Financial Risk Management Zvi Wiener Following P. Jorion, Financial Risk Manager Handbook
Transcription
Financial Risk Management Zvi Wiener Following P. Jorion, Financial Risk Manager Handbook
Financial Risk Management Zvi Wiener Following P. Jorion, Financial Risk Manager Handbook http://pluto.huji.ac.il/~mswiener/zvi.html FRM 972-2-588-3049 Chapter 17 VaR Methods Following P. Jorion 2001 Financial Risk Manager Handbook http://pluto.huji.ac.il/~mswiener/zvi.html FRM 972-2-588-3049 Risk Factors There are many bonds, stocks and currencies. The idea is to choose a small set of relevant economic factors and to map everything on these factors. • Exchange rates • Interest rates (for each maturity and indexation) • Spreads • Stock indices Ch. 17, Handbook Zvi Wiener slide 3 How to measure VaR • Historical Simulations • Variance-Covariance • Monte Carlo • Analytical Methods • Parametric versus non-parametric approaches Ch. 17, Handbook Zvi Wiener slide 4 Historical Simulations • Fix current portfolio. • Pretend that market changes are similar to those observed in the past. • Calculate P&L (profit-loss). • Find the lowest quantile. Ch. 17, Handbook Zvi Wiener slide 5 Example Assume we have $1 and our main currency is SHEKEL. Today $1=4.30. Historical data: P&L 4.00 4.20 4.30*4.20/4.00 = 4.515 0.215 4.20 4.30*4.20/4.20 = 4.30 0 4.10 4.30*4.10/4.20 = 4.198 -0.112 4.15 4.30*4.15/4.10 = 4.352 0.052 Ch. 17, Handbook Zvi Wiener slide 6 USD NIS 2000 100 -120 2001 200 100 2002 -300 -20 2003 20 30 today Ch. 17, Handbook 100 200 300 20 2 3 1 0.06 (1 0.061) (1 0.062) (1 0.063) 4 120 100 20 30 2 3 1 0.1 (1 0.11) (1 0.12) (1 0.13) 4 Zvi Wiener slide 7 today 100 200 300 20 2 3 1 0.06 (1 0.061) (1 0.062) (1 0.063) 4 120 100 20 30 2 3 1 0.1 (1 0.11) (1 0.12) (1 0.13) 4 Changes in IR USD: NIS: +1% +1% +1% 0% +1% -1% +1% -1% 100 200 300 20 2 3 1 0.07 (1 0.071) (1 0.072) (1 0.073) 4 120 100 20 30 2 3 1 0.11 (1 0.11) (1 0.11) (1 0.12) 4 Ch. 17, Handbook Zvi Wiener slide 8 Returns year 1% of worst cases Ch. 17, Handbook Zvi Wiener slide 9 VaR 1 0.8 0.6 0.4 VaR1% 1% 0.2 Profit/Loss -3 Ch. 17, Handbook -2 -1 1 Zvi Wiener 2 3 slide 10 Variance Covariance • Means and covariances of market factors • Mean and standard deviation of the portfolio • Delta or Delta-Gamma approximation • VaR1%= P – 2.33 P • Based on the normality assumption! Ch. 17, Handbook Zvi Wiener slide 11 Variance-Covariance VaR1% V 2.33 V 1% 2.33 -2.33 Ch. 17, Handbook Zvi Wiener slide 12 Monte Carlo 1 0.5 -1 0.5 -0.5 1 -0.5 -1 Ch. 17, Handbook Zvi Wiener slide 13 Monte Carlo • Distribution of market factors • Simulation of a large number of events • P&L for each scenario • Order the results • VaR = lowest quantile Ch. 17, Handbook Zvi Wiener slide 14 Monte Carlo Simulation 15 10 5 10 20 30 40 -5 -10 -15 Ch. 17, Handbook Zvi Wiener slide 15 Weights Since old observations can be less relevant, there is a technique that assigns decreasing weights to older observations. Typically the decrease is exponential. See RiskMetrics Technical Document for details. Ch. 17, Handbook Zvi Wiener slide 16 Stock Portfolio • Single risk factor or multiple factors • Degree of diversification • Tracking error • Rare events Ch. 17, Handbook Zvi Wiener slide 17 Bond Portfolio • Duration • Convexity • Partial duration • Key rate duration • OAS, OAD • Principal component analysis Ch. 17, Handbook Zvi Wiener slide 18 Options and other derivatives • Greeks • Full valuation • Credit and legal aspects • Collateral as a cushion • Hedging strategies • Liquidity aspects Ch. 17, Handbook Zvi Wiener slide 19 Credit Portfolio • rating, scoring • credit derivatives • reinsurance • probability of default • recovery ratio Ch. 17, Handbook Zvi Wiener slide 20 Reporting Division of VaR by business units, areas of activity, counterparty, currency. Performance measurement - RAROC (Risk Adjusted Return On Capital). Ch. 17, Handbook Zvi Wiener slide 21 Backtesting Verification of Risk Management models. Comparison if the model’s forecast VaR with the actual outcome - P&L. Exception occurs when actual loss exceeds VaR. After exception - explanation and action. Ch. 17, Handbook Zvi Wiener slide 22 Backtesting Green zone - up to 4 exceptions OK Yellow zone - 5-9 exceptions increasing k Red zone - 10 exceptions or more intervention Ch. 17, Handbook Zvi Wiener slide 23 Stress Designed to estimate potential losses in abnormal markets. Extreme events Fat tails Central questions: How much we can lose in a certain scenario? What event could cause a big loss? Ch. 17, Handbook Zvi Wiener slide 24 Local Valuation Worst dP ( D * P) (Worst dy ) Simple