Find the effective rate of interest corresponding to the nominal rate of 8% p.a. if it is converted to half yearly
To find the effective rate of interest corresponding to a nominal rate of 8% p.a. compounded half-yearly, we can use the formula:
Effective Rate = (1 + (Nominal Rate / Number of Compounding Periods)) ^ Number of Compounding Periods - 1
In this case, since the nominal rate is 8% p.a., we can calculate the effective rate as follows:
Number of Compounding Periods = 2 (since it is compounded half-yearly)
Nominal Rate = 8% p.a.
Effective Rate = (1 + (0.08 / 2)) ^ 2 -1
= (1 + 0.04) ^ 2 - 1
= (1.04) ^ 2 - 1
= 1.0816 - 1
= 0.0816
Therefore, the effective rate of interest corresponding to a nominal rate of 8% p.a. compounded half-yearly is 8.16%.
To find the effective rate of interest corresponding to a nominal rate of 8% per annum, when it is converted to a half-yearly compounding period, you can use the formula:
Effective Interest Rate = (1 + (Nominal Rate / Number of Compounding Periods))^(Number of Compounding Periods) - 1
In this case, the nominal rate is 8% per annum, and it is compounded semi-annually.
First, we need to determine the number of compounding periods. Since the nominal rate is converted to a half-yearly period, the number of compounding periods per year is 2 (2 half-yearly periods).
Using the formula, we substitute the values:
Effective Interest Rate = (1 + (8% / 2))^2 - 1
Now, we can calculate the effective interest rate:
1. Calculate the value inside the parentheses:
(8% / 2) = 0.04
2. Add 1 to the value:
(1 + 0.04) = 1.04
3. Raise the value to the power of the number of compounding periods:
(1.04)^2 = 1.0816
4. Subtract 1 from the result:
1.0816 - 1 = 0.0816
So, the effective interest rate corresponding to a nominal rate of 8% per annum when converted to a half-yearly compounding period is 0.0816 or 8.16%.
let the effective semi-annual rate be j
(1+j)^2 = 1.08
1+j = √1.08 = 1.03923
j = .03923
2j = .07846
so the effective annual rate compounded semi-annually
= 7.846%
check:
on $1000, the amount after 1 year at 8% per annum
= 1000(1.08) = 1080
on $1000, the amount after 1 year at 7.846% compounded semi-annually
= 1000(1.03923)^2 = 1080