Market Risk
READING 37
MEASURING AND MANAGING MARKET RISK
EXAM FOCUS
This reading reviews approaches to measuring market risk and the tools used to manage and control it. Value at risk (VaR) is a central risk metric; you should understand its definition, the principal methods for computing it, how to interpret results, and the advantages and limitations of each method. Also study extensions of VaR such as conditional VaR (CVaR), incremental VaR (IVaR), and marginal VaR (MVaR). The material on sensitivity analysis and scenario analysis focuses primarily on qualitative interpretation and application. Finally, learn which risk measures are most relevant to different types of institutions (banks, pension funds, insurers, asset managers, hedge funds) and how limits and capital allocation decisions use these measures.
MODULE 37.1: VALUE AT RISK (VaR)
LOS 37.a: Explain the use of value at risk (VaR) in measuring portfolio risk.
Video covering this content is available online.
Value at risk (VaR) is a statistic that quantifies the potential downside loss of a portfolio over a specified time horizon at a specified probability level. VaR has three essential components:
- Loss size - the monetary amount or percentage decline in portfolio value.
- Probability (confidence level) - the probability that losses will be at least as large as the loss size (often expressed as a tail probability such as 1%, 5% or 16%).
- Time horizon - the period over which the loss is measured (daily, weekly, monthly, annual, etc.).
For example, the statement "There is a 5% probability that the company will experience a loss of $25,000 or more in any given month" is equivalent to saying the monthly 5% VaR is $25,000. Equivalently, a 5% monthly VaR of 3% can be stated as: 5% of the time the monthly portfolio value will fall by at least 3%, or we are 95% confident that the monthly loss will be no more than 3%.
Estimating VaR requires specifying the probability level and the time horizon; either the loss size is fixed and we estimate the probability of losses at least that large, or the probability is fixed and we estimate the minimum loss size corresponding to that probability. These choices involve judgment and can materially affect the estimated VaR.
Conceptually, VaR is the boundary of the specified left tail of the return distribution. For a 5% VaR, the left-hand 5% tail of the distribution is bounded by the VaR value (the smallest loss in that left tail).
VaR is commonly reported at tail probabilities of 1%, 5%, and 16% (16% corresponds to one standard deviation below the mean for a normal distribution). Time horizons vary according to the application.
LOS 37.b / 37.c: Compare and compute VaR by the parametric (variance-covariance), historical simulation, and Monte Carlo simulation methods.
The initial step in any VaR estimation is identifying the relevant risk factors that determine portfolio returns (market returns, interest rates, exchange rates, volatilities, credit spreads, and so on).
Parametric (variance-covariance) method
The parametric method estimates the distribution of portfolio returns from estimated means, variances, and covariances (or correlations) of the risk factors. It commonly assumes normality for risk-factor returns; alternative distributions require estimating higher moments such as skewness and kurtosis and add complexity.
Assuming normality, we compute the portfolio mean and portfolio variance from the component statistics and identify the VaR as the percentile of the resulting normal distribution that bounds the left-hand tail. For a portfolio with two assets, A and B, with portfolio weights wA and wB, variances σA2 and σB2, and covariance Cov(A,B), the portfolio variance is:
Var(portfolio) = (wA)2 σA2 + (wB)2 σB2 + 2 wA wB Cov(A,B)
Choice of the lookback period for estimating means, variances, and covariances is important: parameter estimates should reflect the behaviour expected over the VaR horizon. Shorter lookback periods reflect recent volatility; longer lookback periods give more stable, long-term averages. Analysts often adjust recent estimates toward longer-term averages if they expect a reversion.
Worked example - parametric VaR (two-asset portfolio)
Imagine two securities, Security A and Security B, with the following statistics inferred from the working below:
- Mean daily returns: A = 0.0004, B = 0.0003
- Standard deviations of daily returns: A = 0.0158, B = 0.0112
- Covariance between A and B: 0.000106
- Portfolio weights: wA = 0.6, wB = 0.4
How to estimate the 5% daily and annual VaR for a portfolio worth $10,000,000 that is 60% in A and 40% in B.
