Portfolio Optimization with Spectral Measures of Risk

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Portfolio Optimization with Spectral
Measures of Risk Carlo Acerbi and Prospero
Simonetti
Torino January 30, 2003
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Outline of the talk

• What is a Spectral Measure of Risk ?
• Coherency and Convexity
• Minimization of Expected Shortfall (ES)
• Minimization of Spectral Measures
• Risks and Rewards are they really two orthogonal
  dimensions ?

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Part 1 What is a Spectral Measure of Risk ?
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Coherent Measures of Risk
In Coherent measures of Risk (Artzner et al.
Mathematical Finance, July 1999) a set of axioms
was proposed as the key properties to be
satisfied by any coherent measure of risk.
(Monotonicity) if then (Positive Homogeneity)
if then (Translational Invariance)
(Subadditivity)
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The VaR vs ES debate
Value at Risk (for a chosen x confidence level
and time horizon) is defined as The VaR of a
portfolio is the minimum loss that a portfolio
can suffer in the x worst cases Expected
Shortfall is defined as The ES of a portfolio
is the average loss that a portfolio can suffer
in the x worst cases
The debate arises since ES, turns out to be a
Coherent Measure of Risk while VaR is well known
to be a not-Coherent Measure of Risk
VaR the best of worst cases ES the average of
worst cases
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Spectral Measures of Risk

• Lets consider the class of Spectral Measures of
  Risk defined as
• where the Risk Spectrum is an
  arbitrary real function on 0,1
• They include in particular
• ES with Heavyside Step Function
• VaR with Dirac Delta Function

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Spectral Measures of Risk

• Theorem the Spectral Measure of Risk
• Is coherent if and only if its Risk Spectrum
  satisfies
• is positive
• is decreasing

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The Risk Aversion Function ?(p)
Any admissible ?(p) represents a possible
legitimate rational attitude toward risk A
rational investor may express her own subjective
risk aversion through her own subjective ?(p)
which in turns give her own spectral measure M?
?(p) Risk Aversion Function
It may thought of as a function which weights
all cases from the worst to the best
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Risk Aversion Function ?(p) for ES and VaR
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Estimating Spectral Measures of Risk
It can be shown that any spectral measure has the
following consistent estimator
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Part 2 Coherency and Convexity
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Coherency and Convexity in short
Coherency of the Risk Measure
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An interesting prototype portfolio
Consider a portfolio made of n risky bonds all of
which have a 2 default probability and suppose
for simplicity that all the default probabilities
are independent of one another. Portfolio
100 Euro invested in n independent identical
distributed Bonds Bond payoff Nominal (or 0
with probability 2) Question lets choose n in
such a way to minimize the risk of the
portfolio Lets try to answer this question with
a 5 VaR, ES and TCE ( ES (old)) with a time
horizon equal to the maturity of the bond.
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risk versus number of bonds in the portfolio
The surface of risk of ES has a single global
minimum at n? and no fake local minima. ES just
tell us buy more bonds you can
VaR and TCE suggest us NOT TO BUY the 6th, 36thor
83rd bond because it would increase the risk of
the portfolio .... (?)
Are things better for large portfolios ???...
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Big n ... same pattern
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...maybe theres really some tricky risk in the
36th bond !
If we use a 3 VaR instead of a 5 VaR, the
dangerous bond is not the 36th anymore, but the
28th.... (!?)
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Coherency and convexity
The lack of coherence of VaR and TCE is the
reason why their risk surfaces are wrinkled
displaying meaningless local minima. In such a
situation, an optimization process always selects
a wrong local solution, irrespectively of the
starting point. In real life finance, on large
and complex portfolios, such local minima for VaR
surfaces are the rule rather than the exception.
In other words, the local minima are not due to
the simplicity of our chosen portfolio. They
always surround a diversified global optimal
solution. Even though manifest VaR subadditivity
violations are very rare to happen when adding
large (quasi-gaussian) portfolios, it is
nevertheless true that on the same large
portfolios marginal VaR systematically fails to
properly assess the change of risk associated to
buying or selling a single asset.
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Part 3 Minimization of Expected Shortfall
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Minimizing the Expected Shortfall
Let a portfolio of M assets be a function of
their weights wj1....M and let XX(wi ) be
its Profit Loss. We want to find optimal
weights by minimizing its Expected Shortfall
In the case of a N scenarios estimator we
have
Notice also in the case of non parametric VaR a
SORTING operation is needed in the estimator and
the same problem appears
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The Pflug-Uryasev-Rockafellar solution
Pflug, Uryasev Rockafellar (2000, 2001)
introduce a function which is analytic, convex
and piecewise linear in all its arguments. It
depends on X(w) but also on an auxiliary variable
? In the discrete case with N scenarios it
becomes
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Properties of ? the Pflug-Uryasev-Rockafellar
theorem
Minimizing ? in its arguments (w,?) amounts to
minimizing ES in (w) only Moreover the ?
parameter in the extremum takes the value of
VaR(X(w)).
?(w) and ES(w) coincide but just in the minimum !
The auxiliary parameter in the minimum becomes
the VaR
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Properties of ? - linearizability of the
optimization problem
A convex, piecewise linear function is the
easiest kind of function to minimize for any
optimizator. Its optimization problem can also be
reformulated as a linear progamming problem It
is a multidimensional faceted surface ... some
kind of multidimensional diamond with a unique
global minimum
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The role of the auxiliary variable ?
The auxiliary variable is introduced to SPLIT the
5 worst scenarios from the remaining 95. It
is thanks to this variable that the data SORTING
disappears.
In the minimization process ? places on the
specified quantile and gets the value of VaR.
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Application unconstrained ES minimization
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Part 4 Minimization of Spectral Measures
of Risk
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Minimizing a general Spectral Measure M?
The SORTING problem appears in the minimization
of any Spectral Measure
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Generalization of the solution of
Pflug-Uryasev-Rockafellar
Acerbi, Simonetti (2002) generalize the function
of P-U-R to any spectral measure. Also in this
case it is analytic, convex and piecewise linear
in all arguments. In general it depends however
on N auxiliary variables ?i In the discrete
case it becomes
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Properties of the generalized ??
Minimizing ?? in all parameters (w,?) amounts to
minimizing M? in (w) Moreover, in the
extremal, ?k takes the value of VaR(X(w))
associated to the quantile k/N.
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The role of ?
The N auxiliary variables are needed to separate
completely from one another all the ordered
scenarios XiN In the case of a general spectral
measures in fact, splitting the data sample into
TWO SUBSET is not enough (as in the case of ES)
In the minimization any ?k goes to the quantile
k/N. The ? vector separates all scenarios X.
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Part 5 Risks and Rewards are they really
two orthogonal axis ?
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An elementary observation ...
A generic Spectral Measure weights all scenarios
of a portfolio from the worst to the best with
decreasing weights (decreasing risk aversion
function). It therefore weights at the same time
RISKS and REWARDS in an integrated way.
.... -5.72 -4.94 -3.21 1.23 2.34 3.03 4.
92 ....
.... F(n) F(n1) F(n2) F(n3) F(n4)
F(n5) F(n6) ....
- Sneg.F(i) X(i)
- Spos. F(i) X(i)


