Value at Risk P. Sercu, International Finance: Theory into Practice Overview Chapter 14 Value at Risk: Quantifying Overall Net Market Risk Overview Risk Budgeting—a Factor-based, Linear Approach Value at Risk Why work with factors not assets P. Sercu, Linking Bonds to Interest-rate Factors International Stock-market Risk Finance: Theory into Practice Currency Forwards Options Overview Swaps The Risk Budget The Linear/Normal VaR Model: Potential Flaws & Corrections A Zero-Drift (“Martingale”) Process A Constant-Variance Process Constant Correlations Between Factors. Linearizations in the Mapping f ! dV Choice of the factors Normality of dV Assets can be Liquidated in one Day Backtesting, Bootstrapping, Monte Carlo, and Stress Testing Backtesting Bootstrapping Monte Carlo Simulation Stress Testing Closing remarks CDOs/CDSs are hard to price Moral Hazard type risks Overview Risk Budgeting—a Factor-based, Linear Approach Value at Risk Why work with factors not assets P. Sercu, Linking Bonds to Interest-rate Factors International Stock-market Risk Finance: Theory into Practice Currency Forwards Options Overview Swaps The Risk Budget The Linear/Normal VaR Model: Potential Flaws & Corrections A Zero-Drift (“Martingale”) Process A Constant-Variance Process Constant Correlations Between Factors. Linearizations in the Mapping f ! dV Choice of the factors Normality of dV Assets can be Liquidated in one Day Backtesting, Bootstrapping, Monte Carlo, and Stress Testing Backtesting Bootstrapping Monte Carlo Simulation Stress Testing Closing remarks CDOs/CDSs are hard to price Moral Hazard type risks Overview Risk Budgeting—a Factor-based, Linear Approach Value at Risk Why work with factors not assets P. Sercu, Linking Bonds to Interest-rate Factors International Stock-market Risk Finance: Theory into Practice Currency Forwards Options Overview Swaps The Risk Budget The Linear/Normal VaR Model: Potential Flaws & Corrections A Zero-Drift (“Martingale”) Process A Constant-Variance Process Constant Correlations Between Factors. Linearizations in the Mapping f ! dV Choice of the factors Normality of dV Assets can be Liquidated in one Day Backtesting, Bootstrapping, Monte Carlo, and Stress Testing Backtesting Bootstrapping Monte Carlo Simulation Stress Testing Closing remarks CDOs/CDSs are hard to price Moral Hazard type risks Overview Risk Budgeting—a Factor-based, Linear Approach Value at Risk Why work with factors not assets P. Sercu, Linking Bonds to Interest-rate Factors International Stock-market Risk Finance: Theory into Practice Currency Forwards Options Overview Swaps The Risk Budget The Linear/Normal VaR Model: Potential Flaws & Corrections A Zero-Drift (“Martingale”) Process A Constant-Variance Process Constant Correlations Between Factors. Linearizations in the Mapping f ! dV Choice of the factors Normality of dV Assets can be Liquidated in one Day Backtesting, Bootstrapping, Monte Carlo, and Stress Testing Backtesting Bootstrapping Monte Carlo Simulation Stress Testing Closing remarks CDOs/CDSs are hard to price Moral Hazard type risks What are we after? (in this chapter) Value at Risk VaR = maximal loss that could be suffered on the P. Sercu, International current portfolio with 99% confidence Finance: Theory into Practice VaR: losses worse than VaR should occur only one day in 100 Risk Budgeting Linear/Normal VaR: B basis for a bank’s Capital Requirement calculations (“Market Discussion Risk Charge”), which is set at three times VaR or more Other Appraches B can be computed ... Closing remarks – from a normality model if σp is known – with normal factors, linearly related to returns (Riskmetrics) – from a reconstructed history of portfolio returns (backtesting) – from a simulated series of possible future events with ... – resampled actual returns (bootstrapping), or – normal or non-normal computer-generated factors, mapped into returns (Monte Carlo) What are we after? (in this chapter) Value at Risk VaR = maximal loss that could be suffered on the P. Sercu, International current portfolio with 99% confidence Finance: Theory into Practice VaR: losses worse than VaR should occur only one day in 100 Risk Budgeting Linear/Normal VaR: B basis for a bank’s Capital Requirement calculations (“Market Discussion Risk Charge”), which is set at three times VaR or more Other Appraches B can be computed ... Closing remarks – from a normality model if σp is known – with normal factors, linearly related to returns (Riskmetrics) – from a reconstructed history of portfolio returns (backtesting) – from a simulated series of possible future events with ... – resampled actual returns (bootstrapping), or – normal or non-normal computer-generated factors, mapped into returns (Monte Carlo) What are we after? (in this chapter) Value at Risk