1 Helsinki University of Technology Systems Analysis Laboratory London Business School Management Science and Operations Dynamic Risk Management of Electricity Contracts with Contingent Portfolio Programming Janne Kettunen, Ahti Salo, and Derek Bunn Systems Analysis Laboratory Helsinki University of Technology Management Science and Operations London Business School

Helsinki University of Technology Systems Analysis Laboratory 2 London Business School Management Science and Operations Background and Motivation n Electricity market deregulation has increased competition and uncertainties n Uniqueness of electricity market (Bunn, 2004) –Non-storable, stakeholders bear price and load risk –Correlation between price and load (exponentially increasing in load) –Mean reversion –Spikes and seasonal variations –Volatility clustering –High and volatile risk premiums in futures n How should an electricity generator or distributor hedge its risks using futures? n Requirements on model formulation –Correlation, arbitrage free, mean reversions, volatility clustering  scenario tree (Ho, et. al., 1995) –Risk management  Conditional Cash Flow at Risk and risk constraint matrix (Kettunen and Salo, 2006) –Path dependencies  Contingent Portfolio Programming (Gustafsson and Salo, 2005

Helsinki University of Technology Systems Analysis Laboratory 3 London Business School Management Science and Operations Scenario Tree with Two Example Paths Highlighted Time 012 Ho, Stapleton, Subrahmanyam (1995), Peterson Stapleton (2002) s p = spot price s l = load

Helsinki University of Technology Systems Analysis Laboratory 4 London Business School Management Science and Operations VAR Maximum Loss CVAR Portfolio loss Probability Probability 1-β f(x,y) = loss function y = uncertainty p(y) = probability density function β = confidence level x = portfolio decision strategy  = threshold value (=VAR) (Rockafeller and Uryasev, 2000) Conditional Value-At-Risk

Helsinki University of Technology Systems Analysis Laboratory 5 London Business School Management Science and Operations n CCFAR can be derived from CVAR –A discrete CVAR –Portfolio loss framed using cash position beyond the threshold level s.t. Definitions Conditional-Cash-Flow-At-Risk (CCFAR) Computation (Kettunen and Salo, 2006)

Helsinki University of Technology Systems Analysis Laboratory 6 London Business School Management Science and Operations Electricity Contract Portfolio Optimization Risk management constraints for conditional cash flow at risk (CCFAR) … cash position and trading constraints such that, Maximize expected terminal cash position

Helsinki University of Technology Systems Analysis Laboratory 7 London Business School Management Science and Operations n Electricity distributor: uncertain load and price and can use futures to hedge risks n Price data (€/MWh) from Nordpool and futures seen on n Load data (GWh) from Finnish Energy Industries (used 1% of actual) –Conditional volatilities (fitting GARCH(1,1) for filtered data) –Premiums (fitting linear equation) –Mean reversions c P =0.2 and c L =0.4 (fitting linear equation) –Correlation: N=0.08 and λ=0.1 (fitting linearized version of ) –Risk free interest rate 2% –Trade fee 0,03€/MWh Computational Experiments

Helsinki University of Technology Systems Analysis Laboratory 8 London Business School Management Science and Operations Comparison of Contingent Optimization, Periodic Optimization and Fixed Allocation Methods 5,6% cost reduction Figures in million euros

Helsinki University of Technology Systems Analysis Laboratory 9 London Business School Management Science and Operations Uncertainty in Premium and Correlation Risk averse player Competitive player Figures in million euros

Helsinki University of Technology Systems Analysis Laboratory 10 London Business School Management Science and Operations Uncertainty in Premium and Correlation No correlation vs. correlation Risk averse player Competitive player Figures in million euros

Helsinki University of Technology Systems Analysis Laboratory 11 London Business School Management Science and Operations Uncertainty in Mean Reversion and Volatility of Load Risk averse player Competitive player Figures in million euros

Helsinki University of Technology Systems Analysis Laboratory 12 London Business School Management Science and Operations Uncertainty in Mean Reversion and Volatility of Spot Price Risk averse player Competitive player Figures in million euros

Helsinki University of Technology Systems Analysis Laboratory 13 London Business School Management Science and Operations Expected Cost with 6 Weeks and 4 Weeks 95% CCFAR Constraints Competitive player Risk averse player B CCFAR 6wks,95% <€0.6M Figures in million euros

Helsinki University of Technology Systems Analysis Laboratory 14 London Business School Management Science and Operations n Model –Correlation important to include –Optimal strategies robust (remain close to efficient frontiers) –Contingent optimization consistently more efficient than periodic optimization or fixed allocation methods n Risk management perspective –Competitive player: most concern about price related uncertainties –Risk averse player: most concern about premiums –Both players bear load related risk (swing-option contracts) Conclusions 1/2

