1. 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

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

3. 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

4. VaR-alternative definitions1. 2.

5. 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

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

7. VaR/CVaR Comparison

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 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

9. 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

10. 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

11. 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.

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

13. 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

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 percent
of 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

18. Data

19. Data

20. Substitution error

21. 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%

22. Substitution error

23. CVaR Efficient Frontier Without Transaction Costs

24. CVaR Efficient Frontier Without Transaction Costs

25. 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

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.

27. The impact of transaction costs

28. Transaction Costs

29. Transaction Costs

30. The restructured portfolios

31. 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%

32. The restructured portfolio

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-Risk
Risk measures consistent with third or higher order stochastic dominance criteria

34. References
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