What are the stated rates that correspond to an effective annual rate of 10 percent given the compounding frequencies below?

Effective Annual Rate Stated Annual Rate Compounding Frequency

.10 ? 2

.10 ? 12

.10 ? 52

In general, if the effective annual rate is r_e, and the stated rate is r_s, compounded t times per year, then then the periodic rate is (r_s/t) and the effective annual rate is

(1 + (r_s/t) )^t - 1 = r_e (*)

To find r_s given r_e and t, solve equation (*) for r_s. That is (1+(r_s/t) )^t = 1+r_e => (1 + r_s/t) = (1 + r_e)^(1/t) => r_s/t = (1 +r_e)^(1/t) -1 => r_s = t [(1 + r_e)^(1/t) 1].