Mean daily portfolio return = 0.6(0.0004) + 0.4(0.0003) = 0.00036
Variance of portfolio return = (0.6)2(0.0158)2 + (0.4)2(0.0112)2 + 2(0.6)(0.4)(0.000106) = 0.000161
Standard deviation of portfolio returns = √0.000161 = 0.012682
For a 5% VaR (left-tail probability 5%), use the z-value ≈ 1.65 for one-sided 5% tail under normality.
5% daily VaR (as a proportion of portfolio) = [0.00036 - 1.65(0.012682)] × (-1) = 0.0206
Assume 250 trading days in a year and daily returns are independent.
Annual mean return = 250 × 0.00036 = 0.09
Annual standard deviation = 0.012682 × √250 = 0.20052
5% annual VaR (as a proportion of portfolio) = [0.09 - 1.65(0.20052)] × (-1) = 0.2409
For a portfolio value of $10,000,000:
5% daily VaR = 10,000,000 × 0.0206 = $206,000
5% annual VaR = 10,000,000 × 0.2409 = $2,409,000
Notes on the parametric method:
- The method is straightforward when normality is plausible, but parameter estimates (means, variances, covariances) are crucial and sensitive to the lookback period and estimation method.
- Covariance estimates strongly affect portfolio variance and VaR; mistakes or stale estimates can misstate risk materially.
- The method is limited when returns are non-normal (fat tails, skewness) or when the portfolio contains nonlinear instruments such as options, where return distribution is not adequately captured by mean and variance alone.
Historical simulation method
The historical simulation method computes the change in the current portfolio value for each day (or period) in a historical lookback sample using the actual historical changes in risk factors. Order the resulting profit and loss (P&L) outcomes from best to worst. For a 5% VaR, the 95th percentile from the top (or the 5th percentile from the bottom) identifies the VaR: the smallest loss among the worst 5% of outcomes.
Advantages of historical simulation:
- No distributional assumption is required.
- Works naturally for portfolios with nonlinear instruments (options), since historical P&L outcomes reflect realized option value changes.
Limitations:
- VaR estimates depend on the choice of lookback period and cannot easily adjust for differences between historical sample characteristics and expected future conditions.
- Unusual historical periods (very volatile or calm) can lead to over- or underestimates of true future VaR.
Monte Carlo simulation method
Monte Carlo simulation specifies a distribution for each risk factor and the correlations between them, then uses random number generation and pricing models to produce many simulated scenarios of risk-factor changes and resulting portfolio values. Order the simulated outcomes and take the percentile corresponding to the chosen tail probability to estimate VaR.
Advantages and caveats:
- Monte Carlo is very flexible and can accommodate complex instruments and non-normal factor distributions.
- It requires specifying factor distributions and correlations; results depend on these assumptions and on the number of simulations.
- With the same distributional assumptions and sufficiently many simulations, Monte Carlo will converge to results equivalent to the parametric method.
MODULE QUIZ 37.1
1.
Weekly 5% VaR of £1 million indicates:
A.
a maximum allowable loss of £1 million in 5% of weeks.
B.
that the largest weekly loss is £1 million or 5% of portfolio value.
C.
a 5% probability of a loss greater than £1 million in any given week.
2.
A lookback period is least likely to be specified when estimating VaR using:
A.
historical simulation.
B.
the parametric method.
C.
Monte Carlo simulation.
3.
A portfolio manager expects to earn a return of 6.5% over the next year with a standard deviation of 9%. The portfolio is currently valued at $6.4 million. What is the 5% annual VaR of the portfolio?
A.
$83,500.
B.
$160,000.
C.
$534,400.
MODULE 37.2: USING VaR
LOS 37.d: Describe advantages and limitations of VaR.
