How is the stressed expected shortfall (ES) calculated under FRTB, and why did Basel replace VaR with ES?

The FRTB replaced Value-at-Risk with Expected Shortfall as the primary risk measure for the Internal Models Approach (IMA), addressing VaR's well-known failure to capture tail risk. The stressed ES adds a calibration to the most severe historical period.\n\nWhy ES Replaced VaR:\n\nVaR answers: 'What is the most I can lose at the 99th percentile?' But it says nothing about the magnitude of losses beyond that threshold. ES (also called CVaR) answers: 'Given that I'm in the worst 2.5% of outcomes, what is my expected loss?'\n\nUnder FRTB, the confidence level shifts from 99% VaR to 97.5% ES, which produces roughly equivalent capital for normal distributions but significantly higher capital for fat-tailed distributions.\n\nThe Stressed ES Formula:\n\n`mermaid\ngraph TD\n A[\"FRTB Capital: ES Component\"] --> B[\"ES_{R,S}
Reduced set of risk factors
Stressed period calibration\"]\n A --> C[\"ES_{R,C}
Reduced set of risk factors
Current period\"]\n A --> D[\"ES_{F,C}
Full set of risk factors
Current period\"]\n B --> E[\"Scaling Ratio
ES_{R,S} / ES_{R,C}\"]\n D --> F[\"Base ES
ES_{F,C}\"]\n E --> G[\"Stressed ES = ES_{F,C} x (ES_{R,S} / ES_{R,C})\"]\n F --> G\n`\n\nIMES = ES_{F,C} x (ES_{R,S} / ES_{R,C})\n\nwhere:\n- ES_{F,C} = ES using the full set of risk factors over the current (recent) period\n- ES_{R,S} = ES using a reduced set of risk factors over the stressed period\n- ES_{R,C} = ES using the reduced set of risk factors over the current period\n\nThe ratio ES_{R,S}/ES_{R,C} acts as a stress multiplier applied to the full-risk-factor current ES.\n\nLiquidity Horizon Adjustment:\n\nFRTB assigns different liquidity horizons (LH) to risk factor categories:\n\n| Risk Factor Category | Liquidity Horizon |\n|---|---|\n| Large-cap equities, major FX | 10 days |\n| Small-cap equities, corporate bonds | 20 days |\n| Emerging market rates, volatilities | 40 days |\n| Structured credit, exotic positions | 60 days |\n| Illiquid positions, bespoke products | 120 days |\n\nThe aggregated ES accounts for varying LH through a cascading formula:\n\nES = sqrt(sum over j of: (ES_j(LH_j))^2 x ((LH_j - LH_{j-1})/LH_j))\n\nwhere positions are grouped by increasing liquidity horizon buckets.\n\nWorked Example:\nSummitPoint Bank's trading desk holds a portfolio with two risk factor groups:\n\n- Group A (large-cap equity delta): ES_{10d} = $4.2M, LH = 10 days\n- Group B (equity vega, small-cap): ES_{20d} = $3.1M, LH = 20 days\n\nES_aggregated = sqrt(($4.2M)^2 + ($3.1M)^2 x (20-10)/20)\n= sqrt($17.64M^2 + $9.61M^2 x 0.5)\n= sqrt($17.64M^2 + $4.805M^2)\n= sqrt($22.445M^2) = $4.74M\n\nStress multiplier (ES_{R,S}/ES_{R,C}) = 1.65 (based on GFC stress period)\n\nIMES = $4.74M x 1.65 = $7.82M\n\nExplore FRTB capital requirements in our FRM Part II course.