# 2021 FRM Learning Objectives – Part 2

## Introduction

In this post, we continue talking about the 2021 Learning Objectives and showing the details of the various segments.

## Quantitative Analysis

This area tests a candidate’s knowledge of basic probability and statistics, regression and time series analysis and various quantitative techniques useful in risk management.

The broad knowledge points covered in Quantitative Analysis include the following:

Discrete and continuous probability distributions

Estimating the parameters of distributions

Population and sample statistics

Bayesian analysis

Statistical inference and hypothesis testing

Measures of correlation

Linear regression with single and multiple regressors

Time series analysis and forecasting

Simulation methods

## Fundamentals of Probability

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.

## Random Variables

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.

## Common Univariate Random Variables

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

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

## Multivariate Random Variables

Explain how a probability matrix can be used to express a probability mass function.

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 independent and identically distributed (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.

## Sample Moments

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.

## Hypothesis Testing

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

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.

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.

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.

## Linear Regression

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

Interpret the results of an ordinary least squares (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 a linear regression.

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

## Regression with Multiple Explanatory Variables

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 a regression.

## Regression Diagnostics

Explain how to test whether a 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 trade-off.

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.

## Stationary Time Series

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.

## Non-Stationary Time Series

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.

## Measuring Returns, Volatility, and Correlation

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.

## Simulation and Bootstrapping

Describe the basic steps to conduct a Monte Carlo simulation.

Describe ways to reduce 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 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.