Determining the probability distribution of an investment is equal to determining the risk inherent in that investment. By comparing the expected return to the expected risk and overlaying that with an investor's risk tolerance, you may be able to make better decisions about whether to invest in a prospective business venture. This article will present some simple examples of various ways to conduct scenario analysis and provide rationale for their use. (To learn more about probability distributions, read

*Find The Right Fit With Probability Distributions*.)

**Historical performance data is required to provide some insight into the variability of an investment's performance and to help investors understand the risk that has been borne by shareholders in the past. By examining periodic return data, an investor can gain insight into an investment's past risk. For example, because variability equates to risk, an investment that provided the same return every year is deemed to be less risky than an investment that provided annual returns that fluctuated between negative and positive. Although both investments may provide the same overall return for a given investment horizon, the periodic returns demonstrate the risk differentials in these investments. (For more insight, read**

Overview

Overview

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*Measure Your Portfolio's Performance*Strict regulations over the calculation and presentation of past returns ensure the comparability of return information across securities, investment managers and funds. However, past performance does not provide any guarantee about an investment's future risk or return. Scenario analysis attempts to understand a venture's potential risk/return profile. By performing an analysis of multiple pro-forma estimates for a given venture and denoting a probability for each scenario, one begins to create a probability distribution (risk profile) for that particular business enterprise.

**Examples**

Scenario analysis can be applied in many ways. The most typical method is to perform multi-factor analysis (models containing multiple variables) in the following ways:

- Creating a Fixed Number of Scenarios
- Determining the High - Low Spread
- Creating Intermediate Scenarios

- Random Factor Analysis
- Numerous to Infinite Number of Scenarios
- Monte Carlo Analysis

Risk analysis is concerned with trying to determine the probability that a future outcome will be something other than the mean value. One way to show variation is to calculate an estimate of the extreme and the least probable outcomes on the positive and negative side of the mean. The simplest method to forecast potential outcomes of an investment or venture is to produce an upside and downside case for each outcome and then to speculate the probability that it will occur. Figure 1 uses a three scenario method evaluating a base case (B) (mean value), upside case (U) and a downside case (D).

Figure 1 |

For example a simple two factor analysis:

Value V= Variable A + Variable B, where each variable value is not constrained.

50% for Value (B) = 200

25% for Value (U) = 300

25% for Value (D) =1 00

When assigning probabilities the sum of the probabilities assigned must equal 100%. By graphing these values and their probabilities we can infer a rather crude probability distribution (the distribution of all calculated values and the probability of those values occurring). By forming the upside and downside cases we begin to get an understanding of other possible return outcomes, but there are many other potential outcomes within the set bounded by the extreme upside and downside already estimated.

Figure 2 presents one method for determining the fixed number of outcomes between the two extremes. Assuming that each variable acts independently, that is, its value is not dependent on the value of any other variable, we can conduct an upside, base and downside case for each variable. In the simplistic two factor model, this type of analysis would result in a total of nine outcomes. A three-factor model using three potential outcomes for each variable would end up with 27 outcomes, and so forth. The equation for determining the total number of outcomes using this method is equal to

**(**Y

^{X}**)**, where Y= the number of possible scenarios for each factor and X= the number of factors in the model. (For more, see

*Modern Portfolio Theory Stats Primer*.)

Figure 2 |

In Figure 2, there are nine outcomes but not nine separate values. For example, the outcome for BB could be equal to the outcome DU or UD. To finalize this study, the analyst would assign the probabilities for each outcome and then add those probabilities for any like values. We would expect that the value corresponding to the mean, in this case being BB would appear the most times since the mean is the value with the highest probability of occurring. The frequency of like values occurring increases the probability of occurrence. The more times values do not repeat, especially the mean value, the higher the probability that future returns will be something other than the mean. The more factors one has in a model and the more factor scenarios one includes, the more potential scenario values are calculated resulting in a robust analysis and insight into the risk of a potential investment.

**Drawbacks of Scenario Analysis**

The major drawback for these types of fixed outcome analyses are the probabilities estimated and the outcome sets bounded by the values for the extreme positive and negative events. Although they may be low probability events, most investments, or portfolios of investments, have the potential for very high positive and negative returns. Investors must remember that although they don't happen often,these low probability events do happen and it is risk analysis that helps determine whether these potential events are within an investor's risk tolerance. (For related reading, see

*Personalizing Risk Tolerance*and

*Risk Tolerance Only Tells Half The Story*.)

A method to circumvent the problems inherent in the previous examples is to run an extreme number of trials of a multivariate model. Random factor analysis is completed by running thousands and even hundreds of thousands of independent trials with a computer to assign values to the factors in a random fashion. The most common type of random factor analysis is called "Monte Carlo" analysis, where factor values are not estimated but are chosen randomly from a set bounded by the variables own probability distribution. (To learn more about this analysis, read

*Introduction To Monte Carlo Simulation*.)

**Conclusion**

Standards set for reporting investment performance ensure that investors are provided with the risk profile (variability of performance) for past performance of investments. Because past performance does not have any bearing on future risk or return, it is up to the investor or business owners to determine the future risk of their investments by creating pro-forma models. The output of any forecast will only produce the expected or mean value of that initiative; the outcome that the analyst believes has the highest probability of occurrence. By conducting scenario analysis an investor can produce a risk profile for a forecasted investment and create a basis for comparing prospective investments.