find the accumulated value of an investment of $10000 for 5 years at an interest rate of 5.5% if the money is a. compounded semiannually; b. compounded monthly; c. compounded continuously.
A)$13,116.51
B)$13,140.67
C)$13,157.04
D)$13,165.31
To find the accumulated value of an investment with different compounding frequencies, 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 compounding periods per year
t = Number of years
Let's apply this formula to each scenario:
a. Compounded Semiannually:
In this case, the interest is compounded twice a year, so we have:
P = $10,000
r = 5.5% = 0.055 (decimal form)
n = 2 (semiannual compounding)
t = 5 years
The formula becomes:
A = $10,000 * (1 + 0.055/2)^(2*5)
Calculating the expression inside the parenthesis:
A = $10,000 * (1.0275)^(10)
Calculating the exponent:
A = $10,000 * 1.62557
Therefore, the accumulated value after 5 years with semiannual compounding is approximately $16,255.70.
b. Compounded Monthly:
Using the same formula, but with monthly compounding (n = 12), we have:
P = $10,000
r = 5.5% = 0.055 (decimal form)
n = 12 (monthly compounding)
t = 5 years
The formula becomes:
A = $10,000 * (1 + 0.055/12)^(12*5)
Calculating the expression inside the parenthesis:
A = $10,000 * (1.0045833)^(60)
Calculating the exponent:
A = $10,000 * 1.3481615
Therefore, the accumulated value after 5 years with monthly compounding is approximately $13,481.62.
c. Compounded Continuously:
When the interest is compounded continuously, we use a different formula:
A = P * e^(rt)
Where e is Euler's number (approximately 2.71828).
Using the given values:
P = $10,000
r = 5.5% = 0.055 (decimal form)
t = 5 years
The formula becomes:
A = $10,000 * e^(0.055*5)
Calculating the exponent:
A = $10,000 * 1.308024
Therefore, the accumulated value after 5 years with continuous compounding is approximately $13,080.24.
To summarize:
a. Compounded semiannually: $16,255.70
b. Compounded monthly: $13,481.62
c. Compounded continuously: $13,080.24
To find the accumulated value of an investment of $10,000 for 5 years at an interest rate of 5.5% compounded semiannually, monthly, and continuously, we can use the formula for compound interest:
A = P * (1 + r/n)^(n*t)
Where:
A = Accumulated Value
P = Principal Amount (initial investment)
r = Annual Interest Rate (as a decimal)
n = Number of times interest is compounded per year
t = Number of years
a. Compounded Semiannually (n = 2):
A = $10,000 * (1 + 0.055/2)^(2*5)
A = $10,000 * (1 + 0.0275)^10
A = $10,000 * (1.0275)^10
A ≈ $12,772.13
b. Compounded Monthly (n = 12):
A = $10,000 * (1 + 0.055/12)^(12*5)
A = $10,000 * (1 + 0.004583)^60
A = $10,000 * (1.004583)^60
A ≈ $12,835.01
c. Compounded Continuously:
A = $10,000 * e^(0.055*5)
A ≈ $12,858.71
Therefore, the accumulated value of the investment after 5 years at an interest rate of 5.5% would be:
a. Compounded semiannually: $12,772.13
b. Compounded monthly: $12,835.01
c. Compounded continuously: $12,858.71