The Valuation Performance of Mathematically-Optimised, Equity-Based Composite Multiples

Purpose – This paper aims to examine the valuation precision of composite models in each of six key industries in South Africa. The objective is to ascertain whether equity-based composite multiples models produce more accurate equity valuations than optimal equity-based, single-factor multiples models. Design/methodology/approach – This study applied principal component regression and various mathematical optimisation methods to test the valuation precision of equity-based composite multiples models vis-à-vis equity-based, single-factor multiples models. Findings – The findings confirmed that equity-based composite multiples models consistently produced valuations that were substantially more accurate than those of single-factor multiples models for the period between 2001 and 2010. The research results indicated that composite models produced up to 67 per cent more accurate valuations than single-factor multiples models for the period between 2001 and 2010, which represents a substantial gain in valuation precision. Research implications – The evidence, therefore, suggests that equity-based composite modelling may offer substantial gains in valuation precision over single-factor multiples modelling. Practical implications – In light of the fact that analysts’ reports typically contain various different multiples, it seems prudent to consider the inclusion of composite models as a more accurate alternative. Originality/value – This study adds to the existing body of knowledge on the multiples-based approach to equity valuations by presenting composite modelling as a more accurate alternative to the conventional single-factor, multiples-based modelling approach.


Introduction
This paper examines the valuation precision of composite models in each of six key industries in South Africa. The objective is to ascertain whether equity-based composite multiples models produce more accurate equity valuations than optimal equity-based, single-factor multiples models. The analysis will be conducted on an industry basis, as it is anticipated that different composite multiples models will be best suited to different industries (Abukari et al., 2000;Barker, 1999;Fernández, 2001;Goedhart et al., 2005;Liu et al., 2002a;Nel, 2009a;2009b, 2013bSchreiner, 2007). Optimal equity-based composite multiples models will be constructed for each of the six South African industries, and their valuation precision will be compared to that of eight optimal equity-based, single-factor multiples models. The aim is to establish whether industry-specific, equity-based, composite multiples models offer higher degrees of valuation precision vis-á-vis industry-specific, equity-based, single-factor multiples models.
First, the proper composition of the composite models requires investigation. To this end, one has to determine the optimal weight allocations of each of the components of the composite models. This is achieved by using mathematical optimisation algorithms with the goal of minimising the sum of the absolute valuation errors (SAVE), the median valuation errors (MVE) and the sum of the squared valuation errors (SSVE). Second, the increase in valuation precision that composite multiples models may offer over single-factor multiples models is estimated. Third, the consistency of the results is assessed for the period 2001-2010. Multiples are used extensively in practice, which is why analyst reports are typically inundated with various different single-factor multiples (Asquith et al., 2005;Damodaran, 2009Damodaran, , 2006bEfthimios et al., 2004;Fernández, 2002;PricewaterhouseCoopers (PwC), 2015, 2012Roosenboom, 2007). Therefore, there seems to be a case for compiling a composite of these single-factor multiples.
However, multi-factor modelling is not a new phenomenon in financial literature. Ross (1976), for example, presents evidence that a two-factor arbitrage pricing theory model explains asset prices better than the traditional capital asset pricing model. Similarly, Fama and French (1996) document evidence in support of a three-factor capital asset pricing model that encapsulates many of the anomalies that are not explained by the traditional single-factor capital asset pricing model. Although a multi-factor approach may not seem new in the field of finance, it is a novel application in respect of multiples-based valuations. International literature offers very little guidance in this regard, and the evidence from emerging markets, in particular, is limited in scope and seems rather lacklustre (Nel et al., 2014b). It is hoped that the findings from this paper will offer a new perspective for the composition of composite multiples models in emerging markets and in South Africa, in particular.

