2. Basic Properties of Ratios

2.1. Functional Form of Financial Ratios

The traditionally stated major purpose of using financial data in the ratio form is making the results comparable across firms and over time by controlling for size. This basic assertion gives rise to one of the fundamental trends in financial ratio analysis (or FRA for short, in this paper). The usually stated requirement in controlling for size is that the numerator and the denominator of a financial ratio are proportional.

The seminal paper is this field is Lev and Sunder (1979). They point out, using theoretical deduction, that in order to control for the size effect, the financial ratios must fulfill very restrictive proportionality assumptions (about the error term, existence of the intercept, linearity, and dependence on other variables in the basic financial variables relationship models Y = bX + e and its ratio format Y/X = b + e/X). It is shown that the choice of the size deflator (the ratio denominator) is a critical issue. Furthermore, Lev and Sunder bring up the problems caused in multiple regression models where the explaining variables are ratios with the same denominator. This is a fact that has been discussed earlier in statistics oriented literature like in Kuh and Meyer (1955).

Two interrelated trends are evident. Theoretical discussions about the ratio format in FRA and empirical testing of the ratio model. While mostly tackling the former Whittington (1980) independently presents illustrative results finding the ratio specification inappropriate in a sample of U.K. firms. Whittington also discusses the usage of a quadratic form in FRA. Significant instability in the results was reported.

The proportionality considerations have implications on various facets of FRA. Barnes (1982) shows how the non-normality of financial ratios can result from the underlying relationships of the constituents of the financial ratios. He is thus able to tie in the ratio format aspects with the distributional properties of financial ratios (to be discussed later in this review). In the discussion on Barnes's paper (Horrigan, 1983, Barnes, 1983), Horrigan puts forward that financial ratio research should be more interested in the role of the financial ratios themselves than in "the nature of the ratios' components or to the ratios' incidental role as data size deflators".

To extrapolate from Horrigan's critique, in our own interpretation the validity of financial ratio analysis should be determined by its usefulness to the decision making process of the different interested parties (owners, management, personnel,...). To illustrate, consider the potential impact of economics of scale. To assess the efficiency of management a direct comparison of financial ratios of small and big firms would have to be adjusted for the size effect. On the other hand, an investor evaluating different investment targets might be more interested in the level of profitability regardless whether or not it is a result of the size effect.

McDonald and Morris (1984, 1985) present the first extensive empirical studies of the statistical validity of the financial ratio method. The authors use three models with two samples, one with a single industry the other with one randomly selected firm from each (four-digit SIC) industry branch to investigate the implications of homogeneity on proportionality. The first model is the traditional model for replacement of financial ratios by bivariate regression, with intercept
. Y(i) = a + bX(i) + e(i).
The above model is central in this area. It is characteristic that the testing for proportionality is considered in terms of testing the hypothesis H0: a = 0. Barnes (1986) points out for statistical testing that the residual is typically heteroscedastic. For a discussion also see Garcia-Ayuso (1994). The second model in McDonald and Morris is
. Y(i) = b'X(i) + e'(i)
that is without the intercept to tackle heteroscedasticity. Dropping the intercept from the model is not always enough to treat the heteroscedasticity (see Berry and Nix, 1991). The third model applies a (Box-Cox) transformation on the first model to tackle non-linearities. While they find support for financial ratio analysis for comparisons within industry branches, in inter-industry comparisons proportionality of financial ratios is not supported.

Berry and Nix (1991), however, cast doubt on the generality of McDonald and Morris results over time, over ratios and over industries. Similar results was obtained for Finnish data in Perttunen and Martikainen (1989) and for Spanish data by Garcia-Ayuso (1994). By comparing value and equal weighted aggregate financial ratios McLeay and Fieldsend (1987) find evidence based on samples of French firms that "the departure from proportionality varies from ratio to ratio, from size class to size class and from sector to sector".

Research on financial ratio proportionality has close connections to distributional questions. Testing the statistical significance of the parameters of the previous models involves, at least implicitly, assumptions of normality (see Ezzamel, Mar-Molinero and Beecher, 1987, p. 467). Fieldsend, Longford and McLeay (1987) draw on the fact that a number of accounting variables are expected to be lognormally distributed because of technical zero lower bounds. Consequently they test empirically a lognormal regression model
. lnY(ij) = a + blnX(ij) + g(j) + e(ij)
where the industry effect g(j) is explicitly specified in the model. Their empirical results on a single financial ratio (the current ratio) are in line with the earlier results supporting proportionality only if industry effects are included.

