**4. Application of probabilistic modelling using Monte Carlo simulation in business decision making**

## **4.1 The problem**

CEDFA Ltd. deals in new and used cars. The firm has secured a franchise with a major car manufacturer and ordered two car models—saloon and hatchback models. Due to changes in market forces, the unit costs and the demand for the cars are unknown. The firm estimated the demands for each model based on the previous year's data. According to the franchise agreement, any vehicles not sold by the end of the year will need to be discounted to ensure they are sold. The size of the discount is affected by various factors, including the presence of alternative brands in the market. The firm anticipates that the annual franchise fee could increase this year as the contract expires and has estimated the probability of this increase to be 0.25. If it does, the cost will be \$5,000,000. Based on the historical data, the firm's sales and market analysts have come up with the base-case estimates shown in **Table 1** below.


#### **Table 1.**

*Base case estimates.*

The company is worried about how the uncertainty in the business environment would affect the demands for their cars and the subsequent discount rates to be used in discounting the unsold vehicles as per their franchise agreement. Based on the collected data, the company's analysts estimate that the unit cost for each car and the demand for each car follow the *triangular probability distribution* and the discount rates follow the *lognormal probability distribution*. The analysts suggest the *Bernoulli probability distribution* to model the potential increase in the franchise fee. These probability distributions represent the uncertainty in these variables, the full range of


#### **Table 2.**

*Probability distribution parameters.*

possibilities and how likely they are. **Table 2** below shows the parameters for the probability distributions.

Given that the company will have to discount any leftover car as per the franchise agreement, the company wants to know the effects of the different discount rates on the net profit. Based on historical experience and the current market environment, the firm anticipates that the minimum discount it can give for unsold saloon and hatchback cars would be 10% and 15%, respectively. At the same time, the maximum discount rate is estimated as 30% and 40%, respectively. As part of the firm's strategic business decisions, the company executives want to understand the impact of the unit costs, the potential increase in the franchise fee, the demand for the cars and the discount rates on the firm's net profit. Also, the firm wants to decide the minimum and maximum discount rates to apply to the unsold cars and the variable with the most significant impact on the total revenue and net profit.

For this example, we are required to do the following:


#### **4.2 Decision scenarios**

The company wants to decide the discount rate to apply to unsold cars at the end of the year. To make this decision, the firm created two scenarios, namely, *strategy one* -

allowing the discount rate to be as high as possible (untruncated or unbounded), and *strategy two* - truncating (or bounded) the discount rate with minimum and maximum rates, as shown in **Table 2** above. Evaluate the effects of these two different strategies on the firm's net profit.

### **4.3 Solutions**

As you can see, most business decisions depend on several different uncertain factors/variables. Suppose we assume that we can determine or estimate the probability distributions of the individual variables. We can then aggregate these into the probability distribution of the output variable. A commonly used approach is Monte Carlo simulation. In the Monte Carlo simulation method, the probability distribution of the output variable is determined by using a large sample of randomly generated values for the individual variables. These values are drawn from the known or estimated probability distributions of the different input variables or directly from corresponding historical data. Monte Carlo simulations do not make distributional assumptions for the individual probability distributions, and correlations between these variables can be easily considered. In Monte Carlo methods, we perform many simulations and iterations. This is because of the principle of the Central Limit Theorem<sup>2</sup> .

To account for the uncertainties in the input variables, the company's analysts estimate that the unit cost for each car and the demand for each car follow the triangular probability distribution. The discount rates follow the lognormal probability distribution as shown in **Table 2**. The analysts suggest the Bernoulli probability distribution to model the potential increase in the franchise fee. Furnished with these pieces of information, we start the modelling process.