Literature review
Most of the existing international literature focuses on a composite of market price (P) to earnings per share (EPS) and book value of equity (BVE) per share. The use of a composite of P/EPS and P/BVE stems from the multitude of researchers who have attempted to investigate the nature of the relationship between accounting data and company value by focusing on these two multiples (Ohlson, 1995;Ohlson and Juettner-Nauroth, 2005;Penman, 1998). Cheng and McNamara (2000) compared the P/EPS, P/BVE and an equally weighted combination of P/EPS and P/BVE over a period of 20 years from 1973 to 1992 by extracting data from the Industrial Compustat database. Cheng and McNamara (2000) found that a combination of P/EPS and P/BVE outperforms the individual P/EPS and P/BVE multiples. In a similar study conducted in the USA and Europe, Schreiner (2007) tested the valuation precision of a two-factor composite model consisting of P/BVE and other earning-based multiples. He found that a significant valuation performance improvement occurred when opting for a two-factor valuation model vis-á-vis a single-factor valuation model. seemingly as a result of data limitations, which may have obscured their results. In addition, Sehgal and Pandey included R as a value driver in an equity-based valuation analysis, which is conceptually flawed. The matching principle is often neglected by analysts and academic researchers alike, i.e. they fail to distinguish between equity-and company-based valuations (Nel et al., 2013b).
Regrettably, the limited scope of the study by Sehgal and Pandey prohibits a more detailed analysis. Consequently, this paper aims to broaden the scope of the South African case study, in particular, by including eight equity-based single-factor multiples, based on value drivers representing all of the major equity-based value driver categories, namely earnings, assets, dividends and cash flows.

Empirical design 3.1 Data
The composite models constitute equity-based compilations of the eight equity-based, single-factor multiples models, as contained in Table I. The equity-based focus of this paper stems, first, from the objective of this study, which is to investigate the valuation precision of equity-based composite multiples models, in particular and, second, from the finance literature that suggests that equity-based multiples outperform company-based multiples in terms of valuation precision (Nel et al., 2013b;Schreiner, 2007)[1].
The data items were extracted from the McGregor BFA database, one of the leading data houses in South Africa, for the period 2001-2010(PwC, 2015. Note that the matching principle is applied for the selection of the equity-based value drivers, i. e. the value drivers represent claims to equity holders in particular (Damodaran, 2009(Damodaran, , 2006aNel et al., 2014e, 2013bSchreiner, 2007). Although one may be tempted to incorporate company-based, single-factor multiples into the equity-based composite model, this will result in model inconsistencies, which may obscure the interpretation of the results. The number of observations varied for each equity-based multiple, depending on the specific industry in question. The population sizes per industry varied between 242 and 1,248 observations. The focus in this paper is primarily on equity-based multiples and their behaviour in each of six key industries, namely basic materials, consumer goods, consumer services, financials, industrials and technology. As a result of data limitationsa common phenomenon in developing markets (Omran, 2003;Sehgal and Pandey, 2009)a sectorbased approach was not possible. Instead, an industry-based approach was adopted. Although ten industries are demarcated on the McGregor BFA database, insufficient data is Notes: MPVmarket price variable; Pmarket price; PBTprofit before tax; PAT -Profit after tax; HEheadline earnings; BVEbook value of equity; ODordinary dividends; NCIfOAnet cash inflow from operating activities; NCIfIAnet cash inflow from investment activities; FCFEfree cash flow to equity. Own elaboration available for four of these industries, namely healthcare, oil and gas, telecommunications and utilities. Consequently, these four industries are omitted from the analysis, and the focus is on the six key industries, for which sufficient data is available.