As was discussed in Introduction financial ratios can be extended to include market based data. We concentrate mainly on "pure" financial ratios with both the numerator and the denominator originating from the income statement and/or the balance sheet. Nevertheless, concomitant research has been presented with market based ratios. For example, Booth, Martikainen, Perttunen and Yli-Olli (1994) report deviations from proportionality in the E/P ratio.

2.2. Distributional Characteristics of Financial Ratios

It is typical of FRA research that there are several distinct lines with research traditions of their own. In some cases there is little link to the other FRA fields. The distributional characteristics of financial ratios have induced a research line of their own, but part of this research is intertwined with the proportionality research discussed above. In fact some of the papers reviewed tackle both the areas either separately or within the same framework.

The recurring motivation for looking into the distributional properties of financial ratios is that the normal distribution of the financial ratios is often assumed in FRA. This is because the significance tests in parametric methods prevalent in FRA research, such as regression analysis and discriminant analysis, rely on the normality assumption.

In the history of FRA it is common that professional journals and academic papers do not recognize each other. An early paper on financial ratio distributions was published in Management Accounting by Mecimore (1968). It is interesting to recognize that all ingredients of modern distribution analysis already appear incumbent in Mecimore's paper. Using descriptive statistical measures (average and relative deviations from the median) he observes cross-sectional non-normality and positive skewness for twenty ratios in a sample of randomly selected forty-four Fortune-500 firms.

The paper most often referred to in literature as the seminal paper in this field is, however, the much later published article by Deakin (1976). His chi-square findings reject (with one exception) the normality of eleven financial ratios in a sample of 1114 Compustat companies for 1954-72. Less extreme deviations from normality were observed when square-root and logarithmic transformations were applied, but normality was still not supported. Likewise, while not statistically significantly, industry grouping made the distributions less non-normal. Concomitant results are obtained by Lee (1985) using a stronger test (Kolmogorov-Smirnov) for a different set of data.

Bird and McHugh (1977) adopt an efficient Shapiro-Wilk small-sample test for the normality of financial ratios for an Australian sample of five ratios over six years. Like Deakin they find in their independent study that normality is transient across financial ratios and time. They also study the adjustment of the financial ratios towards industry means which is a different area of FRA research. Bougen and Drury (1980) also suggest non-normality based on a cross-section of 700 UK firms.

The results indicating non-normality of financial ratio distributions have led researchers into looking for methods of restoring normality to warrant standard parametric statistical analyses. Frecka and Hopwood (1983) observe that removing outliers and applying transformations in a large Compustat sample covering 1950-79 restored normality in the same financial ratios as tackled by Deakin (1976). They point out that if the ratios follow the gamma distribution, the square root transformation makes the distribution approximately normal. The gamma distribution is compatible with ratios having a technical lower limit of zero. There is, however, a certain degree of circularity in their approach, since instead of identifying the underlying causes of the outliers they employ a mechanistic statistical approach to identify and remove the outliers from the tails of the financial ratio distributions.

A varying and often a considerable number of outliers has to be removed for different financial ratios in order to achieve normality. The empirical results are supported by later papers such as So (1987). Ezzamel, Mar-Molinero and Beecher (1987) and Ezzamel and Mar-Molinero (1990) review and replicate the earlier analyses on UK firms with a particular emphasis on small samples and outliers, respectively. One of the avenues taken is to study new industries. Kolari, McInish and Saniga (1989) take on the distribution of financial ratios in banking. Buckmaster and Saniga (1990) report on the shape of the distributions for 41 financial ratios in a Compustat sample of more than a quarter million observations.

