Abstract: Financial models implemented in corporate finance mostly rely on the discounted cash flow valuation approach. The basic concepts behind various forms of a discounted cash flow valuation are the present value of an asset and the net present value of an asset. Both concepts use a discount rate as a required rate of return for investors or broadly investors and lenders together depending on a particular analytical situation and the type of free cash flow stream. The Monte Carlo simulation is the ultimate solution for considering nearly all possible scenarios in presumably any discounted cash flow valuation. The Author of this paper argues that a discount rate expresses an investor’s cur-rent requirement and should be respectively perceived as a parameter only. The consequences of qualifying a required rate of return (a discount rate) as a risk factor in a dis-counted cash flow valuation are depicted in the paper using a free cash flow financial model of an asset being a hypothetical publicly traded enterprise. The case study is a discounted cash flow valuation using the Monte Carlo simulation for risk analysis. The various sets of assumptions are considered to explain the consequences of qualifying a required rate of return in a discounted cash flow model as a risk factor. As has been in-dicated in the paper, the discount rate as an additional risk factor with an attributed probability distribution increases the volatility of a risk variable. The distribution of a risk variable becomes more flattened then. In previous studies, some Authors indicated that a discount rate could be considered a risk factor in the Monte Carlo simulation (Krysiak 2000, Damodaran 2018).

<a href="https://dx.doi.org/10.15611/fins.2023.2.01">DOI: 10.15611/fins.2023.2.01</a>
<p>JEL Classification: G32</p>
<p>Keywords: corporate finance, valuation, DCF, riskanalysis, Monte Carlo simulation</p>
<h2>1. Introduction </h2>
<p>Financial models implemented in corporate finance mostly rely on the
discounted cash flow valuation approach. The basic concepts behind
various forms of a discounted cash flow valuation are the present value
of an asset and the net present value of an asset. The author used the
present value concept to estimate the value of an asset (e.g. in an
enterprise valuation, a bond or a credit valuation, a derivative
valuation, etc.) whereas we specifically focus on a net present value to
assess the profitability of an asset (e.g. in an investment project
profitability appraisal). Both concepts use a discount rate as a
required rate of return for investors or broadly investors and lenders
together depending on a particular analytical situation and the type of
free cash flow stream.</p>
<p>The traditional approach to discounted cash flow valuation results in
a single scenario built on the analyst’s specific assumptions. However,
a volatile economic environment suggests intuitively an infinite number
of scenarios that might happen in the real world. The ultimate solution
for considering nearly all possible scenarios in presumably any
discounted cash flow valuation is the Monte Carlo simulation (Brealey et
al., 2014; Golden and Golden, 1987; Hertz, 1964; Kroese et al., 2014).
In terms of risk analysis, risk factors are those financial model inputs
that are subject to the economic environment and its volatility. The
others are only parameters. Nonetheless, there is an infinite number of
risk factor scenarios, while there is no more than one scenario of an
input being a parameter.</p>
<p>Some studies indicated that a discount rate could be considered a
risk factor (Damodaran, 2018; Krysiak, 2000), whereas this author argues
that a discount rate expresses an investor’s current requirement and
should be respectively perceived as a parameter. The paper shows the
consequences of taking up an assumption in the Monte Carlo simulation
which would qualify a discount rate as a risk factor.</p>
<h2>2. Theoretical Background </h2>
<p>Although the difference between present value and net present value
is technically very simple, it is conceptually truly significant. The
present value concept is based on a cash flow stream that excludes a
possible payment or a series of possible payments for an asset.
