issues in statistical trading and quantitative portfolio ... · attilio meucci lehman brothers,...

Attilio Meucci

Lehman Brothers, Inc., New York

Issues in Statistical Trading and Quantitative Portfolio ManagementModeling, Estimation Risk, and Robust Allocation

A. Meucci, Risk and Asset Allocation, Springer Finance (2005), www.symmys.com

*

* for a thorough introduction to these and related issues and for references see:

personal website: www.symmys.com

AGENDA

Quantitative recipes

Estimation vs. modeling

Classical optimization

Robust optimization

Robust Bayesian optimization

AGENDA




Robust optimization


• Statistical trading: fixed income PCA• Portfolio management: funds of funds

1. consider N series of T observations of homogeneous forward rates

2. define (NxN positive definite matrix)

3. run PCA (eigenvectors-eigenvalues-eigenvectors)

X

≡S E E'Λ

STATISTICAL TRADING RECIPE – fixed-income PCA

≡S XX'

(TxN panel)




4. analyze the series of the last factor i z-score: structural bandsi“juice”: b.p. from mean i roll-down/slide-adjusted prospective Sharpe ratioi reversion timeframei market events (e.g. Fed, Thursday “numbers”,…)i relation with other series (e.g. oil prices)

5. convert basis points to PnL/risk exposure by dv01

variations: transform series, include mean, support series (PCA-regression),…

X

≡S E E'Λ( )y ≡ NXe

“big picture”

≡S XX'

“small picture”

(TxN panel)


estim

atio

nm

odel

ing

• estimation (backward-looking) and projection/modeling (forward-looking) overlap• non-linearities not accounted for

proj

ectio

n




4. analyze the series of the last factor i z-score: structural bandsi“juice”: b.p. from mean i roll-down/slide-adjusted prospective Sharpe ratioi reversion timeframei market events (e.g. Fed, Thursday “numbers”,…)i relation with other series (e.g. oil prices)

5. convert basis points to PnL/risk exposure by dv01

X

≡S E E'Λ( )y ≡ NXe

≡S XX'

(TxN panel)


AGENDA




Robust optimization


• Statistical trading: fixed income PCA• Portfolio management: funds of funds

1. consider N series of T observations of fund prices (TxN panel)

2. consider the compounded returns

3. estimate covariance (e.g. the sample non-central)

4. define the expected values (e.g. risk-premium)

PORTFOLIO OPTIMIZATION RECIPE – fund of funds

P

( ) ( ), , 1,ln lnt n t n t nC P P−≡ −

1

1 'T

t ttT =

≡ ∑C CΣ

( )diagγ≡ Σµ





5. solve mean-variance:

6. choose the most suitable combination among according to preferences

( )

( )

{ }'

argmax 'i

i

v∈

≤

≡ww w

w wCΣ

µinvestment constraintsgrid of significant variances

( )iw

P

( ) ( ), , 1,ln lnt n t n t nC P P−≡ −

1

1 'T

t ttT =

≡ ∑C CΣ

( )diagγ≡ Σµ






5. solve mean-variance:

6. choose the most suitable combination among according to preferences

( )

( )

{ }'

argmax 'i

i

v∈

≤

≡ww w

w wCΣ

µinvestment constraintsgrid of significant variances

( )iw

P

( ) ( ), , 1,ln lnt n t n t nC P P−≡ −

1

1 'T

t ttT =

≡ ∑C CΣ

( )diagγ≡ Σµ

estim

atio

nm

odel

ing

&

optim

izat

ion

• estimation (backward-looking) and modeling (forward-looking) overlap• projection (investment horizon) not accounted for

• non-linearities of compounded returns not accounted for


AGENDA




Robust optimization


ESTIMATION VS. MODELING – general conceptual framework

investment decision

time series analysis investment horizon

estimation projection

P&L

modeling & optimization

iEstimation: compounded returns

compounded returns are more symmetric, in continuous time they can be modeled (in first approximation) as a Brownian motion

