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## Value at Risk

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**Value at Risk**Presented to: VaR Task Force**Meeting Agenda**• VaR Overview Frank Hayden 9:00-9:50 • Enron VaR Systems Overview Ganapathy Ramesh & Winston Jia 9:50-10:15 • Question & Answer Session Task Force 10:15-10:30 • Related topics Group 10:30-11:00 • RiskTrac Roles • VaR Reporting • VaR Validation**Agenda**• The Role of the Market Risk Management Group • Value at Risk • Component VaR • Performance Measures • VaR Quiz**Enron’s Market Risk Philosophy**The Risks The Process The Result The Goal Market Senior management can instill optimal capital allocations Identify, monitor, and analyze risk Maximize Shareholder Value Performance Measurement As a result ... And thus ... Credit Operational Enron’s primary business is to inter-mediate transactions between customers. This activity creates net positions subject to many forms of risk. These net positions are managed by portfolio within limits allocated by the Board. Market Risk Management Group is responsible for implementing a comprehensive enterprise-wide risk management framework for defining, measuring, evaluating and communicating market risk.**Technology**Trading Senior Management Research Global Credit Board of Directors Market Risk Risk Management Risk Committee Legal Structuring Internal Audit Market Risk Infrastructure Market Risk Management Group is structured to provide independent oversight of risk taking functions to ensure that the overall business objectives and risk appetite, as dictated by the Board of Directors, Risk Committee, and senior management, is adhered to.**Why Calculate Value at Risk ?**• External Reasons • SEC disclosure requirements in annual reports and quarterly reports • Pressure from regulatory and credit rating agencies • Pressure from creditors, customers and investors • Internal Reasons • A summary of risks for senior management and the Board. • Tool for setting trading limits and for performance measurement. • New deal analysis utilizing V@R delta**Value-at-Risk Basics**Value-at-Risk (VaR) is the amount of loss in the value of a portfolio that it will sustain with a certain probability over a fixed period of time. ENA calculates VaR for a 1-day period, given a 5% confidence level (probability). Example: Value-at-Risk of $10 million means that there is a 5% chance that the overnight loss for the portfolio will exceed $10 million, or a loss of $10 million or more should be expected once every 20 trading days. ENA utilizes the Monte Carlo simulation approach for calculating VaR for a majority of the trading portfolios. 0.5 0.4 0.3 0.2 0.1 0 5% 0.0 Return**Value at Risk Example**VaR = Position * Price * Volatility * 1.645 Position: 5,000/day June NYMEX Price: $4.5 Annualized Vol: 40%, Daily Vol: 2.5% VaR= (5,000*30) *4.50 * 0.025 * 1.645 = $27,759**Value-at-Risk Inputs**• Price Curves (Traders) • Volatility Curves (Options Traders) • Positions (transferred from Trading/Risk Management Systems) • Curve Clustering - Natural Gas (statistically simulated) • Factor Loadings (statistically calculated and validated by traders) • Price jumps (statistically simulated) • Diversification - Regional correlation (statistically calculated and validated by traders)**Value-at-Risk Process**• Simulate the entire forward curve for all portfolios/delivery locations/positions • parallel shift • steepening • twisting • Calculate P&L from change in forward curves • Perform calculation 1000 times to get 1000 P&L results • Sort P&L data in increasing order • Pick the 50th number as the potential loss in a portfolio given a 5% confidence interval ENA currently utilizes a Delta-Gamma approximation. This calculation is exact for swaps, but is only an approximation for the option positions.**Value-at-Risk- Monte Carlo Simulation**• Simulates the forward curve as a unified object (HJM model). Commodity module directly simulates a small set of “primary” curves, then defines a mapping for “secondary” curves. Secondary commodities vary proportionally with the primary commodities. • The HJM model is a generaliziation of the equation used to model the dynamics of a single price based on the assumption of Geometric Brownian Motion and is very similar in nature to well known Black-Scholes model. • Treats each price on the forward curve as a distinct point capable of change, subject to the constraints imposed by each month’s volatility and cross correlations. Simulated curve shifts include changes in level, slope and/or curvature. • Each simulated curve is then used to revalue the portfolio. MTM change = Position Delta* (P new - P old) + 0.5*Position Gamma * (P new - P old)2 • The resulting portfolio valuations creates a distribution of possible P&L returns. The gamma term above introduces skewness into the P&L distribution, implying that the 95% confidence interval can no longer be found using 1.645 standard deviations. • V@R at the appropriate confidence interval is found based on the cumulative frequency of P&L results.