# Greenwicher's WikiFRM - Valuation and Risk Models 2017-11-07

[Last Update: Nov 10, 2017]

## VaR Methods

### Defining VaR

### Calculating VaR

### VaR Conversions

### The VaR Methods

- Linear methods (the delta-normal valuation)
- Full valuation (Monte Carlo and historic simulation models)

### Comparing the Methods

## Quantifying Volatility in VaR Models

#### LO 52.1: Explain how asset return distributions tend to deviate from the normal distribution.

- fat-tailed
- skewed
- unstable

#### LO 52.2: Explain reasons for fat tails in a return distribution and describe their implications.

#### LO 52.3: Distinguish between conditional and unconditional distributions.

### Market Regimes and Conditional Distributions

#### LO 52.4: Describe the implications of regimes switching on quantifying volatility.

### Value at Risk

#### LO 52.5: Explain the various approaches for estimating VaR

#### LO 52.6: Compare and contrast different parametric and non-parametric approaches for estimating conditional volatility

#### LO 52.7: Calculate conditional volatility using parametric and non-parametric approaches

### Parametric Approaches for VaR

### GARCH

### Nonparametric vs. Parametric VaR Methods

### Nonparametric Approaches for VaR

- Historical Simulation Method
- Hybrid Approach
- Multivariate Density Estimation (MDE)

### Return Aggregation

#### LO 52.8: Explain the process of return aggregation in the context of volatility forecasting methods.

### Implied Volatility

#### LO 52.9: Evaluate implied volatility as a predictor of future volatility and its shortcomings.

### Mean Reversion and Long Time Horizons

*LO 52.10: Explain long horizon volatility/VaR and the process of mean reversion according to an AR(1) model.*

*LO 52.11: Calculate conditional volatility with and without mean reversion.*

#### LO 52.12: Describe the impact of mean reversion on long horizon conditional volatility estimation.

### Backtesting VaR Methodologies

## Putting VaR to Work

### Linear vs. Non-linear Derivatives

#### LO 53.1: Explain and give examples of linear and non-linear derivatives.

#### LO 53.2: Describe and calculate VaR for linear derivatives.

### Taylor Approximation

#### LO 53.3: Describe the delta-normal approach for calculating VaR for non-linear derivatives.

#### LO 53.4: Describe the limitations of the delta-normal method.

### The Delta-Normal and Full Revaluation Methods

#### LO 53.5: Explain the full revaluation method for computing VaR.

#### LO 53.6: Compare delta-normal and full revaluation approaches for computing VaR.

### The Monte Carlo Approach

#### LO 53.7: Explain structured Monte Carlo, stress testing, and scenario analysis methods for computing VaR, and identify strengths and weaknesses of each approach.

#### LO 53.8: Describe the implications of correlation breakdown for scenario analysis.

### Correlations during Crisis

### Stress Testing

### Worst Case Scenario Measure

#### LO 53.9: Describe worst-case scenario (WCS) analysis and compare WCS to VaR.

## Measures of Financial Risk

### Mean-Variance Framework

#### LO 54.1: Describe the mean-variance framework and the efficient frontier.

### Mean-Variance Framework Limitations

#### LO 54.2: Explain the limitations of the mean-variance framework with respect to assumptions about the return distributions.

### Value at Risk

#### LO 54.3: Define the Value-at-Risk (VaR) measure of risk, describe assumptions about return distributions and holding period, and explain the limitations of VaR.

### Coherent Risk Measures

#### LO 54.4: Define the properties of a coherent risk measure and explain the meaning of each property.

### Expected Shortfall

#### LO 54.5: Explain why VaR is not a coherent risk measure.

#### LO 54.6: Explain and calculate expected shortfall (ES), and compare and contrast VaR and ES.

*LO 54.7: Describe spectral risk measures, and explain how VaR and ES are special cases of spectral risk measures.*

### Scenario Analysis

#### LO 54.8: Describe how the results of scenario analysis can be interpreted as coherent risk measures.

## Binomial Trees

### A One-step Binomial Model

#### LO 55.1: Calculate the value of an American and a European call or put option using a one-step and two-step binomial model.

