Skip to main content
Log in

Risk management of power portfolios and valuation of flexibility

  • Regular Article
  • Published:
OR Spectrum Aims and scope Submit manuscript


Risk management by applying operational flexibility is becoming a key issue for production companies. This paper discusses how a power portfolio can be hedged through its own production assets. In particular we model operational flexibility of a hydro pump storage plant and show how to dispatch it to hedge against adverse movements in the portfolio. Moreover, we present how volume risk, which is not hedgeable with standard contracts from power exchanges, can be managed by an intelligent dispatch policy. Despite the incompleteness of the market we quantify the value of this operational flexibility in the framework of coherent risk measures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

Similar content being viewed by others


  1. i.e. non-risk-neutral.

  2. Evaluated in this context means to decide on the technical investment or its zero-coupon equivalent. Additionally the optimal level of operations can be determined.

  3. Ramp-up times are within 10 s.

  4. Spot prices are determined for each hour of the next day. This is done on hourly auctions.

  5. Subadditivity is probably the most important property to be a good risk measure for portfolios.

  6. These results have been numerically shown by Acerbi and Tasche (2002) and Frey and McNeil (2002)

  7. See Rockafellar and Uryasev (2002) for the mathematical assumptions.

  8. i.e. we assume that the strategy of a single power generation company will not influence spot markets.

  9. In our case for every scenario j, ω j corresponds to a joint path of the stochastic values, spot price S t , demand D t and inflow I t , over all periods t=1,...,T.

  10. A more rigorous mathematical analysis is presented in Rockafellar and Uryasev (2000).

  11. All marginal production costs exclude costs of electricity and neglect Swiss water taxes.

  12. Monthly futures contracts have the highest liquidity in the EEX market.

  13. Using Monte-Carlo simulation techniques.

  14. The risk constraint is no longer a binding restriction.

  15. Note that ΔV is always negative. A smaller end level of water leads to a higher volume as more water can now be used to produce electricity.

  16. Assuming once again non-degeneracy. For degenerated optimal points directional derivatives have to be applied. This additional step will not change the methodology we suggest.

  17. Any coherent or convex risk measure could have been used.


  • Acerbi C, Tasche D (2002) On the coherence of expected shortfall. In: Szegö G (ed) “Beyond VaR” (Special Issue). Journal of Banking & Finance 26:1505–1518

    Google Scholar 

  • Arrow KJ (1971) Essays on the theory of risk-bearing. Markham, Chicago

    Google Scholar 

  • Artzner P, Delbaen F, Eber J-M, Heath D (1999) Coherent risk measures. Math Financ 9:203–228

    Article  Google Scholar 

  • Bertsimas D, Lauprete GJ, Samarov A (2000) Shortfall as a risk measure: properties, optimization and applications. Working paper, MIT

  • Borenstein S, Bushnell J (1999) An empirical analysis of the potential for market power in California’s electricity industry. J Ind Econ 47(3):285–323

    Google Scholar 

  • Burger M, Klar B, Müller A, Schindlmayr G (2004) A spot market model for pricing derivatives in electricity markets. Quantitative Finance 4:109–122

    Article  Google Scholar 

  • Carmona R, Dayanik S (2004) Optimal multiple-stopping of linear diffusions and swing options. Preprint, Princeton University

  • Clewlow L, Strickland C (2001) Energy derivatives – pricing and risk management. Lacima Publications, London

    Google Scholar 

  • Delbaen F (2000) Coherent risk measures, lecture notes at Cattedra Galileiana. Scuola Normale di Pisa, Pisa

    Google Scholar 

  • Deng S, Johnson B, Sogomonian A (2001) Exotic electricity options and the valuation of electricity generation and transmission assets. Decis Support Syst 30(3):383–392

    Article  Google Scholar 

  • Eberlein E, Stahl G (2003) Both sides of the fence: a statistical and regulatory view of electricity risk. Energy Power Risk Manag 8(6):34–38

    Google Scholar 

  • Eydeland A, Geman H (2000) Fundamentals of electricity derivatives in energy modeling and the management of uncertainty. Risk Books, New York

