Quantitative Management of Business Disbursements by Portfolio
Optimization
TODOR STOILOV, KRASIMIRA STOILOVA
Institute of Information and Communication Technologies,
Bulgarian Academy of Sciences,
“Acad. G. Bonchev” str. bl.2, Sofia,
BULGARIA
Abstract: - The paper aims to solve the problem of reallocating financial payments between disbursements and
outflows by increasing the efficiency of working capital. The reallocation of financial payments is a different case
from the reallocation of investments. The disbursements can be regarded as a set of assets, which generate negative
returns in comparison with the assets in the portfolio theory. The purpose of this study is to derive a formal model,
which gives quantitative solutions for the reallocation of resources between disbursements. Thus, the
disbursements can indirectly influence positively the business profit of an economic entity. The reallocation of
payments between a set of disbursements can improve the financial outcome of a business entity. Such
redistribution plays an important role in the business management of the manufacturing units. The paper derives a
quantitative model for the assessment and decision-making payment redistribution between payments. The
quantitative solution is based on the application of a portfolio model. The latter is modified by minimizing
disbursements in the portfolio problem. The empirical application of this model is applied to dairy farm payment
management cases. Comparisons applied to the model with the actual set of payments show that the derived model
is better at reducing the total values of the disbursements.
Key-Words: - management of cash flows, disbursement minimization, portfolio theory, decision-making, maximal
return, inventory costs
Received: March 23, 2023. Revised: July 3, 2023. Accepted: July 13, 2023. Published: July 21, 2023.
1 Introduction
A disbursement is a payment event for a specific
resource and/or service. For a business and
production entity, it is mandatory to maintain and
condition the normal rules for production results.
Disbursements are an important part of cash outflows
that can be claimed over a short time horizon. This
makes it important for business management to plan
and consider optimally the allocation of inflows or
available cash to disbursement requests.
Disbursements are considered negative flows for the
financial balance of the business. Negative
complementarity of payments added to financial
flows can lead to losses, cash shortages, and a lack of
profit. Thus, payments are seen as an important
prerequisite for effective business management.
Disbursement management is considered a
prerequisite for maintaining the liquidity of small and
medium-sized enterprises, [1]. Managing financial
flows is not easy, because the production and sale of
products do not give immediate cash flows, [2]. Made
disbursements are critical for supply chain
management, [3].
Thus, disbursements to maintain a continuous
supply chain are not always eligible due to the
difference between the timing of cash inflows and
outflows. This delay can reduce the working capital.
Payment scheduling solutions are one possible way to
increase working capital productivity. An alternative
solution to the scheduling is the reallocation of
financial resources during the year, [4]. In, [5], are
considered cases that deal with the reallocation of
resources in business entities. The reallocation of
financial resources is frequently applied for
investment by means to achieve high returns and low
risk following the dynamic changes in market
behavior, [6]. This means that the target is to obtain
positive profit from the reallocation.
The case of disbursements and their reallocation
not only directly influence the business profit. They
are related to the production program of the business
entity and they cannot be overcome or ignored, [7].
Thus, payment and disbursements have to be
WSEAS TRANSACTIONS on BUSINESS and ECONOMICS
DOI: 10.37394/23207.2023.20.143
Todor Stoilov, Krasimira Stoilova
E-ISSN: 2224-2899
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Volume 20, 2023
considered as a “necessary evil” for the normal
production operation. The management of the
disbursement has an important role in business
decision-making.
Payment and disbursement optimization is targeted
by developing quantitative and optimization models.
The need for optimal management of financial
disbursements insists definition of optimization
problems for cost accounting, [8]. The reasons for the
usage of quantitative decision-making are motivated
in [9]. In, [10], the reduction and reallocation of
payment resources are based on the definition and
solution of an optimization problem. The latter is
based on formal modeling from network graph
theories. The issue allows for an increase in payment
transactions and, in general, this improves the
competitiveness, well-being, and production activities
of the economic entity. The problem of covering
payout payments in case of resource constraints is
solved by applying a simulated annealing algorithm.
Backward scheduling is implemented to complete
payments and maximize payment flow.
This study applies the redistribution of financial
resources between material resources and services
that are carried out for the production policy of the
economic entity. The added value of this research is
that the redistribution is performed not for financial
investments but for disbursements, which can be
regarded as inverse tools, which decrease the business
profit. Redistribution is carried out in the absence of
additional resources other than those of investments,
which gives benefits to the business management. The
redistribution is carried out with means to preserve
the production profit. Redistribution is quantified by
defining and solving an optimization problem. The
problem is based on the portfolio theory, but the
problem is a modification of the portfolio problems.
The modification is necessary because portfolio
theory reallocates investment resources to obtain
maximum returns. In our case, we seek to redistribute
payments by reducing their total value, but
maintaining a high level of production profit. The
problem gives optimal values of resource allocation.
The derived problem is applied to the case of animal
husbandry with the production of milk and milk
products.
