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1 Noisy Portfolios Imre Kondor Collegium Budapest and Eötvös University SPHINX Econophysics Workshop, Oxford, 27-29 September, 2004

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Collegium Budapest 2 Contents 1.Background and motivation 2.The model/simulation approach 3.Filtering and results 4.Beyond the Gaussian case: non-stationarity 5.Beyond the variance as risk measure: absolute deviation, CVaR (expected shortfall), and maximal loss 6.The minimax problem

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Collegium Budapest 3 Coworkers Szilárd Pafka and Gábor Nagy (CIB Bank, Budapest) Marc Potters (Science & Finance) Richárd Karádi (Institute of Physics, Budapest University of Technology)

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Collegium Budapest 4 Background Correlations of returns play central role in financial theory and applications The covariance matrix is determined from empirical data – it contains a lot of noise Markowitz portfolio theory suffered from the curse of dimensions from the very outset Economists have developed a number of dimension reduction techniques Recent contribution from random matrix theory (RMT)

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Collegium Budapest 5 Our purpose To develop a model/simulation-based approach to test and compare previous methods

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Collegium Budapest 6 Initial motivation: a paradox According 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 useless Yet they are in widespread use and banks still survive

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Collegium Budapest 7 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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Collegium Budapest 8 Optimization vs. risk management 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

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Collegium Budapest 9 Optimization 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.

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Collegium Budapest 10 Risk measurement – management - regulatory capital calculation Assessing risk in a given portfolio – no need to invert σ – the problem of measurement error is much less serious

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Collegium Budapest 11 Dimensional reduction techniques in finance Impose some structure on σ. This introduces bias, but beneficial effect of noise reduction may compensate for this. Examples: -single-index models (βs)All these help. -multi-index modelsStudies are based -grouping by sectorson empirical data -principal component analysis -Baysian shrinkage estimators, etc.

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Collegium Budapest 12 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 σ were just noise How come σ is used in the industry at all ?

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Collegium Budapest 13 Market data are noisy themselves – non-stationary process If 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 series We wanted to separate this part of the problem, too.

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Collegium Budapest 14 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 >> 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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Collegium Budapest 15 Our approach Choose model correlation matrix Cº Generate finite time series with Cº Apply various filtering methods and compare their efficiency Models: 1. Unit matrix 2. Single-index model 3. Market + sectors model 4. Semi-empirical (bootstrap) model

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Collegium Budapest 16 Simplified portfolio optimization Go for the minimal risk portfolio (apart from the riskless asset) (constraint on return omitted)

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Collegium Budapest 17 Measure of the effect of noise where w* are the optimal weights corresponding to and, resp.

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Collegium Budapest 18 Numerical results before filtering

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Collegium Budapest 19 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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Collegium Budapest 20 Results for the market + sectors model

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Collegium Budapest 21 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/N=1 much improved (filtering technique matches 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

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Collegium Budapest 22 Filtering is very powerful in supressing noise, particularly when it matches the underlying structure.

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Collegium Budapest 23 One step towards reality: Non- stationary case Volatility clustering ARCH, GARCH, integrated GARCHEWMA in RiskMetrics (finite memory) t – actual time T – window α – attenuation factor ( T eff ~ -1/log α )

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Collegium Budapest 24 RiskMetrics: α optimal = 0.94 memory of a few months, total weight of data preceding the last 75 days is < 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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Collegium Budapest 25 Exponentially weighted Wishart matrices

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Collegium Budapest 26 Density of eigenvalues: where v is the solution to:

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Collegium Budapest 27 Spectra of exponentially weighted and standard Wishart matrices

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Collegium Budapest 28 The RMT filtering wins again – better than plain EWMA and better than plain MA. There is an optimal α (too long memory will include nonstationary effects, too short memory looses data). The optimal α (for N= 100 ) is 0.996 >>RiskMetrics α.

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Collegium Budapest 29 Absolute deviation as a risk measure Some 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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Collegium Budapest 30 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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Collegium Budapest 31 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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Collegium Budapest 32 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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Collegium Budapest 33 CVaR optimization CVaR 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 a group of 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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Collegium Budapest 34 A pessimistic risk measure: maximal loss Select the worst return in time and minimize this over the weights: This risk measure is subadditive and homogeneous, hence convex. Budget constraint: For T < N there is no solution The existence of a solution for T > N is a probabilistic issue, depending on the time series sample

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Collegium Budapest 35 Why is the existence of an optimum a random event? 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 elliptically distributed s the probability of there being a solution is ½.

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Collegium Budapest 36 Conjectured probability distribution for the existence of an optimum For T>N the probability of a solution is conjectured to be - cumulative binomial distribution For T infinity, p 1.

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Collegium Budapest 37 Probability of the existence of a solution under maximum loss. F is the standard normal distribution.

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Collegium Budapest 38 Probability of the existence of an optimum under CVaR. F is the standard normal distribution. The optimization of CVaR behaves similarly

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Collegium Budapest 39 Some references Physica 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/0305475 submitted to Quantitative Finance, e-print: cond-mat/0402573

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Collegium Budapest 40 Model 1 Spectrum λ = 1, N-fold degenerate Noise will split this into band 1 0 C =C =

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Collegium Budapest 41 The economic content of the single-index model return market 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.

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Collegium Budapest 42 Model 2: single-index Singlet: λ 1 =1+ρ(N-1) ~ O(N) eigenvector: (1,1,1,…) λ 2 = 1- ρ ~ O(1) (N-1) – fold degenerate ρ 1

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Collegium Budapest 43 Model 4: Semi-empirical 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.

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Collegium Budapest 44 How to generate time series? Given independent standard normal Given Define L (real, lower triangular) matrix such that (Cholesky) Get: Empirical covariance matrix will be different from. For fixed N, and T,

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Collegium Budapest 45 Model 3: market + sectors This structure has also been studied by economists 1 singlet - fold degenerate

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Collegium Budapest 46 Risk measurement Given fixed w i s. Choose to generate data. Measure from finite T time series. Calculate It can be shown, for.

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Collegium Budapest 47 Filtering Single-index filter: Spectral decomposition of correlation matrix: to be chosen so as to preserve trace

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Collegium Budapest 48 Random matrix filter to be chosen to preserve trace again where and - the upper edge of the random band.

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Collegium Budapest 49 Covariance estimates after filtering we get and We compare these on the following figure

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Collegium Budapest 50 Results for the single-index (market) model

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Collegium Budapest 51 Results for the semi-empirical model

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