FRM Part 1 - Book 2 - Quantitative Analysis

After completing this reading, you should be able to:

• Describe an event and an event space.

• Describe independent events and mutually exclusive events.

• Explain the difference between independent events and conditionally independent events.

• Calculate the probability of an event for a discrete probability function.

• Define and calculate a conditional probability.

• Distinguish between conditional and unconditional probabilities.

• Explain and apply Bayes’ rule.

After completing this reading, you should be able to:

• Describe and distinguish a probability mass function from a cumulative distribution function and explain the relationship between these two.

• Understand and apply the concept of a mathematical expectation of a random variable.

• Describe the four common population moments.

• Explain the differences between a probability mass function and a probability density function.

• Characterize the quantile function and quantile-based estimators.

• Explain the effect of a linear transformation of a random variable on the mean, variance, standard deviation, skewness, kurtosis, median, and interquartile range.

After completing this reading, you should be able to:

• Distinguish the key properties among the following distributions: uniform distribution, Bernoulli distribution, Binomial distribution, Poisson distribution, normal distribution, lognormal distribution, Chi-squared distribution, Student’s t, and F-distributions, and identify common occurrences of each distribution.

• Describe a mixture distribution and explain the creation and characteristics of mixture distributions.

After completing this reading you should be able to:

• Explain how a probability matrix can be used to express a probability mass function (PMF).

• Compute the marginal and conditional distributions of a discrete bivariate random variable.

• Explain how the expectation of a function is computed for a bivariate discrete random variable.

• Define covariance and explain what it measures.

• Explain the relationship between the covariance and correlation of two random variables and how these are related to the independence of the two variables.

• Explain the effects of applying linear transformations on the covariance and correlation between two random variables.

• Compute the variance of a weighted sum of two random variables.

• Compute the conditional expectation of a component of a bivariate random variable.

• Describe the features of an iid sequence of random variables.

• Explain how the iid property is helpful in computing the mean and variance of a sum of iid random variables.

After completing this reading, you should be able to:

• Estimate the mean, variance, and standard deviation using sample data.

• Explain the difference between a population moment and a sample moment.

• Distinguish between an estimator and an estimate.

• Describe the bias of an estimator and explain what the bias measures.

• Explain what is meant by the statement that the mean estimator is BLUE.

• Describe the consistency of an estimator and explain the usefulness of this concept.

• Explain how the Law of Large Numbers (LLN) and Central Limit Theorem (CLT) apply to the sample mean.

• Estimate and interpret the skewness and kurtosis of a random variable.

• Use sample data to estimate quantiles, including the median.

• Estimate the mean of two variables and apply the CLT.

• Estimate the covariance and correlation between two random variables.

• Explain how coskewness and cokurtosis are related to skewness and kurtosis.

After completing this reading you should be able to:

• Construct an appropriate null hypothesis and alternative hypothesis and distinguish between the two.

• Construct and apply confidence intervals for one-sided and two-sided hypothesis tests, and interpret the results of hypothesis tests with a specific level of confidence.

• Differentiate between a one-sided and a two-sided test and identify when to use each test.

• Explain the difference between Type I and Type II errors and how these relate to the size and power of a test.

• Understand how a hypothesis test and a confidence interval are related.

• Explain what the p-value of a hypothesis test measures.

• Interpret the results of hypothesis tests with a specific level of confidence.

• Identify the steps to test a hypothesis about the difference between two population means.

• Explain the problem of multiple testing and how it can bias results.

After completing this reading you should be able to:

• Describe the models that can be estimated using linear regression and differentiate them from those which cannot.

• Interpret the results of an OLS regression with a single explanatory variable.

• Describe the key assumptions of OLS parameter estimation.

• Characterize the properties of OLS estimators and their sampling distributions.

• Construct, apply, and interpret hypothesis tests and confidence intervals for a single regression coefficient in a regression.

• Explain the steps needed to perform a hypothesis test in linear regression.

• Describe the relationship between a t-statistic, its p-value, and a confidence interval.

After completing this reading you should be able to:

• Distinguish between the relative assumptions of single and multiple regression.

• Interpret regression coefficients in a multiple regression.

• Interpret goodness of fit measures for single and multiple regressions, including R2 and adjusted R2.