Minimizing a Spectral Measure amounts to a
certain MINIMIZATION of RISKS captured by the
negative contribute and a certain MAXIMIZATION of
REWARDS captured by the positive contribute.
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The simplest example ...
Take for instance the case of a Spectral Mesure
obtained as convex combination of ES?(X) and
ES100(X) - average(X)

• This is a particular family of Spectral Measures
  with parameter ? between 0 and 1
• for ?1 it is ES with confidence level ?
• for ?0 it reduces to (- return)
• Minimizing this spectral measure already amounts
  to minimizing Expected Shortfall at confidence
  level ? and maximizing the return at the same
  time.

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Optimal portfolios for different ? values
l1
l0
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Integrated Markowitz problem
One shows in fact that (Acerbi, Simonetti, 2002)
minimizing
with no constraints, for any ?
amounts to minimizing
with constrains, for any specified return value
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Risk Minimization and Return Maximization cannot
be disentangled. Given an optimal portfolio there
always exist a Spectral Measure for which that
portfolio is a minimal risk portfolio.
General result
More generally one can show that any point in the
efficient frontier in the Markowitz plane of
abscissa M? represents also the unconstrained
minimal solution for another spectral measure
M?,?.

• constrained optimal prtf for M?
• unconstrained optimal prtf for M?,?

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Conclusions

• The minimization problem of a Spectral Measure
  of Risk is a convex problem, but dramatic
  computational problems are encountered if a
  straightforward approach is adopted.
• An extension of the P.R.U. methodology however
  allows to exactly convert the minimization
  problem into the minimization of an analytic,
  piecewise linear and convex functional.
• Complexity can be further reduced by an exact
  linearization of the problem.
• Standard linear optimizators (say CPLEX) allow
  to face in an efficient way the optimization
  problem of any Spectral Measure, under any
  probability distribution function for large size
  portfolios.
• Splitting an optimization problem in Risk
  Minimization and Returns Maximization is
  arbitrary. All optimal portfolios in
  Markowitz-like efficient frontiers are in fact
  absolute unconstrained minima of other Spectral
  Measures. The trade-off between risks and reward
  is already taken into account in the choice of
  the risk measure itself.

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References
P. Artzner, F. Delbaen, J.M. Eber and D. Heath,
1999, Coherent Measures of Risk R.T
Rockafellar and S.Uryasev, 2000, Optimization of
Conditional Value-at-Risk C. Acerbi, 2001,
Risk Aversion and Coherent Risk Measures a
Spectral Representation Theorem C. Acerbi, P.
Simonetti, 2002 Portfolio Optimization with
Spectral Measures of Risk