Optimizing Cash-Flow-at-Risk in Construction Projects:
A cost reduction approach
Masoud mohammad Sharifi
Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran;
Email:[email protected]
Morteza Bagherpour
Department of Industrial and systems Engineering, University of Pretoria, Pretoria, South Africa;
Email:[email protected]
Abstract: Project managers normally are facing with difficulties behind management of project cash flow, which
requires distinguished methods and appropriate tools to overcome negative cash flows. Cash-Flow-at-Risk (CFaR)
model is an efficient approach to predict cash flow trend. In this study, all risk factors affecting project management
environment have incorporated to predict an accurate project cash flow. Then, a response surface method (RSM) is
applied to determine optimal level of risk. The results have successfully implemented through a case study to
demonstrate the applicability of the proposed method.
Keywords: Cash-Flow-at-Risk, Gaussian Mixture Model, Project Management, Response Surface Method
1.
Introduction
The Heart of a project is cash flow management especially when the project is going on to hand over or in the middle
of execution phase. Most of projects may face with negative total cash flows through project life cycle. Top level
managers, based on cash flow forecast will make managerial decisions. Therefore, inaccurate cash flow prediction
and inadequate cash management causes the financial distresses [1]. organizations with different sizes may face such
problems which requires distinguished approaches and suitable tools according to the nature and complexity of the
projects [2]. The cash flow of the project normally consists of a complete history of all payments, costs, and all
revenues. Furthermore, cash flow prediction needs to be effective and fast enough, according to the inadequate time
and costs especially at the tendering phase. Contractors scarcely prepare a detailed construction plan at this phase, and
usually wait until the contract being awarded. So a fast and effective approach for prediction of cash flow is highly
desirable [2].
On the other hand, Value at risk is a measure of losses because of “normal” market movements. VaR is proposed as a
tool to determine the total market risks of a financial institution. VaR accumulates different exposed risks into a single
number, and it is easy to communicate, calculate and interpret as well as it is a basis for decision making [3]. In a lost
case, the VaR can be interpreted in the following statement: ‘‘with α percent certainty, the loss is not more than V
dollars in the next N days’’. We can use conditional probability function (CPF) for VaR calculation[4].
Pr  P  N   VaR   F  P  VaR  
VaR
 f  P  x dx  1  
(1)

VaR is inappropriate for non-financial firms which
are not concerned with the value of stocks and
securities of assets and liabilities. Moreover, CFaR represents at-Risk framework to set cash flow' target variable.
Whereas, VaR measures the maximum total value which a firm can be expected to lose under special situations in
future, and CFaR measures the maximum shortage of cash which firm being tolerated [5]. The CFaR is used for
longer time horizon than VaR measure which is common in project environment.
2.
Literature review & research gap
Stein et al. (2001) supposed prices and firm’s cash flow are dependent from all the condition and the top-down
approach is proposed which it focuses on the total cash flows fluctuations. They proposed the comparable-based
approach to predict cash flow and their model included a statistical methodology for transforming data into peer-
benchmarked risk measurements [6]. Because of the inadequacy above methods, Andrén et al. (2005) proposed a
different method, called exposure-based CFaR. They estimated a company's cash flow fluctuation by considering
corporate macroeconomic exposure and the several factors may effect firm’s cash flow [5].
Al-Joburi et. al. (2012) investigated negative cash flow patterns and their effect on project performance. They reviewed
scheduling and financial data for 40 projects. Results of data analysis represented negative cash flow for 30 to 70% of
projects in each of the selected projects [7]. Maravas and Pantouvakis (2012) developed a cash flow calculation
methodology for projects including activities with fuzzy costs and durations. They indicated project cash flow by an
S-surface ensuing by connecting S-curves at different risk possibility levels [8]. Fink and Homberger (2013)
considered a multi-agent extension of the non-preemptive single-mode resource-constrained project scheduling
problem with discounted cash flow objectives. Also, they proposed a general decentralized negotiation approach
which uses ideas from ant colony optimization [9].
Using CFaR managers enable to estimate how much cash flow can change by the predetermined probabilities with
considering volatility in market prices (such as commodity prices, interest rate, and exchange rate). Therefore, CFaR
is applied to estimate how much expected future distribution of cash flow can change over time.
