Energy Risk, Framework Risk, and FloVaR

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1 Energy Risk,, and FloVaR Two Case-Studies Andrea Roncoroni c Energy Finance - INREC 2010 University of Duisgurg - Essen, Germany October 6, 2010 Energy Risk,, and FloVaR

2 Risk Sources FloVaR Methodology Energy Risk and FloVaR Energy Risk,, and FloVaR

3 Risk Sources FloVaR Methodology Notion of Risk Question: What is risk? Risk = Exposure to uncertainty. Examples: 1 Uncertainty with no exposure: Temperature in Santiago is uncertain, yet a London resident has no exposure to it; 2 Positive exposure: The gain of a gambler who knows that his lottery ticket has been selected is a source of uncertainty to which he is favorably exposed. Energy Risk,, and FloVaR

4 Risk Sources FloVaR Methodology Risk Matrix Accounting Compliance Raw data Proxy Curve building Mapping Modelling Knowledge Concentration Credit Liquidity Market Political Settlement Volumetric Taxation Index Legal Force majeure Geographical Legal Procedural Reputation Operational Regulatory Sovereign Technological (source: Leppard: Energy Risk Management. Risk Publications) Energy Risk,, and FloVaR

5 Risk Sources FloVaR Methodology Target of Analysis: Dynamics of Risk Market dynamics determine the time evolution of risk. Variance, VaR, and Exp.shortfall o er a static picture of risk exposure. Time acts on risk through channels: 1 Contract tenor (=payment times); 2 Increase of uncertainty about the future; 3 Periodicity features (e.g., seasonal vol.). Energy Risk,, and FloVaR

6 Risk Sources FloVaR Methodology P&L Distribution The Mark-To-Market Value V (t) of a position (or portfolio) is the assessment resulting from pricing each component at the standing market quote (or fair value) at time t. Let us consider a time lag in the future. (e.g., 1/250 = trading day). The MTM variation over [t, t + ]: P&L (t, ) = V (t + t) V (t) is a random variable representing the position Pro t&loss (P&L). Risk measure = descriptive statistics of P&L(t, ) distribution. Energy Risk,, and FloVaR

7 Risk Sources FloVaR Methodology Value-at-Risk It is a synthetic measure of the riskiness of a complex portfolio. Responds to the issue: We are con dent at a level of α% that the portfolio will not experience a loss exceeding VaR Euros during the next days. Input: Con dence level α + Time horizon. (e.g., α = 99%, = 10d). Output = VaR = (1 α)-quantile of the P&L distribution: Energy Risk,, and FloVaR

8 Risk Sources FloVaR Methodology Application: FloVaR Methodology FloVaR! Estimation of the time evolution of the distribution of the mark-to-market value of a portfolio gathering energy and commodity positions. Input Portfolio composition (=position), Quoted forward curves + Estimated risk factors, Future cash ow dates t 1,..., t n = last maturity. Output! FloVaR The MTM value distributions at dates t 1,..., t n are n functions assigning estimated probability of occurrence to a number of MTM values, one function per date in the future. Energy Risk,, and FloVaR

9 Risk Sources FloVaR Methodology Example: Contract and Method Load contract: V (t k, S k ) = Ek n e r (t i t 0 ) [K! S i ] L i i =k +1 = n e r (t i t 0 ) [K Ek (S i )) ]L i. i =k +1 {z } =F (t k,t i )=f (S k ) Parameters: T = 2 years, tenor = monthly, n.simul. = 20, 000, r = 4% p.a., α = 95%. Energy Risk,, and FloVaR

10 Risk Sources FloVaR Methodology Experiment: Setting Model for S(t): 1-factor Schwartz (1997) model: Futures price: d ln(s(t)) = κ[µ ln(s(t)) 1 2 σ2 S ]dt + σ S dw S (t). lg F = lg S(t) e κ(t t) + (1 e κ(t t) )(µ Model for L(t) = L 0 + L 1 cos(2πt). Parameters: Oil [2/1/1985,17/2/1995] Price Load κ µ σ S λ S(0) Constant L 0 = σ 2 S 2 Variable L 0 =1000 L 1 =800 λ) + σ2 S (1 e κ(t t) ). 4κ Energy Risk,, and FloVaR

11 Potential Exposure (US $) Energy Risk and FloVaR Risk Sources FloVaR Methodology Frequency(%) Experiment: Results under Constant Load 0 x 104 Frequency of the contract % % 7 15% time (year) time (years) V (U S $) 2 4 x 10 5 FloVaR shape of an energy swap is driven by: 1. Uncertainty about future spot/fwd prices: this gure increases with time horizon T as p T ; 2. Load/Volume to deliver: this gure linearly decreases with time horizon. Energy Risk,, and FloVaR

12 Potential Exposure (US $) Energy Risk and FloVaR Risk Sources FloVaR Methodology Frequency(%) Experiment: Results under Variable Load 4 x 104 Frequency of the contract % % 15% time (year) 0.5 time (years) V (U S $) x 10 5 FloVaR shape of an energy swap is additionally driven by: 3. Seasonal e ects (periodicity features). Remark: portfolio contains a bullet bond maturing at position s expiration. Energy Risk,, and FloVaR

13 Assessment in the Oil Market Energy Risk,, and FloVaR

14 Assessment in the Oil Market Target of Analysis: Mapping Risk Model risk may refer to distinct aspects: Structural representativeness; Parameters uncertainty; Framework appropriateness = mapping risk. Mapping risk can be assessed by comparing alternative formulations (=distinct primitives) of a common underlying model. Example: A commodity price model can be equivalently cast under a spot-convenience yield framework or a forward curve setting (Roncoroni-Id Brik (2010)). Energy Risk,, and FloVaR

