To Infinity and Beyond Dealing with the Mathematical Oddities of Ratio Analysis (2024)

Journal Article:

Among the few things I detest more than reality-based TV shows aredenominators that approach zero. The former are insufferable; the latter areinscrutable. During the course of formulating or reviewing a disclosurestatement or business plan, restructuring professionals invariably carry outsome form of benchmarking analysis, typically in testing the reasonableness ofa debtor's operating projections or capital structure relative to thoseof designated peers. Much of this effort boils down to ratio analysis—auseful tool because financial ratios are unaffected by size discrepancies amongfirms or across time.

Thanksto providers of financial statement data in electronic form, products such asStandard & Poor's Compustat allow the restructuring professional todownload enormous amounts of financial and market-based data directly into spreadsheetswithin a matter of seconds. This empowers the analyst to choose dozens of peercompanies from among hundreds of candidates based on user-specified selectioncriteria and to then perform relevant financial ratio benchmarking. Anyone whohas ever done this work is no doubt familiar with those annoying, exceedinglylarge, approaching-infinity calculated values (resulting from extremely small,approaching-zero denominators) that wreak havoc on summary statistics.Furthermore, there are other mathematical oddities, such as ratios withnegative values, ratios with positive values resulting from a negativenumerator and denominator, or other calculated ratio values that have noobvious meaning or interpretation. The analyst must decide how to effectivelydeal with all these quirks without compromising the integrity of the analysis.

Near-zero Denominators

Howshould the analyst best deal with extreme ratio values caused by denominatorsapproaching zero? Deleting the particular observation or company from theanalysis is the obvious and most tempting option, but may produce summarystatistics that are incomplete or non-representative of the peer group. InExhibit 1, we calculated two coverage ratios, EBITDA-to-interest expense andEBIT-to-interest expense, for a swath of manufacturing companies with issuercredit ratings of single-A. Companies' DNA and COL have negligibleinterest expense, and consequently, pull up the average for the entiregroup—unfairly so. Removing these two companies from the data sets wouldresult in coverage ratios of 13.4 and 9.2, respectively—certainly morerealistic values than the unadjusted arithmetic means in Exhibit 1. However, byomitting these four extreme observations, we are effectively removing twocompanies from the group that choose to employ minimal leverage. This seemssomewhat arbitrary, as we are depriving the group of two representativecompanies for no other reason than difficulty in interpreting their ratiovalues. Generally speaking, deleting an extreme observation is considered anappropriate measure only when it represents a true outlier—a value thatis wholly inconsistent with other data points, is not representative of theunderlying characteristic and cannot be explained in any logical way. Is therea meaningful way to include these minimally leveraged companies in thecalculation of the group's coverage ratios without using their distorted(and distorting) calculated values?

Winsorizing

Theanalyst can "winsorize" these calculated values—that is,adjust the computed value of the four extreme observations to the next closest"reasonable value," thereby reining in these runaway values. Forexample, the analyst can use Microsoft Excel's If function to create a command that calculates a ratio but caps theratio value at n if the calculated value exceedsn. In this instance, we could have specifiedthat the calculated EBITDA coverage ratio not exceed, say, 25. This ensuresthat our two companies with extreme values are reasonably represented in thegroup. Lastly, we could have extended this command to companies in the groupwith no leverage at all—that is, zero interest expense. (There was one inour group, HDI, whose calculated coverage ratios in Exhibit 1—#DIV/0! dueto the zero denominator—were omitted.) Whether to delete or winsorize,particularly in the last instance, is a judgment call by the analyst: To omitminimally leveraged and unleveraged companies from our ratio calculations (dueto near-zero or zero denominators) would be to overlook those companies withthe most conservative capital structures, but imposing a subjective adjustmentto a calculated financial ratio whose computed value cannot be easilyinterpreted might appear too manipulative. (There are statistical softwarepackages that winsorize a data set more rigidly, such as by taking thoseobservations in the bottom and top deciles and changing their computed valuesto the decile values immediately above and below them, respectively.) If toomany data points in a data set require winsorizing due to denominator issues,the analyst should consider an alternative ratio that measures a similarcharacteristic. In this case, EBITDAR-to-Fixed Charges would likely have been afine substitute since broadening the definition of the denominator to include rentexpense lessens the likelihood that it will contain a zero or near-zero valuefor any company.

Trimming the Data Set

Analternative to winsorizing individual data points in order to control theimpact of extreme values is to trim the data set. Trimming a data set requiresthe omission of n percent of the calculatedvalues and then computing the mean of the remaining (1-n) percent of the data set. For example, if the analyst decides anappropriate trim percentage (n) is 20 percentfor a data set consisting of 200 data points, then 40 data points are omittedfrom the set—the top 20 and bottom 20 calculated ratio values—andthe mean of the remaining 160 values is computed. The TRIMMEAN calculation is astandard Excel function. Trimming a data set with the TRIMMEAN functioneffectively removes the most extreme values at each end of a data set when calculating summary statistics. It spares the analyst thebother of having to explicitly scan data sets and delete individual data pointsor companies from the sets. If we apply the TRIMMEAN function to our twocoverage ratios in Exhibit 1, we get ratio values of 17.2 and 12.5,respectively, for the group. Any time the analyst uses Excel's AVERAGEfunction, the TRIMMEAN function should be run on the data set as well, and theanalyst should note the degree to which these two averages agree or diverge.

