Noise sensitivity of portfolio selection under various risk measures

1
Noise sensitivity of portfolio selection under
various risk measures

• Imre Kondor
• Collegium Budapest and Eötvös
• University, Budapest, Hungary
• Risk Measurement and Management, Rome, June 9-17,
  2005

2
Contents

• I. Preliminaries
• the problem of noise, risk measures, noisy
  covariance matrices
• II. Random matrices
• Spectral properties of Wigner and Wishart
  matrices
• III. Filtering of normal portfolios
• optimization vs. risk measurement,
  model-simulation approach, random-matrix-theor
  y-based filtering
• IV. Beyond the stationary, Gaussian world
• non-stationary case, alternative risk measures
  (mean absolute deviation, expected shortfall,
  worst loss), their sensitivity to noise, the
  feasibility problem

3
Coworkers

• Szilárd Pafka and Gábor Nagy (CIB Bank, Budapest,
  a member of the Intesa Group), Marc Potters
  (Capital Fund Management, Paris)
• Richárd Karádi (Institute of Physics, Budapest
  University of Technology, now at ProcterGamble)
• Balázs Janecskó, András Szepessy, Tünde Ujvárosi
  (Raiffeisen Bank, Budapest)
• István Varga-Haszonits (Eötvös University,
  Budapest)

4
I. PRELIMINARIES
5
Preliminary considerations

• Portfolio selection vs. risk measurement of a
  fixed portfolio
• Portfolio selection a tradeoff between risk and
  reward
• There is a more or less general agreement on what
  we mean by reward in a finance context, but the
  status of risk measures is controversial
• For optimal portfolio selection we have to know
  what we want to optimize
• The chosen risk measure should respect some
  obvious mathematical requirements, must be
  stable, and easy to implement in practice

6
The problem of noise

• Even if returns formed a clean, stationary
  stochastic process, we only could observe finite
  time segments, therefore we never have sufficient
  information to completely reconstruct the
  underlying process. Our estimates will always be
  noisy.
• Mean returns are particularly hard to measure on
  the market with any precision
• Even if we disregard returns and go for the
  minimal risk portfolio, lack of sufficient
  information will introduce noise, i. e. error,
  into correlations and covariances, hence into our
  decision.
• The problem of noise is more severe for large
  portfolios (size N) and relatively short time
  series (length T) of observations, and different
  risk measures are sensitive to noise to a
  different degree.
• We have to know how the decision error depends on
  N and T for a given risk measure

7
Some elementary criteria on risk measures

• A risk measure is a quantitative characterization
  of our intuitive risk concept (fear of
  uncertainty and loss).
• Risk is related to the stochastic nature of
  returns. It is a functional of the pdf of
  returns.
• Any reasonable risk measure must satisfy
• - convexity
• - invariance under addition of risk free asset
• - monotonicity and assigning zero risk to a zero
  position
• The appropriate choice may depend on the nature
  of data (e.g. on their asymptotics) and on the
  context (investment, risk management,
  benchmarking, tracking, regulation, capital
  allocation)

8
A more elaborate set of risk measure axioms

• Coherent risk measures (P. Artzner, F. Delbaen,
  J.-M. Eber, D. Heath, Risk, 10, 33-49 (1997)
  Mathematical Finance,9, 203-228 (1999)) Required
  properties monotonicity, subadditivity, positive
  homogeneity, and translational invariance.
  Subadditivity and homogeneity imply convexity.
  (Homogeneity is questionable for very large
  positions. Multiperiod risk measures?)
• Spectral measures (C. Acerbi, in Risk Measures
  for the 21st Century, ed. G. Szegö, Wiley, 2004)
  a special subset of coherent measures, with an
  explicit representation. They are parametrized by
  a spectral function that reflects the risk
  aversion of the investor.

