Noise sensitivity of portfolio selection under various risk measures

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Noise sensitivity of portfolio selection under
various risk measures

• Imre Kondor
• Collegium Budapest and Eötvös University
• Application of Random Matrices to Economy and
  Other Complex Systems, Cracow, May 25-28, 2005

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Contents

• Background and motivation risk measures and
  noise
• The noise sensitivity of variance, random matrix
  filtering
• Convexity, coherence
• Risk measures in practice VaR, regulatory
  measures
• Mean absolute deviation (MAD), expected
  shortfall (ES) and worst loss (WL)
• The effect of noise
• The feasibility problem

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Coworkers

• Szilárd Pafka and Gábor Nagy (CIB Bank, Budapest)
• Richárd Karádi (Institute of Physics, Budapest
  University of Technology)
• Balázs Janecskó, András Szepessy, Tünde
  Ujvárosi (Raiffeisen Bank, Budapest)
• István Varga-Haszonits (Eötvös University,
  Budapest)

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Background

• Portfolio selection a tradeoff between risk and
  reward
• There is a more or less general agreement on what
  we mean by reward, but the status of risk
  measures is controversial
• For optimal portfolio selection we have to know
  what we want to optimize
• We also have to be able to implement the chosen
  risk measure in practice

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Motivation

• Expected returns are 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 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

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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. For
  long-tailed distributions this may be grossly
  misleading minimazing the variance may actually
  increase risk rather than decreasing it.
• Minimizing the variance of a portfolio without
  considering return does not, in general, make
  much sense. In index tracking, or benchmarking,
  however, this is precisely what one has to do.

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Variance 2

• Assume that the underlying process is close to
  normal and we want to determine the minimal
  variance portfolio.
• Then we need the covariance matrix. 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. This
  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.

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Variance 3

• Analytic results and simulations confirm this
  scaling. For N/T ?1-0, the error diverges.
• The covariance matrix is positive definite. The
  rank of the estimated covariance matrix is the
  smaller of N and T. For TltN the estimated
  covariance matrix develops zero eigenvalues and
  the task loses its meaning.
• For TgtN, however, the optimization problem always
  has a solution, even if it is not very precise
  for T close to N.

Scaling of the error due to noise
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Variance 4

• In order to reduce error, we have to reduce the
  effective dimension of the problem. This can be
  achieved by a number of filtering techniques
  existing in the literature. Since the root of the
  problem is lack of sufficient information, we
  have to inject knowledge from some external
  source (expert analysis, knowledge about the
  structure of market, etc.)
• This will introduce bias into the estimate. One
  has to chose a filtering method so that to be
  able to control the bias.

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Variance 5

• Factor analysis is one of the standard filtering
  methods. The choice of factors depends on
  economic intuition, their number is not
  determined by any objective criterion. The latter
  holds also for principal components.
• L.Laloux, P. Cizeau, J.-P. Bouchaud, M. Potters,
  PRL 83 1467 (1999) and Risk 12 No.3, 69 (1999),
  and V. Plerou, P. Gopikrishnan, B. Rosenow,
  L.A.N. Amaral, H.E. Stanley, PRL 83 1471 (1999),
  observed that the spectra of empirical covariance
  matrices are infested with noise and proposed a
  filtering method based on random matrix theory
  (RMT).

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Variance 6

• Their method can be regarded as a systematic
  version of principal component analysis, with an
  objective criterion on the number of principal
  components.
• Aspects of RMT-based filtering are the subject of
  a couple of papers at this conference and will
  not be further discussed in my contribution.
  Instead, I will look into the noise sensitivity
  of various other risk measures.

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Some elementary criteria on risk measures

• A risk measure is a quantitative characterization
  of our intuitive risk concept.
• Any reasonable risk measure must satisfy
• - convexity
• - invariance under addition of risk free asset
• - assigning zero risk to a zero position
• The appropriate choice may depend on the nature
  of data and on the context (investment, risk
  management, benchmarking, tracking, regulation,
  capital allocation)

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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.
  (Homogeneity is questionable for very large
  positions.)
• 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.

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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
• - cannot serve as a basis for a consistent limit
  system
• In short, a non-convex risk measure is not a risk
  measure at all.

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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.
• It is not convex!

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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

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An 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
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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.
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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
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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 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.
• CVaR-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, CVaR optimization is not always
  feasible!

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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 maximal 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

• 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 suffer
  from feasibility problems under noise.
• This may make them problematic in a risk
  management or regulatory context.

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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

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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. The whole covariance philosophy corresponds
  to a Gaussian world, so our first risk measure
  will be the variance.

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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

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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.

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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

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Dimensional reduction techniques in finance

• Impose 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.
• multi-index models Studies are based
• grouping by sectors on empirical data
• principal component analysis
• Baysian shrinkage estimators, etc.

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Contribution from econophysics

• Random matrices first appeared in a finance
  context in G. Galluccio, J.-P. Bouchaud, M.
  Potters, Physica A 259 449 (1998)
• Then came the two PRLs with the shocking result
  that most of the eigenvalues of s were just noise
• How come s is used in the industry at all ?

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Main source of error

• Lack of sufficient information
• input data N T ( N -
  size of portfolio,
• required info N N T - length of
  time series)
• Quality of estimate is measured by Q T/N
• Theoretically, we need Q gtgt 1.
• Practically, T is bounded by 500-1000 (2-4 yrs),
• whereas N can be several hundreds or thousands.
• Dimension (effective portfolio size) must be
  reduced

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Simplified portfolio optimization

• Go for the minimal risk portfolio (apart from the
  riskless asset)
• (constraint on return omitted)

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Measure of the effect of noise

• where w are the optimal weights corresponding
  to
• and , resp.

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Numerical results
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Analytical result

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

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Results for the market sectors model
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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

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• Filtering is very powerful in supressing noise,
  particularly when it matches the underlying
  structure.

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One step towards reality Non-stationary case

• Volatility clustering ?ARCH, GARCH, integrated
  GARCH?EWMA in RiskMetrics
• t actual time
• T window
• a attenuation factor ( Teff -1/log a)

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• RiskMetrics aoptimal 0.94
• memory of a few months, total weight of data
  preceding the last 75 days is lt 1.
• Filtering is useful also here. Carol Alexander
  applied standard principal component analysis.
  RMT helps choosing the number of principal
  components in an objective manner.
• We need upper edge of RMT spectrum for
  exponentially weighted random matrices

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Exponentially weighted Wishart matrices
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• Density of eigenvalues
• where v is the solution to

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Spectra of exponentially weighted and standard
Wishart matrices
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• 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.

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Model 1

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

1
0
C
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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.

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Model 2 single-index

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

?
1
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Model 4 Semi-empirical

• 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.

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Why do we use simulated data?

• In order to be able to compare the sensitivity of
  risk measures to noise (i.e. to lack of
  sufficient information) 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