Hva er Conditional VaR (CVaR/ES)?
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Conditional Value at Risk (CVaR), også kjent som Expected Shortfall (ES) eller Expected Tail Loss (ETL), er et risikomål som kvantifiserer det gjennomsnittlige tapet i de verste scenariene som overskrider terskelen for Value at Risk (VaR). Mens VaR svarer "hva er det maksimale tapet jeg forventer ikke å overskride ved et gitt konfidensnivå," svarer CVaR "gitt at jeg har overskredet VaR-terskelen, hva er det gjennomsnittlige tapet jeg bør forvente?" Dette gjør CVaR til et overlegent risikomål for å fange opp halerisiko - de katastrofale tapene som skjer sjelden, men med ødeleggende innvirkning. For eksempel, hvis 95 % 1-dagers VaR for en portefølje er $1 million, betyr det at det er en 5% sjanse for å tape mer enn $1 million på en dag. 95 % CVaR vil da være gjennomsnittet av alle tap i de verste 5 % av scenariene – kanskje 1,8 millioner dollar. CVaR er alltid større enn eller lik VaR på samme konfidensnivå, og gir kritisk informasjon om alvorlighetsgraden (ikke bare sannsynligheten) for ekstreme tap. CVaR har blitt det foretrukne risikomålet i moderne risikostyring og regulatoriske rammer av flere grunner. For det første er det sammenhengende: i motsetning til VaR, tilfredsstiller CVaR den matematiske egenskapen til subadditivitet (CVAR for en kombinert portefølje er ikke større enn summen av individuelle CVaR), noe som betyr at diversifisering alltid belønnes. VaR mangler denne egenskapen. For det andre gir CVaR mer informasjon om halen av tapsfordelingen. For det tredje er CVaR konveks og kan optimaliseres ved hjelp av lineær programmering, noe som gjør det praktisk for porteføljeoptimalisering. Basel-komiteen for banktilsyn (BCBS) endret formelt rammeverket for markedsrisiko fra VaR til Expected Shortfall i Fundamental Review of the Trading Book (FRTB), som fikk full effekt i 2023. Under FRTB må bankene beregne ES på et 97,5 % konfidensnivå (tilsvarer i haledekning til 99 % Vatur-risiko), med mer risiko. Dette reguleringsskiftet erkjenner at VaR systematisk undervurderer risiko under markedskriser, som demonstrert av den globale finanskrisen i 2008 og påfølgende stressperioder. CVaR kan beregnes analytisk (for normalfordelt avkastning), fra historisk simulering (gjennomsnitt av observerte tap i halen), eller ved å bruke Monte Carlo-simulering (gjennomsnitt av simulerte haletap). Hver metode har styrker og svakheter, og sofistikerte risikosystemer bruker vanligvis alle tre som krysssjekker.
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Formel
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Parametrisk CVaR (normal): CVaR_α = −μ + σ × φ(Φ⁻¹(α)) / (1−α)
Historisk CVaR: Gjennomsnitt av alle observerte tap som overstiger VaR_α
Parallell: CVaR = E[Tap | Tap > VaR_α]Variabelbeskrivelse
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| Symbol | Navn | Enhet | Beskrivelse |
|---|---|---|---|
| α | Konfidensnivå | % | Sannsynlighetsterskel: 95 % eller 99 % for VaR; 97,5 % for Basel III ES. CVaR måler gjennomsnittlig tap utover dette nivået. |
| VaR_α | Value at Risk | USD | Tapsterskelen ved konfidensnivå α; CVaR er gjennomsnittet av tap som overstiger VaR. |
| CVaR_α | Betinget VaR / forventet mangel | USD | Gjennomsnittlig tap betinget av å være i (1−α) verste hale; CVaR ≥ VaR alltid. |
| σ | Porteføljevolatilitet | %/day or %/year | Standardavvik for porteføljeavkastning; primær inngang for parametrisk CVaR-beregning. |
| ES_ratio | ES/VaR-forhold | ratio | CVaR delt på VaR på samme konfidensnivå; for normalfordelinger, 99 % ES/VaR ≈ 1,14; høyere for fetthalefordelinger. |
Slik Conditional VaR (CVaR/ES)
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- 1Spesifiser konfidensnivå (α): typisk 95 % eller 99 % for intern risikostyring; 97,5 % for Basel III FRTB regulatorisk kapital.
