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Find the APR, or stated rate, in each of the following cases:
a. An effective interest of 6% compounded semiannually
b. An effective interest of 6% compounded monthly
c. An effective interest of 10% compounded weekly
d. An effective interest of 13% with continuous compounding
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♦ Relevant knowledge
The annual percentage rates (APR) and the effective annual rate (EAR) are two methods of quoting rates of interest on loans. The difference between these two can be explained by the compounding effect of interest. Suppose the effective periodic rate is r, and interest compounds T times a year, APR is simply T*r, but effective annual rate is (1+r)T-1.
a). Annual rate of 6%
Thus, APR = 1+EAR (1/n) – 1.
Needed, APR compounded semi-annually, so here, N = 2
Therefore, the APR = 2*(1.06(1/2),) – 1 = 5.91%
b). Effective annual rate = 6 %
Thus, APR = 1+EAR (1/n) – 1.
Needed, APR compounded every month, so here, N = 12
Therefore, the APR = 12*(1.06 (1/12)) – 1 = 5.84%
c). 10% annual effective rate
Thus, APR = 1+EAR (1/n) – 1.
Needed, APR compounded every other week, so here, N = 52
Therefore, the APR = 52*(1.10(1/522)) – 1 = 9.54%
d). Annual rate of 13%
APR compounded continuously
APR = ln (1+EAR)
Thus, the APR = ln(1)3. = 12.22%