9
Convexity

• Convexity is extremely important.
• A non-convex risk measure
• - penalizes diversification (without convexity
  risk
• can be reduced by splitting the portfolio in
  two
• or more parts)
• - does not allow risk to be correctly aggregated
• - cannot provide a basis for rational pricing of
  risk
• (the efficient set may not be not convex)
• - cannot serve as a basis for a consistent limit
  
• system
• In short, a non-convex risk measure is really not
  a risk measure at all.

10
A classical risk measure the variance

• When we use variance as a risk measure we assume
  that the underlying statistics is essentially
  multivariate normal or close to it.

11
Portfolios

• Consider a linear combination of returns
• with weights . The
  weights add up to unity . The
  portfolios expectation value is
  with variance ,
• where is the covariance matrix, and
  the standard deviation of return .

12
Level surfaces of risk measured in variance

• The covariance matrix is positive definite. It
  follows that the level surfaces (iso-risk
  surfaces) of variance are (hyper)ellipsoids in
  the space of weights. The convex iso-risk
  surfaces reflect the fact that the variance is a
  convex measure.
• The principal axes are inversely proportional to
  the square root of the eigenvalues of the
  covariance matrix.
• Small eigenvalues thus correspond to long
  axes.
• The risk free asset would correspond to and
  infinite axis, and the correspondig ellipsoid
  would be deformed into an elliptical cylinder.

13
The Markowitz problem

• According to Markowitz classical theory the
  tradeoff between risk and reward can be realized
  by minimizing the variance
• over the weights, for a given expected return
• and budget

14

• Geometrically, this means that we have to blow up
  the risk ellipsoid until it touches the
  intersection of the two planes corresponding to
  the return and budget constraints, respectively.
  The point of tangency is the solution to the
  problem.
• As the solution is the point of tangency of a
  convex surface with a linear one, the solution is
  unique.
• There is a certain continuity or stability in the
  solution A small miss-specification of the risk
  ellipsoid leads to a small shift in the solution.

15

• Covariance matrices corresponding to real markets
  tend to have mostly positive elements.
• A large, complicated matrix with nonzero average
  elements will have a large (Frobenius-Perron)
  eigenvalue, with the corresponding eigenvector
  having all positive components. This will be the
  direction of the shortest principal axis of the
  risk ellipsoid.
• Then the solution also will have all positive
  components. Even large fluctuations in the small
  eigenvalue sectors may have a relatively mild
  effect on the solution.

16
The minimal risk portfolio

• Expected returns are hardly possible (on
  efficient markets, impossible) to determine with
  any precision.
• In order to get rid of the uncertainties in the
  returns, we confine ourselves to considering the
  minimal risk portfolio only, that is, for the
  sake of simplicity, we drop the return
  constraint.
• Minimizing the variance of a portfolio without
  considering return does not, in general, make
  much sense. In some cases (index tracking,
  benchmarking), however, this is precisely what
  one has to do.

17
Benchmark tracking

• The goal can be (e.g. in benchmark tracking or
  index replication) to minimize the risk (e.g.
  standard deviation) relative to a benchmark
• Portfolio
• Benchmark
• Relative portfolio

18

• Therefore the relevant problems are of similar
  structure but with returns relative to the
  benchmark
• For example, to minimize risk relative to the
  benchmark means minimizing the standard deviation
  of
• with the usual budget contraint (no condition on
  expected returns!)

19
The weights of the minimal risk portfolio

• Analytically, the minimal variance portfolio
  corresponds to the weights for which
• is minimal, given .
• The solutions is .
• Geometrically, the minimal risk portfolio is the
  point of tangency between the risk ellipsoid and
  the plane of he budget constraint.

20
Empirical covariance matrices

• The covariance matrix has to be determined from
  measurements on the market. From the returns
  observed at time t we get the estimator
• For a portfolio of N assets the covariance matrix
  has O(N²) elements. The time series of length T
  for N assets contain NT data. In order for the
  measurement be precise, we need N ltltT. Bank
  portfolios may contain hundreds of assets, and it
  is hardly meaningful to use time series longer
  than 4 years (T1000). Therefore, N/T ltlt 1 rarely
  holds in practice. As a result, there will be a
  lot of noise in the estimate, and the error will
  scale in N/T.