### The Replicating Portfolio

### Using the Hedge Ratio to Develop the Replicating Portfolio

### Synthetic Call Replication

### Risk-neutral Valuation

### Two-step Binomial Model

### Assessing Volatility

#### LO 55.2: Describe how volatility is captured in the binomial model.

### Modifying the Binomial Model

#### LO 55.4: Explain how the binomial model can be altered to price options on: stocks with dividends, stock indices, currencies, and futures.

### American Options

#### Increasing the Number of Time Periods

#### LO 55.3: Describe how the value calculated using a binomial model converges as time periods are added.

## The Black-Scholes-Merton Model

#### LO 56.1: Explain the lognormal property of stock prices, the distribution of rates of return, and the calculation of expected return.

#### LO 56.2: Compute the realized return and historical volatility of a stock.

### Lognormal Stock Prices

### Expected Value

### Black-Scholes-Merton Model Assumptions

#### LO 56.3: Describe the assumptions underlying the Black-Scholes-Merton option pricing model.

### Black-Scholes-Merton Option Pricing Model

#### LO 56.4: Compute the value of a European option using the Black-Scholes-Merton model on a non-dividend-paying stock.

### Black-Scholes-Merton Model with Dividends

#### LO 56.8: Compute the value of a European option using the Black-Scholes-Merton model on a dividend-paying stock.

### European Options

#### LO 56.7: Explain how dividends affect the decision to exercise early for American call and put options.

### American Options

### Valuation of Warrants

#### LO 56.5: Compute the value of a warrant and identify the complications involving the valuation of warrants.

### Volatility Estimation

#### LO 56.6: Define implied volatilities and describe how to compute implied volatilities from market prices of options using the Black-Scholes-Merton model.

## Greek Letters

### Naked and Covered Call Options

#### LO 57.1: Describe and assess the risks associated with naked and covered option positions.

- naked call option: without owning the underlying asset
- covered call option: a short call option and the writer owns the underlying asset

### A Stop-loss Strategy

#### LO 57.2: Explain how naked and covered option positions generate a stop loss trading strategy.

### Delta Hedging

#### LO 57.3: Describe delta hedging for an option, forward, and futures contracts.

#### LO 57.4: Compute the delta of an option.

### Dynamic Aspects of Delta Hedging

#### LO 57.5: Describe the dynamic aspects of delta hedging and distinguish between dynamic hedging and hedge-and-forget strategy.

### Maintaining the Hedge

### Other Portfolio Hedging Approaches

#### LO 57.6: Define the delta of a portfolio

### Theta, Gamma, Vega, and Rho

#### LO 57.7: Define and describe theta, gamma, vega, and rho for option positions.

#### LO 57.8: Explain how to implement and maintain a delta-neutral and a gamma-neutral position.

#### LO 57.9: Describe the relationship between delta, theta, gamma, and vega.

### Theta

### Gamma

### Relationship Among Delta, Theta, and Gamma

### Vega

### Rho

### Hedging in Practice

#### LO 57.10: Describe how hedging activities take place in practice, and describe how scenario analysis can be used to formulate expected gains and losses with option positions.

### Portfolio Insurance

#### LO 57.11: Describe how portfolio insurance can be created through option instruments and stock index futures.

## Prices, Discount Factors, and Arbitrage

### Fundamentals of Bond Valuation

### Calculating the Value of a Coupon Bond

### Price-Yield Curve

### Bond Price Quotations

### Discount Factors

#### LO 58.1: Define discount factors and use a discount function to compute present and future values.

### Determining Value using Discount Functions

#### LO 58.2: Define the “law of one price”, explain it using an arbitrage argument, and describe how it can be applied to bond pricing.

