2021 FRM Learning Objectives – Part 4

Estimated reading time: 9 minutes



We now look at part 4 of the learning outcomes, as prescribed by GARP.


Valuation and Risk Models

This area will test a candidate’s knowledge of valuation techniques and risk models.

The broad knowledge points covered in Valuation and Risk Models include the following:

Value-at-Risk (VaR)

Expected shortfall (ES)

Estimating volatility and correlation

Economic and regulatory capital

Stress testing and scenario analysis

Option valuation

Fixed income valuation


Country and sovereign risk models and management

External and internal credit ratings

Expected and unexpected losses

Operational risk


Measures of Financial Risk

Describe the mean-variance framework and the efficient frontier.

Explain the limitations of the mean-variance framework with respect to assumptions about return distributions.

Compare the normal distribution with the typical distribution of returns of risky financial assets such as equities.

Define the VaR measure of risk, describe assumptions about return distributions and holding periods, and explain the limitations of VaR.

Explain and calculate ES and compare and contrast VaR and ES.

Define the properties of a coherent risk measure and explain the meaning of each property.

Explain why VaR is not a coherent risk measure.


Calculating and Applying VaR

Explain and give examples of linear and non-linear derivatives.

Describe and calculate VaR for linear derivatives.

Describe and explain the historical simulation approach for computing VaR and ES.

Describe the delta-normal approach for calculating VaR for non-linear derivatives.

Describe the limitations of the delta-normal method.

Explain the structured Monte Carlo method for computing VaR and identify its strengths and weaknesses.

Describe the implications of correlation breakdown for scenario analysis.

Describe worst-case scenario (WCS) analysis and compare WCS to VaR.


Measuring and Monitoring Volatility

Explain how asset return distributions tend to deviate from the normal distribution.

Explain reasons for fat tails in a return distribution and describe their implications.

Distinguish between conditional and unconditional distributions.

Describe the implications of regime switching on quantifying volatility.

Compare and contrast different parametric and non-parametric approaches for estimating conditional volatility.

Calculate conditional volatility using parametric and non-parametric approaches.

Evaluate implied volatility as a predictor of future volatility and its shortcomings.

Apply the exponentially weighted moving average (EWMA) approach and the GARCH (1,1) model to estimate volatility.

Explain and apply approaches to estimate long horizon volatility/VaR and describe the process of mean reversion according to a GARCH (1,1) model.

Calculate conditional volatility with and without mean reversion.

Describe the impact of mean reversion on long horizon conditional volatility estimation.

Describe an example of updating correlation estimates.


External and Internal Credit Ratings

Describe external rating scales, the rating process, and the link between ratings and default.

Describe the impact of time horizon, economic cycle, industry, and geography on external ratings.

Define and use the hazard rate to calculate the unconditional default probability of a credit asset.

Define recovery rate and calculate the expected loss from a loan.

Explain and compare the through-the-cycle and point-in-time internal ratings approaches.

Describe alternative methods to credit ratings produced by rating agencies.

Compare external and internal ratings approaches.

Describe and interpret a ratings transition matrix and explain its uses.

Describe the relationships between changes in credit ratings and changes in stock prices, bond prices, and credit default swap spreads.

Explain historical failures and potential challenges to the use of credit ratings in making investment decisions.


Country Risk: Determinants, Measures, and Implications

Identify sources of country risk.

Explain how a country’s position in the economic growth life cycle, political risk, legal risk, and economic structure affects its risk exposure.

Evaluate composite measures of risk that incorporate all major types of country risk.

Compare instances of sovereign default in both foreign currency debt and local currency debt and explain common causes of sovereign defaults.

Describe the consequences of sovereign default.

Describe factors that influence the level of sovereign default risk; explain and assess how rating agencies measure sovereign default risks.

Describe the characteristics of sovereign credit spreads and sovereign credit default swaps (CDS) and compare the use of sovereign spreads to credit ratings.


Measuring Credit Risk

Evaluate a bank’s economic capital relative to its level of credit risk.

