Standard risk metrics often fail exactly when they are needed most. While Value at Risk (VaR) has been the industry standard for decades, its mathematical shortcomings and blindness to extreme tail events have led to a regulatory paradigm shift.
I. Defining VaR and CVaR
Risk measurement fundamentally asks a simple but crucial question: "How bad can things get?"
- Value at Risk (VaR): Represents the maximum expected loss over a given time horizon at a specific confidence level α. It acts as a strict threshold but completely ignores the severity of the losses beyond that threshold.
- Conditional VaR (CVaR): Also known as Expected Shortfall (ES), it answers a much more critical question: "If the tail event occurs, how much will we lose on average?"
II. Mathematical Coherence
In 1999, Artzner et al. formalized the axioms of a Coherent Risk Measure. In portfolio theory, the most crucial of these axioms is Subadditivity: ρ(X + Y) ≤ ρ(X) + ρ(Y).
The fundamental flaw of VaR is that it is not subadditive. Combining two portfolios can theoretically result in a combined VaR that is higher than the sum of the individual VaRs, mathematically penalizing diversification. Conversely, CVaR mathematically satisfies subadditivity, making it a coherent, theoretically sound, and superior metric.
III. Regulatory Applications (FRTB)
The limitations of VaR became painfully obvious during the 2008 financial crisis. Global banks suffered massive, catastrophic tail losses that their 99% VaR models had deemed statistically "impossible."
In response, the Basel Committee introduced the Fundamental Review of the Trading Book (FRTB). A central mandate for internal market risk models is the structural shift from a 99% VaR to a 97.5% Expected Shortfall (CVaR).