var optimization

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