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Vishwadeep Khatri

Monte Carlo Simulation

Monte Carlo Simulation is a technique that can be used to model the probability of different outcomes without actual data collection. It helps in understanding the risk associated with the different outcomes and aids in decision making in the fields of finance, supply chain, project management, engineering etc.


An application-oriented question on the topic along with responses can be seen below. The best answer was provided by Ram Rajagopalan and Shashikant Adlakha.

Applause for all the respondents - Satyajit Das, Pradeepan Sekar, Selva Mariappan Subramanian, Ram Rajagopalan, Alpesh Gorasia, Shashikant Adlakha, V P Singh, Ram Kumar Chaudhary


Also review the answer provided by Mr Venugopal R, Benchmark Six Sigma's in-house expert.


Q 256. How does Monte Carlo Simulation help in Risk Management? Provide a few examples to explain your views. 


Note for website visitors - Two questions are asked every week on this platform. One on Tuesday and the other on Friday.

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Monte Carlo is a Decision Science approach that helps in improved decision making under uncertainty. It helps in seeing all possible scenarios, outcomes and helps in accessing the Risk. The technique was first used by scientists working on the atom bomb; it was named for Monte Carlo, the Monaco resort town renowned for its casinos. Since its introduction in World War II, Monte Carlo simulation has been used to model a variety of physical and conceptual systems.


Risk is there in any and every part of our lives. There is always uncertainty, ambiguity and variability that makes it difficult to predict future, even with more information available today.

For agriculture - every season the farmers have to make calculated risks on the crops they need to sow. And the risks they face are below, many that are not even remotely under his control

  • There will be adequate monsoons for the water required for the crops
  • Input prices for fertilizers don’t shoot up too much as they are dependent on global oil markets
  • Pest attacks this season will be reasonable
  • Overall crop harvests will be near demand, so that the prices will be under control
  • Government doesn’t ban exports and block sales avenues


Monte Carlo simulations provides the decision makers with a range of possible outcomes and probabilities they will occur for any choice of action.


Monte Carlo is applicable in a wide range of industries and functions

  • Project Management - Every task involved has a inherent risk of completion ahead or behind schedule. The duration for each task follows a probabilistic model, and hence a sequence of tasks will have cumulative effects
  • Manufacturing - A flow manufacturing is a sequence of events/ activities for a job to be completed. Uncertainties include availability of raw materials, machine breakdowns etc.
  • Stock Market - Probably one of the largest applications of Monte Carlo Simulations, the whole underlying aspect of stocks is that over a period of years it grows and its only due to the underlying uncertainties.
  • Energy - Power consumption could be dependent on the weather conditions
  • Insurance - Car premiums always factor in the risk components based on vehicle, the profile of the driver, the city of residence, type of use (commercial/ personal) etc.


The underlying concept uses randomness to solve problems that look deterministic. The approach is

  1. State the target variable (Project Completion Time, Energy Consumption at a particular date etc)
  2. List the underlying input variables
  3. Model the input variables using probability distribution. This is done using historical data
  4. Model the Target variable as a function of the input variables
  5. Repeat multiple time
    1. Calculate the Target variable using Deterministic values of the inputs
    2. Aggregate multiple times


The results are computed over repeated sampling and statistical analysis. Monte Carlo Simulations can take thousands and maybe ten thousands of iterations to produce results. Sawilowsky lists the characteristics of a high-quality Monte Carlo simulation

  • the (pseudo-random) number generator has certain characteristics (e.g. a long "period" before the sequence repeats)
  • the (pseudo-random) number generator produces values that pass tests for randomness
  • there are enough samples to ensure accurate results
  • the proper sampling technique is used
  • the algorithm used is valid for what is being modeled
  • it simulates the phenomenon in question.


Advantages of Monte Carlo Simulation over traditional What If Analysis include

  • Flexbile, considers wide range of parameters and outcomes, reduces uncertainty
  • Ability to model outcome with its probabilities
  • Scenario modeling, sensitivity analysis
  • Visual descriptions
  • Correlation of input variables


The limitations of the model, are more due to the nature of the problem/ the domain its used

  • Extending randomness - In real time many are time bound, like life expectancy etc, while probability models don’t have those
  • Incorporating reality: Since the models are done using underlying distributions, it may predict numbers that are not realistic like interest rates in certain geos (like US which is almost 0). Brining reality into model is a tough task
  • Use errors on the model, parameters and distributions






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The Monte Carlo- model is a popular tool and  commonly used in risk analysis in a number of scenarios. This model is properly executed by Simulation software- The software runs a large number of what-if scenarios, using random number generator (RNG) to choose a random value for each input value and summarizes the results using relevant statistics and charts. Defining assumptions of input variables with type of distribution, define forecast and correlation if any between variables are required to be filled. All combinations of input parameters are incorporated into  the results, using relative frequencies of   different  type of distributions of the parameters.