#### LO 58.5: Identify arbitrage opportunities for fixed income securities with certain cash flows.

### Treasury Coupon Bonds and Treasury STRIPS

#### LO 58.3: Identify the components of a U.S. Treasury coupon bond, and compare and contrast the structure to Treasury STRIPS, including the difference between P-STRIPS and C-STRIPS.

### Constructing a Replication Portfolio

#### LO 58.4: Construct a replicating portfolio using multiple fixed income securities to match the cash flows of a given fixed income security.

### Computing Price between Coupon Dates

#### LO 58.6: Differentiate between “clean” and “dirty” bond pricing and explain the implications of accrued interest with respect to bond pricing.

#### LO 58.7: Describe the common day-count conventions used in bond pricing.

### Accrued Interest

### Day-Count Convention

### Clean and Dirty Bond Pricing

## Spot, Forward, and Par Rates

### Annual Compounding vs. Semiannual Compounding

#### LO 59.1: Calculate and interpret the impact of different compounding frequencies on a bond’s value.

### Holding Period Return

### Deriving Discount Factors from Swap Rates

*LO 59.2: Calculate discount factors given interest rate swap rates.*

### The Spot Rate Curve

#### LO 59.3: Compute spot rates given discount factors.

### Forward Rates

#### LO 59.4: Interpret the forward rate, and compute forward rates given spot rates.

### Par Rates

#### LO 59.5: Define par rate and describe the equation for the par rate of a bond.

### Pricing a Bond using Spot, Forward, and Par Rates

#### LO 59.6: Interpret the relationship between spot, forward and par rates.

### Effect of Maturity on Bond Prices and Returns

#### LO 59.7: Assess the impact of maturity on the price of a bond and the returns generated by bonds.

### Yield Curve Shapes

*LO 59.8: Define the “flattening” and “steepening” of rate curves and describe a trade to reflect expecations that a curve will flatten or steepen.*

## Returns, Spreads, and Yields

### Realized Return

#### LO 60.1: Distinguish between gross and net realized returns, and calculate the realized return for a bond over a holding period including reinvestments.

### Bond Spread

#### LO 60.2: 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.

### Yield to Maturity

#### LO 60.3: Define, interpret, and apply a bond’s yield-to-maturity (YTM) to bond pricing.

#### LO 60.4: Compute a bond’s YTM given a bond structure and price.

### The Limitations of Traditional Yield Measures

#### LO 60.5: Calculate the price of an annuity and a perpetuity.

### Calculating the Price of an Annuity

### Calculating the Price of a Perpetuity

### Spot Rates and YTM

#### LO 60.6: Explain the relationship between spot rates and YTM.

### The Relationship between YTM, Coupon Rate, and Price

#### LO 60.7: Define the coupon effect and explain the relationship between coupon rate, YTM, and bond prices.

### Coupon Effect

### Return Decomposition

#### LO 60.8: Explain the decomposition of P&L for a bond into separate factors including carry roll-down, rate change, and spread change effects.

### Carry-Roll-Down Scenarios

#### LO 60.9: Identify the most common assumptions in carry roll-down scenarios, including realized forwards, unchanged term structure, and unchanged yields.

## One-Factor Risk Metrics and Hedges

### Interest Rate Factors

#### LO 61.1: Describe an interest rate factor and identify common examples of interest rate factors.

### Dollar Value of a Basis Point

#### LO 61.2: Define and compute the DV01 of a fixed income security given a change in yield and the resulting change in price.

- DV01: absolute change in bond price for every basis point change in yield, which is essentially a basis point’s price value

### DV01 Application to Hedging

*LO 61.3: Calculate the face amount of bonds required to hedge an option position given the DV01 of each.*

### Duration

#### LO 61.4: Define, compute and interpret the effective duration of a fixed income secruity given a change in yield and the resulting change in price

### DV01 vs Duration

#### LO 61.5: Compare and contrast DV01 and effective duration as measures of price sensitivity.

- DV01 works better for hedgers, while duration is more convenient for traditional investors.

