What annual nominal rate compounded semiannually has the sam

What annual nominal rate compounded semi-annually has the same APY as 2.6% compounded quarterly? Express your answer as a percentage correct to three decimal places.

Solution

Say that you initial invest $1. Then after 1 year at 2.6% interest compounded quarterly, your $1 will be worth
A = 1*(1 + 0.026/4)^4 = $1.0262546 …………..(1)

Now assume we instead compound semi-annually, at an interest rate of r%. Then at the end of one year your money will be worth

A = 1*(1 + r/2)^2 ………… (2)

We would like quantity (2) to be the same as quantity (1), and we will achieve this by specifying r. So the equation to solve (for r) is

(1 + r/2)^2 = 1.0262546

==> (1 + r/2) = 1.01304225

==> r = 0.0260845