VaR optimization
- 1. Alternative! Risk Measures beyond Markovitz E Value at Risk ! Expected ShortfallFilippo Perugini
- 2. Portfolio Optimization downside risk measures - presentation structure 1 Value at Risk - VaR ! • definition! • portfolio optimization ! • pro - cons 2 Expected Shortfall - CVaR ! • definition! • portfolio optimization ! • pro - cons 3 Implementation! • efficient frontier! • portfolio weights ! • performances 4 Conclusion! which measure to use?
- 3. Downside Risk Measure Roy’s safety first principle Objective! maximization of the probability that the portfolio return is above a certain minimal acceptable level, often also referred to as the bench- mark level or disaster level.E Advantage! • classical portfolio: trade-off between risk and return and allocation depends on utility function! • Roy’s safety first: an investor first wants to make sure that a certain amount of the principal is preserved.
- 4. Value at Risk definition • The VaR of a portfolio is the minimum loss that a portfolio can suffer in x days in the α% worst cases when the absolute portfolio weights are not changed during these x days • VaR of a portfolio is the maximum loss that a portfolio can suffer in x days in the (1-α)% best cases, when the absolute portfolio weights are not changed during these x days. • α small VaRα (W ) = inf{l ∈! :P(W > l) ≤1−α}
- 5. Value at Risk portfolio optimization min w VaRα (w) wT µ ≥ µtarget wT 1= 1 s.t.
- 6. Value at Risk pro - cons Pro! • used by Regulators (Basel)! • risk aversion embedded in the confidence level α! • no distributional assumption needed! • easy estimation (because not dependent on tails) Ã Â Cons! • no sub-additive : violates diversification principle" • best case in worst case scenario: disregards the tail! • non smooth, non convex function of weights: multiple stationary points, difficult to find global optimum
- 7. Expected Shortfall or CVaR definition • The CVaR of a portfolio is the average loss that a portfolio can suffer in x days in the α% worst cases (when the absolute portfolio weights are not changed during these x days) • Average of all worst cases: takes into account the entire tail CVaRα (W ) = 1 α 0 α ∫ VaRγ (W )dγ
- 8. Expected Shortfall or CVaR portfolio optimization wT µ ≥ µtarget wT 1= 1 s.t. min w CVaRα (w)
- 9. Expected Shortfall or CVaR pro - cons Pro! • coherent risk measure: it is sub-additive!! • convex function: optimization is well defined! • takes into account the entire tail: better risk control à  Cons! • estimation accuracy affected by tail modelling ! • historical scenarios may not provide enough tail info
- 10. Numeric! Implementation how theory affects reality Ñ Portfolio Optimization • α= 0.01 fixed • different α’s • performances
- 12. Historical Returns histogram - pathological CVaR
- 13. Mean Variance Frontier VaR - Markovitz
- 14. VaR Frontier VaR - Markovitz
- 15. Portfolio Weights VaR - Markovitz
- 16. Mean Variance Frontier CVaR - Markovitz
- 17. CVaR Frontier CVaR - Markovitz
- 18. Portfolio Weights CVaR - Markovitz
- 19. Mean Variance Frontier VaR - CVaR
- 20. VaR - CVaR Frontier VaR - CVaR
- 21. Portfolio Weights CVaR - VaR
- 22. Different! Confidence Levels a comparison ( • frontiers • weights
- 23. VaR Frontier VaR - Markovitz
- 24. Mean Variance Frontier VaR - Markovitz
- 25. CVaR Frontier CVaR - Markovitz
- 26. Mean Variance Frontier CVaR - Markovitz
- 27. VaR - CVaR Frontier VaR - CVaR
- 33. Performances! out of sample • different time horizon • different portfolios
- 34. Time Frame optimization after crisis
- 39. Time Frame optimization before crisis
- 42. Conclusion: VaR or CVaR ? not a definitive answer • VaR may be better for optimizing portfolios when good models for tails are not available." • CVaR may not perform well out of sample when portfolio optimization is run with poorly constructed set of scenarios! • Historical data may not give right predictions of future tail! • CVaR has superior mathematical properties and can be easily handled in optimization and statistics! • It is the portfolio manager that has to take decision considering all the aspect of portfolio optimisation. Different situation may require different measures.