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Book 4: Valuation and Risk Models - Coggle Diagram

- - - - The traditional mean-variance model estimates the amount of financial risk for portfolios in terms of the portfolio’s expected return (mean) and risk (standard deviation or variance)
      - A necessary assumption for this model is that return distributions for the portfolios are elliptical distributions
      - The efficient frontier is the set of portfolios that dominate all other portfolios in the investment universe of risky assets with respect to risk and return
      - When a risk-free security is introduced, the optimal set of portfolios consists of a line from the risk-free security that is tangent to the efficient frontier at the market portfolio
    - - Assumptions
        
        (1) all investors assume the same means, standard deviations, and correlation coefficients for investments
        
        (2) mean and standard deviation are all that matter
        
        (3) all investors can borrow at the risk-free rate
      - unreliable when the underlying return distribution is not normal or elliptical
      - The standard deviation is not an accurate measure of risk and does not capture the probability of obtaining undesirable return outcomes when the underlying return density function is not symmetrical
      - The standard normal distribution has a mean of zero and a standard deviation of one
      - The z-score can be used to determine the cumulative probability of a return landing at or below a specific value
      - Returns on risky financial assets tend to exhibit a more peaked distribution with fatter tails, implying that the probability of more extreme events is higher than would be expected with the normal distribution
    - - Value at risk (VaR) is a risk measurement that determines the probability of an occurrence in the left-hand tail of a return distribution at a given confidence level
      - VaR can be defined as [µ – (z)(σ)]. The underlying return distribution, arbitrary choice of confidence levels and holding periods, and the inability to calculate the magnitude of losses result in limitations in implementing the VaR model when determining risk
  - - - ρ(βX) = βρ(X) for h > 0
      - portfolio size impacts risk
    - - Y ≥ X ⇒ ρ(Y) ≤ ρ(X)
      - greater future returns imply less risk
    - - : ρ(c + X) = ρ(X) – c
      - adding cash reduces risk by the amount of cash
    - - Expected shortfall (ES) is a more accurate risk measure than VaR
      - ES satisfies all the properties of coherent risk measurements including subadditivity
      - The portfolio risk surface for ES is convex since the property of subadditivity is met. Thus, ES is more appropriate for solving portfolio optimization problems than the VaR method
      - ES gives an estimate of the magnitude of a loss for unfavorable events. VaR provides no estimate of how large a loss may be.
      - ES has less restrictive assumptions regarding risk/return decision rules
      - Subadditivity, the most important property for a coherent risk measure, states that a portfolio made up of subportfolios will have equal or less risk than the sum of the risks of each individual subportfolio
      - VaR violates the property of subadditivity
    - - ES is a special case of the risk spectrum measure where the weighting function is set to 1 / (1 – confidence level) for tail losses that all have an equal weight, and all other quantiles have a weight of zero
      - The VaR is a special case where only a single quantile is measured, and the weighting function is set to one when p-value equals the level of significance, with all other quantiles having a weight of zero
    - - If regulators use non-subadditive risk measures to set capital requirements, then a financial firm might be tempted to break itself up to reduce its regulatory capital requirements
      - ρ(X + Y) ≤ ρ(X) + ρ(Y)
      - portfolio risk ≤ sum of individual asset risks
      - Non-sub-additive risk measures can tempt agents trading on an organized exchange to break up their accounts with separate accounts for separate risks in order to reduce their margin requirements
      - If risks are sub-additive, adding risks together would give us an overestimate of combined risk
- - - - Historical simulation involves identifying risk factors, collecting data on these factors, and using scenario analysis to simulate daily portfolio changes to calculate daily gains or losses of a portfolio
      - The returns are then ranked to calculate the VaR. The expected shortfall is the average of the losses greater than the VaR level
    - - The delta-normal method can be used for portfolios that are linearly dependent on the underlying market variables
      - Assuming the returns on these variables is multivariate normal, then portfolio value changes will be normally distributed
      - The central limit theorem states that the portfolio can be approximately normal with many nonnormal variables
      - The delta-normal method is only an approximation for nonlinear products, because delta is a linear parameter that works well only for small price changes but does not work well for large price changes
      - The gamma parameter helps adjust for the delta’s linearity through nonlinearity
      - Therefore, the delta-normal method understates the probability of high option values and overstates the probability of low option values, creating problems for portfolio managers
    - - The full revaluation approach calculates the VaR of the derivative by valuing the derivative based on the underlying value of the index after the decline corresponding to an x% VaR of the index
      - This approach is accurate, but can be highly computational
      - Both the delta-normal and full revaluation methods measure the risk of nonlinear securities
      - The full revaluation approach can be highly computational, therefore, it does not work well for portfolios of more complex derivatives such as mortgage-backed securities, swaptions, or options with embedded features
      - The delta-normal method calculates risk using either the delta approximation, which is linear or the delta-gamma approximation, which adjusts for the curvature of the underlying relationship
      - This approach simplifies the calculation of more complex securities by approximating the change
  - - - The Monte Carlo approach simulates thousands of possible movements in the underlying asset and then uses those outcomes to estimate the VaR for a portfolio of derivatives
      - 6 Steps
        
        Using the sampled values of Δxi, determine the values of the risk factors at the end of the period.
        
        Revalue the portfolio using the updated risk factor values.
        
        Apply sampling techniques from the multivariate normal probability distribution for the change in x (Δxi).
        
        Subtract the revalued portfolio value from the current value. This will determine the amount of loss.
        
        Using current values of risk factors, value the portfolio today.
        
