## Monte Carlo method

Excel Ideas. Excel Help. This is the second article in a series. The first article is, Introduction to Probabilistic Simulations in Excel.

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During more than three decades of working with spreadsheets in business, most of the spreadsheet models and forecasts I've seen have used what statisticians call a deterministic method. To illustrate, if I were forecasting profits for a period, a deterministic model would use one number for my forecast of sales, another number for my forecast of operating expenses, and so on. Instead, a more useful method would take a probabilistic approach, supported by the Monte Carlo method.

This figure Illustrates a normal probability distribution, which probably is the best approach for most business use. To use this method, you first set up your key assumptions to be defined by this curve. The models in your workbook then randomly select values from this normal distribution, use those values in their calculations, record key results, repeat the process many times, and then summarize your record of results. This process is known as the Monte Carlo method. In this article, I'm going to show you how to do this using Excel Data Tables to record the results produced by each calculation.

First, whenever you open a Monte Carlo analysis that uses data tables, make sure that the Monte Carlo workbook is the only workbook open. This is because it will need to recalculate many times, and if you have other workbooks open they also will recalculate, needlessly. If you work in Finance, you'll probably grumble that the following examples vastly oversimplify what is typically a complex financial-modeling process. In this case, I'll respond that I'm making the model very simple so you can understand the changes I'm suggesting to your standard modeling process.

And if you don't work in Finance, you'll probably grumble that I should have used an example from your own specialty, not Finance.

In this case, I'll respond that I'm using a simple income statement as the example because even if you work in engineering, or operations, or marketing, or wherever, you understand a simple income statement. Therefore, you'll be able to understand what the model is doing, and you'll be able to adapt my techniques to your own models and forecasts.

This figure illustrates a deterministic forecast Click here to get a copy of this workbook with all the Monte Carlo tables and reports described on this page. This will allow you to concentrate on how to adapt the Monte Carlo method to your own company.The Monte Carlo model allows researchers to run multiple trials and define all potential outcomes of an event or investment. Together, they create a probability distribution or risk assessment for a given investment or event.

Monte Carlo analysis is a multivariate modeling technique. All multivariate models can be thought of as complex "what if? Research analysts use them to forecast investment outcomes, to understand the possibilities surrounding their investment exposures, and to better mitigate risks. In the Monte Carlo method, the results are compared against risk tolerance.

That helps a manager decide whether to proceed with an investment or project. Users of multivariate models change the value of multiple variables to ascertain their potential impact on the project being evaluated. The models are used by financial analysts to estimate cash flows and new product ideas. Portfolio managers and financial advisors use them to determine the impact of investments on portfolio performance and risk. Insurance companies use them to estimate the potential for claims and to price policies.

Some of the best-known multivariate models are those used to value stock options. Multivariate models also help analysts determine the true drivers of value. Monte Carlo analysis is named after the principality made famous by its casinos. With games of chance, all the possible outcomes and probabilities are known, but with most investments the set of future outcomes is unknown.

It's up to the analyst to determine the outcomes and the probability that they will occur. In Monte Carlo modeling, the analyst runs multiple trials, sometimes thousands of them, to determine all the possible outcomes and the probability that they will take place.

Monte Carlo analysis is useful because many investment and business decisions are made on the basis of one outcome. In other words, many analysts derive one possible scenario and then compare it to the various hurdles to decide whether to proceed. Most pro forma estimates start with a base case. By inputting the highest probability assumption for each factor, an analyst can derive the highest probability outcome.

However, making any decisions on the basis of a base case is problematic, and creating a forecast with only one outcome is insufficient because it says nothing about any other possible values that could occur. It also says nothing about the very real chance that the actual future value will be something other than the base case prediction.

It is impossible to hedge against a negative occurrence if the drivers and probabilities of these events are not calculated in advance. Once designed, executing a Monte Carlo model requires a tool that will randomly select factor values that are bound by certain predetermined conditions.

By running a number of trials with variables constrained by their own independent probabilities of occurrence, an analyst creates a distribution that includes all the possible outcomes and the probabilities that they will occur. There are many random number generators in the marketplace. Both of these can be used as add-ins for spreadsheets and allow random sampling to be incorporated into established spreadsheet models.

The art in developing an appropriate Monte Carlo model is to determine the correct constraints for each variable and the correct relationship between variables. For example, because portfolio diversification is based on the correlation between assets, any model developed to create expected portfolio values must include the correlation between investments.Analysts can assess possible portfolio returns in many ways.

However, investors shouldn't stop at this. Monte Carlo simulations can be best understood by thinking about a person throwing dice. What are the odds of rolling two threes, also known as a "hard six?