Common uses of Monte Carlo analysis are:-


-       Project schedule risk estimation

-       Project Cost risk estimation

-       Best combinations of projects or activities selection to maximize  the profit

-        Portfolio construction and management 

-        Volatility of stocks, derivatives

-        Expected value of investment

-       Airlines, hotel industries, real estate, oil companies to decide best price of product/service to maximize profits, using OptQuest tool.

-       Sales projection of different products, using different assumptions/trends.

-       DFSS for  six sigma projects in manufacturing





The following example denotes  average statistics,  pertaining to a sale of a good. We want to determine the probability of  atleast a profit of 80$ each day, also we want to determine the risk of having loss rather than profit.








After defining assumptions of different variables and running 1000 simulations, we get the following results:





So, the probability of having minimum 8o$ profit every day is 47.36%






So the chances of sustaining no profit and rather loss is 22.36%

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All projects have many issues and these issues or events may have negative impact as well as positive impact to the project. These unknown negative and positive impacts are known as Risks. Every time Risk is always not bad for the project, there are some positive risks also known as opportunity for the project.

Moreover, we handle these risks with help of Risk Management Plan. It is a process of identifying risks, analyzing it, response it and implement those responses. So, to analyse the project we  have lot of tools and techniques and one such technique is known as Monte Carlo Analysis.

When we analyse risk with the available data, we tend to use a model that simulates the combined effects of every single project risks and sources of uncertainty to evaluate their potential impact on achieving the project objectives. Therefore, the simulations are done by using a Monte Carlo Analysis. It is a technique used to understand the impact of risk in forecasting models. Now a days computer software is used to repeat the risk analysis model several thousand times and the output of the analysis can be shown graphically by a histogram or a S-Curve.

There are lot more uses of Monte Carlo Simulation beyond the project domain, but for me known a bit application at analysis for schedule risk and cost risk at project except project risk analysis.

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Monte Carlo simulation is a simplified powerful mathematical computation  on distribution of the  possible outcomes with probability where the inputs are having inherent uncertainty.  


How it works?


In order to explore the possible outcomes of any process/ model which is having uncertain inputs, we need to derive a mathematical model which computes the effect of input in the output. As a general practice, we use mean or average as an input in order to compute the outcome, but it is more unrealistic as we know average is not going to happen again and again and the variation from an average is having a big impact on the outcome.


In Monte carlo simulation, Based on the distribution we are fitting to the inputs, it generates a random value ( as per the parameters of the distribution) for each input and compute the outcome and this computation will be repeated for 1000, 10000 or million times with the help of software support. Then we will have a wide range of outcome which will give us a result of the probability distribution of the outcome. 


1. It is very important to build a perfect mathematical model which can be done by simple mathematical equation  with inputs or regression equation based on experimental data (or Historical data if history represents the future) and  2. it is also important to fit the range of input data in the adequate distribution with proper estimation of the parameters.


Monte Carlo Simulation in Risk Management:


Risk is always an outcome of uncertainty.  In other words, certain inputs will bring out expected results, but it is rare to have certain inputs and which leads to risk in each and every day -day activity we are carrying out. 


For an instance, risk of India losing the final match in cricket is depend on so many uncertain inputs, starting from tossing the coin to choose first team to bat, whether, Players in the team, their batting order, bowlers performance.  In that case if we are able to build a mathematical model of runs scored  in a ball for a batsman ( who is left only with last over ) based only on the bowling speed and bowling style keeping all other factors as constant and we have fit  the bowling speed in to distribution for the bowler who bowls, we will able to get the distribution of runs scored in last over as an outcome. Average score of the batsman or strike rate will not support as  much as the probable score taken in last over, to find the probability of winning the match.


So Monte carlo simulation can be applied in all aspects where you are able to build a mathematical model for an outcome as a function of uncertain inputs to get probability distribution of outcome which in turns helps you to identify the risk involved in the process.