Explain the distinctions between economic capital and regulatory capital and describe how economic capital is derived.

Identify and describe important factors used to calculate economic capital for credit risk: probability of default, exposure, and loss rate.

Define and calculate expected loss (EL).

Define and explain unexpected loss (UL).

Estimate the mean and standard deviation of credit losses assuming a binomial distribution.

Describe the Gaussian copula model and its application.

Describe and apply the Vasicek model to estimate default rate and credit risk capital for a bank.

Describe the CreditMetrics model and explain how it is applied in estimating economic capital.

Describe and use the Euler’s theorem to determine the contribution of a loan to the overall risk of a portfolio.

Explain why it is more difficult to calculate credit risk capital for derivatives than for loans.

Describe challenges to quantifying credit risk.


Operational Risk

Describe the different categories of operational risk and explain how each type of risk can arise.

Compare the basic indicator approach, the standardized approach, and the advanced measurement approach for calculating operational risk regulatory capital.

Describe the standardized measurement approach and explain the reasons for its introduction by the Basel Committee.

Explain how a loss distribution is derived from an appropriate loss frequency distribution and loss severity distribution using Monte Carlo simulation.

Describe the common data issues that can introduce inaccuracies and biases in the estimation of loss frequency and severity distributions.

Describe how to use scenario analysis in instances when data are scarce.

Describe how to identify causal relationships and how to use Risk and Control Self-assessment (RCSA),

Key Risk Indicators (KRIs), and education to measure and manage operational risks.

Describe the allocation of operational risk capital to business units.

Explain how to use the power law to measure operational risk.

Explain how the moral hazard and adverse selection problems faced by insurance companies relate to insurance against operational risk.


Stress Testing

Describe the rationale for the use of stress testing as a risk management tool.

Explain key considerations and challenges related to stress testing, including choice of scenarios, regulatory specifications, model building, and reverse stress testing.

Describe the relationship between stress testing and other risk measures, particularly in enterprise-wide stress testing.

Describe stressed VaR and stressed ES and compare the process of determining stressed VaR and ES to that of traditional VaR and ES.

Identify the advantages and disadvantages of stressed risk metrics.

Describe the responsibilities of the board of directors, senior management, and the internal audit function in stress testing governance.

Identify elements of clear and comprehensive policies, procedures, and documentations for stress testing.

Identify areas of validation and independent review for stress tests that require attention from a governance perspective.

Describe the Basel stress testing principles for banks regarding the implementation of stress testing.


Pricing Conventions, Discounting, and Arbitrage

Define discount factor and use a discount function to compute present and future values.

Define the “law of one price,” explain it using an arbitrage argument, and describe how it can be applied to bond pricing.

Identify arbitrage opportunities for fixed income securities with certain cash flows.

Identify the components of a US Treasury coupon bond and compare the structure to Treasury STRIPS, including the difference between P-STRIPS and C-STRIPS.

Construct a replicating portfolio using multiple fixed income securities to match the cash flows of a given fixed-income security.

Differentiate between “clean” and “dirty” bond pricing and explain the implications of accrued interest with respect to bond pricing.

Describe the common day-count conventions used in bond pricing.


Interest Rates

Calculate and interpret the impact of different compounding frequencies on a bond’s value.

Define spot rate and compute discount factors given spot rates.

Interpret the forward rate and compute forward rates given spot rates.

Define par rate and describe the equation for the par rate of a bond.

Interpret the relationship between spot, forward, and par rates.

Assess the impact of maturity on the price of a bond and the returns generated by bonds.

Define the “flattening” and “steepening” of rate curves and describe a trade to reflect expectations that a curve will flatten or steepen.

Describe a swap transaction and explain how a swap market defines par rates.

Describe overnight indexed swaps (OIS) and distinguish OIS rates from LIBOR swap rates.


Bond Yields and Return Calculations

Distinguish between gross and net realized returns and calculate the realized return for a bond over a holding period including reinvestments.

