Market Risk Measurement and Management is one of the six broad topics that GARP tests in its FRM Part 2 exam. This broad topic has 20% weight in the exam. This means out of 80 questions asked, you may expect 16 questions from this section. This area focuses on market risk measurement and management techniques. The broad knowledge points include the following:

- VaR and other risk measures
- Parametric and non-parametric methods of estimation
- VaR mapping
- Backtesting VaR
- Expected shortfall (ES) and other coherent risk measures
- Extreme Value Theory (EVT)
- Modeling dependence: correlations and copulas

- Term structure models of interest rates
- Volatility: smiles and term structures
- Fundamental Review of the Trading Book

There are sixteen chapters or readings in this section. If you go through the GARP specified learning objectives (LOs) for this section, you will find a good mix of computational and non-computational LOs. As GARP generally asks tricky questions from the non-computational LOs, non-computational LOs are to be equally emphasized to score well in this section.

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

## Chapter 1: Estimating Market Risk Measures: An Introduction and Overview

This chapter focuses on the estimation of market risk measures, such as value at risk (VaR). VaR indicates the probability that losses will be greater than a pre-specified threshold level. GARP might test your ability to evaluate and calculate VaR using historical simulation and parametric models on the exam day. One limitation of VaR is that it does not indicate losses in the tail of the return’s distribution. Unlike VaR, expected shortfall (ES) indicates the loss in the tail (i.e., after the VaR threshold has been crossed) by averaging loss levels at different confidence levels. Coherent risk measures factor in personal risk aversion across the entire distribution and are more generic than the expected shortfall. Quantile-quantile (QQ) plots visually inspect whether an empirical distribution matches a theoretical distribution.

## Chapter 2: Non-parametric Approaches

This chapter broadly non-parametric estimation and bootstrapping (i.e., resampling). The main difference between these approaches and parametric approaches discussed in the previous chapter is that with non-parametric approaches the underlying distribution is not stated, and it is a data-driven, not assumption-driven analysis. For instance, historical simulation is restricted by the discreteness of the data, but non-parametric analysis “smoothes” the data points to allow for any VaR confidence level between observations. On the exam day, GARP might test your understanding of the bootstrap historical simulation approach as well as the various weighted historical simulation approaches.

## Chapter 3: Parametric Approaches (II): Extreme Value

Extreme values are critical for risk management as they relate to catastrophic events such as the failures of large institutions and market crashes. As they are rare, modeling such events is always a daunting task. This reading talks about the generalized extreme value (GEV) distribution, and the peaks-over-threshold approach and explains how the peaks-over-threshold converges to the generalized Pareto distribution.

## Chapter 4: Backtesting VaR

Value at Risk (VaR) methodologies are used to model risk. VaR models can be used to approximate the changes in value that a portfolio would experience in response to changes in the underlying risk factors. Model validation incorporates several methods that are used to determine how close approximations are to actual changes in value. Model validation helps determine what confidence to place in a model. GARP might test your ability to validate approaches that measure how close VaR model approximations are to actual changes in value on the exam day. Also, be thorough with your understanding of how the log-likelihood ratio (LR) is used to test the validity of VaR models for Type I and Type II errors for both unconditional and conditional tests. You might also be tested on Basel Committee outcomes that require banks to backtest their internal VaR models and penalize banks by imposing higher capital requirements for excessive exceptions.

## Chapter 5: VaR Mapping

This chapter introduces the concept of mapping a portfolio and explains how the risk of a complex, multi-asset portfolio can be decomposed into risk factors. On the exam day, GARP might test your ability to:

- Explain the mapping process for several types of portfolios, including fixed-income portfolios and portfolios consisting of linear and nonlinear derivatives
- Describe how the mapping process simplifies risk management for large portfolios
- Distinguish between general and specific risk factors, and understand the various inputs required for calculating undiversified and diversified value at risk (VaR).

## Chapter 6: Messages from the Academic Literature on Risk Measurement for the Trading Book

This reading discusses the tools for risk measurement, including value at risk (VaR) and expected shortfall. GARP might test your understanding of VaR implementation over different time horizons and VaR adjustments for liquidity costs on the exam day. Also, be thorough with academic studies related to integrated risk management and discusses the importance of measuring interactions among risks due to risk diversification. Note that several concepts introduced in this reading, such as liquidity risk, stressed VaR, and capital requirements, are discussed in more detail in operational and integrated risk management and the Basel Accords.

## Chapter 7: Correlation Basics: Definitions, Applications, and Terminology

This chapter emphasizes the role correlation plays as input for quantifying risk in multiple areas of finance. It also explains how correlation changes the value and risk of structured products such as credit default swaps (CDSs), collateralized debt obligations (CDOs), multi-asset correlation options, and correlation swaps. On the exam day, GARP might test your understanding of how:

- Correlation risk is related to market risk, systemic risk, credit risk, and concentration ratios, and be familiar with how changes in correlation impact implied volatility, the value of structured products, and default probabilities.
- Correlation contributed to the financial crisis of 2007 to 2009.

