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Imre Kondor Collegium Budapest and Eötvös University, Budapest, Hungary Risk Measurement and Management, Rome, June 9-17, 2005. Noise sensitivity of portfolio selection under various risk measures. Contents. I. Preliminaries

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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

Noise sensitivity of portfolio selection under various risk measures


Contents
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-theory-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


Coworkers
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 Procter&Gamble)

  • Balázs Janecskó, András Szepessy, Tünde Ujvárosi (Raiffeisen Bank, Budapest)

  • István Varga-Haszonits (Eötvös University, Budapest)



Preliminary considerations
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


The problem of noise
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


Some elementary criteria on risk measures
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)


A more elaborate set of risk measure axioms
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.


Convexity
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.


A classical risk measure the variance
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.


Portfolios
Portfolios

  • Consider a linear combination of returns

    with weights : . The weights add up to unity: . The portfolio’s expectation value is: with variance: ,

    where is the covariance matrix, and the standard deviation of return .


Level surfaces of risk measured in variance
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.


The markowitz problem
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


Noise sensitivity of portfolio selection under various risk measures

  • 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.


Noise sensitivity of portfolio selection under various risk measures

  • 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.


The minimal risk portfolio
The minimal risk portfolio have mostly positive elements.

  • 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.


Benchmark tracking
Benchmark tracking have mostly positive elements.

  • 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”:


Noise sensitivity of portfolio selection under various risk measures


The weights of the minimal risk portfolio
The weights of the minimal risk portfolio with returns relative to the benchmark:

  • 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.


Empirical covariance matrices
Empirical covariance matrices with returns relative to the benchmark:

  • The covariance matrix has to be determined from measurements on the market. From the returns observed at time twe 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 <<T. Bank portfolios may contain hundreds of assets, and it is hardly meaningful to use time series longer than 4 years (T~1000). Therefore, N/T << 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.


Fighting the curse of dimensions
Fighting the curse of dimensions with returns relative to the benchmark:

  • 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 σ. This introduces bias, but beneficial effect of noise reduction may compensate for this.

  • Examples:

    • single-index models (β’s) 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.


An intriguing observation
An intriguing observation with returns relative to the benchmark:

  • L.Laloux, P. Cizeau, J.-P. Bouchaud, M. Potters, PRL83 1467 (1999) and Risk12 No.3, 69 (1999)

    and to

    V. Plerou, P. Gopikrishnan, B. Rosenow, L.A.N. Amaral, H.E. Stanley, PRL83 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 ?!


Laloux et al 1999
Laloux et al. 1999 with returns relative to the benchmark:

The spectrum of the covariance matrix obtained from the time

series of S&P 500

with N=406, T=1308, 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.


Remarks on the paradox
Remarks on the paradox with returns relative to the benchmark:

  • 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.


A historical remark
A historical remark with returns relative to the benchmark:

  • Random matrices first appeared in a finance context in G. Galluccio, J.-P. Bouchaud, M. Potters, PhysicaA 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)


A filtering procedure suggested by rmt
A filtering procedure suggested by RMT with returns relative to the benchmark:

  • 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.


Noise sensitivity of portfolio selection under various risk measures

  • 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)


Ii random matrices
II. RANDOM MATRICES the „market”, the other big eigenvalues correspond to the main industrial sectors.


Origins of random matrix theory rmt
Origins of random matrix theory (RMT) the „market”, the other big eigenvalues correspond to the main industrial sectors.

  • Wigner, Dyson 1950’s

  • 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.


Noise sensitivity of portfolio selection under various risk measures
RMT the „market”, the other big eigenvalues correspond to the main industrial sectors.

  • 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


Wigner semi circle law
Wigner semi-circle law the „market”, the other big eigenvalues correspond to the main industrial sectors.

  • 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→∞ with prob. 1:

    ,

    , otherwise


Remarks on the semi circle law
Remarks on the semi-circle law the „market”, the other big eigenvalues correspond to the main industrial sectors.

