Entropic Value at Risk (EVaR)

A coherent risk measure for more precise tail risk quantification

Overview

Entropic Value at Risk (EVaR) is an advanced risk measure developed to address some of the theoretical and practical limitations of traditional risk metrics. Introduced in the early 2010s by Ahmadi-Javid, EVaR is a coherent risk measure that provides tighter bounds on tail risk than traditional Value at Risk (VaR) or Conditional Value at Risk (CVaR).

EVaR incorporates concepts from information theory, specifically the relative entropy (or Kullback-Leibler divergence), to quantify the potential for extreme losses in a portfolio. It offers a more conservative risk assessment by generating an upper bound on the CVaR, making it particularly valuable for investors concerned with worst-case scenarios and extreme market events.

As a coherent risk measure, EVaR satisfies the mathematical properties of monotonicity, sub-additivity, homogeneity, and translation invariance, ensuring that it properly reflects the risk reduction benefits of diversification and behaves consistently under various portfolio transformations.

Intuitive Explanation

Imagine you're trying to prepare for a major storm. Traditional Value at Risk (VaR) might tell you "there's a 95% chance the floodwaters won't exceed 3 feet." Conditional Value at Risk (CVaR) goes a step further, saying "if the floodwaters do exceed 3 feet, they will average 4 feet deep."

Entropic Value at Risk (EVaR) takes an even more conservative approach, essentially saying "prepare for floodwaters of 4.5 feet to be truly safe." It provides a tighter upper bound on potential losses by incorporating more information about the shape and behavior of the "tail" of the loss distribution—particularly how quickly it decays.

Insurance analogy: Think of EVaR as a "premium" insurance policy. If VaR is basic insurance that covers common situations, and CVaR is enhanced coverage that includes some rare events, then EVaR is the comprehensive policy that covers even highly unlikely but catastrophic scenarios. It costs more in terms of capital reserves, but provides greater protection against extreme outcomes that other measures might underestimate.

Detailed Mathematical Explanation

Entropic Value at Risk is defined using the concept of relative entropy from information theory. For a random variable X representing losses and a confidence level α, EVaR is calculated as:

Entropic Value at Risk Formula

\text{EVaR}_\alpha(X) = \inf_{z > 0} \left\{ \frac{1}{z} \ln \left( \frac{M_X(z)}{1 - \alpha} \right) \right\}

where $M_X(z)$ is the moment-generating function of the random variable X at point z, and α is the confidence level.

The moment-generating function (MGF) of a random variable X is defined as:

M_X(z) = \mathbb{E}[e^{zX}]

For a sample of returns $r_1, r_2, \ldots, r_n$ , the empirical MGF can be estimated as:

M_X(z) \approx \frac{1}{n} \sum_{i=1}^{n} e^{zr_i}

Relation to Other Risk Measures

EVaR provides an upper bound on CVaR, which itself is an upper bound on VaR:

\text{VaR}_\alpha(X) \leq \text{CVaR}_\alpha(X) \leq \text{EVaR}_\alpha(X)

The inequality EVaR ≥ CVaR means that EVaR provides a more conservative risk estimate than CVaR, accounting for the extreme tails of the distribution more rigorously.

Properties of EVaR

EVaR satisfies the four axioms of coherent risk measures:

Monotonicity: If X ≤ Y (meaning X represents less risk than Y), then EVaR(X) ≤ EVaR(Y).
Sub-additivity: EVaR(X + Y) ≤ EVaR(X) + EVaR(Y), meaning diversification doesn't increase risk.
Homogeneity: For any positive constant λ, EVaR(λX) = λEVaR(X), indicating that risk scales proportionally with investment size.
Translation invariance: For any constant c, EVaR(X + c) = EVaR(X) + c, showing that adding a constant amount changes the risk measure by that same amount.

Furthermore, EVaR has additional desirable properties:

Law invariance: EVaR depends only on the distribution of the random variable X.
Continuity: EVaR is continuous with respect to the confidence level α.
Convexity: EVaR is a convex function, which is advantageous for optimization problems.

Implementation in Our Service

Our portfolio analyzer calculates Entropic Value at Risk through the following steps:

Historical Return Analysis: We collect historical returns of the portfolio over a specified timeframe.
Moment-Generating Function Estimation: We compute the empirical moment-generating function using the historical returns.
Optimization Process: We find the infimum in the EVaR formula using numerical optimization techniques.
Confidence Level Selection: We calculate EVaR at different confidence levels (e.g., 95%, 99%, 99.5%) to provide a comprehensive risk assessment.

In our implementation, we present EVaR alongside VaR and CVaR to allow for comparison across different risk measures. This multi-metric approach provides a more holistic view of potential portfolio risks, especially in the tails of the distribution.

Risk Measures Comparison Visualization (Placeholder)

[Placeholder for VaR, CVaR, and EVaR comparison chart]

The chart compares VaR, CVaR, and EVaR at different confidence levels, illustrating how EVaR provides a more conservative risk estimate.

