Bachelors Program in Accounting and Finance
Risk Management & Derivatives
•
Bachelor
Market Risk Measurement
Introduction
Market risk is the possibility of loss caused by changes in interest rates, exchange rates, equity prices, commodity prices, volatilities, correlations and related market variables. It affects banks, investment funds, manufacturers, exporters, importers and any organisation whose financial position depends on market prices.
Market risk measurement begins with exposure identification. Sensitivity measures then estimate how value changes when one risk factor moves. Duration, convexity and beta provide specialised measures for fixed-income and equity positions. Value at Risk (VaR) and Expected Shortfall (ES) summarise portfolio-level loss distributions, while backtesting, stress testing and scenario analysis evaluate model performance and vulnerability under adverse conditions.
No single measure is sufficient. Sensitivities may miss nonlinear effects, VaR does not describe losses beyond its threshold, and statistical models can fail during structural breaks. Sound risk management therefore combines complementary measures with governance and professional judgement.
Learning Objectives
After studying this chapter, students should be able to:
Identify interest-rate, foreign-exchange, equity and commodity exposures.
Calculate and interpret first-order sensitivity measures.
apply duration and convexity to fixed-income positions.
Use beta to estimate systematic equity exposure.
Calculate parametric, historical and simulation-based VaR.
Calculate ES using a confidence-level-consistent tail definition.
Conduct and interpret VaR backtesting.
Design stress tests and scenario analyses.
Evaluate the assumptions, limitations and governance requirements of market risk models.
1. From Market Positions to Risk Exposure
A market position is an asset, liability or contractual cash flow whose value depends on one or more risk factors. Exposure is the amount by which value changes when those factors move.
Examples include:
Position
Principal risk factors
Typical measures
Fixed-rate bond
Yield levels and curve shape
DV01, duration, convexity
Foreign-currency receivable
Spot exchange rate
Currency delta, net open position
Equity portfolio
Share prices and market index
Equity delta, beta
Commodity inventory
Spot and futures prices
Price delta, basis sensitivity
Option
Price, volatility, time and rates
Delta, gamma, vega and theta
Let portfolio value be V = f(x1,x2,\ldots,xn), where each xi is a market risk factor. For small changes, a first-order approximation is:
∆V ≈ ∑i=1n δi∆xi
Here, δi = ∂V/∂xi is the portfolio's sensitivity to risk factor xi. The units of each sensitivity depend on the factor. For example, foreign-exchange delta may be measured in foreign-currency units, while interest-rate sensitivity may be expressed as currency value per basis point.
If nonlinear effects are material, a second-order approximation is:
∆V ≈ ∑i=1n δi∆xi + ½∑i=1n∑j=1n Γij∆xi∆xj
The element Γij = ∂2V/(∂xi∂xj) measures curvature or interaction between risk factors. This approximation assumes sufficiently small movements and a value function that is twice differentiable in the relevant region.
2. Interest-Rate Risk
Interest-rate risk arises when market rates affect the present value of future cash flows. Fixed-rate instruments generally lose value when yields rise because their contractual cash flows are discounted at higher rates.
2.1 Bond pricing
For a fixed-cash-flow bond with yield compounded m times per year:
P = ∑t CFt(1 + y/m)-mt
where:
P = bond price in currency units;
CFt = cash flow received at time t;
y = annual nominal yield expressed as a decimal;
m = compounding frequency per year;
t = time in years.
This representation assumes known contractual cash flows. Securities with embedded options require models that allow cash flows to change with rates.
2.2 Macaulay and modified duration
Macaulay duration is the present-value-weighted average time to receive the bond's cash flows:
DMac = \frac{∑t t × PV(CFt)}{P}
It is measured in years. Modified duration converts this timing measure into approximate price sensitivity:
Dmod = \frac{DMac}{1+y/m}
For a small parallel change in yield:
\frac{∆P}{P} ≈ -Dmod∆y
The yield change ∆y must be expressed as a decimal. Thus, 50 basis points equals 0.0050. Modified duration has units of years, so its product with a decimal annual yield change is dimensionless.