        Repeat steps two–five to create a loss distribution.
      - An advantage of the Monte Carlo approach is that it can assume any distribution type, and is able to address multiple risk factors by generating correlated scenarios based on a statistical distribution
      - The main disadvantages are that the process is slow and computationally intensive
    - - Crisis events cause diversification benefits to deteriorate due to a contagion effect that occurs when a rise in volatility and correlation result in a different return generating process for the underlying asset
      - This creates problems when using simulations for traditional scenario
    - - The worst-case scenario (WCS) extends VaR risk measurement by estimating the extent of the loss given the worst result within a specific timeframe
      - WCS indicates the maximum a portfolio can lose over a given period
- - - - Contrary to when volatilities change slowly, volatilities may also change quickly and abruptly
      - The abrupt change in volatilities is referred to as regime switching
  - - - parametric approach
        
        typically assumes asset returns are normally or lognormally distributed with time-varying volatility
        
        historical standard deviation
        
        exponential smoothing
        
        delta-normal method
      - nonparametric approach
        
        less restrictive in that there are no underlying assumptions of the asset returns distribution
        
        historical simulation
        
        the multivariate density estimation method
        
        a weight is assigned to the historical data depending on how similar it is to the current data
        
        Advantages
        
        MDE allows for weights to vary based on how relevant the data is to the current market environment, regardless of the timing of the most relevant data
        
        MDE is very flexible in introducing dependence on economic variables (called state variables or conditioning variables)
        
        Fat tails, skewness, and other deviations from some assumed distribution are no longer a concern in the estimation process for nonparametric methods
        
        Nonparametric models do not require assumptions regarding the entire distribution of returns to estimate VaR
        
        Disadvantages
        
        Therefore, large sample sizes are required to precisely estimate volatility using historical simulation
        
        Separating the full sample of data into different market regimes reduces the amount of usable data for historical simulations
        
        Data is used more efficiently with parametric methods than nonparametric methods
        
        MDE may lead to data snooping or over fitting in identifying required assumptions regarding the weighting scheme identification of relevant conditioning variables and the number of observations used to estimate volatility
        
        MDE requires a large amount of data that is directly related to the number of conditioning variables used in the model
        
        σt 2 = (rt–K,t–K+1 2 + ... + rt–3,t–2 2 + rt–2,t–1 2 + rt–1,t 2 ) /K
    - - A value at risk (VaR) method for estimating risk can be based either on historical-based approach or an implied-volatility-based approach
      - Historical-based approaches of measuring VaR typically fall into two subcategories: parametric and nonparametric
  - - - implied from options prices, which are dependent on volatilities
      - less than annual volatilities are adjusted by the square root of time
      - Chicago Board Options Exchange (CBOE) Volatility Index (VIX)
    - - generates volatility estimates based on weightings of the last estimate of volatility and the latest current price change information
      - λ = weight on previous volatility estimate
        (λ is a positive constant between zero and one)
      - adaptive volatility estimation
      - The exponentially weighted moving average (EWMA) model
      - was used by RiskMetrics, formerly a division of JPMorgan
      - The day before would have weight equivalent to (1 – λ)λ
    - - not only incorporate the most recent estimates of volatility and return, but also incorporate a long-run average level of variance
      - α = weighting on the previous period’s return
      - β = weighting on the previous volatility estimate
      - ω = weighted long-run variance = γVL
      - estimates of volatility have a better theoretical justification than the EWMA model
      - In the event that model parameter estimates indicate instability, however, EWMA volatility estimates may be used
    - - Volatility tends to exhibit a mean-reverting characteristic, meaning that if current volatility is high, it is expected to decline; if it is low, it is expected to increase
      - If volatility is forecast to decline, we will overestimate volatility if we multiply the standard deviation of the return by the square root time
      - GARCH (1,1) provides a pull back toward the long-term average mean, EWMA does not
      - If mean reversion exists, the long horizon risk (and resulting VaR calculation) will be smaller than the square root volatility
      - Correlations can be estimated from covariances
        
        Correlation between two assets is their covariance divided by the product of their standard deviations
      - Persistence
        
        It’s the model with the highest persistence that takes the longest time to revert to its mean
        
        α + β
        
        If we assume that the long-run estimated variance remains unchanged as α and β increase, the volatility term structure predicted by the model reverts to the long-run estimated variance more slowly.
      - for GARCH, When σt is lower than the long-run average, the volatility term structure would go up
  - - - there is a 5% chance that on any given day, the portfolio will experience a loss of
        $10,000 or more
- - - - the rate at which defaults happen at a specific time and can be used to calculate unconditional default probabilities
      - Assuming that h is the average hazard rate between two time periods, and that h1 and h2 are the average hazard rates between times t 1 and t 2
      - unconditional default probability between t 0 and t: 1 − e^(−ht)
      - probability of survival (also known as the survival rate): e^(−ht)
      - unconditional probability of default between times t 1 and t 2: e^(−h1t1) − e^(−h2t 2)
    - - The recovery rate (for a bond) is equal to its value just after default, expressed as a percentage of face value
      - The loss given default is equal to 100 – the percentage recovery rate
      - So over a period of time, the expected loss is equal to either the probability of default × the loss given default, or the probability of default × (1 – recovery rate)
      - EL = PD * LGD
    - - A rating agency has two options for how they will rate issuances
      - they can either rate through-the-cycle or at a point-in-time
      - Through-the-cycle will evaluate creditworthiness over several years
      - thereby insulating it from the inevitable ups and downs of business cycles
      - Point-in-time is the current best estimate of the probability of future default
      - Given the goal of ratings stability, the through-the-cycle approach tends to be the one used by agencies
    - - Moody’s
      - Standard and Poor’s (S&P)
      - Fitch
      - serve as external sources of credit risk data, providing independent opinions on credit risk
      - The focus of these agencies is on money market instruments and bonds issued by governments and corporations
      - Because rating agencies usually only focus on firms with publicly traded instruments, banks who lend money to borrowers without external ratings may rely on their own internal rating systems
    - - typically issued for an instrument issued by an entity, as opposed to a rating on the entity itself
      - The rating is established for the purpose of assessing how likely it is that an entity will default on its obligations
      - For all three agencies, the probability of default increases down the scale
      - A key dividing line is where investments are considered investment grade or noninvestment grade (also known as speculative grade or junk bonds)
      - investment grade (Baa3 and BBB–)
      - noninvestment grade
      - the impact of time horizon, economic cycle, industry and
        geography on external ratings
        
        Although external ratings have had a fairly good record in indicating relative rates of default, they are designed to be relatively stable over the business cycle
        
        using an average cycle approach
        
        A goal of rating agencies is to have ratings which reflect the same thing across different countries and different industries, such that a BB rating in one country is equivalent to a BB rating in another
        
        The probability of default given any rating at the beginning of a cycle increases with the horizon
        
