Mathematical modelling and the empirical validation of organisational financial performance – conceptual insights Into the inferential focus of the analytical perspectives in the finance

International Journal of Management (IJM)
Volume 11, Issue 3, March 2020, pp. 399–407, Article ID: IJM_11_03_042
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MATHEMATICAL MODELLING AND THE
EMPIRICAL VALIDATION OF
ORGANISATIONAL FINANCIAL
PERFORMANCE – CONCEPTUAL INSIGHTS
INTO THE INFERENTIAL FOCUS OF THE
ANALYTICAL PERSPECTIVES IN THE
FINANCE DISCIPLINE
Dr. P. Raghunadha Reddy
Professor and Head, Department of Management Studies,
Sri Venkateswara University, Tirupati, India
V.G. Siva Sankara Reddy
Research Scholar, Department of Management Studies,
Sri Venkateswara University, Tirupati, India
ABSTRACT
Purpose of the Study:
This study traces the evolution of analytical methods in building Finance Theory
with a view to strike an ‘optimal’ balance between the analytical rigour and the realworld inferential insights.
Methodology:
The theoretical developments in the latter half of the 20th century, in the field of
Finance, have focused, extensively, on the Analytical basis of sound theory building and
its Empirical validation using the Statistical tools.
The pioneering work of Miller and Modigliani that analytically established the
relationship between the firm’s financial leverage (Debt component) and the Value of

the firm, under varying assumptions marked the beginning of analytical approaches to
building Finance Theory. The corporate bankruptcy model developed by Altman was
also studied.
The subsequent empirical studies have also been examined to assess the practical
validity and relevance of their findings.

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Dr. P. Raghunadha Reddy and V.G. Siva Sankara Reddy
Main Findings:
a) While the analytical modelling, by virtue of its elegance of logic and causation,
has been widely acclaimed as the most efficient tool of theory building, it is beset with
certain inherent limitations. More specifically, the field of Social sciences, which
includes several functional areas of management, is intrinsically determined by
behavioral factors and therefore, the stand-alone mathematical modelling (that
overlooks the ‘unpredictability’ of behavioral parameters) is fraught with the danger of
erroneous conclusions.
b) The Behavioral parameters are, in turn, determined by the psycho-sociological,
ethnic, geographic and other factors; this makes the ‘analytical’ handling of the
behavioral parameters more cumbersome and therefore inefficient.
c) The ‘percolation’ of Statistical analysis into conceptually deterministic models
has blurred the researcher’s distinction between the Stochastic and Deterministic
(tautological) models thereby, resulting in ‘proving’ the obvious.
d) Finally, the article concludes with the observation that the utility of mathematical
modelling can be enhanced by articulating the broad contours of causal relationships
among the various parameter so as to gain tangible insights into the real-life decision

situations and also by suitably modifying the rigidities of the model to suit the ‘nuances’
of the specific situation. In other words, the researcher should stress more on the ‘spirit’
of the model as opposed to its elegantly framed ‘structure’ of equations.
Applications of this Study:
This study is expected to make the Finance researchers to focus on the inferential
insights into the ‘quantitative’ parameters emerging from the analytical models so as
to enhance the utility of Analytical methods employed in Finance.
Key words: Mathematical modelling, behavioral factors, Inferential Focus, Stochastic
versus Deterministic models, Theory building.
Cite this Article: Dr. P. Raghunadha Reddy and V.G. Siva Sankara Reddy,
Mathematical Modelling and the Empirical Validation of Organisational Financial
Performance – Conceptual Insights into the Inferential Focus of the Analytical
Perspectives in the Finance Discipline, International Journal of Management (IJM), 11
(3), 2020, pp. 399–407.
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1. INTRODUCTION TO ANALYTICAL PERSPECTIVES IN FINANCE
The Finance discipline has its foundations in the Economic Sciences and therefore, the most of
the analytical approaches in Finance have been adopted from Economics. In the earlier days,
when reasoning in Economics was based on the Subjective Knowledge (as opposed to
Objective) that was accumulated through past experiences and observations, the greatest of
economic thinkers applied the non-mathematical logic in their deductive approaches to
Knowledge building. We may trace the beginnings of calculus-based reasoning approaches in
Economics to Alfred Marshall, who believed that analytical methods lead to precision of
definition that leads to the ‘accuracy’ of the deductions and conclusions that emanate from it.
The Nobel Laureate Paul Samuelson may be credited with popularizing the analytic formalism
in the basic courses in Economics.
The Finance discipline has been essentially based on the numerical inputs and therefore the
use of Quantitative methods in Finance are obvious. However, as Accounting (in the earlier
days) was considered the prime objective of the Finance function, the earlier theories in Finance
were mostly confined to elementary analysis of the numerical figures generated by the