Worked example: duration effect
A bond portfolio is worth $4,000,000 and has modified duration of 5.5 years. If yields rise by 40 basis points:
∆P ≈ -5.5 × 0.0040 × $4,000,000 = -$88,000
The estimated portfolio value is therefore $3,912,000. The calculation assumes a small, parallel yield-curve shift and unchanged cash flows.
2.3 DV01
Dollar value of one basis point, also called DV01 or PV01, is the approximate absolute change in value for a one-basis-point yield movement:
DV01 = Dmod × P × 0.0001
DV01 is normally reported as a positive magnitude. For an ordinary fixed-rate bond, a one-basis-point increase in yield produces an approximate loss equal to DV01.
For the preceding portfolio:
DV01 = 5.5 × $4,000,000 × 0.0001 = $2,200
2.4 Convexity
Duration gives a linear approximation, although the bond price–yield relationship is curved. Convexity measures this curvature:
C = \frac{1}{P}\frac{d2P}{dy2}
When yield is measured as an annual decimal, convexity has units approximately equal to years squared. Combining duration and convexity gives:
\frac{∆P}{P} ≈ -Dmod∆y + ½C(∆y)2
Suppose a bond has modified duration 6.0, convexity 35 and value $2,000,000. For a 1% yield increase:
The estimated loss is $2,000,000 × 0.05825 = $116,500. Without convexity, duration alone would estimate a $120,000 loss. Positive convexity moderates the estimated loss.
Duration and convexity based on yield to maturity are most defensible for fixed cash flows and approximately parallel curve changes. Key-rate duration is preferable when changes in different maturity segments must be measured separately.
3. Foreign-Exchange Risk
Foreign-exchange risk arises when assets, liabilities, revenues or expenses are denominated in a currency different from the reporting currency.
Suppose S is quoted as domestic currency per unit of foreign currency and the organisation holds X foreign-currency units. Its domestic value is:
V = XS
Therefore:
∆V = X∆S
The currency delta is X. A positive foreign-currency asset gains value when the foreign currency appreciates against the domestic currency.
Worked example
A UK company expects to receive €800,000. If the spot rate falls from £0.860 to £0.835 per euro, then:
∆V = €800,000 × (£0.835 - £0.860) = -£20,000
The company loses £20,000 of sterling value. A forward contract could lock in a sterling conversion rate, although counterparty, liquidity and forecast-transaction risks remain.
Organisations should measure net exposure by currency and maturity rather than merely adding gross foreign-currency assets and liabilities. Offsetting positions may have different settlement dates or contractual conditions.
4. Equity Risk and Beta
A direct equity position has approximately unit price delta: a 1% fall in the share price causes a 1% loss in the position, before considering currency effects or derivatives.
Beta measures an equity's sensitivity to broad market returns:
βi = \frac{Cov(Ri,RM)}{Var(RM)}
where Ri is the asset return and RM is the market return. Both returns must be measured over consistent intervals and in consistent currency terms.
Under a single-factor approximation:
Ri ≈ αi + βiRM + εi
Here, εi represents asset-specific return. For short-horizon scenario estimation, the market-driven value change is approximately:
∆Vmarket ≈ βiViRM
A $3 million equity portfolio with beta 1.3 is expected to lose approximately:
1.3 × $3,000,000 × 0.06 = $234,000
if the market falls by 6%, ignoring alpha and idiosyncratic return. Beta is estimated from historical data and may change during crises. It measures systematic exposure, not maximum loss or total risk.
