Market Risk Measurement and Management: FRM® Part II

It is 20% of the 2026 Part II exam. 100+ learning outcomes across 18 learning modules.

610+ practice questions and 131 formula cards on this topic in the app.

Estimating Market Risk Measures: An Introduction and Overview

• Work out VaR from historical simulation of past data.
• Compute VaR with a parametric method, covering both normally distributed and lognormally distributed asset prices.
• Derive expected shortfall (ES) from a sample of profit-and-loss (P&L) or return observations.
• Obtain risk measures by estimating the relevant quantiles of the distribution.
• Judge how reliable risk measure estimators are by working out their standard errors.
• Read quantile-quantile (QQ) plots to infer the features of a distribution, such as its tails.

Non-Parametric Approaches

• Obtain estimates of coherent risk measures by resampling the data in a bootstrapped historical simulation.
• Explain how a smoothed, non-parametric estimate of the density can underpin a historical simulation.
• Set the age-weighted, volatility-weighted, correlation-weighted and filtered historical simulation methods side by side, noting how they differ and what they share.
• Pinpoint the strengths and weaknesses of non-parametric estimation techniques.

Parametric Approaches (II): Extreme Value

• Account for why extreme values matter in risk management and what makes dealing with them difficult.
• Characterise extreme value theory (EVT) and how risk managers put it to use.
• Explain the POT technique, which models only the observations that exceed a high cutoff.
• Weigh the generalized extreme value (GEV) method against POT when it comes to quantifying extreme risk, noting what separates them.
• Examine the role the generalized Pareto (GP) distribution plays when POT is used.
• Show how EVT extends to several variables at once and why that helps risk management.

Backtesting VaR

• Set out what backtesting and exceptions are, and show why backtesting matters for VaR models.
• Account for the main obstacles that make a VaR model hard to backtest.
• Judge a VaR model's accuracy by applying a model verification test to its exceptions or failure rates.
• Recognise Type I and Type II errors as they arise when backtesting, and say what each one means.
• Show why backtesting should take conditional coverage into account.
• Summarise the Basel requirements that govern VaR backtesting.

VaR Mapping

• Show the reasoning behind VaR mapping and walk through the steps of the mapping procedure.
• Explain why mapping reflects market-wide as well as position-specific risk, and quantify each for a portfolio given the underlying primitive risk factors.
• Tell apart the trio of techniques available for mapping bond portfolios.
• Summarise the way a bond portfolio gets expressed through a set of standard benchmark instruments.
• Show in what ways stress tests can draw on the mapping of risk factors.
• Show how VaR may be computed and used when measured against a performance benchmark.
• Explain how options, forwards, forward rate agreements (FRAs) and interest rate swaps are each handled when they are mapped.

Validating Bank Holding Companies’ Value-at-Risk Models for Market Risk

• Identify what a bank needs to consider as it appraises whether a VaR model is conceptually sound in the course of validating it.
• Show how a VaR model's sensitivity analysis is carried out and weigh its potential benefits against its difficulties.
• Characterise the difficulties an institution may meet in computing confidence intervals around VaR.
• Talk through the difficulties of benchmarking VaR models and the remedies suggested for them.

Forecasting Distribution Using Probability Integral Transform

• Spot which features of an exceedance-based backtest signal an accurate VaR model, and show how a PIT-based backtest captures those same features.
• Describe how a validator computes probability integral transforms (PITs) from a VaR model's forecasts.
• Explain how the shape of the PIT distribution signals how good a VaR model is.
• Lay out PIT-based backtesting and set the goodness-of-fit tests for judging PIT distributions against each other: Kolmogorov-Smirnov, Anderson-Darling and Cramér-von Mises.

Correlation Basics: Definitions, Applications, and Terminology

• Characterise financial correlation risk and name the areas of finance where it shows up.
• Explain how correlation helped bring about the 2007-2009 global financial crisis.
• Describe what correlation does to the prices of quanto options, and of exotic options written on several assets.
• Lay out how a correlation swap is built, what it is used for and what it pays.
• Work out how varying the correlations among trading book assets changes the VaR capital charge.
• Show what correlation risk contributes to both market risk and credit risk.
• Explain the links between correlation risk and, on one side, systemic risk and, on the other, concentration risk.

Empirical Properties of Correlation: How Do Correlations Behave in the Real World?

• Characterise the behaviour of equity correlations and of correlation volatilities across different states of the economy.
• Use a standard regression to work out a mean reversion rate, and derive the matching autocorrelation.
• Determine which statistical distribution gives the closest fit for each of equity correlations, bond correlations and default correlations.

Financial Correlation Modeling: Bottom-Up Approaches

• Show what copula functions are for and how they are used in finance.
• Characterise the Gaussian copula and demonstrate its use in deriving the chance that a pair of assets both default.
• Summarise how the Gaussian copula produces a default time for an asset whose defaults are correlated with all the other names held in a portfolio.