One can compare multiple future outcomes and customize the model to various assets and portfolios under review. A Monte Carlo simulation can accommodate a variety of risk assumptions in many scenarios and is therefore applicable to all kinds of investments and portfolios. The Monte Carlo simulation has numerous applications in finance and other fields. The result is a range of net present values NPVs along with observations on the average NPV of the investment under analysis and its volatility.

How Does A Monte Carlo Simulation Work?

Monte Carlo is used for option pricing where numerous random paths for the price of an underlying asset are generated, each having an associated payoff. These payoffs are then discounted back to the present and averaged to get the option price. It is similarly used for pricing fixed income securities and interest rate derivatives.

But the Monte Carlo simulation is used most extensively in portfolio management and personal financial planning. She factors into a distribution of reinvestment ratesinflation rates, asset class returns, tax ratesand even possible lifespans. The result is a distribution of portfolio sizes with the probabilities of supporting the client's desired spending needs. The analyst next uses the Monte Carlo simulation to determine the expected value and distribution of a portfolio at the owner's retirement date.

The simulation allows the analyst to take a multi-period view and factor in path dependency ; the portfolio value and asset allocation at every period depend on the returns and volatility in the preceding period.

The client's different spending rates and lifespan can be factored in to determine the probability that the client will run out of funds the probability of ruin or longevity risk before their death.

A client's risk and return profile is the most important factor influencing portfolio management decisions. The client's required returns are a function of her retirement and spending goals; her risk profile is determined by her ability and willingness to take risks. More often than not, the desired return and the risk profile of a client are not in sync with each other. For example, the level of risk acceptable to a client may make it impossible or very difficult to attain the desired return.

Moreover, a minimum amount may be needed before retirement to achieve the client's goals, but the client's lifestyle would not allow for the savings or the client may be reluctant to change it. Let's consider an example of a young working couple who works very hard and has a lavish lifestyle including expensive holidays every year. None of the above alternatives higher savings or increased risk are acceptable to the client.

Thus, the analyst factors in other adjustments before running the simulation again. The resulting distribution shows that the desired portfolio value is achievable by increasing allocation to small-cap stock by only 8 percent. With the available insight, the analyst advises the clients to delay retirement and decrease their spending marginally, to which the couple agrees. A Monte Carlo simulation allows analysts and advisors to convert investment chances into choices.

Another great disadvantage is that the Monte Carlo simulation tends to underestimate the probability of extreme bear events like a financial crisis. It is, however, a useful tool for advisors.Monte Carlo methods are used in corporate finance and mathematical finance to value and analyze complex instrumentsportfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining the distribution of their value over the range of resultant outcomes.

The advantage of Monte Carlo methods over other techniques increases as the dimensions sources of uncertainty of the problem increase. Monte Carlo methods were first introduced to finance in by David B. Hertz through his Harvard Business Review article,  discussing their application in Corporate Finance. InPhelim Boyle pioneered the use of simulation in derivative valuation in his seminal Journal of Financial Economics paper.

This article discusses typical financial problems in which Monte Carlo methods are used. It also touches on the use of so-called "quasi-random" methods such as the use of Sobol sequences. The Monte Carlo method encompasses any technique of statistical sampling employed to approximate solutions to quantitative problems. In financethe Monte Carlo method is used to simulate the various sources of uncertainty that affect the value of the instrumentportfolio or investment in question, and to then calculate a representative value given these possible values of the underlying inputs. Although Monte Carlo methods provide flexibility, and can handle multiple sources of uncertainty, the use of these techniques is nevertheless not always appropriate.

In general, simulation methods are preferred to other valuation techniques only when there are several state variables i. See below. Many problems in mathematical finance entail the computation of a particular integral for instance the problem of finding the arbitrage-free value of a particular derivative. In many cases these integrals can be valued analyticallyand in still more cases they can be valued using numerical integrationor computed using a partial differential equation PDE.

However, when the number of dimensions or degrees of freedom in the problem is large, PDEs and numerical integrals become intractable, and in these cases Monte Carlo methods often give better results. For more than three or four state variables, formulae such as Black—Scholes i. In these cases, Monte Carlo methods converge to the solution more quickly than numerical methods, require less memory and are easier to program.

For simpler situations, however, simulation is not the better solution because it is very time-consuming and computationally intensive. Monte Carlo methods can deal with derivatives which have path dependent payoffs in a fairly straightforward manner.

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Monte-Carlo methods are harder to use with American options. This is because, in contrast to a partial differential equationthe Monte Carlo method really only estimates the option value assuming a given starting point and time. However, for early exercise, we would also need to know the option value at the intermediate times between the simulation start time and the option expiry time. In the Black—Scholes PDE approach these prices are easily obtained, because the simulation runs backwards from the expiry date.