The paper contains five sections. The introduction
tells the negative role of the disbursements on the
financial balance. The second section motivates the
opportunity for usage of the formal portfolio problem
to disbursement management. Correspondence
between the portfolio modeling and disbursement
management formalized inappropriate optimization
problems. In section three the defined problem for
disbursement reallocation is applied to the case of
animal husbandry, in which accounting data are taken
from their financial records. Section four derives a
business management policy for the reallocation of
disbursements. This policy is empirically applied to
the accounting data of husbandry. The final section
makes a comparison and conclusion that the derived
formal problem for disbursement reallocation gives
benefit for the business profit of animal husbandry.
Further trends of improvements are mentioned for the
integration of the disbursements with the production
of the husbandry.
2 Portfolio Problem for Disbursement
Reallocations
Portfolio theory is a powerful basis for the optimal
reallocation of financial resources for investment. The
practical problem at which portfolio optimization is
directed is to estimate the shares of the investment
that should be invested in various assets to achieve
future maximization of returns and to maintain low or
minimal risk to the investment. Portfolio theory has
its achievements and proper applications, [11]. Trends
and improvements are discussed in, [12]. An
overview of applied optimization technologies can be
found in, [13]. Portfolio theory raises two important
criteria for optimizing investments: risk and return.
Risk is defined as a range of real portfolio return
values that are close to the portfolio’s average return.
This range is quantified by the standard deviation of
the portfolio’s real returns that can occur around the
mean. The risk and return of a portfolio are functions
of the respective risks and returns of the assets
participating in the portfolio. The inputs to the
portfolio problem are the historically obtained returns
on the assets that are commonly traded on the
exchange
,
(1)
where N is the number of assets in the portfolio,
n is the number of historical returns,
is the real return on asset i at time j.
The average return of asset, i is the average sum
The risk of the asset t is quantified by the standard
deviation of the series of values or
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Todor Stoilov, Krasimira Stoilova
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(3)
Formally, the portfolio return is the weighted
sum of asset returns
where is the relative amount of the
investment that is allocated to purchase asset i.
Portfolio returns are a bit more complicated to
estimate. It depends not only on the risks of the
individual assets , but takes into account the
influence of the covariance between the time series
of each pair of assets
These values can be positive and negative
depending on whether the time series of asset returns
is increasing or decreasing simultaneously for a
positive covariance or vice versa for a negative one.
The set of covariance coefficients forms the
covariance matrix , which is symmetric,
and its diagonal contains the variances of the asset
returns
The portfolio risk is estimated as a quadratic
form
Relations (4) and (7) quantify the values of
portfolio risk and return depending on the same
parameters for the assets and their relative
participation in the portfolio.
The portfolio problem is defined as minimizing
portfolio risk and maximizing portfolio return, [14].
i
where the coefficient 0 ≤ λ ≤ 1 gives the weighting
preferences in the risk or return objective function of
the portfolio. The coefficient λ must be determined
by the investor, depending on his intention to take or
not risk. To overcome this subjective choice, the
broad practice is to choose such a value of that
gives the minimum relation.
This research applies the methodological
formalization of the portfolio theory but for a
different set of assets. This set is constituted of
disbursements, which play an opposite role in the
portfolio return. The innovativeness of this approach
is that it can formalize an optimization problem in the
reallocation of the disbursements. A particular case is
that this reallocation can be provided without
additional financial resources. Thus quantitative
solutions in an optimal manner can be evaluated for
the reallocation of the disbursement. The formal
background for the definition of the optimization
problem is motivated below, based on the portfolio
theory.
These general relations (1)-(9) of portfolio theory
apply to the case of redistribution of payments.
Historical data is assumed to be available for the
various payment categories
,
where N is the different payout category and n is
historical data available for them. The corresponding
mean and covariance matrix are calculated according
to (4-6).
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The payment redistribution portfolio problem is of
the form (8) but modified to minimize the two
components of the mean and risk of the payments.
For simplicity, we keep the notation for the weights
by w
i
Problem (12) generates many solutions for
different values of λ. To choose the optimal one, we
define a criterion that corresponds to (9), but the
point is to reduce both the portfolio payoff values
and the corresponding risk
.
The
new distribution criterion REL is defined as the
minimum value of the sum
The solutions of (12) will give the relative
mode of payments according to their total amount.
For the case when
≤0.5, payoff i should be increased by the value .
In the opposite case, when ≥ 0.5, the payoff i must
decrease to fraction of its current value.
Problem (12) gives a set of solutions for the
disbursement reallocations, which are sensitive to the
value of the parameter . The last gives a reference
for the manager to choose this reallocation w, which
minimizes the loss or the risk from the
redistribution . The criteria (13) give unique
solutions, which minimize both components of the
goal function (12).
Ratios (12) and (13) are applied to estimate the
costs of animal husbandry from the central region of
Bulgaria.
3 Assessment of Disbursements in
Animal Husbandry
The input data are taken from the 2021 ledger of a
livestock farm. Three categories of payments are
considered that have a relatively large impact on
payments: electricity, staff wages, and fleet fuel.
Payments are recorded for each month of the year and
their values are given in Table 1.
