = $5,000 * (1 + 0.10)^1
= $5,000 * (1 + 0.10)
= $5,500
For the second investment at the end of the second year, the future value would be:
Future Value2 = $5,000 * (1 + 0.10)^2
= $5,000 * (1 + 0.10)^2
= $5,500 * (1 + 0.10)
= $5,500 * (1.10)
= $6,050
We can continue this calculation for each of the six investments and sum up the future values.
Future Value = Future Value1 + Future Value2 + Future Value3 + Future Value4 + Future Value5 + Future Value6
Future Value = $5,500 + $6,050 + Future Value3 + Future Value4 + Future Value5 + Future Value6
To calculate the future values for the remaining four investments, we can use a similar calculation:
Future Value3 = $5,000 * (1 + 0.10)^3
Future Value4 = $5,000 * (1 + 0.10)^4
Future Value5 = $5,000 * (1 + 0.10)^5
Future Value6 = $5,000 * (1 + 0.10)^6
After calculating these values, we can sum them up to find the final future value of the annuity.
I'll calculate these future values for you:
Future Value3 = $5,000 * (1 + 0.10)^3 = $5,000 * 1.331 = $6,655
Future Value4 = $5,000 * (1 + 0.10)^4 = $5,000 * 1.4641 = $7,320.50
Future Value5 = $5,000 * (1 + 0.10)^5 = $5,000 * 1.61051 = $8,052.55
Future Value6 = $5,000 * (1 + 0.10)^6 = $5,000 * 1.771561 = $8,857.80
Finally, summing up all the future values:
Future Value = $5,500 + $6,050 + $6,655 + $7,320.50 + $8,052.55 + $8,857.80 = $42,436.85
Therefore, the future value of this annuity, assuming the first $5,000 is invested at the end of the first year, is approximately $42,436.85.
1. $5000 @ end of first yr.
2. 5000*1.1 + 5000=10500 @ end of 2nd yr.
3. 10500*1.1 + 5000 = 16,550. @ end of 3rd yr.
4. 16,550*1.1 + 5000 = 23,205 @ end of
fourth yr.
5. 23,205*1.1 + 5000 = 30,525.50 @ end
of 5th yr.
6. 30,525.50*1.1 + 5000 = $38,578.05 @
end of 6th yr.