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Filled-In CIMA Certification Examination Preparation Checklist -- DRAFT (November 20, 2023) Certification Exam Detailed Content Outline % of Exam Notes Formulas Examples I. FUNDAMENTALS 15% A. Statistics and Methods 5% 1. Basic statistical measures Mean Mean is a measure of central tendency. It is the arithmetic average of a sample or population, or the expected value where each possible value if weighted by the respective probability of occurrence. Median Median is a measure of central tendency. It is the value at the 50th percentile (midpoint) of a sample or population. Mode Mode is a measure of central tendency. It is the value that occurs most frequently in a sample or population. Variance Variance is a measure of dispersion or spread of a sample, population, or set of probabilities. It is standard deviation squared. It cannot be negative. Standard deviation Standard deviation is a measure of dispersion or spread of a sample, population, or set of probabilities. It is the square root of variance. The empirical rule says that, for mound- shaped data, approximately 68% of observations are within +/- 1 standard deviation of the mean, approximately 95% of observations are within +/- 2 standard deviations of the mean, and approximately 99% of observations are within +/- 3 standard deviations of the mean. A z- score tells us how many standard deviations an observation is from the mean. Covariance Covariance is a measure of how variables move together. It is just a directional measure (a positive value means that things move in the same direction and a negative value means that things move in opposite directions). SSSSSSSSSSSS: RR = ∑ii=11 nn RRii nn PPPP PP PP PP : μμ = ∑ii=11 nn RRii nn EEEE EE EE: EE(RR) = � ii=11 nn PPiiRRii SSSSSSSSSSSS: ss22 = ∑ii=11 nn RRii − RR 22 nn − 11 Population: σσ22 = ∑ii=11 nn RRii − μμ 22 nn Expected: VV VV RR = � ii=11 nn PPii(RRii−EE(RR))22 SSSSSSSSSSSS: ss = ∑ii=11 nn RRii − RR 22 nn − 11 = ss22 PPPP PP PP PP : σσ = ∑ii=11 nn RRii − μμ 22 nn = σσ22 EEEE EE EE: SS(RR) = � ii=11 nn PPii(RRii−EE(RR))22 = VV VV(RR) z−score = RRii − μμ σσ SSSSSSSSSSSS: ssAAAA = ∑ii=11 nn RRAA,ii − RRAA RR ,ii − RR nn − 11 PPPP PP PP PP : σσAAAA = ∑ii=11 nn RRAA,ii − μμAA RR ,ii − μμ nn EEEE EE EE:CC CC RRAA,RR = � ii=11 nn PPii[RRAA,ii − EE RRAA ][RR ,ii − EE RR ] Chicago Booth CIMA Education Program Filled-In CIMA Certification Exam Preparation Checklist -- DRAFT (Nov 20, 2023) Page 1
Certification Exam Detailed Content Outline % of Exam Notes Formulas Examples Correlation Correlation is a measure of how variables move together. Its values range between -1 and +1. It indicates both the direction and the strength of the relationship between variables, but captures only linear relationships, as the examples illustrate. For instance, in the above line, correlation between the returns of A and B is always 1 because the returns to B are always half those of A (one is a linear function of the other). Skewness Skewness is a departure from the normal distribution. A left or negatively skewed distribution is one where mean < median < mode. A right or positively skewed distribution is one where mean > median > mode. Stock returns are positively skewed due to survivorship bias. Standard deviation underestimates risk for negatively skewed distributions and overestimates risk for positively skewed distributions. Kurtosis Kurtosis of a normal distribution = 3. A distribution with positive kurtosis has kurtosis > 3 (i.e., excess kurtosis > 0), and one with negative kurtosis has kurtosis < 3 (i.e., excess kurtosis < 0). Kurtosis helps assess the risk of extreme outcomes for an investment. A distribution with positive excess kurtosis has more of its observations in the tails (i.e., "fat tails") and fewer in the center in comparison to a normal distribution, meaning that the probability of an extreme outcome is higher than in a normal distribution. A distribution with negative excess kurtosis has fewer of its observations in the tails and more in the center in comparison to a normal distribution, meaning that the probability of an extreme outcome is lower than in a normal distribution. Range Range is the difference between the largest and smallest value in a sample or population. Absolute deviation Average absolute deviation is the average of the absolute values of differences between each value and a measure of central tendency. The measures of central tendency used can include