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Mathematical Modelling and the Empirical Validation of Organisational Financial
Performance – Conceptual Insights into the Inferential Focus of the Analytical Perspectives in
the Finance Discipline
Accounting System. As a result, until the mid-twentieth century, the Finance discipline was
considered to be a ‘number crunching’ job that was not ‘intellectually’ demanding. However,
the pioneering work of Modigliani and Miller in 1958, which was popularly known as the MMHypothesis, initiated the development of the Analytical approaches in the field of Finance.
Today, we are witnessing a phenomenal growth in Financial Modelling that is threatening to
replace the human wisdom by the Algorithmic wisdom in several areas of Finance. Therefore,
this is the time for the researchers in Finance to ‘reflect’ on the likely ‘erroneous’ conclusions
flowing out of the ‘excessive’ mathematical modelling in Finance that may lead to several
dysfunctional consequences.

2. DuPont Chart and the evolution of Financial Performance Modelling
The DuPont chart illustrates the components that contribute to the Return on Capital Employed
that forms the basis of a simple equation that links the Total Asset Turnover ratio and the
Profitability ratio to the Return on Total Assets. The basic equation is given below.
ROA = (Total Asset Turnover ratio)*(Profit Margin) or (Net Income/Total Assets) =
(Sales/Total Assets)*(Net Income/Sales).
This leads to the basic deduction that the ROA can be increased by either increasing the
Total Asset Turnover or by increasing the Profit margin (or both). In other words, the DuPont
chart based analysis provided the Finance manager with the basic tools for achieving the ROA
objective.
The researchers in the Finance field (since the end of 1960s) attempted to bring in a greater

analytical focus into the theoretical foundations of Finance. This approach is clearly evidenced
in the landmark paper published by Edward Altman titled, “Financial Ratios, Discriminant
Analysis and the Prediction of Corporate Bankruptcy” in the Journal of Finance (1968). In his
introductory remarks, the author stated as follows. “Can we bridge the gap, rather than sever
the link, between traditional Ratio “analysis” and the more rigorous statistical techniques which
have become more popular among academicians in the recent years?”
This work by Altman has made the Z-score metric developed by him very popular amongst
Finance professionals to evaluate the credit strength of the concerned organizations. While it is
not the main purpose of this article to make a critical appraisal of the Z-score, the Altman’s
work has been referred to trace the analytic evolution in Finance Research. The researchers in
their enthusiasm to add ‘rigor’ to their research work have ended up being ‘dominated’ by the
predictions of the man-made ‘mathematical/statistical’ models; this has made the decisionmakers increasingly ‘algorithmic’ driven during the present times. The point is that the
researchers should attempt to promote a ‘symbiotic’ relation-ship between the ‘model-driven’
logic and the ‘practical’ wisdom of the decision-makers. Such an approach is needed to factorin the ‘behavioral’ parameters which are subject to ‘erratic’ fluctuations; this phenomena
‘appears’ to be adequately captured by the introduction of a ‘statistical’ random variable
governed by a ‘mathematically’ defined probability distribution. For instance, the Nobel
laureates who popularized the extensive use of the Black-Scholes option pricing model towards
the end of the 20th century, based their rigorous analysis assuming the ‘log normal property’ of
Stock prices (that is ‘too’ idealistic, if not unrealistic). The researchers in Finance should note
that ‘behavioral’ variables do not fit a standard stochastic process that is valid across different
time horizons (unlike the random variables encountered in the physical sciences; for instance
the Brownian motion.
Most of the present day research on financial performance is focused on Multivariate linear
Regression models that are designed to explain ROA or ROE in terms of the various other
financial ratios such as Debt-ratio and Total Asset Turnover.

