In this paper, we have presented a corporate valuation model. The model combine several valuation methods in order to get more accurate results. To determine the corporate asset value we have used the Gordon-like two-stage asset valuation model based on the calculation of the free cash flow to the firm. We have used the free cash flow to the firm to determine the corporate market value, which was calculated with use of the Black-Scholes option pricing model in frame of the two-dimensional Monte Carlo simulation method. The combined model and the use of the two-dimensional simulation model provides a better opportunity for the corporate value estimation.corporate valuation, free cash flow to the firm, real options, Black-Scholes option pricing model, two-dimensional Monte Carlo simulation, R statistics

Fenyves Veronika

Tarnóczi Tibor

Tóth Réka

English

Research Papers in Economics

  788
CORPORATE VALUATION USING TWO-DIMENSIONAL MONTE CARLO 
SIMULATION 
Tarnóczi Tibor 
University of Debrecen, Faculty of Applied Economics and Rural Development 
Institute of Accounting and Finance, Department of Finance 
Fenyves Veronika 
University of Debrecen, Faculty of Applied Economics and Rural Development 
Institute of Accounting and Finance, Department of Controlling 
Tóth Réka 
University of Debrecen, Faculty of Applied Economics and Rural Development 
Institute of Accounting and Finance, Department of Accounting 
 
In  this  paper,  we  have  presented  a  corporate  valuation  model.  The  model  combine  several 
valuation methods in order to get more accurate results. To determine the corporate asset value 
we have used the Gordon-like two-stage asset valuation model based on the calculation of the 
free cash flow to the firm. We have used the free cash flow to the firm to determine the corporate 
market value, which was calculated with use of the Black-Scholes option pricing model in frame 
of the two-dimensional Monte Carlo simulation method. The combined model and the use of the 
two-dimensional  simulation  model  provides  a  better  opportunity  for  the  corporate  value 
estimation. 
 
Keywords corporate valuation, free cash flow to the firm, real options, Black-Scholes option 
pricing model, two-dimensional Monte Carlo simulation, R statistics 
JEL code: G32, C15 
 
1. Corporate valuation 
There  is  an  ever-increasing  need  for  financial  valuation  services  pertaining  to  nonpublic 
companies and subsidiaries, divisions, or segments of public companies. Nowadays, the main 
purpose of the companies to increase their corporate value. In order to measure the value creation 
of a company, namely the increase in value compared to previous one, it is necessary that we can 
properly estimate the company's current value. The new reality presents a challenge to business 
managers: the need to manage value and to focus as never before on the value their corporate-
level strategies are creating. Copeland-Koller-Murrin (2000) write in their book: „We believe that 
clear  thinking  about  valuation  and  skill  in  using  valuation  to  guide  business  decisions  are 
prerequisites for success in today's competitive environment. … Underlying it is our basic belief 
that managers who focus on building shareholder value will create healthier companies than 
those  who  do  not.  We  also  think  that  healthier  companies  will,  in  turn,  lead  to  stronger 
economies, higher living standards, and more career and business opportunities for individuals.” 
The valuation plays a key role in many areas of finance - in corporate finance, mergers and 
acquisitions and portfolio management. In general, there are three approaches to valuation. The 
first, discounted cashflow valuation, relates the value of an asset to the present value of expected 
future cashflows on that asset. The second, relative valuation, estimates the value of an asset by 
looking at the pricing of 'comparable' assets relative to a common variable such as earnings, 
cashflows, book value or sales. The third, contingent claim valuation, uses option pricing models 
to measure the value of assets that share option characteristics. (Damodaran, 2002) 
To build our company valuation model we have used a kind of the combination of the methods 
above-described  by  Damodaran.  The  combined  methods  were  designed  to  obtaining  more 
accurate results and to taking into account the forecast uncertainty during the calculations. Still 
combining of the methods are needing, because the company valuation is a complicated process.   789 
As business structures become more complicated, business valuations become more complicated. 
As  transparency  in  corporate  financial reporting  becomes  more  desirable and less  attainable, 
business valuations become more complicated. And, as securities market cycles become more 
exaggerated  and  less  predictable,  business  valuations  become  more  complicated.  (Reilly-
Schweihs, 2004)￿
The corporate valuation is also important to measure the performance of the management. Value 
creation  is  the  ultimate  measure  of  performance  for  a  management  team.  (Copeland-Koller-
Murrin, 2000) 
 
