48
Noisy Portfolios Imre Kondor Collegium Budapest and Eötvös University EXYSTENCE Thematic Institute, Budapest, June 2, 2004

Noisy Portfolios

  • Upload
    enid

  • View
    54

  • Download
    0

Embed Size (px)

DESCRIPTION

Noisy Portfolios. Imre Kondor Collegium Budapest and Eötvös University EXYSTENCE Thematic Institute, Budapest, June 2, 2004. Contents. Background and motivation The model/simulation approach Filtering and results Beyond the Gaussian case: non-stationarity - PowerPoint PPT Presentation

Citation preview

Page 1: Noisy Portfolios

Noisy Portfolios

Imre Kondor

Collegium Budapest and Eötvös University

EXYSTENCE Thematic Institute, Budapest, June 2, 2004

Page 2: Noisy Portfolios

Collegium Budapest

Contents

1. Background and motivation2. The model/simulation approach3. Filtering and results4. Beyond the Gaussian case: non-stationarity5. Beyond the Gaussian case: absolute

deviation and CVaR 6. Complex portfolios: capital allocation,

regulation, systemic risk

Page 3: Noisy Portfolios

Collegium Budapest

Coworkers

• Szilárd Pafka (CIB Bank, Budapest, and, Department of Physics of Complex Systems, Eötvös University)

• Marc Potters (Science & Finance)• Richárd Karádi (Institute of Physics, Budapest

University of Technology)• Balázs Janecskó, András Szepessy, Tünde Ujvárosi

(Raiffeisen Bank, Budapest)

Page 4: Noisy Portfolios

Collegium Budapest

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)

Page 5: Noisy Portfolios

Collegium Budapest

Our purpose

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

Page 6: Noisy Portfolios

Collegium Budapest

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

Page 7: Noisy Portfolios

Collegium Budapest

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.

Page 8: Noisy Portfolios

Collegium Budapest

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

Page 9: Noisy Portfolios

Collegium Budapest

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.

Page 10: Noisy Portfolios

Collegium Budapest

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

Page 11: Noisy Portfolios

Collegium Budapest

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 (β’s) All these help.- multi-index models Studies are based- grouping by sectors on empirical data- principal component analysis- Baysian shrinkage estimators, etc.

Page 12: Noisy Portfolios

Collegium Budapest

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 PRL’s with the shocking result that most of the eigenvalues of σ were just noise

• How come σ is used in the industry at all ?

Page 13: Noisy Portfolios

Collegium Budapest

Market data are noisy themselves –

non-stationary processIf we want to assess noise reduction techniques we’d 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.

Page 14: Noisy Portfolios

Collegium Budapest

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

Page 15: Noisy Portfolios

Collegium Budapest

Our approach

• Choose model correlation matrix Cº• Generate finite time series with Cº• Apply various filtering methods and compare

their efficiency• Models:

1. Unit matrix2. Single-index model3. Market + sectors model4. Semi-empirical (bootstrap) model

Page 16: Noisy Portfolios

Collegium Budapest

Model 1

Spectrum

λ = 1,

N-fold degenerate

Noise will split this

into band

1

0

C =

Page 17: Noisy Portfolios

Collegium Budapest

Model 2: single-index

Singlet: λ1=1+ρ(N-1) ~ O(N)

eigenvector: (1,1,1,…)

λ2 = 1- ρ ~ O(1)

(N-1) – fold degenerate

ρ

1

Page 18: Noisy Portfolios

Collegium Budapest

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.

iMiii rr

0

0

,0

i

Mi

ji

r

ji

22

iijMjiij

Page 19: Noisy Portfolios

Collegium Budapest

Model 3: market + sectors

This structure has also been studied by economists

1

1

0)(~)()1(1 10111 NONNN

)(~)1(1 110112 NONN singlet

1

1N

N - fold degenerate

)1(~1 13 O

1N

NN - fold degenerate

Page 20: Noisy Portfolios

Collegium Budapest

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’.