Helsinki University of Technology Systems Analysis Laboratory 15 London Business School Management Science and Operations n Standard risk management intuitions supported –Increase in volatilities increase risks –Decrease in mean reversions increase risks –Increase in premiums increase cost n Risk constraint matrix for concurrent time periods and confidence levels –Re-run model when new information arrives (rolling horizon) –Regulatory requirements –Financially tight situation Conclusions 2/2

Helsinki University of Technology Systems Analysis Laboratory 16 London Business School Management Science and Operations n Applications –Optimal hedging strategies –Price non-market tradable contracts n Own production possible to incorporate (futures at the production cost) n Develop model for other stakeholders in energy market, natural resources, or financial contracts n Include market power related phenomena (dynamic scenario tree) Current and Future Work

Helsinki University of Technology Systems Analysis Laboratory 17 London Business School Management Science and Operations Thank You – Questions?

Helsinki University of Technology Systems Analysis Laboratory 18 London Business School Management Science and Operations References »Bunn, D. W Modelling Prices in Competitive Electricity Markets. Wiley, London. »Doege, J., H.-J. Lüthi, P. Schiltknecht Risk management of power portfolios and valuation of flexibility. OR Sectrum »Gustafsson, J., A. Salo Contingent portfolio programming for the management of risky projects. Operations Research 53(6) »Kettunen, J. A. Salo Dynamic Risk Management in Contingent Portfolio Programming. Submitted. »Kettunen, J. A. Salo. D. Bunn Dynamic risk management of electricity contracts with contingent portfolio programming. Manuscript. »Peterson, S. J., R. C. Stapleton The pricing of bermudan-style options on correlated assets. Review of Derivatives Research »Rockafeller, R. T., S. Uryasev Optimization of conditional value-at-risk. The Journal of Risk 2(3) »Eichhorn, A., W. Römisch, I. Wegner Polyhedral risk measures in electricity portfolio optimization. PAMM Proc. Appl. Math. Mech. Minisymposium MA1 4(1) Available at www3.interscience.wiley.com/.

Helsinki University of Technology Systems Analysis Laboratory 19 London Business School Management Science and Operations Scenario Tree 1/3 Expected spot prices derived from market observed futures Upward and downward movement amounts for spot price and load Nodal values for spot price and load Following Ho, Stapleton, Subrahmanyam (1995) and its extension Following Ho, Stapleton, Subrahmanyam (1995) and its extension Peterson Stapleton (2002)

Helsinki University of Technology Systems Analysis Laboratory 20 London Business School Management Science and Operations Scenario Tree 2/3 Conditional upward moving probabilities of spot price and load Correlation (exponentially increasing in load)

Helsinki University of Technology Systems Analysis Laboratory 21 London Business School Management Science and Operations Scenario Tree 3/3 Scenario specific unconditional probability Scenario specific future price

Helsinki University of Technology Systems Analysis Laboratory 22 London Business School Management Science and Operations Use Asset ReturnsUse Cash Flows Measures maximum loss with certain confidence level VAR (U.S. SEC ~1980) CFAR (Linsmeier and Pearson 2000) Measures expected loss with certain confidence level conditional tail event occurred CVAR (Rockafeller and Uryasev 2000 and Artzner et. al. 1999) CCFAR (Kettunen and Salo 2006) Value-at-Risk Approaches

Helsinki University of Technology Systems Analysis Laboratory 23 London Business School Management Science and Operations CVAR - a Coherent Risk Measure n X and Y are random returns –Translation invariance: –Subadditivity: –Positive homogeneity : –Positivity:

Helsinki University of Technology Systems Analysis Laboratory 24 London Business School Management Science and Operations n CCFAR can be derived from CVAR (Uryasev 2000, and Artzner et. al. 1999) –A discrete CVAR –Portfolio loss framed using cash position beyond the threshold level s.t. Definitions Conditional-Cash-Flow-At-Risk (CCFAR) Computation (Kettunen and Salo 2006)

Helsinki University of Technology Systems Analysis Laboratory 25 London Business School Management Science and Operations = risk tolerances at different time periods and confidence levels = Risk Constraint Matrix = risk measurements with a given measure (e.g., CCFAR ) Definitions n Rationale –Compact presentation of risk requirements –Allows for dynamic risk management of  different time periods  different percentiles –Risk mgmt using complementary risk measures Risk Contraint Matrix (RCM)

Helsinki University of Technology Systems Analysis Laboratory 26 London Business School Management Science and Operations Cash position constraints Electricity Contract Portfolio Optimization with All Constraints Risk management constraints s.t., Trading constraints

Helsinki University of Technology Systems Analysis Laboratory 27 London Business School Management Science and Operations Computational Experiments n Compare performance of contingent optimization to periodic optimization and fixed allocation methods n Analyze effects of increased uncertainty, two perspectives –competitive player –risk averse player n Robustness of suggested strategies n Concurrent risk constraints