2. The corporate valuation model 
There are only three approaches to value any asset, business or business interest: 
1.  the income approach; 
2.  the market approach; 
3.  the asset approach. 
There are no other approaches to value. However, there are numerous methods within each one 
of the approaches that the analyst may consider in performing a valuation. All three approaches 
should be considered in each valuation. However, it is not common to use all three approaches in 
each valuation. (Hitchner, 2003) 
To determine the market value of the company we have developed a combined model whose 
main stages are presented in the Figure 1. On behalf of the more efficient estimation we had 
combined more known methods and the corporate value as the final result was calculated using 
the so-called two dimensional Monte Carlo methods with utilization of intermediate results of the 
specific  methods.  The  raw  data  necessary  to  the  modeling  was  collected  from  the  financial 
statements some of the companies listed on the Budapest Stock Exchange. We chose companies 
belonging to the stock exchange with the intention to compare the calculated results with the 
stock market capitalization of the firms. 
The value determination was carried out for five companies. However, due to size limitations of 
the article the details of the calculation are only presented for one company (Danubius). 
 
2.1. Calculation of free cash flow to the firm 
To preparation of forecasting which is necessary to determine the asset value of a company we 
have used the last five years data (2004-2008) of the firms downloaded from the Budapest Stock 
Exchange web-page. The calculations have been performed in Microsoft Excel 2007 and the R 
statistical  system.  R  statistical  system  is  an  integrated  suit  for  software  facilities  for  data 
manipulation, calculation and graphical display. R is an environment within which many classical 
and  modern  statistical  techniques  have  been  implemented.  (Venables-Smith,  2009)  The 
commands of R can be executed from the Microsoft Excel, and we have used this possibility. 
During the forecasting we have calculated those values which were necessary to determine the 
free cash flow to the firm (FCFF). We used the company's growth ratio to make the forecast of 
the data. The forecast was based on the company growth rate
528 and marginal taxation rate
529. To 
generate  the  latter rates  we  have  used the normal  distribution  with  parameters  based  on the 
averages and standard deviations of the previous years’ data. (Table 1) 
The forecast results are in the Table 1. The FCFF values were calculated using the after-tax 
operating profit, as follows 
FCFF = After-tax operating profit + Depreciation - Investments 
- Changing in non-cash working capital                (1) 
                                                       
528 mean=9.14, sd=2.23 
529 mean=22.55, sd=6.93   790
In  case  of  Formula  1,  the  cash  flow  does  not  include  the  benefits  arising  from  the  interest 
payments, since the tax-saving effects of the credits has been taken into consideration in case of 
the determination of the cost of capital. 
The corporate asset value was calculated by using of the Gordon-like two-stage valuation model, 
in consideration of the following relations: 
n
n
n
1 t
t
t
A r) (1
P
r) (1
FCFF
V
+
+
+
= ￿
=
          (2) 
g) (r
g) (1 * FCFF
P
n
n −
+
=
            (3)
 
 
Figure 1: The process and the raw data of the corporate valuation 
 
 
The growth rate (g) used to the determination of the terminal value (Pn) was calculated as 
an average of the forecasted growth rates (the growth rate from 6th year - Table 1). 
(Formula 3) The weighted average cost of capital (r – required rate of return – Table 1) 
was calculated as an average of the yearly weighted average costs of capitals. (Formula 
2, 3) 
 
