Properties and Computation of CoVaR efficient portfolio Shin, Dong Wook SOLAB Industrial and System Engineering Department, KAIST

Overview of portfolio optimization

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

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

Overview of risk measures

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. >

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)

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

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

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

Financial data

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

Portfolio minimizing CoVaR

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 }

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

Comparison of efficient frontiers

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 ( ) 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

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

Backtesting

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

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

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

Conclusion

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………………