First, we develop a deterministic model using the base-case estimates of the input variables provided by the firm. A deterministic model is a model that does not account for randomness/uncertainty in the input variables that are used in forecasting the output variable of interest<sup>3</sup> [12]. Consequently, a deterministic model will always produce the same output from a given starting condition. The output from the deterministic model (**Table 3**) assumes that the unit cost, demand and the discount rate applied to sell leftover cars for both the saloon and hatchback models are constant over time. However, we know that these variables are outside the control of the decision-maker and can change over time. For instance, the demand for these cars is not known with certainty. Some of the factors that affect demand could include the presence of substitutes in the market, the business cycle<sup>4</sup> and the advertisement budget, amongst others. So based on a discount rate of 15% and 25% applied to leftover saloon and hatchback, the firm will end up with total revenue of \$110,175,000 and a profit of \$11,175,000.

<sup>2</sup> The central limit theorem posits that the probability distribution of a sum of independent random variables can be approximated by a normal distribution if the number of individual random variables is large enough.

<sup>3</sup> The variables of interest in this example are the total revenue and the profit.

<sup>4</sup> Business cycles are intervals of expansion (economic booms) followed by a recession (burst) in economic activity [13]. Business cycles have implications for the welfare of the wider population and businesses.


#### **Table 3.**

*Total revenue and profit: deterministic model.*

It is easy to observe the challenges of relying on this model to make business decisions:


Why is this important? This is important because we may be able to evaluate the potential downside risk of our decision. That is the probability of making a loss.

We will illustrate how to use probability distributions to propagate uncertainty in the input variables. We use Monte Carlo simulation to produce information and insights from our model and its assumptions. As alluded to in the previous section, the precision of a probabilistic model relies heavily on the appropriate use of probability distributions to accurately represent the problem's uncertainty, randomness and variability. In practice, inappropriate use of probability distributions is a common failure of probabilistic models. We start by replacing the 'fixed' estimates of the input variables in the model in table three with the parameters for the probability distributions in table two. Second, we will mark the variables of interest (total revenue and net profit) as the 'output' cells. These two variables are the variables we want to analyze. The simulation<sup>5</sup> exercise allows us to sample all the input distributions randomly and recalculates the spreadsheet repeatedly, keeping track of the resulting output values. Each separate recalculation in the simulation process is known as an 'iteration'. A single iteration represents a possible future set of circumstances in the model. So, two iterations represent two possible future sets of circumstances in the model. Since the sampling is random, commonly occurring input ranges and combinations of inputs would appear more frequently in the simulation data. On the other hand, rarer scenarios will be less likely. We present the output from the probabilistic Monte Carlo simulation models for the two strategies. The probability density graphs with some statistics are reported in **Figures 1** and **2**.

<sup>5</sup> We used the Palisade @Risk software for the probabilistic modeling.

*Perspective Chapter: Application of Monte Carlo Methods in Strategic Business Decisions DOI: http://dx.doi.org/10.5772/intechopen.106201*

**Figure 1.** *Profit strategy one (Untruncated Discount Rates).*

**Figure 2.** *Profit strategy two (Truncated Discount Rates).*

The primary purpose of quantifying uncertainties is to obtain a sound basis for our decisions. As you can see, all available information on an uncertain (random) variable is, in principle, contained in the corresponding probability distributions (see **Figures 1** and **2**). In decision analytics, a well-informed decision is based on comparing the decision-maker's risk appetite or risk tolerance threshold to the outcomes they are exposed to. For instance, we may be interested in evaluating the probability of

incurring a loss or reaching a defined target. The outputs from our model allow us to achieve these goals. Statistics such as the mean or standard deviation of the output variable are also computed. These statistics are used to describe various future outcomes. For **strategy one**, we see that the effect of an **untruncated** discount rate on the company's bottom line could result in a loss of (\$4,900,126) at a 95% confidence interval. The probability of realizing this loss for this company is 2.5%. The same strategy could result in a potential net profit of \$12,780,912 with a probability of 97.5%.

