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**The Efficient Conditional Value-at-Risk/Expected Return Frontier**

THE ACADEMY OF ECONOMIC STUDIES BUCHAREST DOCTORAL SCHOOL OF FINANCE AND BANKING The Efficient Conditional Value-at-Risk/Expected Return Frontier Student: Stan Anca Mihaela Supervisor:Professor Moisa Altar

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Contents Objectives VaR, CVaR, ER-properties and optimization algorithms Methodology Empirical Application Concluding remarks

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**Objectives Construct the efficient CVaR/Expected Return frontier**

Analyze CVaR’s performance as a proxy variable for VaR Use CVaR as a risk tool in order to efficiently restructure portfolios Analyze the impact of transaction costs

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**VaR-alternative definitions**

1. 2.

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CVaR The expected losses exceeding VaR calculated with a precise confidence level: In terms of lower partial moments, CVaR can be defined as a lower partial moment of order one with

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Expected Regret The mean value of the loss residuals, the differences between the losses exceeding a fixed threshold and the threshold itself.

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VaR/CVaR Comparison

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**VaR/CVaR Comparison VaR CVaR**

Simple convenient representation of risks (one number) Measures downside risk (compared to variance which is impacted by high returns) Applicable to nonlinear instruments, such as options, with non-symmetric (non-normal) loss distributions Easily applied to backtesting Established as a standard risk measure Consistent with first order stochastic dominance Simple convenient representation of risks (one number) Measures downside risk Applicable to nonlinear instruments, such as options, with non-symmetric (non-normal) loss distributions Not easily applied to efficient backtesting methods Consistent with second order stochastic dominance

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**VaR/CVaR Comparison does not measure losses exceeding VaR**

reduction of VaR may lead to stretch of tail exceeding VaR non-sub-additive and non-convex: non-sub-additivity implies that portfolio diversification may increase the risk - incoherent in the sense of Artzner, Delbaen, Eber, and Heath1 - difficult to control/optimize for non-normal distributions: VaR has many extremums accounts for risks beyond VaR (more conservative than VaR) convex with respect to portfolio positions coherent in the sense of Artzner, Delbaen, Eber and Heath: (translation invariant, sub-additive, positively homogeneous, monotonic w.r.t. Stochastic Dominance1) continuous with respect to confidence level α, consistent at different confidence levels compared to VaR consistency with mean-variance approach: for normal loss distributions optimal variance and CVaR portfolios coincide easy to control/optimize for non-normal distributions, by using linear programming techniques

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**CVaR optimization . Notations:**

x = (x1,...xn) = decision vector (e.g., portfolio positions) y = (y1,...yn) = random vector yj = scenario of random vector y , ( j=1,...J ) f(x,y) = loss functions =CVaR at confidence level =VaR at confidence level

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**CVaR Optimization Rockafellar and Uryasev (1999) have shown that both**

can be characterized in terms of the function defined on by: By solving the optimization problem we find an optimal portfolio x* , corresponding VaR, which equals to the lowest optimal , and minimal CVaR, which equals to the optimal value of the linear performance function.

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CVaR Optimization When the function F is approximated using scenarios, the problem is reduced to LP with the help of a dummy variable:

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ER Optimization If the function G is approximated using scenarios, the problem can be reduced to a linear programming problem, having the same constraints as the CVaR optimization problem and with the objective function

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**Optimization problem The constraint on return takes the form:**

The balance constraint that maintains the total value of the portfolio less transaction costs:

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**Optimization problem We impose bounds on the position changes:**

We also consider that the positions themselves can be bounded: We do not allow for an instrument i to constitute more than a given percent of the total portfolio value:

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**Optimization Problem Size of LP**

For n instruments and J scenarios, the formulation of the LP problem presented above has n+2 equalities, 3n+J+1 variables and n+J inequalities.

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**Empirical Application-Data**

Portfolio consisting of 5 Romanian equities traded on Bucharest Stock Exchange – ATB, AZO, OLT, PCL, TER, selected by taking into account the most actively trading securities in the analyzed period. 450 daily closing prices between 03/05/2001 to 12/18/02

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Data

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Data

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Substitution error

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**Substitution error 200 scenarios 300 scenarios Portfolio 37.58% -8.08%**

10.69% -7.07 1.00% -1.01% 0% 17.90% 4.04% Portfolio 4.93% 7.07% 6.06% -1.01% 0%

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Substitution error

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**CVaR Efficient Frontier Without Transaction Costs**

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**CVaR Efficient Frontier Without Transaction Costs**

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**Restructuring the initial portfolio**

Expected Return Alpha 7,500 VaR 27,429.48 CVaR 37,302.43 ER 12,428.36 Markowitz 18,729.29

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**Restructuring the initial portfolio**

However, the restructured portfolios are not efficient with respect to their return level, they lie on the “inefficient”, lower section of the boundary. For a CVaR of 33, we can find, for instance, on the CVaR efficient frontier a portfolio (x1=16.68, x2=43.49, x3=183.63, x4=68.82, x5=92.58) that has an expected return of (instead of ) – this suggests that the” efficient” portfolio, offering maximum return for a given minimal risk level can be achieved by lowering the position in the first asset (ATB) that is the most risky one and has a negative expected return and by investing more in the second (AZO) and fifth asset (TER) that have the highest expected return.