Model specification
Multiples-based equity valuations assume that the actual equity value (V e it ) of a company (i ) at a given point in time (t) is equal to the product of a multiple (l e t ) and a specific value driver (a it ) at that specific point in time, so that Refer to Nel, Bruwer and Le Roux (2014a) for more details pertaining to single-factor multiples. In this paper, the market-based approach, as modelled in Equation (1), is adopted.
To investigate the valuation precision of composite multiples models, Equation (1) is extended to accommodate composite modelling: whereV e it is the predicted equity value of company i at time t andl e jpt Á a jit represents each single-factor equity value prediction (j) that is included in the composite multiples. The aim of the research is to establish the ability of valuations based on Equation (2) to approximate actual share values. Functions for the calculation ofV e it and the statistical analysis thereof were developed in the R-package, an open source programming language that lends itself to statistical analysis and graphics (R Core Team, 2015). The optimal number (k) of single-factor multiples models that is catered for in each composite model will depend on the optimal weightings (b jt ), as obtained from the optimisation applications. It is envisaged that these multiples will be drawn from various value driver categories, which may include earnings, assets, dividends and cash flows. A high level of multicollinearity was anticipated amongst the respective value drivers. Therefore, principal component analysis (PCA) was applied to transform the initial multi-variable data set into uncorrelated combinations (principal components) of the original independent variables, which nullified kappa readings (measure of multicollinearity) to insignificant numbers. All of the principal components were, therefore, independent of each other after transformation. However, there were assumption violations in our application pertaining to principal component regression (PCR), which constitutes a linear regression approach. Therefore, as a result of these violations, we adopted a direct constraint optimisation approach, aimed at optimising the median absolute valuation errors and not the sum (or mean) of the squared evaluation errors. The b -value refers to the corresponding weightings for each of the single-factor multiples, which will be determined by mathematical optimisation applications in the R-package. The assumptions regarding b are that 0 # b 1t , b 2t , . . ., b kt , # 1 and P k j51 b jt 51. The composite multiples models' predicted equity values will, therefore, encapsulate the weighted average of the predicted values of the respective single-factor multiples. Subtracting Equation (2) from the actual equity value (V e it ) of a company (i) at a given point in time (t) produces the valuation error margin: As the valuation error margin will be size-dependent, the standardised absolute deviation (« it ) is expressed proportionally to the actual equity value, V e it ; therefore: The market-based approach that is adopted in this paper was introduced to the finance literature by Alford (1992) in a joint research effort between the Massachusetts Institute of Technology and corporate financiers from Ernst & Young. It has since been refined by various scholars (Berkman et al., 2000;Cheng and McNamara, 2000;Dittmann and Weiner, 2005;Gilson et al., 2000;Kaplan and Ruback, 1995;Liu et al., 2007Liu et al., , 2002a2002b;Minjina, 2008;Nissim, 2011). From the literature review, it is evident that the initial research conducted on the construction of composite multiples models focused on equally weighted models, which required no optimisation procedure. However, subsequent studies found that when these weightings were not restricted, i.e. when the single-factor multiples models were not allocated an equal weighting, the valuation precision of the composite multiples models increased vis-á-vis equally weighted composite multiples models. The objective of the resulting optimisation process in composite-based modelling is the minimisation of the valuation error, as per Equation (4). Various methods were considered for determining the optimal weight allocations of the components of the composite models. Among the alternatives considered were R-based PCA, PCR and three mathematical optimisation applications, namely, lpSolve, Rsolnp, as well as Quadprog. Unfortunately, the nature of the data rendered some of these alternatives unsuitable for the purposes of this study. Consequently, the components of the composite models were weighted based on the three mathematical optimisation applications, namely, lpSolve, Rsolnp, as well as Quadprog.

Mathematical optimisation
As the objective of the optimisation process is to determine the optimal weightings that should be allocated to the single-factor multiples models contained in each composite model, the problem is essentially one of mathematical optimisation. However, given the nature of the minimisation objective of the optimisation function, there is no closed-form algebraic solution to the optimisation objective. Consequently, it was deemed prudent to use more than one optimisation method, namely, SAVE, MVE and SSVE. Two restrictions were imposed on all three methods. The first was that the weightings had to add up to one and the second was that all the weightings had to be positive.
The first application, lpSolve, optimises the weight allocations based on the objective of minimising the SAVE. The lp function, which is an integer programming application in the R-package lpSolve, was used to apply the SAVE method. The objective of the lp function was to produce optimal weightings to be allocated to each of the single-factor multiples models included in the composite multiples models, to minimise the SAVE. To this end, the R function SAVE was written to effect the optimisation of the objective function [Equation (5) where y i is the i th actual equity value, while m 0 i represents a vector of equity value estimates corresponding to y i and a denotes the weight allocation to each single-factor multiple. The vector a is of size p À the number of single-factor multiples.
However, a key focus point in the international literature is the minimisation of the MVE (Schreiner, 2007). Consequently, an R function, namely, MinMed3, which focuses on the minimisation of the MVE, was written to implement the following: The median of the values (d i ), as defined in function [Equation (6)], is minimised by MinMed3.
The output of MinMed3 contains the optimal weightings of the various single-factor multiples models contained in the composite multiples models. The MVE approach was affected via the solnp function, a non-linear optimisation function based on the Lagrange method, in the R-package Rsolnp.
The third application, namely, Quadprog, optimises the weight allocations, based on the objective of minimising the SSVE. However, the underlying principle of the SSVE approach is similar to that of linear regression, which academic researchers generally favour due to its simplicity and the ample software programmes available in support of it. Valuation theory, however, suggests that very few, if any, relationships among multiples are linear (Damodaran, 2006a;Yee, 2005). Therefore, despite the popularity thereof, the SSVE-based results were deemed less reliable.
Consequently, the lpSolve application, which optimises the weight allocations, based on the objective of minimising the SAVE, was used as the main mathematical optimisation tool. The second application, Rsolnp, which optimises the weight allocations based on the objective of minimising the MVE, was used to validate the results that were obtained from the lpSolve application. The latter results also afford one the opportunity to compare the results with those of studies which applied median-based valuation errors in the US and European markets.
As with any mathematical optimisation method, the solnp function in the R-package Rsolnp requires the specification of starting parameter vectors. The solution offered by solnp, or any other optimisation function, is dependent on these starting parameter vectors. When the starting parameter vectors are omitted, the solnp function assumes equally weighted starting parameter vectors by default. However, omitting the starting parameter vectors may potentially increase the risk of encountering local minimums.

