How to work out weights for a portfolio based on an inverse ratio with positive and negative values?

I am trying to work out how to determine weights for the assets in order to form a portfolio. The ratio I am using is EV/EBIT, hence the smaller the better. The problem is I don't know how to handle it when EV < 0. Obviously that is kind of a 'free lunch' mathematically speaking and I realise the discontinuity at x/0 is what messes things up in a way. Would anyone be able to suggest something?

Thanks!

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You mean when EBIT<0? Then just take EBIT/EV instead. – John Jan 31 '13 at 17:34
No, I don't select any assets with EBIT < 0. It is for a few when market cap < net cash that you get EV < 0. – ArturoP Jan 31 '13 at 19:02

what I usually do for my calculations is the following. Lets say that we want a neutral EV/EBITDA. And we have our companie's EV/EBITDA: A,B,C. We want to invest 100% and our weights will be called x,y,z.

Therefore we have a system formed with 2 equations and 3 unknown variables: x+y+z=100% A*x+B*y+C*z=0 and 3 constraints: x>0,y>0,z>0.

Just set this on Excel, apply the solver and you will get an easy and fast solution :)

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Hi arodrisa, welcome to Quant.SE! I don't see how this helps with the case that $\mathrm{EV} \leq 0$ or $\mathrm{EBIT} = 0$. Can you elaborate? – Bob Jansen Sep 18 '14 at 9:22
If he only wants to calculate the weights, in case that there is any of the values negative, or 0, if you don't set any constrain it will be compensated by the other factors. Let's say that EV/EBITDA of the different cases is: A=1,B=-1, C=0; and we want a total EV/EBITDA of 1. If there is no constrains we have as solution x=100%, y=z=0; or x=200%, y=100%,z=0. You can add more constrains depending on your problem. – arodrisa Sep 18 '14 at 9:37

An EV<0 is an "ideal" situation (for a value investor). When you find such a rare bird, give it the maximum portfolio weight allowed by your investment policy.

(Given your statement, "the smaller the better," I'm assuming that your portfolio weights are some "reciprocoal" of your calculated ratio.)

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ArturoP, as John said, instead of minimizing EV/EBIT, you could as well maximize the inverse ratio EBIT/EV, thus eliminating the division by 0.

You could think of the ratio (e.g. EBIT/EV) as a utility function, i.e. how well you evaluate a company based on the 2 variables, such as U(EBIT,EV) = EBIT/EV.

You'll notice that the ratio above works well when EBIT >= 0. But it is less intuitive when EBIT < 0, since presumably expansive non-profitable companies are the worst case. A workaround for that could be splitting your domain into 2, and define a negative utility as the example below:

\begin{align*} W \sim U(EBIT,EV) = \begin{cases} \frac{EBIT}{EV}, & \text{ if } EBIT >= 0\\ EBIT.EV, & \text{ if } EBIT < 0 \end{cases} \end{align*}

So that your allocation weights W are somehow proportional to your utility.

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