5. Commodity Risk
Commodity exposure may arise from inventory, anticipated purchases, sales contracts, futures or options. If Q is the quantity of a commodity and S is its price per unit, then for a direct position:
V = QS, \qquad ∆V = Q∆S
A company holding 30,000 tonnes of copper loses $1.5 million if copper prices fall by $50 per tonne:
∆V = 30,000 × (-$50) = -$1,500,000
Hedging with futures introduces basis risk: the spot exposure and futures contract may not move identically because of differences in grade, location, delivery date or market liquidity. Commodity exposures may also be nonlinear when production options, storage decisions or minimum-purchase clauses are present.
6. Value at Risk
VaR estimates a loss threshold over a specified horizon and confidence level. Let L denote portfolio loss, with positive values representing losses. VaR at confidence level α is the loss quantile:
VaR_α(L) = inf{l : Pr(L ≤ l) ≥ α}
A one-day 99% VaR of $2 million means that, according to the model, loss should not exceed $2 million on approximately 99% of trading days. It does not mean that the maximum possible loss is $2 million.
Every VaR report should identify:
confidence level;
holding period;
valuation date;
currency and units;
data window and weighting method;
treatment of nonlinear instruments;
assumptions concerning distributions and correlations.
6.1 Parametric or variance–covariance VaR
If loss is normally distributed with mean μL and standard deviation σL, then:
VaR_α = μL + z_ασL
Here, z_α is the standard normal quantile. For short horizons, mean loss is often assumed to be zero, giving VaR_α ≈ z_ασL.
For linear factor exposures represented by vector δ and factor covariance matrix Σ:
σL = √(δTΣδ)
The covariance matrix must use factor changes measured over the same horizon and units as the sensitivities.
Worked example: parametric VaR
A $10 million portfolio has daily return volatility of 1.5%. Assuming zero mean, normal returns and linear exposure, 99% one-day VaR is:
VaR0.99 = 2.326 × 0.015 × $10,000,000 = $348,900
Normal VaR may underestimate risk when returns are skewed, heavy-tailed or affected by abrupt liquidity changes.
6.2 Historical simulation
Historical simulation revalues the current portfolio using observed past changes in risk factors. The losses are sorted, and the required empirical quantile is selected. It does not impose normality but assumes the historical sample is relevant to current exposure.
Its principal limitations are limited tail observations, sensitivity to the selected window and inability to represent events absent from the data. Volatility-weighted historical simulation can adjust old observations to current volatility, but this adds modelling assumptions.
6.3 Monte Carlo simulation
Monte Carlo VaR follows four broad steps:
Specify joint stochastic models for the risk factors.
Generate many simulated factor changes.
Revalue the portfolio under each simulated state.
Estimate VaR from the simulated loss distribution.
Monte Carlo simulation can accommodate options, nonlinear payoffs and complex dependencies. Its results nevertheless depend on model specification, parameter estimation and the number of simulations.
7. Expected Shortfall
Expected Shortfall measures the average loss in the worst 1-α proportion of outcomes. A definition that remains valid for discrete distributions and probability masses is:
ES_α(L) = \frac{1}{1-α}∫α1 VaRu(L)du
For a continuous loss distribution, this simplifies to:
ES_α(L) = E[L \mid L ≥ VaR_α(L)]
The conditional-expectation expression should not be applied mechanically to discrete distributions because observations at the VaR threshold may require fractional weighting. The quantile-integral definition is the more general rule.
7.1 Tail-consistent empirical ES
For n equally weighted losses sorted as l(1) ≤ ... ≤ l(n), each observation occupies a probability interval of width 1/n. Empirical ES averages exactly the upper 1-α probability mass. If n(1-α) is an integer k, ES is the arithmetic mean of the largest k losses. If it is not an integer, the observation at the tail boundary receives only the fractional weight needed to complete the tail probability.
Worked example: discrete ES
Consider five equally weighted losses:
Ordered outcome
Loss
1
−$200
2
$0
3
$100
4
$400
5
$1,000
At 80% confidence, the upper tail probability is 20%. Each observation has probability 20%, so the tail contains only the $1,000 loss:
ES0.80 = $1,000
Averaging $400 and $1,000 would cover 40% of the distribution and would therefore not be an 80% ES.