However, there are many sources of uncertainties throughout the project execution such as rework, change order, price
fluctuation, delay etc. They can significantly effect on project cash flow and ultimately makes it impossible to forecast
project cash flow. Regarding to sources of uncertainty, we propose novel CFaR in project environment. Although,
CFaR is used in manufacturing companies however it has not been applied in project management environment. Also,
there is no comprehensive research in project management area considering uncertainty sources for predicting final
project cash flow.
3.
Research methodology
Here research methodology is explained. First, prediction horizon should be determined. This step is important
because in case of using short time prediction, many parameters are deterministically known and some of them would
be un-deterministic. In the next step, Work Breakdown Structure (WBS) is developed and then cost elements related
to an individual activity is identified. Due to the effect of each risk factor on project cash flow, it is important to
identify all the influencing factors significantly. After identifying such factors, they would be divided into two
categories: deterministic and probabilistic. Deterministic items are ones which have not been associated with a major
risk and normally deterministically known. But probabilistic items are ones that they exposed a variety of risks and
they should predict through a proper approach. Following steps have developed for probabilistic analysis.
Figure1. Research methodology
Risk identification is the first stage in risk management process and it is the basis for next steps. Appropriate risk
identification ensures efficiency of risk management. If risk managers fail in identifying all possible risk factor, these
non-identified risks will become non-manageable and the organization will not overcome such issues and will not take
any corrective action.
This phase is basically includes two steps. The First step is to identify the risks and the next stage is the classification
of the identified risks. There are many forms of uncertainty which effect on cash flow. Risks inherently exist in any
project. Obviously interaction among expert judgments, experiences and individuals creativity is playing an important
role in identification of risks.
The next step thorough risk management process is risk analysis. The purpose of risk analysis is measuring the impact
of the identified risks on a project cash flow. Depends on available data, risk analysis can be carried out qualitatively
or quantitatively or semi quantitatively. Many studies have been applied risk analysis techniques to a project.
Furthermore, sophisticated simulation techniques can be adapted to multi-dimensional risk [10] .
Risk analysis involves the analysis of risk factors may normally occur in projects. There is no doubt, strong
relationship between usefulness of the results of the risk analysis and reliability of input data exist, thus appropriate
analysis of input data into risk analysis is crucial. Two types of data or information can be considered as input including
objective and subjective data, however different terminology may be utilized.
The risks such as raw material price, inflation rate and exchange are predicted by time series models and for this type
of data; variance-covariance matrix should be developed due to interaction between factors. In this study, we propose
a new approach for prediction of delay by Gaussian mixture models. More information is presented in section 3.1.
Subjective data are extracted from expert judgments or expert opinions. Perhaps the best-known and most commonly
used approach for estimating amount of risk is subjective estimation. Probability distribution can be used for this type
of data. Then, the most appropriate distribution is fitted to the identified risks. As mentioned above, a better fitted
distribution for each risk may result in accurate predictions.
Then by considering all obtained results, Cash flow is simulated for the considered periods. In this step, CFaR can be
specified in determined confidence level (usually 95% or 99%) but sometimes this CFaR is not acceptable for project
manager, in this situation they can use risk management tools and techniques to reduce risk and determine best known
condition to optimize profitability index and trend.
In this step, response surface method (RSM) is applied:
1. To establish a relationship between several explanatory variables (raw materials price and currency rates) and
response variable (cash flow) this can be used for prediction of response values.
2. To determine significance of the influencing factors on cash flow.
3. To determine the optimal settings of inputs and outputs.
By considering the effect of explanatory variables on cash flow, managers can decide which variable has un-desirable
and prevent subsequent results. They therefore may settle optimal conditions. More explanations are presented in
section 6.1.
In case of using financial derivative, it is possible some non-deterministic data turn into deterministic data till an
acceptable result is found. When acceptable condition is determined, CFaR can be then specified.
4. Risk identification:
4.1.
Delay
In general, claims are common in construction projects and may happen due to several reasons. This may lead to delay
in a project and construction disputes. Delay means the time overrun or more time than the planned completion date
[11]. Because of the importance of claims in projects and also there isn’t a applicable model for measurement of claims
impact on cost overrun we here have applied a methodology for prediction impact of claim through cost overrun as
described in the following.
The data related to occurrence claims in previous projects have collected and they represent a historical data from all
different types of projects. We need to subdivide the data into subsamples, where each subsample is composed of
projects with similar characteristics. We thus develop an idea of how to predict claim in a project. Four characteristics
have settled which these characteristics seem to be most strongly associated with patterns in predict claim.