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Dr. P. Raghunadha Reddy and V.G. Siva Sankara Reddy
The researcher’s focus on the mathematical/statistical ‘nuances’ of the model is diverting
the attention of the readers from the ‘validity’ aspects of the underlying parameters that the
Accounting System provides. The point is that the ‘validity’ of the parameters are intrinsically
dependent on the ‘objectivity’ and ‘standardization’ of the Accounting outputs and surely, we
are not anywhere near the goal of ‘international convergence of accounting principles’ that is
vital for cross-border comparison of financial performance. Further, the Accounting policies
are varying across business sectors due to the peculiarities of the environment surrounding the
specific sector. Therefore, the rigorous model building approach to Theory-building in Finance
may satisfy the ‘intellectual’ self-actualization needs of the researcher but may not be
‘practically’ beneficial.
3. REGRESSION MODELS AND THE BOUNDARY CONDITIONS
The primary limitation of fitting the Regression model is that it is based on an ‘implied’
assumption that the data fit is described by an ‘elegantly’ defined mathematical function (be it
Linear, Polynomial, Logarithmic, trigonometric or Differential). In reality, a given set of data
may be fitted into a ‘standard’ mathematical function using the ‘Least Squares’ argument. But,
the truth is that such regressions are valid only within a specified range of values of the
independent variables; these are commonly known as the Boundary conditions (a term used in
Physical Sciences).
With respect to the linear regression models frequently used in financial performance
analysis, it should be noted that the partial derivatives are ‘constant’ terms; this is not a
‘realistic’ assumption. For instance, if Y = f(x1, x2...xN) is a linear regression function, then
the ‘Beta’ coefficients βj, for j=1 to N, are all constants. Mathematically speaking βj is the
partial derivative of Y with respect to xj or we may write (∂Y/∂xj) = βj; but, in a real-life
situation, not all the partial derivatives are constant. This is the one serious limitation of using
the elegantly defined mathematical functions for Theory building in the Social Sciences.

4. STOCHASTIC VERSUS DETERMINISTIC MODELS

Conceptually, the researchers in Finance should clearly appreciate the basic distinction between
the Stochastic and Deterministic models. The mathematical relationship that is defined by
parameters that take ‘specific’ values (as opposed to ‘random’ values) is known as a
Deterministic model. In contrast, when the parametric inputs defining the mathematical
relationship are ‘random’ variables, we refer to them as stochastic models. A Stochastic model
is based on the underlying probability distribution that is generally governed by the two
important parameters called the ‘mean’ and ‘standard deviation’. In other words, a simple
deterministic Revenue model may be represented by “R=p*Q” where ‘R’ is the sales revenue
and ‘p’ is the price and ‘Q’ is the sales quantity. However, the same equation becomes a
stochastic model when ‘Q’ is considered as a ‘random’ variable and such models call for
statistical validation. However, some researchers attempt to validate a deterministic model
using the statistical tests of significance which are ‘superfluous’. Thus, such validations end up
proving the ‘obvious’.
Such redundancies occur when the researcher attempts to prove the significance of the
‘slope’ of a deterministic linear model. For instance, consider a deterministic Total cost model,
“C=f0+v*Q” where ‘C’ is the Total cost, ‘f0’ is the fixed cost and ‘v’ is the unit variable cost
and ‘Q’ is the output quantity. We need not use elaborate Statistical model building tools to
develop this relationship and prove the statistical significance of the ‘slope’ of the cost function
which is obviously (tautologically) equal to ‘v’ (a ‘determinable’ parameter, as opposed to
‘random’ parameter).

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Mathematical Modelling and the Empirical Validation of Organisational Financial
Performance – Conceptual Insights into the Inferential Focus of the Analytical Perspectives in
the Finance Discipline

5. GLOBAL STOCK-MARKET INDICES AS A FACTOR IN
EVALUATING FINANCIAL PERFORMANCE
Some researchers have used the ratio, (market price of the share/book value) as an independent
variable in the Regression model determining the ROE (Return on Equity) which has certain
‘inherent’ flaws. The Stock market price of the shares is a highly volatile variable whose
‘stability’ is limited to a very short time interval. This is basically due the market
sentiment/expectations of the investors (including speculators) which makes the Regression
model a ‘Time’ dependent function that has only a ‘momentary’ utility. Today’s Stock markets
have grown much beyond their original mandate of providing a market for ‘liquidity and price
discovery’ to encourage the retail investor participation in the capitalist economy. However,
the ‘excessive’ speculation prevalent in the international markets has made ‘Stock Market
Price’ a fairly ‘unreliable’ measure of the firm’s economic value (due to the Stock market
volatility). The researchers may reflect on this aspect of ‘erratic’ randomness which leads to
‘time-variant’ probability distributions. As a result, the models built on the assumption of a
‘precisely’ defined ‘time-invariant’ probability distribution is fraught with the danger of
‘erroneous’ inferences.