Table 1. monthly disbursements per category
CATEGORY
[BGN}
ELECTRICITY,
D1
SALARIES,
D2
FUEL, D3
TOTAL
JAN
5940
1530
7354
14824
FEB
5290
1596
8354
15240
MARCH
6089
1558
8433
16080
APR
5549
1463
7362
14374
MAY
4858
1498
6602
12958
JUNE
5931
1447
7521
14899
JULY
8440
1395
8087
17922
AUG
7455
1153
7911
16519
SEPT
7302
1153
8905
17360
OCT
10617
1275
7688
19580
NOV
11643
1280
7049
19972
DEC
13688
1312
6996
21996
The means DE and covariance matrix of
payments have values
DE = [0.0773 0.0139 0.0769]x105
= [8.0002 -0.2420 -0.4585;
-0.2420 0.0231 -0.0136;
-0.4585 -0.0136 0.4546] x106.
The solution of the modified portfolio problem
(12) is solved for the parameter set .
These solutions are represented graphically in Figure
1, where on the horizontal axis are the risk estimates
and on the vertical axis are the payoffs
.
Fig. 1: Graphical representation of the results of the
modified portfolio problem
Applying (13) to the choice of the minimum relation
REL, the values are obtained:
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The solution says that payments D1 should
increase by 3%, D2 should decrease to 97% of their
current value, and D3 should remain at their current
value.
To verify the effectiveness of this payment
adaptation and redistribution policy, we will apply a
predictive control model approach and compare the
results of applying the modified portfolio problem
(12) and the evaluation criteria (13).
4 Implementing the Modified Portfolio
Solution in Sliding Mode
The portfolio model is applied and evaluated with a
sliding mode approach to evaluate portfolio decisions.
The reallocation of the disbursements is made with
the accounting data of previous months. Then, this
reallocation is applied for the current month, for
which the accounting data are not yet available. At the
end of the current month, a comparison is made for
the real obtained profit from the accounting records
and the protentional profit, which the applied
reallocation of disbursement would be obtained. The
comparison assesses whether the results from the
derived problem give benefits according to the real
recorded state of payments. Hence this management
policy performs adaptation and reallocation of
disbursement funds. It is an example of the usage of
problems (12) and (13) for the real management of
the disbursements. Figure 2 illustrates the
management approach.
Historical accounting data for three initial months
are used for the evaluation of the reallocation of the
disbursements. For this historical period, the
parameters of the portfolio problem (12) are
calculated for the average values of total payments
DE(t1) and the corresponding covariance matrix
. The definition and solution of the portfolio
problem and by applying the REL criteria of (13)
give the minimal solution
. Since our main interest concerns the total
payments for this period , this value is
compared to the actual total payments
TOTAL(t1) for this period. Thus, through a
comparison between the recommendations of the
problem (12) and the real one, it will identify the
benefit of the proposed management procedure.
Fig. 2: A sliding mode procedure for evaluating
modified portfolio solutions
The sliding procedure is applied from April until
the end of the year, which gives nine values for
comparison, t = t1,…,t9. The results of this
comparison are presented graphically in Figure 3.
Current levels of total payments are higher on the
schedules. The values recommended by the portfolio
problem are lower, giving advantages to the
derivative problem. The results of the portfolio
problem are relatively constant. This means that the
farm can plan approximately equal total resources for
all its payments. Current payment behavior is
incremental, requiring additional financial resources
to always be planned to cover payments.
Fig. 3: Comparison between classical and
optimization model payoff
5 Conclusions
This study derived a quantitative model for the
redistribution of payments that a business entity must
cover for its normal operations. Especially for the
production case, the level and consistency of
payments play an important role in maintaining the
liquidity of the enterprise. A possible way to support
liquidity is to synchronize income flows with
outgoing financial flows. This means appropriate
rejection of payments. Our approach was to
redistribute payouts by timing and payout category.
This is done by defining and solving a modified
portfolio problem, which meaning is a reallocation of
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
Months
History
History
. .
.
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the financial resources between the disbursements.
Such reallocation is performed without the need for
additional financial resources for covering the
disbursements. Thus, the disbursements indirectly
influence the business return in its increase. An
optimization problem was defined, in which formal
relations are based on the portfolio theory. The
problem applies new optimization criteria for the
selection of a unique solution to the portfolio
problem. It is aimed at minimizing the sum of the
average value of the total payments as their potential
risk. This risk indicates a relationship for an increase
or decrease in a mutually applied payment. Through
the defined problem, the reallocation of payments
between the different categories of payments is
evaluated numerically. The problem is defined and
solved with real data from the accounting
documentation of the animal breeding activity with
the production of milk and milk products. Empirical
comparisons between the results of the defined
problem and the real records of livestock activities
give preference to the derived model. A promising
trend and improvement of this quantitative
expression can be found by integrating both payment
tasks and livestock production. In this way, payments
can be closely linked to production volumes and
financial flows.
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The authors equally contributed to the present
research, at all stages from the formulation of the
problem to the final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The research leading to these results has received
funding from the Ministry of Education and Science
under the National Science program INTELLIGENT
ANIMAL HUSBANDRY, grant agreement N Д01-
62/18.03.2021).
Conflict of Interest
The authors have no conflicts of interest to declaree.
Creative Commons Attribution License 4.0
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Creative Commons Attribution License 4.0
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