We turn our attention to the second scenario, **strategy two**, where we **truncate/limit** the amount of discount that could be used in selling the leftover cars. The idea is that no matter how many cars are left unsold; we cannot go beyond the maximum discount rates for the cars. From the probability density graph in **Figure 2**, we can see that the upper and lower bounds for the net profit are \$414,780 and \$12,623,794. The probability of ending up with a lower profit is 2.5% and ending up with a higher profit is 97.5%.

Let us briefly consider how we can make decisions using the results from our probabilistic modelling using Monte Carlo methods. Observe the statistics reported in **Figures 1** (truncated model) and **2** (untruncated model). For the untruncated strategy, the highest possible loss that the firm can incur is (£28,055,462), and the maximum profit is £17,279,472. While for the truncated strategy, the maximum loss is (£5,795,584), and the maximum potential profit is £15,496,276. Although the firm may earn a slightly higher profit if they allow market forces to determine how much discount they can apply to unsold cars, it is clear from the reported statistics that this strategy is riskier than the bounded discount rate strategy. **Table 4** shows a comparison of the two strategies.

Observe from **Table 4** that the bounded/truncated strategy outperforms the untruncated strategy in key metrics like maximum loss (\$5,795,584, vs. \$28,055,462) and expected profit (\$7,323,010 vs. \$5,286,830). On the other hand, the unbounded strategy outperforms the bounded strategy in terms of the maximum net profit (\$17,279,472 vs. \$15,496,276). It is also important to mention that the unbounded/ untruncated strategy carries more risk than the bounded/truncated strategy. In this example, we use the standard deviation as a measure of risk. Specifically, we see a standard deviation of \$3,224,254.35 for the truncated strategy versus \$4,551,470 for the untruncated strategy. The next thing we need to consider is to identify our key inputs using sensitivity analysis. Sensitivity analyses study how various sources of uncertainty in a model contribute to the model's overall uncertainty or volatility. Our primary goal is to identify the most 'critical' inputs, the inputs to concentrate on most when making decisions. We will use the regression coefficient, regression mapped values, correlation and contributions to variance tornado charts to explore the inputs we need to prioritize in our decision-making. The results are shown in **Figures 3**–**6**.

We will discuss the sensitivity graphs from strategy one (untruncated discount rate) for brevity. The tornado graphs allow us to analyse how the inputs in our


#### **Table 4.**

*Comparison of the two Strategies (unbounded and Bounded Strategies).*

*Perspective Chapter: Application of Monte Carlo Methods in Strategic Business Decisions DOI: http://dx.doi.org/10.5772/intechopen.106201*

**Figure 3.**

*Sensitivity tornado graph (Estimated Regression Coefficient).*

decision model drive the variations/behaviour of our output variables (in this case, the net profit). Variables with the most considerable impact on the output distribution have the longest bars in the graph.

Looking at the regression coefficients (**Figure 3**), we see that increase in the franchise fee has the biggest impact on the net profit. Specifically, if the franchise fee is increased by 1%, that increase will result in a 0.47% decrease in the net profit. The demand for the hatchback model is another variable with a great impact on the net profit. If the demand increases by 1%, this will result in a 0.44% increase in the net profit, while a similar increase in the demand for the saloon model will result in a 0.29% increase in profit<sup>6</sup> . Also, increasing the discount rates for both hatchback and saloon models by one per cent would result in a reduction in net profit of 0.38% and 0.27%, respectively. The company should also pay attention to the unit costs of both models. Specifically, a one per cent increase in the unit cost of saloon and hatchback cars will result in 0.35% and 0.32%, respectively.

In monetary terms, the firm may lose up to \$2,163,404.38 for a 1% increase in the franchise fee while holding all the other variables constant - see **Figure 4** below. Similarly, an increase in the demand for the hatchback model will lead to about a \$2,023,527.94 increase in the net profit. In addition, increasing the discount rates will result in a reduction in profit. For the hatchback cars, increasing the discount rate by 1% will lead to about \$1,736,223.46 loss in profit; a similar increase in the discount rate for the saloon cars will lead to a \$1,226,373.03 loss in profit. Based on these pieces of information, the firm can make an informed decision while taking its risk appetite and equity capital into consideration.