  • 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


Noise sensitivity of portfolio selection under various risk measures

Convergence to the semi-circle as the „market”, the other big eigenvalues correspond to the main industrial sectors.N increases


Noise sensitivity of portfolio selection under various risk measures

N the „market”, the other big eigenvalues correspond to the main industrial sectors.=20

Elements of M are distributed normally


Noise sensitivity of portfolio selection under various risk measures

N the „market”, the other big eigenvalues correspond to the main industrial sectors.=50


Noise sensitivity of portfolio selection under various risk measures

N the „market”, the other big eigenvalues correspond to the main industrial sectors.=100


Noise sensitivity of portfolio selection under various risk measures

N the „market”, the other big eigenvalues correspond to the main industrial sectors.=200


Noise sensitivity of portfolio selection under various risk measures

N the „market”, the other big eigenvalues correspond to the main industrial sectors.=500


Noise sensitivity of portfolio selection under various risk measures

N the „market”, the other big eigenvalues correspond to the main industrial sectors.=1000


Noise sensitivity of portfolio selection under various risk measures

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


Noise sensitivity of portfolio selection under various risk measures

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

N=1000


Noise sensitivity of portfolio selection under various risk measures

N mean, one large eigenvalue breaks away, the rest stay in the semi-circle=1000


Noise sensitivity of portfolio selection under various risk measures

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


Noise sensitivity of portfolio selection under various risk measures

Sample from Student t (freedom=3) distribution theorem still holds, but the convergence is slow

N=20


Noise sensitivity of portfolio selection under various risk measures

N theorem still holds, but the convergence is slow=50


Noise sensitivity of portfolio selection under various risk measures

N theorem still holds, but the convergence is slow=100


Noise sensitivity of portfolio selection under various risk measures

N theorem still holds, but the convergence is slow=200


Noise sensitivity of portfolio selection under various risk measures

N theorem still holds, but the convergence is slow=500


Noise sensitivity of portfolio selection under various risk measures

N theorem still holds, but the convergence is slow=1000


Noise sensitivity of portfolio selection under various risk measures

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


Noise sensitivity of portfolio selection under various risk measures

Illustration of the rotation of eigenvectors taking place in the bulkinstability of the eigenvectors, although the distribution of the eigenvalues is the same.

Sample 1

Matrix elements normally distributed

N=1000


Noise sensitivity of portfolio selection under various risk measures

Sample 2 rotation of eigenvectors taking place in the bulk


Noise sensitivity of portfolio selection under various risk measures

Sample k rotation of eigenvectors taking place in the bulk



Noise sensitivity of portfolio selection under various risk measures

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


Noise sensitivity of portfolio selection under various risk measures

Illustration of the there is one) is much more stable. The larger the eigenvalue, the more so. stability of the largest eigenvector

Sample 1

Matrix elements are normally distributed, but the sum of the elements in the rows is not zero.

N=1000


Noise sensitivity of portfolio selection under various risk measures

Sample 2 there is one) is much more stable. The larger the eigenvalue, the more so.


Noise sensitivity of portfolio selection under various risk measures

Sample k there is one) is much more stable. The larger the eigenvalue, the more so.


Noise sensitivity of portfolio selection under various risk measures

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.


The eigenvector components
The eigenvector components eigenvalue of the matrix. The larger the first eigenvalue, the closer the scalar products to 1 or -1.

  • 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


Wishart matrices random sample covariance matrices
Wishart matrices – random sample covariance matrices eigenvalue of the matrix. The larger the first eigenvalue, the closer the scalar products to 1 or -1.

  • Let Aij NxT matrix with i.i.d. elements (0 mean and finite second moment)

  • σ =1/T AA’ where A’ is the transpose

  • Wishart or Marchenko-Pastur spectrum (eigenvalue distribution):

    where


Remarks
Remarks eigenvalue of the matrix. The larger the first eigenvalue, the closer the scalar products to 1 or -1.