Estimating the present value of an asset gives an analyst a possible
value of an asset – a hypothetical payment. If an investor paid the
present value for an asset and received the exact estimated cash flow
stream in the considered time horizon, they would certainly achieve
their required rate of return being here the internal rate of return of
the transaction. One can conclude that the present value is a value of
an asset that resembles both its ability to generate a certain cash flow
stream and the investor’s profitability requirement. Net present value
additionally considers a payment or a series of payments for an asset
(e.g. a market price of a stock or a bond, a capital expenditure or
expenditures related to an investment project, etc.) and depicts the
profitability of a transaction. The basic rule here is widely known and
accepted – if a payment for an asset was lower (higher) than its present
value, the net present value would be positive (negative) and the
investor would realize an internal rate of return higher (lower) than
their required rate of return. The concepts of present value, net
present value, and internal rate of return are quite dated (e.g.
Fischer, 1930; Gordon, 1955; Lorie and Savage, 1955; Williams, 1938) but
they remain truly relevant for modern finance. The present value concept
has successfully been the core of the value-based management idea so far
(Rappaport, 1999; Rappaport, 2006), whereas the net present value
concept is deservedly classified as one of the most important concepts
in modern finance (Brealey et al., 2014).</p>
<p>An analyst who decides to perform a discounted cash flow valuation
must estimate 1) a cash flow stream and 2) a required rate of return.
The question is whether cash flow stream components and a required rate
of return are all risk factors. Beginning with the definition of a risk
factor, a risk factor is technically a financial model input variable
that may vary from its forecast value due to the economic environment
and its volatility. Assuming that a required rate of return was a risk
factor would mean that an investor would be uncertain about the level of
the satisfying required rate of return. An investor is actually always
uncertain about the final payment or the series of payments – not
uncertain about their own required rate of return at the moment of
valuation. Therefore, a discounted cash flow valuation is performed to
find the present value which could assure the investor’s required rate
of return; thus the investor may decide whether the offered payment is
acceptable.</p>
<p>In terms of own equity, one thinks about shareholders’ required rates
of return. These rates were successfully estimated using the dividend
model (Gordon and Shapiro, 1956) or the CAPM (Lintner, 1965; Mossin,
1966; Sharpe, 1964; Treynor 1962) for assets being stocks of publicly
traded enterprises so far. For non-publicly traded companies, usually
the CAPM with Hamada’s correction was involved (Damodaran, 2012; Hamada,
1972). The dividend model derives a required rate of return as the
internal rate of return of a cash flow stream including the current
price of a stock and an infinite series of dividends. Through the CAPM a
required rate of return is the sum of a risk-free rate and the
difference between a market risk rate and the risk-free rate adjusted by
systemic risk. Hamada’s equation changes the required rate of return to
reflect the financial leverage of a particular enterprise. A required
rate of return derived this way is mostly affected by the historical
period considered or the interval chosen. Therefore, it may differ
depending on subjective assumptions of a particular financial analyst
and may be even, easily manipulated – but it still helps to find a
reasonable and widely accepted level of a required rate of return in a
particular analytical situation. The fact that a required rate of return
derived through the mentioned or similar models may differ due to the
period and/or the interval of the historical data taken into account,
does not mean that a required rate of return of an investor is somehow
volatile at the moment of valuation. It is just an estimated investor’s
requirement, broadly – a minimum value of a required rate of return they
could have accepted.</p>
<p>A cash flow stream is simultaneously affected by numerous risk
factors and parameters. Therefore, as there is no single scenario of a
cash flow stream that may happen in the future – there is logically an
infinite number of possibilities due to infinite combinations of
possible values of risk factors – due to infinite future states, the
economic environment may fall into. The solution for considering nearly
all possible scenarios of cash flows in a discounted cash flow valuation
is the Monte Carlo simulation (Hertz, 1964; Kroese et al., 2014). In
fact, this simulation is not quite new and was not even designed
specifically for finance (Ulam et al., 1947) but is very relevant and
useful. The scenarios of cash flows are the direct source of the present
value and net present value scenarios respectively. As a consequence,
the outcome of the Monte Carlo simulation is then the probability
distribution of possible present values or net present values (or the
other cash flow-based profitability measures).</p>
<p>The Monte Carlo simulation results in a probability distribution of a
risk variable. Typically, risk factors are financial or non-financial
figures like unit prices, unit costs, exchange rates, interest rates,
fixed costs, unit demands, etc. Whenever the Monte Carlo simulation is
involved, every input variable being a risk factor must be depicted with
a probability distribution chosen objectively, quasi-objectively, or
subjectively. Due to interdependencies between them, a correlation
matrix must be specified too (Kaczmarzyk, 2016; compare Hull, 2018;
Vose, 2008). The result of the Monte Carlo simulation is the probability
distribution of a risk variable which reflects simultaneous,
interdependent, and non-linear changes in risk factors. The concept of
qualifying the discount rate as an additional risk factor (Damodaran,
2018; Krysiak, 2000,) results in attributing an additional probability
distribution. This could be a uniform probability distribution (e.g.