Foster (1978) points out the outlier problem in FRA. Later, he presented in Foster (1986) a list of alternatives for handling outliers in FRA. The list includes deleting true outliers, retaining the outlier, adjusting the underlying financial data, winsorizing that is equating the outliers to less extreme values, and trimming by dropping the tails. Foster also puts forward accounting, economic and technical reasons for the emergence of outliers in FRA. While improving the statistical results trimming and transformations can pose a problem for the theoretical rigor in FRA research. Instead of deleting or adjusting the observations McLeay (1986a) proposes using a better fitting distribution with fat tails for making statistical inferences in FRA. He seeks for a best fitting t-distribution for a cross-section of 1634 UK and Irish firms. Also his empirical results confirm non-normality. The best-fitting (in the maximum-likelihood sense) t-distribution varies across financial ratios (the t-distribution can be considered a family of distributions defined by its degrees of freedom). McLeay (1986b) also tackles the choice between equally weighted and value weighted aggregated financial ratios in terms of ratio distributions on a sample of French firms. Also the results by Martikainen (1991) demonstrate that normality can be approached by other procedures than removing outliers. In a sample of 35 Finnish firms, four ratios and fifteen years about half of the non-normal distributions became normal if economy-wide effects were first controlled for using the so-called accounting-index model. Martikainen (1992) uses a time-series approach to 35 Finnish firms in turn observing that controlling for the economy factor improves normality.

Typically, many later papers tackle the same basic question of ratio distributions using different samples and expanding on the methodologies. Buijink and Jegers (1986) study the financial ratio distributions from year to year from 1977 to 1981 for 11 ratios in Belgian firms corroborating the results of the earlier papers in the field. Refined industry classification brings less extreme deviation from normality. They also point to the need of studying the temporal persistence of cross-sectional financial ratio distributions and suggest a symmetry index for measuring it. Virtanen and Yli-Olli (1989) studying the temporal behavior of financial ratio distributions observe in Finnish financial data that the business cycles affect the cross-sectional financial ratio distributions.

The question of the distribution of a ratio format variable (financial ratio) has been tackled mathematically as well as empirically. Barnes (1982) shows why the ratio of two normally distributed financial variables does not follow the normal distribution (being actually skewed) when ratio proportionality does not hold. Tippett (1990) models financial ratios in terms of stochastic processes. The interpretation in terms of implications to financial ratio distributions are not, however, immediately evident, but the general inference is that "normality will be the exception rather than the rule".

Because of these results bringing forward the significance of the distributional properties of financial ratios many later papers report routinely about the distributions of financial ratios in connection with some other main theme. Often these themes are related to homogeneity and industry studies such as Ledford and Sugrue (1983). The distributional properties of the financial ratios also have a bearing in testing proportionality as can be seen, for instance, in McDonald and Morris (1984). In a bankruptcy study Karels and Prakash (1987) put forward that in applying the multivariate methods (like discriminant analysis) the multivariate normality is more relevant than the (univariate) normality of individual financial ratios. They observe that deviations from the multivariate normality is not as pronounced as the deviations in the earlier univariate studies.

Watson (1990) examines the multivariate distributional properties of four financial ratios from a sample of approximately 400 Compustat manufacturing firms for cross-sections of 1982, 1983 and 1984. Multivariate normality is rejected for all the four financial ratios. Multivariate normality is still rejected after applying Box's and Cox's modified power transformations. However, when multivariate outliers are removed, normality is confirmed. Multivariate normality has particular bearing on research using multivariate methods, for example on bankruptcy prediction. It also has implications on univariate research, since while univariate normality does not imply multivariate normality, the opposite is true.

2.3. Classification of Financial Ratios

A central question both in FRA research and practice is finding a parsimonious set of financial ratios to cover the activities of the firm. The main approaches in this area are fairly clearcut. They are pragmatical empiricism (a term coined by Horrigan 1968), a data oriented classification approach, a deductive approach, and lately, the combination of the last two. An interesting early paper on financial ratios which has many of the later issues in a embroynic form can be seen in Horrigan (1965).

2.3.1. Pragmatical Empiricism

Several accounting and finance text-books present a subjective classification of financial ratios based on the practical experience or views of the authors. It is common that the classifications and the ratios in the different categories differ between the authors as pointed out in a tabulation by Courtis (1978, p. 376). In very general terms three categories of financial ratios are more or less common: profitability, long-term solvency (capital structure) and short-term solvency (liquidity). Beyond that there is no clear consensus. Pragmatical empiricism is exemplified by the text-books of Weston and Brigham (1972), Lev (1974a), Foster (1978, 1986), Tamari (1978), Morley (1984), Bernstein (1989), White, Sondhi and Fried (1994), Brealey and Myers (1988, Ch. 27), and handbook chapters such as Beaver (1977), and Holmes and Sugden (1990, Ch 24).

