"Quantitative Analysis" is one of the four broad topics that GARP tests in its FRM Part 1 exam. This topic has a 20% weight in the exam. This means out of a total of 100 questions asked, you may expect 20 questions from this section. 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. There are thirteen chapters or readings in this section. If you go through the GARP specified learning objectives (LOs) for this section, you will find many non-computational LOs. As this section is named "Quantitative Analysis," FRM candidates generally bias computational LOs while preparing for this section. However, be informed that GARP often asks tricky questions on non-computational LOs. Therefore, to score well, non-computational LOs are to be equally emphasized.

Let's go through the essence of each of the thirteen chapters or readings and identify concepts that GARP might test on the exam day.

**Chapter 1: Fundamentals of Probability**

This reading discusses important terms and concepts relating to probability theory. Do pay special attention to the difference between independent and mutually exclusive events, discrete probability functions, and the difference between unconditional and conditional probabilities. Bayes' rule, used to update a given set of prior probabilities, is a hot topic and is tested invariably by GARP almost every time. Make sure you are comfortable with the calculation of conditional probabilities, joint probabilities, and probabilities based on a probability function. Also, have clarity on when and how to apply the Bayes' formula.

**Chapter 2: Random Variables**

This reading covers the concepts of expected value, variance, skewness, and kurtosis including their characteristics and calculations. GARP often tests your ability to distinguish among a probability mass function, a cumulative distribution function, and a probability density function. Also, there could be questions on the computation of expected value and identification of the four common population moments of a statistical distribution.

**Chapter 3: Common Univariate Random Variables**

This reading relates to the following common probability distributions: uniform, Bernoulli, binomial, Poisson, normal, lognormal, chi-squared, Student's t-, F-, exponential, and beta, including their properties, parameters, and common occurrences. From this reading, your focus should be mainly on the binomial, normal, and Student's t-distributions. GARP might test you on how to standardize a normally distributed random variable, how to use a z-table, and how to construct confidence intervals.

**Chapter 4: Multivariate Random Variables**

This reading discusses the dependency of multivariate random variables. GARP might test your ability to explain and calculate the mean and variance for bivariate random variables. The dependence between the components is vital, and you should be thorough with calculating covariance and correlation. The marginal and conditional distributions are used to transform bivariate distributions and offer valuable insights for finance and risk management. GARP might also test your ability to use these distributions to compute a conditional expectation and conditional moments that summarize the conditional distribution of a random variable.

**Chapter 5: Sample Moments**

This reading talks about how to sample moments (mean, variance, skewness, and kurtosis) are used to estimate the true population moments for data generated from independent and identically distributed (i.i.d.) random variables. GARP often tests a candidate's ability to estimate these sample moments and explain the differences from population moments. Also, GARP would like you to be thorough in understanding what makes estimators biased, unbiased, and consistent. In addition, you need to have absolute clarity on the law of large numbers (LLN) and the central limit theorem (CLT). Lastly, you might be tested on your ability to contrast the advantages of estimating quantiles to traditional measures of dispersion.

**Chapter 6: Hypothesis Testing**

This reading discusses how risk managers deal with portfolio decisions based on a statistical analysis of samples of investment returns or other random economic and financial variables. Hypothesis testing procedures used to conduct tests concerned with population means and population variances are explained lucidly in this reading. Specific tests discussed include the z-test and the t-test. On the exam day, GARP might test your ability to construct and interpret a confidence interval and determine when and how to apply each of the test statistics in hypothesis testing.

**Chapter 7: Linear Regression**

Linear regression talks about the process of representing relationships with linear equations where there is one dependent variable being explained by one or more independent variables. Typically, a regression equation is estimated using ordinary least squares (OLS), which minimizes the sum of squared errors in the sample data. From this reading, GARP might test your ability to conduct hypothesis tests, calculate confidence intervals, and recall the assumptions underlying the regression model. Also, be thorough in understanding how a regression equation is interpreted.

**Chapter 8. Regression with Multiple Explanatory Variables**

The regression model is generalized to factor in multiple explanatory variables in this reading. On the exam day, you might be tested on your ability to evaluate and calculate goodness-of-fit measures such as R2 and adjusted R2. Also, get yourself conversant with hypothesis testing related to these concepts. Hypothesis testing of individual slope coefficients in a multiple regression model and confidence intervals of those coefficients is also often tested by GARP in the exam.

**Chapter 9. Regression Diagnostics**

This reading covers model-specification issues and the determination of whether the assumptions underlying multiple regression are violated. On the exam day, your ability to explain the effects of heteroskedasticity and multicollinearity on a regression might get tested. Also, your understanding of the bias-variance trade-off and the consequences of including an irrelevant explanatory variable versus excluding a relevant explanatory variable can be tested by GARP.

**Chapter 10: Stationary Time Series**

This reading discusses how to model the cyclical component of a time series using autoregressive (AR), moving average (MA), and autoregressive moving average (ARMA) processes. The time series must be stationary. Otherwise, the past values of a time series would fail to guide its future values. On the exam day, you might get questions on the difference between an AR process and an MA process and how some series can be modelled best with a combination of the two. Also, you might get tested on your understanding of the model evaluation using residual autocorrelations.

**Chapter 11: Non-Stationary Time Series**

This reading addresses non-stationary time series and talks about three sources of non-stationarity namely, time trends, seasonality, and unit-roots (random walks). GARP might test your ability to distinguish among these sources and identify the recommended approach to resolving them on the exam day. Time series with time trends can often be turned into stationary series by estimating and eliminating the trend component. Seasonality is often modelled with dummy variables or by analysing year-on-year changes. Time series with unit roots are generally analysed in terms of their change from the previous period.

**Chapter 12: Measuring Returns, Volatility, and Correlation**

Traditionally, the terms "volatility" and "risk" are often used interchangeably. Thus, the precise volatility estimation is critical to understanding possible risk exposure. On the exam day, GARP might test you on how to calculate simple and continuously compounded returns and recognize differences between definitions of volatility. As financial returns tend to follow non-normal distributions, it is important to understand the properties of these distributions, how to test for these types of distributions, and what the tails look like in these distributions. This reading concludes with discussions on concepts of correlations and dependence and how to test them using various methods statistically.

**Chapter 13: Simulation and Bootstrapping**

Simulation methods are popular in modelling uncertainty by generating random inputs that follow a suitable probability distribution. This reading covers the basic steps for conducting a Monte Carlo simulation and compares this simulation method and the bootstrapping technique. From this reading, GARP might test your ability to explain ways to minimize Monte Carlo sampling error, including the use of antithetic and control variates. Also, you might get tested on your understanding of the pseudo-random number generation method. In addition, there could be questions on the advantages and disadvantages of the bootstrapping technique compared to the traditional Monte Carlo approach.

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