Damodaran, 2018, p. 35) or a triangle probability distribution (e.g.
Krysiak, 2000, p. 68). The variability of a discount rate will affect
the volatility of a risk variable and blur the influence of the other
risk factors. The aim of the paper was to show how a required rate of
return qualified as a risk factor affects the result of a DCF
analysis.</p>
<h2>3. Methodology</h2>
<p>The consequences of qualifying a required rate of return (a discount
rate) as a risk factor in a discounted cash flow valuation are shown in
the paper using a free-cash-flow-to-firm financial model of an asset
being a hypothetical publicly traded enterprise. The model estimates the
intrinsic value per share and is based directly on the present value
concept (Figure 1).</p>
<p>
<img src="/articles/2023/Kaczmarzyk_sklad/media/image1.png" />
</p>
<p><strong>Fig. 1.</strong> The financial model
Source: own elaboration.</p>
<p>The model starts with the revenue forecast (Row 4) with a certain
level of revenue growth rate (Row 3). The operational costs less
depreciation (Row 5), fixed assets (Row 10), and net working capital
(Row 13) are derived from the respective ratios indicating their
relation to the revenues (Rows 5, 7, and 12). Then, the cash flow stream
(Row 23) is calculated. The required rate of return in the model is the
weighted average cost of capital (Row 28). The cash flow stream is
affected by two undisputable risk factors: the revenue growth rate and
the ratio of operational costs less depreciation as well as by the
questionable risk factor – the required rate of return. The revenue
growth rate and the ratio of operational costs less depreciation are
assumed to be the same in every single period of the detailed
projection. The model output is the intrinsic value of a single
stock.</p>
<p><strong>Table 1.</strong> Sets of assumptions</p>
<table class="table table-bordered">
<colgroup>
<col></col>
<col></col>
<col></col>
<col></col>
</colgroup>
<thead>
<tr><th>Set 1</th>
<th>Fixed required rate</th>
<th>Fixed revenue growth rate</th>
<th>Fixed operational cost ratio</th>
</tr>
</thead>
<tr><td>Set 2</td>
<td>Uniform required rate</td>
<td>Fixed revenue growth rate</td>
<td>Fixed operational cost ratio</td>
</tr>
<tr><td>Set 3</td>
<td>Triangular required rate</td>
<td>Fixed revenue growth rate</td>
<td>Fixed operational cost ratio</td>
</tr>
<tr><td>Set 4</td>
<td>Normal required rate</td>
<td>Fixed revenue growth rate</td>
<td>Fixed operational cost ratio</td>
</tr>
<tr><td>Set 5</td>
<td>Fixed required rate</td>
<td>Uniform revenue growth rate</td>
<td>Fixed operational cost ratio</td>
</tr>
<tr><td>Set 6</td>
<td>Fixed required rate</td>
<td>Triangular revenue growth rate</td>
<td>Fixed operational cost ratio</td>
</tr>
<tr><td>Set 7</td>
<td>Fixed required rate</td>
<td>Normal revenue growth rate</td>
<td>Fixed operational cost ratio</td>
</tr>
<tr><td>Set 8</td>
<td>Fixed required rate</td>
<td>Fixed revenue growth rate</td>
<td>Uniform operational cost ratio</td>
</tr>