VaR = maximal loss that could be suffered on the P. Sercu, International current portfolio with 99% confidence Finance: Theory into Practice VaR: losses worse than VaR should occur only one day in 100 Risk Budgeting Linear/Normal VaR: B basis for a bank’s Capital Requirement calculations (“Market Discussion Risk Charge”), which is set at three times VaR or more Other Appraches B can be computed ... Closing remarks – from a normality model if σp is known – with normal factors, linearly related to returns (Riskmetrics) – from a reconstructed history of portfolio returns (backtesting) – from a simulated series of possible future events with ... – resampled actual returns (bootstrapping), or – normal or non-normal computer-generated factors, mapped into returns (Monte Carlo) Outline Risk Budgeting—a Factor-based, Linear Approach Value at Risk Why work with factors not assets P. Sercu, Linking Bonds to Interest-rate Factors International Stock-market Risk Finance: Theory into Practice Currency Forwards Options Risk Budgeting Swaps Why factors not assets? The Risk Budget The interest-rate factors Stock-market Risk The Linear/Normal VaR Model: Potential Flaws & Corrections Currency Forwards A Zero-Drift (“Martingale”) Process Options A Constant-Variance Process Swaps The Risk Budget Constant Correlations Between Factors. Linear/Normal VaR: Linearizations in the Mapping f ! dV Discussion Choice of the factors Other Appraches Normality of dV Assets can be Liquidated in one Day Closing remarks Backtesting, Bootstrapping, Monte Carlo, and Stress Testing Backtesting Bootstrapping Monte Carlo Simulation Stress Testing Closing remarks CDOs/CDSs are hard to price Moral Hazard type risks Why “Risk Budgeting”? Value at Risk Suppose we know std(~rp): VaR follows easily: P. Sercu, International Finance: Theory into Example Practice – data: – current value 100m, Risk Budgeting – expected return 10%, stdev 30%, both p.a. Why factors not assets? – all assets are liquid The interest-rate factors ) horizon is 1 trading day, i.e. 1/260 year Stock-market Risk Currency Forwards – Computations – expected value tomorrow: Options “ 0:10 ” Swaps 100m × 1 + 260 = 100:04m The Risk Budget – variance is linear in time, so the 1-day std is Linear/Normal VaR: q 1 Discussion 100m × 0:30 × 260 = 1:9m Other Appraches – maximal loss below 100.04 (with 99% confidence): Closing remarks 1:9m × 2:33 = 4:13m over one day How to get std? B portfolio theory: need full varcov matrix of all assets B Need Nobs N of assets, otherwise “linear dependencies” B )factor-covariance/normality solution: “map” return as functions of far fewer underlying factors Why “Risk Budgeting”? Value at Risk Suppose we know std(~rp): VaR follows easily: P. Sercu, International Finance: Theory into Example Practice – data: – current value 100m, Risk Budgeting – expected return 10%, stdev 30%, both p.a. Why factors not assets? – all assets are liquid The interest-rate factors ) horizon is 1 trading day, i.e. 1/260 year Stock-market Risk Currency Forwards – Computations – expected value tomorrow: Options “ 0:10 ” Swaps 100m × 1 + 260 = 100:04m The Risk Budget – variance is linear in time, so the 1-day std is Linear/Normal VaR: q 1 Discussion 100m × 0:30 × 260 = 1:9m Other Appraches – maximal loss below 100.04 (with 99% confidence): Closing remarks 1:9m × 2:33 = 4:13m over one day How to get std? B portfolio theory: need full varcov matrix of all assets B Need Nobs N of assets, otherwise “linear dependencies” B )factor-covariance/normality solution: “map” return as functions of far fewer underlying factors More on sources of risk and factors) Value at Risk P. Sercu, Sources of uncertainty: (in standard software) International Finance: Theory into B exchange risks Practice B stock price risks (by country, in local currency) Risk Budgeting interest rate risk (by currency and time to maturity; say 13 Why factors not assets? B The interest-rate factors rates) Stock-market Risk Currency Forwards B commodity price risk Options Swaps The Risk Budget A factor: the unexpected percentage change in the Linear/Normal VaR: source-of-risk variable. Discussion (C) Other Appraches B J.P. Morgan’s Riskmetrics provides a covariance matrix for Closing remarks hundreds of these factors. Mapping: link between factor and asset return ∗ “ ∆V∗ ∆S ” B E.g. a 5-year FC bond: V = V S so ∆V ≈ V × V∗ + S ∆V∗ with V∗ a function of the foreign 5-year interest rate More on sources of risk and factors) Value at Risk P. Sercu, Sources of uncertainty: (in standard software) International Finance: Theory into B exchange risks Practice B stock price risks (by country, in local currency) Risk Budgeting interest rate risk (by currency and time to maturity; say 13 Why factors not assets? B The interest-rate factors rates)