Find the accumulated value of an investment of $10,000 for 3 years at an interest rate of 6% if the money is a)compounded seiannually,b)compounded quarterly, c) compounded monthly, d) compounded continually
To find the accumulated value of an investment, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Accumulated value
P = Principal amount (initial investment)
r = Annual interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Number of years
Now, let's calculate the accumulated value for each scenario:
a) Compounded semiannually:
In this case, we will compound the interest twice a year.
A = 10,000(1 + 0.06/2)^(2*3)
A = 10,000(1 + 0.03)^6
A = 10,000(1.03)^6
A ≈ $11,895.74
b) Compounded quarterly:
Here, we will compound interest four times a year.
A = 10,000(1 + 0.06/4)^(4*3)
A = 10,000(1 + 0.015)^12
A = 10,000(1.015)^12
A ≈ $11,948.68
c) Compounded monthly:
For monthly compounding, the interest is compounded twelve times a year.
A = 10,000(1 + 0.06/12)^(12*3)
A = 10,000(1 + 0.005)^36
A = 10,000(1.005)^36
A ≈ $11,977.25
d) Compounded continually:
To calculate the continuously compounded interest, we can use the formula:
A = Pe^(rt)
Where e represents Euler's number and is approximately equal to 2.71828.
A = 10,000e^(0.06*3)
A = 10,000 * 2.71828^(0.18)
A ≈ $11,914.92
Therefore, the accumulated values for each scenario are approximately:
a) $11,895.74
b) $11,948.68
c) $11,977.25
d) $11,914.92