If the effective annual interest rate is 6%, what is the nominal rate or APR (Annual Percentage Rate), given monthly compounding?

Question

BlackTom AI · Accepted Answer

The question asks for the nominal APR when the effective annual rate is 6% with monthly compounding. 
First, recall the relationship between APR (monthly compounded) and EAR: (1 + APR/12)^12 = 1 + EAR. Here, EAR = 0.06, so (1 + APR/12)^12 = 1.06. 
Next, solve for APR/12 by taking the 12th root: 1 + APR/12 = 1.06^(1/12). Numerically, 1.06^(1/12) ≈ 1.004867. Therefore APR/12 ≈ 0.004867. 
Multiplying by 12 yields APR ≈ 0.05840, or about 5.84%. 
Now examine the options: 
- Option a: 5.84% – This matches the calculated nominal APR that, when compounded monthly, produces a 6% EAR. 
- Option b: 6.02% – This is slightly higher than the correct APR; using 6.02% monthly would yield an EAR a bit above 6%, not exactly 6%. 
- Option c: 7.02% – This is far too high; even with monthly compounding, such a nominal rate would produce a noticeably larger EAR than 6%. 
- Option d: 3.43% – This is too low; it would yield an EAR well below 6% with monthly compounding. 
- Option e: 5.12% – This is in the neighborhood of the correct value but underestimates the APR; plugging it back into (1 + 0.0512/12)^12 would give an EAR less than 6%. 
In summary, the math shows that a nominal APR of approximately 5.84% is the value that compounds monthly to achieve a 6% EAR, while the other options fail to satisfy the (1 + APR/12)^12 = 1.06 relationship.

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