<tr><td>Set 9</td>
<td>Fixed required rate</td>
<td>Fixed revenue growth rate</td>
<td>Triangular operational cost ratio</td>
</tr>
<tr><td>Set 10</td>
<td>Fixed required rate</td>
<td>Fixed revenue growth rate</td>
<td>Normal operational cost ratio</td>
</tr>
<tr><td>Set 11</td>
<td>Fixed required rate</td>
<td>Normal revenue growth rate</td>
<td>Normal operational cost ratio</td>
</tr>
<tr><td>Set 12</td>
<td>Uniform required rate</td>
<td>Normal revenue growth rate</td>
<td>Normal operational cost ratio</td>
</tr>
<tr><td>Set 13</td>
<td>Fixed required rate (lower)</td>
<td>Normal revenue growth rate</td>
<td>Normal operational cost ratio</td>
</tr>
<tr><td>Set 14</td>
<td>Fixed required rate (higher)</td>
<td>Normal revenue growth rate</td>
<td>Normal operational cost ratio</td>
</tr>
</table>
<p>Source: own elaboration.</p>
<p>The case study is a discounted cash flow valuation using the Monte
Carlo simulation for risk analysis. The various sets of assumptions
(Table 1) are considered to explain the consequences of qualifying a
required rate of return in a discounted cash flow model as a risk
factor. The model was developed using Microsoft Excel 365 and the
simulations performed using Palisade @RISK 8.2.1.</p>
<h2>4. Case study</h2>
<p>The first set of assumptions (Figure 2) qualifies only the required
rate of return as a risk factor to depict how a fixed required rate of
return may affect the valuation. Obviously, the fixed required rate of
return level results in the fixed intrinsic value level (Set 1). The
higher (lower) the required rate of return level, the lower (higher) the
intrinsic value of an enterprise would occur.</p>
<table class="table table-bordered">
<colgroup>
<col></col>
<col></col>
<col></col>
</colgroup>
<thead>
<tr><th>Set 1</th>
<th><img src="/articles/2023/Kaczmarzyk_sklad/media/image2.png" /></th>
<th><img src="/articles/2023/Kaczmarzyk_sklad/media/image3.png" /></th>
</tr>
</thead>
</table>
<p><strong>Fig. 2.</strong> Fixed required rate of return, fixed revenue
growth rate, and fixed operational cost ratio</p>
<p>Source: own elaboration.</p>
<p>If the required rate of return of the investor was assumed to be a
risk factor, it would reflect a very hypothetical situation in which the
investor would be uncertain about an adequate required rate of return
level (Figure 3). The investor could assume then, e.g. that their
required rate of return belonged to a certain range but every level
within this range had the same probability. To reflect such a
phenomenon, a uniform probability for the required rate of return should
have been chosen (Set 2). The probability distribution of the intrinsic
value precisely reflects the way how a certain level of the required
rate of return affects the intrinsic value. Namely, the required rate of
return affects the intrinsic value in a nonlinear way. For the same
relative increase or decrease of the initial required rate of return
level there occurs an incommensurate change in the intrinsic value.