21
Fighting the curse of dimensions

• Economists have been struggling with this problem
  for ages. Since the root of the problem is lack
  of sufficient information, the remedy is to
  inject external info into the estimate. This
  means imposing some structure on s. This
  introduces bias, but beneficial effect of noise
  reduction may compensate for this.
• Examples
• single-index models (ßs) All these help
  to
• multi-index models various degrees.
• grouping by sectors Most studies are
  based
• principal component analysis on
  empirical data
• Baysian shrinkage estimators, etc.

22
An intriguing observation

• L.Laloux, P. Cizeau, J.-P. Bouchaud, M. Potters,
  PRL 83 1467 (1999) and Risk 12 No.3, 69 (1999)
• and to
• V. Plerou, P. Gopikrishnan, B. Rosenow, L.A.N.
  Amaral, H.E. Stanley, PRL 83 1471 (1999)
• noted that there is such a huge amount of noise
  in empirical covariance matrices that it may be
  enough to make them useless.
• A paradox Covariance matrices are in widespread
  use and banks still survive ?!

23
Laloux et al. 1999
The spectrum of the covariance matrix obtained
from the time series of SP 500 with N406,
T1308, i.e. N/T 0.31, compared with that of a
completely random matrix (solid curve). Only
about 6 of the eigenvalues lie beyond the random
band.
24
Remarks on the paradox

• The number of junk eigenvalues may not
  necessarily be a proper measure of the effect of
  noise The small eigenvalues and their
  eigenvectors fluctuate a lot, indeed, but perhaps
  they have a relatively minor effect on the
  optimal portfolio, whereas the large eigenvalues
  and their eigenvectors are fairly stable.
• The investigated portfolio was too large compared
  with the length of the time series.
• Working with real, empirical data, it is hard to
  distinguish the effect of insufficient
  information from other parasitic effects, like
  nonstationarity.

25
A historical remark

• Random matrices first appeared in a finance
  context in G. Galluccio, J.-P. Bouchaud, M.
  Potters, Physica A 259 449 (1998). In this paper
  they show that the optimization of a margin
  account (where, due to the obligatory deposit
  proportional to the absolute value of the
  positions, a nonlinear constraint replaces the
  budget constraint) is equivalent to finding the
  ground state configuration of what is called a
  spin glass in statistical physics. This task is
  known to be NP-complete, with an exponentially
  large number of solutions.
• Problems of a similar structure would appear if
  one wanted to optimize the capital requirement of
  a bond portfolio under the rules stipulated by
  the Capital Adequacy Directive of the EU (see
  below)

26
A filtering procedure suggested by RMT

• The appearence of random matrices in the context
  of portfolio selection triggered a lot of
  activity, mainly among physicists. Laloux et al.
  and Plerou et al. proposed a filtering method
  based on random matrix theory (RMT) subsequently.
  This has been further developed and refined by
  many workers.
• The proposed filtering consists basically in
  discarding as pure noise that part of the
  spectrum that falls below the upper edge of the
  random spectrum. Information is carried only by
  the eigenvalues and their eigenvectors above this
  edge. Optimization should be carried out by
  projecting onto the subspace of large
  eigenvalues, and replacing the small ones by a
  constant chosen so as to preserve the trace. This
  would then drastically reduce the effective
  dimensionality of the problem.

27

• Interpretation of the large eigenvalues The
  largest one is the market, the other big
  eigenvalues correspond to the main industrial
  sectors.
• The method can be regarded as a systematic
  version of principal component analysis, with an
  objective criterion on the number of principal
  components.
• In order to better understand this novel
  filtering method, we have to recall a few results
  from Random Matrix Theory (RMT)

28
II. RANDOM MATRICES
29
Origins of random matrix theory (RMT)

• Wigner, Dyson 1950s
• Originally meant to describe (to a zeroth
  approximation) the spectral properties of (heavy)
  atomic nuclei
• - on the grounds that something that is
  sufficiently complex is almost random
• - fits into the picture of a complex system, as
  one with a large number of degrees of freedom,
  without symmetries, hence irreducible, quasi
  random.
• - markets, by the way, are considered stochastic
  for similar reasons
• Later found applications in a wide range of
  problems, from quantum gravity through quantum
  chaos, mesoscopics, random systems, etc. etc.