The first one is the relevant experience of previous projects conducted by contractors. The second key characteristic
is relevant experience of previous projects being completed by the owner. The third one is similar projects from the
perspective of scope and type of projects. Eventually, last factor is the relevant to projects with the same prices. A
distribution for each mentioned factors was fitted. Due to heavy intensive data and central limit theorem, we suppose
the distribution of all four characteristic follow Gaussian distribution.
Now four Gaussian distribution is assumed and there is a need for a method to merge them altogether to achieve an
individual distribution for predicting claim through a specified project. Here, a Gaussian Mixture Model (GMM) is
used which is a weighted sum of M component Gaussian densities [12]. Here, Maximum-likelihood estimation is one
of the most widely used methods for estimation of the parameters of GMM model. As well as, ExpectationMaximization (EM) algorithm is one of popular and common algorithm among existing maximum-likelihood
estimation methods [13].
M
p  x |     wi g ( x | i , i )
(2)
i 1
x is a D-dimensional continuous-valued data vector (i.e. measurement or features), 𝑤𝑖
, 𝑖 = 1. . . M, are the mixture weights, and g ( x | i , i ) ,𝑖 = 1. . . M, are the component Gaussian densities. Every
component
density
g ( x | i , i ) 
is
1
(2 )
D /2
i
With covariance matrix i
1/2
a
Gaussian
function
as
given
using
 1

exp  ( x  i ) / i 1  x  i 
 2

equation
(3):
(3)
M
and mean vector 𝜇𝑖 , the mixture weights satisfy
 w  1 . The completed Gaussian
i 1
i
mixture model is applied mixture of weights, covariance matrices and the mean vectors from all components. These
parameters are represented by the notation as used in Fatma & Çetin 1999,
  wi , i , i  , 𝑖 = 1, … , 𝑀.
Fig.2 Mixed distribution for claims
Same weights have been considered for estimation of four characteristics in equation (3), in other word we considered
four characteristics with the same impact on project claims. It is possible for project manager to consider different
weights for claim affecting factors or Add (or subtract) other characteristics based on his/her judge. The resultant
distribution for claim factors is shown in Figure 2. This distribution is used in CFaR model to predict impact of claims
on cash flow.
4.2.
Raw material cost & exchange rate
The problem of cost overrun, especially in the construction industry is a worldwide event and its consequences are
normally a source of disagreement between client and contractor. If this disagreement is not appropriately handled,
this may lead to project failure.
In construction projects, normally a significant portion of total costs is being consumed for purchasing of required
procurements. Here, price fluctuations during project horizon and especially in large scale project can be potentially
happened. This may generate turbulence in budgetary estimate.
In this paper using time series analysis, price fluctuations incorporated in our model to study effect of raw material
fluctuations on project cash flow.
5. Time series
Time series analysis is a mathematical method using parameter estimation and curve fitting for the observed data and
forecast the future trend. Time series analysis is the low cost and an accurate method. We here applied time series
analysis for model fitting for Commodity price, exchange rate and inflation rate trend [14]. Applications of the
GARCH and ARIMA approaches are widely applied in situations where the price volatility is a main issue. These
models, particularly in financial applications, are important tools in the analysis of time series. They are especially
useful when the goal of the study is to analyze and forecast volatility. GARCH(p, q) model is proper to capture the
dynamics of a time series conditional variance.
GARCH (p, q) model is expressed as following:
t
 t 1
N (0,  t )
 t   t ut , ut
(4)
N (0,1)
p
q
t 1
j 1
(5)
 t    i t2i    j t  j
(6)
Where 𝑝 ≥ 0, 𝑞 ≥ 0, 𝜔 > 0, 𝛼𝑖 ≥ 0(𝑖 = 1, 2, … 𝑝), 𝛽𝑗 ≥ 0 (𝑗 = 1, 2, … 𝑞), 𝑝 is the order of GARCH terms 𝛿 and 𝑞 is
the order of the terms 𝜀 2 . ARIMA model is used widely in the area of non-stationary time series forecasting, which is
expressed as [14]:
∅(𝐵)(1 − 𝐵)𝑑 𝑋𝑡 = 𝜃(𝐵)𝜀𝑡
(7)
Where 𝑋𝑡 is a non-stationary time series at time t, 𝜀𝑡 is a white noise (zero mean and constant variance), 𝑑 is the order
of differencing, B is a backward shift operator defined by 𝐵𝑋𝑡 = 𝑋𝑡−1 , ∅(𝐵) = 1 − ∅1 𝐵 − ∅2 𝐵2 − ⋯ ∅𝑝 𝐵𝑝 and
𝜃(𝐵) is the moving average operator defined as: 𝜃(𝐵) = 1 − 𝜃1 𝐵 − 𝜃2 𝐵2 − ⋯ − 𝜃𝑞 𝐵𝑞 [14].
Price time series of steel, aluminum, copper and zinc and currency rates of EUR/USD, GBP/USD and inflation rate
have been collected as inputs and time series analysis is then used for prediction of price trend in future. AIC measure
is then applied to evaluate the adequacy of the model by choosing an appropriate model among possible alternatives.
6.
Variance-covariance matrix
Covariance matrix summarizes the volatilities and correlations of the returns on a set of assets, risk factors or interest
rates. It contains all the necessary information for simulation of the correlated values, for estimation of the volatility
of a portfolio for its risk factors. Covariance matrix is applied because it tries to continue the existence correlation
among existing risk factors (such as commodity prices and exchange rates) in future through simulation analysis. Risk
and asset managers therefore require covariance matrix to identify risk factors. For instance, in a risk management
system relevant to a large organization, all the main foreign exchange rates and commodity prices will encompass in
one large dimensional covariance matrix [15]. However, generating the covariance matrix depends on the number of
exchange rates and commodity prices.
cov( 1 ,  2 )
 var( 1 )

cov(  2 , 1 )
var(  2 )
variance-covariance matrix  


 cov(  k , 1 ) cov(  k ,  2 )
cov( 1 ,  k ) 

cov(  2 ,  k ) 


var(  k ) 
(8)
Here, Covariance matrix is used for raw material prices and currency rates to establish a relation among the potential
risk factors.
7.
Case study
In this section, a case study is examined. A WBS is developed for a project which all activities have been recognized.
Manpower, raw material, currency are assigned to each individual activity. We suppose different metals such as steel,
copper, zinc and aluminum are used during the construction process as well as the currencies such as EUR/USD and
GBP/USD are used to Supply equipment. In this stage, the risks effects on cost are determined. For example, price
fluctuation is an effective parameter on raw material price, exchange rate or inflation rate also can effect manpower,
or unpredicted can effect on rework, change order as well as claims. In this stage, predictable risk factors such as price
fluctuation are determined using time series analysis; we fitted the most appropriate model for each factor to predict
the future trend. Price fluctuations for twelve subsequent months related to commodity price, exchange rate have been
collected. Here, covariance matrix is a 6*6 one because in this case study 4 commodity prices (steel, copper, zinc and
aluminum) and 2 exchange rate (EUR/USD and GBP/USD) have been considered.
Figure3. Results of project cash flow simulation
The probability of reworks and change order are assigned to an individual activity. These risks are subjective data are
predicted by expert judgments. Claim is assigned to the whole of project according to what presented in section 3.1.The
simulated data is used for prediction of claim, change order & rework. Then, after running simulation study, the results
of cash flow distribution is presented in figure3.
In Figure 3, project cash flow is shown. Simulation results show cash flow range falls between 82,362 (minimum) and
116,641 (maximum) and mean is equal to 99588. The CFaR in prediction period is equal to difference between mean
and 5th percentile of cash flow, therefore CFaR is equal to 7470. This means, with 95% confidence interval Cash flow
will not fall shorter than the expected amount of 99588.
Figure4. Tornado graph for affecting cost risk factors
In Figure 4 tornado chart for the cash flow affecting factors is shown. The longer bars in tornado chart indiate the
greatest effects on project cash flow. As seen, commodities price have the greatest effect on cash flow and it can
change cash flow between 94,794 and 104,468.
7.1.
Simulation based Optimization
In this section we need a methodology assisting top level managers to evaluate the impact of different hedging
strategies on cash flow variability or to determine optimum price of parameters. In order to measure cash flow and
determine coefficients of risky factors here optimization model is developed. A lot of necessary information for
deciding the size of the hedge position is included. A Response Surface Method (RSM) is applied to determine the
best known scenario of the affecting risk factors. This method has ability to determine an approximate function with
a smaller number of runs. The RSM is a set of statistical and mathematical methods, which enables researchers to
determine optimum condition of a risk. A Central Composite Design (CCD) here selected in order to study the
effective factors. Six major factors are determined to study by the RSM-CCD and the values are studied in five levels
(-α, -1, 0, +1, + α) through 96 experiments. Some risk sources in project environment is introduced in section 3 which
may have a significant impact on cash flow however only raw material prices and currency rates can be controlled
and other risk sources are normally uncontrollable. Therefore, these factors include raw material prices (copper, steel,
zinc and aluminum) and currency rates (EUR/USD and GBP/USD). Response function is chosen in order to increase
cash flow [16]. Model fitting is executed by RSM and it proposed which a linear model suited the best fit.