the mean, median, or mode. Mean absolute deviation and median absolute deviation are both abbreviated as "MAD". Semi-variance Semi-variance, like variance, is a measure of dispersion or spread of a sample, population, or set of probabilities. As opposed to variance, the semi-variance formula only includes observations below the mean or other target value (such as MAR or minimum acceptable return). Semi-deviation Semi-deviation is also known as "downside deviation". Like standard deviation, it is a measure of dispersion or spread of a sample, population, or set of probabilities. As opposed to standard deviation, the semi-deviation formula only includes observations below the mean or other target value (such as MAR or minimum acceptable return). It is used in calculating the Sortino ratio. Coefficient of variation Coefficient of variation is the ratio of standard deviation to the mean. SSSSSSSSSSSS: AAAA = ssAAAA ssAAss PPPP PP PP PP : ρρAAAA = σσAAAA σσAAσσ EEEE EE EE:CC RRAA,RR = CC CC RRAA,RR SS RRAA SS(RR ) DDDD = ∑ii=11 nn RRAAii − MM ,00 22 nn − 11 ∑ii=11 nn RRAAii − MM ,00 22 nn − 11 ∑ii=11 nn RRii − mm nn , wwww mm ii oo tt ss CC = σσ μμ ss RR SSSS SS SS oo = SS = σσ22 μμ22 Chicago Booth CIMA Education Program Filled-In CIMA Certification Exam Preparation Checklist -- DRAFT (Nov 20, 2023) Page 2
Certification Exam Detailed Content Outline % of Exam Notes Formulas Examples 2. Basic statistical concepts Normal distribution Normal distributions are symmetrical probability distributions, in which mean = median = mode, and the empirical rule applies. The empirical rule says that, for mound-shaped data, approximately 68% of observations are within +/- 1 standard deviation of the mean, approximately 95% of observations are within +/- 2 standard deviations of the mean, and approximately 99% of observations are within +/- 3 standard deviations of the mean. Probability Probability is the likelihood that a scenario will occur. In several examples for "Basic statistical measures" above we have denoted the probability of each possible value occurring by Pi . Probabilities always sum to 100%, as the examples also show. Expected value Expected value is a probability-weighted average. The example for "Mean" above illustrates the calculation of an Expected Value. Sampling from a population Sampling from a population involves taking random samples of observations from a population. With a large sample, the sample mean approximates the population mean (or expected value), and the sample standard deviation approximates the population standard deviation. The Central Limit Theorem states that with several large and independent samples, the average of those samples approximately follows a normal distribution with mean equal to the population mean and standard deviation proportional to the population standard deviation. In the examples for "Basic statistical measures" we showed both the population (with 7 possible values) and a sample containing 5 observations. It illustrates how the sample mean approximates the population mean (or expected value), although this is not perfect due to sampling error. Significance testing Significance testing checks for whether we can invalidate the null hypothesis with a given level of confidence (denoted α in the example), using t-statistics and p-values. There are one- tailed and two-tailed hypothesis tests. For example, say we have a sample from a normal distribution and want to test whether we can invalidate the null hypothesis that the population mean is bigger than some value μ0 at α=5%. Then we can do a one-tailed test (top left graph in the example figure). First we calculate the z-score for the sample mean, assuming the true mean is μ0 and using the sample standard error (this z-score is sometimes also called the t-statistic). If this z-score is smaller than the z-score for 5% (shown in the example), then we can reject the null hypothesis (in other words, the likelihood that we would have gotten this sample mean if the population mean was μ0 - this likelihood is the p-value - is below 5%, so if our confidence cutoff is 5%, we reject any null hypothesis that implies a p-value lower than 5%). A standard error is the standard deviation of a sample divided by the square root of the number of observations n and is a measure of the likelihood that our sample represents the population. Non-normal distribution Non-normal distributions include distributions that are negatively skewed and positively skewed, distributions with negative and positive excess kurtosis, and uniform distributions. See the examples for kurtosis and skewness above. Decision trees Decision trees provide visual representations of sets of branching scenarios. A tree splits into different branches based on decisions points or event probabilities, with each branch leading to a possible outcome (e.g., if A, then B, else C). Decision trees are similar to binomial distribution pricing models. 