### Convexity

#### LO 61.6: Define, compute, and interpert the convexity of a fixed income security given a change in yield and the resulting change in price

### Price Change using both Duration and Convexity

### Portfolio Duration and Convexity

#### LO 61.7: Explain the process of calculating the effective duration and convexity of a portfolio of fixed income securities.

### Negative Convexity

#### LO 61.8: Explain the impact of negative convexity on the hedging of fixed income securities.

### Constructing a Barbell Portfolio

#### LO 61.9: 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

## Multi-Factor Risk Metrics and Hedges

### Weakness of Single-factor Approaches

#### LO 62.1: Describe and assess the major weakness attributable to single-factor approaches when hedging portfolios or implementing asset liability techniques.

### Key Rate Exposures

#### LO 62.2: Define key rate exposures and know the characteristics of key rate exposure factors including partial ‘01s and forward-bucker ‘01s.

### Key Rate Shift Technique

#### LO 62.3: Describe key-rate shift analysis

### Key Rate ‘01 and Key Rate Duration

*LO 62.4: Define, calculate, and interpret key rate ‘01 and key rate duration*

### Hedging Applications

#### LO 62.5: Describe the key rate exposure technique in multi-factor hedging applications; summarize its advantages and disadvantages

#### LO 62.6: Calculate the key rate exposures for a given security, and compute the appropriate hedging positions given a specific key rate exposure profile.

### Partial ‘01s and Forward-Bucket ‘01s

#### LO 62.7: Relate key rates, partial ‘01s and forward-bucket ‘01s, and caluclate the forward bucket ‘01 for a shift in rates in one or more buckets.

### Hedging Across Forward-Bucket Exposures

#### LO 62.8: Construct an appropriate hedge for a position across its entire range of forward bucket exposures.

### Estimating Portfolio Volatility

#### LO 62.9: Apply key rate and multi-factor analysis to estimating portfolio volatility.

## Country Risk: Determinants, Measures and Implications

### Sources of Country Risk

#### LO 63.1: Identify sources of country risk

### Country Risk Exposure

#### LO 63.2: Explain how a country’s position in the economic growth life cycle, political risk, legal risk, and economic structure affect its risk exposure.

### Evaluating Country Risk

#### LO 63.3: Evaluate composite measures of risk that incorporate all types of country risk and explain limitations of the risk services.

### Sovereign Defaults

#### LO 63.4: Compare instances of sovereign default in both foreign curency debt and local currency debt, and explain common causes of sovereign defaults.

### Consequences of Sovereign Default

#### LO 63.5: Describe the consequences of sovereign defaults.

### Factors Influencing Sovereign Default Risk

#### LO 63.6: Describe factors that influence the level of sovereign default risk; explain and assess how rating agencies measure sovereign default risks.

### The Sovereign Default Spread

#### LO 63.7: Describe the advantages and disadvantages of using the sovereign default spread as a predictor of defaults.

## External and Internal Ratings

### External Credit Ratings

#### LO 64.1: Describe external rating scales, the rating process, and the link between ratings and default.

#### LO 64.6: Describe a ratings transition matrix and explain its uses.

#### LO 64.2: Describe the impact of time horizon, economic cycle, industry, and geography on external ratings.

#### LO 64.3: Explain the potential impcat of ratings changes on bond and stock prices.

### Evolution of Internal Credit Ratings

#### LO 64.4: Compare external and internal ratings approaches.

### Internal Credit Ratings.

#### LO 64.5: Explain and compare the through-the-cycle and at-the-point internal ratings approaches.

#### LO 64.7: Describe the process for and issues with building, cailbrating and backtesting and internal rating system.

#### LO 64.8: Identify and describe the biases that may affect a rating system.

## Capital Structure in Banks

### Credit Risk Factors

#### LO 65.2: Identify and describe important factors used to calculate economic capital for credit risk: probability of default, exposure, and loss rate.