Local minimums when optimising beta
The risk with local minimums is that the b -values offered by the optimisation applications may not be optimal, i.e. they could differ substantially from global minimums (Le Roux et al., 2014d). One method of addressing the risk of local minimums vis-á-vis global minimums is by altering the starting parameter vectors, i.e. by using various different (random) starting parameter vectors, and by repeating the optimisation process. The optimal solution set would be the one that produces the lowest valuation error, which, if repeated often enough, should be very close to the global minimum, or at least immaterially different from it. Intuitively, then, one could use the optimal output of a previous run of the same method or the optimal output of a different optimisation method as starting parameter vectors. The latter approach was adopted in this study. The optimised output, i.e. the weight allocations in the composite models that produced the most accurate valuations, from the SAVE method, was used as the set of starting parameter vectors for the MVE method. Table II, for example, illustrates the results of the optimisation process for 2010. Note that all the single-factor multiples originally start with an equal weighting of 0.125 in SAVE, after which the optimal output of SAVE becomes the starting parameter vectors in

Own elaboration
Equity-based composite multiples MinMed3. As is evident from Table II, the output from SAVE is optimised further via MinMed3 to eventually reach the optimal MVE-based weightings. A substantial improvement in the valuation precision of the new composite-based model vis-á-vis the original run/method would imply that one has moved substantially closer to the global minimum (Le Roux et al., 2014d).
Although it is impossible to determine conclusively whether the final solution constitutes the global minimum, the aim of this study is not to establish whether the valuation error is at the global minimum. The objective is merely to establish whether composite multiples models produce more accurate valuations vis-á-vis single-factor multiples models and, as the results in the next section will indicate, the latter was confirmed without the knowledge of the actual global minimum valuation errors.

Empirical results and discussion
The empirical analysis initially focuses on the correlation matrices of market capitalisation (MCap) and all eight equity-based value drivers for the period between 2001 and 2010. The focus then shifts to the correlation matrices of the eight equity-based value drivers over the market as a whole, as well as within the six key industries, including a discussion of the occurrence and mitigation of multicollinearity. This is followed by a framework for composite multiples models for each of the six key industries in South Africa. The valuation performance of these composite models is then compared to that of the single-factor multiples models, as contained in Table I, to determine the magnitude of the increase in valuation precision, if any. Lastly, the consistency of the results is investigated for the 10-year period between 2001 and 2010 and compared to evidence from the developed market literature and the only other emerging market study of this kind.

Consistency of the market price variable and value drivers over time
An analysis of the observed relationships between MCaps for the period 2001-2010 is contained in Table III. All the MCaps were positively correlated, and very strongly so, with correlation coefficients ranging between 0.8472 and 0.9813. Therefore, a high MCap in any particular year for the period 2001-2010 is likely to be accompanied by a high MCap in the other nine years as well.
A similar conclusion can be drawn from the value drivers contained in Table III, i.e. all the observed relationships were positive, and, with the exception of OD, net cash inflow from investment activities (NCIfIA) and free cash flow to equity (FCFE), these relationships were very strong, with correlation coefficients ranging between 0.7409 and 0.9699. Even among the three value drivers mentioned above, only a few pairwise combinations of years exhibit a relatively poor correlation coefficient compared to the other value drivers.
The  have produced a high estimate of MCap in the other nine years as well. However, given the selection of value drivers, the existence of a high degree of multicollinearity is also likely. Table IV contains the pair-wise Pearson correlations of all eight value drivers of the market as a whole for 2010. All eight value drivers exhibit positive and very strong relationships. Overall, the correlation coefficients range between 0.6220 and 0.9912, which may suggest that not all the value drivers share the same information content.

