*"And where two waging fires meet together*

*They do consume the thing that feeds their fury.*

*Though little fire grows great with little wind,*

*Yet extreme gusts will blow out fire and all."*

- Shakespeare

One of the great tools available to evaluate any idea is to examine the results of using extremely high and low values. Those with a quantitative background may recognize this as the concept of limits from calculus, but you can intuitively use extremes (i.e. limits) to evaluate ideas without any heavy math. All you need is arithmetic skills and a little creativity.

**An Example**
A great example investing example is the use of the PEG ratio. I often read articles that select stocks primarily on the basis of the PEG ratio, but let me first explain the basics of the PEG ratio in case you've never heard of it. The P/E ratio (price/earnings) is probably the most basic and widely used valuation indicator and is simply the price of a stock divided by its earnings. For example, if the price of a stock is $50 and its earnings are $5, then the P/E is 10. The growth rate is the yearly percentage growth in earnings, so if earnings are $5 this year and $5.50 next year, the growth rate is 10. The PEG ratio is the P/E divided by the growth. For example, if a stock has a P/E of 10 and a growth rate of 10, the PEG ratio is 1.

Now if Company A and Company B are priced the same relative to current earnings, but Company A is growing much faster, then Company A is cheaper because it will soon be a much larger company. Thus, companies with higher growth rates will tend to have higher P/E ratios to equalize the valuation. This makes intuitive sense and we also see that in practice faster growing stocks generally do have higher P/E ratios.

The PEG ratio is an attempt to adjust the P/E with the growth rate to arrive at a single number that can tell you whether the stock is overvalued or undervalued. Most PEG strategies postulate that a PEG of one is fair value. Thus, if the PEG ratio is greater than one, the stock is overvalued, and if it's less than one, it's undervalued. Some people screen for stocks with a PEG of less than one. However, in my opinion, investment indicators often start to go sour when we try to reduce things to one number and one boundary condition.

**Extremely Low Values**
So let's use extreme values to examine whether blindly selecting stocks with a PEG of less than one makes sense. By typical PEG criteria, if a stock has a growth rate of 10, then the P/E must be less than 10 for us to select it. If the P/E is more, then the PEG ratio will be above one and so it would be considered overvalued. Now let's drop the growth rate to 5. Now we're saying that the P/E must be less than 5 or it's overvalued. (The strategy is already looking dubious.) Now let's choose an extremely low value. For a growth rate of 1, the P/E must be less than 1 to be considered a good investment. Intuitively, this does not make sense, right? At a growth rate of zero, the investment is always considered overvalued at any price. Clearly something is not right with this formula. It implies that a company that is not growing is worth nothing.

**Extremely High Values**
Now let's try the other extreme. If a company has a P/E of 20, it's still considered a good value if the growth rate is 21. For a P/E of 50 and growth rate of 51, it's still undervalued by the formula. Now how about a P/E of a million? Not possible you say?

*It is possible.* A development stage company may lose money for a few years until it generates enough sales to move into the black. It's entirely possible that the breakeven year could be some very low number for earnings followed by a significant amount. So if total earnings are $1 a year ago and then $1 million the next year, then the growth rate is one million. So by the PEG criteria, it's still undervalued if the P/E is less than one million. Hence, the company is (supposedly) still attractive at one million times one million = one trillion dollars! So clearly the formula does not make sense at this extreme either.

**What did we learn?**
1. The formula undervalues slower growing stocks, and the slower the growth, the less the formula makes sense.

2. The formula overvalues faster growing stocks, and the faster the growth, the less the formula makes sense.

3. The formula is not meaningful for investments that don't grow at all (e.g. bonds).

4. The formula is clearly an oversimplification. The general theory that faster growing stocks deserve higher valuations has merit, but the PEG approach is too simple.

**Implications**
If you still want to use P/E ratios and growth rates, you might look for another formula that handles high and low values much better. Such formulas do exist, but are more complicated. You might also choose to apply the PEG formula only in a narrow range (e.g. growth between 9 and 12 percent), or only as a relative measure between two very similar stocks. Or perhaps (like I advocate), you might want to simply examine the two numbers separately and not try to reduce things to a single number.

**A General Evaluation Framework**
So here is the general pattern you can use to evaluate any financial idea.

1. Think about exactly what is being stated in the idea.

2. Try to pick out the key variables that can be different and how they relate to each other.

3. Try to think of the lowest and highest possible values. Don't worry about whether these values are common. The idea is not to plan for likely scenarios, but to poke holes in the theory you're examining.

4. Does the idea still make sense for these high and low values? If not, it's telling you something important. Maybe it's an oversimplified rule of thumb that doesn't apply in many cases. Maybe it's just plain wrong!

6. Think about how you can improve the idea to work with the extreme values you used. This may lead to a deeper understanding of the problem and the solutions to it.