what are the units for a rate of return?

Question

My understanding of "rate" is more physics oriented. For example, distance/time is understandable for me and something I can explain. However, a rate of return:

"The return, or rate of return, can be calculated over a single period, or where there is more than one time period, the return and rate of return over the overall period can be calculated, based upon the return within each sub-period."

http://en.wikipedia.org/wiki/Rate_of_return#Calculation

I run into a problem of units. There's an initial value and final value, the rate being defined as that difference divided by the initial value.

Specifically, if a car travels so far in such amount of time, that rate has units: kilometers and hours. In a word problem, those units are quite important.

However a "rate" of twelve percent return has, seemingly, no units. Yet in a word problem, there will be units.

I'm unclear on how a rate of return has, seemingly, no units. Can it be expressed as a ratio with units?

Answer 1

You need to look at the definition of rate: it can mean a quantity of something based on another quantity, or the amount of charge with reference to calculation. Your question is about the second definition. Aside from that, the rate refers to some amount of currency, but is a ratio which is dimensionless.

Answer 2

You may be missing something in the terminology here.

From your Wikipedia link:

The change in value is 1,030 USD - 1,000 USD = 30 USD, so the return is 30 / 1,000 = 3%.

Note that the return is expressed as a percent. A return of 1 is often meant to be 100% as this is where the final value is twice the initial value if you look at the formula.

Answer 3

The rate of return has no diemension since it is a ratio of two numbers expressed using the same units. Looking at the Wikipedia page you refer to, the definition which is given is $$r=\frac {V_f-V_i} {V_i}$$ where $V_f$ is the final value and $V_i$ the initial value.

Answer 4

This brings up a good point.

In general, we define ‘rate’ as - a comparison of two different kinds of quantities (by division).

And we define ‘ratio’ as - a comparison of two quantities of the same kind (by division).

If such distinction is valid, then ‘rate’ should have a unit attached while ‘ratio’ has not.

Thus, if [V(f) – V(i)] / V(i) is need to find the “(?) of return”, then (?) should not be quoted as rate. However, if all things are clearly defned, we can “borrow” the term ‘rate’ and use it to define “the rate of return”.

Additional note: The “borrowing” act is not uncommon. For example, “interest ‘rate’” is in % and has no unit involved.