Multicollinearity
Two exceptions are noted, namely, the pairwise combinations of ordinary dividends (OD) and net cash inflow from operating activities (NCIFOA), and OD and NCIfIA. These two, cashflow-based combinations are the only value drivers that exhibit correlation coefficients of less than 0.70. This might suggest that OD, NCIfOA and NCIfIA carry incremental information content, not only relative to the other cash-flow-based value drivers but also across all the value drivers, i.e. including those that were extracted from other types of financial statements.
From the evidence presented by Nel et al. (2014e, 2013d, one would be inclined to argue that the construction of a composite multiples model should incorporate headline earnings (HE) as an independent variable. From the correlation coefficient matrix in Table IV, it   2010  2009  2008  2007  2006  2005  2004  2003  2002  Notes: Note that the correlation matrices contain the logged correlation coefficients. There were numerous outliers in this study, which decreased the correlation coefficients. Consequently, a logged analysis was deemed more appropriate as it diminished the impact of these outliers. Own elaboration Equity-based composite multiples seems prudent to consider OD or NCIfIA as a second independent variable. However, a carte blanche application of such a composite model is not warranted. Each of the six industries should be considered in isolation and a composite model consisting of a combination of HE, OD and NCIfIA may not be the de facto best choice for inclusion in every composite model. From a financial statement perspective, all value drivers that were extracted from the same type of financial statement have high correlation coefficients, i.e. they share considerable information content. Value drivers that were extracted from the statement of comprehensive income, in particular, exhibit very high correlation coefficientsin the vicinity of 0.94 or higher. Similarly, value drivers that were extracted from the cash flow statement share considerable information content, which is evident from their respective correlation coefficients of around 0.90 or higher. This suggests a high likelihood of encountering a fair amount of multicollinearity when using regression analysis to the data. The correlation matrices discussed thus far were based on the market as a whole, while the focus of this paper is on the construction of industry-specific composite multiples models. Consequently, it is equally important to compare the correlation coefficients of the equity-based value drivers on an industry basis as this forms the basis of the composite modelling. Table V contains these matrices for 2010.
The correlation coefficients contained in Table V indicate that the basic materials and financials industries also exhibit positive and very high correlations among the equity-based value drivers, on which the composite modelling in this paper is based. Although the majority of the pair-wise correlations in the Consumer Goods industry are highly positive, NCIfIA and OD exhibit a pairwise correlation of 0.5467, which is poor. In the consumer services industry, NCIfIA is poorly correlated with all the earning-based value drivers, indicating pairwise correlation coefficients between 0.5216 and 0.5875. Similarly, OD is poorly correlated with all the cash-flowbased value drivers, indicating pair-wise correlation coefficients of between 0.2182 and 0.6325. OD is particularly poorly correlated with NCIfIA, which is indicated by a correlation coefficient of 0.2182. In the industrials industry, NCIfIA is poorly correlated with all the non-cash-flow-based value drivers, which is reflected by correlation coefficients of between 0.3653 and 0.6277, while OD is poorly correlated with BVE (0.6301) and all the cash-flow-based value drivers, which is reflected by correlation coefficients of between 0.3653 and 0.6453. In the technology industry, it is evident that OD is poorly correlated with all the other value drivers, reflecting correlation coefficients of around 0.40, or less, while NCIfIA is poorly correlated with BVE, indicating a correlation coefficient of 0.5303.