• Construct, apply, and interpret joint hypothesis tests and confidence intervals for multiple coefficients in regression.

After completing this reading you should be able to:

• Explain how to test whether regression is affected by heteroskedasticity.

• Describe approaches to using heteroskedastic data.

• Characterize multicollinearity and its consequences; distinguish between multicollinearity and perfect collinearity.

• Describe the consequences of excluding a relevant explanatory variable from a model and contrast those with the consequences of including an irrelevant regressor.

• Explain two model selection procedures and how these relate to the bias-variance tradeoff.

• Describe the various methods of visualizing residuals and their relative strengths.

• Describe methods for identifying outliers and their impact.

• Determine the conditions under which OLS is the best linear unbiased estimator.

After completing this reading you should be able to:

• Describe the requirements for a series to be covariance stationary.

• Define the autocovariance function and the autocorrelation function.

• Define white noise; describe independent white noise and normal (Gaussian) white noise.

• Define and describe the properties of autoregressive (AR) processes.

• Define and describe the properties of moving average (MA) processes.

• Explain how a lag operator works.

• Explain mean reversion and calculate a mean-reverting level.

• Define and describe the properties of autoregressive moving average (ARMA) processes.

• Describe the application of AR, MA, and ARMA processes.

• Describe sample autocorrelation and partial autocorrelation.

• Describe the Box-Pierce Q-statistic and the Ljung-Box Q statistic.

• Explain how forecasts are generated from ARMA models.

• Describe the role of mean reversion in long-horizon forecasts.

• Explain how seasonality is modeled in a covariance-stationary ARMA.

After completing this reading you should be able to:

• Describe linear and nonlinear time trends.

• Explain how to use regression analysis to model seasonality.

• Describe a random walk and a unit root.

• Explain the challenges of modeling time series containing unit-roots.

• Describe how to test if a time series contains a unit root.

• Explain how to construct an h-step-ahead point forecast for a time series with seasonality.

• Calculate the estimated trend value and form an interval forecast for a time series.

After completing this reading you should be able to:

• Calculate, distinguish, and convert between simple and continuously compounded returns.

• Define and distinguish between volatility, variance rate, and implied volatility.

• Describe how the first two moments may be insufficient to describe non-normal distributions.

• Explain how the Jarque-Bera test is used to determine whether returns are normally distributed.

• Describe the power law and its use for non-normal distributions.

• Define correlation and covariance and differentiate between correlation and dependence.

• Describe properties of correlations between normally distributed variables when using a one-factor model.

After completing this reading you should be able to:

• Describe the basic steps to conduct a Monte Carlo simulation.

• Describe ways to reduce the Monte Carlo sampling error.

• Explain the use of antithetic and control variates in reducing Monte Carlo sampling error.

• Describe the bootstrapping method and its advantage over the Monte Carlo simulation.

• Describe pseudo-random number generation.

• Describe situations where the bootstrapping method is ineffective.

• Describe the disadvantages of the simulation approach to financial problem-solving.

After completing this reading you should be able to:

• Discuss the philosophical and practical differences between machine-learning techniques and classical econometrics.

• Differentiate among unsupervised, supervised, and reinforcement learning models.

• Use principal components analysis to reduce the dimensionality of a set of features.

• Describe how the K-means algorithm separates a sample into clusters.

After completing this reading you should be able to:

• Understand the differences between and consequences of underfitting and overfitting and propose potential remedies for each.

• Explain the differences among the training, validation, and test data sub-samples, and how each is used.

• Explain how reinforcement learning operates and how it is used in decision-making.

• Be aware of natural language processing and how it is used.

After completing this reading you should be able to:

• Explain the role of linear regression and logistic regression in prediction.

• Understand how to encode categorical variables.

• Discuss why regularization is useful and distinguish between the ridge regression and LASSO approaches.

After completing this reading you should be able to:

• Show how a decision tree is constructed and interpreted.

• Describe how ensembles of learners are built.

• Outline the intuition behind the K nearest neighbors and support vector machine methods for classification. - Understand how neural networks are constructed and how their weights are determined.

• Evaluate the predictive performance of logistic regression models and neural network models using a confusion matrix.