What is the effective rate of the interest rate of the interest when the rate is 6 percent compounded continuously?
Find the effective rate of interest that corresponds to an annual rate of
8% compounded continuously
To calculate the effective rate of interest for an interest rate compounded continuously, you can use the formula:
Effective interest rate = e^r - 1
where:
- "e" is the mathematical constant approximately equal to 2.71828
- "r" is the nominal interest rate
In this case, let's calculate the effective interest rate for a 6% interest rate compounded continuously:
Step 1: Convert the percentage to a decimal:
6% = 0.06
Step 2: Plug the decimal value into the formula:
Effective interest rate = e^(0.06) - 1
Step 3: Calculate the result using a calculator or software that supports exponentiation with e:
Effective interest rate ≈ 1.0618 - 1
Step 4: Simplify the expression:
Effective interest rate ≈ 0.0618 or 6.18%
Therefore, the effective interest rate for a 6% interest rate compounded continuously is approximately 6.18%.
To find the effective rate of interest when the interest is compounded continuously at a given rate, you can use the formula for continuous compound interest:
A = P * e^(rt)
Where:
A = the final amount
P = the principal (initial amount)
e = Euler's number (approximately 2.71828)
r = the interest rate (in decimal form)
t = the time (in years)
In this case, you want to find the effective rate, so the final amount would be the same as the principal.
Let's calculate it for an interest rate of 6 percent compounded continuously.
Step 1: Convert the interest rate to decimal form:
6 percent = 6/100 = 0.06
Step 2: Substitute the values into the formula:
A = P * e^(rt)
P = A (the final amount) since it's the effective rate
r = 0.06
t = 1 (assuming 1 year)
So, the formula becomes:
A = A * e^(0.06*1)
Step 3: Simplify the equation:
1 = e^(0.06)
Step 4: Solve for A:
Using a natural logarithm (ln) on both sides:
ln(1) = ln(e^(0.06))
0 = 0.06
This means that the final amount (A) is equal to the initial amount (P). Therefore, the effective rate of interest when the interest is compounded continuously at 6 percent is 6 percent.