# Noisy Portfolios

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Noisy PortfoliosImre KondorCollegium Budapest and Etvs UniversityEXYSTENCE Thematic Institute, Budapest, June 2, 2004

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ContentsBackground and motivationThe model/simulation approachFiltering and resultsBeyond the Gaussian case: non-stationarityBeyond the Gaussian case: absolute deviation and CVaR Complex portfolios: capital allocation, regulation, systemic risk

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CoworkersSzilrd Pafka (CIB Bank, Budapest, and, Department of Physics of Complex Systems, Etvs University)Marc Potters (Science & Finance)Richrd Kardi (Institute of Physics, Budapest University of Technology)Balzs Janecsk, Andrs Szepessy, Tnde Ujvrosi(Raiffeisen Bank, Budapest)

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BackgroundCorrelations of returns play central role in financial theory and applicationsThe covariance matrix is determined from empirical data it contains a lot of noiseMarkowitz portfolio theory suffered from the curse of dimensions from the very outsetEconomists have developed a number of dimension reduction techniques Recent contribution from random matrix theory (RMT)

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Our purposeTo develop a model/simulation-based approach to test and compare previous methods

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Initial motivation: a paradoxAccording to 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)there is a huge amount of noise in empirical covariance matrices, enough to make them uselessYet they are in widespread use and banks still survive

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Some key pointsLaloux 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 managementThere is a fundamental difference between the two kinds of uses of the covariance matrix for optimization resp. risk measurement.Where do people use for portfolio selection at all?- Goldman&Sachs technical document- tracking portfolios, benchmarking, shrinkage- capital allocation (EWRM)- hidden in softwares

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OptimizationWhen is used for optimization, we need a lot more information, because we are comparing different portfolios.To get optimal portfolio, we need to invert , 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 the problem of measurement error is much less serious

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Dimensional reduction techniques in financeImpose some structure on . This introduces bias, but beneficial effect of noise reduction may compensate for this.Examples:single-index models (s)All these help.multi-index modelsStudies are basedgrouping by sectorson empirical dataprincipal component analysisBaysian shrinkage estimators, etc.

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Contribution from econophysicsRandom 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 were just noiseHow come is used in the industry at all ?

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Market data are noisy themselves non-stationary processIf we want to assess noise reduction techniques wed better use well-controlled data, such as those generated by a known stochastic process Expected returns are hard to estimate from time seriesWe wanted to separate this part of the problem, too.

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Main source of errorLack of sufficient informationinput data: N T ( N - size of portfolio, required info: N N T - length of time series)Quality of estimate is measured by Q = T/NTheoretically, we need Q >> 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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Our approachChoose model correlation matrix CGenerate finite time series with CApply various filtering methods and compare their efficiencyModels:1. Unit matrix2. Single-index model3. Market + sectors model4. Semi-empirical (bootstrap) model

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Model 1Spectrum = 1, N-fold degenerateNoise will split thisinto band10C =

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Model 2: single-indexSinglet: 1=1+(N-1) ~ O(N)eigenvector: (1,1,1,)

2 = 1- ~ O(1)(N-1) fold degenerate

1

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The economic content of the single-index model return market return with standard deviation

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 3: market + sectorsThis structure has also been studied by economists1singlet- fold degenerate- fold degenerate

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Model 4: Semi-empirical Very long time series (T) for many assets (N).Choose N < N time series randomly and derive C from these data. Generate time series of length T >RiskMetrics .

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Absolute deviation as a risk measureSome methodologies (e.g. Algorithmics) choose the absolute deviation rather than the standard deviation to characterize the fluctuation of portfolios. The objective function to minimize is then:

instead of

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

We get:

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The result scales in T/N again. The optimal portfolio is more risky than in the variance-based optimization.Geometrical interpretation: in the original Markowitz case the optimal portfolio is found as the point where the ellipsoid corresponding to a fixed variance first touches the plane corresponding to the budget constraint. In the abs. 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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The abs. deviation-based portfolios can be filtered again, by associating a covariance matrix with the time series, then filtering this matrix, and generating a new time series via this reduced matrix. This procedure significantly reduces the noise in the abs. deviation.Note that this risk measure can be used in the case of non-Gaussian portfolios as well.

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CVaR optimizationCVaR is the conditional expectation beyond the VaR quantile. For continuous pdfs it is a coherent risk measure and as such it is strongly promoted by academics. In addition, Uryasev showed that its optimizaton can be reduced to linear programming for which extremely fast algorithms exist. CVaR-optimized portfolios tend to be much noisier than any of the previous ones. One reason is the instability related to the linear risk measure, the other is that a high quantile sacrifices most of the data.

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Generalized portfoliosCapital allocation problems require that a whole banking group be considered as a single large portfolio. Unconventional correlations and constraints may arise in this context.Regulatory capital requirements impose unusual (nonlinear, incoherent, sometimes non-convex) constraints on the trading book and, with Basel II entering into force in 2006-07, on the banking book as well.Studies of systemic risk have to consider huge portfolios, consisting of all the banks of a country or the whole world.

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Some referencesPhysica A 299, 305-310 (2001) European Physical Journal B 27, 277-280 (2002)Physica A 319, 487-494 (2003)To appear in Physica A, e-print: cond-mat/0305475submitted to Quantitative Finance, e-print: cond-mat/0402573

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