30
RMT

• Has developed into a rich field with a huge set
  of results for the spectral properties of various
  classes of random matrices
• They can be thought of as a set of central limit
  theorems for matrices

31
Wigner semi-circle law

• Mij symmetrical NxN matrix with i.i.d. elements
  (the distribution has 0 mean and finite second
  moment)
• ?k eigenvalues of Mij
• The density of eigenvalues ?k (normed by N) goes
  to the Wigner semi-circle for N?8 with prob. 1
• ,
• , otherwise

32
Remarks on the semi-circle law

• Can be proved by the method of moments (as done
  originally by Wigner) or by the resolvent method
  (Marchenko and Pastur and countless others)
• Holds also for slightly dependent or
  non-homogeneous entries (e.g. for the association
  matrix in networks theory)
• The convergence is fast (believed to be of 1/N,
  but proved only at a lower rate), especially what
  concerns the support

33

• Convergence to the semi-circle as N increases

34
N20
Elements of M are distributed normally
35
N50
36
N100
37
N200
38
N500
39
N1000
40

• If the matrix elements are not centered but have
  a common mean, one large eigenvalue breaks away,
  the rest stay in the semi-circle

41
If the matrix elements are not centered
N1000
42
N1000
43

• For fat-tailed (but finite variance)
  distributions the theorem still holds, but the
  convergence is slow

44
Sample from Student t (freedom3) distribution
N20
45
N50
46
N100
47
N200
48
N500
49
N1000
50

• There is a lot of fluctuation, level crossing,
  random rotation of eigenvectors taking place in
  the bulk

51
Illustration of the instability of the
eigenvectors, although the distribution of the
eigenvalues is the same. Sample 1 Matrix elements
normally distributed N1000
52
Sample 2
53
Sample k
54
Scalar product of the eigenvectors assigned to
the j. eigenvalue of the matrix.
55

• The eigenvector belonging to the large
  eigenvalue (when there is one) is much more
  stable. The larger the eigenvalue, the more so.

56
Illustration of the stability of the largest
eigenvector Sample 1 Matrix elements are normally
distributed, but the sum of the elements in the
rows is not zero. N1000
57
Sample 2
58
Sample k
59
Scalar product of the eigenvectors belonging to
the largest eigenvalue of the matrix. The larger
the first eigenvalue, the closer the scalar
products to 1 or -1.
60
The eigenvector components

• A lot less is known about the eigenvectors.
• Those in the bulk have random components
• The one belonging to the large eigenvalue (when
  there is one) is completely delocalized

61
Wishart matrices random sample covariance
matrices

• Let Aij NxT matrix with i.i.d. elements (0 mean
  and finite second moment)
• s 1/T AA where A is the transpose
• Wishart or Marchenko-Pastur spectrum (eigenvalue
  distribution)
• where

62
Remarks

• The theorem also holds when EA is of finite
  rank
• The assumption that the entries are identically
  distributed is not necessary
• If T lt N the distribution is the same with and
  extra point of mass 1 T/N at the origin
• If T N the Marchenko-Pastur law is the squared
  Wigner semi-circle
• The proof extends to slightly dependent and
  inhomogeneous entries
• The convergence is fast, believed to be of 1/N ,
  but proved only at a lower rate

63

• Convergence in N, with T/N 2 fixed

64
N20 T/N2
The red curve is the limit Wishart distribution
65
N50 T/N2
66
N100 T/N2
67
N200 T/N2
68
N500 T/N2
69
N1000 T/N2
70

