Nominal Rates/Effective rate computation, confusion.

Question

Given a nominal rate of 6% per annum. Change it to an effective rate per month.

What I do is:

$$(1+\frac {0.06}{12})^{12}=(1+i)^{12}$$ where $i$ is the effective interest rate per month.

Now what if the question had said that it was the effective annual rate that was 6%, what would I do then?

Is it simply

$$1+{0.06}=(1+i)^{12}$$ where $i$ is the effective montly rate?

If so, what is the diffence between nominal rate and effective rate?

Answer 1

The effective rate of interest is the amount of money that one unit invested at the beginning of a period will earn during the period, with interest being paid at the end of the period; when we speak of the effective rate of interest we mean interest is paid once per measurement period.

An interest rate is called nominal if the frequency of compounding (e.g. a month) is not identical to the basic time unit (normally a year): interest is paid more than once per measurement period.

When interest is paid (i.e., reinvested) more frequently than once per period, we say it is payable (convertible, compounded) each fraction of a period, and this fractional period is called the interest conversion period. A nominal rate of interest $i^{(m)}$ payable $m$ times per period, where $m$ is a positive integer, represents $m$ times the effective rate of compound interest used for each of the $m$-th of a period. In this case, $\frac{i^{(m)}}{m}$ is the effective rate of interest for each $m$-th of a period.

Thus, for a nominal rate of $6\%$ compounded monthly, the effective rate of interest per month is $0.5\%$ since there are twelve months in a year.

If $i$ denotes the effective rate of interest per one measurement period equivalent to $i^{(m)}$ then we can write $$ 1 + i =\left(1 +\frac{i^{(m)}}{m}\right)^m$$ since each side represents the accumulated value of a principal of 1 invested for one year.

For any $t> 0$ we have $$ (1 + i)^t =\left(1 +\frac{i^{(m)}}{m}\right)^{mt}.$$

So in your case $(m=12)$

• $i^{(m)}=i^{(12)}=6\%$ is the nominal interest rate compounded monthly;
• $\frac{i^{(m)}}{m}=\frac{i^{(12)}}{12}=0.5\%$ is the monthly effective rate of interest:
• $i= \left(1 +\frac{i^{(m)}}{m}\right)^m-1=\left(1 +\frac{i^{(12)}}{12}\right)^{12}-1=6.17\%$ is the annual effective rate of interest

Answer 2

Finance, as practiced, is littered with legacy jargon and imprecise shortcuts. For an interest rate $r$ not too far from zero, we have the approximation:

$$ (1 + r)^n \approx 1 + r \cdot n $$

For example: $$(1+.01)^{12} = 1.1268 \approx 1.12$$

That's close enough for the marketing department (and not close enough for accounting department)!

So a bank might say it's charging you "a nominal interest rate of 12 percent compounded monthly" instead of saying an "annual effective interest rate of 12.68 percent."

In this context, the effective interest rate is the mathematically relevant concept, and the nominal interest rate is all about marketing, back of the envelope shortcuts, legacy terminology etc...

In some sense, this is like measuring length in feet or meters, the important thing is not to mix up what numbers are in what units.