Define and interpret the spread of a bond and explain how a spread is derived from a bond price and a term structure of rates.

Define, interpret, and apply a bond’s yield-to-maturity (YTM) to bond pricing.

Compute a bond’s YTM given a bond structure and price.

Calculate the price of an annuity and a perpetuity.

Explain the relationship between spot rates and YTM.

Define the coupon effect and explain the relationship between coupon rate, YTM, and bond prices.

Explain the decomposition of the profit and loss (P&L) for a bond position or portfolio into separate factors including carry roll-down, rate change, and spread change effects.

Explain the four common assumptions in carry roll-down scenarios.


Applying Duration, Convexity, and DV01

Describe a one-factor interest rate model and identify common examples of interest rate factors.

Define and compute the DV01 of a fixed income security given a change in yield and the resulting change in price.

Calculate the face amount of bonds required to hedge an option position given the DV01 of each.

Define, compute, and interpret the effective duration of a fixed income security given a change in yield and the resulting change in price.

Compare and contrast DV01 and effective duration as measures of price sensitivity.

Define, compute, and interpret the convexity of a fixed income security given a change in yield and the resulting change in price.

Explain the process of calculating the effective duration and convexity of a portfolio of fixed income securities.

Describe an example of hedging based on effective duration and convexity.

Construct a barbell portfolio to match the cost and duration of a given bullet investment and explain the advantages and disadvantages of bullet versus barbell portfolios.


Modeling Non-Parallel Term Structure Shifts and Hedging

Describe principal components analysis and explain its use in understanding term structure movements.

Define key rate exposures and know the characteristics of key rate exposure factors, including partial ‘01s and forward-bucket ‘01s.

Describe key-rate shift analysis.

Define, calculate, and interpret key rate ‘01 and key rate duration.

Compute the positions in hedging instruments necessary to hedge the key rate risks of a portfolio.

Relate key rates, partial ‘01s, and forward-bucket ‘01s and calculate the forward-bucket ‘01 for a shift in rates in one or more buckets.

Apply key rate and multi-factor analysis to estimating portfolio volatility.


Binomial Trees

Calculate the value of an American and a European call or put option using a one-step and two-step binomial model.

Describe how volatility is captured in the binomial model.

Describe how the value calculated using a binomial model converges as time periods are added.

Define and calculate delta of a stock option.

Explain how the binomial model can be altered to price options on stocks with dividends, stock indices, currencies, and futures.


The Black-Scholes-Merton Model

Explain the lognormal property of stock prices, the distribution of rates of return, and the calculation of expected return.

Compute the realized return and historical volatility of a stock.

Describe the assumptions underlying the Black-Scholes-Merton option pricing model.

Compute the value of a European option on a non-dividend-paying stock using the Black-Scholes-Merton model.

Define implied volatilities and describe how to compute implied volatilities from market prices of options using the Black-Scholes-Merton model.

Explain how dividends affect the decision to exercise early for American call and put options.

Compute the value of a European option using the Black-Scholes-Merton model on a dividend-paying stock, futures, and exchange rates.

Describe warrants, calculate the value of a warrant, and calculate the dilution cost of the warrant to existing shareholders.


Option Sensitivity Measures: The “Greeks”

Describe and assess the risks associated with naked and covered option positions.

Describe the use of a stop loss hedging strategy, including its advantages and disadvantages, and explain how this strategy can generate naked and covered option positions.

Describe delta hedging for options as well as for forward and futures contracts.

Compute the delta of an option.

Describe the dynamic aspects of delta hedging and distinguish between dynamic hedging and hedge-and forget strategies.

Define and calculate the delta of a portfolio.

Define and describe theta, gamma, vega, and rho for option positions and calculate the gamma and vega for a portfolio.

Explain how to implement and maintain a delta-neutral and a gamma-neutral position.

Describe the relationship between delta, theta, gamma, and vega.

Describe how portfolio insurance can be created through option instruments and stock index futures.


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