( ) ( )ln lnt t tC P Pττ−

≡ −

ESTIMATION VS. MODELING – fund of funds

estimation interval




≡ −


( )1t tt J t JC C C Cτ τ τ ττ τ− − −

= + + +iProjection to investment horizon

compounded returns can be easily projected to the investment horizon because they are additive (“accordion” expansion)

investment horizon



/ 1t t tL P Pττ−≡ −


≡ −


iModeling: linear returns

linear returns are related to portfolio quantities (P&L):

compounded returns are NOT related to portfolio quantities (P&L):


= + + +iProjection to investment horizon

compounded returns can be easily projected to the investment horizon because they are additive (“accordion” expansion)

LΠ = w'Lportfolio return

securities’ relative weightssecurities’ returns

CΠ ≠ w'C




iProjection to investment horizon

{ }

{ }

12

,

1 12 2

, ,, 1

n nn

n nn m mm nm

n t n

nm t n t m

m E L e

S Cov L L e e

τ τ

τ τ τ τ τ

τ µτ τ

τ τµ µτ τ τ τ

⎛ ⎞+ Σ⎜ ⎟⎝ ⎠

⎛ ⎞+ Σ + + Σ Σ⎜ ⎟⎝ ⎠

≡ =

⎛ ⎞≡ = −⎜ ⎟

⎝ ⎠

( )1

1 ' diagT

t ttT

ττ ττ τ γ=

≡ ≡∑C CΣ Σµ

τ ττ ττ ττ τ

≡ ≡Σ Σ µ µ


≡ −


≡ + + +


“square root rule”:

sample/risk-premium:

Black-Scholesassumption: (log-normal)




iProjection to investment horizon

( )1

1 ' diagT

t ttT

ττ ττ τ γ=

≡ ≡∑C CΣ Σµ

τ ττ ττ ττ τ

≡ ≡Σ Σ µ µ


≡ −


≡ + + +


“square root rule”:

sample/risk-premium:

Black-Scholesassumption: (log-normal)

the mean - variance optimization can be fed with the appropriate inputs

{ }

{ }

12

,

1 12 2

, ,, 1

n nn

n nn m mm nm

t n

t n

n

nm t m

E L e

Cov L L

m

S e e

τ τ

τ τ τ τ τ

τ µτ τ

τ τµ µτ τ τ τ

⎛ ⎞+ Σ⎜ ⎟⎝ ⎠

⎛ ⎞+ Σ + + Σ Σ⎜ ⎟⎝ ⎠

≡ =

⎛ ⎞≡ = −⎜ ⎟

⎝ ⎠

AGENDA




Robust optimization


( ) { }argmax 'i ≡w

w w m

{ }E tτ

τ+≡m L

{ }Cov tτ

τ+≡ LS

w : relative portfolio weights

C : set of investment constraints

( )iv : significant grid of target variances

CLASSICAL OPTIMIZATION – MV in theory …

subject to ( )' iv

∈

≤S

w

w w

C


w w m

{ }E tτ

τ+≡m L

{ }Cov tτ

τ+≡S L




CLASSICAL OPTIMIZATION – … MV in practice

subject to ( )' iv

∈

≤

Cw

w Sw

m

S

: estimate of

: estimate of

m

S


w w m

subject to( )' iv

∈

≤S

w

w w

C

AGENDA




Robust optimization


• Theory• Practice: mean-variance

ROBUST OPTIMIZATION – estimation risk: problem

The true optimal allocation is determined by a set of parameters that are estimated with some error:

point estimate

true (unknown)space of

possible parameters

values

( ) ( ),≡ ≡m S m, Sθ θ ≠

θθ

ROBUST OPTIMIZATION – estimation risk: problem

The true optimal allocation is determined by a set of parameters that are estimated with some error:

• The classical “optimal” allocation based on point estimates is sub-optimal

• More importantly, the sub-optimality due to estimation error is large(Jobson & Korkie (1980); Best & Grauer (1991); Chopra & Ziemba (1993))

( ),≡θ m S

θθ

point estimate


possible parameters

values

( ) ( ),≡ ≡m S m, Sθ θ ≠

ROBUST OPTIMIZATION – estimation risk: solution

• The point estimate for the parameters must be replaced by an uncertainty region that includes the true, unknown parameters:

uncertainty region Θ

θθ

point estimate


possible parameters

values

( ),≡ m S Θθ

ROBUST OPTIMIZATION – estimation risk: solution

• The point estimate for the parameters must be replaced by an uncertainty region that includes the true, unknown parameters:

• The allocation optimization must be performed over all the parameters in the uncertainty region:

( )

( ) ( ){ }

, ,

argmax ...i

∈≡

≡

m S m Sw

wC

( )

( )

{ }argmax ...i

∈∈

≡wm S

wC, Θ

uncertainty region Θ

θθ

point estimate


possible parameters

values

( ),≡ m S Θθ

AGENDA




Robust optimization


• Theory• Practice: mean-variance

m




S

: (point) estimate of


m

S

ROBUST OPTIMIZATION – from the standard MV …


w w m

subject to( )' iv

∈

≤S

w

w w

C

( ) { }margmax i 'ni

∈≡

mw mw w m

Θ


C( )iv : significant grid of target variances

subject to{ } ( )max ' iv

∈

∈

≤SS

w

w Sw

C

Θ

ROBUST OPTIMIZATION – … to a conservative MV approach

mΘ

ΘS

: uncertainty set for


m

S


w w m

subject to( )' iv

∈

≤

w

w Sw

C

m

S



m

S

: set of investment constraints

11S12S

22S

ΘS

positivity boundary

ROBUST OPTIMIZATION – uncertainty regions

Trade-off for the choice of the uncertainty regions:• Must be as large as possible, in such a way that the true, unknown parameters (most likely) are captured• Must be as small as possible, to avoid trivial and nonsensical results

EXAMPLE: 2x2 COVARIANCE MATRIX

AGENDA




Robust optimization

Robust Bayesian optimization• Theory• Practice: mean-variance• An example

ROBUST BAYESIAN OPTIMIZATION – Bayesian estimation

experience posterior density

historical information

( )p of θ

( )p rf θprior density

The Bayesian approach to estimation of the generic market parameters differs from the classical approach in two respects:

• it blends historical information from time series analysis with experience

• the outcome of the estimation process is a (posterior) distribution, instead of a number

( )≡θ m, S

θ0θ

space of possible parameters values

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

in the Bayesian approach the expected values of the returns are a random variable

ROBUST BAYESIAN MV – Bayesian estimation

1m

2m

more likely estimates

less likely estimatesEXAMPLE: 2-dim EXPECTED VALUES

11S12S

22S

positivity boundary

ROBUST BAYESIAN MV – Bayesian estimation

in the Bayesian approach the covariance matrix of the returns isa random variable

more likely estimates

less likely estimates

EXAMPLE: 2x2 COVARIANCE MATRIX

Robust allocations are guaranteed to perform adequately for all the markets withinthe given uncertainty ranges.

However…

• the uncertainty regions for the market parameters are somewhat arbitrary

• the investor’s experience, or prior knowledge, is not considered

ROBUST BAYESIAN OPTIMIZATION – uncertainty regions

Bayesian approach to parameter estimation within the robust framework:

ROBUST BAYESIAN OPTIMIZATION – uncertainty regions

Robust allocations are guaranteed to perform adequately for all the markets within the given uncertainty ranges.

However…

• the uncertainty regions for the market parameters are somewhat arbitrary

• the investor’s experience, or prior knowledge, is not considered

• naturally identifies a suitable uncertainty regions for the market parameters

• includes within a sound statistical framework the investor’s prior experience

uncertainty regionq

Θ

ROBUST BAYESIAN OPTIMIZATION – Bayesian ellipsoids

( ) ( ) 2: 'q

q− − ≤ce ceSΘ θ-1 θθθθ

The Bayesian posterior distribution defines naturally a self-adjusting uncertainty region for the market parameters

This region is the location-dispersion ellipsoid defined by

• a location parameter: the classical-equivalent estimator

• a dispersion parameter: the positive symmetric scatter matrix

• a radius factor

Sθ

ceθ

q

posterior density

space of possible parameters values

qΘ

( )p of θ

ROBUST BAYESIAN OPTIMIZATION – Bayesian ellipsoids

Standard choices for the classical equivalent and the scatter matrix respectively:

