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# Continuous compounding - Miller - Mathematics and Statistics

#### tkvfrm

##### New Member
I have a question here. in Miller's book page 17 - problems..Question 2: Nominal monthly rate 5% - calculate equivalent in continuous. The answer given is 4.99%.
Question: When nominal rate is 5% compounding has to be more than 5%. My calculation using the 'e' function in BA II plus gives 5.12% . Using ICONV also gives 5.12. Am I doing something wrong here? Will be happy to have a response from the group.

#### brian.field

##### Well-Known Member
Subscriber
Nominal Monthly Rate (k) is 5.0%. Nominal means that you need to use the following formula to transform in into an effective annual rate (i): (1 + k/m)^(mt) = 1 + i.

Then, we use the following formula to arrive at the equivalent continuously compounded rate (d): exp^d = 1 + i which gives ln(1 + i) = d. Now, plug and chug.

k = .05
m = 12

(1 + k/m)^(mt) = (1 + 0.05/12) ^ 12 = 1 + 0.051162 = 1.051162. This implies that i = 0.051162.

Now, ln(1 + i) = d implies ln(1.051162) = 0.049896 = d = the equivalent continuously compounded rate.

#### brian.field

##### Well-Known Member
Subscriber
"Question: When nominal rate is 5% compounding has to be more than 5%."

This is somewhat true but missing the mark slightly. The continuous compounded rate that would be equiavelent to a 5% nominal rate compounded monthly must be LESS than the nominal rate as the effect of compounding increases as the number of periods in which to compound increases.

5% compounded annually is 1.05.
5% compounded semi-annually is (1 + 0.05/2 )^2 = 1.0506025 which is greater than 0.05 so you are correct in that intuition.

But, the question is asking for the continuous rate that grows to equate to a 5.0% nominal rate compounded monthly which would be the d in the following equation.

e^d = (1 + k/m)^(m*t)

and since the number of compounded periods is greater than 12 (infinite number of periods) the equivalent rate must be LESS than the corresponding nominal rate compounded monthly.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
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!

#### tkvfrm

##### New Member
Thank you very much gentlemen! Much obliged at the indepth interest my thread has generated.I missed the subtlety in the question. A valuable lesson learnt. Thank you/

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