# Limitation of ratios ?

1. ### musicgold

183
Hi,

Consider a hypothetical. An investor short* sells Stock A for $100 and uses that money to buy Stock B for$100. After one month, stock A drops in value by 5% to 95 and stock B rises 10% to 110. The investor closes his investment ( sells stock B for $110, uses$95 of that to buy a stock A and returns it to its owner) and has made a profit of $15 ($5 on stock A and $10 on stock B). The conventional return on investment (ROI) calculation would give an infinite return. ROI = returns / Investment = 15 / 0 = infinity My argument is that the ROI formula is not a suitable measure in this situation. 1. Is my argument correct? 2. Is there a physical process that is equivalent to the above example? I am trying to understand if there are any natural phenomena that generate something from nothing? ( my guess is it is impossible) Thanks, MG. Short selling a stock means selling a borrowed stock. Our investor borrowed the stock from another investor and sold it for$100. The borrowed stock is returned when the position was closed, i.e. our investor bought back one stock A from the market and return it to its owner. You can read more on short selling here.

Last edited: Jan 19, 2011
2. ### sin2beta

28
Why is your investment 0? I would say that one invested $100 dollars. The initial stock sold was still worth$100 and was an investment. Does that make sense?

3. ### musicgold

183
Please see my explanation in the original post. Thanks.

4. ### sin2beta

28
I believe that the investment whether borrowed or not is still considered an investment.

But if I am wrong (which is quite possible, I don't know finance or economics) and short selling is not an investment, then the ROI formula makes no sense to use. It is return / investment. When the investment is 0, ROI = return / 0. This does not equal infinity. Dividing by zero is undefined. This makes sense. Without an investment, one cannot have a return on it. Thus the return on investment is undefined.

5. ### yuiop

Your stockbroker will probably have a good idea that your good for the $100 and that he will get it back one way or another, so the$100 is probably backed by assets. You could just as easily borrow 100 from your bank and invest it on the stock market. The bank will ensure that you able to repay the debt. They probably would not lend you a million dollars if they did not think you had the assets (house, income etc) to pay it back. So the maximum amount you borrow is related to your worth so it is not exactly a free investment. In an extreme example, a homeless destitute hobo would not be able to short sell a $100 of stocks despite the fact it appears he needs to invest nothing in the venture. Sin2beta is right that the$100 is the investment and it makes mathematical sense too (but he is wrong that x/0 is undefined if $x \ne 0$). If you made a bad investment and lost the \$100 you would be expected to pay it back, so you are risking your own assets.

6. ### sin2beta

28
This may be hijacking the thread, but why is x/0 when $x \ne 0$ not undefined. I'm not saying your wrong. I'm just wondering if there is something I am thinking about incorrectly since I haven't really revisited that aspect of math since middle school. However, for anything I can think of, in elementary arithmetic and algebra under the number sets in question for ROI, it would be undefined. Could you shed some light on what you mean?

7. ### yuiop

I am not sure how to prove this in a formal way, but here is a simple illustration.

Let us assume that $0*\infty$ is undefined, then all the following statements are valid:

$$\infty*0 = -2 \rightarrow -2/0 = \infty$$
$$\infty*0 = -1 \rightarrow -1/0 = \infty$$
$$\infty*0 = 0 \rightarrow 0/0 = \infty$$
$$\infty*0 = +1 \rightarrow +1/0 = \infty$$
$$\infty*0 = +2 \rightarrow +2/0 = \infty$$

So x/0 defined as infinity is consistent with $0*\infty$ being undefined.

Note that 0/0 is an exception since:

$$-2*0 = 0 \rightarrow 0/0 = -2$$
$$-1*0 = 0 \rightarrow 0/0 = -1$$
$$0*0 = 0 \rightarrow 0/0 = 0$$
$$+1*0 = 0 \rightarrow 0/0 = +1$$
$$+2*0 = 0 \rightarrow 0/0 = +2$$

So clearly 0/0 is undefined as it can take on any value.

This definition of $x/0 = \infty$ is often used in physics, but in that context it usually means that in the limit that y goes to zero, x/y goes to infinite. In the strictly mathematical sense, it seems most mathematicians would disagree with me and strongly agree with your assertion that x/0 is undefined or indeterminate. See http://mathforum.org/dr.math/faq/faq.divideby0.html ... I usually play around with physics more than mathematics so I might well be wrong :(

P.S. In the "Ask Dr Math" forum linked to above, Dr Margaret gives this argument:
Her statement that "anything times 0 is always zero" is not entirely true. Call that rule 1. We can also say that "anything times $\infty$ is always infinity". Call that rule 2. There is now a contradiction when we ask what is the result of 0*$\infty$? Rule 1 says the answer is zero and rule 2 says the answer is infinity. So we might conclude that zero times infinity is indeterminate and can answer Dr Margaret's question with x = $\infty$, because $\infty$*0 = 5 is one possible solution (among many) to $\infty$*0.

Last edited: Jan 20, 2011
8. ### sin2beta

28
I don't think that works. If nothing else just because one can not divide by 0 in that way. If one could then we would have the classic 1=2 proof as valid. http://mathforum.org/library/drmath/view/55792.html

9. ### yuiop

Well I have stated that 0/0 is indeterminate so in the 1=2 proof at step 4 we have:

4) (x + y)*0 = y*0 (factor)

and at step 5 we have:

5) (x + y)*0/0 = y*0/0 (divide out (x - y))

so step 6 should be:

6) indeterminate = indeterminate (0/0 = 0/0)

so no 1=2 proof.

Last edited: Jan 20, 2011
10. ### yuiop

Actually I just found the flaw in own arguments.

If we start with $\infty*0 = x$ then the next step is

$$\infty*0/0 = x/0$$

so it appears that x/0 is indeterminate and I withdraw my earlier claim that $x/0=\infty$ and concede sin2beta is correct.

11. ### AlephZero

7,300
Your example is too hypothetical. In reality there will be extra costs incurred in carrying out the transactions, depositing money to cover possible margin calls (which incurrs the cost of not being able to use that money for something else), etc. There is also the time value of the money involved in the transactions (inflation and interest rates).

When you include those factors, the "net investment" is not zero and you will get a sensible "real world" ROI, which may be large on a leveraged investment strategy like this, but it will not be infinite.