VaR vs. Expected Shortfall: Coherence & Sub-additivity

1. What Makes a Risk Measure "Coherent"?

When we summarize the risk of an entire distribution of potential outcomes into a single number, we want that number to behave in a sensible and consistent way. A set of four properties, known as the axioms of coherence, formalizes this "sensible behavior." A risk measure $\rho (X)$ that satisfies all four is called a coherent risk measure.

The following resource from AnalystPrep clearly outlines these four properties.

As you've just read, the four properties of a coherent risk measure $\rho$ for any two portfolios (or random loss variables) $X$ and $Y$ are:

1. Monotonicity: If portfolio $Y$ consistently produces worse outcomes (larger losses) than portfolio $X$ in every possible state of the world (i.e., $Y\ge X$), then its risk must be higher: $\rho (Y)\ge \rho (X)$.
2. Translation Invariance: If we add a risk-free amount of cash, $h$, to a portfolio, its risk should decrease by exactly that amount: $\rho (X-h)=\rho (X)-h$.
3. Positive Homogeneity: If we scale our portfolio by a positive factor $k$, the risk should scale by the same factor: $\rho (kX)=k\rho (X)$ for $k>0$.
4. Sub-additivity: The risk of a combined portfolio should be no greater than the sum of the risks of the individual portfolios: $\rho (X+Y)\le \rho (X)+\rho (Y)$.

This last property, sub-additivity, is arguably the most important from a practical standpoint. It is the mathematical expression of the principle of diversification. The risk of the whole should not be greater than the sum of its parts. A risk measure that violates this property might discourage diversification, which goes against a cornerstone of modern finance.

2. The Failure of VaR: Violating Sub-additivity

Value at Risk is not a coherent risk measure because it can fail the sub-additivity test. This isn't just a theoretical curiosity; it has real-world implications. It means VaR can, in certain situations, make a diversified portfolio appear riskier than its individual components, which is counter-intuitive and misleading.

To see exactly how this happens, we'll work through a classic example involving a portfolio of corporate bonds. The following video provides an exceptionally clear, step-by-step demonstration.

Let's break down the logic from the video. The setup is perfect for leveraging your probability background.

Scenario:

• We have two identical, independent corporate bonds, Bond A and Bond B.
• Each bond has a 4% probability of defaulting over the next year.
• If a bond defaults, the loss is $100. If it does not default, the loss is $0.
• We want to calculate the 95% VaR. This means we are interested in the loss that will not be exceeded with 95% confidence (or, equivalently, the loss threshold at the worst 5% of outcomes).

Analysis:

1. Risk of a Single Bond (VaR(A))

   • The probability of no default is $1-0.04=0.96$, or 96%.
   • The 95% VaR asks for the loss we are 95% confident will not be exceeded. Since there is a 96% chance of a $0 loss, the 95% quantile falls in the "no default" scenario.
   • Therefore, VaR_95%(A) = $0.
   • Similarly, VaR_95%(B) = $0.
   • The sum of the individual risks is: VaR(A) + VaR(B) = $0 + $0 = $0.

2. Risk of the Combined Portfolio (VaR(A+B))

   • Now, let's consider a portfolio holding both Bond A and Bond B. Since they are independent, we can calculate the probabilities of joint events:
     • Prob(0 defaults) = $0.96\times 0.96=0.9216$ (92.16%)
     • Prob(2 defaults) = $0.04\times 0.04=0.0016$ (0.16%)
     • Prob(1 default) = $1-0.9216-0.0016=0.0768$ (7.68%)
   • What is the 95% VaR of this portfolio? We look at the cumulative loss distribution. There's a 92.16% chance of a $0 loss. This means the 5% tail (from 95% to 100%) must contain events with losses greater than $0.
   • Specifically, the event of "at least one default" has a probability of $1-0.9216=7.84\%$. Since 7.84% is greater than our significance level of 5%, a loss event is "captured" by our 95% VaR. The smallest possible loss is $100 (from one default).
   • Therefore, VaR_95%(A+B) = $100.

3. The Violation

   • We have found that VaR(A+B) = $100.
   • We also found that VaR(A) + VaR(B) = $0.
   • Clearly, $100 > $0.
   • This demonstrates the failure of sub-additivity: $\rho (A+B)>\rho (A)+\rho (B)$. The risk of the diversified portfolio is measured to be greater than the sum of the individual risks.

The issue is that VaR is "myopic." It only cares about the loss at a specific quantile ($\alpha$). It is completely blind to the nature and magnitude of losses beyond that point. In the single-bond case, the 4% default risk was just outside the 95% VaR's field of view. By combining the bonds, we increased the probability of at least one loss event, pulling it into the 95% VaR's view.

The two distributions in the image below illustrate this point well. Two portfolios can have the exact same VaR, but one can be far riskier because of a fatter tail. VaR doesn't capture this distinction. Combining assets can change the shape of the tail in a way that VaR incorrectly penalizes.

3. Expected Shortfall to the Rescue

Expected Shortfall, by its very definition, is designed to overcome this myopia. Because it is the average of all losses in the tail, it is not blind to what happens beyond the VaR threshold. It considers the entire tail of the distribution.

Let's briefly revisit our bond example and consider the 95% ES.

• For a single bond: The worst 5% of outcomes consists of the 4% chance of a $100 loss and 1% of the 96% chance of a $0 loss. The average loss across this 5% slice of probability is $\frac{0.04\times \$100}{0.05}=\$80$. So, ES_95%(A) = $80. The sum is ES(A) + ES(B) = $160.
• For the portfolio: The worst 5% of outcomes consist of the 0.16% chance of a $200 loss and a 4.84% slice of the 7.68% chance of a $100 loss. The expected loss in this tail is $\frac{(0.0016\times \$200)+(0.0484\times \$100)}{0.05}=\frac{\$0.32+\$4.84}{0.05}=\$103.20$.
• Conclusion: ES_95%(A+B) = $103.20, which is less than ES(A) + ES(B) = $160. Sub-additivity holds.

The formal proof that ES is always sub-additive is complex, but the intuition is straightforward. Since ES is a conditional expectation, and the expectation operator itself is sub-additive, this property carries through. By averaging over all possible tail events, ES naturally accounts for diversification benefits.

The fact that ES is a coherent risk measure, while VaR is not, is a major reason why regulators (for example, in the Basel III framework) have moved towards using ES for measuring market risk in banks' trading books.

Conclusion

This lesson concludes our exploration of fundamental risk measures. We have moved from simple metrics like variance to more sophisticated tail measures like VaR and ES, and we have now established a theoretical basis for preferring ES.

Here are the key takeaways from today's lesson:

• A coherent risk measure satisfies four axioms: monotonicity, translation invariance, positive homogeneity, and sub-additivity.
• Sub-additivity is the mathematical representation of the diversification principle: the risk of a whole should not exceed the sum of its parts.
• Value at Risk (VaR) is not coherent because it can violate sub-additivity. Its focus on a single quantile makes it blind to the shape of the tail, sometimes leading it to penalize diversification.
• Expected Shortfall (ES) is a coherent risk measure. Because it averages all losses within the tail, it properly captures the effects of diversification and provides a more robust measure of risk.

We have now built a solid foundation in understanding and quantifying investment risk. In our next module, we will begin our study of Mean-Variance Portfolio Theory. We will see how these concepts of risk and return are used to construct optimal portfolios, directly applying the principle of diversification that was so central to today's discussion.