6. THE SPECTRUM OF GLOBAL RESEARCH FINDINGS BASED ON
MATHEMATICAL/STATISTICAL MODELLING:
1. Hansen and Mowen (2005) - “Firm performance measurement is vital for effective
management”
2. Fleming and Heany (2005) - “Asset dis-utilization may increase Agency costs”
3. Katja (2009) - “performance measures are used to evaluate the success of Economic
units”
4. Okwo (2012) – “Fixed Assets to profitability showed a positive correlation but not
statistically significant.” This is because, fixed assets are only one of the factors affecting
profitability. Further, the other external factors contributing to profitability are generally
excluded from the Regression model that may get incorporated in the ‘intercept’ which is the
constant parameter β0.

5. However, the study by Xu and Xu (2013) titled, “A study on optimal allocation of asset
structure and business performance”, has found that the relationship between Fixed assets and
Profitability is significant. From this, the reader has to realize that there is nothing ‘universal’
about the statistically validated theoretical results based on empirical studies. In order to
understand the variance between two research findings attempting to build the same conceptual
foundations, the astute reader should delve into the ‘specifics’ of the researcher’s empirical
settings. For instance, the significance of the fixed assets in the manufacturing setting may more
than that in the Service sector.
6. The other studies that have agreed with the findings of Xu and Xu are, Jose et al (2010),
Wu et al (2010) and Seema et al (2011).

7. THE EMPIRICAL RESULTS ON LEVERAGE TO ROE
RELATIONSHIP:
1. Acquino (2010) studied the Capital structure of listed and unlisted Filipino firms and
concluded that High Debt rate is positively associated with the firm’s growth and profitability.
Similar conclusions were drawn by Joshua (2005).

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Dr. P. Raghunadha Reddy and V.G. Siva Sankara Reddy
2. On the other hand, Aivazania et al (2005) examined the impact of leverage on investment
decisions and found it to be negative. This may be because the ‘riskiness’ of a highly levered
firm is rather high and so the prospective investor may rate such investments low
(notwithstanding its prospects of posting higher ROE).
3. Ahna et al (2006) found that the negative relation-ship between the Debt ratio and
Investment decisions is more significant in the non-critical sector than in the critical sector.

H. the Statistical significance of ‘Tautologies’ - its ‘Redundancy’: A ‘tautology’ is basically
a logical equivalent of an established truth. For instance, if we use two regression equations for
two different dependent variables Y1 and Y2 which are expressed as a linear combination of
several independent variables x1, x2, ...xN. We know that Y2=λ*Y1 where λ is a positive
constant. Thus Y2 is derived from Y1 by multiplying with a constant λ. Further, if we find that
the regression coefficient with respect to x1 (β1) is significant in the case of Y1, it need not be
again separately mentioned for Y2 as these two are linked by a positive constant multiple λ.
There are many instances when the significance tests are repeated to arrive at the same truth.

8. INFERENTIAL FOCUS – A PRACTICAL ILLUSTRATION.
The following Regression model developed by a researcher, Mou Hu of the University if Thai
Chamber of Commerce, titled, “Factors affecting Financial Performance of firms listed on
Shanghai Stock Exchange” which studied the impact of factors like Liquidity, Asset Utilization,
Leverage and Firm Size on the Financial performance has considered two Dependent Variables,
ROA (Return on Assets) and ROE (Return on Equity). It was found that the ROA and ROE are
significantly impacted by Debt ratio (leverage) with a negative slope and Total Asset Turnover
(Asset Utilization) with a positive slope. The Regression equations are given below:
ROA= 26.94+2.115(CR) + 2.294(TAT)-35.452(DR)-2.926(FSD)
ROE=23.12+1.162(CR) + 2.493(TAT)-19.325(DR)-2.91(FSD)
Where CR is the current ratio, TAT is the Total Asset Ratio, DR is the Debt ratio and FSD
is the Firm size discriminant.
From a practical perspective, the research work could have examined the factors
contributing to the difference between the beta-coefficients in respect of DR (-35.452 in the 1st
equation and -19.325 in the 2nd equation).
Using the basic definition of ROA and ROE, we obtain the following mathematical
relationships.
ROA=EBIT/A, where A is the value of total assets financed by ‘D’ and ‘E’ where ‘D’ is
the debt component and ‘E’ is the equity component. We may also use the function “E=(1λ)*A” and “D=λ*A” , where ‘λ’ is the Debt ratio or the debt component of the asset-base.
We may state that ROE= (EBIT- INT on debt- TAX)/E where ‘E’ is the equity. Let‘t’ be
the tax rate and ‘rD’ be the interest rate on Debt. Therefore, the above equation gets rewritten