Video covering this content is available online.
Advantages of VaR
- Intuitive concept: VaR is straightforward to explain: the minimum loss at a given probability level over a specified horizon.
- Comparability: VaR provides a single-number summary that allows comparison of risk across portfolios, asset classes, and business lines.
- Performance evaluation: VaR facilitates risk-adjusted performance measures (returns per unit of VaR) and aids capital allocation and risk budgeting.
- Regulatory acceptance: Global banking regulators accept VaR as a measure of financial risk (they do not mandate a specific estimation method or maximum VaR).
- Backtesting: VaR models can be validated by backtesting - comparing predicted VaR exceptions to realized losses.
Limitations of VaR
- Model and parameter sensitivity: VaR depends on choices such as probability level, time horizon, lookback period, distributional assumptions, and parameter estimates; different choices can produce very different VaRs.
- Underestimation of tail risk: Assuming normality tends to underestimate the frequency and magnitude of extreme losses when return distributions have fat tails.
- Liquidity effects: VaR that ignores changes in market liquidity during stress understates realized losses when positions cannot be liquidated at quoted prices.
- Correlation breakdown in stress: Correlations between assets often increase in market stress, reducing diversification benefits and causing VaR based on normal correlations to understate risk.
- Single-number limitation: VaR summarizes risk into one number and thus omits information about tail severity beyond the VaR threshold and about upside potential. Users must understand its limited scope.
LOS 37.e: Describe extensions of VaR.
Conditional VaR (CVaR), also called expected shortfall or expected tail loss, is the expected loss given that the loss is equal to or greater than the VaR. When VaR is estimated by historical simulation or Monte Carlo simulation, CVaR is straightforward to compute as the average of losses beyond the VaR cutoff. For a parametric VaR, computing CVaR requires knowledge of the tail distribution beyond the VaR point and is mathematically more complex.
Incremental VaR (IVaR) is the change in portfolio VaR resulting from a change in the portfolio allocation to a particular security. For example, if increasing a security's weight by 2% raises VaR from $1,345,600 to $1,562,400, then the IVaR for that 2% increase equals $1,562,400 - $1,345,600 = $216,800.
Marginal VaR (MVaR) is the derivative (slope) of the VaR function with respect to a security's weight at the current weight. It approximates the effect on VaR of a small (infinitesimal) change in a security's weight and is commonly interpreted as the change in VaR per 1% change in weight (an approximation, since the slope is local).
Ex ante tracking error (relative VaR) measures the VaR of the difference between a portfolio's return and its benchmark return. A 5% monthly relative VaR of 2.5% implies that 5% of the time the portfolio will underperform the benchmark by at least 2.5% over one month. This relative VaR can be calculated by forming a portfolio that is long the subject portfolio and short the benchmark portfolio and computing the VaR of that combined position.
LOS 37.f: Describe sensitivity risk measures and scenario risk measures and compare these measures to VaR.
Because VaR has limitations, it should be complemented by other measures.
Sensitivity analysis estimates the change in portfolio value for a small change in a single risk factor. It identifies exposures and the relative importance of different risk factors but does not assign probabilities to outcomes. Sensitivity analysis is useful for hedging decisions and for identifying excessive exposures that should be reduced.
Scenario analysis estimates the effect on portfolio value of a set of simultaneous changes in multiple risk factors, typically of significant magnitude. Scenarios may be:
- Historical scenarios: actual past sets of factor changes (for example, the 1987 crash or the 2008 subprime crisis).
- Hypothetical scenarios: plausible but not historical combinations of factor moves, which can be more extreme than anything seen before.
Stress tests are scenario analyses that examine extreme but plausible changes to determine effects on value or solvency. Reverse stress testing starts with an unacceptable outcome (for instance, insolvency) and searches for scenarios of factor moves that would produce that outcome - useful for identifying vulnerabilities and contingency planning.