### Expected Loss

#### LO 65.3: Define and calculate expected loss (EL).

- EL: EA
*PD*LR

### Unexpected Loss

#### LO 65.4: Define and calculate unexpected loss (UL).

- UL represents the variation in expected loss: $UL = EA \times \sqrt{PD \times \sigma
*{LR}^{2} + LR^{2} \times \sigma*{PD}^{2}}$

#### LO 65.5: Estimate the variance of default probability assuming a binomial distribution.

### Portfolio Expected and Unexpected Loss

*LO 65.6: Calculate UL for a portfolio and the risk contribution of each asset.*

### Exonomic Capital

#### LO 65.1: Evaluate a bank’s economic capital relative to its level of credit risk.

#### LO 65.7: Describe how economic capital is derived.

### Modeling Credit Risk

#### LO 65.8: Explain how the credit loss distribution is modeled.

- beta distribution is commonly used to model credit risk

#### LO 65.9: Describe challenges to quantifying credit risk.

## Operational Risk

### Defining Operational Risk

### Operational Risk Capital Requirements

#### LO 66.1: Compare three approaches for calculating regulatory capital.

### Operational Risk Categories

#### LO 66.2: Describe the Basel Committee’s seven categories of operational risk.

- clients, products, and business practices
- internal fraud
- external fraud
- damage to physical assets
- execution, delivery, and process management
- business disruption and system failures
- employment practices and workplace safety

### Loss Frequency and Loss Severity

#### LO 66.3: Derive a loss distribution from the loss frequency distribution and loss severity distribution using Monte Carlo simulations.

### Data Limitations

#### LO 66.4: Describe the common data issues that can introduce inaccuracies and biases in the estimation of loss frequency and severity distributions.

#### LO 66.5: Describe how to use scenario analysis in instances when data is scarce.

### Forward-Looking Approaches

#### LO 66.6: Describe how to identify causal relationships and how to use risk and control self assessment (RCSA) and key risk indicators (KRIs) to measure and manage operational risks.

### Scorecard Data

#### LO 66.7: Describe the allocation of operational risk capital to business units.

### The Power Law

#### LO 66.8: Explain how to use the power law to measure operational risk.

### Insurance

#### LO 66.9: Explain the risks of moral hazard and adverse selection when using insurance to mitigate operational risks.

## Governance Over Stress Testing

### Effective Governance and Controls over Stress Testing

#### LO 67.1: Describe the key elements of effective governance over stress testing.

### Responsibilities of the Board and Senior Management

#### LO 67.2: Describe the responsibilities of the board of dirctors and senior management in stress testing activities.

### Policies, Procedures, and Documentation

#### LO 67.3: Identify elements of clear and comprehensive policies, procedures, and documentations on stress testing.

### Valuation and Independent Review

#### LO 67.4: Identify areas of validation and independent review for stress tests that require attention from a governance perspective.

### Role of Internal Audits

#### LO 67.5: Describe the important role of the internal audit in stress testing governance and control.

### Key Aspects of Stress Testing Governance

#### LO 67.6: Identify key aspects of stress testing governance, including stress testing coverage, stress testing types and approaches, and capital and liquidity stress testing.

## Stress Testing and Other Risk Management Tools

### The Role of Stress Testing

#### LO 68.1: Describe the relationship between stress testing and other risk measures, particularly in enterprise-wide stress testing.

### Complementing Stress Tests with VaR Models

#### LO 68.2: Describe the various approaches to using VaR models in stress tests.

### Stressed Input and Stressed VaR

#### LO 68.3: Explain the importance of stressed inputs and their importance in stressed VaR.

### Stressed Risk Metrics Advantages and Disadvantages

#### LO 68.4: Identify the advantages and disadvantages of stressed risk metrics.

## Principles for Sound Stress Testing Practices and Supervision

### Stress Testing in Risk Management

#### LO 69.1: Describe the rationale for the use of stress testing as a risk management tool.

#### LO 69.2: Describe weakness identified and recommendations for improvement in:

- The use of stress testing and integration in risk governance
- Stress testing methodologies
- Stress testing scenarios
- Stress testing handling of specific risks and products