  • The theorem also holds when E{A}is of finite rank

  • The assumption that the entries are identically distributed is not necessary

  • If T < 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


Noise sensitivity of portfolio selection under various risk measures

Convergence in eigenvalue of the matrix. The larger the first eigenvalue, the closer the scalar products to 1 or -1.N, with T/N = 2 fixed


Noise sensitivity of portfolio selection under various risk measures

N eigenvalue of the matrix. The larger the first eigenvalue, the closer the scalar products to 1 or -1.=20

T/N=2

The red curve is the limit Wishart distribution


Noise sensitivity of portfolio selection under various risk measures

N eigenvalue of the matrix. The larger the first eigenvalue, the closer the scalar products to 1 or -1.=50

T/N=2


Noise sensitivity of portfolio selection under various risk measures

N eigenvalue of the matrix. The larger the first eigenvalue, the closer the scalar products to 1 or -1.=100

T/N=2


Noise sensitivity of portfolio selection under various risk measures

N eigenvalue of the matrix. The larger the first eigenvalue, the closer the scalar products to 1 or -1.=200

T/N=2


Noise sensitivity of portfolio selection under various risk measures

N eigenvalue of the matrix. The larger the first eigenvalue, the closer the scalar products to 1 or -1.=500

T/N=2


Noise sensitivity of portfolio selection under various risk measures

N eigenvalue of the matrix. The larger the first eigenvalue, the closer the scalar products to 1 or -1.=1000

T/N=2


Noise sensitivity of portfolio selection under various risk measures

Evolution of the distribution with T/N, with N = 1000 fixed eigenvalue of the matrix. The larger the first eigenvalue, the closer the scalar products to 1 or -1.


Noise sensitivity of portfolio selection under various risk measures

The quadratic limit eigenvalue of the matrix. The larger the first eigenvalue, the closer the scalar products to 1 or -1.N=1000

T/N=1


Noise sensitivity of portfolio selection under various risk measures

N eigenvalue of the matrix. The larger the first eigenvalue, the closer the scalar products to 1 or -1.=1000

T/N=1.2


Noise sensitivity of portfolio selection under various risk measures

N eigenvalue of the matrix. The larger the first eigenvalue, the closer the scalar products to 1 or -1.=1000

T/N=2


Noise sensitivity of portfolio selection under various risk measures

N eigenvalue of the matrix. The larger the first eigenvalue, the closer the scalar products to 1 or -1.=1000

T/N=3


Noise sensitivity of portfolio selection under various risk measures

N eigenvalue of the matrix. The larger the first eigenvalue, the closer the scalar products to 1 or -1.=1000

T/N=5


Noise sensitivity of portfolio selection under various risk measures

N eigenvalue of the matrix. The larger the first eigenvalue, the closer the scalar products to 1 or -1.=1000

T/N=10


Noise sensitivity of portfolio selection under various risk measures

Scalar product of the eigenvectors belonging to the j eigenvalue of the matrices for different samples.


Eigenvector components
Eigenvector components eigenvalue of the matrices for different samples.

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



Noise sensitivity of portfolio selection under various risk measures

Market model eigenvalue exists.

Underlying distribution is Wishart

N=100

T/N=2

Rho=0.1


Noise sensitivity of portfolio selection under various risk measures

N eigenvalue exists. =200

T/N=2


Noise sensitivity of portfolio selection under various risk measures

N eigenvalue exists. =500

T/N=2


Noise sensitivity of portfolio selection under various risk measures

N eigenvalue exists. =1000

T/N=2


Noise sensitivity of portfolio selection under various risk measures

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.


Noise sensitivity of portfolio selection under various risk measures

Distribution of the eigenvector components, if no dominant eigenvalue exists.

N=1000

T/N=2

Rho=0.1


Noise sensitivity of portfolio selection under various risk measures

Distribution of the eigenvector components, if one of the eigenvalues is not typical for random matrixes.

N=1000

T/N=2

Rho=0.1


Noise sensitivity of portfolio selection under various risk measures

Distribution of the eigenvector components, if one of the eigenvalues is not typical for random matrixes.

N=1000

T/N=2

Rho=0.1


Noise sensitivity of portfolio selection under various risk measures

N eigenvalues is not typical for random matrixes. =1000

T/N=2

Rho=0.5


Noise sensitivity of portfolio selection under various risk measures

The interval becomes narrower as correlation increases. eigenvalues is not typical for random matrixes.

N=1000

T/N=2

Rho=0.9


Iii filtering of normal portfolios
III. FILTERING OF NORMAL PORTFOLIOS eigenvalues is not typical for random matrixes.


Some key points
Some key points eigenvalues is not typical for random matrixes.

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.