        From an industry perspective, ratings consistency is more difficult to capture. Financial corporations have had higher default rates than nonfinancial corporations with the same rating, although past experiences are certainly not indicative of the future
  - - - A rating transition matrix is designed to show the probability that a bond issuer may go from one rating to another during a one-year time period
      - Based on historical data, investment grade bonds had a significantly higher chance of maintaining their ratings over the course of the year than noninvestment grade bonds
      - The ratings matrix phenomenon reflects the fact that a downgrade (upgrade) in one year has a higher likelihood of being followed by another downgrade (upgrade) in the next year
    - - Ratings downgrades tend to have a significant impact on bond and stock prices, whereas upgrades have had much less of an impact
      - Also, bonds and stocks do not seem to contain any information regarding the future outlook for ratings changes
    - - Rating agencies have been fairly successful in the area of money market instruments and bonds
      - Ratings for these products were based on modeling, which tended to be overly optimistic and easily solvable by issuers looking to create products that would garner higher ratings
      - A critical failure was in the structured products market, which was one of the drivers of the economic crisis in 2007–2009
      - Rating agency independence has also been questioned, although more significant regulations are in place now to prevent some of the issues identified from the past
    - - These estimates are point-in-time, and because the models are more responsive to changing circumstances, the goal of stability is not a factor. Internal ratings may also be developed and used to evaluate the creditworthiness of potential borrowers
      - In contrast to external ratings, there are organizations which use models that incorporate factors such as firm equity market value, equity volatility, and debt in the firm’s capital structure to estimate default probabilities
        
        Kamakura and KMV
    - - Internal rating systems used to evaluate potential borrowers are developed by financial institutions for three primary reasons
        
        (2) accounting standards require banks to account for default probabilities when they value loans on their balance sheet
        
        (3) probabilities of default drive regulatory credit risk capital
        
        (1) there may not be external ratings available
- - - - more mature markets and companies within those markets are less risky than those firms and countries in the early stages of growth
    - - corruption in the country
      - the occurrences of physical violence due to wars or civil unrest
      - the possibility of nationalization and expropriations
      - the continuity and severity of risks versus discontinuous risks
    - - the protection of property rights and the speed with which disputes are settled affect default risk
    - - the level of economic diversification and competitive advantage are important factors in assessing country risk
      - A disproportionate reliance on a single commodity or service in an economy increases a country’s risk exposure
    - - Political Risk Services (PRS)
        
        evaluated using 22 variables to measure risks
        
        Political risk
        
        financial risk
        
        economic risk
      - Media outlets
        
        Euromoney survey of 400 economists
        
        The Economist
        
        assesses currency risk, sovereign debt risk, and banking risks to develop country risk scores
      - The World Bank (assess six areas)
        
        government effectiveness
        
        political stability
        
        the level of corruption
        
        rule of law
        
        accountability
        
        regulatory quality
      - Some are critical of these composite risk measures because not all measures are equally important to all investors
      - they are not readily comparable with each other due to a lack of standardization across the information providers
      - Also, the scores are better used as rankings than as a way to interpret the relative risk of countries
  - - - Countries are often without the foreign currency to meet the debt obligation and are unable to print money to repay the debt
    - - (1) Gold reserves, the use of the gold standard prior to 1971 made it more difficult for some countries to print money
      - (2) Currency union, shared currencies, such as the euro, make it impossible for countries to control their own monetary policy
      - (3) Currency debasement, some counties must conclude that the costs of currency debasement and potentially higher inflation are greater than the costs of default
    - - Sovereign defaults may be preceded by difficult economic conditions
      - Defaulting countries typically suffer reputational loss, increasing future borrowing costs and reduced investment in stock and bond markets
      - Countries may also be faced with an economic downturn and political instability as a result of a default
    - - Several factors determine a country’s sovereign default risk
      - its social security commitments
      - The country’s level of indebtedness
      - the level and stability of tax receipts
      - political risks
      - implicit guarantees from other countries all impact a country’s likelihood of defaulting on sovereign debt
      - Rating agencies consider several factors when evaluating default risk
        
        These factors are related to the economic, political, and institutional characteristics of a country with respect to its ability to repay debtThese factors are related to the economic, political, and institutional characteristics of a country with respect to its ability to repay debt
        
        Rating agencies also assess rating transitions, looking at the probability that a country would maintain its current rating or transition to a different rating category
    - - The sovereign credit spread reflects the sovereign bond yield compared to a riskless investment in the same currency and maturity
      - Advantages of the sovereign risk spreads relative to sovereign bond ratings are that changes occur in real time and risk premiums adjust to new information more quickly
      - Credit default swaps (CDSs), which are similar to insurance contracts, provide important information about sovereign credit spreads
- - - - economic capital measures a bank’s own estimate of capital
      - regulatory capital is the capital regulators require banks to keep
      - Basel II requires that banks calculate credit risk capital either under the standardized approach or the internal ratings-based (IRB) approach
      - Current measures used to evaluate credit risk
        
        loss given default (LGD), which represents the likely percentage loss if the borrower defaults
        
        exposure at default (EAD), which can be stated as a dollar amount (e.g., the loan balance outstanding) or as a percentage of the nominal amount of the loan or the maximum amount available on a credit line
        
        probability of default (PD), which is the likelihood that a borrower will default
        