- 2Samle porteføljeavkastningsdata: historisk avkastning (historisk simulering) eller modellparametere μ og σ (parametrisk).
- 3Beregn VaR på nivå α: tapet overskredet med sannsynlighet (1−α). For normal: VaR = −μ + σ × Φ⁻¹(α).
- 4For historisk CVaR: sorter alle historiske tap fra verst til best; gjennomsnitt tapene i de verste (1−α)% av observasjonene.
- 5For parametrisk CVaR (normal): CVaR = −μ + σ × [φ(Φ⁻¹(α)) / (1−α)], hvor φ er standard normal PDF og Φ⁻¹ er invers CDF.
- 6For Monte Carlo CVaR: simuler tusenvis av porteføljeavkastningsscenarier; sortere etter tap; gjennomsnitt de verste (1−α)% av simulerte utfall.
- 7Rapporter CVaR sammen med VaR og ES/VaR-forholdet. Sammenlign med kapitalreserver og risikogrenser. Under Basel FRTB erstatter ES VaR som det primære regulatoriske kapitalmålet.
Løste eksempler
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CVaR is 15% larger than VaR — captures severity of extreme losses
For a normal distribution: VaR₉₉% = σ × Φ⁻¹(0.99) = 1.5% × 2.326 = 3.489% → $348,900. CVaR₉₉% = σ × φ(2.326) / 0.01 = 1.5% × (0.02665 / 0.01) = 1.5% × 2.665 = 3.998% → $399,800. CVaR of $400K represents the average daily loss in the worst 1% of days. This means on the very worst days, the portfolio expects to lose $400,000 on average — not just the $348,900 VaR threshold. Risk managers use CVaR to size capital buffers for tail events.
Historical tail is fatter than normal — CVaR/VaR ratio=1.47x
With 2,520 observations at 95% confidence: worst 5% = 126 observations. Sort all returns from worst to best; the 126th worst is the VaR cutoff (loss = 1.5% × $50M = $750K). Average all 126 worst losses: if the average worst-5% loss is 2.2%, CVaR = 2.2% × $50M = $1,100,000. The CVaR/VaR ratio of 1.47 reflects fat tails in actual equity returns — real distributions have more extreme events than normal distribution predicts. This is why historical simulation often produces higher CVaR than parametric methods.
FRTB uses ES at 97.5% over stressed historical scenarios — replacing old 99% VaR
Under Basel III FRTB (effective 2023), market risk capital for trading books uses Expected Shortfall at 97.5% confidence over a 10-day liquidity horizon, calibrated to a stressed historical period. A 97.5% ES under normality is approximately equal to a 99% VaR in tail area, but ES captures the magnitude of losses beyond the threshold. Banks must identify the worst one-year stress period in their history and calculate ES using that data. Capital = ES_stressed × liquidity adjustment factors for different asset classes.
CVaR is subadditive — VaR is not (VaR fails this property)
The combined portfolio CVaR of $650K is less than the sum of individual CVaRs ($500K + $400K = $900K), demonstrating CVaR's subadditivity property. The $250K difference is the diversification benefit — assets A and B don't perfectly co-crash, so tail losses are partially offset. VaR does not always exhibit this property (VaR of combined can exceed the sum in some cases), which is why VaR is not a coherent risk measure while CVaR is. This mathematical coherence makes CVaR preferable for portfolio optimization and regulatory capital.