2.2. Real options in corporate valuation 
The  calculation  is  essentially  based  on  the  assumption  that  the  market  capitalization  of  a 
company determines the market value of a joint stock company. Two essential factors play an 
important role in general to elaborate the market price: the actual performance of the company 
and the investor expectations. The riskiness of a company reflects in the volatility of the stock 
prices. 
The new economy provides a challenge for the corporate decision-maker. Corporate valuation 
may no longer depends on traditional fundamentals but rather on future expectations. Investment 
strategies  with  high  risks  and  uncertainty  or  irreversible  corporate  decisions  coupled  with 
managerial flexibility provide the best candidates for real options. Real options is a systematic 
approach and integrated solution using financial theory, economic analysis, management science, 
decision sciences, statistics and econometric modeling in applying options theory in valuing real   791 
physical assets, as opposed to financial assets, in a dynamic and uncertain business environment. 
(Mun, 2002) 
To determine the corporate market value we have used the Black-Scholes options pricing model. 
The value of a real options depends on five basic variables: 
1.  the value of underlying risky asset (S) – corporate asset value; 
  2.  the exercise price (K) – face value of the outstanding loans; 
  3.  the time to expiration of the option (t) – weighted average maturity of loans; 
  4.  the standard deviation of the value of the undelying risky asset (￿) – variance of the 
firm’s share; 
  5.  the risk-free rate of interest over the life of the option(r) - treasury bond yield rate 
adjusted to the option lifetime. (Copeland-Antikarov, 2003) 
 
Table 1: The corporate asset value calculation based on forecasted data 
     million HUF 
 
 
The weighted average maturity of loans was estimated with the using of the notes of the accounts 
of corporate annual reports. The variance of the firm’s shares was calculated with the using the 
closing price of the company's shares what was downloaded from the web-page of the Budapest 
Stock Exchange. (Table 2) 
To  calculate the corporate  value  we  have  used the following  formulas  of the  Black-Scholes 
model: 
value of the call option = S * N(d1) - K * e
-rt * N(d2)      (4) 
t
t r
K
S
d
*
*
2
ln
2
1
σ
σ
￿ ￿
￿
￿
￿ ￿
￿
￿
+ + ￿
￿
￿
￿
￿
￿
=
   
t d d * 1 2 σ − =     (5) 
 
2.3. Two dimensional Monte Carlo simulation 
A  two-dimensional  (or  second-order)  Monte  Carlo  simulation  is  useful  to  estimate  the 
“uncertainty” in the risk estimates stemming from parameter uncertainty. A two-dimensional 
Monte-Carlo  simulation  is  a  Monte-Carlo  simulation  where  the  distributions  reflecting 
”variability”  and  the  distributions  representing  ”uncertainty”  are  sampled  separately  in  the 
simulation, so that ”variability” and ”uncertainty” in the output may be estimated separately. 
1 2 3 4 5
Growth rate 10,57% 8,49% 9,44% 8,22% 7,42%
Assets 91 363 99 124 108 482 117 398 126 111
Sales 49 417 53 615 58 677 63 499 68 212
Operating profit 2 324 2 521 2 759 2 986 3 208
Marginal taxation rate 32,35% 18,82% 29,02% 11,66% 28,10%
After-tax operating profit 1 572 2 047 1 958 2 638 2 306
Invested assets 83 767 90 882 99 462 107 636 115 626
Depreciation 4 937 5 357 5 863 6 344 6 815
Investments 4 685 5 083 5 563 6 020 6 467
Changing in non-cash working capital 274 297 325 351 378
Free Cash Flow to the Firm 1 551 2 023 1 933 2 610 2 277
Terminal price 38 625
Present value of FCFF 1 340 1 512 1 248 1 457 1 098
Corporate asset value 45 279
growth rate from the 6. year  9,26%
required rate of return 15,70%
Forecasted years
Titles of data  792
The developers of the two dimensional simulation model write in their modell application guide 
(Pouillot, R. et al., 2010) the followings: „a QRA (Quantitative Risk Assessment) should reflect 
the „variability” in the risk and calculate the“uncertainty”associated with the risk estimate. 
The“variability”represents temporal, geographical and/or individual heterogeneity of the risk 
for  a  given population.  The  “uncertainty”is  understood  as  stemming  from  a  lack  of  perfect 
knowledge about the QRA model structure and associated parameters.” On basis of Cullen and 
Frey book (1999, p.3.), what can be regarded as the theoratical background of two dimensional 
simulation, the previous definition is supplemented by the followings: „In general, variability 
can not be reduced by additional study and measurement. Random and systematic measurement 
errors as well as reliance on models and surrogate indicators, all sources of uncertainty. … In 
confronting variability and uncertainty a decision-maker stands to better understand the degree 
of variance int he full distribution of exposure or risk, the impact of various assumptions, data 
gaps, and model choice on decision making, and the most fruitful avenues for further study.” 
The input parameters of the simulation model are presented in Table 2. 
 