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**The impact of transaction costs**

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Transaction Costs

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Transaction Costs

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**The restructured portfolios**

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**The restructured portfolios**

Rest CVaR x1 x2 x3 x4 x5 With transaction costs 8.97% 2.78% 27.79% 33.22% 27.25% Without transaction costs 9.938% 3.403% 26.916% 32.586% 27.157%

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**The restructured portfolio**

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**Concluding Remarks CVaR is a conceptually superior risk measure to VaR**

It can be used to efficiently manage and restructure a portfolio (other applications include the hedging of a portfolio of options, credit risk management (bond portfolio optimization) and portfolio replication). Direction for further development: Conditional Drawdown-at-Risk Risk measures consistent with third or higher order stochastic dominance criteria

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References 1. Acerbi, A., (2002), “Spectral Measures of Risk: A Coherent Representation of Subjective Risk Aversion”, in Journal of Banking & Finance, vol. 26, n. 7. 2. Acerbi, A. and D. Tasche, (2002), “On the Coherence of Expected Shortfall”, in Journal of Banking& Finance, vol. 26, n. 7. 3. Andersson, F. and S. Uryasev, (1999), “Credit risk optimization with Conditional Value-at-Risk criterion”, Research Report #99-9, Center for Applied Optimization, Dept. of Industrial and Systems Engineering, Univ. of Florida 4. Artzner, P., F. Delbaen, J.M. Eber and D. Heath (1999), “Coherent Measures of Risk” in Mathematical Finance 9 (July) p 5. Artzner, P., F. Delbaen, J. M. Eber and D. Heath, (1997), “Thinking Coherently,” Risk, Vol. 10, No. 11, pp , November 1997. 6. Basak, S. and A. Shapiro, (1998), “Value-at-Risk Based Management: Optimal Policies and Asset Prices”, Working Paper, Wharton School, University of Pennsylvania 7. Bawa, Vijay S., (1978), “Safety-First, Stochastic Dominance and Optimal Portfolio Choice”, in: Journal of Financial and Quantitative Analysis, vol. 13, p 8. Di Clemente, Annalisa (2002), “The Empirical Value-at-Risk/Expected Return Frontier: A Useful Tool of Market Risk Managing” 9. Fishburn, Peter C., (1977), “Mean-Risk Analysis with Risk Associated with Below-Target Returns”, in: American Economic Review, vol. 57, p 10. Gaivoronski, A.A. and G. Pflug, (2000), “Value at Risk in portfolio optimization: properties and computational approach”, NTNU, Department of Industrial Economics and Technology Management, Working paper.

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References 11. Grootweld H. and W.G. Hallerbach, (2000), “Upgrading VaR from Diagnostic Metric to Decision Variable: A Wise Thing to Do?”, Report 2003 Erasmus Center for Financial Research. 12. Guthoff, A., A. Pfingsten and J. Wolf, (1997), “On the Comapatibility of Value at Risk, Other Risk Concepts, and Expected Utility Maximization” in Beiträge zum 7. Symposium Geld, Finanzwirtschaft, Banken und Versicherungen an der Universität Karlsruhe vom Dezember 1996, Karlsruhe 1997, p 13. Hadar, Josef, and William R. Russell, (1969), “Rules for ordering uncertain prospects”, American Economic Review 59, 14. Hanoch, Giora, and Haim Levy, (1969), “The efficiency analysis of choices involving risk”, Review of Economic Studies 36, 15. Jorion, P. (1997), “Value at Risk: The New Benchmark for Controlling Market risk”, Irwin Chicago 16. Larsen, N., Mausser, H. and S. Uryasev, (2002), “Algorithms for Optimization of Value-At-Risk” Algorithms for Optimization of Value-At-Risk. Research Report, ISE Dept., University of Florida 17. Levy, Haim, (1992), “Stochastic dominance and expected utility: Survey and analysis”, Management Science 38, 18. Markowitz, H.M., (1952), “Portfolio Selection”, Journal of Finance. Vol.7, 1, 19. Mausser, H. and D. Rosen, (1998), “Beyond VaR: from measuring risk to managing risk”, in Algo Research Quaterly, vol. 1, no.2. 20. Ogryczak, W. and A. Ruszczynski, (1997), “From Stochastic Dominance to Mean-Risk Models: Semideviations as Risk Measures,”Interim Report , International Institute for Applied Systems Analysis, Laxenburg, Austria.

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References 21. Pflug, G.Ch., (2000), “Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk” In.”Probabilistic Constrained Optimization: Methodology and Applications”, Ed. S.Uryasev, Kluwer Academic Publishers, 2000. 22. (1999b), “How to Measure Risk?” Modelling and Decisions in Economics. Essays in Honor of Franz Ferschl, Physica-Verlag, 1999. 23. Rockafellar, R.T. and S. Uryasev (2000a), “Conditional Value-at-Risk for General Loss Distribution”, Journal of Banking&Finance, vol. 26, n. 7. 24. (2000b), “Optimization of Conditional Value-at-Risk”, The Journal of Risk, vol. 2, no. 3 25. Roy, A. D., (1952), "Safety First and the Holding of Assets.” Econometrica, no. 20: 26. Rothschild, M., and J. E. Stiglitz, (1970), “Increasing Risk: I. A Definition,” Journal of Economic Theory, Vol. 2, No. 3, 1970, pp 27. Tasche, D., (1999), “Risk contributions and performance measurement.”, Working paper, Munich University of Technology. 28. Testuri, C.E. and S. Uryasev (2000), “On Relation Between Expected Regret and Conditional Value-at-Risk”, Working Paper, University of Florida 29. The MathWorks Inc. Matlab 5.3, 1999 30. Von Neumann, J., and O. Morgenstern, (1953), “Theory of Games and Economic Behavior”, Princeton University Press, Princeton, New Jersey, 1953. 31. Whitmore, G. Alexander, 1970, Third-degree stochastic dominance, American Economic Review 60, 32. Yamai, Y. and T. Yoshiba, (2001), “On the Validity of Value-at-Risk: Comparative Analyses with Expected Shortfall”. Institute for Monetary and Economic Studies. Bank of Japan. IMES Discussion Paper 2001-E-4.

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