approach based on linear approximation. Full Valuation Worst dP P( y0 Worst dy) P( y0 ) Requires repricing of assets. Ch. 17, Handbook Zvi Wiener slide 25 Delta-Gamma Method 2 dP 1d P 2 dP dy (dy) 2 dy 2 dy dP D * Pdy 0.5CP(dy) 2 The valuation is still local (the bond is priced only at current rates). Ch. 17, Handbook Zvi Wiener slide 26 FRM-97, Question 13 An institution has a fixed income desk and an exotic options desk. Four risk reports were produced, each with a different methodology. With all four methodologies readily available, which of the following would you use to allocate capital? A. Simulation applied to both desks. B. Delta-Normal applied to both desks. C. Delta-Gamma for the exotic options desk and the delta-normal for the fixed income desk. D. Delta-Gamma applied to both desks. Ch. 17, Handbook Zvi Wiener slide 27 FRM-97, Question 13 An institution has a fixed income desk and an exotic options desk. Four risk reports were produced, each with a different methodology. With all four methodologies readily available, which of the following would you use to allocate capital? A. Simulation applied to both desks. B. Delta-Normal applied to both desks. C. Delta-Gamma for the exotic options desk and the delta-normal for the fixed income desk. D. Delta-Gamma applied to both desks. Ch. 17, Handbook Zvi Wiener slide 28 Mapping Replacing the instruments in the portfolio by positions in a limited number of risk factors. Then these positions are aggregated in a portfolio. Ch. 17, Handbook Zvi Wiener slide 29 Delta-Normal method Assumes • linear exposures • risk factors are jointly normally distributed The portfolio variance is (returns) x x 2 T Forecast of the covariance matrix for the horizon Ch. 17, Handbook Zvi Wiener slide 30 Delta-normal Valuation linear Distribution normal Extreme events low prob. Ease of comput. Yes Communicability Easy VaR precision Bad Major pitalls nonlinearity fat tails Ch. 17, Handbook Zvi Wiener Histor. full actual recent intermed. Easy depends unstable MC full general possible No Difficult good model risk slide 31 FRM-97, Question 12 Delta-Normal, Historical-Simulations, and MC are various methods available to compute VaR. If underlying returns are normally distributed, then the: A. DN VaR will be identical to HS VaR. B. DN VaR will be identical to MC VaR. C. MC VaR will approach DN VaR as the number of simulations increases. D. MC VaR will be identical to HS VaR. Ch. 17, Handbook Zvi Wiener slide 32 FRM-97, Question 12 Delta-Normal, Historical-Simulations, and MC are various methods available to compute VaR. If underlying returns are normally distributed, then the: A. DN VaR will be identical to HS VaR. B. DN VaR will be identical to MC VaR. C. MC VaR will approach DN VaR as the number of simulations increases. D. MC VaR will be identical to HS VaR. Ch. 17, Handbook Zvi Wiener slide 33 FRM-98, Question 6 Which VaR methodology is least effective for measuring options risks? A. Variance-covariance approach. B. Delta-Gamma. C. Historical Simulations. D. Monte Carlo. Ch. 17, Handbook Zvi Wiener slide 34 FRM-98, Question 6 Which VaR methodology is least effective for measuring options risks? A. Variance-covariance approach. B. Delta-Gamma. C. Historical Simulations. D. Monte Carlo. Ch. 17, Handbook Zvi Wiener slide 35 FRM-99, Questions 15, 90 The VaR of one asset is 300 and the VaR of another one is 500. If the correlation between changes in asset prices is 1/15, what is the combined VaR? A. 525 B. 775 C. 600 D. 700 Ch. 17, Handbook Zvi Wiener slide 36 FRM-99, Questions 15, 90 2 A B 2 A B 2 A B 2 A 300 500 300 500 2 15 2 Ch. 17, Handbook 2 B 2 2 AB 600 Zvi Wiener 2 slide 37 Example On Dec 31, 1998 we have a forward contract to buy 10M GBP in exchange for delivering $16.5M in 3 months. St - current spot price of GBP in USD Ft - current forward price K - purchase price set in contract ft - current value of the contract rt - USD risk-free rate, rt* - GBP risk-free rate - time to maturity Ch. 17, Handbook Zvi Wiener slide 38 1 1 * Pt PV ($1) , Pt PV (1GBP ) * 1 rt 1 rt St K * ft St Pt KPt * 1 rt 1 rt df t df t df t * df t dS dP * dP dS dP dP * * P dS SdP KdP Ch. 17, Handbook Zvi Wiener slide 39 * dS dP * dP df SP SP * KP S P P * The forward contract is equivalent to a long position of SP* on the spot rate a long position of SP* in the foreign bill a short position of KP in the domestic bill Ch. 17, Handbook Zvi Wiener slide 40 GBP10M St $16.5M Vt Qf t * 1 rt 1 rt On the valuation date we have S = 1.6595, r = 4.9375%, r* = 5.9688% Vt = $93,581 - the current value of the contract Ch. 17, Handbook Zvi Wiener slide 41