LOS 37.g: Demonstrate how equity, fixed-income, and options exposure measures are used in measuring and managing market and volatility risk.
Risk factors and exposure measures differ across asset classes:
- Equity risk: commonly measured by beta, the sensitivity of a security's (or portfolio's) returns to market returns. The capital asset pricing model (CAPM) relates expected return to beta as:
E(Ri) = Rf + βi [E(RMKT) - Rf] - Fixed-income risk: measured primarily by duration (first-order sensitivity of price to yield changes) and convexity (second-order effect capturing curvature). For reasonably small yield changes ΔY, the approximate percentage change in price is given by:
Change in price ≈ -Duration × (ΔY) + ½ × Convexity × (ΔY)2 - Options risk: captured by the Greeks: delta (sensitivity to changes in underlying price), gamma (sensitivity of delta to changes in underlying price - a second-order effect), and vega (sensitivity to changes in implied volatility). For call options, a first- and second-order approximation of price change is:
Change in call price ≈ delta × (ΔS) + ½ × gamma × (ΔS)2 + vega × (ΔV), where ΔS is the change in underlying price and ΔV is the change in volatility.
Duration and delta measure first-order effects; convexity and gamma capture second-order adjustments and improve estimates for larger changes in yields or underlying prices.
Professor's note: If Macaulay duration is used in the fixed-income change-in-price formula rather than modified duration, replace ΔY by ΔY/(1 + Y) in the duration term.
MODULE QUIZ 37.2
1.
Which of the following is a limitation of VaR?
A.
VaR focuses on downside risk.
B.
Use of VaR is discouraged by banking regulators.
C.
Estimates of VaR for different asset classes are not comparable.
2.
The expected amount of a loss, given that it is equal to or greater than the VaR, is the:
A.
marginal VaR.
B.
conditional VaR.
C.
incremental VaR.
3.
The sensitivity of an option value to changes in volatility of the underlying asset price is measured by:
A.
beta.
B.
vega.
C.
gamma.
MODULE 37.3: SENSITIVITY AND SCENARIO RISK MEASURES
LOS 37.h: Describe the use of sensitivity risk measures and scenario risk measures.
Video covering this content is available online.
Sensitivity measures quantify exposures of the portfolio to individual risk factors and help managers identify and hedge excessive exposures. Eliminating all risk is not the goal; removing all systematic exposure would leave only the risk-free return.
When portfolios include instruments with nonlinear payoffs, such as options or bonds with embedded options, sensitivity measures (delta, duration) are only reliable for small changes in risk factors. For larger changes, pricing models must be used to revalue each instrument under the scenario's factor values. Scenario analysis therefore typically combines model-pricing with specified factor moves to compute scenario P&L.
Scenario analysis can be performed assuming instantaneous moves in factors, which gives conservative estimates because it ignores the ability of portfolio managers to adjust positions. Alternatively, scenarios can be modeled incrementally, allowing for manager actions (e.g., reduction of positions or dynamic hedging) as factor moves occur; incremental approaches can be less conservative but possibly more realistic in some contexts.
Reverse stress testing identifies the most vulnerable exposures, specifies an unacceptable outcome for the firm (often insolvency or breach of a capital trigger), and searches for plausible scenarios that would produce that outcome. Assessing the likelihood of such scenarios helps managers and boards decide on mitigation actions.
Scenario analysis is typically the final validation in a risk management process after VaR and sensitivity analysis: it tests the portfolio's resilience to sequences or combinations of extreme factor moves and to changes in correlations.
LOS 37.i: Describe advantages and limitations of sensitivity and scenario risk measures.
Sensitivity and scenario analyses complement VaR but have distinct limitations:
- Sensitivity analysis provides only local (small-change) estimates and does not provide probabilities of factor moves; differences in factor volatilities across instruments can make identical sensitivities imply very different probabilities of losses.