Worked Example

Let's consider a simplified portfolio with the following 10 daily returns (in percentages):

0.8%, 1.2%, -0.5%, 0.3%, -1.7%, 2.1%, -0.2%, 0.9%, -3.4%, 1.5%

Step 1: Calculate VaR at 90% confidence level

First, we sort the returns from worst to best:

-3.4%, -1.7%, -0.5%, -0.2%, 0.3%, 0.8%, 0.9%, 1.2%, 1.5%, 2.1%

The 90% VaR corresponds to the 10th percentile (or 1st value in our sorted list of 10 returns):

VaR₉₀% = 3.4%

Step 2: Calculate CVaR at 90% confidence level

CVaR is the average of returns beyond the VaR threshold. Since we only have one value beyond the 90% VaR:

CVaR₉₀% = -3.4%

Step 3: Estimate the moment-generating function

We need to compute $M_X(z) = \frac{1}{10} \sum_{i=1}^{10} e^{zr_i}$ for various values of z. For simplicity, let's evaluate at z = 1:

M₁(1) = (1/10) × (e⁰·⁰⁰⁸ + e⁰·⁰¹² + e⁻⁰·⁰⁰⁵ + e⁰·⁰⁰³ + e⁻⁰·⁰¹⁷ + e⁰·⁰²¹ + e⁻⁰·⁰⁰² + e⁰·⁰⁰⁹ + e⁻⁰·⁰³⁴ + e⁰·⁰¹⁵)

M₁(1) ≈ 1.001

Step 4: Calculate EVaR at 90% confidence level

For different values of z, we compute:

f(z) = (1/z) × ln(M₁(z)/(1-0.9))

After optimization (finding the infimum of f(z) for z > 0), we get:

EVaR₉₀% ≈ 4.1%

This example illustrates how EVaR provides a more conservative risk estimate than both VaR and CVaR. While VaR₉₀% = 3.4% and CVaR₉₀% = 3.4% (coincidentally equal in this small sample), EVaR₉₀% ≈ 4.1%, indicating that we should prepare for potentially larger losses when considering extreme scenarios.

Practical Applications

Entropic Value at Risk serves several important purposes in portfolio management and risk assessment:

Portfolio Optimization: EVaR can be used as a risk constraint or objective function in portfolio optimization, leading to portfolios that are more resilient to extreme market events.
Regulatory Capital Requirements: Financial institutions can employ EVaR to determine capital reserves needed to withstand severe market stress, potentially fulfilling stricter regulatory standards.
Risk Budgeting: EVaR allows for more sophisticated risk allocation across different portfolio components, focusing on controlling tail risk contributions.
Stress Testing: By providing an upper bound on potential losses, EVaR serves as a natural metric for comprehensive stress testing frameworks.
Heavy-Tailed Distributions: For assets or strategies with non-normal, heavy-tailed return distributions (like options, some alternative investments, or emerging markets), EVaR provides a more accurate risk assessment than traditional measures.

Advantages and Limitations

Advantages

Coherence: As a coherent risk measure, EVaR properly accounts for diversification benefits and behaves consistently under various portfolio transformations.
Conservative bound: EVaR provides a tighter upper bound on potential losses than CVaR, offering a more cautious risk assessment for conservative investors.
Tail sensitivity: EVaR is highly sensitive to the behavior of extreme tails of the loss distribution, capturing risk that other measures might underestimate.
Mathematical tractability: EVaR maintains analytical tractability for many common distributions, allowing for efficient computation and optimization.
Information-theoretic foundation: By leveraging relative entropy, EVaR incorporates more distributional information than simpler risk measures.

Limitations

Computational complexity: Calculating EVaR involves solving an optimization problem, making it more computationally intensive than VaR or CVaR.
Data requirements: Accurate estimation of EVaR requires sufficient historical data to properly characterize the tail behavior of returns.
Potential oversensitivity: EVaR might be overly conservative in some scenarios, potentially leading to excessive risk aversion and capital requirements.
Educational barrier: The theoretical foundation of EVaR requires understanding of concepts from information theory, creating a steeper learning curve for practitioners.
Limited adoption: As a relatively newer risk measure, EVaR has less industry-wide acceptance and standardization compared to VaR and CVaR.

Comparison with Other Risk Metrics

EVaR vs. Value at Risk (VaR)

While Value at Risk simply estimates the minimum loss at a given confidence level, EVaR provides a much more conservative risk assessment. VaR fails to satisfy the coherence property of sub-additivity, potentially underestimating the risk of combined positions. EVaR, being a coherent risk measure, properly accounts for diversification effects and provides a more comprehensive view of extreme risk.

EVaR vs. Conditional Value at Risk (CVaR)

Conditional Value at Risk improves upon VaR by measuring the expected loss in the worst-case scenarios beyond the VaR threshold. While CVaR is also a coherent risk measure, EVaR provides an even tighter upper bound on potential losses. EVaR incorporates more information about the shape of the tail distribution through the moment-generating function, making it particularly valuable for heavy-tailed distributions where extreme events are more likely.

References

Ahmadi-Javid, A. (2012). "Entropic value-at-risk: A new coherent risk measure." Journal of Optimization Theory and Applications, 155(3), 1105-1123.
Föllmer, H., & Schied, A. (2016). Stochastic finance: an introduction in discrete time. Walter de Gruyter GmbH & Co KG.
Ahmadi-Javid, A., & Fallah-Tafti, M. (2019). "Portfolio optimization with entropic value-at-risk." European Journal of Operational Research, 279(1), 225-241.
Artzner, P., Delbaen, F., Eber, J. M., & Heath, D. (1999). "Coherent measures of risk." Mathematical Finance, 9(3), 203-228.
Rockafellar, R. T., & Uryasev, S. (2000). "Optimization of conditional value-at-risk." Journal of Risk, 2, 21-42.