Table 2. Result of RSM method
Source
Model
Sum of
Squares
1.43E9
df
6
Mean
Square
2.39E8
Residual
8.44E8
81
1.04E7
Lack of fit
2.98E8
70
4.26E6
Pure error
5.46E8
11
4.26E6
Cor total
2.35E9
95
Std. Dev.
3229.02
R-Squared
0.6296
Mean
99577.91
adjusted R-Squared
0.6022
C.V%
3.24
predicted R-Squared
0.5367
PRESS
1.05E9
Adequate precision
72.15.602
F
Value
22.95
p-value
Prob > F
< 0.0001
The results of ANOVA are given in Table 2. The adequacy (statistical significance) of a linear model is significantly
tested through F- and p-values statistics. large F-value statistic indicates a regression equation can explained most of
the variation whereas low p-value (<0.05) indicates the model is considered to be statistically significant [16]. Here
F-value is 22.95 and p <0.0001 indicates linear model is enough significant. The goodness of the model can be
confirmed by the coefficient of determination R2 (0.6296) and the adjusted𝑅2 . Both values are not somehow closed
to 1, which are fairly high and indicate correlation between the observed and the predicted values. According to
evaluation of ANOVA results, the statistical significance of linear model for the response is confirmed and the model
can be used for more analysis in order to discover the effect of variables (risk factors) on project cash flow. The
obtained equation is shown below:
𝐶𝑎𝑠ℎ 𝐹𝑙𝑜𝑤 = 3.93𝐸5 − 63417.3 ∗ 𝑌(𝐸𝑈𝑅 ) − 57207.5 ∗ 𝑌(𝐺𝐵𝑃 ) − 9.36 ∗ 𝑋𝐶𝑜𝑝𝑝𝑒𝑟 − 10.11 ∗ 𝑋𝐴𝑙𝑢𝑚𝑖𝑛𝑖𝑢𝑚
𝑈𝑆𝐷
−9.7 ∗ 𝑋𝑍𝑖𝑛𝑐 − 36.1 ∗ 𝑋𝑆𝑡𝑒𝑒𝑙
Where 𝑋 is price per ton and 𝑌 is exchange rate.
𝑈𝑆𝐷
(9)
Figure 5. 3D response surface curve. the optimum value of cash flow is obtainable when the parameters be equal to
(EUR/USD=1.25, GBP/USD=1.51, COPPER=7553, zinc=1925, aluminum=1940, steel=239.7).
The equation (5) is an important part for project managers. Using this equation, managers can assess the impact of
price fluctuations on cash flow. Also it leads to a better hedging decision. The 3D response surface curve is plotted
to observe the interaction of factors and the optimum settings of each factor required for optimum cash flow
configuration. The 3D surface graph for the response is shown in figure 6.
Figure 6. Cash flow simulation after optimization
After determining optimum situation conducted for all parameters, cash flow is optimally simulated to determine cash
flow distribution under the suggested condition. The Result is shown in Figure 6. As shown in Figure 6 cash flow is
between 93354 and 122879 and the mean is 109603. In Figure 6, CFaR has improved and reached to 6617, this means
CFaR is improved 11.4%. In other word, this means the exposed risk is decrease. Moreover, the cash flow mean is
improved %11.
8.
Conclusion remark and further recommendations
In this paper, the obtained results successfully confirmed cash flow at risk model is a powerful approach to the project
environment and this model can be applied for prediction of project cash flow as well as in cost estimation of projects
considering all of the affecting risk factors. Additionally, in this paper, we proposed a new method for cost estimation
to predict project cash flows.
Here a simulation based optimization method to predict cash flow is applied and the results determined distribution
of cash flow through project life cycle. After specifying cash flow distribution, calculation of CFaR in accordance
with level of confidence is therefore possible. Then, in order to improve the obtained results and making managerial
decisions about financing strategies, response surface method is applied and the optimal level of risk factors has been
determined. The results showed using RSM, CFaR is improved 11.4%. Also the mean is improved %10. The approach
therefore hopefully reduces cost overruns.
Further research can be conducted on applying multi- objective simulation based optimization for project cash flow
management.
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