1. Scenario Optimization, part 2

2. Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Contents CVAR portfolio optimization Demo of VAR and CVAR optimization Put-call efficient frontiers

3. Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Value at Risk in portfolio optimization Loss function Probability that loss does not exceed some threshold Probability of losses strictly greater than some threshold

4. Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Value at Risk in portfolio optimization Relation between different quantities

5. Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Value at Risk in portfolio optimization Conditional Value at Risk

6. Financial Optimization and Risk Management Professor Alexei A. Gaivoronski CVAR portfolio models Loss function CVAR for  being 100%  VAR: Or for discrete scenarios Assuming denominator equal 1-  We need to simplify this keeping in mind our objective of using CVAR in portfolio risk management models

7. Financial Optimization and Risk Management Professor Alexei A. Gaivoronski CVAR portfolio models General CVAR portfolio model Two possible advantages of this model  It takes into account the losses incurred if abnormal scenarios materialize  CVAR is convex function of portfolio as opposed to VAR and for this reason it is easier to compute  In order to take advantage of this it is necessary to look more carefully into CVAR formulation

8. Financial Optimization and Risk Management Professor Alexei A. Gaivoronski VaR and CVaR: comparison CVaR may give very misleading ideas about VaR VaR/CVaR fraction of portfolio 2

9. Financial Optimization and Risk Management Professor Alexei A. Gaivoronski LP formulation of CVAR portfolio model Introduce auxilliary variables Or in case of discrete scenarios Averaging these with respect to scenarios

10. Financial Optimization and Risk Management Professor Alexei A. Gaivoronski LP formulation of CVAR portfolio model Which gives Dividing this by 1-  and rearranging we get And recalling expression for CVAR we get finally

11. Financial Optimization and Risk Management Professor Alexei A. Gaivoronski LP formulation of CVAR portfolio model LP CVAR portfolio model This is linear model if losses are linear

12. Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Portfolio optimization with CVAR constraints

13. Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Put-call efficient frontiers Portfolio performance is measured against random target g: liabilities, benchmark, index, competition, etc Reward: portfolio exceeds target, risk: portfolio is below target Integrated view of financial management process Upside potential: payoff of a call option on the future portfolio value relative to target Downside potential: short position in a European put option on the future portfolio value relative to the target Portfolio call value: expected upside, put value: expected downside Put/call efficient portfolios and put/call efficient frontiers

14. Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Put-call efficient frontiers Tracing put/call efficient frontier Start with LP without constraints Portfolio value Target portfolio value

15. Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Put-call efficient frontiers Constraint which connects portfolio value with upside and downside Put/Call efficient portfolio

16. Financial Optimization and Risk Management Professor Alexei A. Gaivoronski Dual problem Helps to obtain insight into the nature of solution Solution does not depend on  !! This means that efficient frontier is the straight line