1. Properties and Computation of CoVaR efficient portfolio Shin, Dong Wook 20030282 SOLAB Industrial and System Engineering Department, KAIST oogino1@kaist.ac.kr

2. Overview of portfolio optimization

3. System Optimization Lab Overview of portfolio optimization Efficient diversification of investment >H. Markowitz (1952): Mean-variance approach >Return vs. Risk (Variance) Source : http://Financial-edge.blogspot.com ① ②

4. System Optimization Lab General formulation for portfolio optimization Overview of portfolio optimization

5. Overview of risk measures

6. System Optimization Lab VaR (Value-at-Risk) overview of risk measures >Measure of downside risk >VaR of portfolio (Historical Simulation) : > Threshold value such that the probability that the mark-to-market loss on the portfolio over the given time horizon exceed this value. >

7. System Optimization Lab VaR (Value-at-Risk) overview of risk measures >Problems : >Difficult to approach non-parametrically >Risk measure for individual asset (or portfolio) >What if individual assets are significantly correlated? Source : Uryasev(2000)

8. System Optimization Lab CoVaR overview of risk measures >Contagion, or Co-movement Value-at-risk >Tobias Adrian & Markus Brunnermeier, 2008 >Measure for spillover risk >(ex)Spillover risk from MS to GS >Definition : The VaR of insitutition j conditional on. >We focus on the case i=system, in order to capture systemic risk of portfolio when the system is in distress. ***Systemic risk : a.k.a. market risk or un-diversifiable risk

9. System Optimization Lab CoVaR overview of risk measures >Estimation of CoVaR : Quantile regression >calculate alpha and beta of the equation : >Linear Programming for quantile regression >Minimize sum of weighted absolute residuals of the quantile equation!! > : Coefficients of quantile regression, > : Positive(negative) quantile errors > : Return of a stock(or a portfolio) at period t > : Return of an index at period t

10. System Optimization Lab >then the CoVaR of j conditional on i is : CoVaR overview of risk measures >Estimation of CoVaR : Quantile regression

11. Financial data

12. System Optimization Lab Financial data >Daily stock price data of the 100 companies in the S&P100 index >From 2008/09/02 to 2009/06/17 (200 business days) >The date of the first 100 days :constructing and drawing efficient portfolios >The data of the last 100 days : backtesting >During the period, S&P100 fell 50 % from its 2007 high

13. Portfolio minimizing CoVaR

14. System Optimization Lab Portfolio minimizing CoVaR Linear Bi-level Programming problem (BLP) >Upper(F(x,y)) and Lower(f(x,y) level problems >Upper(G(x,y)) and Lower(g(x,y)) level constraints >Formal formation of linear BLP is : >The set of solutions for the lower level problem is called {Inducible region } >Thus, the upper level optimizer searches optimal values within the { Inducible region }

15. System Optimization Lab Portfolio minimizing CoVaR Mixed Integer Programming (MIP)

16. Comparison of efficient frontiers

17. System Optimization Lab Comparison of Efficient Frontiers Comparison of 3 efficient frontiers >Mean-variance model : Quadratic programming problem >Mean-CVaR model : Linear programming problem >Mean-CoVaR model : Bi-level linear programming problem (MIP) Softwares >MOSEK Optimization software (5.0.0.127) for large-scale quadratic programming >ILOG Cplex 11.0 for Linear and Mixed Integer programming We put different efficient frontiers on return/variance, return/CVaR, return/CoVaR space

18. System Optimization Lab Comparison of Efficient Frontiers Efficient frontiers in return/CoVaR space. CoVaR CVaR variance

19. Backtesting

20. System Optimization Lab Backtesting Design for backtesting >Financial Data from 2009/01/26 to 2009/06/17 (100 business days) >Initial investment : $1,000,000 >Portfolio revision period : every 25 days/ every 50 days >4 different R exp (expected return) values : >{,,, } > : average return of S&P100 index for past 100 days

21. System Optimization Lab Backtesting Result of backtesting (revision period = every 25 days)

22. System Optimization Lab Backtesting Result of backtesting (revision period = every 50 days)

23. Conclusion

24. System Optimization Lab Conclusion Why should we use CoVaR as Risk Measure?? >CoVaR-efficient portfolio outperforms the other portfolios >CoVaR-efficient portfolio prevents systemic risk of the entire market to be transferred to the portfolio >For MIP problem, computational complexity goes up exponentially >’big M’ value should be minimized to relieve it >efficient branch-and-cut algorithm may help! >Financial Market has changed >Globalized >Largely Interconnected (Sub-prime Mortgage Crisis 2008) ***To be more efficient………………