Optimization vs risk management
Optimization vs. risk management eigenvalues is not typical for random matrixes.

  • There is a fundamental difference between the two kinds of uses of the covariance matrix σ for optimization resp. risk measurement.

  • Where do people use σ for portfolio selection at all?

    - Goldman&Sachs technical document

    - tracking portfolios, benchmarking, shrinkage

    - capital allocation (EWRM)

    - hidden in softwares


Optimization
Optimization eigenvalues is not typical for random matrixes.

  • When σ is used for optimization, we need a lot more information, because we are comparing different portfolios.

  • To get optimal portfolio, we need to invert σ, and as it has small eigenvalues, error gets amplified.


Risk measurement management regulatory capital calculation
Risk measurement – management - regulatory capital calculation

Assessing risk in a given portfolio – no need to invert σ – the problemofmeasurement error is much less serious


A measure of the effect of noise
A measure of the effect of noise calculation

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.


W e will mostly use simulated data
W calculatione 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


The model simulation approach
The model-simulation approach calculation

  • Our strategy is to choose various model covariancematrices 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 .


Noise sensitivity of portfolio selection under various risk measures

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


Model 1 the unit matrix
Model 1: the unit matrix characteristic features of real markets. Four simple models of slightly increasing complexity will be considered

Spectrum

λ = 1, N-fold degenerate

Noise will split this

into band

1

0

C=


Model 2 single index
Model 2: single-index characteristic features of real markets. Four simple models of slightly increasing complexity will be considered

Singlet: λ1=1+ρ(N-1) ~ O(N)

eigenvector: (1,1,1,…)

λ2 = 1- ρ~ O(1)

(N-1) – fold degenerate

ρ

1


The economic content of the single index model
The economic content of the single-index model characteristic features of real markets. Four simple models of slightly increasing complexity will be considered

returnmarket return with

standard deviation σ

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.


Model 3 market sectors
Model 3: market + sectors characteristic features of real markets. Four simple models of slightly increasing complexity will be considered

singlet

- fold degenerate

1

This structure has also been studied by economists

- fold degenerate


Model 4 s emi empirical
Model 4: characteristic features of real markets. Four simple models of slightly increasing complexity will be consideredSemi-empirical

Suppose we have very long time series (T’) for many assets (N’).

Choose N<N’ time series randomly and derive Cº from these data. Generate time series of length T<<T’ from Cº.

The error due to T is much larger than that due to T’.


How to generate time series
How to generate time series? characteristic features of real markets. Four simple models of slightly increasing complexity will be considered

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 → ,


Noise sensitivity of portfolio selection under various risk measures

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


We get numerically for model 1 the following scaling result
We get numerically for Model 1 the following scaling result the empirical covariances and determine the measure


This confirms the expected scaling in n t the corresponding analytic result
This confirms the expected scaling in the empirical covariances and determine the measureN/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.


The same in a risk measurement context
The same in a risk measurement context the empirical covariances and determine the measure

Given fixed wi’s. Choose to generate data. Measure from finite T time series.

Calculate

It can be shown that , for


Filtering
Filtering the empirical covariances and determine the measure

Single-index filter:

Spectral decomposition of correlation matrix:

to be chosen so as topreserve trace


Random matrix filter
Random matrix filter the empirical covariances and determine the measure

whereto be chosen to preservetrace again

and - the upper edge of the random band.


Covariance estimates
Covariance estimates the empirical covariances and determine the measure

after filtering we get

and

Silarly for the other models. We compare results on the following figures


Results for the market sectors model
Results for the market + sectors model the empirical covariances and determine the measure


Results for the semi empirical model
Results for the semi-empirical model the empirical covariances and determine the measure


Comments on the efficiency of filtering techniques
Comments on the efficiency of filtering techniques the empirical covariances and determine the measure

  • Results depend on the model used for Cº.

  • Market model: still scales withT/N, singular at T/N=1

    much improved (filtering techniquematches structure), can go even below T=N.

  • Market + sectors: strong dependence on parameters

    RMT filtering outperforms the other two

  • Semi-empirical: data are scattered, RMT wins in most cases


Noise sensitivity of portfolio selection under various risk measures

  • 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.)