Table 2: The input value of the simulation model 
 
 
To estimate the stochastic variables marked U (uncertainty) and V (variability) letters we have 
used the triangular distribution with the parameter values in Table 2. (Kotz-van Dorp, 2004) 
 
3. Results of simulation 
To execute the simulation model we have used the ’mc2d’
530 package of R statistical system. 
The simulation statistical results are presented in Table 3. The statistical indicators of Table 3 
show that the value of the coefficient of variation is very low, and the interquartile range
531 is 
relatively small, which show that the estimated values are located in a relatively narrow interval 
around the mean and median around. All these can be easily seen in the boxplot diagram. 
To determine a confidence interval for the mean and median the program was used the one 
sample t-test and the Wilcoxon signed rank test with continuity correction. The calculation results 
are shown the Table 4. We can see that the intervals have a very small scale. 
To measure how the value of an option will change when one of the input variables changes 
while the others remain the same, we have used the „Greeks” (delta=0.7021, gamma=6.86*10
-7). 
 
Table 3: The statistical results of the simulation 
                                                       
530 Monte-Carlo à Deux Dimensions 
531 interquartile range = 3rd quartile – 1st quartile 
Corporate asset value 45 279 million forint
Average maturity of loans 5,54 év
Variance of the corporate shares U
                         minimum 8,85%
                         mean 9,78%
                         maximum 10,70%
Outstanding loans V
                         minimum 21 310 million forint
                         mean 23 966 million forint
                         maximum 27 511 million forint
Five-year treasury bond yield U
                         minimum 4,50%
                         mean 6,00%
                         maximum 8,50%  793 
 
 
Table 4: Confidence interval estimation for mean and median 
 
Figure 3: The diagram results of simulation
532 
   
 
4. Conclusions 
The performed calculations show that the constructed model can be applied in determining the 
market value of companies. The stock market capitalization of the company's shares was HUF 
29,330 at the end of 2008, which is 2.49% less than the estimated average value. The model 
slightly over estimates the current market value. 
The sensitivity tests carried out have confirmed the applicability of real options. 
Probably we get better results of the corporate value if we would have a more accurate internal 
company information. We have obtained similar values in case of the other analyzed companies 
too. In each case, the estimate is well approximated in the current market value. 
We believe that the presented model is useful to determine the value of the companies whose 
shares are traded on the stock exchange. 
 
Bibliography 
1. Copeland, T.-Antikarov, V. (2003): Real Options: A Practioner’s Guide. TEXERE-Thomson 
Co., New York 
2. Copeland, T.-Koller, T.-Murrin, J. (2000): Valuation. Measuring and Managing the Value of 
Companies, [McKinsey & Company, Inc.] John Wiley & Sons, Inc. 
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book for Dealing Variability and Uncertainty in Models and Inputs, Plenum Press, New York 
4. Damodaran, A. (2002): Investment Valuation. Tools and Techniques for Determeining the 
Value of Any Assets, John Wiley&Sons Inc., New York 
                                                       
532 The result variable contains 1 million piece of the estimated value. 
Statistical indicators Minimum
1st 
quartile
Median Mean
3rd 
quartile
Maximum
Standard 
deviation
Coeff. of 
variation
Inter-
quartile 
range
Kurtosis Skewness
Value of the company 27 290 30 060 30 720 30 680 31 350 33 510 939 3,06% 1 290 -0,2228 -0,1972
lower bound upper bound
mean 30 683 30 687 30 685
median 30 700 30 704 30 702
confidence interval (95% ) Statistical 
indicators
estimated 
value  794
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Inc., Hoboken 
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CORPORATE VALUATION USING TWO-DIMENSIONAL MONTE CARLO SIMULATION

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CORPORATE VALUATION USING TWO-DIMENSIONAL MONTE CARLO SIMULATION

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