Title: Noise sensitivity of portfolio selection under various risk measures
1Noise 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
2Contents
- 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
3Coworkers
- 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)
4I. PRELIMINARIES
5Preliminary 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
6The 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
7Some 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)
8A 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.
9Convexity
- 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.
10A 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.
11Portfolios
- 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 .
12Level 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.
13The 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.
16The 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.
17Benchmark 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!)
19The 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.
20Empirical 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.
21Fighting 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.
22An 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 ?!
23Laloux 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.
24Remarks 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.
25A 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)
26A 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)
28II. RANDOM MATRICES
29Origins 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.
30RMT
- 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
31Wigner 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
32Remarks 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
34N20
Elements of M are distributed normally
35N50
36N100
37N200
38N500
39N1000
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
41If the matrix elements are not centered
N1000
42N1000
43- For fat-tailed (but finite variance)
distributions the theorem still holds, but the
convergence is slow
44Sample from Student t (freedom3) distribution
N20
45N50
46N100
47N200
48N500
49N1000
50- There is a lot of fluctuation, level crossing,
random rotation of eigenvectors taking place in
the bulk
51Illustration of the instability of the
eigenvectors, although the distribution of the
eigenvalues is the same. Sample 1 Matrix elements
normally distributed N1000
52Sample 2
53Sample k
54Scalar 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.
56Illustration 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
57Sample 2
58Sample k
59Scalar 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.
60The 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
61Wishart 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
62Remarks
- 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
64N20 T/N2
The red curve is the limit Wishart distribution
65N50 T/N2
66N100 T/N2
67N200 T/N2
68N500 T/N2
69N1000 T/N2
70- Evolution of the distribution with T/N, with N
1000 fixed
71The quadratic limit
N1000
T/N1
72N1000 T/N1.2
73N1000 T/N2
74N1000 T/N3
75N1000 T/N5
76N1000 T/N10
77Scalar product of the eigenvectors belonging to
the j eigenvalue of the matrices for different
samples.
78Eigenvector components
- The same applies as in the Wigner case the
eigenvectors in the bulk are random, the one
outside is delocalized
79Distribution of the eigenvector components, if no
dominant eigenvalue exists.
80Market model
Underlying distribution is Wishart
N100 T/N2 Rho0.1
81N200 T/N2
82N500 T/N2
83N1000 T/N2
84Scalar product of the eigenvectors belonging to
the largest eigenvalue of the matrix. The larger
the first eigenvalue, the closer the scalar
products to 1.
85Distribution of the eigenvector components, if no
dominant eigenvalue exists.
N1000 T/N2 Rho0.1
86Distribution of the eigenvector components, if
one of the eigenvalues is not typical for random
matrixes.
N1000 T/N2 Rho0.1
87N1000 T/N2 Rho0.1
Distribution of the eigenvector components, if
one of the eigenvalues is not typical for random
matrixes.
88N1000 T/N2 Rho0.5
89N1000 T/N2 Rho0.9
The interval becomes narrower as correlation
increases.
90III. 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.
92Optimization 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
93Optimization
- 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.
94Risk 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
95A 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.
96We 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
97The 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
99Model 1 the unit matrix
- Spectrum
-
- ? 1, N-fold degenerate
- Noise will split this
- into band
1
0
C
100Model 2 single-index
- Singlet ?11?(N-1) O(N)
- eigenvector (1,1,1,)
- ?2 1- ? O(1)
- (N-1) fold degenerate
?
1
101The 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.
102Model 3 market sectors
singlet
- fold degenerate
1
This structure has also been studied by economists
- fold degenerate
103Model 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.
104How 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
106We get numerically for Model 1 the following
scaling result
107This 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.
108The 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
-
109Filtering
- Single-index filter
- Spectral decomposition of correlation matrix
-
to be chosen so as to
preserve trace
110Random matrix filter
-
- where to be chosen to preserve trace
again - and - the upper edge of
the random band.
111Covariance estimates
-
- after filtering we get
- and
- Silarly for the other models. We compare results
on the following figures
112Results for the market sectors model
113Results for the semi-empirical model
114Comments 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.)
116IV. 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.
118One 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
120Exponentially weighted Wishart matrices
121Sz. Pafka, M. Potters, and I.K. submitted to
Quantitative Finance, e-print cond-mat/0402573
- Density of eigenvalues
- where v is the solution to
122Spectra 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.
124Alternative risk measures
125Risk 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
126Risk 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
127An 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
128Another 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
129Mean 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.
130Effect 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
131Noise 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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133Filtering 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.
134Expected 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!
135Before 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.
136The 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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141- the suboptimality (q0) scales in T/N (for large N
and T)
142Risk 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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145The 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
146Feasibility 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.
147A 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
148Why 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.
149Probability 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.
150Probability 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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165Concluding 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.
166Some 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