### Ranking Functions

Tuesday, September 15th, 2009In order to investigate cause and effect relationships, we need to rank companies at the start of a longitudinal study and then evaluate them as time progresses. In some cases, ranking functions can be simple, such as a ratio (net margin, for example) or a fractional change (percentage change in revenue, for example).

But when we look at changes over time between two values that could be zero or negative—such as EPS—we are faced with a problem. If we use a simple ratio of earnings for a growth rate ranking function, then a company going from losing money to being profitable gets a negative ranking value, even though it would seem that this kind of company is showing very positive signs. Worse, a company doing exactly the opposite—going from profitable to losing money—may well get exactly the same ranking value! Clearly these two simple company examples would be very different in most investors’ opinions and should get quite different rankings.

So we will have to spend a little effort looking at ranking functions. A good ranking function should:

- Use data that is readily available for a large part of our company universe
- Provide justifiable and sensical relative rankings between companies (in other words, the actual values the function produces do not need to have any direct financial meaning; they just need to provide the right relative ranking between companies)
- Be a smooth function, with no singularities (division by zero) or high sensitivity to any input parameters, including when parameters are at or near zero
- Be efficient to compute

Note that we are talking only about the form of the ranking function. A separate question, specific to each analysis we are doing, is what particular company properties are used to rank it: EPS, return on capital, price momentum, etc. In developing an analysis idea, we would first decide what we will use to rank the companies (EPS, etc.), and we would then turn to selecting the appropriate type of ranking function.

We will start by setting up an example of four companies as shown in the following table. We will be ranking these companies on 5-year EPS growth and now need to find the right type of ranking function.

COMPANY | Prev | Cur |
---|---|---|

A | 0.10 | 1.00 |

B | -0.50 | 0.50 |

C | -4.00 | -2.50 |

D | 2.00 | 1.00 |

Company A’s EPS 5 years ago was $0.10 and now it’s $1.00. Its EPS has grown by 10 times, or by $0.90.

Company B’s EPS 5 years ago was ($0.50) and today it’s $0.50. Its EPS has grown by $1.00, better than Company A’s. But we can’t make a sensical statement about its ratio of EPS—it’s -1. Which company shows better EPS growth?

Company C lost ($4.00) per share 5 years ago and now is “only” losing ($2.50) per share. It has improved by $1.50/share. Is it the best company of the three, by EPS growth?

Company D has been profitable but is earning less per share now than it did before. Where should it lie in the ranking?

It turns out there is no single “right” answer. We will have to choose from a set of functions for the specific criteria and purposes we are interested in. Let’s look at some of the candidates.

### Simple Ratio

For some criteria, a simple ratio is a perfectly adequate ranking function. If company Q has $100,000 revenue per employee, and company R has $500,000 revenue per employee, clearly company R is better than company Q on this measure.

But for our four example companies, a ratio ranking function is not going to do a very good job:

COMPANY | Prev | Cur | ratio |
---|---|---|---|

A | 0.10 | 1.00 | 10.000 |

B | -0.50 | 0.50 | -1.000 |

C | -4.00 | -2.50 | 0.625 |

D | 2.00 | 1.00 | 0.500 |

Company C, which has been losing money hand over fist, comes out the second-best ranked company out of four, higher than Company B which made great strides over the last 5 years and is now profitable. This is clearly not right. Companies A and D are reasonably well ranked, but both B and C are not correctly ranked by any reasonable interpretation, either against A and D or even against each other. A ratio ranking function does not work well for EPS growth. It also fails requirement #3, that it be a smooth function without singularities.

### Normalized Difference

A difference between EPS values might do better than a ratio, except that we will need to normalize the result to make the values reasonably comparable and remove the effects of other factors such as outstanding shares and company size.

Such a “normalized difference” function can be very useful for a large number of cases where we must deal with positive, zero, or negative underlying values. This function divides, or normalizes, the difference between two quantities by the “magnitude” of those values:

norm_diff = (current - prev) / sqrt(current^2 + prev^2)

The full range of values for `norm_diff`

is -1.414 to 1.414. The function is smooth, has no singularities (or even discontinuities), and meets all the requirements set out earlier. A company that is improving its EPS always gets a higher ranking than a company that has falling EPS. By these simple measures, the normalized difference ranking function is a good candidate.

For oiur four example companies, the normalized difference function gives us:

COMPANY | Prev | Cur | norm_diff |
---|---|---|---|

A | 0.10 | 1.00 | 0.896 |

B | -0.50 | 0.50 | 1.414 |

C | -4.00 | -2.50 | 0.318 |

D | 2.00 | 1.00 | -0.447 |

This ranking function achieves a reasonable ranking among these three companies. However, notice that Company B is ranked much higher than Company A. In fact, Company B has the highest possible value of this ranking function, namely 1.414 (the square root of 2). The very highest ranking is given to companies which exactly reverse their measured values from negative to positive. This may or may not fit the type of company behavior we are looking for.

Furthermore, Company C is ranked substantially higher than Company D, because even though Company C’s EPS is negative (it is currently losing money, and maybe has been for at least 5 years), it is improving. This is where the ranking function’s selection is not so obvious. Some investors may be attracted to Company C because it may be showing promise of EPS growing into strong positive territory. On the other hand, Company D may be a more conservative choice. There is no “right” answer on this issue, although most likely this is not quite the behavior we want from a function to rank EPS growth.

This is a crucial point: *the ranking function has to be selected based on the interplay between the values and the meaning we attach to them*. The ranking function must at least roughly match a ranking we would assign among companies, given the same data.

We will end this post here and continue with other candidate ranking functions in the next post.