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".

Examples
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 Investopedia.com
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 Investopedia.com
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.

Conclusion
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. Investing Basics

    5 Common Mistakes Young Investors Make

    Missteps are common whenever you’re learning something new. But in investing, missteps can have serious financial consequences.
  2. Fundamental Analysis

    5 Basic Financial Ratios And What They Reveal

    Understanding financial ratios can help investors pick strong stocks and build wealth. Here are five to know.
  3. Options & Futures

    4 Equity Derivatives And How They Work

    Equity derivatives offer retail investors opportunities to benefit from an underlying security without owning the security itself.
  4. Mutual Funds & ETFs

    The 4 Best American Funds for Growth Investors in 2016

    Discover four excellent growth funds from American Funds, one of the country's premier mutual fund families with a history of consistent returns.
  5. Products and Investments

    A Guide to DIY Portfolio Management

    These are some of the pillars needed to build a DIY portfolio.
  6. Investing

    What Investors Need to Know About Returns in 2016

    Last year wasn’t a great one for investors seeking solid returns, so here are three things we believe all investors need to know about returns in 2016.
  7. Options & Futures

    Five Advantages of Futures Over Options

    Futures have a number of advantages over options such as fixed upfront trading costs, lack of time decay and liquidity.
  8. Mutual Funds & ETFs

    The Top 5 Buffalo Funds for Retirement Diversification in 2016

    Discover the top five Buffalo Funds for retirement diversification in 2016, with a summary of each fund, including manager and performance information.
  9. Term

    What is Pegging?

    Pegging refers to the practice of fixing one country's currency to that of another country. It also describes a practice in which investors avoid purchasing security shares underlying a put option.
  10. Home & Auto

    Understanding Pre-Qualification Vs. Pre-Approval

    Contrary to popular belief, being pre-qualified for a mortgage doesn’t mean you’re pre-approved for a home loan.
RELATED FAQS
  1. What is a derivative?

    A derivative is a contract between two or more parties whose value is based on an agreed-upon underlying financial asset, ... Read Full Answer >>
  2. What is finance?

    "Finance" is a broad term that describes two related activities: the study of how money is managed and the actual process ... Read Full Answer >>
  3. What is after-hours trading? Am I able to trade at this time?

    After-hours trading (AHT) refers to the buying and selling of securities on major exchanges outside of specified regular ... Read Full Answer >>
  4. What is the difference between positive and normative economics?

    Positive economics is objective and fact based, while normative economics is subjective and value based. Positive economic ... Read Full Answer >>
  5. Does mutual fund manager tenure matter?

    Mutual fund investors have numerous items to consider when selecting a fund, including investment style, sector focus, operating ... Read Full Answer >>
  6. Why do financial advisors dislike target-date funds?

    Financial advisors dislike target-date funds because these funds tend to charge high fees and have limited histories. It ... Read Full Answer >>
Hot Definitions
  1. Discouraged Worker

    A person who is eligible for employment and is able to work, but is currently unemployed and has not attempted to find employment ...
  2. Ponzimonium

    After Bernard Madoff's $65 billion Ponzi scheme was revealed, many new (smaller-scale) Ponzi schemers became exposed. Ponzimonium ...
  3. Quarterly Earnings Report

    A quarterly filing made by public companies to report their performance. Included in earnings reports are items such as net ...
  4. Dark Pool Liquidity

    The trading volume created by institutional orders that are unavailable to the public. The bulk of dark pool liquidity is ...
  5. Godfather Offer

    An irrefutable takeover offer made to a target company by an acquiring company. Typically, the acquisition price's premium ...
Trading Center