Iv beyond the stationary gaussian world
IV. BEYOND THE STATIONARY GAUSSIAN WORLD when it matches the underlying structure.


Noise sensitivity of portfolio selection under various risk measures

  • 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.


One step towards reality non stationary case
One step towards reality: Non-stationary case clustering, changing economic or legal environment, etc.), nor Gaussian (fat tails)

  • Volatility clustering →ARCH, GARCH, integrated GARCH→EWMA (Exponentially Weighted Moving Averages) in RiskMetrics

    t – actual time

    T – window

    α – attenuation factor ( Teff~-1/log α), the rate of

    forgetting


Noise sensitivity of portfolio selection under various risk measures

  • RiskMetrics: clustering, changing economic or legal environment, etc.), nor Gaussian (fat tails)αoptimal= 0.94

    memory of a few months, total weight of data preceding the last 75 days is < 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


Exponentially weighted wishart matrices
Exponentially weighted Wishart matrices clustering, changing economic or legal environment, etc.), nor Gaussian (fat tails)


Sz pafka m potters and i k submitted to quantitative finance e print cond mat 0402573
Sz. Pafka, M. Potters, and I.K.: clustering, changing economic or legal environment, etc.), nor Gaussian (fat tails)submitted to Quantitative Finance, e-print: cond-mat/0402573

Density of eigenvalues:

where v is the solution to:



Noise sensitivity of portfolio selection under various risk measures


Alternative risk measures
Alternative risk measures better than plain MA.


Risk measures in practice var
Risk measures in practice: VaR better than plain MA.

  • 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


Risk measures implied by regulation
Risk measures implied by regulation better than plain MA.

  • 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


An example specific risk of bonds
An example: better than plain MA.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


An other example foreign exchange
An better than plain MA.other 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


Mean absolute deviation mad
Mean absolute deviation (MAD) better than plain MA.

Some methodologies (e.g. Algorithmics) use the mean absolute deviationrather 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.


Effect of noise on absolute deviation optimized portfolios
Effect of noise on absolute deviation-optimized portfolios better than plain MA.

We generate artificial time series (say iid normal), determine the true abs. deviation and compare it to the „measured” one:

We get:


Noise sensitivity of mad
Noise sensitivity of MAD better than plain MA.

  • 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.


Filtering for mad
Filtering for MAD (? better than plain MA.?)

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.


Expected shortfall es optimization
Expected shortfall (ES) optimization better than plain MA.

ES is the mean loss beyond a high threshold defined in probability (not in money). For continuous pdf’s 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!


Noise sensitivity of portfolio selection under various risk measures

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.


The next slides show
The next slides show let us

  • 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)


Noise sensitivity of portfolio selection under various risk measures


Risk measures in risk measurement as opposed to portfolio optimization
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 (T=500,1000), the mean is 1 and the estimation error is usually within 5-10%, i.e. negligible if compared to the portfolio optimization context


The essence of the feasibility problem
The essence of the feasibility problem optimization)

  • For T < N, there is no solution to the portfolio optimization problem under any of the risk measures considered here.

    For T > 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 > 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


Noise sensitivity of portfolio selection under various risk measures

Feasibility of optimization under ES optimization)

Probability of the existence of an optimum under CVaR.

F is the standard normal distribution. Note the scaling in N/√T.


A pessimistic risk measure worst loss
A pessimistic risk measure: worst loss optimization)

  • 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 Acerbi’s spectral measures.

  • For T < N there is no solution

  • The existence of a solution for T > N is a probabilistic issue again, depending on the time series sample


Why is the existence of an optimum a random event
Why is the existence of an optimum a random event? optimization)

  • To get a feeling, consider N=T=2.

  • 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.


Probability of the feasibility of the minimax problem
Probability of the feasibility of the minimax problem optimization)

  • For T>N 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/√T.

  • For T→ infinity, p →1.


Noise sensitivity of portfolio selection under various risk measures

Probability of the existence of a solution under maximum loss.

F is the standard normal distribution. Scaling is in N/√T again.


Concluding remarks
Concluding remarks loss.

  • 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 pdf’s.

  • 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 displaylarge sample-to-sample fluctuations and feasibility problems under noise.This may cast a shade of doubt on their applications.


Some references
Some references loss.

  • 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