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 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

VaR-alternative definitions 1. 2.

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.

VaR/CVaR Comparison

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

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

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 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.

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

Data

Data

Substitution error

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%

Substitution error

CVaR Efficient Frontier Without Transaction Costs

CVaR Efficient Frontier Without Transaction Costs

Restructuring the initial portfolio Expected Return 0.001071 Alpha 7,500 VaR 27,429.48 CVaR 37,302.43 ER 12,428.36 Markowitz 18,729.29

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

Transaction Costs

Transaction Costs

The restructured portfolios

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% 76 0.001492 21215.6582 15930.8490 21215.6582 29789.3824 18.63 40.28 153.64 61.98 81.97

The restructured portfolio

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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