Calculate the rate of interest compounded yearly (j) equivalent to 8% pa compounded quarterly. Give your answer as a percentage per annum to 3 decimal places.
let the rate compounded yearly be i
then
1+i = (1 + .02)^4
1 + i = 1.08243
i = .08243 or 8.243%
To calculate the rate of interest compounded yearly equivalent to 8% pa compounded quarterly, we can use the formula:
\(j = (1 + r/n)^(n*t) - 1\)
Where:
- \(j\) is the equivalent rate of interest compounded yearly,
- \(r\) is the annual interest rate (in decimal form), which is 8% or 0.08 in this case,
- \(n\) is the number of compounding periods per year, which is 4 for quarterly compounding, and
- \(t\) is the number of years, which is 1 in this case.
Using the formula, we substitute the values and solve for \(j\):
\(j = (1 + 0.08/4)^(4*1) - 1\)
First, we simplify the fraction:
\(j = (1 + 0.02)^(4) - 1\)
Next, we calculate the value inside the parentheses:
\(j = 1.02^4 - 1\)
Now, we calculate the value of \(1.02^4\) using a calculator:
\(j = 1.082432 - 1\)
Finally, we subtract 1 from the result:
\(j = 0.082432\)
To express the result as a percentage per annum, we multiply by 100:
\(j = 8.2432\% \)
Therefore, the rate of interest compounded yearly equivalent to 8% pa compounded quarterly is 8.2432% per annum (to 3 decimal places).