## Statistics and Risk Measures

Risk Solver provides a wide range of statistics and risk measures, computed for each output of interest across the trials of a Monte Carlo simulation.  You can use measures of central tendency, measures of variation or dispersion, and quantile measures, to assess almost any output in your model.

#### Measures of Central Tendency

Risk Solver provides several measures of central tendency:

• PsiMean, the average of all the values
• PsiPercentile for the median or 50th percentile
• PsiMode, the most frequently occurring single value

#### Measures of Variation

Risk Solver provides several standard measures of variation:

• PsiVariance, which describes the spread of the distribution of values
• PsiStdDev for standard deviation, the square root of variance
• PsiSkewness, which describes the asymmetry of the distribution of values
• PsiKurtosis, which describes the peakedness of the distribution of values
• PsiMin, PsiMax, and PsiRange for the minimum and maximum values, and the difference between them

#### Risk Measures

Risk Solver also provides several risk measure functions that are most often used in quantitative finance applications, but may be used in any model:

• PsiAbsDev for 'MAD', which measures absolute deviations from the mean
• PsiSemiVar for semivariance or lower partial moment, which measures and weights negative deviations from the mean
• PsiSemiDev for semideviation, the square root of semivariance (qth root for the lower partial moment)

PsiSemiVar and PsiSemiDev are useful in situations where 'upward' variation -- for example, higher stock prices or increased profits -- is desirable, but 'downward' variation -- lower prices or losses -- is undesirable.

#### Quantile Measures

To get a complete grasp of the range of outcomes, it's essential to look at quantile measures, such as percentiles and Value at Risk, in addition to measures of central tendency and variation. Quantile measures allow you to answer questions such as 'How much money might we lose, with 5% or 10% probability?' or 'What is the probability that we'll make at least \$100,000?' based on your simulation model. Risk Solver provides:

• PsiPercentile, which provides percentile values from 1% to 99%
• PsiTarget, which returns the proportion of values less than or equal to a target value
• PsiBVaR, which measures standard ('Basel') Value at Risk
• PsiCVaR, which measures Conditional Value at Risk

#### Confidence Intervals

Every Monte Carlo simulation uses a sample of the possible values of your uncertain variables; hence any statistic resulting from the simulation involves some degree of sampling error. For the mean and standard deviation of an output value, Risk Solver provides functions that help you assess this error, and estimate the range or interval in which you can be confident that the true mean or standard deviation lies, at a confidence level that you specify:

• PsiMeanCI, which returns a confidence interval for the mean
• PsiStdDevCI, which returns a confidence interval for the standard deviation
• PsiCITrials, which returns the number of simulation trials needed to obtain a confidence interval of a given size, at a given confidence level

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