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

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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. Contents. Objectives VaR, CVaR, ER-properties and optimization algorithms

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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

Student: Stan Anca Mihaela

Supervisor:Professor Moisa Altar

Contents

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

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

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

Expected Regret

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

Simple convenient representation of risks (one number)

Measures downside risk (compared to variance which isimpacted 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

VaR/CVaR ComparisonVaR

CVaR

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 therisk

- 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, monotonicw.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 lossdistributions optimal variance and CVaR portfolios coincide

easy to control/optimize for non-normal distributions, by using linear programming techniques

VaR/CVaR ComparisonCVaR 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

CVaR Optimization

- Rockafellar and Uryasev (1999) have shown that both

can be characterized in terms of the function

definedon

by:

By solving the optimization problem we find an optimal portfolio x* , corresponding VaR,which equals to the lowest optimal , and minimal CVaR, whichequals to the optimal value of the linear performance function.

CVaR Optimization

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

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

Optimization problem

The constraint on return takes the form:

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

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:

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.

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

Portfolio

37.58%

4.93%

-8.08%

7.07%

10.69%

6.06%

-7.07

-1.01%

0%

1.00%

-1.01%

0%

4.93%

0%

0%

7.07%

17.90%

0%

0%

4.04%

Substitution error200 scenarios

300 scenarios

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 33,239.28 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 0.001492 (instead of 0.001071) – 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.

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%

The restructured portfolios76 0.001492 21215.6582 15930.8490 21215.6582 29789.3824 18.63 40.28 153.64 61.98 81.97

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

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 203-228
- 5. Artzner, P., F. Delbaen, J. M. Eber and D. Heath, (1997), “Thinking Coherently,” Risk, Vol. 10, No. 11, pp. 68-71, 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. 255 - 271.
- 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. 116 - 126.
- 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.

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 11.-13. Dezember 1996, Karlsruhe 1997, p. 591-614.
- 13. Hadar, Josef, and William R. Russell, (1969), “Rules for ordering uncertain prospects”, American Economic Review 59, 25-34.
- 14. Hanoch, Giora, and Haim Levy, (1969), “The efficiency analysis of choices involving risk”, Review of Economic Studies 36, 335-346.
- 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, 555-593.
- 18. Markowitz, H.M., (1952), “Portfolio Selection”, Journal of Finance. Vol.7, 1, 77-91.
- 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 97-027, International Institute for Applied Systems Analysis, Laxenburg, Austria.

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. 20:431-449
- 26. Rothschild, M., and J. E. Stiglitz, (1970), “Increasing Risk: I. A Definition,” Journal of Economic Theory, Vol. 2, No. 3, 1970, pp. 225-243.
- 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, 457-459.
- 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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