Estimated reading time: 6 minutes

## Introduction

We present part 5 of the learning outcomes, as prescribed by GARP.

## Market Risk Measurement and Management

This area focuses on market risk measurement and management techniques.

The broad knowledge points covered in Market Risk Measurement and Management 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

## Estimating Market Risk Measures

Estimate VaR using a historical simulation approach.

Estimate VaR using a parametric approach for both normal and lognormal return distributions.

Estimate the expected shortfall given profit and loss (P/L) or return data.

Define coherent risk measures.

Estimate risk measures by estimating quantiles.

Evaluate estimators of risk measures by estimating their standard errors.

Interpret quantile-quantile (QQ) plots to Identify the characteristics of a distribution.

## Non-parametric Approaches

Apply the bootstrap historical simulation approach to estimate coherent risk measures.

Describe historical simulation using non-parametric density estimation.

Compare and contrast the age-weighted, the volatility-weighted, the correlation-weighted, and the filtered historical simulation approaches.

Identify advantages and disadvantages of non-parametric estimation methods.

## Parametric Approaches

Explain the importance and challenges of extreme values in risk management.

Describe extreme value theory (EVT) and its use in risk management.

Describe the peaks-over-threshold (POT) approach.

Compare and contrast generalized extreme value and POT.

Evaluate the tradeoffs involved in setting the threshold level when applying the generalized Pareto (GP) distribution.

Explain the multivariate EVT for risk management.

## Backtesting VaR

Describe backtesting and exceptions and explain the importance of backtesting VaR models.

Explain the significant difficulties in backtesting a VaR model.

Verify a model based on exceptions or failure rates.

Define and Identify Type I and Type II errors.

Explain the need to consider conditional coverage in the backtesting framework.

Describe the Basel rules for backtesting.

## VaR Mapping

Explain the principles underlying VaR mapping and describe the mapping process.

Explain how the mapping process captures general and specific risks.

Differentiate among the three methods of mapping portfolios of fixed income securities.

Summarize how to map a fixed income portfolio into positions of standard instruments.

Describe how mapping of risk factors can support stress testing.

Explain how VaR can be used as a performance benchmark.

Describe the method of mapping forwards, forward rate agreements, interest rate swaps, and options.

## Correlation Basics: Definitions, Applications, and Terminology

Describe financial correlation risk and the areas in which it appears in finance.

Explain how correlation contributed to the global financial crisis of 2007-2009.

Describe the structure, uses, and payoffs of a correlation swap.

Estimate the impact of different correlations between assets in the trading book on the VaR capital charge.

Explain the role of correlation risk in market risk and credit risk.

Relate correlation risk to systemic and concentration risk.

## Empirical Properties of Correlation

Describe how equity correlations and correlation volatilities behave throughout various economic states.

Calculate a mean reversion rate using standard regression and calculate the corresponding autocorrelation.

Identify the best-fit distribution for equity, bond, and default correlations.

## Financial Correlation Modeling—Bottom-Up Approaches

Explain the purpose of copula functions and the translation of the copula equation.

Describe the Gaussian copula and explain how to use it to derive the joint probability of default of two assets.

Summarize the process of finding the default time of an asset corelated to all other assets in a portfolio using the Gaussian copula.

## Empirical Approaches to Risk Metrics and Hedging

Explain the drawbacks to using a DV01-neutral hedge for a bond position.

Describe a regression hedge and explain how it can improve a standard DV01-neutral hedge.

Calculate the regression hedge adjustment factor, beta.

Calculate the face value of an offsetting position needed to carry out a regression hedge.

Calculate the face value of multiple offsetting swap positions needed to carry out a two-variable regression hedge.

Compare and contrast level and change regressions.

Describe principal component analysis and explain how it is applied to constructing a hedging portfolio.

## The Science of Term Structure Models

Calculate the expected discounted value of a zero-coupon security using a binomial tree.

Construct and apply an arbitrage argument to price a call option on a zero-coupon security using replicating portfolios.

Define risk-neutral pricing and apply it to option pricing.

Distinguish between true and risk-neutral probabilities and apply this difference to interest rate drift.

Explain how the principles of arbitrage pricing of derivatives on fixed income securities can be extended over multiple periods.

Define option-adjusted spread (OAS) and apply it to security pricing.

Describe the rationale behind the use of recombining trees in option pricing.

Calculate the value of a constant maturity Treasury swap, given an interest rate tree and the risk- neutral probabilities.

Evaluate the advantages and disadvantages of reducing the size of the time steps on the pricing of derivatives on fixed-income securities.

Evaluate the appropriateness of the Black-Scholes-Merton model when valuing derivatives on fixed income securities.

## The Evolution of Short Rates and the Shape of the Term Structure

Explain the role of interest rate expectations in determining the shape of the term structure.

Apply a risk-neutral interest rate tree to assess the effect of volatility on the shape of the term structure.

Estimate the convexity effect using Jensen’s inequality.

Evaluate the impact of changes in maturity, yield, and volatility on the convexity of a security.

Calculate the price and return of a zero-coupon bond incorporating a risk premium.

## The Art of Term Structure Models

Construct and describe the effectiveness of a short-term interest rate tree assuming normally distributed rates, both with and without drift.

Calculate the short-term rate change and standard deviation of the rate change using a model with normally distributed rates and no drift.

Describe methods for addressing the possibility of negative short-term rates in term structure models.

Construct a short-term rate tree under the Ho-Lee Model with time-dependent drift.

Describe uses and benefits of the arbitrage-free models and assess the issue of fitting models to market prices.

Describe the process of constructing a simple and recombining tree for a short-term rate under the Vasicek Model with mean reversion.

Calculate the Vasicek Model rate change, standard deviation of the rate change, expected rate in T years, and half-life.

Describe the effectiveness of the Vasicek Model.

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

Describe the short-term rate process under a model with time-dependent volatility.

Calculate the short-term rate change and determine the behavior of the standard deviation of the rate change using a model with time-dependent volatility.

Assess the efficacy of time-dependent volatility models.

Describe the short-term rate process under the Cox-Ingersoll-Ross (CIR) and lognormal models.

Calculate the short-term rate change and describe the basis point volatility using the CIR and lognormal models.

Describe lognormal models with deterministic drift and mean reversion.

## Volatility Smiles

Define volatility smile and volatility skew.

Explain the implications of put-call parity on the implied volatility of call and put options.

Compare the shape of the volatility smile (or skew) to the shape of the implied distribution of the underlying asset price and to the pricing of options on the underlying asset.

Describe characteristics of foreign exchange rate distributions and their implications on option prices and implied volatility.

Describe the volatility smile for equity options and foreign currency options and provide possible explanations for its shape.

Describe alternative ways of characterizing the volatility smile.

Describe volatility term structures and volatility surfaces and how they may be used to price options.

Explain the impact of the volatility smile on the calculation of the Greeks.

Explain the impact of a single asset price jump on a volatility smile.

## Fundamental Review of the Trading Book

Describe the changes to the Basel framework for calculating market risk capital.

Review of the Trading Book (FRTB) and the motivations for these changes.

Compare the various liquidity horizons proposed by the FRTB for different asset classes.

Explain the FRTB revisions to Basel regulations in the following areas:

- Classification of positions in the trading book compared to the banking book
- Backtesting, profit and loss attribution, credit risk, and securitizations

## In closing

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