        The probability of default is a binomial distribution
        
        The uniform distribution assumes that all defaulted borrowers would have the same probability of losing anywhere between 0 percent and 100 percent
    - - EL = EAD × PD × LGD
      - The difference between the actual loss and expected loss is called the unexpected loss (UL)
      - UL typically exceeds EL, but they are both equal to zero when probability of default is zero
    - - A bank’s own estimate of the capital it requires is called economic capital
      - A bank must hold enough economic capital to buffer itself from unexpected losses in order to continue to operate
  - - - σ = √ (PD − PD2 )× [L(1 − RR)]
      - For a portfolio consisting of n loans, the loss on the ith loan default with a face value of L will be Li (1 – RRi )
      - If all loans have the same principal L and the standard deviation of the loss from loan i is the same for all i, then the standard deviation simplifies to
      - PDi × Li(1 – RRi) + (1 – PDi) × 0 = PDi × Li(1 – RRi)
      - α = σ √ [1 + (n − 1) ρ ] / [√ n L ]
    - - a mathematical model that is used to determine regulatory capital
      - assess counterparty risk by defining both EL and UL
      - The model looks at the left tail distribution of defaults
      - UL = (WCDR – PD) × LGD × EAD
      - WCDR = worst case default rate, or the 99.9 percentile of the default rate distribution
      - WCDR
      - to determine regulatory capital
    - - The CreditMetrics model is used to determine economic capital
      - Borrowers are assigned credit ratings and Monte Carlo simulation is used to assess the probability that the borrower’s rating could change during a one-year period
      - The CreditMetrics model differs from the Vasicek model because it not only considers the impact of defaults, but also the impact of rating changes
      - Default correlations are very difficult, if not impossible, to observe
      - If the loan portfolio contains ‘n’ loans, [n(n-1)]/2 pairwise default correlations need to be estimated
      - Default correlations are small, but positive providing considerable benefits to diversification in credit portfolios
    - - Euler’s theorem lets a portfolio’s risk functions to be decomposed into their individual contributions and determines what the contribution of each underlying loan to the overall loan portfolio risk is
    - - A copula creates a joint probability distribution between two or more variables while maintaining their individual marginal distributions.
      - The Gaussian copula model maps the probability distributions of two variables that may have unique shapes to the standard normal distribution
      - The Gaussian copula is used by both banks and regulators to map the probability distributions of two variables that may have unique shapes to the standard normal distribution, where the normal distribution has a mean of zero and a standard deviation of one. Mapping of variables is done on a percentile-to-percentile basis.
  - - - (3) challenges in calculating EAD for derivatives, which may also be subject to wrong-way risk
      - (4) difficulty in estimating correlations
      - (2) economic downturns lead to higher default rates while at the same time increasing LGD
      - (5) other risks including market, operational, liquidity, and strategic risk all impact a bank’s required capital
      - (1) banks’ need to calculate
        both through-the-cycle PD and point-in-time PD
- - - - The Basel definition of operational risk is “the risk of loss resulting from inadequate or failed internal processes, people and systems or from external events.”
      - 7 Types
        
        (4) clients, products, and business practices
        
        (5) damage to physical assets
        
        (3) employment practices and workplace safety
        
        (6) business disruption and system failures
        
        (2) external fraud
        
        (7) execution, delivery, and process management
        
        (1) internal fraud
    - - three methods for calculating operational risk capital requirements
        
        (2) the standardized approach
        
        the bank similarly calculates gross income, but uses eight business lines to calculate a percentage capital charge
        
        Commercial banking 15%
        
        Payment and settlement 18%
        
        Retail banking 12%
        
        Agency services 15%
        
        Trading and sales 18%
        
        Asset management 12%
        
        Corporate finance 18%
        
        Retail brokerage 12%
        
        (3) the advanced measurement approach (AMA)
        
        Large banks are encouraged to move from the standardized approach to the AMA in an effort to reduce capital requirements
        
        it would have 56 different combinations to estimate 1-year losses
        (56 = 8 business lines × 7 risk categories)
        
        Even if a firm is well protected from high-severity operational risk events, the news of such events will still take a toll on its reputation.
        
        To help keep the behavior of the insured in check, insurers use a host of features that include deductibles, policy limits, and coinsurance provisions.
        
        Adverse selection creates a high-risk pool of policies that are likely to result in claims, increasing the cash outflows of the insurer relative to inflows.
        
        Most insurance policies have a cover ceiling, meaning there’s always a cap on the compensation that could be handed to the policyholder.
        
        Under the scorecard approach, each unit manager is subjected to a survey which has questions on matters risk. Each manager’s responses are transformed into a quantitative measure to come up with an overall score. This total score represents the unit’s exposure to risk.
        
        (1) the basic indicator approach
        
        gross income
        = interest earned + noninterest income – interest paid
        
        operational risk capital is based on 15% of the bank’s annual gross income over a 3-year period, where gross income includes both net interest income and noninterest income
  - - - a BI component, and a loss component. For an average bank, the BI component and the loss component should be equal
      - the loss component: 7X + 7Y + 5Z
      - Under the standardized measurement approach (SMA) a bank is required to calculate a business indicator (BI)
      - X, Y, and Z are estimates of 10-year average annual operational risk losses
      - It is used to determine the bank’s internal loss multiplier (ILM)
    - - Operational risk losses can be classified along two dimensions
      - loss frequency
        
        the number of losses over a specific time period
        
        Poisson distribution
        
        a distribution
        that models random events
        
        sing the expected number of losses as λ, the
        probability of n losses during a year is
      - loss severity
        
        loss severity is defined as the size of a loss, should a loss occur
        
        lognormal distribution
    - - Estimation procedures
        
        Banks should use internal data to combine objective data with subjective judgment when estimating the frequency of losses, and utilize both internal and external data when estimating the severity of losses
        
        Regarding external data, banks can use sharing agreements to estimate losses experienced by other financial institutions
      - Potential biases
        
        Vendor data may contain reporting biases, and as a result vendor data is more appropriate when used to determine relative loss severity
        
        It is generally estimated that a β of 0.23 provides a good fit.
        
        Loss severity estimates should be adjusted for inflation and also adjusted for loss size using a scale adjustment that applies the loss event to the financial institution’s own circumstances
    - - Scenario analysis is a method for obtaining additional operational risk data points
      - It is used to estimate losses for high-severity, low-frequency events using a bank’s own previous experience or hypothetical scenarios
      - Regulators encourage the use of scenarios because they allow management to incorporate events that have not yet occurred
  - - - Forward-looking approaches are also used to discover potential operational risk loss events
        
        (2) risk and control self-assessment (RCSA)
        
        to survey those managers directly responsible for the operations of the various business lines, including interviewing managers and staff, conducting questionnaires, and reviewing internal and external risk reports
        
        (3) key risk indicators (KRI)
        
        (1) have a predictive relationship to losses
        
        (2) be accessible and
        measurable in a timely fashion
        
        (1) causes of losses
        
        Educating employees about good business practices is another important step in mitigating operational risk
    - - Allocating operational risk capital to each business unit encourages managers to improve their management of operational risks
      - However, this should be combined with a cost-benefit analysis
    - - The power law is useful in evaluating the nature of the loss tail of a given distribution
      - The use of this law is appropriate because operational risk losses are likely to occur in the tails
      - The power law holds well for the large losses experienced by banks
      - Loss data (internal or external) and scenario analysis are employed to estimate the power law parameters using the maximum likelihood approach
      - When loss distributions are aggregated, the distributions with the heaviest tails tend to dominate
      - The loss with the lowest alpha defines the extreme tails of the total loss distribution.
    - - moral hazard
        
        A moral hazard occurs when an insurance policy causes a company to act differently (that is, riskier behavior) with insurance protection
      - adverse selection
        