Unlike VaR, ES reflects the severity of losses beyond the quantile. ES is also a coherent risk measure under standard mathematical conditions, including subadditivity, making it more suitable for recognising diversification. Its estimate can nevertheless be unstable because tail observations are scarce.
Loss distribution showing Value at Risk and Expected Shortfall tail
Figure: Loss distribution showing Value at Risk and Expected Shortfall tail
Figure: VaR identifies a loss threshold, whereas ES measures average loss across the specified tail probability.
8. Backtesting Market Risk Models
Backtesting compares model forecasts with realised outcomes. For VaR, an exception occurs when actual loss exceeds forecast VaR. The organisation should specify consistently how equality is treated.
For confidence level α, the expected exception probability is p = 1-α. Over T independent observations, the expected number of exceptions is:
E[X] = T(1-α)
For example, a correctly calibrated 99% daily VaR model should produce approximately 2.5 exceptions over 250 days. The actual count will vary randomly.
The Kupiec unconditional-coverage test examines whether the observed exception rate differs significantly from the model rate. If x exceptions occur in T observations:
Under standard large-sample assumptions and the null hypothesis of correct unconditional coverage, the statistic is approximately chi-squared with one degree of freedom. This test assesses frequency, not whether exceptions cluster. Independence or conditional-coverage tests are required to detect clustering.
Backtesting difficulties include changes in portfolio composition, intraday trading, valuation adjustments and differences between hypothetical and actual profit and loss. ES backtesting is more complex because ES is not a single observable quantile threshold.
9. Stress Testing and Scenario Analysis
VaR and ES depend on modelled probability distributions. Stress testing instead asks what would happen under severe but plausible conditions.
9.1 Sensitivity stress tests
A sensitivity test changes one factor at a time—for example, a 200-basis-point rate rise or a 20% equity fall. It is transparent but may ignore interactions among factors.
9.2 Historical scenarios
Historical scenarios reproduce movements from events such as financial crises, currency devaluations or commodity shocks. They are intuitive, but current portfolios and market structures may differ from those of the historical period.
9.3 Hypothetical scenarios
Hypothetical scenarios combine internally consistent movements based on economic narratives. A recession scenario might combine falling equities, widening credit spreads, declining commodity demand and appreciation of safe-haven currencies.
9.4 Reverse stress testing
Reverse stress testing starts with an unacceptable outcome—such as breaching capital or liquidity limits—and identifies scenarios capable of producing it. It reveals hidden concentrations and challenges assumptions about what is plausible.
Good stress tests specify the horizon, risk-factor shocks, dependencies, management actions, liquidity effects and second-round consequences. Adding the worst individual shock for every factor may create an economically inconsistent scenario.
10. Illustrative Integrated Case: Meridian Treasury Portfolio
Background
Meridian Group is a fictional multinational company reporting in US dollars. Its treasury portfolio contains:
$20 million of fixed-rate bonds with modified duration 4.2 and convexity 22;
a €3 million receivable valued at $1.10 per euro;
a $5 million equity portfolio with beta 1.2;
20,000 barrels of oil inventory priced at $78 per barrel.
Problem
Management considers a scenario involving a 75-basis-point interest-rate rise, an 8% depreciation of the euro against the dollar, a 12% equity-market fall and a 15% oil-price fall.
Management could reduce exposure through an interest-rate swap or bond futures, a euro forward, equity-index futures and oil futures. Hedge ratios should reflect the relevant sensitivities rather than merely matching notional amounts. For example, the bond hedge should be based on DV01, while an equity-index hedge should reflect portfolio beta.
A complete decision would also consider hedge cost, accounting designation, counterparty credit risk, liquidity, collateral requirements and basis risk. The $1,835,625 result should not be described as a probability-based loss because the scenario was specified rather than estimated from a distribution.