Medians

Themedian is probably the most common but simplistic summary statistic used tomanage the impact of extreme values on a data set. It is simply the middle valuein a data set and is completely unaffected by extreme values. The median may bea better indicator of central tendency than the arithmetic average, even ifonly a couple of extreme values remain in a data set. The median values of ourtwo coverage ratios were 12.7 and 7.3, respectively, in each case smaller thanthe adjusted arithmetic average (after deleting the extreme values) and thetrimmed mean. In its periodic reports on key industrial financial ratios,S&P only presents median ratio values for each debt-rating category.

Negative Ratios and Other Oddities

Anothercommon problem encountered in ratio analysis is how to deal with negative valueratios. We see in Exhibit 2 that TXN's debt-to-EBIT ratiois a negative value. Deleting a negative data point is the most common remedy,but once again, this should not be done automatically. First, try to make surethe figure in question is "clean." In this instance, the differencebetween TXN's EBITDA and EBIT values is unusually large. Perhaps goodwillwas deemed impaired and written-off, or some other non-cash, non-recurringcharge hit the P&L. The analyst is encouraged to investigate these types ofdiscrepancies and normalize the financial statement data if the informationrequired to do so is available. Retrieving financial statement data for severalsurrounding quarters allows the analyst to eyeball numbers, establish someinformal "range of normalcy" and quickly spot suspicious figures.However, this data-scrubbing might be impractical or overly time-consuming if adata set comprises dozens of companies. When using ratios that requireincome-statement data, it's best to use trailing four-quarter P&Ldata rather than quarterly data, so as to remove the impact of seasonality onthe computed ratio values.

Forcertain ratios, a negative value may be an accurate (albeit undesirable)measurement of the characteristic of interest. For example, negative operatingmargins and subsequent negative returns on equity are not highly unusualobservations for distressed companies or industries, and are perfectlyexplainable within the conceptual definition of the ratio. While negativevalues for these two ratios cannot persist indefinitely without eventuallyresulting in financial ruination, they can endure for several quarters andshould be left intact within a data set if they are reflective of underlyingbusiness conditions during that time period. For other financial ratios, anegative value has no discernible meaning, and the data point should be deleted.If return on equity were negative due to negative shareholders' equity(as opposed to a net loss), then this ratio value would have no obviousinterpretation and should be deleted from the data set. As with ratio valuesinvolving zero and near-zero denominators, it might be preferable to identifyan alternative ratio that measures a related characteristic but avoids the mathquirk, such as return on total assets instead of return on equity.

Unfortunately,it isn't always obvious whether a negative value is an acceptable valuefor a particular ratio. The analyst must first discern whether a negative valueis consistent or inconsistent with the natural direction of the ratio. Forexample, the larger the debt-to-EBIT ratio, the more leveraged a firm is consideredto be. However, by including a company with negative EBIT, as we did with TXNin Exhibit 2, we (improperly) reduced the group's average leverage ratio.By this logic, a group that contains some firms with operating losses wouldhave a lower debt-to-EBIT ratio than if those firms had operating income ceterisparibus. This conclusion is counterintuitive andnonsensical. Therefore, any negative value data points should be excluded fromthe group for this particular ratio. (Winsorizing these negative data pointswould be a large and subjective adjustment here.) Conversely, the analyst mightdecide to include a negative value data point for the coverage ratioEBIT-to-interest expense, since such a reading is not inconsistent with thenormal direction of the ratio (i.e., smaller is"worse" and negative means smaller). In Exhibit 1, leavingTXN's negative EBIT-to-interest expense ratio value in the data set doesnot distort the group average as its negative debt-to-EBIT ratio value does inExhibit 2.

Anothermathematical subtlety is a positive ratio caused by two negative numbers, suchas a positive return on equity resulting from a net loss and negativeshareholders' equity. Without question, it would be improper to allowthis calculation to remain in the group. The trick here is simply to identifythese instances, which are easy to miss if the data service provider'ssoftware calculates the ratio directly. As a general rule, it's best todownload the raw financial statement data that underlie a ratio and scan themto ensure that two negatives don't inadvertently produce a positive valueratio that stays in the data set. Excel's MAX and MIN functions areextremely helpful in quickly locating oddball numbers within a large dataseries.

Hopefullyit is clear by now that ratio analysis, when carried out thoughtfully, can betedious work that requires lots of attention to detail. The second-worst sinthe analyst could commit in this exercise (first place goes to deliberateselection bias—that is, picking a sample that will produce a desiredoutcome) is to ignore the insidious subtleties of working with fractions. Withratio analysis, the formulas may all be fine, strictly speaking, but theconclusions can still be way off the mark.