        Adverse selection occurs when an insurance company cannot differentiate between good (low risk) and bad (high risk) insurance risks
- - - - Selecting a time horizon is the first step in creating a stress-test scenario, with the most common periods ranging from three months to two years
      - Scenarios can be chosen internally or set by outside regulators and, if based on historical data, are predicated on the assumption that variables will behave in the future how they have in the past
      - As an entity creates scenarios for stress testing, it may choose to focus on large changes in critical variables, including equity prices, interest rates, GDP, unemployment rates, overall volatility, commodity prices, default rates, and exchange rates
      - The most basic models typically only stress a few key variables. However, a more comprehensive approach to modeling captures the potential behavior of a range of variables. While the immediate impact of adverse scenarios must be captured, models should also reflect knock-on effects, which reflect the consequences of other firms’ reactions to adverse events
      - While stress testing involves creating scenarios and evaluating their impacts, reverse stress testing looks at the combination of situations that could result in the demise of a financial institution
    - - Stressed VaR and stressed ES use data from periods that are particularly stressed
      - allows an institution to understand how it might perform if that period were to occur again
    - - Advantages
        
        forward-looking
        
        lend themselves to hypothetical (what-if) scenarios
        
        focus on relatively long time horizons
      - Disadvantages
        
        risk metrics will not necessarily respond to current market conditions
        
        do not produce a probability distribution for losses
        
        cannot feasibly be backtested
  - - - Board of directors
        
        The board of directors is accountable for the entire organization and must be sufficiently knowledgeable about the organization’s stress-testing activities
        
        Stress-test results inform the board of the institution’s risk appetite, risk profile, and operating and strategic decisions. The results may serve as early warning signs of upcoming pressures
        
        Boards should actively challenge the results of stress tests and supplement them with other tests as well as both quantitative and qualitative information
      - Senior management
        
        Senior management, with oversight from the board, is responsible for establishing robust policies for stress tests, which could supplement other risk, capital, and adequacy measures
        
        Management is also responsible for reviewing and coordinating stress-testing activities, assigning competent staff, challenging results and assumptions, and incorporating remedies to potential problems.
        
        Senior management should ensure there is a sufficient range of stress-testing activities to evaluate risks.
        
        Results should be benchmarked and regularly updated given that risks, data sources, and the operating environment can change
        
        An independent auditor should verify test results
        
        Senior management should regularly report to the board of directors on stress-test results in a clear and concise way, highlighting the key elements of the stress-testing activities and any limitations
    - - Stress-testing activities should be governed by clear and comprehensive policies and procedures that are kept up to date and documented in an appropriate manner
      - Appropriate documentation allows senior management to track and analyze results over time
      - address a range of issues
        
        Determining frequency and priority of stress-testing activities
        
        Tracking for how stress-test results change over time
        
        Outlining the stress-testing process, including scenario design and selection
        
        Establishing consistent and adequate stress-testing practices, including how independent reviews will function
        
        Indicating the use of stress tests and who uses the tests
        
        Providing documentation for how third-party models and software acquisitions function
        
        Describing roles and responsibilities
        
        Providing transparency to third parties regarding the stress-testing process, which allows third parties to evaluate the tests and their components
        
        Describing the overall purpose of stress tests
        
        Updating and reviewing policies and procedures to remain consistent with the institution’s risk appetite, risk exposures, and changing market conditions
    - - Ongoing validation and independent review of stress-testing activities is an important component of an institution’s governance
      - Validation and independent review should be unbiased and critical, and it should form part of the institution’s overall validation and review processes
      - Reviews should be based on sound theory, acknowledge limitations, cover subjective elements, and monitor results on a routine basis.
      - Reviews should also account for the reality of increased variable correlations during stressed market conditions, with multiple models used together typically providing a more accurate means of assessing potential losses.
    - - The internal audit assesses the integrity, consistency, and reliability of policies and procedures
      - It should verify that stress tests are conducted thoroughly and as intended, by staff with the relevant expertise
      - The internal audit should also review the procedures relating to the documentation, review, and approval of stress tests
      - Any deficiencies and areas for potential improvement should be identified and communicated to senior management
    - - principles
        
        facilitating risk mitigation strategies
        
        supporting communications
        
        driving risk tolerance levels
        
        overcoming modeling limitations
        
        assessing risk in the futur
        
        helping to drive liquidity
        
        capital planning procedures
      - Specifically, stress testing should have clear and formal objectives, include an , , and, ,, include , , and
        
        be adequately resourced
        
        capture material and relevant risks
        
        be used as a risk management tool
        
        effective governance structure
        
        be subject to review and challenge
        
        be communicated within and across jurisdictions
        
        appropriate models and methodologies
        
        be supported by granular data and robust IT systems
    - - While institutions should have governance processes in place for conducting and utilizing stress testing, external regulatory bodies have stress-testing requirements in place that must be followed depending on the size of the institution
      - Key elements of effective governance and controls over stress testing include the governance structure, policies and procedures, documentation, validation and independent review, and internal audit
- - - - (1) the coupon payments
      - (2) the return of principal
  - - - Instruments with identical future cash flows should sell for the same price, which is known as the law of one price
      - Arbitrage opportunities allow investors to exploit mispricings when two securities have identical future cash flows but different current prices.
      - By short selling a rich security and using the proceeds to purchase a similar cheap security, an investor can take advantage of an arbitrage opportunity and make a riskless profit with no investment
      - Short positions are important considerations for arbitrage, but if the security pays income (coupon for bonds or dividends for equities), the investor must pay this income to the lender of the security
  - - - Shorter-term STRIPS tend to trade rich while longer-term STRIPS tend to trade cheap
      - Shorter-term C-STRIPS tend to trade at a premium, while longer-term C-STRIPS tend to trade at a discount.
    - - To create a replicating portfolio using multiple fixed-income securities, we must determine the face amounts of each fixed-income security to purchase, which match the cash flows from the bond we are trying to replicate.
    - - Clean and dirty bond pricing
        
        clean price = dirty price – accrued interest
      - Accrued interest
        
        Accrued interest is an important consideration when trading fixed-income securities. It is the interest earned between coupon dates, calculated as the coupon interest times the ratio of time passed since the last coupon and time between coupon dates.
        
