Computational Management Science

, Volume 12, Issue 2, pp 221–242 | Cite as

On variance reduction of mean-CVaR Monte Carlo estimators

  • Václav KozmíkEmail author
Original Paper


We formulate an objective as a convex combination of expectation and risk, measured by the \(\mathrm{CVaR }\) risk measure. The poor performance of standard Monte Carlo estimators applied on functions of this form is discussed and a variance reduction scheme based on importance sampling is proposed. We provide analytical solution for random variables based on normal distribution and outline the way for the other distributions, either by analytical computation or by sampling. Our results are applied in the framework of stochastic dual dynamic programming algorithm. Computational results which validate the previous analysis are given.


Importance sampling Risk-averse optimization Monte Carlo sampling Stochastic dual dynamic programming  

Mathematics Subject Classification

65C05 90C15 91G60 



The research was partly supported by the project of the Czech Science Foundation P/402/12/G097 ‘DYME/Dynamic Models in Economics’.


  1. Artzner P, Delbaen F, Eber J-M, Heath D (1999) Coherent measures of risk. Math Financ 9:203–228CrossRefGoogle Scholar
  2. Bayraksan G, Morton DP (2011) A sequential sampling procedure for stochastic programming. Oper Res 59:898–913CrossRefGoogle Scholar
  3. Blum M, Floyd RW, Pratt V, Rivest RL, Tarjan RE (1972) Linear time bounds for median computations. In: Proceedings of the fourth annual ACM symposium on theory of computing. pp 119–124Google Scholar
  4. Denneberg D (1990) Premium calculation: why standard deviation should be replaced by absolute deviation. ASTIN Bull Int Actuar Assoc 20:181–190CrossRefGoogle Scholar
  5. Hesterberg TC (1995) Weighted average importance sampling and defensive mixture distributions. Technometrics 37:185–194CrossRefGoogle Scholar
  6. Knopp R (1966) Remark on algorithm 334 [G5]: normal random deviates. Commun ACM 12:281CrossRefGoogle Scholar
  7. Kozmík V, Morton D (2014) Evaluating policies in risk-averse multi-stage stochastic programming. Math Program. doi: 10.1007/s10107-014-0787-8
  8. Kozmík V (2012) Multistage risk-averse asset allocation with transaction costs. In: Proceedings of 30th international conference on mathematical methods in economics. pp 455–460Google Scholar
  9. Krokhmal P, Zabarankin M, Uryasev S (2011) Modeling and optimization of risk. Surv Oper Res Manag Sci 16:49–66Google Scholar
  10. Markowitz HM (1952) Portfolio selection. J Financ 7:77–91Google Scholar
  11. Markowitz HM (1959) Portfolio selection: efficient diversification of investments. Wiley, New YorkGoogle Scholar
  12. Pereira MVF, Pinto LMVG (1991) Multi-stage stochastic optimization applied to energy planning. Math Program 52:359–375CrossRefGoogle Scholar
  13. Philpott AB, de Matos VL, Finardi EC (2013) On solving multistage stochastic programs with coherent risk measures. Oper Res 61:957–970CrossRefGoogle Scholar
  14. Rockafellar RT, Uryasev S (2002) Conditional value-at-risk for general loss distributions. J Bank Financ 26:1443–1471CrossRefGoogle Scholar
  15. Rudloff B, Street A, Valladao D (2014) Time consistency and risk averse dynamic decision models: definition, interpretation and practical consequences. Eur J Oper Res 234(3):743–750Google Scholar
  16. Ruszczynski A (2010) Risk-averse dynamic programming for Markov decision processes. Math Program 125:235–261CrossRefGoogle Scholar
  17. Shapiro A (2009) On a time consistency concept in risk averse multistage stochastic programming. Oper Res Lett 37:143–147CrossRefGoogle Scholar
  18. Shapiro A (2011) Analysis of stochastic dual dynamic programming method. Eur J Oper Res 209:63–72CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Probability and Mathematical Statistics, Faculty of Mathematics and PhysicsCharles University in PraguePragueCzech Republic

Personalised recommendations