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Risk assessment is a crucial step in any business process. Therefore, it is important to perform risk analysis before making a business decision. However, the uncertainties and risk factors involved in the business are often difficult to calculate precisely and hence estimating the desired outcome is a tedious process. For example, an investment decision may involve combination of more than 10 unique factors and each factor may have different underlying assumptions and probabilistic outcomes associated with it. In such cases, the process of deciding whether the investment is worthwhile or not is cumbersome. There may be hundreds of thousands of outcomes in this case, given the complexity of possible combinations of different factors to compute manually.

Monte Carlo simulation helps in assessing the risk by modelling the different factors and calculating them. It then presents the likely outcomes with a certainty by performing numerous iterations of factors by substituting the values within its defined probabilistic functions. The probability of desired and optimal outcomes can be obtained from the uncertainties and risks.

MC Simulation not only predicts the range of possible outcomes in a given scenario but also the critical areas that impact the outcome most. Sensitivity analysis for instance depicts the impact of variations within the most important factors and its correlation to the outcomes. By doing so, the stakeholder can focus on the most crucial elements or factors in order to minimize the overall risk.

There are many applications of MC simulation in different sectors and industries in assessing the risk. Ranging from individual’s choice of products or services among alternatives to government’s policies at the macro level, MC simulations are of practical significance in almost any area.

One such example can be seen in the world of capital market. A fund manager who manages clients’ portfolios will have to assess the risk and invest according to the clients’ expectations. There are several risks and uncertainties such as funds’ historical returns, macroeconomic factors, unsystematic risks peculiar to the funds, overall market volatility and so on. In this scenario, the fund manager can model the risks using his/her knowledge and available information and identify the optimal portfolio using the Monte Carlo simulation to mitigate the risks before actually investing in the funds.

Another application of Monte Carlo simulation can be witnessed in predicting the success rate of oil exploration. It is a highly risky business in which the investment might far outweigh the benefits most times in terms of project cost, long pay off period etc., In such cases, MC simulation can be of great advantage in predicting the likelihood of success, identifying the high-risk areas and deciding the ways to maximise success.

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Monte Carlo simulation is used for problem solving technique for identifying proximity the probability of outcomes by running multiple trails through random sampling.


It generates possible outcomes of your decisions and assesses the impact of risk and allowing better decision making under uncertainty.

In Monte Carlo simulation model is built using the simulation in which we assign the assumption variable through wide range of possible value, also define the decision variable with wide range of possible value and see the impact on forecast variable through running of simulation with probability distribution at any point with certainty % value to success it.Also through Optimizer we can minimize the risk through identify factors value of assumption variable and decision variable.


Sensitivity chart is another main feature of Monte Carlo simulation for risk assessment where we can identify impact of different assumption variable on forecast value which help to minimize risk.


Below are some examples through which we can identify & mitigate risk through mote Carlo simulation.


1> Selection of projects from multiple project list having limitation on investment & will provide maximum project savings.

2> Selection of correct product CTQ's from multiple product dimensions 

3> Identify Critical activities in executing project to mitigate or reduce project delay risk. 


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Monte carlo simulation is used to know the forecasting/ risk  due to delays (which are  unpredictable in actual schedules or standard cycle times) of the project or activity by making data iterations. 

let us  consider  batch manufacturing cycle time example:

  operation 1: 8-10 hrs

 operation  2: 12-14 hrs

 operation 3: 25-28 hrs.

operation 4: 14-18 hrs.

 To complete the batch manufacturing and to achieve the output we need to add all four operations i.e 1+2+3+4 operations.


min hrs max hrs Average
8 10 9
12 14 13
25 28 26.5
14 18 16
59 70 64.5

based on the above data, 

 we can complete the batch in 59 hrs as best case and worst case is 70 hrs and an average performance 64.5 hrs.

by reducing the cycle time at operation 3 at half of the time by dividing the operation in to sub ,  batch cycle times will be differed and output/ day will increased.

OPERATION  min hrs max hrs Average(hrs)
1 9 10 9.5
2 12 14 13
3 14 18 16
4 14 18 16
total 49 60 54.5


min max average
8 10 9
12 14 13
25 35 30
14 18 16
59 77


Based on the above iterations, cycle time reduction  at operation no3  will result to improve the  Batch output/ day.