## Chapter 8: Empirical Properties of Correlation: How Do Correlations Behave in the Real World?

This chapter discusses how equity correlations and correlation volatility change during different economic states and uses a standard regression model to estimate the mean reversion rate and autocorrelation. On the exam day, GARP might test your ability to:

- Calculate the mean reversion rate
- Compare and contrast the nature of correlations and correlation volatility for equity, bond, and default correlations.
- Identify the best fit distribution for these three types of correlation distributions.

## Chapter 9: Financial Correlation Modeling—Bottom-Up Approaches

A copula is a joint multivariate distribution that shows how variables from marginal distributions come together. Copulas are useful as an alternative measure of dependence between random variables that is not subject to the same limitations as correlation in applications such as risk measurement. On the exam day, GARP might test your understanding of how:

- A correlation copula is created by mapping two or more unknown distributions to a known distribution that has well-defined properties.
- The Gaussian copula is used to estimate joint probabilities of default for specific time periods and the default time for multiple assets.

The material in this chapter is quite advanced, so your emphasis here should be on having a general understanding of how a copula function is put to use.

## Chapter 10: Empirical Approaches to Risk Metrics and Hedging

This chapter talks about how the dollar value of a basis point (DV01)-style hedges can be improved. Regression-based hedges improve DV01-style hedges by examining yield changes over time. Principal components analysis (PCA) significantly eases bond hedging techniques. On the exam day, GARP might test your understanding of:

- The drawbacks of a standard DV01-neutral hedge
- How to compute the face value of an offsetting position using DV01
- How to adjust this position using regression-based hedging techniques

## Chapter 11: The Science of Term Structure Models

This chapter focuses on the valuation of interest rate derivative contracts using a risk-neutral binomial model. The valuation process for interest rate derivatives necessitates rigorous calculations and is very tedious. However, the relationship turns out to be straightforward when it is modeled to support risk neutrality. On the exam day, GARP might test your understanding of:

- The concepts of backward induction
- How the addition of time steps will increase the accuracy of any bond pricing model
- The price-yield relationship of both callable and putable bonds.

Note that this chapter builds on the elements of material from the FRM Part I curriculum relating to the valuation of options with binomial trees.

## Chapter 12: The Evolution of Short Rates and the Shape of the Term Structure

This chapter talks about how the decision tree framework is used to estimate the price and returns of zero-coupon bonds. This decision tree framework shows how interest rate expectations determine the shape of the yield curve. On the exam day, GARP might test your understanding of:

- How current spot rates and forward rates are determined by the expectations, volatility, and risk premiums of short-term rates.
- The use of Jensen’s inequality to demonstrate how maturity and volatility increase the convexity of zero-coupon bonds.

## Chapter 13: The Art of Term Structure Models: Drift

This chapter discusses different term structure models for estimating short-term interest rates, including models that have no drift (Model 1), constant drift (Model 2), time-deterministic drift (Ho-Lee), and mean-reverting drift (Vasicek). On the exam day, GARP might test your understanding of:

- the differences between different short rate models
- how to construct a two-period interest rate tree using model predictions
- how the limitations of each model impact model effectiveness.
- how to convert a nonrecombining tree into a combining tree in the case of the Vasicek model

## Chapter 14: The Art of Term Structure Models: Volatility and Distribution

This chapter factors in non-constant volatility into term structure models. The generic time-dependent volatility model is quite flexible and mainly suitable for valuing multiperiod derivatives like interest rate caps and floors. The Cox-Ingersoll-Ross (CIR) mean-reverting model advocates that the term structure of volatility rises with the level of interest rates and never becomes negative. The lognormal model also has non-negative interest rates that proportionally rise with the level of the short-term rate. On the exam day, GARP might test your understanding of:

- How these models impact the short-term rate process
- How a time-dependent volatility model (Model 3) differs from the models discussed in the previous chapter
- The differences between the CIR and the lognormal models, as well as the differences between the lognormal models with drift and mean reversion

## Chapter 15: Volatility Smiles

This chapter brings out some of the reasons for the existence of volatility smiles, and how volatility affects options pricing as well as other option characteristics. For the exam, your focus should be on the explanation of why volatility smiles exist in currency and equity options. Also, be thorough with your understanding of how volatility smiles influence the Greeks and how to interpret price jumps.

## Chapter 16: Fundamental Review of the Trading Book

This reading talks about the new banking capital requirements that will greatly change the way the capital for the market risk is calculated. Several crucial innovations will bring about this change. For the exam day, be thorough with the following innovations:

- First, banks will be required to replace using the 99% confidence interval VaR measure with 97.5% confidence interval expected shortfall measure. This change will help better capture the potential dollar loss (i.e., tail risk) that a bank could suffer in a given time window. Many risk managers have already started using expected shortfall in practice for internal audits.
- Second, risk assets will be classified into liquidity horizons that better replicate the volatility in specific asset categories.
- The third innovation is a rule-based criterion that will help distinguish a trading book asset from a banking book asset. This step will help mitigate the potential for regulatory arbitrage.

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