<sup>6</sup> The key to interpreting the graphs is to consider both the signs (positive or negative) and the magnitude of the estimated coefficients.

**Figure 4.** *Regression mapped values.*

Understanding the interdependency of the input variables in the model is very important. Correlation measures the extent to which two variables vary together, including the strength and direction of their relationship. Exploring the correlation between variables is an essential part of exploratory data analysis. A high correlation indicates a strong relationship, and a weak correlation indicates that they are not closely related. **Figure 5** reveals a negative correlation between the increase in franchise fee, unit cost and discount rate and a positive correlation between demand and net profit.

We look at the contribution to the variance tornado graph (**Figure 6**) below. This graph allows us to explore the contributions of each of the input variables to changes in the output variable. Again, we see that changes in the franchise fee cause 22.7% of

**Figure 5.** *Correlation analysis.*

*Perspective Chapter: Application of Monte Carlo Methods in Strategic Business Decisions DOI: http://dx.doi.org/10.5772/intechopen.106201*

the variation in the output variable, while the demand for hatchback cars caused 19.2% amongst others.

Finally, we look at scenario analysis. We are particularly interested in examining a combination of input variables that contribute significantly towards reaching a specific goal, also referred to as the **target scenario** associated with the output values. In this case, we want to examine the combination of input variables that could result in the following:


The results from the three scenarios analysis are reported in **Figures 7**–**9** below. Our first desired scenario is to explore the possibility of having a net profit in the 75th percentile<sup>7</sup> . In the net profit probability distribution, we locate the 75th percentile as \$8,624,840.69. The scenario analysis identifies the input combinations that have the most significant effect in achieving this net profit. Looking at **Figure 7** above, we can see that the strategy that would allow the firm to achieve the desired scenario is to increase the demand for hatchback cars at full price by 0.54% and reduce the unit cost of saloon cars to 0.50%. Essentially, scenario analysis helps you identify the input combinations that have the most significant effect in achieving a net profit value at

<sup>7</sup> Percentiles indicate the percentage of values that fall below a particular value. They tell you where a value stands relative to other values. The general rule is that if value *Y* is at the Nth percentile, then *Y* is greater than *N*% of the values.

the 75th percentile. Given that the unit cost of the saloon cars is outside the firm's control, the company may try and negotiate a discount with the manufacturer by increasing the number of vehicles ordered.

**Figure 8** below provides the second scenario, which is the worst-case scenario. Observe that increasing the franchise fee by 2.3%, a drop in demand for the hatchback of 0.66%, and an increase in the discount rate applied to leftover hatchbacks will result in a loss. We are always concerned with the worst-case scenario in risk and decision analytics. At this stage, the firm needs to decide whether to continue with this franchise or look for alternative manufacturers. Nevertheless, they must block any attempt to increase the franchise fee by more than 2%.

The final scenario shown in **Figure 9** below, which is the best, achieves a net profit value in the 90th percentile. To achieve the desired target, the firm will have to

**Figure 8.** *Net loss (Worst-Case Scenario).*

*Perspective Chapter: Application of Monte Carlo Methods in Strategic Business Decisions DOI: http://dx.doi.org/10.5772/intechopen.106201*

**Figure 9.** *Net profit at the 90th percentile.*

increase the demand for hatchbacks and saloons by 0.85% and 0.60%, respectively. At the same time, negotiating discounts for the saloons and hatchbacks by 0.70% and 0.60%, respectively. Using probabilistic Monte Carlo methods as an exploratory decision-making tool can improve the decision maker's understanding of the significant business value drivers. This method allows us to appreciate the most relevant uncertain input variables and their sensitivities to the variable of interest. As you can see, much of the value in probabilistic modelling comes from its role in structuring a more constructive management discussion.
