Research analysts use multivariate models to forecast investment outcomes to understand the possibilities surrounding their investment exposures and to better mitigate risks. Monte Carlo analysis is one specific multivariate modeling technique that allows researchers to run multiple trials and define all potential outcomes of an event or investment. Running a Monte Carlo model creates a probability distribution or risk assessment for a given investment or event under review. By comparing results against risk tolerances, managers can decide whether to proceed with certain investments or projects. (To learn more about Monte Carlo basics, see Introduction To Monte Carlo Simulation and Monte Carlo Simulation With GBM.)

Multivariate Models
Multivariate models can be thought of as complex, "What if?" scenarios. By changing the value of multiple variables, the modeler can ascertain his or her impact on the estimate being evaluated. These models are used by financial analysts to estimate cash flows and new product ideas. Portfolio managers and financial advisors use these models to determine the impact of investments on portfolio performance and risk. Insurance companies use these models 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
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 is up to the analyst to determine the set of outcomes and the probability that they will occur. In Monte Carlo modeling, the analyst runs multiple trials (often thousands) to determine all the possible outcomes and the probability that they will take place.

Monte Carlo analysis is useful for analysts 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 return 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 actually 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 or insure against a negative occurrence if the drivers and probabilities of these events are not calculated in advance. (To learn more about how to manage the risk in your portfolio, see our Risk and Diversification tutorial.)

Creating the Model
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 probability of occurrence, an analyst creates a distribution that includes all the possible outcomes and the probability that they will occur. There are many random number generators in the marketplace. The two most common tools for designing and executing Monte Carlo models are @Risk and Crystal Ball. 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. (To learn more, read The Importance of Diversification.)

In order to choose the correct distribution for a variable, one must understand each of the possible distributions available. For example, the most common one is a normal distribution, also known as a bell curve. In a normal distribution, all the occurrences are equally distributed (symmetrical) around the mean. The mean is the most probable event. Natural phenomena, people's heights and inflation are some examples of inputs that are normally distributed.

In the Monte Carlo analysis, a random-number generator picks a random value for each variable (within the constraints set by the model) and produces a probability distribution for all possible outcomes. The standard deviation of that probability is a statistic that denotes the likelihood that the actual outcome being estimated will be something other than the mean or most probable event. Assuming a probability distribution is normally distributed, approximately
68% of the values will fall within one standard deviation of the mean, about 95% of the values will fall within two standard deviations and about 99.7 % will lie within three standard deviations of the mean. This is known as the "68-95-99.7 rule" or the "empirical rule".

Let us take for example two separate, normally distributed probability distributions derived from random-factor analysis or from multiple scenarios of a Monte Carlo model.

Copyright © 2008
Figure 1

In both of the probability distributions (Figure 1), the expected value or base cases both equal 200. Without having performed scenario analysis, there would be no way to compare these two estimates and one could mistakenly conclude that they were equally beneficial. (To learn more, read Scenario Analysis Provides Glimpse of Portfolio Potential.)

In the two probability distributions, both have the same mean but one has a standard deviation of 100, while the other has a standard deviation of 200. This means that in the first scenario analysis there is a 68% chance that the outcome will be some number between 100 and 300, while in the second model there is a 68% chance that the outcome will be between 0 and 400. With all things being equal, the one with a standard deviation of 100 has the better risk-adjusted outcome. Here, by using Monte Carlo to derive the probability distributions, the analysis has given an investor a basis by which to compare the two initiatives.

Monte Carlo analysis can also help determine whether certain initiatives should be taken on by looking at the risk and return consequences of taking certain actions. Let us assume we want to place debt on our original investment.

Copyright © 2008
Figure 2

The distributions in Figure 2 show the original outcome and the outcome after modeling the effects of leverage. Our new leveraged analysis shows an increase in the expected value from 200 to 400, but with an increased financial risk of debt. Debt has increased the expected value by 200 but also the standard deviation. Before 1 standard deviation was a range from 100 to 300. Now with debt, 68% of values (1 standard deviation) fall between 0 and 400. By using scenario analysis an investor can now determine whether the additional increase in return equals or outweighs the additional risk (variability of potential outcomes) that comes with taking on the new initiative.