3. Interpretation of potential investment outcomes of statistical results from probabilistic models μμ = ∑ii=11 nn XXii nn ss = ∑ii=11 nn XXii − μμ 22 nn − 11 = ss22 SS = ss nn z−score (left−tail) = − μμ − μμ00 SS z−score (right−tail) = μμ − μμ00 SS z−score (two−tails) = |μμ − μμ00| SS Chicago Booth CIMA Education Program Filled-In CIMA Certification Exam Preparation Checklist -- DRAFT (Nov 20, 2023) Page 3
Certification Exam Detailed Content Outline % of Exam Notes Formulas Examples Monte Carlo simulation Monte Carlo simulation calculates multiple (e.g., hundreds of thousands of) iterations of scenarios with slightly different inputs to show a range of possible estimated outcomes. An advantage of Monte Carlo is that it generates lots of data points, while a disadvantage is that the usefulness of those data points is highly dependent on the inputs to the model (i.e., garbage in, garbage out). Data tables and graphs You need to be able to read and interpret various kinds of data tables and graphs. Correlation analysis Definition Correlation analysis is a statistical technique for assessing the direction and strength of the relationship between variables. It can be evaluated visually, using scatterplots, and quantitatively, by calculating a correlation coefficient. Scatterplots Depicts the relationship between observations of two variables (e.g. returns of stocks x and y each day of a year) in a plot: each observation is a dot positioned such that the x-axis shows the value of that observation for variable x and the y-axis shows the value of that observation for variable y. Correlation coefficient Correlation coefficient is a statistical metric that captures the value of a correlation between variables, ranging from -1 to +1. Correlation is a measure of how variables move together. It indicates both the direction and the strength of the linear relationship between variables. A positive value means that the variables move in the same direction, while a negative value means that the variables move in opposite directions. A high absolute value means that variables have a strong linear relationship with each other, while a low absolute value means that variables have a weak linear relationship with each other. See example above in "Basic statistical measures". Linear regression Assumptions of the linear regression model Linear regression is a regression model that assumes a linear functional form, fixed independent variables, independent observations, a representative sample and proper specification of the model (no omitted variables), normality of the residuals or errors, equality of variance of the errors (homogeneity of residual variance), no multicollinearity, no autocorrelation of the errors, and no outlier distortion. Independent and dependent variables Dependent variables are something that one tries to explain or predict as a function of other available data. Independent variables are the factors in your data that may be affecting the dependent variable. Regression analysis evaluates the correlation between the dependent variable and the independent variables, as well as the confidence with which one can assert that a correlation exists. Standard error Standard error is a measure of dispersion or spread of a sample (but not of an entire population). Standard error tells us how the degree to which a sample statistic (e.g., sample mean) differs from the actual population parameter (e.g., population mean). Standard errors are smaller for larger sample sizes, and they are larger for smaller sample sizes. See example above for "Significance testing" and the example below for "Hypothesis testing" for how Standard errors are used in practice. 4. Correlation, regression, and multiple regression concepts, methods, and interpretation SS = ss nn SSSSSSSSSSSS: AAAA = ssAAAA ssAAss PPPP PP PP PP : ρρAAAA = σσAAAA σσAAσσ EEEE EE EE:CC RRAA,RR = CC CC RRAA,RR SS RRAA SS(RR ) Chicago Booth CIMA Education Program Filled-In CIMA Certification Exam Preparation Checklist -- DRAFT (Nov 20, 2023) Page 4