        Valuation of bonds between coupon payment dates requires the calculation of accrued interest and a modification to the bond pricing formula. Values derived between coupon dates will include accrued interest. This is also known as the dirty price. Subtracting the accrued interest from the dirty price gives the clean price of the bond.
    - - Actual/360
        
        U.S. corporate and municipal bonds
      - 30/360
        
        money market securities in Canada, New Zealand, and Australia
      - Actual/actual (in period)
        
        U.S. government bonds pay coupons semiannually
- - - - The spot rate is the rate earned on an investment when it is received at a single point in the future
      - The future value received is linked to the current value today via the spot rate
      - The spot rate and the discount factor are related in that either can be used to equate the present and future value of an investment over a period of time
    - - Forward rates are future spot rates based on current spot rates
      - The forward rate is the rate earned on a future-year investment that should make an investor indifferent between shorter and longer investment options for the same instrument
      - Spot rates are equal to forward rates compounded for successive periods
      - A forward rate agreement (FRA) guarantees a specific rate to be paid or earned during a future period
      - The value of the FRA is equal to the present value of the difference between the guaranteed rate and the current forward rate
      - When the guaranteed rate is higher than the current forward rate, the value of the FRA is positive
      - when the guaranteed rate is lower than the current forward rate, the value of the FRA is negative
    - - The par rate at maturity is the rate at which the present value of a bond equals its par value
      - Par rates represent bond coupon payments when a bond’s price is equal to its par value
      - Annuity factors, which represent the sum of each period’s discount factor, can be used to calculate the par rate.
      - The par rate and annuity factors can also be used to value bonds with other coupon rates.
    - - When the term structure is upward sloping, forward rates will be the highest, with spot rates in the middle and par rates as the lowest rates
      - When the term structure is downward sloping, par rates will be the highest, with spot rates in the middle and forward rates as the lowest rates.
      - When the term structure of rates is flat, all spot rates are equal and par and forward rates will both equal spot rates
    - - If the forward rate for the final period is less than the coupon rate, the value of the bond will fall
      - Investors make borrowing and investing decisions based on their expectations of forward rates relative to the actual forward rates implied by looking at investment alternatives that span different maturities
      - The value of a bond will rise if the forward rate for the final period is greater than its coupon rate, which is what tends to happen in an upward-sloping term structure
    - - The London Interbank Offered Rate, otherwise known as LIBOR, is an estimated unsecured borrowing rate between banks
      - quoted for many time periods and for many currencies
      - it is often used as a reference rate for derivatives and other transactions
      - A swap is a derivatives transaction where two parties agree to exchange payments based on the movement of an underlying asset
      - A fixed rate for floating rate swap involves one party making payments based on a fixed rate and receiving payments based on a floating rate, while the counterparty has the opposite position
      - The notional principal is the amount that the interest rates are applied to in order to determine the net payment each period
      - The swap market is used to define par rates, in that an n-year swap rate defines an n-year bond selling for par
    - - The overnight indexed swap (OIS) is a swap where every period a fixed rate is exchanged for a floating rate (where the floating rate is the geometric average of the overnight federal funds rates during the period)
      - The fixed rate in the OIS is known as the OIS rate
- - - - The present value of an annuity (equal future cash flows) can be found by using the YTM and cash flows
      - The present value, or price, of a perpetuity can be found by dividing the coupon payment by the YTM.
    - - When pricing a bond, YTM or spot rates can be used. The YTM will be a blend of the spot rates for the bond
      - The YTM declines as the coupon rate increases. If the term structure of interest rates is downward sloping, the YTM increases as the coupon rate increases
    - - The coupon effect describes a scenario where two bonds with identical maturities but different coupons will have different yields to maturity
      - The bond with the
        smaller coupon will be more sensitive to interest rate changes
      - When a bond is trading above par, the coupon rate is greater than the YTM, and the bond then trades at a premium
      - When the bond is trading at par, the coupon rate is equal to the YTM
      - When the bond is trading below par, the coupon rate is less than the YTM, and is said to trade at a discount
      - Japanese Yields
  - - - the realized return when this realized return is different from
        what was assumed under the carry roll-down
    - - component accounts for price changes due to changes in the
        bond’s spread relative to other bonds
    - - Consider the impact of financing
        
        If financing is considered, the cost of financing should be added as a fourth component of the P&L, and both gross and net returns would be calculated.
      - Consider accrued interest on both the initial and final valuation dates
      - P&L is generated by price appreciation plus cash-carry, which consists of explicit cash flows like coupon payments and financing costs
        
        P&L due to carry is meant to convey how much a position earns due to the passage of time, holding everything else constant.
        
        P&L due to roll-down is meant to convey how much a position earns due to the fact that. as a security matures, its cash flows are priced at earlier points on the term structure
        
