VaR tells you where the loss tail begins.CVaR tells you how deep that tail goes.
"If the 95% VaR barrier is broken, what is my average loss on those worst days?"
Because it captures magnitude as well as frequency, CVaR is recognized by regulators (Basel/FRTB) and academics as a coherent risk measure—unlike VaR, it satisfies sub-additivity.
For confidence level (e.g., 95%):
This formula calculates the average of all returns that fall below the VaR threshold. In simpler terms, it tells us the average loss we can expect when a VaR breach occurs. The result is typically expressed as a positive number representing the magnitude of expected loss.
var_95 = np.percentile(port_returns, 5)
cvar_95 = port_returns[port_returns <= var_95].mean()
var_90 = np.percentile(port_returns, 10)
cvar_90 = port_returns[port_returns <= var_90].mean()| Aspect | Detail |
|---|---|
| Data | Daily simple returns (port_returns) |
| Horizon | 1-day (matching VaR) |
| Technique | Purely empirical (no distributional assumptions) |
| Sign | Stored negative (loss) → displayed as e.g. –3.2% |
Both cvar_95 and cvar_90 are returned in every optimization card.
| Metric | Meaning |
|---|---|
| CVaR 95% | Average loss on the worst 5% of days |
| CVaR 90% | Average loss on the worst 10% of days |
CVaR ≈ VaR → loss tail is thin.
CVaR ≫ VaR → tail is fat; draw-downs escalate once VaR is breached.
Think of VaR and CVaR as two complementary ways of understanding portfolio risk:
What it tells you: "I am X% confident my losses won't exceed this threshold on a given day."
Real-world analogy: Like a flood warning level on a river - it tells you when to start worrying, but not how bad the flooding might get.
Mathematical property: Not coherent (adding two portfolios can increase risk)
What it tells you: "When worst-case scenarios occur, this is the average loss I can expect."
Real-world analogy: Like measuring the average depth of flooding after the river exceeds the warning level - it tells you how severe problems might be.
Mathematical property: Coherent (diversification always reduces risk)
VaR is intuitive and answers "How often might I face significant losses?", while CVaR answers the follow-up question "How bad could those losses be?" Together, they provide a more complete picture of portfolio risk.
This is why modern risk management frameworks and regulations increasingly require both measures.
| Variant | Formula / Method | Use-case |
|---|---|---|
| Parametric | (Normal) | Quick estimate for near-Gaussian assets |
| Cornish-Fisher | Adjust z-score using skew/kurtosis | Mildly non-normal tails |
| Monte Carlo | Simulate paths; average worst fraction | Options & non-linear pay-offs |
| Filtered ES | GARCH-scaled historical returns | Volatility-clustering markets |
Your platform currently runs the Historical ES (transparent and distribution-free).
Report alongside VaR. (Already done.)
Use rolling CVaR. Tail thickness evolves; a 1-year rolling plot flags build-ups.
Stress back-tests: compare crisis-year CVaR vs. normal periods.
Scenario scaling: for multi-day horizon under i.i.d., scale by — but beware volatility clustering.
Metric rows: "CVaR 95%: –3.2%, CVaR 90%: –2.5%".
Histogram overlay: solid red line for CVaR complementing dashed VaR.
Tooltip: "Average loss given the worst X% days." → link here.
Risk colour coding: highlight CVaR values that exceed user-set draw-down tolerances.
By surfacing CVaR 95% and 90% next to VaR, your platform equips users with a full picture of tail risk—frequency and severity—critical for robust risk management and capital planning.
The relationship between VaR and CVaR can be visualized on a return distribution:
X-axis: Portfolio returns
Y-axis: Frequency
Left tail shows VaR as the cutoff point, and CVaR as the average of all returns beyond that point
This illustration helps investors understand that while VaR represents a threshold (a single point in the distribution), CVaR represents the average of the entire extreme tail beyond that threshold, providing a more comprehensive view of potential extreme losses.
CVaR isn't just a risk measurement tool—it can be directly used as an optimization objective:
Directly minimize the expected shortfall of a portfolio to create allocations specifically designed to protect against tail events. These portfolios typically have different compositions than minimum-variance portfolios, especially when returns are not normally distributed.
Add CVaR constraints to other optimization objectives (like maximum return or Sharpe ratio) to ensure portfolios don't exceed specific tail risk thresholds, creating more resilient allocations that can withstand extreme market conditions.
The mathematical advantage of CVaR in optimization is that it can be reformulated as a linear programming problem, making it computationally tractable even for large portfolios with many constraints.
Coherent risk measure: Unlike VaR, CVaR satisfies the mathematical property of subadditivity, meaning diversification always reduces risk.
Tail sensitivity: Provides insight into the severity of extreme losses, not just their frequency, allowing for better worst-case scenario planning.
Regulatory alignment: Increasingly preferred by financial regulators (Basel Committee) for its more comprehensive risk assessment capabilities.
Optimization friendly: Can be formulated as a linear programming problem, making it tractable for complex portfolio optimization problems.
Distribution agnostic: Historical CVaR makes no assumptions about return distributions, capturing non-normal behavior common in financial markets.
Complexity: Less intuitive for non-technical users compared to VaR, requiring additional explanation to communicate its meaning effectively.
Data requirements: Needs more historical data to estimate accurately, as it focuses on tail events which are, by definition, rare.
Estimation sensitivity: Highly sensitive to outliers in the historical dataset, as these directly affect the tail average calculation.
Computational intensity: Some advanced CVaR calculation methods (like Monte Carlo) require significantly more computational resources than simple VaR.
Over-conservatism: May lead to excessively conservative portfolio allocations when used as the sole optimization objective, potentially sacrificing returns.
Basel Committee on Banking Supervision (2019) – FRTB: Expected Shortfall at 97.5% replaces VaR.
Artzner et al. (1999) – "Coherent Measures of Risk." Mathematical Finance, 9(3), 203-228.
Rockafellar & Uryasev (2002) – "Conditional Value-at-Risk for General Loss Distributions." Journal of Banking & Finance, 26, 1443-1471.