Issues in Statistical Trading and Quantitative Portfolio ... · Attilio Meucci Lehman Brothers,...
Transcript of 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
Quantitative recipes
Estimation vs. modeling
Classical optimization
Robust optimization
Robust Bayesian 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)
1. consider N series of T observations of homogeneous forward rates
2. define (NxN positive definite matrix)
3. run PCA (eigenvectors-eigenvalues-eigenvectors)
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)
STATISTICAL TRADING RECIPE – fixed-income PCA
estim
atio
nm
odel
ing
• estimation (backward-looking) and projection/modeling (forward-looking) overlap• non-linearities not accounted for
proj
ectio
n
1. consider N series of T observations of homogeneous forward rates
2. define (NxN positive definite matrix)
3. run PCA (eigenvectors-eigenvalues-eigenvectors)
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)
STATISTICAL TRADING RECIPE – fixed-income PCA
AGENDA
Quantitative recipes
Estimation vs. modeling
Classical optimization
Robust optimization
Robust Bayesian 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
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)
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
PORTFOLIO OPTIMIZATION RECIPE – fund 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)
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
PORTFOLIO OPTIMIZATION RECIPE – fund of funds
AGENDA
Quantitative recipes
Estimation vs. modeling
Classical optimization
Robust optimization
Robust Bayesian 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
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
( )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
iEstimation: compounded returns
compounded returns are more symmetric, in continuous time they can be modeled (in first approximation) as a Brownian motion
/ 1t t tL P Pττ−≡ −
( ) ( )ln lnt t tC P Pττ−
≡ −
ESTIMATION VS. MODELING – fund of funds
iModeling: linear returns
linear returns are related to portfolio quantities (P&L):
compounded returns are NOT related to portfolio quantities (P&L):
( )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)
LΠ = w'Lportfolio return
securities’ relative weightssecurities’ returns
CΠ ≠ w'C
iEstimation: compounded returns
ESTIMATION VS. MODELING – fund of funds
iModeling: linear returns
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
( ) ( )ln lnt t tC P Pττ−
≡ −
( )1t tt J t JC C C Cτ τ τ ττ τ− − −
≡ + + +
/ 1t t tL P Pττ−≡ −
“square root rule”:
sample/risk-premium:
Black-Scholesassumption: (log-normal)
iEstimation: compounded returns
ESTIMATION VS. MODELING – fund of funds
iModeling: linear returns
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
( ) ( )ln lnt t tC P Pττ−
≡ −
( )1t tt J t JC C C Cτ τ τ ττ τ− − −
≡ + + +
/ 1t t tL P Pττ−≡ −
“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
Quantitative recipes
Estimation vs. modeling
Classical optimization
Robust optimization
Robust Bayesian 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
( ) { }argmax 'i ≡w
w w m
{ }E tτ
τ+≡m L
{ }Cov tτ
τ+≡S L
w : relative portfolio weights
C : set of investment constraints
( )iv : significant grid of target variances
CLASSICAL OPTIMIZATION – … MV in practice
subject to ( )' iv
∈
≤
Cw
w Sw
m
S
: estimate of
: estimate of
m
S
( ) { }argmax 'i ≡w
w w m
subject to( )' iv
∈
≤S
w
w w
C
AGENDA
Quantitative recipes
Estimation vs. modeling
Classical optimization
Robust optimization
Robust Bayesian 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
true (unknown)space of
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
true (unknown)space of
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
true (unknown)space of
possible parameters
values
( ),≡ m S Θθ
AGENDA
Quantitative recipes
Estimation vs. modeling
Classical optimization
Robust optimization
Robust Bayesian optimization
• Theory• Practice: mean-variance
m
w : relative portfolio weights
C : set of investment constraints
( )iv : significant grid of target variances
S
: (point) estimate of
: (point) estimate of
m
S
ROBUST OPTIMIZATION – from the standard MV …
( ) { }argmax 'i ≡w
w w m
subject to( )' iv
∈
≤S
w
w w
C
( ) { }margmax i 'ni
∈≡
mw mw w m
Θ
w : relative portfolio weights
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
: uncertainty set for
m
S
( ) { }argmax 'i ≡w
w w m
subject to( )' iv
∈
≤
w
w Sw
C
m
S
: (point) estimate of
: (point) estimate of
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
Quantitative recipes
Estimation vs. modeling
Classical optimization
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
Quantitative recipes
Estimation vs. modeling
Classical optimization
Robust optimization
Robust Bayesian optimization• Theory• Practice: mean-variance• An example
m
w : relative portfolio weights
C( )iv : significant grid of target variances
S
: (point) estimate of
: (point) estimate of
m
S
ROBUST BAYESIAN MV – from the standard MV …
( ) { }argmax 'i
∈≡
wmw w
C
subject to( )' iv≤w Sw
: set of investment constraints
( ) { }argmax min 'i
∈∈≡
mmww w m
C Θ
w : relative portfolio weights
C( )iv : significant grid of target variances
subject to { } ( )max ' iv∈
≤SS
w SwΘ
ROBUST BAYESIAN MV – … to the conservative robust MV …
mΘ
ΘS
( ) { }argmax 'i
∈≡
ww w m
C
subject to( )' iv≤w Sw
m
S
: (point) estimate of
: (point) estimate of
m
S
: uncertainty set for
: uncertainty set for
m
S
: set of investment constraints
( ), 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
w : relative portfolio weights
C( )iv
m
S
: (point) estimate of
: (point) estimate of
m
S
: set of investment constraints
: significant grid of target variances
( ) { }argmax 'i
∈≡
ww w m
C
subject to( )' iv≤w Sw
( ),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
Quantitative recipes
Estimation vs. modeling
Classical optimization
Robust optimization
Robust Bayesian optimization• Theory• Practice: mean-variance• An example
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
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Risk andAsset Allocation
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