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Helsinki University of Technology Systems Analysis Laboratory

London Business SchoolManagement Science and Operations

Dynamic Risk Management of Electricity Contracts with

Contingent Portfolio Programming

Janne Kettunen, Ahti Salo, and Derek Bunn

Systems Analysis LaboratoryHelsinki University of Technology

Management Science and OperationsLondon Business School

Helsinki University of Technology Systems Analysis Laboratory

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London Business SchoolManagement Science and Operations

Background and Motivation Background and Motivation Electricity market deregulation has increased competition and uncertainties 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

How should an electricity generator or distributor hedge its risks using futures?

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

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London Business SchoolManagement Science and Operations

Scenario Tree with Two Example Paths HighlightedScenario Tree with Two Example Paths Highlighted

1

1

1 1

1 1

1 1 1 1

1 0 0 1

1

0

0

0

0

1

1 1

0 0

1 0 0 1

1 1 1 1

1 0 0 1

0 1

0 1

1 0 1 0

1 0

0 1

0 0

1 1

0 0 0 0

1 0 0 1

0 0

0 0

1 0 0 1

0 0 0 0

Time0 1 2

Ho, Stapleton, Subrahmanyam (1995), Peterson Stapleton (2002)

1,11

1,1

2 2,1 ,22

2 2,1 ,2

1 1 0 0, , ,

1 0 1 0

1 1 1 1 1 1 0 0, , ,...,

1 1 1 0 0 1 0 0

P

L

P P

L L

s

s

s s

s s

S

S

sp = spot pricesl = load

Helsinki University of Technology Systems Analysis Laboratory

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London Business SchoolManagement Science and Operations

VAR Maximum LossCVARPortfolioloss

Probability

Probability 1-β

( , )

1( ) min ( , ) ( )

1 f

CVAR f p d

x y

x x y y y

f(x,y) = loss function y = uncertaintyp(y) = probability density function

β = confidence levelx = portfolio decision strategy = threshold value (=VAR)

(Rockafeller and Uryasev, 2000)

Conditional Value-At-RiskConditional Value-At-Risk

Helsinki University of Technology Systems Analysis Laboratory

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London Business SchoolManagement Science and Operations

CCFAR can be derived from CVAR– A discrete CVAR – Portfolio loss framed using cash position beyond the threshold level ( )tRT CP s

( ) ( ) ( ( ) )t t ts p s RT CP s

( ) 0ts

, ( )

1( ) min ( )

1tt t

t

ss S

CCFAR X s

s.t.

Definitions

= threshold value (=CFAR at optimum)

= confidence level

X = portfolio decision strategy

p( ) = probability of scenario

RT = reference target amount

CP( ) =

t t

t

s s

s cash position in scenario ts

Conditional-Cash-Flow-At-Risk (CCFAR)Conditional-Cash-Flow-At-Risk (CCFAR)

Computation

(Kettunen and Salo, 2006)

Helsinki University of Technology Systems Analysis Laboratory

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London Business SchoolManagement Science and Operations

Electricity Contract Portfolio OptimizationElectricity Contract Portfolio Optimization

Risk management constraints for conditionalcash flow at risk (CCFAR)

… cash position and trading constraints

such that,

Maximize expected terminal cash position

Helsinki University of Technology Systems Analysis Laboratory

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London Business SchoolManagement Science and Operations

Electricity distributor: uncertain load and price and can use futures to hedge risks Price data (€/MWh) from Nordpool 1999-2005 and futures seen on 24.3.2006 Load data (GWh) from Finnish Energy Industries 1999-2005 (used 1% of actual)

– Conditional volatilities (fitting GARCH(1,1) for filtered data)

– Premiums (fitting linear equation)

– Mean reversions cP=0.2 and cL=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 ExperimentsComputational Experiments

Helsinki University of Technology Systems Analysis Laboratory

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London Business SchoolManagement Science and Operations

Comparison of Contingent Optimization, Periodic Comparison of Contingent Optimization, Periodic Optimization and Fixed Allocation Methods Optimization and Fixed Allocation Methods

5,6% cost reduction

Figures in million euros

Helsinki University of Technology Systems Analysis Laboratory

9

London Business SchoolManagement Science and Operations

Uncertainty in Premium and CorrelationUncertainty in Premium and Correlation

Risk averse player

Competitive player Figures in million euros

Helsinki University of Technology Systems Analysis Laboratory

10

London Business SchoolManagement Science and Operations

Uncertainty in Premium and CorrelationUncertainty in Premium and CorrelationNo correlation vs. correlation

Risk averse player

Competitive player Figures in million euros

Helsinki University of Technology Systems Analysis Laboratory

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London Business SchoolManagement Science and Operations

Uncertainty in Mean Reversion and Volatility of LoadUncertainty 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 SchoolManagement Science and Operations

Uncertainty in Mean Reversion and Volatility of Spot PriceUncertainty in Mean Reversion and Volatility of Spot Price

Risk averse player

Competitive player Figures in million euros

Helsinki University of Technology Systems Analysis Laboratory

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London Business SchoolManagement Science and Operations

Expected Cost with 6 Weeks and 4 Weeks 95% Expected Cost with 6 Weeks and 4 Weeks 95% CCFAR ConstraintsCCFAR Constraints

Competitive playerRisk averse player

B

CCFAR6wks,95%<€0.6M

Figures in million euros

Helsinki University of Technology Systems Analysis Laboratory

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London Business SchoolManagement Science and Operations

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

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 1/2Conclusions 1/2

Helsinki University of Technology Systems Analysis Laboratory

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London Business SchoolManagement Science and Operations

Standard risk management intuitions supported– Increase in volatilities increase risks

– Decrease in mean reversions increase risks

– Increase in premiums increase cost

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 2/2Conclusions 2/2