• Evolution of the distribution with T/N, with N
  1000 fixed

71
The quadratic limit
N1000
T/N1
72
N1000 T/N1.2
73
N1000 T/N2
74
N1000 T/N3
75
N1000 T/N5
76
N1000 T/N10
77
Scalar product of the eigenvectors belonging to
the j eigenvalue of the matrices for different
samples.
78
Eigenvector components

• The same applies as in the Wigner case the
  eigenvectors in the bulk are random, the one
  outside is delocalized

79
Distribution of the eigenvector components, if no
dominant eigenvalue exists.
80
Market model
Underlying distribution is Wishart
N100 T/N2 Rho0.1
81
N200 T/N2
82
N500 T/N2
83
N1000 T/N2
84
Scalar product of the eigenvectors belonging to
the largest eigenvalue of the matrix. The larger
the first eigenvalue, the closer the scalar
products to 1.
85
Distribution of the eigenvector components, if no
dominant eigenvalue exists.
N1000 T/N2 Rho0.1
86
Distribution of the eigenvector components, if
one of the eigenvalues is not typical for random
matrixes.
N1000 T/N2 Rho0.1
87
N1000 T/N2 Rho0.1
Distribution of the eigenvector components, if
one of the eigenvalues is not typical for random
matrixes.
88
N1000 T/N2 Rho0.5
89
N1000 T/N2 Rho0.9
The interval becomes narrower as correlation
increases.
90
III. FILTERING OF NORMAL PORTFOLIOS
91
Some key points

• Laloux et al. and Plerou et al. demonstrate the
  effect of noise on the spectrum of the
  correlation matrix C. This is not directly
  relevant for the risk in the portfolio. We wanted
  to study the effect of noise on a measure of
  risk.

92
Optimization vs. risk management

• There is a fundamental difference between the two
  kinds of uses of the covariance matrix s for
  optimization resp. risk measurement.
• Where do people use s for portfolio selection at
  all?
• - GoldmanSachs technical document
• - tracking portfolios, benchmarking, shrinkage
• - capital allocation (EWRM)
• - hidden in softwares

93
Optimization

• When s is used for optimization, we need a lot
  more information, because we are comparing
  different portfolios.
• To get optimal portfolio, we need to invert s,
  and as it has small eigenvalues, error gets
  amplified.

94
Risk measurement management - regulatory
capital calculation

• Assessing risk in a given portfolio no need to
  invert s the problem of measurement error is
  much less serious

95
A measure of the effect of noise

• Assume we know the true covariance matrix and
• the noisy one . Then a natural, though not
  unique,
• measure of the impact of noise is
• where w are the optimal weights corresponding
• to and , respectively.

96
We will mostly use simulated data

• The rationale behind this is that in order to be
  able to compare the efficiency of filtering
  methods (and later also the sensitivity of risk
  measures to noise) we better get rid of other
  sources of uncertainty, like non-stationarity.
  This can be achieved by using artificial data
  where we have total control over the underlying
  stochastic process

97
The model-simulation approach

• Our strategy is to choose various model
  covariance matrices and generate N long
  simulated time series by them. Then we cut
  segments of length T from these time series, as
  if observing them on the market, and try to
  reconstruct the covariance matrices from them. We
  optimize a portfolio both with the true and
  with the observed covariance matrix and
  determine the measure .

98

• The models are chosen to mimic at least some of
  the characteristic features of real markets. Four
  simple models of slightly increasing complexity
  will be considered

99
Model 1 the unit matrix

• Spectrum
• ? 1, N-fold degenerate
  
• Noise will split this
• into band

1
0
C
100
Model 2 single-index

• Singlet ?11?(N-1) O(N)
• eigenvector (1,1,1,)
• ?2 1- ? O(1)
• (N-1) fold degenerate

?
1
101
The economic content of the single-index model

• return market return with
• standard deviation s
• The covariance matrix implied by the above
• The assumed structure reduces of parameters to
  N.
• If nothing depends on i then this is just the
  caricature Model 2.

102
Model 3 market sectors

singlet
- fold degenerate
1
This structure has also been studied by economists
- fold degenerate
103
Model 4 Semi-empirical

• Suppose we have very long time series (T) for
  many assets (N).
• Choose N lt N time series randomly and derive Cº
  from these data. Generate time series of length
  T ltlt T from Cº.
• The error due to T is much larger than that due
  to T.

104
How to generate time series?

• Given independent standard normal
• Given
• Define L (real, lower triangular) matrix such
  that
• (Cholesky)
• Get
• Empirical covariance matrix will be different
  from . For fixed N, and T ? ? ,

105

• We look for the minimal risk portfolio for both
  the true and the empirical covariances and
  determine the measure

106
We get numerically for Model 1 the following
scaling result
107
This confirms the expected scaling in N/T. The
corresponding analytic result

• can easily be derived for Model 1. It is valid
  within O(1/N) corrections also for more general
  models.

108
The same in a risk measurement context

• Given fixed wis. Choose to generate data.
  Measure from finite T time series.
• Calculate
• It can be shown that , for

109
Filtering

• Single-index filter
• Spectral decomposition of correlation matrix
• 
  to be chosen so as to
  preserve trace

110
Random matrix filter

• where to be chosen to preserve trace
  again
• and - the upper edge of
  the random band.

111
Covariance estimates

• after filtering we get
• and
• Silarly for the other models. We compare results
  on the following figures

112
Results for the market sectors model
113
Results for the semi-empirical model
114
Comments on the efficiency of filtering techniques

• Results depend on the model used for Cº.
• Market model still scales with T/N,
  singular at T/N1
• much improved (filtering
  technique matches structure), can go even below
  TN.
• Market sectors strong dependence on parameters
• RMT filtering outperforms the other two
• Semi-empirical data are scattered, RMT wins in
  most cases

115

• Filtering is very powerful in supressing noise,
  particularly when it matches the underlying
  structure.
• Is there information buried in the random band?
• With T increasing more and more eigenvalues
  crawl out of from below the upper random band
  edge.
• How to dig out information buried in the random
  band?
• Promising steps by various groups (Z. Burda, A.
  Görlich, A. Jarosz and J. Jurkiewicz,
  cond-mat/0305627 and Z. Burda and J. Jurkiewicz,
  cond-mat/0312496, Jagellonian University, Cracow
  Th. Guhr, Lund University P. Repetowicz, P.
  Richmond and S. Hutzler, Trinity College, Dublin
  G. Papp, Sz. Pafka, M.A. Nowak, and I.K.,
  Budapest and Cracow, etc.)

116
IV. BEYOND THE STATIONARY GAUSSIAN WORLD
117

• Real-life time series are neither stationary
  (volatility clustering, changing economic or
  legal environment, etc.), nor Gaussian (fat
  tails)
• For long-tailed distributions the variance is not
  an appropriate risk measure (even when it
  exists) minimizing the variance may actually
  increase rather than decrease risk.

118
One step towards reality Non-stationary case

• Volatility clustering ?ARCH, GARCH, integrated
  GARCH?EWMA (Exponentially Weighted Moving
  Averages) in RiskMetrics
• t actual time
• T window
• a attenuation factor ( Teff -1/log a), the
  rate of
• forgetting

119

• RiskMetrics aoptimal 0.94
• memory of a few months, total weight of data
  preceding the last 75 days is lt 1.
• Because of the short effective time cutoff,
  filtering is even more important than before.
  Carol Alexander applied standard principal
  component analysis.
• RMT helps choosing the number of principal
  components in an objective manner.
• For the application of RMT we need the upper edge
  of the random band for exponentially weighted
  random matrices

120
Exponentially weighted Wishart matrices
121
Sz. Pafka, M. Potters, and I.K. submitted to
Quantitative Finance, e-print cond-mat/0402573

• Density of eigenvalues
• where v is the solution to

122
Spectra of exponentially weighted and standard
Wishart matrices
123

• The RMT filtering wins again better than plain
  EWMA and better than plain MA.
• There is an optimal a (too long memory will
  include nonstationary effects, too short memory
  looses data).
• The optimal a (for N 100) is 0.996
  gtgtRiskMetrics a.

124
Alternative risk measures
125
Risk measures in practice VaR

• VaR (Value at Risk) is a high (95, or 99)
  quantile, a threshold beyond which a given
  fraction (5 or 1) of the statistical weight
  resides.
• Its merits (relative to the Greeks, e.g.)
• - universal can be applied to any portfolio
• - probabilistic content associated to the
  distribution
• - expressed in money
• Wide spread across the whole industry and
  regulation. Has been promoted from a diagnostic
  tool to a decision tool.
• Its lack of convexity promted search for coherence

126
Risk measures implied by regulation

• Banks are required to set aside capital as a
  cushion against risk
• Minimal capital requirements are fixed by
  international regulation (Basel I and II, Capital
  Adequacy Directive of the EEC) the magic 8
• Standard model vs. internal models
• Capital charges assigned to various positions in
  the standard model purport to cover the risk in
  those positions, therefore, they must be regarded
  as some kind of implied risk measures
• These measures are trying to mimic variance by
  piecewise linear approximants. They are quite
  arbitrary, sometimes concave and unstable

127
An example Specific risk of bonds
Specific ri
CAD, Annex I, 14 The capital requirement of
the specific risk (due to issuer) of bonds is
Iso-risk surface of the specific risk of bonds
128
Another example Foreign exchange
According to Annex III, 1, (CAD 1993, Official
Journal of the European Communities, L14, 1-26)
the capital requirement is given as
,
,
in terms of the gross
.
and the net position
The iso-risk surface of the foreign exchange
portfolio
129
Mean absolute deviation (MAD)
Some methodologies (e.g. Algorithmics) use the
mean absolute deviation rather than the standard
deviation to characterize the fluctuation of
portfolios. The objective function to minimize is
then
instead of
The iso-risk surfaces of MAD are polyhedra again.
130
Effect of noise on absolute deviation-optimized
portfolios
We generate artificial time series (say iid
normal), determine the true abs. deviation and
compare it to the measured one
We get
131
Noise sensitivity of MAD

• The result scales in T/N (same as with the
  variance). The optimal portfolio other things
  being equal - is more risky than in the
  variance-based optimization.
• Geometrical interpretation The level surfaces of
  the variance are ellipsoids.The optimal portfolio
  is found as the point where this risk-ellipsoid
  first touches the plane corresponding to the
  budget constraint. In the absolute deviation case
  the ellipsoid is replaced by a polyhedron, and
  the solution occurs at one of its corners. A
  small error in the specification of the
  polyhedron makes the solution jump to another
  corner, thereby increasing the fluctuation in the
  portfolio.

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Filtering for MAD (??)

• The absolute deviation-optimized portfolios can
  be filtered, by associating a covariance matrix
  with the time series, then filtering this matrix
  (by RMT, say), and generating a new time series
  via this reduced matrix. This (admittedly
  fortuitous) procedure significantly reduces the
  noise in the absolute deviation.
• Note that this risk measure can be used in the
  case of non-Gaussian portfolios as well.

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Expected shortfall (ES) optimization

• ES is the mean loss beyond a high threshold
  defined in probability (not in money). For
  continuous pdfs it is the same as the
  conditional expectation beyond the VaR quantile.
  ES is coherent (in the sense of Artzner et al.)
  and as such it is strongly promoted by a group of
  academics. In addition, Uryasev and Rockefellar
  have shown that its optimizaton can be reduced to
  linear programming for which extremely fast
  algorithms exist.
• ES-optimized portfolios tend to be much noisier
  than either of the previous ones. One reason is
  the instability related to the (piecewise) linear
  risk measure, the other is that a high quantile
  sacrifices most of the data.
• In addition, ES optimization is not always
  feasible!

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Before turning to the discussion of the
feasibility problem, let us compare the noise
sensitivity of the following risk measures
standard deviation, absolute deviation and
expected shortfall (at 95). For the sake of
comparison we use the same (Gaussian) input data
of length T for each, determine the minimal risk
portfolio under these risk measures and compare
the error due to noise.
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The next slides show

• plots of wi (porfolio weights) as a function of i
• display of q0 (ratio of risk of optimal portfolio
  determined from time series information vs full
  information)
• results show that the effect of estimation noise
  can be significant and more advanced risk
  measures are more demanding for information (in
  portfolio optimization context)

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• the suboptimality (q0) scales in T/N (for large N
  and T)

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Risk measures in risk measurement (as opposed to
portfolio optimization)

• in the context of risk measurement of given
  (fixed) portfolios, the estimation error is much
  smaller, it scales usually as
  independently of N !
• see next slides show the histogram of measured
  risk/true risk for different risk measures
  (T500,1000), the mean is 1 and the estimation
  error is usually within 5-10, i.e. negligible if
  compared to the portfolio optimization context

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The essence of the feasibility problem

• For T lt N, there is no solution to the portfolio
  optimization problem under any of the risk
  measures considered here.
• For T gt N, there always is a solution under
  the variance and MAD, even if it is bad for T not
  large enough. In contrast, under ES (and WL to be
  considered later), there may or may not be a
  solution for T gt N, depending on the sample. The
  probability of the existence of a solution goes
  to 1 only for T/N going to infinity.
• The problem does not appear if short selling is
  banned

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Feasibility of optimization under ES
Probability of the existence of an optimum under
CVaR. F is the standard normal distribution. Note
the scaling in N/vT.
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A pessimistic risk measure worst loss

• In order to better understand the feasibility
  problem, select the worst return in time and
  minimize this over the weights
• subject to
• This risk measure is coherent, one of Acerbis
  spectral measures.
• For T lt N there is no solution
• The existence of a solution for T gt N is a
  probabilistic issue again, depending on the time
  series sample

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Why is the existence of an optimum a random event?

• To get a feeling, consider NT2.
• The two planes
• intersect the plane of the budget constraint in
  two straight lines. If one of these is
  decreasing, the other is increasing with ,
  then there is a solution, if both increase or
  decrease, there is not. It is easy to see that
  for elliptical distributions the probability of
  there being a solution is ½.

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Probability of the feasibility of the minimax
problem

• For TgtN the probability of a solution (for an
  elliptical underlying pdf) is
• (The problem is isomorphic to some problems in
  operations research and random geometry.)
• For N and T large, p goes over into the error
  function and scales in N/vT.
• For T? infinity, p ?1.

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Probability of the existence of a solution under
maximum loss. F is the standard normal
distribution. Scaling is in N/vT again.
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Concluding remarks

• Due to the large number of assets in typical bank
  portfolios and the limited amount of data, noise
  is an all pervasive problem in portfolio theory.
• It can be efficiently filtered by a variety of
  techniques from portfolios optimized under
  variance.
• RMT is (one of) the latest of these filtering or
  dimensional reduction techniques. It is quite
  competitive with existing alternatives already,
  shows enhanced performance when applied in
  conjunction with extra information about the
  structure of the market, and holds great promise
  for resolving the spectrum under the upper edge
  of the random band.
• Unfortunately, variance is not an adequate risk
  measure for fat-tailed pdfs.
• Piecewise linear risk measures show instability
  (jumps) in a noisy environment.
• Risk measures focusing on the far tails show
  additional sensitivity to noise, due to loss of
  data.
• The two coherent measures we have studied display
  large sample-to-sample fluctuations and
  feasibility problems under noise. This may cast a
  shade of doubt on their applications.

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Some references

• Physica A 299, 305-310 (2001)
• European Physical Journal B 27, 277-280 (2002)
• Physica A 319, 487-494 (2003)
• Physica A 343, 623-634 (2004)
• submitted to Quantitative Finance, e-print
  cond-mat/0402573