( )( ) ( )po' f d≡ − −∫ ce ceSθ θ θ θ θ θ θ

( )pof d≡ ∫ceθ θ θ θ

• local picture: mode / modal dispersion

( )1

poln'

f−

⎛ ⎞∂⎜ ⎟≡ −⎜ ⎟∂ ∂⎝ ⎠ce

Sθθ

θθ θ

( ){ }poargmax f≡ceθ

θ θ

• global picture: expected value / covariance matrix

AGENDA




Robust optimization


m



S



m

S

ROBUST BAYESIAN MV – from the standard MV …

( ) { }argmax 'i

∈≡

wmw w

C

subject to( )' iv≤w Sw


( ) { }argmax min 'i

∈∈≡

mmww w m

C Θ



subject to { } ( )max ' iv∈

≤SS

w SwΘ

ROBUST BAYESIAN MV – … to the conservative robust MV …

mΘ

ΘS

( ) { }argmax 'i

∈≡

ww w m

C


m

S



m

S



m

S


( ), argmax min '

q

iqp

∈ ∈

⎧ ⎫≡ ⎨ ⎬⎩ ⎭mw m

w w mΘC

subject to { } ( )max 'p

iv∈

≤SS

w SwΘ

qΘm

pΘS

: Bayesian ellipsoid of radius for m

S: Bayesian ellipsoid of radius for

q

p

ROBUST BAYESIAN MV – … to the robust BMV


C( )iv

m

S



m

S


: significant grid of target variances

( ) { }argmax 'i

∈≡

ww w m

C


( ),i

p qwROBUST BAYESIAN MV – 3-d frontier

The robust Bayesian efficient allocations represent a three-dimensional frontier parametrized by:

1. Exposure to market risk represented by the target variance

2. Aversion to estimation risk for the expected returns represented by radius

...indeed, a large ellipsoid corresponds to an investor that is very worried about

poor estimates of

3. Aversion to estimation risk for the returns covariance represented by radius

…indeed, a large ellipsoid corresponds to an investor that is very worried about poor estimates of

( )iv

q

Σ

p

m

Σ

qΘm

pΘS

m

AGENDA




Robust optimization


ROBUST BAYESIAN MV – market model

We make the following assumptions:

• The market is composed of equity-like securities, for which the returns are independent and identically distributed across time

• The estimation interval coincides with the investment horizon

• The linear returns are normally distributed:

( )| , N ,tτ

τ+ ∼L m S m S

We model the investor’s prior as a normal-inverse-Wishart distribution:

11 0

0 00 0

| N , , W ,T

νν

−−⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∼ ∼ SSm S m S

: investor’s experience on( )0 0,m S ( ),m S

: investor’s confidence on( )0 0,T ν

where

( )0 0,m S

ROBUST BAYESIAN MV – posterior distribution of market parameters

Under the above assumptions, the posterior distribution is normal-inverse-Wishart, see e.g. Aitchison and Dunsmore (1975):

where

1

1 T

ttT

τ

=

≡ ∑m l ( )( )1

1 'T

ttT

τ ττ

=

≡ − −∑S l m l m

1 0T T T≡ +

1 0 01

1 T TT

⎡ ⎤≡ +⎣ ⎦m m m

1 0 T≡ +ν ν

( )( )0 01 0 0

1

0

'11 1T

T T

νν

⎡ ⎤⎢ − − ⎥⎢ ⎥≡ + +⎢ ⎥+⎢ ⎥⎣ ⎦

m m m mS S S

11 1

1 11 1

| N , , W ,T

νν

−−⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

∼ ∼ SSm S m S

ROBUST BAYESIAN MV – location-dispersion ellipsoids in practice

The certainty equivalent and the scatter matrix for the posterior (Student t) marginal distribution of are computed in Meucci (2005):m

1ce 1 1

1 1

1,2T

νν

= =−mm m S S

The certainty equivalent and the scatter matrix for the posterior (inverse-Wishart) marginal distribution of are computed in Meucci (2005):S

( )( )( )

2 11 1

ce 1 31 1

2,1 1N N

ν νν ν

−

Ν Ν= = ⊗+ + + +

'D DSS S S S S−1 −11 1

where is the duplication matrix (see Magnus and Neudecker, 1999) and is the Kronecker product