Regression Hedging and Principal Component Analysis

• Account for the shortcomings of hedging a bond position with a DV01-neutral hedge.
• Set out what a regression hedge is and show how it improves on the plain DV01-neutral hedge.
• Work out beta, the adjustment factor in a regression hedge.
• Determine the face amount of the offsetting position that a regression hedge calls for.
• Compute how much face value each of the multiple swaps must carry when a regression hedge uses two explanatory variables.
• Set level regressions and change regressions side by side, noting their similarities and differences.
• Explain the reasons for, and the mechanics of, the difference between a regression hedge and one derived from regressing in the opposite direction.
• Describe principal component analysis (PCA) and how it helps in putting together a portfolio of hedges.

Arbitrage Pricing with Term Structure Models

• Use a binomial tree to arrive at the discounted expected value for a zero-coupon bond.
• Price a call on a zero-coupon bond through an arbitrage argument that rests on a replicating portfolio.
• State the idea of risk-neutral pricing and put it to work in valuing options.
• Show how true probabilities differ from risk-neutral ones and use that gap to handle the drift in interest rates.
• Show how the arbitrage pricing principles for fixed-income derivatives carry over to several periods.
• State what the option-adjusted spread (OAS) is and use it when pricing securities.
• Give the reasoning for using recombining trees when pricing options.
• Given a rate tree together with risk-neutral probabilities, price a constant-maturity Treasury (CMT) swap.
• Weigh up what is gained and lost, when pricing fixed-income derivatives, by shrinking the time steps.
• Assess how fitting the Black-Scholes-Merton model is as a tool for pricing fixed-income derivatives.

Expectations, Risk Premium, Convexity, and the Shape of the Term Structure

• Show how expectations about future interest rates help determine the shape of the term structure.
• Employ a risk-neutral rate tree to assess how the term structure is shaped by volatility.
• Use Jensen's inequality to estimate the convexity effect.
• List the components of a bond's return, then calculate what a risk-averse investor can expect to earn from holding it.

The Art of Term Structure Models: Drift

• Build a short-rate tree for normally distributed rates, with and without drift, and describe how well it works.
• Work out the short-rate change and the standard deviation of that change under a normal-rates model with zero drift.
• Describe how term structure models can cope with short rates possibly going below zero.
• Build the Ho-Lee short-rate tree, in which drift is a function of time.
• Describe what arbitrage-free models are good for, and evaluate the difficulty of making a model match observed market prices.
• Outline how a simple, recombining short-rate tree is built under the mean-reverting Vasicek Model.
• Using the Vasicek Model, calculate the change in rate and its standard deviation, along with the rate expected T years out and the half-life.
• Characterise how well the Vasicek Model performs.

The Art of Term Structure Models: Volatility and Distribution

• Characterise how the short rate evolves in a model whose volatility depends on time.
• Work out the short-rate change under a time-dependent volatility model and establish how the standard deviation of that change behaves.
• Weigh up how well time-dependent volatility models work.
• Characterise the evolution of the short-term rate in the CIR model of Cox, Ingersoll and Ross, and in lognormal models.
• Work out the short-rate change under the CIR and lognormal models, and characterise the basis point volatility they imply.
• Characterise lognormal rate models extended with a drift that is deterministic or with mean-reverting behaviour.

The Vasicek and Gauss+ Models

• Outline how the Gauss+ model is structured and discuss what that structure means for its capacity to reproduce interest rate behaviour seen in practice.
• Contrast the Vasicek and Gauss+ models in terms of their dynamics, characteristics and uses.
• Calculate the movements in the Gauss+ model's three rate factors, namely the short, medium and long ones.
• Show how empirical data can be used to estimate the Gauss+ model parameters.

Volatility Smiles and Volatility Surfaces

• Explain what is meant by a smile and by a skew in implied volatility.
• Show what put-call parity implies for the implied volatilities of calls and puts.
• Link the form taken by a smile or skew to the price distribution it implies for the underlying, and to the resulting option prices.
• Characterise how currency returns are distributed, and trace what that implies for implied volatility and the pricing of FX options.
• Describe how the smile looks for options on stocks and for options on currencies, and put forward plausible explanations of why.
• Set out other ways in which the volatility smile can be described.
• Characterise volatility term structures and volatility surfaces, and show how option pricing can make use of them.
• Show how an option's Greek letter risk measures are affected by the volatility smile.
• Show how one jump in an asset's price affects a volatility smile.

Fundamental Review of the Trading Book

• Account for why the FRTB reform was undertaken, and outline how it modifies the way Basel requires banks to compute capital against market risk.
• Contrast the liquidity horizons the FRTB assigns across asset classes, and show how a bank derives its expected shortfall using those different horizons.
• Explain how the FRTB amends Basel rules for deciding between trading and banking book, for backtesting, for P&L attribution, for credit risk, and for securitizations.

Other topics

• Credit Risk Measurement and Management
• Operational Risk and Resilience
• Liquidity and Treasury Risk Measurement and Management
• Risk Management and Investment Management
• Current Issues in Financial Markets
• All of FRM Part II