Moreover, the probability of the intrinsic value shows precisely that
the lowest intrinsic value has the highest probability of occurrence,
whereas the highest intrinsic value has the lowest probability. The
phenomenon of the nonlinear type of interdependency between the required
rate of the return level and the respective intrinsic value would also
be clearly visible if the investor applied another type of theoretical
probability distribution to reflect their beliefs. For example, they
could use a simple triangular probability distribution to reflect their
belief that an adequate required rate of return belonged to a certain
range in which extreme values had the lowest probability and there was a
certain level of the required rate of return between them with the
highest chance of occurrence from the investor’s point of view (Set
3).</p>
<table class="table table-bordered">
<colgroup>
<col></col>
<col></col>
<col></col>
<col></col>
</colgroup>
<thead>
<tr><th>Set 2</th>
<th><img src="/articles/2023/Kaczmarzyk_sklad/media/image4.png" /></th>
<th><img src="/articles/2023/Kaczmarzyk_sklad/media/image5.png" /></th>
</tr>
</thead>
<tr><td>Set 3</td>
<td><img src="/articles/2023/Kaczmarzyk_sklad/media/image6.png" /></td>
<td><img src="/articles/2023/Kaczmarzyk_sklad/media/image7.png" /></td>
</tr>
<tr><td>Set 4</td>
<td><img src="/articles/2023/Kaczmarzyk_sklad/media/image8.png" /></td>
<td><img src="/articles/2023/Kaczmarzyk_sklad/media/image9.png" /></td>
</tr>
</table>
<p><strong>Fig. 3.</strong> Variable required rate of return, fixed
revenue growth rate, and fixed operational cost ratio</p>
<p>Source: own elaboration.</p>
<p>The way in which a required rate of return affects an intrinsic value
would be a little bit less visible but still relevant if a normal
probability distribution was chosen (Set 4) to reflect investor’s
requirements. It should be emphasised that a symmetrical required rate
of return probability distribution generally results in an asymmetrical
probability distribution of the intrinsic value.</p>
<p>The interpretation of the intrinsic value probability distribution
would change significantly when the investor had a fixed required rate
of return but assumed that the growth of revenues could be the only risk
factor (Figure 4). This would be a reasonable assumption because, at the
moment of valuation, the investor would not know how the economic
environment could develop and how this all might affect the
entrepreneurial activity of their enterprise. An infinite number of
possible revenue growth paths results in the respective infinite number
of intrinsic value scenarios. Different probability distributions
represent different types of possible investor’s assumptions (Sets 5, 6
and 7). The impact of the growth rate level on the intrinsic value is
also nonlinear which is worth emphasising here.</p>
<table class="table table-bordered">
<colgroup>
<col></col>
<col></col>
<col></col>
</colgroup>
<thead>
<tr><th>Set 5</th>
<th><img src="/articles/2023/Kaczmarzyk_sklad/media/image10.png" /></th>
<th><img src="/articles/2023/Kaczmarzyk_sklad/media/image11.png" /></th>
</tr>
</thead>
<tr><td>Set 6</td>
<td><img src="/articles/2023/Kaczmarzyk_sklad/media/image12.png" /></td>
<td><img src="/articles/2023/Kaczmarzyk_sklad/media/image13.png" /></td>
</tr>
<tr><td>Set 7</td>
<td><img src="/articles/2023/Kaczmarzyk_sklad/media/image14.png" /></td>
<td><img src="/articles/2023/Kaczmarzyk_sklad/media/image15.png" /></td>
</tr>
</table>
<p><strong>Fig. 4</strong>. Fixed required rate of return, variable
revenue growth rate, and fixed operational costs ratio</p>
<p>Source: own elaboration.</p>
<p>Assuming that the ratio of the operational costs was a risk factor
would also be reasonable. The probability distribution of the intrinsic
value is symmetrical due to possible changes in the ratio of the
operational costs (Figure 5).</p>
<table class="table table-bordered">
<colgroup>
<col></col>
<col></col>
<col></col>
</colgroup>
<thead>
<tr><th>Set 8</th>
<th><img src="/articles/2023/Kaczmarzyk_sklad/media/image16.png" /></th>
<th><img src="/articles/2023/Kaczmarzyk_sklad/media/image17.png" /></th>
</tr>
</thead>
<tr><td>Set 9</td>
<td><img src="/articles/2023/Kaczmarzyk_sklad/media/image18.png" /></td>
<td><img src="/articles/2023/Kaczmarzyk_sklad/media/image19.png" /></td>
