var optimization
TRANSCRIPT
Alternative!Risk Measuresbeyond Markovitz
E
Value at Risk !
Expected ShortfallFilippo Perugini
Portfolio Optimizationdownside 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?
Downside Risk MeasureRoy’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.
Value at Riskdefinition
• 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}
Value at Riskportfolio optimization
minwVaRα (w)
wTµ ≥ µtarget
wT1= 1
s.t.
Value at Riskpro - 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
ÂCons!• 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
Expected Shortfall or CVaRdefinition
• 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 ) =1α 0
α
∫ VaRγ (W )dγ
Expected Shortfall or CVaRportfolio optimization
wTµ ≥ µtarget
wT1= 1
s.t.
minwCVaRα (w)
Expected Shortfall or CVaRpro - cons
Pro!• coherent risk measure: it is sub-additive!!• convex function: optimization is well defined!• takes into account the entire tail: better risk control
Ã
ÂCons!• estimation accuracy affected by tail modelling !• historical scenarios may not provide enough tail info
Numeric!Implementation
how theory affects reality
ÑPortfolio
Optimization • α= 0.01 fixed • different α’s
• performances
Historical Returnshistogram
Historical Returnshistogram - pathological CVaR
Mean Variance FrontierVaR - Markovitz
VaR FrontierVaR - Markovitz
Portfolio WeightsVaR - Markovitz
Mean Variance FrontierCVaR - Markovitz
CVaR FrontierCVaR - Markovitz
Portfolio WeightsCVaR - Markovitz
Mean Variance FrontierVaR - CVaR
VaR - CVaR FrontierVaR - CVaR
Portfolio WeightsCVaR - VaR
Different!Confidence Levels
a comparison
(
• frontiers • weights
VaR FrontierVaR - Markovitz
Mean Variance FrontierVaR - Markovitz
CVaR FrontierCVaR - Markovitz
Mean Variance FrontierCVaR - Markovitz
VaR - CVaR FrontierVaR - CVaR
Portfolio Weightsα=0.1
Portfolio Weightsα=0.05
Portfolio Weightsα=0.01
Portfolio Weightsα=0.005
Portfolio Weightsα=0.001
Performances!out of sample
!
• different time horizon • different portfolios
Time Frameoptimization after crisis
Portfolio Weightsportfolio number 30
Portfolios Performanceportfolio number 30
Portfolio Weightsportfolio number 10
Portfolios Performanceportfolio number 10
Time Frameoptimization before crisis
Portfolio Weightsportfolio number 30
Portfolios Performanceportfolio number 30
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.
YOUTHANKfor your attention