Composite model framework
To compile the composite multiples models, it was necessary to obtain the optimal weightings for each of the components to be included in each model. All eight equity-based single-factor multiples contained in Table I, namely, P/PBT, P/PAT, P/HE, P/BVE, P/OD, P/NCIfOA, P/ NCIfIA and P/FCFE, were considered for inclusion in the composite models. These eight singlefactor multiples emanate from four different value driver categories, namely, earnings, assets, dividends and cash flow. The inclusion of value drivers from four different value driver categories ensures that each value driver category potentially carries incremental information content as all four value driver categories originate from different financial statements. PAT, for example, was extracted from the statement of comprehensive income, and, while it is an indication of a company's profitability, it does not represent cash in the bank for shareholders, i.e. profit after tax (PAT) is unlikely to culminate in an equally valued cash dividend. In this case, OD would be a more realistic value driver from an equity holder's perspective. Equity-based composite multiples From these eight single-factor multiples models, composite multiples models were constructed for each of the six key industries in the South African market. The breakdown of the composite models, based on the SAVE optimisation process, is contained in Table VI.
The following can be gleaned from the composite models: First, composite models do not perform the most accurate equity valuations across the board. The evidence suggests that in the consumer services industry, a single-factor multiple, specifically P/HE, is the optimal choice of multiple in 2008 and 2010. Similarly, P/HE is the optimal choice of multiple in the technology industry in 2003.
Second, the evidence suggests that there is no one-size-fits-all composite framework across all six industries, or even consistently so within any single industry. For example, while the composite multiples model in the basic materials industry in 2009 consists of five different single-factor multiples models, the composite multiples model in 2001 consists of just two.
Third, note that with the exception of 2007 in the basic materials industry, which consists of six single-factor multiples models, none of the composite multiples models consists of more than five single-factor multiples models, despite the availability of eight single-factor multiples models. The composite multiples models predominantly consist of two to four single-factor multiples models and the most common number of single-factor multiples models included in the composite multiples models is three. This suggests that an ad hoc addition of single-factor multiples models will not necessarily increase the valuation precision of the composite models.
Fourth, note how earning-based single-factor multiples models dominate the composition of the composite multiples models over all six industries. On average, earning-based value drivers, as a category, comprise between 40.90 and 89.68 per cent of the composite models, which confirms the superior valuation performance of earning-based multiples evident in the literature (Nel et al., 2014e, 2013d. Earning-based multiples carried a particularly heavy weighting in the consumer services and technology industries, comprising, on average, 89.68 and 78.05 per cent of the composite models respectively. On an individual value driver basis, on average, the weighting assigned to HE is between 27.29 and 78.55 per cent, confirming its superiority among the individual value drivers selected for this study. HE comprised, on average, more than half the composition of the composite models in three industries, namely, consumer services (78.55 per cent), technology (62.67 per cent) and industrials (56.62 per cent). Profit before tax (PBT) managed to secure weightings of, on average, between 1.99 and 14.27 per cent over all the industries, with the exception of the consumer services industry, where it failed to secure a weighting. Similarly, PAT carried an average weighting of between 0.87 and 19.96 per cent over all six industries.

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These findings concur with earlier evidence in the literature regarding the valuation performance of the P/HE, P/PBT and P/PAT single-factor multiples (Nel et al., 2014e). Fifth, note the cash-flow-based value driver category's unexpected contribution to the composition of the composite models. As a value driver category, cash flows generally produce the least accurate valuations, even less so than revenue (Nel et al., 2014e;2013d). However, on an individual value driver basis, two of the cash-flow-based value drivers, namely, NCIfOA and NCIfIA, occupied, on average, between 0.01 and 13.40 per cent and between 1.11 and 14.95 per cent component shares, respectively, over five of the six industries. NCIfOA failed to occupy a weighting in the financials industry and NCIfIA failed to occupy a weighting in the consumer goods industry. NCIfIA, in particular, when combined in a composite model with value drivers from other value driver categories, seems to contribute to a greater extent, in comparison to its isolation as a single-factor multiple. This suggests that NCIfIA carries incremental information content, in addition to that offered by HE, for example. FCFE, the third cash-flow-based value driver, had the lowest component share of all eight value drivers, occupying, on average, less than five per cent of the composite models across all six industries. The latter concurs with the relatively poor valuation performance of P/FCFE as a single-factor multiple (Nel et al., 2014e, 2013d. Sixth, the asset-based value driver category, on average, occupied similar weightings to the cash-flow-based value driver category. Although these two value driver categories, on average, on a per industry basis, managed to outperform each other interchangeably, their average weightings over all six industries were similar. The contribution of BVE to the composite models varied between 0.84 and 25.84 per cent and was particularly prevalent in the basic materials and financials industries, where it occupied, on average, 25.84 and 20.03 per cent component shares, respectively. However, the contribution of BVE in the consumer services (0.84 per cent) and technology (3.05 per cent) industries were insubstantial. It is of interest to note that, on average, BVE occupied a marginally smaller component share than the cash-flowbased value driver category over all six industries. This is in stark contrast with findings in developed markets, where BVE is frequently included as a second most well-weighted constituent in composite modelling (Penman, 1998;Schreiner, 2007;Yoo, 2006). Seventh, the dividend-based value driver category, which, on average, over all six industries, occupied the smallest component share of all four value driver categories, contributed slightly less in a composite structure than when isolated as a single-factor multiple, culminating in component shares of between an average of 2.43 and 14.90 per cent. OD's weightings in the consumer services (3.65 per cent) and technology (2.43 per cent) industries were insubstantial. OD carried its highest weighting in the financials (14.90 per cent), industrials (14.88 per cent) and consumer goods (14.55 per cent) industries.
These results suggest that composite multiples models offer superior explanatory power as compared to single-factor multiples models [2]. The question, however, is whether the increase in valuation precision that is offered by composite modelling, as compared to the more traditional single-factor modelling, is substantial.