        The P&L due to the passage of time excluding cash-carry is called carry-roll-down
    - - For an expected term structure, no-change scenarios include realized forward, unchanged term structure, and unchanged yields
      - the estimated return from bond price movements and
        coupon payment assuming no change to interest rate expectations
      - The realized forward scenario assumes that forward rates are equal to expected future spot rates
      - The unchanged term structure scenario assumes that the term structure will remain unchanged
      - The unchanged yields scenario assumes that bond yields remain unchanged.
- - - - Effective duration measures the percentage change in a security’s value for a particular unit’s change in rates
      - It can be interpreted as the percentage change in a bond’s price for a 100 basis point change in rates
    - - the bond’s duration can be calculated as if the bond would not be called in the future
      - An alternative approach is to calculate duration as the weighted probability that the bond would not be called and the probability that it would be called
      - calculates duration valuing the bond’s price today, revaluing the price for an assumed 1 basis point parallel change in interest rates
      - calculating duration from the percentage change in the bond’s
        price
      - The effective duration of a putable bond is calculated similar to callable bonds
    - - DV01 works better for hedgers, while duration is more convenient for traditional investors
    - - Convexity is the sensitivity of duration to changes in interest rates
      - Convexity is positive and will always increase the price calculated by duration only
      - Convexity is the second derivative of the price rate function
      - Convexity explains why the price of bond falls less and rises more in the case of changes in interest rates
      - Convexity enhances the bond’s return
    - - Hedging an investment with a value V and a duration DV can be done with a bond with value P and duration of DP
      - For the investment to be fully hedged, it must be that ΔV = ΔP. We can then rearrange the equation to state
      - An investor could also hedge an investment using both duration and convexity
      - This could be done by using two bonds with prices P1 and P2, durations D1 and D2, and convexities C1 and C
      - An investor could also hedge an investment using both duration and convexity. This could be done by using two bonds with prices P1 and P2, durations D1 and D2, and convexities C1 and C2
    - - A barbell strategy is typically used when an investment manager uses bonds with short and long maturities, thus forgoing any intermediate-term bonds
      - A bullet strategy is used when an investment manager buys bonds concentrated in the intermediate maturity range
  - - - The goal of a hedge is to produce a combined investment + hedge position that will not change in value for a small change in yield
    - - (1) assuming a parallel
        shift in the interest rate term structure
      - (2) using yields by taking the YTM and recalculate the bond prices by increasing and decreasing the YTM by the same basis point change, and averaging the two prices
      - Analysts can also distinguish between yield-based DV01, and changes in bond prices from changes in spot rates (DVDZ or DPDZ) or forward rates (DVDF or DPDF)
- - - - Principal components analysis is a technique used to analyze term structure movements in historical data
      - The technique identifies multiple factors that impact the movements in the term structure, with the first two to three typically having the biggest impact
      - Factor scores are variable values relating to a specific data point, with their standard deviations aligned with the relative importance of each factor.
      - The sum of the variance of all of the factor scores equals the total variance of all rate movements
      - The factors themselves may reflect situations where all rates move in the same direction (similar to a parallel shift)
      - situations where short-term and long-term rates move in different directions
      - situations where intermediate rates move in the opposite direction to short- and long-term rates
    - - Key rate exposures help describe how the risk of a bond portfolio is distributed along the term structure, and they assist in setting up a proper hedge for a bond portfolio
      - Partial ‘01s are used for measuring and hedging risk in swap portfolios (or a portfolio with a combination of bonds and swaps)
      - These partial ‘01s are derived from the most liquid money market and swap instruments for which a swap curve is usually constructed
      - Forward-bucket ‘01s are also used in swap and combination bond/swap contexts and measure risk based on changes in the shape of the yield curve. Partial ‘01s and forward-bucket ‘01s are similar to key rate approaches but use more rates, which divide the term structure into many more regions
    - - Key rate shift analysis makes the simplifying assumption that all rates can be determined as a function of a few key rates
      - f a key rate shifts by one basis point, it is called a key rate shift
      - The key rate shift technique is an approach to nonparallel shifts in the yield curve, which allows for changes in all rates to be determined by changes from selected key rates
      - the key rate approach is appealing because
        
        (2) key rates are mostly affected by the closest key rate
        
        (3) key rate effects are smooth (do not jump across maturity)
        
        (1) key rates are affected by a combination of rates closest to them
        
        (4) a parallel shift across the yield curve results
    - - A key rate ‘01 is the effect of a dollar change of a one basis point shift around each key rate on the value of the security
      - Key rate duration is a percentage change calculation that measures the sensitivity of a portfolio’s value to a 100 basis point change in yield for a specific maturity
  - - - DV01, which is equal to the impact of a one basis point shift in all spot rates on the value of a portfolio, can be divided into several key rate exposures
      - DV01 can be used to hedge against parallel shifts in the interest rate term structure, while key rate shifts allow for better hedging of a bond position given nonparallel shifts in the curve
    - - Key rate exposure analysis is very useful for measuring bond price sensitivity, but it assumes that the rate of a given term is affected only by the key rates that surround it
    - - Key rate exposures can be calculated using changes in portfolio values for given spot rate maturities, assessing the shifts of the various spot rates, and calculating partial ‘01s
    - - Hedging involves setting KR01s to zero and algebraically solving for the combination of hedging investments that will produce perfect hedges
  - - - Instead of using individual key rates, buckets can be used as segments of interest rates covering the term structure
      - The dollar impact on the portfolio value is then derived from changing every spot rate in each bucket by one basis point
    - - Forward rates can also be used in buckets
      - A forward-bucket ‘01 represents the decrease in portfolio value for a one basis point increase in all forward rates within a bucket
    - - Banks must calculate the impact of one basis point shifts in 10 KR01s, including spot rates at three and six months, along with rates at 1, 2, 3, 5, 10, 15, 20, and 30 years
      - Banks must also calculate expected shortfall and VaR measures using KR01 exposures as they relate to the standard deviations and correlations between the rates
      - Key rate shifts can be defined using spot or par rates.
      - Because market prices for actively traded instruments are typically used to derive the term structure of spot rates, par yields ultimately reflect spot rates
      - Similar to how DV01 can be segmented into spot rates, par rates, or forward rates, the same concept can be applied to duration calculations
      - In place of DV01, an ‘01 measure can be substituted such that the duration calculation is expressed as
- - - - The delta of a stock option, ∆, tells us how many units of the stock to hold per call option to be shorted to make a hedge work
    - - call price = delta × [stock price – PV(borrowing)]
  - - - The binomial option pricing model can be altered to value a stock that pays a continuous dividend yield, q
      - rDC = domestic currency risk-free rate of interest
        
        rFC = foreign currency risk-free rate of interest
      - The binomial model can also incorporate the unique characteristics of options on futures by replacing the numerator of the upward probability equation with 1 – D
    - - Valuing American options with a binomial model requires the consideration of the ability of the holder to exercise early
      - In the case of a two-step model, that means
        determining whether early exercise is optimal at the end of the first period
      - If the payoff from early exercise (the intrinsic value of the option) is greater than the option’s value (the present value of the expected payoff at the end of the second period), then it is optimal to exercise early.
    - - As the period covered by a binomial model is divided into arbitrarily small, discrete time periods, the model results converge to those of the continuous-time model (the Black-Scholes-Merton model).
- - - - Estimating Historical Volatility
        
        The annualized volatility is the estimated volatility multiplied
        by the square root of the number of trading days in a year
        
        In using historical data to calculate volatility, the best approach is to remove stock price changes on ex-dividend dates from the dataset
      - Expected Value
        
        The realized return for a portfolio is computed using a geometric return (i.e., chain-linking returns)
        