⊗ΝD

ROBUST BAYESIAN MV – optimal portfolios in practice

Under the above assumptions the robust Bayesian mean-variance problem:

( ) ( ) { }, 1argmax ' 'ip q λ λ

∈⊂ ≡ −

ww w w m w S w

C1

• The three-dimensional frontier collapses to a line

• The efficient frontier is parametrized by the exposure to overall risk, which includes

market risk, estimation risk for and estimation risk for m S

subject to

…simplifies as follows:

( ), argmax min '

q

iqp

∈ ∈

⎧ ⎫≡ ⎨ ⎬⎩ ⎭mw m

w w mΘC

{ } ( )max 'p

iv∈

≤SS

w SwΘ

ROBUST BAYESIAN MV – Bayesian self-adjusting nature

• When the number of historical observations is large the uncertainty regions collapse to classical sample point estimates:

• When the confidence in the prior is large the uncertainty regions collapse to the prior parameters:

robust Bayesian frontier = classical sample-based frontier

robust Bayesian frontier = “a-priori” frontier (no information from the market)

( ) { }1argmax ' 'λ λ∈

≡ − 1w

w w m w S wC

( ) { }0 0argmax ' 'λ λ∈

≡ −w

w w m w S wC

market & estimation risk

1

0

1

0

0 0, νT T0 0, νT T

portf

olio

w

eigh

ts

prior frontier

robust Bayesian frontier

sample-based frontier

1

0

ROBUST BAYESIAN MV – Bayesian self-adjusting nature

Springer FinanceSpringer Finance

Risk and Asset Allocation

1

A. MeucciThis encyclopedic, detailed exposition spans all the steps of one-periodallocation from the foundations to the most advanced developments.

Multivariate estimation methods are analyzed in depth, including non-para-metric, maximum-likelihood under non-normal hypotheses, shrinkage, robust,and very general Bayesian techniques. Evaluation methods such as stochasticdominance, expected utility, value at risk and coherent measures arethoroughly discussed in a unified setting and applied in a variety of contexts,including prospect theory, total return and benchmark allocation. Portfoliooptimization is presented with emphasis on estimation risk, which is tackledby means of Bayesian, resampling and robust optimization techniques.

All the statistical and mathematical tools, such as copulas, location-dispersionellipsoids, matrix-variate distributions, cone programming, are introducedfrom the basics.

At symmys.com the reader will find freely downloadable complementarymaterials: the Exercise Book; a set of thoroughly documented MATLAB®applications; and the Technical Appendices with all the proofs.More materials and complete reviews can also be found at symmys.com.

This exciting new book takes a fresh look at asset allocation and offers up a masterlyaccount of this important subject. The quantitative emphasis and included MATLABsoftware make it a must-read for the mathematically-oriented investment professional.

Peter Carr, Head of Quantitative Research, Bloomberg LP.,Director of Masters in Mathematical Finance, NYU

Meucci’s Risk and Asset Allocation is one of those rare books that take a completely freshlook at a well-studied problem, optimal financial portfolio allocation based onstatistically estimated models of risk and expected return. Designed for graduatestudents or quantitatively-oriented asset managers, Meucci provides a sophisticatedand integrated treatment (…). This is rigorous and relevant !

Darrell Duffie, Professor of Finance, Stanford University

A wonderful book ! Mathematically rigorous and yet practical, heavily illustrated withgraphs and worked examples, Attilio Meucci has written a comprehensive treatment ofasset allocation (…).

Bob Litterman, Head of Quantitative Resources, Goldman Sachs Asset Management

This book takes the reader on a journey through portfolio management starting withthe basics and reaching some fascinating terrain. Attilio Meucci shows a real talent forexplaining the most difficult of subjects in a very clear manner.

Paul Wilmott (wilmott.com)

S F

Meucci

Attilio Meucci

Dieser Farbausdruck/pdf-file kann nur annähernddas endgültige Druckergebnis w

iedergeben !60573

HKS 39HKS 4

27.4.05 designandproduction GmbH

– Bender

› springeronline.com

783540 2221329

ISBN 3-540-22213-3

1 23

Risk andAsset Allocation

22213-8 Meucci U. 27.04.2005 8:07 Uhr Seite 1

see www.symmys.com

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