Comparison between composite models and traditional models
The relative valuation performance of the composite multiples models and single-factor multiples models for the entire period from 2001 to 2010 is displayed in Table VII. The evidence suggests that composite multiples models carry incremental information content vis-á-vis single-factor multiples models. The impact of the incremental information, as encapsulated in the composite models, on the valuation precision of equity-based multiples for the period 2001-2010 is also summarised in Table VII

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Note that the potential percentage increase in valuation precision (IMP) in Table VII indicates the extent to which equity-based composite multiples models outperformed the optimal equity-based, single-factor multiples models (highlighted) in each of the six industries. See Nel et al. (2014e) for a detailed discussion of the construction of optimal single-factor multiples. The description NA refers to industry years where there was insufficient data for comparison. A zero value in the IMP column, as is the case in the consumer services industry in 2008, for example, refers to industry years where specific single-factor multiples models produced the most accurate multiple, i.e. where composite multiples models failed to produce more accurate valuations than single-factor multiples models.
As is evident in Table VII, the results indicate that on average, there are substantial gains to be secured by using composite multiples models instead of single-factor multiples models. The average annual IMPs, i.e. over all six industries, are indicated in the last column in Table VII. The range of average annual IMPs over all six industries for each of the 10 years lies between 20.21 and 44.59 per cent, which is substantial. The consistency of the outperformance of composite multiples models over single-factor multiples models is evident in all the industries except for the Technology industry, where a lack of data obscured a more detailed analysis. Equally substantial gains can be secured on a per industry basis over the 10-year period, with an IMP range, on average, of between 10.12 and 44.11 per cent. With the exception of the consumer services industry, which secured precision gains of 10.12 per cent, all the industries indicate gains in excess of 25 per cent, on average.
Aside from the lp function in the R-package lpSolve, the solnp function (in the R-package Rsolnp), which is particularly adept at handling non-linear optimisations, was also used to determine the optimal weightings. Although not shown here, the results from the solnp function indicated a similar, but higher, average annual IMP range of precision gains of between 12.65 and 66.98 per cent over the 10-year period[3]. On a per industry basis over the 10-year period, the IMPs, on average, ranged between 14.39 and 72.64 per cent. All the industries indicated substantial precision gains of 30 per cent or higher, on average, with the exception of the consumer services industry, which indicated an average gain in valuation precision of 14.39 per cent.
Although the data contained in Table VII reflects the magnitude of the increase in valuation precision that composite modelling may offer over single-factor modelling, the multidimensional nature of the data obscures a comprehensive grasp of the relative valuation performance of composite modelling over time. This is an important consideration for the way in which composite multiples models should be applied in practice. Because the data occupies multi-dimensional spacei.e. it encapsulates multiple coordinate axesthe use of a conventional two-dimensional scatter plot is inappropriate (Gower et al., 2011). However, the use of PCA biplots accommodates higher-dimensional data by approximating it in lower, usually two-dimensional space, thereby enabling the visualisation of multi-dimensional data.

Consistency of the valuation performance of composite models over time
The superior valuation performance of composite multiples models relative to single-factor multiples models can be illustrated more effectively with the help of PCA biplots. Figure 1, for example, depicts the valuation performance of the composite multiples models relative to that of the single-factor multiples models in the basic materials industry for the entire period from 2001 to 2010. The composite models are depicted to the far right of the PCA biplot, confirming their consistent superior valuation performance for the period from 2001 to 2010. For a more detailed discussion on the use of biplots, see Gower et al. (2011).
Note that the axes are colour-coded. The ten pink axes reflect the fact that composite multiples models produced more accurate valuations than single-factor multiples models for all ten years between 2001 and 2010. The quality of display reading of the PCA biplot in Figure 1  In summary, the evidence from the South African market suggests that a composite modelling approach to equity valuations outperforms the traditional single-factor modelling approach. How do these results compare with the results from other emerging markets and developed markets?