In Monte-Carlo this information is harder to obtain, but it can be done for example using the least squares algorithm of Carriere see link to original paper which was made popular a few years later by Longstaff and Schwartz see link to original paper. The fundamental theorem of arbitrage-free pricing states that the value of a derivative is equal to the discounted expected value of the derivative payoff where the expectation is taken under the risk-neutral measure .

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An expectation is, in the language of pure mathematicssimply an integral with respect to the measure. Monte Carlo methods are ideally suited to evaluating difficult integrals see also Monte Carlo method. Today's value of the derivative is found by taking the expectation over all possible samples and discounting at the risk-free rate. Now suppose the integral is hard to compute. We can approximate the integral by generating sample paths and then taking an average. Suppose we generate N samples then.

In finance, underlying random variables such as an underlying stock price are usually assumed to follow a path that is a function of a Brownian motion 2. For example, in the standard Black—Scholes modelthe stock price evolves as. This leads to a sample path of. We obtain the Monte-Carlo value of this derivative by generating N lots of M normal variables, creating N sample paths and so N values of Hand then taking the average. Commonly the derivative will depend on two or more possibly correlated underlyings.

### Creating a Monte Carlo Simulation Using Excel

The method here can be extended to generate sample paths of several variables, where the normal variables building up the sample paths are appropriately correlated. It follows from the central limit theorem that quadrupling the number of sample paths approximately halves the error in the simulated price i.Monte Carlo simulations are used to model the probability of different outcomes in a process that cannot easily be predicted due to the intervention of random variables.

It is a technique used to understand the impact of risk and uncertainty in prediction and forecasting models. Monte Carlo simulation can be used to tackle a range of problems in virtually every field such as finance, engineering, supply chain, and science.

When faced with significant uncertainty in the process of making a forecast or estimation, rather than just replacing the uncertain variable with a single average number, the Monte Carlo Simulation might prove to be a better solution. Since business and finance are plagued by random variables, Monte Carlo simulations have a vast array of potential applications in these fields. Analysts use them to assess the risk that an entity will default and to analyze derivatives such as options.

Insurers and oil well drillers also use them. Monte Carlo simulations have countless applications outside of business and finance, such as in meteorology, astronomy and particle physics. The technique was first developed by Stanislaw Ulam, a mathematician who worked on the Manhattan Project. He became interested in plotting the outcome of each of these games in order to observe their distribution and determine the probability of winning.

After he shared his idea with John Von Neumann, the two collaborated to develop the Monte Carlo simulation. One way to employ a Monte Carlo simulation is to model possible movements of asset prices using Excel or a similar program. By analyzing historical price data, you can determine the drift, standard deviationvarianceand average price movement for a security. These are the building blocks of a Monte Carlo simulation. To project one possible price trajectory, use the historical price data of the asset to generate a series of periodic daily returns using the natural logarithm note that this equation differs from the usual percentage change formula :.

P, and VAR. P functions on the entire resulting series to obtain the average daily return, standard deviation, and variance inputs, respectively. The drift is equal to:. Alternatively, drift can be set to 0; this choice reflects a certain theoretical orientation, but the difference will not be huge, at least for shorter time frames.

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Repeat this calculation the desired number of times each repetition represents one day to obtain a simulation of future price movement. By generating an arbitrary number of simulations, you can assess the probability that a security's price will follow given trajectory.

The frequencies of different outcomes generated by this simulation will form a normal distributionthat is, a bell curve. The most likely return is at the middle of the curve, meaning there is an equal chance that the actual return will be higher or lower than that value. Crucially, Monte Carlo simulations ignore everything that is not built into the price movement macro trendscompany leadership, hype, cyclical factors ; in other words, they assume perfectly efficient markets.

For example, the fact that Time Warner lowered its guidance for the year on November 4 is not reflected here, except in the price movement for that day, the last value in the data; if that fact were accounted for, the bulk of simulations would probably not predict a modest rise in price.Risk analysis is part of every decision we make. We are constantly faced with uncertainty, ambiguity, and variability.

Monte Carlo simulation also known as the Monte Carlo Method lets you see all the possible outcomes of your decisions and assess the impact of risk, allowing for better decision making under uncertainty. Monte Carlo simulation is a computerized mathematical technique that allows people to account for risk in quantitative analysis and decision making.

Monte Carlo simulation furnishes the decision-maker with a range of possible outcomes and the probabilities they will occur for any choice of action. It shows the extreme possibilities—the outcomes of going for broke and for the most conservative decision—along with all possible consequences for middle-of-the-road decisions. 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.