Page 21: Noisy Portfolios

Collegium Budapest

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 → ,

itx

TLL)0(σ

)0(σ

j

jtijit xLy

)0()1( σσ

)0(σ

Page 22: Noisy Portfolios

Collegium Budapest

Simplified portfolio optimization

• Go for the minimal risk portfolio (apart from the riskless asset)

(constaint on return omitted)

ij

jijip ww min2

i

iw 1

jkjk

jij

iw)(

)(

1

1

*

Page 23: Noisy Portfolios

Collegium Budapest

Measure of the effect of noise

where w* are the optimal weights corresponding to

and , resp.

ijjiji

ijjiji

ww

ww

q)*0()0()*0(

)*1()0()*1(

20

)0(σ )1(σ

Page 24: Noisy Portfolios

Collegium Budapest

Numerical results

Page 25: Noisy Portfolios

Collegium Budapest

Analytical result

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

TN

q

1

10

Page 26: Noisy Portfolios

Collegium Budapest

Risk measurement

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

Calculate

It can be shown

, for .

)0(σ)1(σ

)0(

)1(

0

q

TOq

110 T

Page 27: Noisy Portfolios

Collegium Budapest

Filtering

Single-index filter:

Spectral decomposition of correlation matrix:

to be chosen so as to preserve trace

N

k

kj

kiji

marketij

k

kj

kikij

vvvvC

vvC

2

111

NNTrC 11

Page 28: Noisy Portfolios

Collegium Budapest

Random matrix filter

to be chosen to preserve trace again

where

and - the upper edge of the random band.

K

k

N

Kk

kj

ki

kj

kik

randij vvvvC

1 1

1max KK 2

max 1

T

N

Page 29: Noisy Portfolios

Collegium Budapest

Covariance estimates

after filtering we get

and

We compare these on the following figure

T

tjtit

histij yy

T 1

1

marketijji

marketij C )( rand

ijjirand

ij C

histiii

ji

histij

ijC

)(

Page 30: Noisy Portfolios

Collegium Budapest

Results for the single-index (market) model

Page 31: Noisy Portfolios

Collegium Budapest

Results for the market + sectors model

Page 32: Noisy Portfolios

Collegium Budapest

Results for the semi-empirical model

Page 33: Noisy Portfolios

Collegium Budapest

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 parametersRMT filtering outperforms

the other two• Semi-empirical: data are scattered, RMT wins in most

cases

histq0

marketq0

Page 34: Noisy Portfolios

Collegium Budapest

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

• How to dig out information buried in the random band?

Promising steps by Z. Burda, A. Görlich, A. Jarosz and J. Jurkiewicz, cond-mat/0305627

and Z. Burda and J. Jurkiewicz, cond-mat/0312496

Page 35: Noisy Portfolios

Collegium Budapest

One step towards reality: Non-stationary case

• Volatility clustering →ARCH, GARCH, integrated GARCH→EWMA in RiskMetrics

t – actual time

T – window

α – attenuation factor ( Teff ~ -1/log α)

1

0,,1

1 T

kktjkti

kTij rr

Page 36: Noisy Portfolios

Collegium Budapest

• RiskMetrics: αoptimal = 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 # of principal components in an objective manner.

• We need upper edge of RMT spectrum for exponentially weighted random matrices

Page 37: Noisy Portfolios

Collegium Budapest

Exponentially weighted Wishart matrices

0

)1(k

jkikk

ij xx

1,0, 2 ikikik xxiidx

constNthatsoN )1(,01,

Page 38: Noisy Portfolios

Collegium Budapest

Density of eigenvalues:

where v is the solution to:

)1()(

N

0)1()sin(loglog)tan(

N

Page 39: Noisy Portfolios

Collegium Budapest

Spectra of exponentially weighted and standard Wishart matrices

Page 40: Noisy Portfolios

Collegium Budapest

The spectrum for a 400x400 matrix

Page 41: Noisy Portfolios

Collegium Budapest

• 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.

Page 42: Noisy Portfolios

Collegium Budapest

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

ij ij t iiitj

ttjitijiji wx

Twxx

Twww

211

t i

iitabs wxT

1

Page 43: Noisy Portfolios

Collegium Budapest

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

„measured” one:

We get:

t i

iitmeasured wxTiw

1min

N

wq i

i

abs 1

2'

Page 44: Noisy Portfolios

Collegium Budapest

• 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.

Page 45: Noisy Portfolios

Collegium Budapest

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.

Page 46: Noisy Portfolios

Collegium Budapest

CVaR optimization

CVaR is the conditional expectation beyond the VaR quantile. For continuous pdf’s 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.

Page 47: Noisy Portfolios

Collegium Budapest

Generalized portfolios

• Capital 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.

Page 48: Noisy Portfolios

Collegium Budapest

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