Outcome and Learning Points
The analysis shows that equity exposure creates the largest individual loss, but the bond exposure is also material. The case demonstrates that:
exposures must be translated into common currency units;
nonlinear interest-rate effects can matter;
beta-based losses capture market exposure but omit idiosyncratic equity movements;
scenario losses should be reported separately from VaR and ES;
hedges should target sensitivities and maturities, not only notionals.
11. Model Limitations, Governance and Contemporary Practice
Market risk models are vulnerable to model risk. Historical relationships may break down, correlations may rise during crises, markets may become illiquid, and data may contain errors. Linear approximations can be inaccurate for options or large shocks. A low measured risk can therefore reflect an inadequate model rather than a safe portfolio.
Effective governance requires independent model validation, documented assumptions, data-quality controls, version management, limit monitoring and escalation procedures. Traders and managers should not manipulate valuation inputs or reporting windows to conceal exposure. Compensation arrangements should discourage excessive short-term risk taking.
Regulatory practice increasingly emphasises ES, stressed calibration, liquidity horizons and more granular risk-factor treatment. Institutions also use high-frequency data, cloud computation and machine learning to detect changing volatility regimes. Artificial intelligence can help identify anomalies and generate scenarios, but opaque models must not replace explainability, validation or human accountability.
Best practice is to operate a measurement dashboard combining:
current sensitivities and concentration limits;
VaR and ES at clearly specified horizons;
historical and hypothetical stress losses;
backtesting results and model exceptions;
liquidity and basis-risk indicators;
qualitative assessment of emerging risks.
Chapter Summary
Market risk measurement converts positions into economically meaningful exposure estimates. Interest-rate risk can be measured through duration, DV01 and convexity; foreign-exchange and commodity exposures through price sensitivities; and systematic equity exposure through beta. VaR estimates a confidence-level loss threshold, while ES measures average loss over an exact tail probability. For discrete samples, ES must use probability-consistent weighting rather than arbitrarily averaging observations near the tail.
Backtesting evaluates whether forecast losses are consistent with realised outcomes. Stress testing and scenario analysis complement statistical measures by assessing severe conditions, structural changes and interacting risks. Because all models depend on assumptions, reliable market risk management requires multiple measures, independent validation, transparent governance and informed professional judgement.
Key Terms
Basis Risk: Risk that a hedging instrument and the underlying exposure do not move identically.
Beta: Sensitivity of an asset's return to the return on a market portfolio.
Convexity: Measure of curvature in the relationship between a bond's price and yield.
Currency Delta: Change in domestic value for a unit change in an exchange rate.
DV01: Approximate currency change in value resulting from a one-basis-point yield movement.
Expected Shortfall: Average loss across the worst specified proportion of outcomes.
Historical Simulation: Risk estimation using observed historical changes in market factors.
Key-Rate Duration: Sensitivity to a yield change at a selected maturity point.
Macaulay Duration: Present-value-weighted average time to receive fixed-income cash flows.
Market Risk: Potential loss caused by changes in market prices or rates.
Model Risk: Risk of loss or poor decisions arising from defective models or their misuse.
Modified Duration: Approximate percentage price sensitivity to a small yield change.
Monte Carlo Simulation: Risk estimation using repeated draws from specified stochastic models.
Net Open Position: Unhedged exposure remaining after offsetting positions are considered.
Parametric VaR: VaR calculated from an assumed distribution and estimated parameters.
Reverse Stress Test: Analysis that identifies scenarios capable of producing a predetermined adverse outcome.
Scenario Analysis: Evaluation of portfolio effects under a coherent set of risk-factor changes.
Sensitivity Analysis: Measurement of value changes caused by changes in individual risk factors.
Stress Testing: Evaluation of exposure under severe but plausible market conditions.
Value at Risk: Loss quantile for a specified horizon and confidence level.
Review Questions
Short Answer Questions
Distinguish between a market position and a market risk exposure.
What assumptions underlie the use of modified duration?
Why does VaR not represent a maximum possible loss?
What is basis risk in commodity hedging?
