Presentation on theme: "The Efficient Conditional Value-at-Risk/Expected Return Frontier"— Presentation transcript:
1 The Efficient Conditional Value-at-Risk/Expected Return Frontier THE ACADEMY OF ECONOMIC STUDIES BUCHARESTDOCTORAL SCHOOL OF FINANCE AND BANKINGThe Efficient Conditional Value-at-Risk/Expected Return FrontierStudent: Stan Anca MihaelaSupervisor:Professor Moisa Altar
2 ContentsObjectivesVaR, CVaR, ER-properties and optimization algorithmsMethodologyEmpirical ApplicationConcluding remarks
3 Objectives Construct the efficient CVaR/Expected Return frontier Analyze CVaR’s performance as a proxy variable for VaRUse CVaR as a risk tool in order to efficiently restructure portfoliosAnalyze the impact of transaction costs
8 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 distributionsEasily applied to backtestingEstablished as a standard risk measureConsistent with first order stochastic dominanceSimple convenient representation of risks (one number)Measures downside riskApplicable to nonlinear instruments, such as options, with non-symmetric (non-normal) loss distributionsNot easily applied to efficient backtesting methodsConsistent with second order stochastic dominance
9 VaR/CVaR Comparison does not measure losses exceeding VaR reduction of VaR may lead to stretch of tail exceeding VaRnon-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 extremumsaccounts for risks beyond VaR (more conservative than VaR)convex with respect to portfolio positionscoherent 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 VaRconsistency with mean-variance approach: for normal loss distributions optimal variance and CVaR portfolios coincideeasy to control/optimize for non-normal distributions, by using linear programming techniques
10 CVaR optimization . Notations: x = (x1,...xn) = decision vector (e.g., portfolio positions)y = (y1,...yn) = random vectoryj = scenario of random vector y , ( j=1,...J )f(x,y) = loss functions=CVaR at confidence level=VaR at confidence level
11 CVaR Optimization Rockafellar and Uryasev (1999) have shown that both can be characterized in terms of the functiondefined onby: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.
12 CVaR OptimizationWhen the function F is approximated using scenarios, the problem is reduced to LP with the help of a dummy variable:
13 ER OptimizationIf the function G is approximated using scenarios, the problem can be reduced toa linear programming problem, having the same constraints as the CVaRoptimization problem and with the objective function
14 Optimization problem The constraint on return takes the form: The balance constraint that maintains the total value of the portfolio less transaction costs:
15 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 percentof the total portfolio value:
16 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.
17 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
23 CVaR Efficient Frontier Without Transaction Costs
24 CVaR Efficient Frontier Without Transaction Costs
25 Restructuring the initial portfolio Expected ReturnAlpha7,500VaR27,429.48CVaR37,302.43ER12,428.36Markowitz18,729.29
26 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.
33 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-RiskRisk measures consistent with third or higher order stochastic dominance criteria
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