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

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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


Contents

Contents

  • Objectives

  • VaR, CVaR, ER-properties and optimization algorithms

  • Methodology

  • Empirical Application

  • Concluding remarks


Objectives

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


Var alternative definitions

VaR-alternative definitions

1.

2.


The efficient conditional value at risk

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

Expected Regret

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


Var cvar comparison

VaR/CVaR Comparison


Var cvar comparison1

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 Comparison

VaR

CVaR


Var cvar comparison2

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 Comparison


Cvar optimization

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


Cvar optimization1

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 optimization2

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

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

Optimization problem

The constraint on return takes the form:

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


Optimization problem1

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 problem2

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

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


The efficient conditional value at risk

Data


The efficient conditional value at risk

Data


Substitution error

Substitution error


Substitution error1

Portfolio

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 error

200 scenarios

300 scenarios


Substitution error2

Substitution error


Cvar efficient frontier without transaction costs

CVaR Efficient Frontier Without Transaction Costs


Cvar efficient frontier without transaction costs1

CVaR Efficient Frontier Without Transaction Costs


Restructuring the initial portfolio

Restructuring the initial portfolio


Restructuring the initial portfolio1

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.


The impact of transaction costs

The impact of transaction costs


Transaction costs

Transaction Costs


Transaction costs1

Transaction Costs


The restructured portfolios

The restructured portfolios


The restructured portfolios1

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%

The restructured portfolios

76 0.001492 21215.6582 15930.8490 21215.6582 29789.3824 18.63 40.28 153.64 61.98 81.97


The restructured portfolio

The restructured portfolio


Concluding remarks

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

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.


References1

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.


References2

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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