Great points

@brian.field ! At the same time, this is a deceptively difficult question (in my humble opinion). Consider the question itself: "2. The nominal monthly rate for a loan is quoted at 5%. What is the equivalent annual rate? Semiannual rate? Continuous rate?"

Because we can answer the first question (what is the equivalent annual rate?) with (1+5%/12)^12 - 1 = 5.11619%, we are

**tempted** (but wrong) to answer the second question (equivalent semiannual rate?) with (1+5%/2)^2-1 = 5.06250%. This is wrong, the answer is here not 5.06%; it is the answer to a different question, namely, what is the effective annual rate of 5.0% compounded semiannuallly.

The answer to the second question (i.e., what is the semi-annual rate that is equivalent to 5.0% monthly rate?) is correctly given by Miller as 5.05% ~= 5.05237% = (sqrt[(1+5%/12)^12]-1)*2. Because (1+5.05237%/2)^2 = (1+5.0%/12)^12. Continuing, the correct answer the third question (i.e., what is the continuous rate that is equivalent to 5.0% monthly rate?) is given by 12*LN(1+5%/12) = 4.98961% ~= 4.99%

So the key really is

@brian.field 's statement that "

**the question is asking for the continuous rate that grows to equate to a 5.0% nominal rate compounded monthly." **
It's instructive. One key point is that a nominal (aka, stated) rate is always a feature but is insufficient by itself. If we say "the nominal rate for a loan is quoted at 5.0%," we still need compound frequency. Hence the assumption given of "the nominal monthly rate for a loan is quoted at 5%" is technically accurate. The 5.0% is a nominal (aka, stated)

*per annum* rate; but a

*nominal rate of 5.0% per annum* has infinite solutions, it is not specific. The higher the compound frequency, the greater the

*effective *return. When we get the answer to the second question "What is the equivalent semiannual rate [ie, equivalent to the monthly rate of 5.0%]?", we return 5.05% which is

**also a nominal rate**! It just happens to be the nominal rate that, if compounded semiannually, produces an effective (annual) return that is identical to the nominal rate of 5.0% when compounded monthly. Thanks!

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