Official bodies also can give recommendations. For example, in Finland the Committee for corporate analysis (1990) guidelines influence Finnish reporting practices. More generally security exchange commission stipulations influence reporting of financial ratios in many countries.

2.3.2. Deductive Approach

The classic of deductive approach goes back to 1919 to the du Pont triangle system (profits/total assets), (profits/sales), (sales/total assets):

               profits
           sales     total assets
Courtis (1978) returns to the theme. He presents a diagram for a financial ratios framework based on financial ratios used in earlier studies, textbooks, "other sources", deliberation, and visual approximation of relationships in a sample of 79 ratios. Laitinen (1983) presents a model of the financial relationships in the firm with attached financial ratios. The model is based on Laitinen (1980). For the most part empirical evidence based on 43 publicly traded Finnish firms supports the structure of the model. Bayldon, Woods, and Zafiris (1984) evaluate a pyramid scheme of financial ratios. In a case study the pyramid scheme does not function as expected. The deductive approach to establish relevant financial ratio categories has more or less stalled, and this approach has become intermixed with confirmatory approach discussed later.

2.3.3. Inductive Approach

The emphasis on data and statistical methods is characteristic of the inductive approach to financial ratio classification like it is in the proportionality and distribution studies discussed earlier. The empirical rather than the theoretical foundations for grouping the financial ratios are central in this approach.

The seminal paper in empirically-based FRA classifications ("taxonomies") is Pinches, Mingo and Caruthers (1973). They apply factor analysis to classify 51 log-transformed financial ratios of 221 Compustat firms for four cross sections six years apart. The selection of the method was prompted by applications in other behavioral disciplines (e.g. psychology and organizational analysis). They identify seven factors, Return on investment, capital intensiveness, inventory intensiveness, financial leverage, receivables intensiveness, short-term liquidity, and cash position. These factors explain 78-92% (depending on the year) of the total variance of the 51 financial ratios. Moreover, the correlations for the factor loadings, and the differential R-factor analysis indicate that the ratio patterns are reasonably stable over time. The same study is replicated for adjacent years 1966-1969 in Pinches, Eubank, Mingo and Caruthers (1975).

Johnson (1978) runs the factor analysis for a single year 1972, but for two industries based on a sample of 306 primary manufacturing and 61 retail firms. Congruency coefficients of financial ratio patterns indicate a good stability of the nine factors for the two industries. Johnson (1979) repeats the study for a larger sample of firms and for two years.

Chen and Shimerda (1981) present a summary of the financial ratios used in a number of early studies which use the financial ratios for analysis and prediction. They note that there is an abundant 41 different financial ratios which are found useful in the earlier studies. They reconcile by judgement the factors in the earlier studies into financial leverage, capital turnover, return on investment, inventory turnover, receivables turnover, short-term liquidity, and cash position. They identify ten financial ratios which are representative of their seven factors. After a principal component factor analysis of 39 ratios of the Pinches, Eubank, Mingo and Caruthers (1975) they conclude that there is a high instability in always selecting the financial ratio with the highest absolute factor loading as the representative financial ratio for the observed factors.

Cowen and Hoffer (1982) study the inter-temporal stability of financial ratio classification in a single, homogeneous industry. Their findings do not support the Pinches, Mingo and Caruthers results about the stability of the ratio patterns. Cowen and Hoffer's sample consist of 72 oil-crude industry firms for 1967-75. Four or five factors are found for each year for the 13 financial ratios included. As the authors put it "there was little consistency and stability in the factor loadings across all years". The results are only slightly improved with log-transformations. Cowen and Hoffer also find applying cluster analysis that groupings of firms with respect to the financial ratios exist within the industry, but that they are not stable over time. Ezzamel, Brodie and Mar-Molinero (1987) detect instability in the factors of financial ratios for a sample of UK firms. Martikainen and Ankelo (1991) find that instability of financial ratio groups is more pronounced for firms about to fail than for healthy firms in a sample of 40 Finnish firms. Martikainen, Puhalainen and Yli-Olli (1994) observe significant instability of the financial ratio classification patters across industries in a sample typical of bankruptcy research.