Monte Carlo analyses are not only conducted by finance professionals but also by many other businesses. It is a decision-making tool that integrates the concept that every decision will have some impact on overall risk. Every individual and institution has different risk/return tolerances. As such, it is important that the risk/return profile of any investment be calculated and compared to risk tolerances.

The probability distributions produced by a Monte Carlo model create a picture of risk. A picture is an easy way to convey the idea to others, such as superiors or prospective investors. Because of advances in software, very complex Monte Carlo models can be designed and executed by anyone with access to a personal computer.

Related Articles
  1. Mutual Funds & ETFs

    Top 3 Muni California Mutual Funds

    Discover analyses of the top three California municipal bond mutual funds, and learn about their characteristics, historical performance and suitability.
  2. Investing Basics

    What Does Plain Vanilla Mean?

    Plain vanilla is a term used in investing to describe the most basic types of financial instruments.
  3. Investing Basics

    What Does In Specie Mean?

    In specie describes the distribution of an asset in its physical form instead of cash.
  4. Economics

    Calculating Cross Elasticity of Demand

    Cross elasticity of demand measures the quantity demanded of one good in response to a change in price of another.
  5. Professionals

    How to Sell Mutual Funds to Your Clients

    Learn about the various talking points you should cover when discussing mutual funds with clients and how explaining their benefits can help you close the sale.
  6. Professionals

    Fund and ETF Strategies for Volatile Markets

    Looking for short-term fixes in reaction to market volatility? Here are a few strategies — and their downsides.
  7. Investing

    How Diversifying Can Help You Manage Market Mayhem

    The recent market volatility, while not unexpected, has certainly been hard for any investor to digest.
  8. Fundamental Analysis

    Emerging Markets: Analyzing Colombia's GDP

    With a backdrop of armed rebels and drug cartels, the journey for the Colombian economy has been anything but easy.
  9. Investing

    How to Win More by Losing Less in Today’s Markets

    The further you fall, the harder it is to climb back up. It’s a universal truth that is painfully apparent in the investing world.
  10. Options & Futures

    Pick 401(k) Assets Like A Pro

    Professionals choose the options available to you in your plan, making your decisions easier.
  1. What licenses does a hedge fund manager need to have?

    A hedge fund manager does not necessarily need any specific license to operate a fund, but depending on the type of investments ... Read Full Answer >>
  2. Can mutual funds invest in hedge funds?

    Mutual funds are legally allowed to invest in hedge funds. However, hedge funds and mutual funds have striking differences ... Read Full Answer >>
  3. When are mutual funds considered a bad investment?

    Mutual funds are considered a bad investment when investors consider certain negative factors to be important, such as high ... Read Full Answer >>
  4. What fees do financial advisors charge?

    Financial advisors who operate as fee-only planners charge a percentage, usually 1 to 2%, of a client's net assets. For a ... Read Full Answer >>
  5. Can mutual funds invest in options and futures?

    Mutual funds invest in not only stocks and fixed-income securities but also options and futures. There exists a separate ... Read Full Answer >>
  6. What does a high turnover ratio signify for an investment fund?

    If an investment fund has a high turnover ratio, it indicates it replaces most or all of its holdings over a one-year period. ... Read Full Answer >>

You May Also Like

Hot Definitions
  1. Zero-Sum Game

    A situation in which one person’s gain is equivalent to another’s loss, so that the net change in wealth or benefit is zero. ...
  2. Capitalization Rate

    The rate of return on a real estate investment property based on the income that the property is expected to generate.
  3. Gross Profit

    A company's total revenue (equivalent to total sales) minus the cost of goods sold. Gross profit is the profit a company ...
  4. Revenue

    The amount of money that a company actually receives during a specific period, including discounts and deductions for returned ...
  5. Normal Profit

    An economic condition occurring when the difference between a firm’s total revenue and total cost is equal to zero.
  6. Operating Cost

    Expenses associated with the maintenance and administration of a business on a day-to-day basis.
Trading Center
You are using adblocking software

Want access to all of Investopedia? Add us to your “whitelist”
so you'll never miss a feature!