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}
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)
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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
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Cons!
• estimation accuracy affected by tail modelling !
• historical scenarios may not provide enough tail info
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.