        The volatility estimation process involves collecting daily price data and then computing the standard deviation of the series of corresponding continuously compounded returns
    - - ST = stock price at time T
      - S0 = stock price at time 0
      - µ = expected return on stock per year
      - σ = volatility of the stock price per year
      - N[m, s] = normal distribution with mean m and standard deviation
      - The BSM model suggests that stock prices are lognormal over longer time periods, but suggests that stock returns are normally distributed
  - - - The volatility of the underlying asset is constant, known, and always available
      - Markets are frictionless
      - Trading is continuous.
      - The underlying asset generates no cash flows
      - The (continuous) risk-free rate is constant and known.
      - The options are European
      - The price of the underlying asset follows a lognormal distribution.
    - - N(d1) and N(d2) are found in a table of probability values
        (i.e., the z-table)
  - - - To adjust the BSM model for assets with a continuously compounded rate of dividend payment equal to q, S0e−qT is substituted for S0 in the formula
      - Options on foreign currencies use the same formulas as when there are continuously compounded dividends, except that the foreign risk-free rate (rf ) is substituted in place of the continuous dividend yield (q)
      - Options on futures use the same formulas as when there are continuously compounded dividends, except that the stock price S0 is replaced with the futures price (F), and the dividend yield (q) is replaced with the domestic risk-free rate (r).
    - - When no dividends are paid, there is no difference between European and American call options
      - Dividends complicate the early exercise decision for American-style options because a dividend payment effectively decreases the price of the stock
        
        If the call option is unexercised and the dividend is paid, its unexercised value is
      - The closer the option is to expiration and the larger the dividend, the more optimal early exercise will become.
        
        An investor will only exercise when
      - For American put options, early exercise becomes less likely with larger dividends. The value of the put option increases as the dividend is paid
      - At the last dividend date before expiration, tn, the exercised value of the option is
      - the closer the option is to expiration and the larger the dividend, the more optimal early exercise will become. The previous result can be generalized to show that early exercise is not optimal if
      - For American put options, early exercise becomes less likely with larger dividends. The value of the put option increases as the dividend is paid. Early exercise is, therefore, not optimal as long as
    - - Warrants are attachments to a bond issue that give the holder the right to purchase shares of a security at a stated price
      - Warrants can be valued as a separate call
        option on the firm’s shares
      - Assuming there is no benefit to the company from
        issuing warrants, the value of each warrant is computed as
    - - Implied volatility is the volatility that, when used in the BSM formula, produces the current market price of the option
      - There is no closed-form solution for calculating implied volatility, as it must be found by iteration (trial and error)
      - Historical volatility is the standard deviation of a past series of continuously compounded returns for the underlying asset
      - If a value for volatility makes the value of a call calculated from the BSM model lower than the market price, it needs to be increased (and vice versa) until the model value equals market price (remember, option value and volatility are positively related)
      - Volatility varies with strike prices such that graphically it produces a volatility smile
      - Assessing the volatility surface involves looking at implied volatilities relative to strike price and time to maturity
- - - - Stop-loss trading strategies are designed to minimize losses in the event the price of the underlying exceeds the strike price of a short call option position.
  - - - the value of the portfolio does
        not change with changes in the value of the asset.
      - dynamic hedging
        
        Adjusting the hedge on a frequent basis
      - static hedging
        
        the hedge is initially set up but never adjusted
        
        hedge-and-forget strategy
    - - the delta-neutral position only
        holds for very small changes in the value of the underlying stock
      - the delta-neutral portfolio must be frequently (continuously) rebalanced to maintain the hedge
    - - Option Delta
      - Forward Delta
        
        equal to one, implying a one-to-one relationship
        between the value of the forward contract and its underlying asset
      - Futures Delta
        
        Unlike forward contracts, the delta of a futures position is not ordinarily one because of the spot-futures parity relationship.
        
        For example, the delta of a futures position is erT on a stock or stock index that pays no dividends, where r is the risk-free rate and T is the time to maturity
        
        Assets that pay a dividend yield, q, would generate a delta equal to e(r – q)T.
        
        Δfutures contract = erT
      - To completely hedge a long stock or short call position, an investor must purchase the number of shares of stock equal to delta times the number of options sold
      - Another term for being completely hedged is delta neutral
      - A forward/futures contract position can easily be hedged with an offsetting underlying asset position with the same number of securities.
  - - - Gamma is used to correct the hedging error associated with delta-neutral positions by providing added protection against large movements in the underlying asset’s price
      - Gamma-neutral positions are created by matching the gamma of the portfolio with an offsetting option position.
    - - r = the risk-neutral, risk-free rate of interest
        
        Π = the price of the option
        
        Θ = the option theta
        
        S = the price of the underlying stock
        
        Δ = the option delta
        
        Δ = 0
        
        σ2 = the variance of the underlying stock
        
        Γ = the option gamma
    - - ∂t = change in time
        
        ∂c = change in the call price
        
        call
        
        put
      - Theta varies with changes in stock prices and as time passes.
      - Theta affects the value of put and call options in a similar way (e.g., as time passes, most call and put options decrease in value, all else equal).
      - Theta is most pronounced when the option is at the money, especially nearer to expiration.
      - Theta values are usually negative, which means the value of the option decreases as it gets closer to expiration.
      - Theta usually increases in absolute value as expiration approaches.
      - It is possible for a European put option that is in the money to have a positive theta value.
    - - measures an option’s sensitivity to changes in the risk-free rate
    - - (I) All long options, calls and puts, are positive Gamma. All short options, calls and puts, are negative Gamma.
      - (II) The delta value of calls is always positive (somewhere between 0 and 1) and with puts, it's always negative (somewhere between 0 and -1). Stocks effectively have a delta value of 1.
    - - Portfolio insurance is the combination of
      - (1) an underlying instrument
      - (2) either cash or a derivative that generates a floor value of the portfolio in the event that market valuations decline, while allowing for upside potential
    - - the sensitivity of the option’s price to changes in the volatility of the underlying stock
      - For example, a vega of 8 indicates that for a 1% increase in volatility, the option’s price will increase by 0.08
      - example
        
        exhibits high unfavorable sensitivity to an increase in implied volatility
        
        Recall that The theta of a portfolio of options is the rate of change of the value of the portfolio with respect to the passage of time with all else remaining the same
        
        Since Mark has experienced significant daily losses, it is imperative to say the longer the time, the more losses experienced
        
        On the other hand, Vega implies the rate of change in option value with respect to the implied volatility of the underlying asset as time goes by
        
        Intuitively, hedging can be achieved by selling long-term options and buying short-term options.
      - A position in the underlying asset has a Vega equal to zero