Why can beta change during periods of financial stress?
Long Answer Questions
Explain how duration, DV01 and convexity jointly support the management of fixed-income risk.
Compare parametric, historical simulation and Monte Carlo approaches to VaR.
Explain why Expected Shortfall provides more information about tail severity than VaR.
Evaluate the role of backtesting in market risk model governance.
Discuss how stress testing should be designed to complement VaR and ES.
Analytical / Application Questions
A $12 million bond portfolio has modified duration 3.8 and convexity 18. Estimate its value change following a 60-basis-point yield increase.
Construct a coherent recession scenario for a portfolio containing equities, foreign-currency receivables and commodity inventory. Explain the assumed relationships among the shocks.
Seven equally weighted losses are $0, $50, $100, $150, $300, $500 and $1,000. Using the quantile-integral definition, calculate empirical ES at a confidence level of 80%, applying fractional probability weighting where necessary.
Multiple Choice Questions with Answers
Question 1
A $5 million bond portfolio has modified duration 4.0. Ignoring convexity, what is the approximate loss if yields rise by 25 basis points?
A. $5,000
B. $25,000
C. $50,000
D. $500,000
Correct Answer: C. $50,000
Question 2
A bond has modified duration 6 and convexity 30. For a yield increase of 1%, what is its approximate percentage price change?
A. −6.15%
B. −5.85%
C. −3.00%
D. +5.85%
Correct Answer: B. −5.85%
Question 3
A $20 million portfolio has daily volatility of 1%. Assuming zero mean and normal returns, what is its 99% one-day VaR using z0.99=2.326?
A. $200,000
B. $232,600
C. $465,200
D. $2,326,000
Correct Answer: C. $465,200
Question 4
Five equally weighted ordered losses are −$200, $0, $100, $400 and $1,000. Under the tail-probability-consistent empirical definition, what is ES at 80% confidence?
A. $400
B. $700
C. $800
D. $1,000
Correct Answer: D. $1,000
Question 5
A $4 million equity portfolio has beta 1.5. Ignoring idiosyncratic return, what loss is estimated if the market falls by 8%?
A. $320,000
B. $480,000
C. $600,000
D. $800,000
Correct Answer: B. $480,000
Question 6
A US company has a €2 million receivable. The exchange rate falls from $1.12 to $1.07 per euro. What is the change in the receivable's dollar value?
A. A gain of $100,000
B. A loss of $50,000
C. A loss of $100,000
D. A loss of $2,140,000
Correct Answer: C. A loss of $100,000
Question 7
For a correctly calibrated 99% daily VaR model tested over 250 days, what is the expected number of exceptions?
A. 1.0
B. 2.5
C. 5.0
D. 25.0
Correct Answer: B. 2.5
Question 8
Which situation best illustrates basis risk?
A. A copper producer hedges with copper futures whose delivery grade differs from its output
B. A bond loses value when market yields rise
C. An equity portfolio falls when the stock market falls
D. A euro receivable gains value when the euro appreciates
Correct Answer: A. A copper producer hedges with copper futures whose delivery grade differs from its output
Question 9
Which statement correctly distinguishes scenario analysis from VaR?
A. Scenario analysis always assigns an exact probability to each loss
B. VaR measures the maximum possible loss, whereas scenario analysis measures average loss
C. Scenario analysis evaluates specified joint shocks, whereas VaR estimates a distributional loss quantile
D. VaR can be used only for interest-rate risk
Correct Answer: C. Scenario analysis evaluates specified joint shocks, whereas VaR estimates a distributional loss quantile
Question 10
Why should market risk reporting combine VaR with Expected Shortfall and stress testing?
A. The three measures are mathematically identical
B. VaR alone does not show tail severity or vulnerability to specified extreme conditions
C. Expected Shortfall eliminates all model risk
D. Stress testing guarantees that every future crisis is represented
Correct Answer: B. VaR alone does not show tail severity or vulnerability to specified extreme conditions