Based on the above iterations, increase in cycle time in operation No 3 will have the impact on batch output/ day.

such that by using Monte Carlo simulation, we can perform the 12 iterations with different scenarios and we can estimate the risk in daily production run rates. 

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Leveraging Monte Carlo Simulation to reduce Risk in back office operations of finance institutions

Category of risk : Operational ( Non IT related) resulting in operational or reputation loss to client.


1. Back office operations of financial institutions are vulnerable to multiple risks induced into operations due to manual interventions required to complete transactions. Example - transactions detail shared with wrong counterparty ( data breach); maker - checker miss ( resulting in incorrect payment); delayed processing of transaction ( resulting in interest charges) etc. While these events might be few but the impact is high. Mostly in operations manual controls are put in place for want of funds to arrest such events from occurring. Monte Carlo Simulation enables presently a view to leadership from provider as well buyer of services from financial institutions to take a data view of possible scenarios and pick opportune areas to invest in technology to arrest prime categories of these incidents. Usually on ground RCA's are developed for specific incidents and each one gets treated as single incident and unable to draw investments on technology front.


2. There is an extended debate in back office operations that profile of individuals required for financial back office is significantly different than others and so is there learning curve on the domain. When attrition and cost pressure play simultaneously resulting in erosion of tenure & domain knowledge, incidents spike. Talent is scare with relevant domain knowledge; in such situations Monte Carlo Simulation can be leveraged to model talent which optimizes cost while maintaining risk levels  


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Benchmark Six Sigma Expert View by Venugopal R

Monte Carlo Simulation is a statistical method, using computers for quantitative risk analysis. In this method, random sample data can be generated based on known distributions, instead of carrying out actual experiments, which could be impractical for many situations. Monte Carlo simulation methods are used in the field of Finance, Manufacturing, Engineering, Project Management, Medical and many other areas. Interestingly this method evolved based on a study on the outcomes for a gambling game, but the method was applied for studying neutron diffusion!


Monte Carlo simulation methods are not required if we have a deterministic solution based on analytical relationship between two variables. This method is required where there is influence and interaction by complex set of variables. For instance, if we need to evaluate the health risks to children due to the vehicular pollution of cities, multiple factors and conditions would impact the outcome such that, no two iterations of simulation would give exactly same outcomes. Based on large number of trials a collective picture of the outcome is possible.


To explain in simple terms, when a process takes multiple inputs to provide a combined outcome, the outcome will be reasonably predictable if each of the inputs are stable. However, if there is a variation for each of the inputs, it becomes quite complex to predict the outcome. Monte Carlo simulation becomes a useful method for such situations. The method uses computational algorithms to simulate the process for a very large number of times encompassing the entire variability span for each of the inputs. The output is based on a probability distribution, that depicts all outcomes along with the likelihood of occurrence for each outcome. Monte Carlo simulations are considered as remarkably accurate models provided there is good accuracy and randomization in the input data.


The steps involved in a simulation process will include:

  1. Defining the problem
  2. Collect real system data
  3. Formulate and develop a model
  4. Validate the model & document
  5. Design the simulation exercise
  6. Perform simulation runs
  7. Interpret the results
  8. Recommendations based on the results


The simulation outcomes are presented as 'expected ranges' with a confidence level associated with each range. As the number of trials increases, the range of the outcome will reduce.


Consider an example below, where the Project Leader comes up with estimated delivery times for a project with 3 options viz. Relaxed, Normal and Aggressive.




Considering the fact that the tasks for the project depend upon several factors with varying extent of control by the team, the above data is unable to provide the likelihood of each option.


Applying Monte Carlo simulation and running a large number of simulation trials, the range of the each of the tasks will be taken into consideration as random values, and the percentage likely hood for each option is obtained. It may look like the below table.


The above example hopefully gives a broad understanding of how the Monte Carlo simulation tool would help to take decisions for a business situation. The methodology can be used for diverse applications. It is very important to ensure that the inputs provided are realistic and randomization of data to obtain a reliable output.

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The winners for this question are Ram Rajagopalan and Shahikant Adlakha. All published answers are acceptable too. Please go through the response by Benchmark Expert Venugopal as well. 


Kindly note that your anwer may not show as published if it has plagiarism more than 5%. You can do a self check on plagiarism by pasting your response at https://smallseotools.com/plagiarism-checker/ 


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