Suppose an investment account is opened with an intial deposit of $12,000 earning 9.6% interest. Round all answers to the nearest dollar.
a. How much will the account be worth after 20 years if it is compounded monthly? $
b. How much will the account be worth after 20 years if it is compounded continuously? $
a)09.6/12=0.008
12000 x (1.008)^20x12=12000x(1.008)^240=
$81228.60
b)12000 x e^.096x20=12000 x e^1.92=$81851.50
To calculate the future value of an investment account, we can use the formula:
Future Value = Principal * (1 + Interest Rate/Number of Compounding Periods)^(Number of Compounding Periods * Time)
Let's start by calculating the future value of the account if it is compounded monthly.
a. Compounded Monthly:
Interest Rate = 9.6% = 0.096 (convert to decimal)
Number of Compounding Periods per year = 12 (since it is compounded monthly)
Time = 20 years
Using the formula, we have:
Future Value = $12,000 * (1 + 0.096/12)^(12 * 20)
To find the solution, plug the values into a calculator or use a spreadsheet program:
Future Value = $12,000 * (1 + 0.008)^(12 * 20) = $12,000 * (1.008)^240
Calculating this expression, we get:
Future Value = $12,000 * 5.9831 = $71,797 (rounded to the nearest dollar)
So, if the account is compounded monthly, it will be worth $71,797 after 20 years.
b. Compounded Continuously:
To calculate the future value if the account is compounded continuously, we use the formula:
Future Value = Principal * e^(Interest Rate * Time)
Where e is the mathematical constant approximately equal to 2.71828.
Future Value = $12,000 * e^(0.096 * 20) = $12,000 * e^(1.92)
To find the solution, use a calculator or a spreadsheet program to calculate e^(1.92) and multiply it by $12,000:
Future Value ≈ $12,000 * 6.831 = $81,972 (rounded to the nearest dollar)
So, if the account is compounded continuously, it will be worth approximately $81,972 after 20 years.