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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

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

- 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

1.

2.

- 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

- 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

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

- . 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

- 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.

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

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

The constraint on return takes the form:

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

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:

- 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.

- 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

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%

200 scenarios

300 scenarios

- 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.

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

- 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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