- Scenario analysis examines simultaneous large changes in multiple factors and can incorporate historical or hypothetical stress events; however, hypothetical scenarios may be misspecified and scenario analysis typically does not assign an objective probability to the scenario unless an explicit model is used.
- Both approaches produce useful information about vulnerabilities that VaR alone omits, but neither substitutes for probabilistic measures when probability estimates are required.
MODULE 37.4: APPLICATIONS OF RISK MEASURES
LOS 37.l: Describe risk measures used by banks, asset managers, pension funds, and insurers.
Video covering this content is available online.
Risk measures differ across firms because of the nature of their exposures, regulatory environments, investment objectives, and use of leverage. The following summarises typical approaches by institution type.
Banks
Banks usually measure market risk for trading portfolios with VaR, perform sensitivity analysis for nontrading positions (duration, currency exposure), and carry out scenario analysis and stress testing for the entire balance sheet. Banks also measure leverage risk, estimate VaR for economic capital purposes, and disaggregate risk by geography and business unit. Banks pay particular attention to asset-liability mismatches and the risk to their economic capital (the capital required to survive severe losses).
Professor's note: Economic capital is the capital a firm requires to remain solvent through extreme adverse outcomes arising from its business risks.
Traditional (long-only) asset managers
These managers generally focus on relative risk versus a benchmark. Typical measures include position sizes, sensitivity measures, historical and hypothetical scenario analysis, and options risk. A specific measure used by active managers is active share - the sum of absolute differences between portfolio and benchmark weights across securities.
Tracking error is used in two senses:
- Ex-post tracking error - historical volatility of active returns over a lookback period, used to evaluate past performance and skill.
- Ex-ante tracking error - forward-looking estimate of potential active return volatility, used to control current risk relative to the benchmark.
Managers pursuing absolute-return objectives may use VaR instead of relative risk measures.
Hedge funds
Hedge-fund risk measures depend on strategy: common measures include sensitivity analysis, leverage measures, scenario analysis, and stress testing. Funds with both long and short positions report risk for long and short legs and for gross exposure. Hedge funds often target short-term, small-probability VaR limits (e.g., VaR measured over short horizons with probabilities under 10%).
Funds with non-normal return distributions also focus on maximum drawdown - the largest peak-to-trough decline over a specified historical period.
Defined-benefit pension funds
Pension funds focus on the present value of assets relative to liabilities. A common risk measure is surplus-at-risk - a VaR-style measure for the difference between asset value and the present value of liabilities. Pension funds also use hedged and unhedged exposure metrics to separate liability-matching assets from return-seeking investments. A glide path is a multi-year plan to adjust contributions or asset allocations to address overfunded or underfunded positions.
Insurance companies
Insurance firms face both market risk and insurance-specific risks. Property & casualty (P&C) insurers have underwriting exposures that are often uncorrelated with market risk; they manage insurance risk via reinsurance and geographic diversification. P&C insurers use VaR and capital-at-risk measures for their investment portfolios and often combine market and insurance risks in scenario-based stress tests. Regulators may require minimum reserves and may discount risky assets held as reserves.
Life insurers' liabilities (annuities, long-term policies) are more closely correlated with market risk. Life insurers therefore stress both asset portfolios and liability valuations (which are sensitive to discount factors) and monitor the asset-liability mismatch. Scenario analysis for life insurers commonly blends market and insurance risks to assess the impact on surplus.
MODULE QUIZ 37.3, 37.4
1.
Which of the following risk measures is most likely to be used by a traditional asset manager?
A.
Active share.
B.
Surplus at risk.
C.
Maximum drawdown.
2.
The risk measure of volatility of surplus would most likely be used by a:
A.
bank.
B.
pension fund.
C.
life insurance company.
MODULE 37.5: CONSTRAINTS AND CAPITAL ALLOCATION DECISIONS
LOS 37.j: Explain constraints used in managing market risks, including risk budgeting, position limits, scenario limits, and stop-loss limits.
