Financial Risk Management Zvi Wiener Following P. Jorion, Financial Risk Manager Handbook FRM http://pluto.huji.ac.il/~mswiener/zvi.html

Following P. Jorion 2001 Financial Risk Manager Handbook Chapter 17 VaR Methods Following P. Jorion 2001 Financial Risk Manager Handbook FRM http://pluto.huji.ac.il/~mswiener/zvi.html

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 Zvi Wiener

How to measure VaR Historical Simulations Variance-Covariance Monte Carlo Analytical Methods Parametric versus non-parametric approaches Zvi Wiener

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. Zvi Wiener

Example Assume we have $1 and our main currency is SHEKEL. Today $1=4.30. Historical data: 4.00 4.20 4.10 4.15 4.30*4.20/4.00 = 4.515 4.30*4.20/4.20 = 4.30 4.30*4.10/4.20 = 4.198 4.30*4.15/4.10 = 4.352 P&L 0.215 -0.112 0.052 Zvi Wiener

USD NIS 2000 100 -120 2001 200 100 2002 -300 -20 2003 20 30 today Zvi Wiener

today Changes USD: +1% +1% +1% +1% in IR NIS: +1% 0% -1% -1% Zvi Wiener

Returns year 1% of worst cases Zvi Wiener

VaR Profit/Loss 1% VaR1% Zvi Wiener

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! Zvi Wiener

Variance-Covariance  1% 2.33 -2.33 Zvi Wiener

Monte Carlo Zvi Wiener

Monte Carlo Distribution of market factors Simulation of a large number of events P&L for each scenario Order the results VaR = lowest quantile Zvi Wiener

Monte Carlo Simulation Zvi Wiener

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. Zvi Wiener

Stock Portfolio Single risk factor or multiple factors Degree of diversification Tracking error Rare events Zvi Wiener

Bond Portfolio Duration Convexity Partial duration Key rate duration OAS, OAD Principal component analysis Zvi Wiener

Options and other derivatives Greeks Full valuation Credit and legal aspects Collateral as a cushion Hedging strategies Liquidity aspects Zvi Wiener

Credit Portfolio rating, scoring credit derivatives reinsurance probability of default recovery ratio Zvi Wiener

Reporting Division of VaR by business units, areas of activity, counterparty, currency. Performance measurement - RAROC (Risk Adjusted Return On Capital). Zvi Wiener

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. Zvi Wiener

Backtesting OK Green zone - up to 4 exceptions increasing k Yellow zone - 5-9 exceptions Red zone - 10 exceptions or more OK increasing k intervention Zvi Wiener

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? Zvi Wiener

Local Valuation Full Valuation Simple approach based on linear approximation. Full Valuation Requires repricing of assets. Zvi Wiener

Delta-Gamma Method The valuation is still local (the bond is priced only at current rates). Zvi Wiener

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. Zvi Wiener

FRM-97, Question 13 Bad question! 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. Bad question! Zvi Wiener

Mapping Replacing the instruments in the portfolio by positions in a limited number of risk factors. Then these positions are aggregated in a portfolio. Zvi Wiener

Delta-Normal method Assumes linear exposures risk factors are jointly normally distributed The portfolio variance is Forecast of the covariance matrix for the horizon Zvi Wiener

Delta-normal Histor. MC Valuation linear full full Distribution normal actual general Extreme events low prob. recent possible Ease of comput. Yes intermed. No Communicability Easy Easy Difficult VaR precision Bad depends good Major pitalls nonlinearity unstable model fat tails risk Zvi Wiener

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. Zvi Wiener

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. Zvi Wiener

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. Zvi Wiener

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. Zvi Wiener

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 Zvi Wiener

FRM-99, Questions 15, 90 Zvi Wiener

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 Zvi Wiener

Zvi Wiener

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 Zvi Wiener

On the valuation date we have S = 1.6595, r = 4.9375%, r* = 5.9688% Vt = $93,581 - the current value of the contract Zvi Wiener