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Noisy Portfolios

Imre Kondor

Collegium Budapest and Eötvös University

SPHINX Econophysics Workshop, Oxford, 27-29 September, 2004

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Contents

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

absolute deviation, CVaR (expected shortfall), and maximal loss

6. The minimax problem

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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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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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Our purpose

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

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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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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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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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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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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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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.

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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 ?

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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.

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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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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

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Simplified portfolio optimization

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

(constraint on return omitted)

ij

jijip ww min2

i

iw 1

jkjk

jij

iw)(

)(

1

1

*

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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

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Numerical results before filtering

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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

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Results for the market + sectors model

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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

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

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One step towards reality: Non-stationary case

• Volatility clustering →ARCH, GARCH, integrated GARCH→EWMA in RiskMetrics (finite memory)

t – actual time

T – window

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

1

0,,1

1 T

kktjkti

kTij rr

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• 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 number of principal components in an objective manner.

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

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Exponentially weighted Wishart matrices

0

)1(k

jkikk

ij xx

1,0, 2 ikikik xxiidx

constNthatsoN )1(,01,

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Density of eigenvalues:

where v is the solution to:

)1()(

N

0)1()sin(loglog)tan(

N

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Spectra of exponentially weighted and standard Wishart matrices

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• 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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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

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t i

iitabs wxT

1

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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:

t i

iitmeasured wxTiw

1min

N

wq i

i

abs 1

2'

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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 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 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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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

iiti xw

twi

maxmin

1i

iw

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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.

xwf

xwf

ii

i

ii

i

2

2

12

1

2

11

w1

x

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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.

1

11

11

2

T

NkT k

Tp

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

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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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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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Model 1

Spectrum

λ = 1,

N-fold degenerate

Noise will split this

into band

1

0

C =

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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

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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

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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 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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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

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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

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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

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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

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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

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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

)(

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Results for the single-index (market) model

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Results for the semi-empirical model