Praktiske anvendelser
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Bank trading book regulatory capital (Basel III FRTB), enabling practitioners to make well-informed quantitative decisions based on validated computational methods and industry-standard approaches, which requires precise quantitative analysis to support evidence-based decisions, strategic resource allocation, and performance optimization across diverse organizational contexts and professional disciplines
Hedge fund risk monitoring and investor reporting, helping analysts produce accurate results that support strategic planning, resource allocation, and performance benchmarking across organizations, where accurate numerical computation is essential for producing reliable outputs that inform planning, evaluation, and continuous improvement processes in both corporate and individual settings
Insurance company reserve adequacy under Solvency II, allowing professionals to quantify outcomes systematically and compare scenarios using reliable mathematical frameworks and established formulas, demanding systematic calculation approaches that translate raw input data into actionable insights for stakeholders who depend on quantitative rigor in their daily professional activities
Portfolio optimization with tail risk constraints, supporting data-driven evaluation processes where numerical precision is essential for compliance, reporting, and optimization objectives, necessitating robust computational methods that deliver consistent and verifiable results suitable for reporting, auditing, and long-term trend analysis in professional environments
Pension fund asset-liability risk management, which requires precise quantitative analysis to support evidence-based decisions, strategic resource allocation, and performance optimization across diverse organizational contexts and professional disciplines
Spesielle tilfeller
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Under a Student's t-distribution with 4 df, ES/VaR ratios are 50–100% higher than normal, meaning parametric normal CVaR significantly underestimates actual tail risk. Always supplement parametric CVaR with historical simulation."}. Professionals working with conditional var should be especially attentive to this scenario because it can lead to misleading results if not handled properly. Always verify boundary conditions and cross-check with independent methods when this case arises in practice.
Professionals working with conditional var should be especially attentive to this scenario because it can lead to misleading results if not handled properly. Always verify boundary conditions and cross-check with independent methods when this case arises in practice.
Professionals working with conditional var should be especially attentive to this scenario because it can lead to misleading results if not handled properly. Always verify boundary conditions and cross-check with independent methods when this case arises in practice.
ES/VaR Ratios at Various Confidence Levels (Normal Distribution)
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| Konfidensnivå | VaR Multiplier (z) | ES Multiplier | ES/VaR-forhold | Basel Application |
|---|---|---|---|---|
| 90% | 1.282 | 1.755 | 1.370 | Internal risk limit (common) |
| 95% | 1.645 | 2.063 | 1.254 | Standard risk reporting |
| 97.5% | 1.960 | 2.338 | 1.193 | Basel III FRTB primary metric |
| 99% | 2.326 | 2.665 | 1.145 | Legacy Basel II/2.5 standard |
| 99.5% | 2.576 | 2.892 | 1.123 | Insurance / Solvency II |
| 99,9 % | 3.090 | 3.368 | 1.090 | Economic capital / extreme risk |
Ofte stilte spørsmål
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Why is CVaR considered superior to VaR?
CVaR is superior to VaR for several interconnected reasons. First, CVaR is coherent: it satisfies subadditivity (diversification always helps), while VaR does not. Second, CVaR provides information about loss severity beyond the threshold, not just the probability of exceeding it. VaR says 'you'll lose no more than X with probability α' — but gives no information about how bad losses are when they do exceed X. Third, CVaR is convex and can be incorporated into portfolio optimization frameworks using linear programming. Fourth, CVaR is more sensitive to the shape of the tail — fat-tailed distributions produce materially higher CVaR, giving an honest picture of tail risk that VaR can obscure.
What is the Basel III FRTB requirement for Expected Shortfall?
The Fundamental Review of the Trading Book (FRTB), Basel III's market risk framework, replaced 99% VaR with 97.5% Expected Shortfall as the primary capital metric. Key requirements include: ES must be calculated using a 10-business-day liquidity horizon (not the old 10-day scaled from 1-day); different asset classes have different liquidity horizons (from 10 days for large-cap equities to 120 days for some credit products); ES must be calibrated to a stressed historical period; and banks must also calculate a non-stressed ES for comparison. The framework took effect for large internationally active banks in 2023, with Basel III full implementation completed through 2025–2028 in various jurisdictions.
What is the relationship between VaR and CVaR at different confidence levels?
For normally distributed returns, the ES/VaR ratio is a fixed function of confidence level. At 95%, ES is approximately 25% higher than VaR. At 99%, ES is approximately 14% higher than VaR. At 99.9%, ES is about 8% higher. However, for fat-tailed (leptokurtic) distributions — which better describe actual financial returns — the ratio is substantially higher. A Student's t-distribution with 4 degrees of freedom produces ES/VaR ratios 50–100% higher than the normal case. This is why historical or Monte Carlo simulation methods typically produce higher CVaR than parametric normal models.