  • Eydeland A, Wolyniec K (2002) Energy and power risk management, Wiley

  • Fleten S-E, Wallace SW, Ziemba WT (2002) Hedging electricity portfolios via stochastic programming. Decision making under uncertainty: energy and power. Springer, Berlin, Heidelberg New York, pp 71–94

    Google Scholar 

  • Föllmer H, Schied A (2002) Convex measures of risk and trading constraints. Finance Stoch 6(4):429–447

    Article  Google Scholar 

  • Frey R, McNeil A (2002) VaR and expected shortfall in credit portfolios: conceptual and practical insights. In: Szegö G (ed) “Beyond VaR” (Special Issue). J Bank Financ 26:1317–1334

  • Geman H (2001) Spot and derivatives trading in deregulated European electricity markets. Rev Econ Soc 8

  • Gröwe-Kuska N, Römisch W (2002) Stochastic unit commitment in hydrthermal power production planning. Preprint 02-3, Institute for Mathematics, Humboldt-University Berlin

  • Güssow J (2001) Power systems operations and trading in competitive energy markets. PhD thesis, University of St. Gallen, Switzerland

  • Hinz J (2003) Optimizing a portfolio of power-producing plants. Bernoulli 9(4):659–669

    Article  Google Scholar 

  • Hinz J, von Grafenstein L, Verschuere M, Wilhelm M (2004) Pricing electricity risk by interest rate methods, to appear in quantitative finance

  • Jaillet P, Ronn EI, Tompaidis S (2004) Valuation of commodity-based swing options. Manag Sci 50:909–921

    Article  Google Scholar 

  • Kamat R, Oren S (2002) Exotic options for interruptible electricity supply contracts. Oper Res 50(5):835–850

    Article  Google Scholar 

  • Kholodnyi VA (2004) Valuation and hedging of European contingent claims on power with spikes: a non-Markovian approach. J Eng Math 49:233–252

    Article  Google Scholar 

  • Ku A (2003) Risk and flexibility in electricity. Risk Books, London

    Google Scholar 

  • Lüthi H-J, Doege J (2005) Convex risk measures for portfolio optimization and concepts of flexibility. Math Program, Series B 104(2–3):541–559

    Article  Google Scholar 

  • Markowitz HM (1952) Portfolio selection. J Finance 7:77–91

    Article  Google Scholar 

  • Mas-Colell A, Whinston M, Green J (1995) Microeconomic theory. Oxford University Press, New York

    Google Scholar 

  • Pilipovic D (1997) Energy risk: valuing and managing energy derivatives. McGraw-Hill

  • Rockafellar RT, Uryasev S (2000) Optimization of conditional value-at-risk. J Risk 2(3):21–41

    Google Scholar 

  • Rockafellar RT, Uryasev S (2002) Conditional value-at-risk for general loss distributions. J Bank Financ 26:1443–1471

    Article  Google Scholar 

  • Rudin W (1976) Principles of mathematical analysis. McGraw-Hill, Singapore

    Google Scholar 

  • Schwartz ES, Lucia JJ (2002) Electricity prices and power derivatives. Evidence from the Nordic power exchange. Rev Deriv Res 5(1):5–50

    Article  Google Scholar 

  • Stoft S (2002) Power system economics. IEEE, Wiley-Interscience, New York

    Google Scholar 

  • Thompson AC (1995) Valuation of path-dependent contingent claims with multiple exercise decisions over time: the case of take-or-pay. J Financ Quant Anal 30(2):271–293

    Article  Google Scholar 

  • Unger G (2002) Hedging strategy and electricity contract engineering, Ph D thesis, Swiss Federal Institute of Technology – ETH Zürich, unger

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Hans-Jakob Lüthi.

Additional information

This research project is gratefully supported by the Swiss Innovation Promotion Agency KTI/CTI, Berne (CH).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Doege, J., Schiltknecht, P. & Lüthi, HJ. Risk management of power portfolios and valuation of flexibility. OR Spectrum 28, 267–287 (2006).

Download citation

  • Published:

  • Issue Date:

  • DOI: