Which is a better rate of interest, 16% compounded quarterly or 16 1/4% compounded semi-annually?
To determine which rate of interest is better, we need to calculate the effective interest rate for both options.
For the first option, the interest rate is 16% compounded quarterly. This means that the interest is calculated and added to the principal every three months. To calculate the effective interest rate, we can use the formula:
Effective Interest Rate = (1 + (r/n))^n - 1
Where:
r = annual interest rate (16% or 0.16)
n = number of times interest is compounded per year (quarterly, so n = 4)
Plugging in the values, we have:
Effective Interest Rate = (1 + (0.16/4))^4 - 1
= (1 + 0.04)^4 - 1
= (1.04)^4 - 1
= 1.16985856 - 1
= 0.16985856 or 16.985856%
So, the effective interest rate for the first option is approximately 16.986%.
For the second option, the interest rate is 16 1/4% compounded semi-annually. This means that interest is calculated and added to the principal twice a year. To calculate the effective interest rate, we use the same formula as before:
Effective Interest Rate = (1 + (r/n))^n - 1
In this case, the annual interest rate is 16.25% or 0.1625, and the interest is compounded semi-annually, so n = 2.
Plugging in the values, we have:
Effective Interest Rate = (1 + (0.1625/2))^2 - 1
= (1 + 0.08125)^2 - 1
= (1.08125)^2 - 1
= 1.170015625 - 1
= 0.170015625 or 17.0015625%
So, the effective interest rate for the second option is approximately 17.002%.
Comparing the two effective interest rates, we can see that the second option, with an effective interest rate of approximately 17.002%, has a slightly higher rate of interest compared to the first option, which has an effective interest rate of approximately 16.986%. Therefore, the second option, 16 1/4% compounded semi-annually, is the better rate of interest.