Quantitative Analysis: FRM® Part I
It is 20% of the 2026 Part I exam. 120 learning outcomes across 15 learning modules.
Fundamentals of Probability
- State the meaning of an event and of the event space within which events are defined.
- Characterise mutually exclusive events and independent events.
- Distinguish plain independence of events from conditional independence.
- Use a discrete probability function to find how likely a given event is.
- Define a conditional probability, explain it and work it out.
- Tell conditional probabilities apart from unconditional ones.
- State Bayes' rule and use it to update probabilities.
Random Variables
- Contrast a probability mass function with a cumulative distribution function and show how each can be derived from the other.
- Define the mathematical expectation of a random variable and use it in calculations.
- Outline the four population moments that are commonly used.
- Explain what separates a probability density function from a probability mass function.
- Outline the quantile function along with estimators that are based on quantiles.
- Show how a linear transformation of a random variable alters its median, interquartile range, mean, standard deviation, variance, skewness and kurtosis.
Common Univariate Random Variables
- Show the main properties and uses of the Bernoulli, binomial, Poisson, uniform, normal, lognormal, Chi-squared, Student's t, F, exponential and Beta distributions.
- Build mixture distributions and describe how they arise and what features they have.
Multivariate Random Variables
- Express a probability mass function through a probability matrix and explain how this works.
- From a discrete bivariate random variable, obtain both its marginal and its conditional distributions.
- Describe the way an expected value of a function is obtained for a discrete bivariate random variable.
- Give the definition of covariance and say what quantity it captures.
- Describe how covariance and correlation of two random variables relate to each other and to whether the variables are independent.
- Show, with examples, what happens to the covariance and correlation of two random variables when linear transformations are applied to them.
- Find the variance when two random variables are combined using weights.
- For a bivariate random variable, compute the expected value of one of its components given the other.
- Identify what makes a sequence of random variables independent and identically distributed (iid).
- Illustrate why, for iid random variables, the mean and variance of their sum are easier to calculate.
Sample Moments
- Use sample data to estimate a mean, a variance and a standard deviation.
- Describe how a sample moment differs from a population moment.
- Distinguish an estimator, which is a rule, from an estimate, which is a value it produces.
- Explain what the bias of an estimator is and what it indicates about the estimator.
- Interpret the claim that the estimator of the mean is BLUE.
- Define the consistency of an estimator and describe why the idea is useful.
- Apply the Central Limit Theorem (CLT) and the Law of Large Numbers (LLN) to a sample mean, saying what each result implies.
- Use data to estimate the skewness and kurtosis of a random variable and interpret the results.
- Use sample data to estimate quantiles, the median among them.
- Apply the CLT after estimating the means of two variables from sample data.
- From sample data, calculate estimates of how two random variables covary and how they correlate.
- Relate coskewness and cokurtosis back to the skewness and kurtosis of a single variable.
Hypothesis Testing
- Formulate a suitable null hypothesis and alternative hypothesis and set the two apart.
- Tell one-sided tests from two-sided tests and decide which is appropriate in a given situation.
- Distinguish Type I from Type II errors, and relate each to a test's size and power.
- Explain how confidence intervals and hypothesis tests are connected.
- Describe what a p-value measures and compute it from a test statistic.
- Use confidence intervals in one-sided and two-sided tests, and state what a test's result means at a chosen confidence level.
- Identify the sequence of steps for testing whether two population means differ.
- Explain why running many tests together is problematic and how it can distort results.
Linear Regression
- Identify which models linear regression can estimate and set them apart from those it cannot.
- Read the estimates from an ordinary least squares (OLS) regression of a dependent variable on one explanatory variable and say what they mean.
- Summarise the assumptions on which estimating OLS parameters depends.
- Summarise what OLS estimators are like, covering their sampling distributions as well.
- Construct, apply and interpret confidence intervals and hypothesis tests that concern a single regression coefficient.
- Go through the stages of conducting a hypothesis test in a linear regression.
- Explain how the t-statistic, its p-value and the confidence interval are connected.
- With a single explanatory variable, recover the correlation coefficient from the R2 that the linear regression gives.
Regression with Multiple Explanatory Variables
- Contrast the assumptions made in single regression with those made in multiple regression.
- Explain how the coefficients of a multiple regression should be read.
- Interpret the goodness-of-fit measures R2 and adjusted-R2 for both single and multiple regressions.
- Test several regression coefficients together with joint hypothesis tests, build confidence intervals for them, and interpret the findings.
- Compute the regression R2 from the decomposition of the dependent variable's variation into explained, total and residual sums of squares.
Regression Diagnostics
- Describe how to check whether heteroskedasticity affects a regression.