International comparison
Unfortunately, composite-related studies are limited, both in number and scope. In addition, the industries selected in these studies seldom match the six key industries for which sufficient data was available in the South African market. Those studies that do offer a comparative perspective on composite modelling, both concur with, and contradict, the findings from this paper.
The most comparable set of results was produced by Schreiner (2007), who compared a two-factor composite model over three industries in Europe and the USA. Schreiner's overall results showed that two-factor composite multiples models produced, on average, 10.86 per cent more accurate valuations than single-factor multiples models in the USA and 15.32 per cent more accurate valuations in Europe. The South African results, therefore, concur with those of the developed markets, in that composite multiples models in the South African market produce more accurate valuations than single-factor multiples models. From Section 4.4, it is evident that the magnitude of the improvement in valuation precision is more substantial in South Africa's case. Unfortunately, a more detailed comparison is not possible as none of Schreiner's selected industries correspond with any of the six key industries in the South African study.
However, the research results from this paper are in stark contrast with the results produced by Sehgal and Pandey (2010), who found conflicting evidence in South Africa's case. On the basis of the root mean squared errors method, they found that two-factor composite multiples models failed to outperform optimal single-factor multiples models. Then, on the basis of Theil inequality coefficients, they found an insubstantial improvement in valuation precision of 4.17 per cent. Equally insubstantial and inconsistent results were found for the other emerging markets.

Conclusion
The aim of this paper was to determine whether industry-specific, equity-based, composite multiples models offer higher degrees of valuation precision compared to industry-specific, equity-based, singlefactor multiples models. The findings confirmed that equity-based composite multiples models producedvaluationsthatweresubstantiallymoreaccuratethanthoseofsingle-factormultiplesmodels.
The study focused on equity-based multiples, in particular, and the results were tested for the period between 2001 and 2010. On the basis of the SAVE methodthe primary optimisation method that was applied in this studycomposite models, on annual average, produced between 20.21 and 44.59 per cent more accurate valuations than single-factor multiples models for the period 2001-2010 did. Although this already presents a substantial IMP range, the results obtained from the MVE method indicated an even higher average annual IMP range of between 12.65 and 66.98 per cent. However, these results were not equally consistent over all six key industries. The composite multiples models failed to offer higher degrees of valuation precision compared to single-factor multiples models in 2008 and 2010 in the consumer services industry, and in 2003 in the technology industry.
An interesting phenomenon was observed regarding the valuation performance of the dividend-based value driver category within the context of composite modelling. The market-and industry-based research findings obtained from the finance literature suggest that dividends produce fairly accurate valuations. However, the dividend-based value driver category, on average, secured the lowest weighting of all four value driver categories, and had particularly low component shares in the consumer services and technology industries. Equally interesting was that on a value driver category basis, the cash-flow-based value driver category, which the finance literature suggests generally produces poor valuations in terms of valuation precision, managed to secure a higher weighting than the asset-and dividend-based value driver categories, on average.
As was gleaned from the finance literature, earning-based multiples contributed substantially to the valuation precision of the composite multiples models. Accordingly, earning-based multiples did, indeed, dominate the composition of the composite multiples models. Earning-based multiples occupied, on average, between 40.90 and 89.68 per cent of the composite models. The bulk of the earnings weighting was carried by HE, which comprised a component share of between 27.29 and 78.55 per cent, on average. These results concur with the valuation performance of earning-based multiples, and HE as a single-factor multiple, in particular, in the finance literature.
The evidence, therefore, suggests that equity-based composite modelling may offer substantial gains in valuation precision over equity-based, single-factor multiples modelling. These gains are, however, industry-specific and a carte blanche application thereof is ill Equity-based composite multiples advised. Therefore, as analysts' reports typically contain various single-factor multiples, it seems prudent to consider the inclusion of composite models as a more accurate alternative. Although the research results concur with evidence from developed capital markets, they contradict the findings from the only other study conducted on composite modelling in emerging markets. Although it is not entirely clear why the research results from this study differ from those of the other emerging market-related study, it is possible that at least some of the discrepancies can be traced to different designs and methodologies applied in these studies.
The focus of this study was on equity-based modelling in particular. Although one may be inclined to consider company-based variables for inclusion in composite models, these models should be constructed in an internally consistent manner. Failure to do so may result in conceptually flawed models, which may obscure the interpretation of the results. Although it was not the focus of this study, a separate study, focused on company-based composite modelling, may produce interesting results.
Notes 1. Note that incorporating company-based, single-factor multiples into the equity-based composite model will result in model inconsistencies, which may obscure the interpretation of the results.
2. The data were also subjected to PCA on a per industry basis, following which PCR analysis was applied to the resulting two or three principal components, with similar results. Although the composite modelling via PCR indicated R-squared values of between 0.75 and 0.95, with statistically significant coefficients, at least at the 95 per cent confidence level, and of the correct sign (positive), various assumptions of the standard Gauss-Markov theorem were violated. Consequently, the regression results were omitted from the analysis.
3. The SSVE method was also applied via the solve.QP function in the R-package Quadprog, with similar results. However, given the non-linear nature of the data and its other limitations, these results are not shown here.