Market risk Study Guide

📖 Core Concepts Market risk – potential loss from changes in market variables (prices, rates, volatilities). Equity, interest‑rate, currency, commodity risk – specific market‑risk categories tied to the underlying asset class. Margining risk – cash‑flow risk from future margin calls when positions move adversely. Shape risk – risk from changes in the shape of a yield curve or volatility surface (e.g., steepening, flattening). Holding‑period risk – uncertainty about how long a position will be held, affecting exposure duration. Basis risk – imperfect correlation between a hedged instrument and the exposure it protects. Value at Risk (VaR) – a single‑point estimate of the worst‑case loss over a given horizon at a chosen confidence level (e.g., 99%). Conditional Value at Risk (CVaR) – the expected loss beyond the VaR threshold; a coherent, sub‑additive risk measure. Fundamental Review of the Trading Book (FRTB) – 2016 regulatory overhaul that replaced VaR with Expected Shortfall (ES) for capital calculation and added an explicit illiquidity component. --- 📌 Must Remember VaR assumes a static portfolio (no trades) over the horizon. VaR is not coherent → it fails sub‑additivity; CVaR fixes this. Historical correlations can break down in turbulent markets (asymmetric dependence). FRTB: Uses Expected Shortfall instead of VaR. Requires a liquidity horizon per risk factor. Internal Models Approach (IMA) = bank‑specific models (must be approved). Standardized Approach (SA) = formulaic capital calculation when IMA is unavailable. Basis risk arises when hedge and exposure move out of sync → residual exposure remains. --- 🔄 Key Processes Calculating VaR (Variance‑Covariance) Estimate mean vector $\mu$ and covariance matrix $\Sigma$ of risk‑factor returns. Linearize portfolio P/L: $\Delta P \approx \mathbf{w}^\top \Delta \mathbf{X}$ (weights $\mathbf{w}$). Compute portfolio variance: $\sigmaP^2 = \mathbf{w}^\top \Sigma \mathbf{w}$. VaR$\alpha = z\alpha \sigmaP$, where $z\alpha$ is the standard‑normal quantile for confidence level $\alpha$. Historical Simulation VaR Collect a rolling window of past market‑factor changes. Reprice the portfolio for each historical shock. Sort resulting P/Ls; VaR is the $\alpha$‑quantile (e.g., the 1‑% worst loss). Monte‑Carlo VaR Specify a multivariate distribution for risk factors (e.g., Gaussian, Student‑t). Simulate a large number $N$ of factor paths over the horizon. Reprice portfolio for each path → generate loss distribution. Extract VaR/ES from the simulated loss distribution. FRTB Capital Calculation (simplified) Bucket risk factors (e.g., equity, rates, FX) and assign liquidity horizons. Compute Expected Shortfall for each bucket over its horizon. Apply prescribed risk‑weight and correlation matrices. Aggregate across buckets → total market‑risk capital. --- 🔍 Key Comparisons VaR vs. CVaR VaR: single loss quantile; not sub‑additive → can underestimate tail risk. CVaR: average loss beyond VaR; coherent, sub‑additive → better for diversification assessment. Variance‑Covariance vs. Historical Simulation Variance‑Covariance: assumes normality & linearity, fast, but misses fat tails & non‑linear payoffs. Historical Simulation: uses real past moves, captures non‑linearity, but relies on the stability of historic correlations. Internal Models vs. Standardized Approach IMA: tailored, potentially lower capital if model is approved; higher validation burden. SA: fixed formula, easier compliance; often yields higher capital charges. Equity Risk vs. Currency Risk Equity: price & implied volatility of stocks/indices. Currency: FX rates & their implied volatility; also subject to basis risk when hedging with forwards. --- ⚠️ Common Misunderstandings “VaR tells you the maximum loss.” Reality: VaR is a threshold (e.g., 99% VaR = loss not exceeded 99% of the time). Losses can exceed VaR in the remaining 1%. “Historical simulation automatically captures tail risk.” Reality: If the historical window lacks extreme events, the tail will be under‑estimated. “Higher VaR = worse risk.” Reality: VaR is scale‑dependent; a larger portfolio naturally has a larger VaR. Compare risk‑adjusted metrics (e.g., VaR/Notional). “Basis risk is negligible if the hedge ratio is 1.” Reality: Even with a 1:1 hedge, mismatched sensitivities (e.g., delta vs. gamma) create residual exposure. --- 🧠 Mental Models / Intuition “Loss Distribution = Landscape” – Visualize the portfolio’s possible P/L as a hill; VaR is the height of a contour line (e.g., 99% contour), CVaR is the average height beyond that contour. “Liquidity Horizon = Speed Limit” – Longer liquidity horizons mean you must assume the market can move further before you can exit, inflating capital. “Correlation is a Bridge, not a Wall.” – When markets are calm, the bridge (correlation) is sturdy; during stress it may collapse, making diversification less effective. --- 🚩 Exceptions & Edge Cases Non‑linear instruments (options, guarantees) – Variance‑covariance VaR underestimates risk; Monte‑Carlo or full revaluation is required. Discrete loss distributions – CVaR remains sub‑additive, while VaR can be highly sensitive to single data points. Asymmetric dependence – In crises, assets that usually move together may diverge; historical correlation assumptions become invalid. Margining risk – Not captured by standard VaR; needs a cash‑flow stress test for potential margin calls. --- 📍 When to Use Which Quick, back‑of‑the‑envelope check → Variance‑covariance VaR (linear portfolios). Portfolios with options, caps/floors, or other non‑linear payoffs → Monte‑Carlo or full revaluation historical simulation. Regulatory reporting under FRTB → Expected Shortfall with prescribed liquidity horizons (standardized or approved internal model). Assessing diversification benefits → Use CVaR (or ES) because of sub‑additivity. When data window is short or market regime has shifted → Stress‑testing or scenario analysis rather than pure historical simulation. --- 👀 Patterns to Recognize “Flat‑yield‑curve + steepening risk” → Shape risk shows up when long‑duration positions are present. “Large VaR but small CVaR” → Indicates a fat‑tailed loss distribution with few extreme outliers. “Correlation spikes in stress periods” → Expect higher capital under FRTB due to increased ES. “Margin calls coinciding with market moves” → Signals margining risk; look for simultaneous negative P/L and cash‑flow stress. --- 🗂️ Exam Traps Choosing VaR over ES for FRTB questions – The regulator now requires Expected Shortfall; VaR answers will be marked wrong. Assuming historical correlations are always stable – In volatility spikes, they break down; the correct answer will mention asymmetric dependence. Confusing “basis risk” with “basis” in futures pricing – Basis risk is about imperfect hedge correlation, not the price difference (basis) itself. Selecting the variance‑covariance method for an option‑heavy book – The correct method is Monte‑Carlo or a full revaluation approach. Treating CVaR as the same as “average VaR” – CVaR is the expected loss beyond the VaR threshold, not an average of multiple VaR levels. ---