Video covering this content is available online.
Risk constraints must balance protection and profitability. Overly restrictive constraints can impair returns; too lenient constraints can expose the firm to severe losses. Limits imposed at the business-unit level may be overly restrictive if they ignore diversification and offsetting positions across units. Common types of limits include:
- Risk budgeting: Determine an acceptable total risk for the organisation and allocate it across activities, strategies, or asset classes. Example: set a maximum allowable 5% VaR and distribute that VaR across trading desks.
- Position limits: Limits on allocations to single securities, issuers, asset classes, countries, or currencies, expressed as currency amounts, portfolio percentages, or liquidity-based measures (e.g., multiples of average daily volume). These enforce minimum diversification.
- Scenario limits: Limits on expected loss for specified scenarios. These are scenario-specific loss tolerances.
- Stop-loss limits: Rules requiring reduction or closing of exposures when losses exceed a threshold over a given time. Stop-loss can be implemented by forced sales or by dynamic hedging (portfolio insurance), for example by buying index puts as prices fall.
LOS 37.k: Explain how risk measures may be used in capital allocation decisions.
Capital allocation addresses how firm capital funds business units and activities. Introducing risk measures into allocation requires that expected returns be adjusted for risk. One approach is to compute VaR (or another risk metric) for each activity or business unit, set an acceptable aggregate risk budget, and allocate capital to activities in a manner that respects the aggregate budget - analogous to portfolio risk budgeting. This produces a risk-aware allocation rather than a purely return-maximising one.
MODULE QUIZ 37.5
1.
The risk committee of an investment management firm believes high-yield bonds will decrease in value if the economy goes into recession, and the committee decides to limit exposure to this asset class to 10% of assets under management. This constraint is best described as a:
A.
position limit.
B.
scenario limit.
C.
stop-loss limit.
KEY CONCEPTS
LOS 37.a
Value at risk (VaR) is an estimate of the minimum loss that will occur with a given probability over a specified period, expressed as a currency amount or as a percentage of portfolio value.
LOS 37.b
VaR estimation methods:
- Parametric method - uses estimated variances and covariances of portfolio securities (often assuming normality) to estimate the distribution of portfolio returns.
- Historical simulation - uses historical risk-factor changes over a lookback period to derive a distribution of portfolio returns for the current portfolio.
- Monte Carlo simulation - draws risk-factor changes repeatedly from assumed distributions and correlations, prices the portfolio for each draw, and uses the simulated distribution of outcomes to estimate VaR.
LOS 37.c
The x% VaR is the minimum loss for the current portfolio that will occur x% of the time, based on an estimated distribution of portfolio values.
LOS 37.d
Advantages of VaR:
- Accepted by regulators.
- Simple to interpret.
- Provides a single-number summary of risk.
- Useful for comparing risk across portfolios, components, and business units.
Disadvantages of VaR:
- Subjective in choice of time period and probability level.
- Sensitive to estimation method and assumptions.
- Focuses only on left-tail (downside) outcomes and omits tail severity beyond the VaR point.
- Vulnerable to misspecification or manipulation of inputs by the user.
LOS 37.e
- Conditional VaR (CVaR) - expected loss given that loss exceeds the VaR.
- Incremental VaR (IVaR) - estimated change in portfolio VaR from a specified change in a position.
- Marginal VaR (MVaR) - local estimate (slope) of change in VaR for a small change in a position's weight; used to approximate a security's contribution to overall VaR.
- Ex-ante tracking error (relative VaR) - VaR of the difference between portfolio returns and benchmark returns.
LOS 37.f
- Sensitivity analysis estimates the change in security or portfolio value for an incremental change in a risk factor.
- Scenario analysis evaluates the effect on portfolio value of a specified set of changes in relevant risk factors; scenarios may be historical or hypothetical.
- Stress tests examine extreme scenarios to evaluate effects on value or solvency.
