A bank offers a rate of 5.3% compounded semi-annually on its four year GICs(Guaranteed Investment Certificates). What monthly and annually compounded rates should it quote in order to have the same effective interest rate at all three nominal rates?
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first calculate the annual equivalence
(1+.053/2)^2 = 1.0537 ergo, the equivalent annual rate is 5.37%
Let the monthly equivalent rate be x.
So (1+x/12)^12 = 1.0537
12ln(1+x/12) = ln(1.0537) = .0523078
ln(1+x/12) = .00435898
(1+x/12) = e(.00435898) = 1.0043685
x/12 = .0043685
x = .05242
annual rate is 5.242%
To find the monthly and annually compounded rates that would result in the same effective interest rate as a 5.3% compounded semi-annually rate, we need to use the concept of equivalent interest rates.
First, let's determine the effective interest rate for the 5.3% compounded semi-annually rate. The formula to calculate the effective interest rate is:
A = P(1 + r/n)^(nt)
Where:
A = the final amount (principal + interest)
P = the principal (initial amount)
r = annual interest rate (in decimal form)
n = number of compounding periods in one year
t = number of years
For the 5.3% compounded semi-annually rate:
r = 5.3% = 0.053 (annual interest rate in decimal form)
n = 2 (semi-annual compounding)
t = 4 (number of years)
Plugging in these values into the formula, we get:
A = P(1 + 0.053/2)^(2*4)
Next, let's solve for the monthly compounded rate. We need to find the monthly interest rate that, when compounded monthly for 4 years, would give us the same effective interest rate as the 5.3% compounded semi-annually rate.
The formula for the monthly compounded rate is:
r_m = (1 + r/n)^(n/m) - 1
Where:
r_m = monthly interest rate
r = annual interest rate (in decimal form)
n = number of compounding periods in one year
m = number of compounding periods in one month
Plugging in the values from above, we have:
r_m = (1 + 0.053/2)^(2/12) - 1
Now let's solve for the annually compounded rate. We need to find the annual interest rate that, when compounded annually for 4 years, would give us the same effective interest rate as the 5.3% compounded semi-annually rate.
The formula for the annually compounded rate is:
r_a = (1 + r/n)^(nt) - 1
Where:
r_a = annual interest rate
r = annual interest rate (in decimal form)
n = number of compounding periods in one year
t = number of years
Plugging in the values from above, we have:
r_a = (1 + 0.053/2)^(2*4) - 1
By calculating these two rates, we can find the monthly and annually compounded rates that would result in the same effective interest rate as the 5.3% compounded semi-annually rate.