Question

Find the APR, or stated rate, in each of the following cases:

a. An effective interest of 18% compounded semiannually

b.	An effective interest of 11% compounded monthly
c.	An effective interest of 16% compounded weekly

d.	An effective interest of 13% with continuous compounding

Answer 1

The formula for APR is:

EAR = effective annual rate m = number of times the interest is compounding during the ear

a. An effective interest of 18% compounded semiannually

EAR = 18% = 0.18   m = 2

APR = 2 x [ (1 + 0.18)^1/2 -1 ] = 2 x [ (1.18)^1/2 -1 ] = 2 x [ 1.086278 -1 ] = 2 x 0.086278 = 0.1725561

APR = 17.26%

b. An effective interest of 11% compounded monthly

EAR = 11% = 0.11 m = 12

APR = 12 x [ (1 + 0.11)^1/12 -1 ] = 12 x [ (1.11)^1/12 - 1 ] = 12 x [ 1.00873459 - 1 ] = 12 x 0.00873459 = 0.10481513

APR = 10.48%

c. An effective interest of 16% compounded weekly

EAR = 16% = 0.16 m = 52

APR = 52 x [ (1 + 0.16)^1/52 -1 ] = 52 x [ (1.16)^1/52 - 1 ] = 52 x [ 1.00285831 - 1 ] = 52 x 0.00285831 = 0.148632

APR = 14.86%

d. An effective interest of 13% with continuous compounding

The formula for continuous compounding is

APR = ln(1+EAR)

EAR = 13% = 0.13   

APR = ln(1+0.13) = ln(1.13) = 0.1222176 = 12.22%