LOS 37.g
- Equity risk measured by beta (sensitivity to market returns).
- Fixed-income risk measured by duration (sensitivity to yield changes) and convexity (second-order effect).
- Options risk measured by delta, gamma, and vega.
- Market risk is managed by adjusting portfolio holdings to control exposures to these factors.
LOS 37.h
Stress tests use extreme sensitivity or scenario changes to examine effects on firm equity or solvency. Reverse stress testing identifies scenarios that would cause business failure. Sensitivity analysis reveals vulnerabilities to individual risk factors; scenario analysis examines simultaneous factor changes but provides no formal probability for hypothetical scenarios.
LOS 37.i
VaR, sensitivity analysis, and scenario analysis are complementary: VaR provides probabilities of losses; sensitivity analysis provides exposure magnitudes for small changes; scenario analysis evaluates combined large changes but does not, on its own, provide objective probabilities for hypothetical scenarios.
LOS 37.j
- Risk budgeting: determine acceptable total risk and allocate it among positions or business units to maximise return for the permitted risk.
- Position limits: maximum currency amounts or percentages for securities, issuers, or asset classes.
- Stop-loss limits: require reduction or closure of positions when losses exceed given thresholds.
- Scenario limits: require the portfolio to be adjusted so that expected loss under a specified scenario does not exceed a set amount.
LOS 37.k
Firms adjust expected returns for risk in capital allocation decisions. One method is to compute VaR for each business activity and allocate capital so that the firm-wide VaR constraint is satisfied - a risk-aware capital allocation process.
LOS 37.l
- Banks: concern with asset-liability mismatches, market risk of investment portfolios, leverage, duration and convexity of fixed-income holdings, and risk to economic capital.
- Asset managers: focus on return volatility and probability distributions of absolute or benchmark-relative losses.
- Pension funds: concerned with mismatches between assets and liabilities and the volatility of surplus.
- P&C insurers: concerned with investment portfolio sensitivity, VaR of economic capital, and combined market and insurance scenarios as part of stress testing.
- Life insurers: concerned with market risks to assets and liabilities, mismatches and scenarios that could cause large surplus reductions.
ANSWER KEY FOR MODULE QUIZZES
Module Quiz 37.1
1.
C Weekly 5% VaR of £1 million indicates that there is a 5% probability that a loss during any given week will be greater than £1 million. (LOS 37.a)
2.
C Monte Carlo simulation uses estimated statistical properties for each of its risk factors and does not primarily rely on a historical lookback period. The parametric method and historical simulation both use lookback data for parameter estimation. (LOS 37.b)
3.
LOS 37.c - Calculation:
% VaR = [0.065 - 1.65(0.09)] × (-1) = 0.0835
$ VaR = 0.0835 × $6,400,000 = $534,400
Module Quiz 37.2
1.
A Because VaR focuses on negative (left-tail) outcomes, it does not by itself provide a complete view of the risk-return trade-off. Advantages of VaR include regulatory acceptance and usefulness in comparing risk across asset classes. (LOS 37.d)
2.
B Conditional VaR is the expected loss given that the loss is equal to or greater than the VaR. Marginal VaR and incremental VaR are related but different measures. (LOS 37.e)
3.
B Vega measures the sensitivity of an option's value to changes in the volatility of the underlying asset price. (LOS 37.g)
Module Quiz 37.3, 37.4
1.
A Active share is a risk measure particular to traditional asset managers and measures the difference between portfolio and benchmark weights. Maximum drawdown is commonly used by hedge funds, while surplus-at-risk is used by defined benefit pension plans. (Module 37.4, LOS 37.l)
2.
B Pension fund managers are concerned with mismatches between assets and liabilities and with the volatility of the surplus (assets minus liabilities). (Module 37.4, LOS 37.l)
Module Quiz 37.5
1.
A Limiting the allocation to an asset class is an example of a position limit. (LOS 37.j)