**Value-at-Risk Inputs - Factor Loadings**Factor Loadings determine the shocks along the points on the price curve which are simulated for the purposes of calculating a potential loss. Factor loadings represent the price relationships between respective months, i.e. correlations between months used in consistent shifting of all the reference points along the price curves. Currently, statistical simulations are performed for the first 60 months for Natural Gas and Liquids positions and for the first 12 months for Power Positions. The Factor Loadings are calculated by correlating daily log-returns for the major (Primary) curves. The rest of the curves (Secondary) are mapped to a closely related Primary curve. Monthly correlations are periodically validated by the traders and RAC.**Value-at-Risk Inputs - Clusters**For purposes of calculating VaR for Natural Gas positions, we group all basis curves into clusters. The clusters are composed of curves that are highly correlated to a Core curve within a cluster (prices at the locations within a cluster move in tact). The VaR is then simulated for the Core curve and extrapolated to the whole cluster. This allows for more efficient and faster simulation of Value-at-Risk for Natural Gas positions. Currently, the correlations between curves, used in the composition of clusters, are calculated by correlating daily log-returns for the curves. These correlations are periodically validated by the traders and RAC.**Value-at-Risk Inputs - Price Jumps**Price Jumps are described by the Jump Diffusion Process and are calculated in the Jump Diffusion module of the VaR application. Main assumptions behind the Jump Diffusion Process are: • in the electricity markets the intersection of supply and demand produces frequent price spikes • floor reverting process: prices are likely to jump upward from the floor determined by the cost of generation in the most efficient units • lognormal diffusion process models the price changes resulting from arrival of normal information • abnormal information resulting in shocks to the price is modeled through a Poison process Currently, Jump Diffusion factors (Price Jumps) are simulated for intramonth positions only. These factors define the magnitude of a price jump and the probability of a jump occurring. The Jump Diffusion factors are statistically compiled by simulating spot prices at various locations for a sample period of 60 effective dates.**Value-at-Risk Inputs - Diversification**VaR may be calculated at the position, book or portfolio level. Since all prices do not move together, the aggregate VaR is expected to be less than the arithmetic sum of individual VaR amounts. The VaR for two portfolios (MW Region and NE Region) is calculated as follows: VaR = Square Root (VaR1 squared + VaR2 squared + 2*VaR1*VaR2*L) VaR1 is the Value-at-Risk of MW Region, VaR2 is the Value-at-Risk of NE Region, L is the correlation between the two portfolios. Correlations between curves and commodities, used to calculate the respective portfolio and the overall VaR, are statistically compiled by calculating correlations between curve prices at different locations for a sample of 60 effective dates. Correlations are periodically validated by the traders and RAC.**Value-at-Risk Reasonableness Check**Does this really make sense? To ensure the accuracy of the Value-at-Risk calculations we perform the Backtesting analyses regularly. • Backtesting analysis compares VaR data and P&L • Given a 5% confidence level, actual losses are expected to exceed VaR 5% of the time (i.e. 13 times per calendar year - 256 trading days) • P&L should not exceed VaR for a number of consecutive days**Component VaR**Component VaR (CVaR) is additive: sum of trader Component VaRs equals portfolio VaR. Example: Portfolio A CVaR is $5 million (long) Portfolio A CVaR is $12 million (short) Total VaR is $7 million (short) • Component VaR allows more precise decomposition of risks by • time-buckets • traders • sub-portfolios • curves • Useful in identifying risk-contributors and hedges in a complex portfolio.