Monte Carlo simulation performs risk analysis by building models of possible results by substituting a range of values—a probability distribution—for any factor that has inherent uncertainty. It then calculates results over and over, each time using a different set of random values from the probability functions. Depending upon the number of uncertainties and the ranges specified for them, a Monte Carlo simulation could involve thousands or tens of thousands of recalculations before it is complete.

Monte Carlo simulation produces distributions of possible outcome values. By using probability distributions, variables can have different probabilities of different outcomes occurring. Probability distributions are a much more realistic way of describing uncertainty in variables of a risk analysis.

Values in the middle near the mean are most likely to occur. Examples of variables described by normal distributions include inflation rates and energy prices.

Values are positively skewed, not symmetric like a normal distribution. Examples of variables described by lognormal distributions include real estate property values, stock prices, and oil reserves. All values have an equal chance of occurring, and the user simply defines the minimum and maximum. Examples of variables that could be uniformly distributed include manufacturing costs or future sales revenues for a new product.

The user defines the minimum, most likely, and maximum values. Values around the most likely are more likely to occur. Variables that could be described by a triangular distribution include past sales history per unit of time and inventory levels.

The user defines the minimum, most likely, and maximum values, just like the triangular distribution. However values between the most likely and extremes are more likely to occur than the triangular; that is, the extremes are not as emphasized. An example of the use of a PERT distribution is to describe the duration of a task in a project management model.

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The user defines specific values that may occur and the likelihood of each. During a Monte Carlo simulation, values are sampled at random from the input probability distributions. Each set of samples is called an iteration, and the resulting outcome from that sample is recorded. Monte Carlo simulation does this hundreds or thousands of times, and the result is a probability distribution of possible outcomes. In this way, Monte Carlo simulation provides a much more comprehensive view of what may happen. It tells you not only what could happen, but how likely it is to happen. An enhancement to Monte Carlo simulation is the use of Latin Hypercube sampling, which samples more accurately from the entire range of distribution functions.

The advent of spreadsheet applications for personal computers provided an opportunity for professionals to use Monte Carlo simulation in everyday analysis work. First introduced for Lotus for DOS inRISK has a long-established reputation for computational accuracy, modeling flexibility, and ease of use.

What is Monte Carlo Simulation? How Monte Carlo Simulation Works Monte Carlo simulation performs risk analysis by building models of possible results by substituting a range of values—a probability distribution—for any factor that has inherent uncertainty.A Monte Carlo simulation can be developed using Microsoft Excel and a game of dice.

Today, it is widely used and plays a key part in various fields such as finance, physics, chemistry, and economics. The term Monte Carlo refers the administrative area of Monaco popularly known as a place where European elites gamble.

Using the simulation can help provide solutions for situations that prove uncertain. It can also be used to understand how risk works, and to comprehend the uncertainty in forecasting models.

As noted above, the simulation is often used in many different disciplines including finance, science, engineering, and supply chain management —especially in cases where there are far too many random variables in play. For example, analysts may use Monte Carlo simulations in order to evaluate derivatives including options or to determine risks including the likelihood that a company may default on its debts. Here's how the dice game rolls:. It is also recommended to use a data table to generate the results.

Moreover, 5, results are needed to prepare the Monte Carlo simulation. First, we develop a range of data with the results of each of the three dice for 50 rolls. Then, we need to develop a range of data to identify the possible outcomes for the first round and subsequent rounds. There is a three-column data range. In the first column, we have the numbers one to In the second column, the possible conclusions after the first round are included.

As stated in the initial statement, either the player wins Win or loses Loseor they replay Re-rolldepending on the result the total of three dice rolls. In the third column, the possible conclusions to subsequent rounds are registered. We can achieve these results using the "IF" function. In this step, we identify the outcome of the 50 dice rolls. The first conclusion can be obtained with an index function.

This function searches the possible results of the first round, the conclusion corresponding to the result obtained.

### Monte Carlo Simulation Definition

For example, when we roll a six, we play again. One can get the findings of other dice rolls, using an "OR" function and an index function nested in an "IF" function. Now, we determine the number of dice rolls required before losing or winning. We develop a range to track the results of different simulations. To do this, we will create three columns.

In the first column, one of the figures included is 5, In the second column, we will look for the result after 50 dice rolls. In the third column, the title of the column, we will look for the number of dice rolls before obtaining the final status win or lose. In fact, one could choose any empty cell. We can finally calculate the probabilities of winning and losing.

We finally see that the probability of getting a Win outcome is National Center for Biotechnology Information. Tools for Fundamental Analysis. Risk Management. Financial Analysis. Portfolio Management. Retirement Planning.