- Outline the ways of working with heteroskedastic data.
- Describe multicollinearity and set it apart from perfect collinearity.
- Compare the damage done by omitting an explanatory variable that matters with the effects of keeping in a regressor that does not, and illustrate both.
- Explain two procedures for selecting a model and how each connects with the bias-variance trade-off.
- Outline the different ways to visualise residuals and the strengths of each relative to the others.
- Apply methods that identify outliers and describe the effect those outliers have.
- Identify when OLS qualifies as the best linear unbiased estimator by stating the required conditions.
Stationary Time Series
- Outline what a series must satisfy in order to be covariance stationary.
- Give definitions for the autocorrelation function and for the autocovariance function.
- Define white noise and explain how its independent form and its normal (Gaussian) form differ.
- Define autoregressive (AR) processes, describe their properties and compute the series for an AR process from given parameters.
- Define moving average (MA) processes, describe their properties and compute the series for an MA process from given parameters.
- Describe the action of a lag operator, derive a time series process's lag polynomial, and judge stationarity using characteristic equations.
- Explain what mean reversion is and calculate the level a series reverts to.
- Give the definition of autoregressive moving average (ARMA) processes and describe their properties.
- Outline how AR, MA and ARMA processes are applied, and use model selection criteria to pick the time series model that fits best.
- Outline autocorrelation and partial autocorrelation as measured on a sample.
- Outline the Ljung-Box Q statistic and the Box-Pierce Q statistic, including how each is calculated and what its value indicates.
- Describe the way forecasts are produced from ARMA models.
- Outline what part mean reversion plays in forecasts over long horizons.
- Describe how a covariance-stationary ARMA model can represent seasonality.
Non-Stationary Time Series
- Describe time trends that are linear and those that are nonlinear.
- Show how modelling seasonality can be done through regression analysis.
- Outline what a random walk is and what a unit root is.
- Describe the difficulties that arise when modelling time series that contain unit roots.
- Explain how a time series is checked for the presence of a unit root.
- Explain the construction of a point forecast h steps ahead when the series is seasonal.
- Compute the estimated trend value of a time series, and from it construct an interval forecast.
Measuring Returns, Volatility, and Correlation
- Compute simple and continuously compounded returns, distinguish them and convert from one to the other.
- Define volatility, variance rate and implied volatility and set them apart.
- Outline why the first two moments may not be enough to describe distributions that are non-normal.
- Compute the Jarque-Bera test statistic and show how it lets you judge whether a set of returns is normal.
- Describe the power law and explain how it serves for distributions that are not normal.
- Define covariance and correlation, and explain why correlation is not the same thing as dependence.
- In a one-factor model, characterise the correlations among variables that are normally distributed.
- Compare the various measures of correlation that are used to assess dependence, noting how they are alike and how they differ.
Simulation and Bootstrapping
- Illustrate how a Monte Carlo simulation is used to approximate moments or other quantities, and outline the basic steps of carrying it out.
- Outline the ways in which Monte Carlo sampling error can be reduced.
- Illustrate how antithetic variates and control variates are used to cut Monte Carlo sampling error, and explain why they work.
- Describe bootstrapping and the way in which it improves on Monte Carlo simulation.
- Outline how pseudo-random numbers are generated.
- Outline the circumstances in which bootstrapping does not work well.
- Outline the drawbacks of using simulation to solve financial problems.
Machine-Learning Methods
- Discuss how machine-learning techniques differ from classical econometrics in both philosophy and practice.
- Compare the two methods used to rescale variables when preparing data, and apply them.
- Explain what the training, validation and test sub-samples are, and the distinct purpose each serves.
- Examine how underfitting and overfitting differ and what follows from each, and outline possible remedies for both.
- Use principal components analysis to compress a set of features into fewer dimensions, and explain why that works.
- Outline how K-means assigns the observations in a sample to separate clusters.
- Describe how natural language processing works and what it is used for.
- Distinguish unsupervised learning models from supervised ones and from reinforcement learning models.
- Describe the operation of reinforcement learning and calculate the Q-values used when making decisions.
Machine Learning and Prediction
- Say how forecasting a quantity can draw on a linear regression and on a logistic regression, and what each does there.
- Assess how well logistic regression models predict.
- Outline the methods for encoding categorical variables and apply them.
- Explain the usefulness of regularization and set ridge regression alongside LASSO for comparison.
- Show how a decision tree is built and how it should be read.
- Explain how a set of learners is combined into an ensemble.
- Describe the underlying intuition and the working steps of two classification methods, K nearest neighbors and support vector machines.
- Describe how neural networks are put together and how the values of their weights are arrived at.
- Use a confusion matrix to compare the logistic regression and neural network approaches to classification.
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