as follows.
ROE=t*[EBIT – (rD)*(λ*A)]/(1-λ)/A. Upon, simplifying this equation, we arrive at the
following end result.
ROE= [t/(1-λ)]*{ROA- λ*(rD)}. Any researcher using the basic mathematical reasoning
would ‘decipher’ that ROE is ‘Not’ a linear function of ‘λ’ which represents the ‘Debt ratio’.
Clearly, this invalidates the ‘Linear’ Regression modelling assumption. Such ‘unrealistic’
assumptions can be avoided if only, the researchers adopt an inferential focus into the financial
modelling and its validation.

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Mathematical Modelling and the Empirical Validation of Organisational Financial
Performance – Conceptual Insights into the Inferential Focus of the Analytical Perspectives in
the Finance Discipline

9. MODEL VALIDATION – THE INDIAN SETTING
The above model has been fitted to a leading Indian conglomerate using the 5 year financial
results as the base-data for model fitting and the following mathematical relationship has been
established.
ROE = 24.705 + 0.885*(TAT) – 32.21*(DR). In this model, the linear approximation of a
non-linear slope (ROE to Debt ratio) is fairly valid considering the fact that the range of
variation in the debt ratio is confined to the interval 0.425 to 0.440
The above equation has been obtained using the data collected from the Published financial
statements of an Indian conglomerate listed on the Bombay Stock Exchange. The details of the
workings are presented in Appendix – 1.

10. DISCUSSION AND CONCLUSIONS:
From the above, it may be noted that the regressions coefficients are not constant across the
industry spectrum; nor are they comparable across divergent world financial markets. As a
result, the researcher would get better insights into the financial performance metrics by
analytically deducing ‘situation specific’ based on certain reasonable assumptions. For
instance, by assuming that λ=λ0 (a constant debt ratio), we may rewrite the relationship between
ROE and ROA as given below. “ROE = k*(ROA) – (k*λ0)*(rD) where k={t/(1-λ)} and (rD) is
the interest rate on Debt. From the above equation we find that the ‘situation specific’ slope of
the ROE with respect to Debt ratio(λ) or ∂ROE/∂λ is equal to slope of ROA with respect to
debt ratio multiplied by ‘k’ or k*∂ROA/∂λ. Thus, the relationship between the absolute values
of the two slopes are dependent on the factor ‘k’ referred above. This forms the basis for the
inferential conclusions that a researcher can draw from the Regression modelling of the
financial parameters.
The Analytical Models developed in Finance should not just be confined to the testing of
the significance of the Regression coefficients constituting the model. Moreover, the absolute
values of these parameters do not help the researcher in strengthening his conceptual insights.
For this purpose, the researchers have to undertake a critical study of the relevant interconnected factors to ‘deduce’ the ‘complex-maze’ of cause-effect linkages which would add
new insights into the conceptual foundations governing the subject.
Thus, the researchers in Finance would serve the professional community better by
providing practical insights into the ‘numbers’ generated by the Regression models through an
adequate reflection upon the physical reality lying beneath the ‘labyrinth’ of numbers and its
‘modelled’ equations.

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Mathematical Modelling and the Empirical Validation of Organisational Financial
Performance – Conceptual Insights into the Inferential Focus of the Analytical Perspectives in
the Finance Discipline
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APPENDIX-1
Regression ,Model for a BSE-Sensex firm
YEAR
201415
201516
201617
201718
201819

EQUITY roe(Y)

d/edebt
ratio ratio(X1)

tot
assets

(Rs.crores)
Tatratio(X2)

23566

218482

0.108

0.74

0.425

514075

0.756

293298

25171

231566

0.109

0.78

0.438

528689

0.555

330180

29901

263709

0.113

0.75

0.429

614706

0.537

430731

34988

289798

0.121

0.75

0.429

675520

0.638

622809

39588

324644

0.122

0.74

0.425

763868

0.815

total

0.573
∑Y

TUNROVER

PAT

388494

2.146
∑X1

3.301
∑X2

The model to be fitted to the above data
Y=c0+c1*X1+c2*X2
By solving the Normal equations are
the parametric values of the constants
are obtained
Regression Coefficients (Values)
c0=
0.24704917
c1=
-0.3222118
c2=
0.0088521

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