How does CVaR change during market stress periods?
CVaR increases dramatically during stress periods because: (1) portfolio volatility rises, shifting the entire loss distribution rightward; (2) correlations between assets increase toward 1.0 during crises, reducing diversification benefits; (3) return distributions become more negatively skewed and fat-tailed, amplifying tail losses beyond what volatility alone would predict. During the 2008 crisis, typical daily VaR estimates were breached 10–20 times more frequently than the implied 1% probability — demonstrating that historical CVaR estimates calibrated to tranquil periods severely understate tail risk in crises. This is why FRTB requires stressed-period calibration.
Can CVaR be used for portfolio optimization?
Yes, and this is one of CVaR's major practical advantages over VaR. Rockafellar and Uryasev (2000) showed that CVaR minimization can be formulated as a linear program, making it computationally tractable even for large portfolios. The mean-CVaR portfolio optimization framework minimizes CVaR for a given expected return (analogous to mean-variance Markowitz optimization), but incorporates tail risk into the objective function. This produces portfolios that are more robust to extreme events compared to mean-variance optimal portfolios, at a modest cost in expected return. Many institutional risk managers now use mean-CVaR optimization as their primary portfolio construction framework.
What is backtesting for CVaR and why is it difficult?
Backtesting VaR is straightforward: count how many days actual losses exceeded the VaR estimate and compare to the expected frequency (e.g., 1% for 99% VaR). CVaR backtesting is more challenging because CVaR is an average conditional on tail events — and tail events are rare, making statistical inference difficult with limited data. Tests include: (1) conditional coverage tests — do losses beyond VaR average to the predicted CVaR? (2) Acerbi-Szekely spectral tests — comparing the entire tail distribution. Because tail events are rare, CVaR backtests have low statistical power and require many years of data to detect even moderate underestimation. This is an active area of research in quantitative risk management.
How is CVaR different from stress testing?
CVaR is a statistical estimate of the average tail loss given the current distributional model. Stress testing examines specific worst-case scenarios — often historical crises (2008 GFC, COVID-19 crash, 1987 Black Monday) or hypothetical events (geopolitical shock, central bank policy reversal, sector collapse). CVaR is model-dependent and captures average tail behavior under the modeled distribution; stress tests capture specific scenario losses regardless of their statistical probability. Best practice combines both: CVaR for ongoing daily risk monitoring and capital allocation, stress testing for scenario-specific preparedness and board-level risk communication.
Vanlige feil å unngå
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- !Confusing CVaR with VaR — VaR is a threshold, CVaR is an average beyond that threshold.
- !Using normal distribution parametric CVaR as the only method without validating against historical simulation, which typically reveals fatter tails.
- !Not calibrating to a stressed historical period as required by FRTB for regulatory capital purposes.
- !Treating CVaR as additive across business lines without accounting for correlation structure.
- !Ignoring the liquidity horizon: 1-day CVaR is not appropriate for illiquid positions that cannot be sold in one day.
Pro Tips
Always report VaR and CVaR together, along with the CVaR/VaR ratio. A ratio much above 1.3–1.5 at 99% confidence signals fat tails in the return distribution and warns that the VaR underestimates the severity of tail events significantly.
Visste du?
CVaR as a formal risk measure was introduced by Rockafellar and Uryasev in their 2000 paper 'Optimization of Conditional Value-at-Risk' in the Journal of Risk. The same paper showed how CVaR could be computed via a simple linear programming problem, making portfolio CVaR minimization practically feasible for the first time — a major advance in quantitative risk management.
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Referanser
- ›Rockafellar & Uryasev (2000): Optimization of Conditional Value-at-Risk, Journal of Risk
- ›Basel Committee: Minimum Capital Requirements for Market Risk (FRTB, 2019)
- ›McNeil, Frey & Embrechts: Quantitative Risk Management (2nd ed.), Princeton University Press
- ›Investopedia: Conditional Value at Risk (CVaR)
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