**Performance Measurement and Capital Allocation**Goals: • Independent measurement of performance • Develop metrics for measuring risk-adjusted performance • Measure exposure to sources of Risk (Market, Liquidity, Credit, etc.) • Forecast Expected P&L • Allocation of capital between business units/traders • Determine required capital • Determine risk premium needed for profitable product pricing • Optimization of risk-return ratios • Create an optimal portfolio of businesses**Performance Measurement Ratios**• RoVaR Ratio (Return on Value-at-Risk) = P&L / V@R. P&L is standardized by expected (not realized) risk. Strong candidate for ENA capital allocation. Larger, the better. • Sharpe Ratio = P&L / volatility (P&L) Standardizes earnings by risk in order to facilitate performance comparisons across dissimilar markets or commodities. Most widely used performance benchmark used to compare mutual and hedge funds. Larger, the better. • Efficiency Ratio = V@R / vol(P&L) / 1.645 Strong desk management tool. Indicates if a trader is “stepping out” relative to traditional risk tolerance; alternatively, indicates if market risk is “creeping up” and overwhelming a trader relative to traditional risk tolerance. Closer to 1, the better.**Performance Measurement Example**Trader 1 Trader 2 YTD P&L - $800,000 YTD P&L - $650,000 Annualized VaR - $4,000,000 Annualized VaR - $1,625,000 Average Daily VaR - $250,000 Average Daily VaR - $101,562 Return on VaR (Capital) - 20% Return on VaR (Capital) - 40% Annualized P&L Vol. - $3,520,000 Annualized P&L Vol. - $720,000 Daily P&L Volatility - $220,000 Daily P&L Volatility - $45,000 Analysis: Trader 1Trader 2 RoVaR: 20% 40% Sharpe: 23% 90% Efficiency: 0.69 1.37 Conclusion: Although, Trader 1 achieved higher profit, more capital would be allocated to Trader 2 based on higher risk-adjusted returns.**Quiz**Q1: What is the time horizon and confidence interval for ENE’s VaR calculations ?**Quiz**Q1: What is the time horizon and confidence interval for ENE’s VaR calculations ? A: 1-day and 95% confidence interval.**Quiz**Q2: What are the inputs to the VaR model?**Quiz**Q2: What are the inputs to the VaR Model? A: Price curves Volatility curves Positions Correlations Factor Loadings Jump Factors**Quiz**Q3: Calculate VaR, based on the following data: 30 contracts of June NYMEX Jun-01 Price of $4.50/MMBtu Jun-01 Monthly Volatility of 40%**Quiz**Q3: Calculate VaR, based on the following data: 30 contracts of June NYMEX Jun-01 Price of $4.50/MMBtu Jun-01 Monthly Volatility of 40% A: (30*10,000) * $4.50 * (0.4/sqrt(256)) * 1.645 = $55,518**Quiz**Q4: Portfolio A consists of: Scott Neal VaR = $5 million (long position), Brad McKay VaR = $2 million (short position), Correlation between Scott and Brad is “1”, what is the portfolio VaR?**Quiz**Q4: Portfolio A consists of: Scott Neal VaR = $5 million (long position), Brad McKay VaR = $2 million (short position), Correlation between Scott and Brad is “1”, what is the portfolio VaR? A: Portfolio A VaR = $3 (5-2)**Quiz**Q5: Portfolio A consists of: Scott Neal VaR = $3 million (long position), Brad McKay VaR = $4 million (long position), Correlation between Scott and Brad is “.5”, what is the portfolio VaR?**Quiz**Q5: Portfolio A consists of: Scott Neal VaR = $3 million (long position), Brad McKay VaR = $4 million (long position), Correlation between Scott and Brad is “.5”, what is the portfolio VaR? A: Portfolio A VaR is ~$6 million sqrt(32 +42 + 2*3*4*0.5) = sqrt (37) = ~ 6**Quiz**Q6: East portfolio consists of: Scott’s Component VaR = $8 million (short position), Brad’s Component VaR = -$5 million (long position), Sandra’s Component VaR = $20 million (short position), Susan’s Component VaR = -$3 million (long position), What is East VaR?**Quiz**Q6:East portfolio consists of: Scott’s Component VaR = $8 million (short position), Brad’s Component VaR = -$5 million (long position), Sandra’s Component VaR = $20 million (short position), Susan’s Component VaR = -$3 million (long position), What is East VaR? A: East VaR is $20 million (8 - 5 +20 - 3)**RisktRAC Roles**• Risk Controls • Analyze requests • Review for consistency • Post entry problem solving • Commercial/ • Book Admins • Books • Curves • Conv. Factors • Hierarchy Designation • Portfolios RiskTrac request input • RAC • Factor loading • Curve clusters • Price jumps RisktRac**VaR Reporting**RisktRAC DPR rac.enron.com Email notifications Violation notices**VaR Validation**• Model valuation • Back testing - methodology, schedule, review, reporting • Operational validation • Daily change analysis – drill down capability • Day/day change (by portfolio and VaR component – breaking down components) • Procedures/process for communicating problems/issues • Explaining daily VaR movements