15 Assessment in the Oil Market Commodity Price Model Setting I Spot-convenience yield formulation under P: with â = a ds(t)/s(t) = (µ δ(t))dt + σ 1 dw 1 (t) dδ(t) = κ[ˆα δ(t)]dt + σ 2 dw 2 (t) ρ = d dt hw 1, W 2 i t, λσ 2 /κ, λ = market price of convenience risk. Estimation method: Kalman lter. Energy Risk,, and FloVaR

16 Assessment in the Oil Market Commodity Price Model Setting II Forward price formulation under P: " # df T = µ r + λ e κ(t t) 1 dt F T κ 1 e κ(t t) +σ 1 dw 1 σ 2 [ ]dw 2, κ F T (0) = S(0)e (r ˆα+ σ2 2 2κ ρ σ 1 σ 2 κ )T + σ e 2κT κ 3 + Estimation method: GMM (or Exact Likelihood). σ ακ+ρσ 1 σ e κt 2 κ κ Energy Risk,, and FloVaR

17 Price (US $/bbl) Energy Risk and FloVaR Price (US$/bbl) Assessment in the Oil Market Data and Estimation Results Market: NYMEX WTI crude oil futures. Periods: 2005 [Jan.1 - Dec.30, 2005]. 80 Futures Contracts month 9 months Futures Contracts Time To Maturity (month) Jan05 Time Feb Feb05 Jun05 Sep05 Dec05 Time Results: Setting σ 1 σ 2 ρ κ µ λ α S(0) Fwd Sc Energy Risk,, and FloVaR

18 Price (US $/bbl) Energy Risk and FloVaR Price (US $/bbl) Assessment in the Oil Market Price (US $/bbl) Test 1: Trajectorial Properties Simulation Futures Contracts Simulation Time To Maturity (month) Jan05 Time Feb Time To Maturity (month) Jan05 Time Feb Time To Maturity (month) 5 0 Jan05 Time Feb06 Forward Historical Spot - Convenience Yield Energy Risk,, and FloVaR

19 Assessment in the Oil Market Test 2: Re-estimation Stability: Forward Model Description: 1 Sample 50 paths (1 path = 4 ttm s (1, 3, 9, 18m)150d); 2 Estimation on simulated paths; 3 Descriptive statistics of discrepancy re-estimated/initial par. Fwd-2005 σ 1 σ 2 ρ κ λ µ α S(0) Mean Std.Dev Skewness Kurtosis Energy Risk,, and FloVaR

20 Assessment in the Oil Market Test 2: Re-estimation Stability: SC Model σ 1 σ 2 ρ κ λ µ α Mean Std.Dev Skewness Kurtosis Energy Risk,, and FloVaR

21 Assessment in the Oil Market Test 3: Stability to Perturbations: Forward Model Description: Sample 50 paths perturbed at 10 rand pts by N (0, 1). - FD Model σ 1 σ 2 ρ κ λ µ α S(0) Mean SE Skewness Kurtosis SC Model σ 1 σ 2 ρ κ λ µ α Mean Std.Dev Skewness Kurtosis Energy Risk,, and FloVaR

22 Assessment in the Oil Market Test 4: Convergence with Increasing Information Descriptions: Estimation across thicker&thicker term structures. Tenor (Fwd model) σ 1 σ 2 ρ κ λ µ α S(0) 1,3,9, ,3,6,9,12, ,2,3,6,9,12,15, All Tenor (SC model) σ 1 σ 2 ρ κ λ µ α 1,3,9, ,3,6,9,12, ,2,3,6,9,12,15, All Energy Risk,, and FloVaR

23 Assessment in the Oil Market Tests 5-6: Computational Time and Volatility Structure Computations time over increasing tenors: Tenor 1,3,9,18 1,3,6,9,12,18 1,2,3,6,9,12,15,18 all time to maturity SC 50 sec, 100 sec, 120 sec, 380 sec, Fwd 1 sec, 2-3 sec, 2-4 sec, 5-6 sec, Recovery of volatility structure: 0.32 Volatility Struture 0.3 Volatility Struture FD estimation SC estimation Empirical Time to Maturity (month) FD estimation SC estimation 0.15 Empirical Time to Maturity (month) Energy Risk,, and FloVaR

24 Assessment in the Oil Market Conclusions Kalman lter on spot-convenience yield model estimation: Several parameters to estimate, Weak statistical stability and convergence, The optimizing function displays several or even no local maxima, Time intensive computation. GMM/Exact Likelihood on forward model estimation: Rather statistically stable and quick to compute. Energy Risk,, and FloVaR

25 Assessment in the Oil Market The Author Andrea Roncoroni is Professor of Finance at ESSEC Business School (Paris - Singapore) and VP Lecturer at Bocconi University (Milan). He holds PhD s in Applied Mathematics and in Finance. His research interests cover Energy Finance, Financial Econometrics and Commodity-linked Derivative Structuring. He has consulted for private companies (e.g., Gaz de France, Edison Trading, EGL, Dong Energy) and lectured for public institutions (e.g., International Energy Agency, Central Bank of France, Italian Stock Exchange). He regularly published on academic journals (e.g., J. Business, J. Banking and Finance, J. Economic Dynamics) and nancial book series (Implementing Models in Quantitative Finance: Methods and Cases, Springer, 2008, and the Handbook of Multi-Commodity Markets and Products, Wiley, forthcoming in 2011, with G.Fusai). andrea.roncoroni@gmail.com Web page: Energy Risk,, and FloVaR