Question

What rate compounded quarterly is equivalent to 6% compounded semi annually?

Answer 1

Effective Annual Rate(EAR) = (1+r/n)^n - 1

We have a rate of 6% compounded semi-annually. Hence r=6%, n=2

EAR = (1+r/n)^n - 1

= (1+0.06/2)^2 - 1

= (1+0.03)^2 - 1

= (1.03)^2 - 1

= 1.0609 - 1

= 0.0609

We have some unknown interest rate which is x. This value is compounded quarterly to get some effective annual interest rate.

EAR = (1+r/n)^n - 1 = (1+x/4)^4 - 1

Set the two EAR expressions equal to one another.

(1+x/4)^4 - 1 = 0.0609

(1+x/4)^4 - 1+1 = 0.0609+1

(1+x/4)^4 = 1.0609

[(1+x/4)^4]^(1/4) = (1.0609)^(1/4)

1+x/4 = 1.014889

1+x/4 - 1 = 1.014889 - 1

x/4 = 0.014889

4*(x/4) = 4*0.014889

x = 0.059556

Therefore, 5.96% rate compounded quarterly is equivalent to 6% compounded semi annually.