</tr>
<tr><td>Set 10</td>
<td><img src="/articles/2023/Kaczmarzyk_sklad/media/image20.png" /></td>
<td><img src="/articles/2023/Kaczmarzyk_sklad/media/image21.png" /></td>
</tr>
</table>
<p><strong>Fig. 5</strong>. Fixed required rate of return, fixed revenue
growth rate, and variable operational cost ratio</p>
<p>Source: own elaboration.</p>
<p>The Monte Carlo simulation is used in finance especially to examine
the influence of the simultaneous impact of risk factors. The volatility
of the intrinsic value would rise if both risk factors were taken into
account at the same time (Figure 6, Set 11). If the required rate of
return was added as a third risk factor (e.g. uniformly distributed),
the volatility of the intrinsic value would grow again (Figure 6, Set
12). Although technically possible, such an approach leads to a blurred
picture of risk. The volatility of the economic environment is raised by
uncertain expectations of an investor.</p>
<table class="table table-bordered">
<colgroup>
<col></col>
<col></col>
</colgroup>
<thead>
<tr><th><p>Set 11</p>
<p>Set 12</p></th>
<th><img src="/articles/2023/Kaczmarzyk_sklad/media/image22.png" /></th>
</tr>
</thead>
</table>
<p><strong>Fig. 6</strong>. Fixed and variable required rate of return,
variable revenue growth rate, and variable operational cost ratio</p>
<p>Source: own elaboration.</p>
<table class="table table-bordered">
<colgroup>
<col></col>
<col></col>
</colgroup>
<thead>
<tr><th>Set 11 Set 13 Set 14</th>
<th><p><img src="/articles/2023/Kaczmarzyk_sklad/media/image23.png" /></p>
<table class="table table-bordered"><colgroup><col></col><col></col><col></col><col></col></colgroup><thead><tr><th>
</th><th>Set 11, Fixed required rate, Normal revenue growth rate, Normal
operational cost ratio</th>
<th>Set 13, Fixed required rate (lower), Normal revenue growth rate,
Normal operational cost ratio</th>
<th>Set 14, Fixed required rate (higher), Normal revenue growth rate,
Normal operational cost ratio</th>
</tr></thead><tr><td>Expected Value</td>
<td>104.16</td>
<td>202.29</td>
<td>39.01</td>
</tr><tr><td>Standard deviation</td>
<td>41.10</td>
<td>53.05</td>
<td>33.60</td>
</tr><tr><td>Coefficient of variation</td>
<td>0.39</td>
<td>0.26</td>
<td>0.86</td>
</tr></table></th>
</tr>
</thead>
</table>
<p><strong>Fig. 7</strong>. Different fixed required rates of return,
variable revenue growth rate and variable operational cost ratio</p>
<p>Source: own elaboration.</p>
<p>Even though a volatile required rate of return blurs the risk
associated with an asset, it somehow is a volatile input because it may
differ among different investors valuing an asset with different
required rates of return. Having set the same assumptions for all risk
factors besides the required rate of return (Figure 7), the investor
with the lowest required rate of return will face the highest absolute
(lowest relative) volatility of the intrinsic value (Set 13), whereas
the investor with the highest – will face the lowest (highest)
volatility (Set 14).</p>
<h2>5. Conclusion</h2>
<p>The discount rate as a risk factor with an attributed probability
distribution affects a risk variable in a nonlinear way. This can be
easily confirmed by qualifying the discount rate as the only risk factor
during simulation. Regarding a uniformly distributed discount rate,
namely a discount rate with a certain range of possible values with the
same probability of occurrence, one can expect different probabilities
of a risk variable, such as an intrinsic value. Depending on a risk
factor, influence can be nonlinear (e.g. the growth rate of revenues) or
linear (e.g. the share of variable costs). The discount rate as an
additional risk factor with an attributed probability distribution
increases the volatility of a risk variable. Then, the distribution of a
risk variable becomes more flattened.</p>
<p>In the case of different discount rate levels representing different
investors with different expectations, one must emphasise that the
higher the discount rate, the lower the present value, and consequently,
the lower the intrinsic value. At the same time, the lower the absolute
volatility, the higher the relative volatility of the present value, and
consequently, the intrinsic value.</p>
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