Aho (1980) includes also cash-flow based profitability ratios in a factorization study for 24 financial ratios of 57 Finnish firms in 1967-1976. His financial characteristic factors become financial structure, profitability, liquidity, working capital turnover and financial opportunities for investments. Gombola and Ketz (1983) include cash-flow based (adjusted for all accruals and deferrals) financial ratios in their factorization of 40 financial ratios for a sample of 119 Compustat firms for 1962-80. Contrary to the earlier studies, the cash-flow based financial ratios load on a distinct factor. The results are not sensitive to using historical costs vs general price-level adjusted data. Similar results on the empirical distinctiveness of cash flow ratios are later obtained in Salmi, Virtanen and Yli-Olli (1990) in a study that also introduces market-based ratios to the analysis.

Yli-Olli and Virtanen (1986, 1989, 1990) introduce the usage of transformation analysis to study the stability of the financial ratio patterns. After aggregating financial ratios for 1947-75 for the US and 1974-84 for Finland they find that value-weighted aggregation produces ratio patterns that are stable both over time and across countries. The stability is further improved by using first differences of the financial ratios.

Factorization of financial ratios has also been a part in several multivariate studies analyzing the economic features of the firms. Pinches and Mingo (1973) screen a set of 35 financial variables into seven factors in a bond rating study. Likewise, Libby (1975) reduces an original 14-ratio set to five financial factors by a principal component analysis in connection with a bankruptcy study. Another example is Richardson and Davidson (1984). Hutchinson, Meric and Meric (1988) also classify ratios with principal component analysis in a study attempting to identify small firms which have achieved quotation on the UK Unlisted Securities Market. Martikainen (1993) classifies financial ratios and tests their stability with transformation analysis in a study on identifying the key factors which determine stock returns.

2.3.4. Confirmatory Approach

It seems that despite the initial optimism the inductive studies have been unable to agree on a consistent classification of financial ratio factors, at least beyond three to five factors. Consequently a number of later studies hypothesize an a priori classification and then try to confirm the classification with empirical evidence.

A tentative emergence of this idea can be detected in Laurent (1979). As noted earlier Courtis (1978) presents a pyramid scheme of financial ratios based on a mix of experience, deduction and visual approximation of data. This can be considered an a priori classification. Laurent performs a standard principal component factorization for a set of 45 financial ratios presumably for a single year of 63 Hong Kong companies. He compares his results with the deductive classification by Courtis (1978) and finds a good correspondence. With the exception of administration Laurent identifies and locates each of his ten empirical factors in Courtis's framework. Such a comparison has the hallmarks of the basic idea of the confirmatory approach.

Pohlman and Hollinger (1981) test two a priori classification schemes based an a sample of Compustat firms for 1969-78. They call the first the "traditional" scheme. (It practically is Lev's (1974) categorization.) The second is not actually a priori classification but the empirical classification by Pinches, Eubank, Mingo and Caruthers (1975) with seven factors. They use the redundancy indexes produced by canonical correlation analysis to evaluate how well financial ratios fit the relevant factor. They find that the a priori categories are correlated with each other. Thus they caution against using too few financial ratios in FRA.

Luoma and Ruuhela (1991) present five a priori "dimensions" for the financial ratios, profitability, financial leverage, liquidity, working capital, and revenue liquidity. Rather than using cross-sections across firms their data consist of time series of 40 Finnish firms for 1974-84. They apply cluster analysis to group the 15 initial ratios separately for each firm in the sample, and compare the empirical clusters with the a priori dimensions. Profitability and revenue liquidity appear almost invariably as distinct clusters. The other three dimensions turn out more commonly to be interrelated.

Kanto and Martikainen (1991) evaluate Lev's (1974) a priori classification of financial ratios by introducing the usage of confirmatory factor analysis to testing a priori classifications of financial ratios. Confirmatory factor analysis provides statistical significance tests for the existence and stability of the a priori factor structure. Using Compustat firms it is observed for 1947-75 that the a priori financial ratio categories are significantly correlated. Thus Lev's classification is not corroborated. Similar results are observed for a sample of Finnish firms in Kanto and Martikainen (1992).


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Department of Accounting and Finance, University of Vaasa,
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