Meucci Fully Flexible Views - Theory and Practice.ppt · A. Meucci - Fully Flexible Views (Entropy...

A. Meucci - Fully Flexible Views (Entropy Pooling)

Attilio Meucci

Fully Flexible Views:Fully Flexible Views: Entropy Pooling in Theory and Practice

Swissquote Conference on Asset Management

Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerlandy q ( ),

October 21, 2011

ESTIMATION RISK

SCENARIO ANALYSIS

THE BLACK-LITTERMAN APPROACH

ENTROPY POOLINGENTROPY POOLING

CASE STUDIES

REFERENCES AND CONCLUSIONS

A. Meucci - Fully Flexible Views (Entropy Pooling) Estimation risk

E T m R

CovS R

: expected returns

: covariance

argmax ' mw w

Cov T S R : covariance

argmaxvw

subject to ' vw Sw

w : portfolio weights

v : grid of target variances

' 1 01C : investment constraints, e.g. ' 1, 0w 1 w

A. Meucci - Fully Flexible Views (Entropy Pooling) Estimation risk

E T m R

CovS R

: expected returns

: covariance

argmax 'w w m

Cov T S R : covariance

argmaxvw

w w mC

subject to ' vw Sw

efficient frontier

Dell S4S5

0.05 0.1 0.15 0.2 0.25 0.3 0.350

A. Meucci - Fully Flexible Views (Entropy Pooling) N , 1, ,t t T r m S~

Estimation risk

? E T m R

CovS R

: expected returns

: covariance?

argmax 'w w m

Cov T S R : covariance?

argmaxvw

w w mC

subject to ' vw Sw

efficient frontier

Dell S4S5

0.05 0.1 0.15 0.2 0.25 0.3 0.350

N , 1, ,t t T r m S~

Estimation risk

? E T m R

CovS R

: expected returns

: covariance?1

S r m r m

argmax 'w w m

Cov T S R : covariance? 1

S r m r m

argmaxvw

w w mC

subject to ' vw Sw

efficient frontier

Dell S4S5

0.05 0.1 0.15 0.2 0.25 0.3 0.350

N , 1, ,t t T r m S~

Estimation risk

? E T m R

CovS R

: expected returns

: covariance?1

S r m r m

argmax 'w w m

argmax ' mw w

S r m r m

~ argmaxvw

w w mC

subject to ' vw Sw

argmaxv

subject to ' vw Sw~

efficient frontier

Dell S4S5

0.05 0.1 0.15 0.2 0.25 0.3 0.350

N , 1, ,t t T r m S~

Estimation risk

? E T m R

CovS R

: expected returns

: covariance?1

S r m r m

argmax 'w w m

argmax ' mw w

S r m r m

~ argmaxvw

w w mC

subject to ' vw Sw

argmaxv

subject to ' vw Sw~

efficient frontier

LIVES10

LIVEDell S4S5

0.05 0.1 0.15 0.2 0.25 0.3 0.350

N , 1, ,t t T r m S 1~

Estimation risk

? E T m R

CovS R

: expected returns

: covariance?1

S r m r m

argmax 'w w m

argmax ' mw w

S r m r m

~ argmaxvw

w w mC

subject to ' vw Sw

argmaxv

subject to ' vw Sw~

efficient frontier efficient frontier

Dell S4S5

0.1 0.15 0.2 0.25 0.3 0.350

v0.05 0.1 0.15 0.2 0.25 0.3 0.350

A. Meucci - Fully Flexible Views (Entropy Pooling) N , 1, ,t t T r m S

Estimation risk

? E T m R

CovS R

: expected returns

: covariance?1

S r m r m

argmax 'w w m

argmax ' mw w

S r m r m

~ argmaxvw

w w mC

subject to ' vw Sw

argmaxv

subject to ' vw Sw~

Dell S4S5

0.05 0.1 0.15 0.2 0.25 0.3 0.350

v0.05 0.1 0.15 0.2 0.25 0.3 0.350

A. Meucci - Fully Flexible Views (Entropy Pooling) N , 1, ,t t T r m S

Estimation risk

? E T m R

CovS R

: expected returns

: covariance?1

S r m r m

argmax 'w w m

argmax ' mw w

S r m r m

~ argmaxvw

w w mC

subject to ' vw Sw

argmaxv

subject to ' vw Sw~

Dell S4S5

0.05 0.1 0.15 0.2 0.25 0.3 0.350

v 0.1 0.15 0.2 0.25 0.3 0.350

ESTIMATION RISK

SCENARIO ANALYSIS

CASE STUDIES

CONCLUSIONS AND REFERENCES

Market distribution

Scenario analysis / stress-testing

returns on asset classes/funds

Market distribution

Scenario analysisMarket distribution

FULL specification:

vQ = identity

FOCUS: Q = portfolios

Conditional formula

Optimization

ESTIMATION RISK

SCENARIO ANALYSISSCENARIO ANALYSIS

THE BLACK-LITTERMAN APPROACH - Estimation risk

- Views

- Discussion

ENTROPY POOLING

CASE STUDIESCASE STUDIES

Market distribution

Black-Litterman – estimation risk

Market distribution

estimation risk?

Market distribution

estimated by exponential smoothing

Market distribution

mean-variance

Market distribution

w Σ πλ λ

mean-variance

Market distribution

w Σ πλ λ

mean-variance

Market distribution

equilibrium portfolio

equilibrium returns

Market distribution

equilibrium returns

Market distribution

efficient frontier

0.05 0.1 0.15 0.2 0.25 0.3 0.350 v IBM

Optimization

ESTIMATION RISK

- Views

- Discussion

ENTROPY POOLING

Market distribution Views

Black-Litterman – views

scenario analysis with uncertainty

FOCUS: Q = portfolio

ViewsMarket distribution

Bayes’ formulaBayes formula

small uncertainty

Bayes’ formula conditional formulaBayes formula

conditional formula

Ω large

uncertainty

Bayes’ formulaBayes formula

Bayes’ formula

Optimization Optimization

LIVELIVEOptimization Optimization

Optimization0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.05 0.1 0.15 0.2 0.25 0.3 0.350v v

Optimization

ESTIMATION RISK

- Views

- Discussion

ENTROPY POOLING

Views/ScenariosMarket distribution

Black-Litterman – pro’s

Views/Scenarios

+ uncertainty

Market distribution

Optimization

Black-Litterman – con’s

Views/Scenarios

+ uncertainty

Market distribution

Market is not only returns: implied volatilities (derivatives)rates paths (mortgages)implied correlations (CDO’s)….

Optimization

Views/Scenarios

+ uncertainty

Market distribution

Market is not only returns

Market is not only normal: fat tails skewness tail-risk codependenceMarket is not only normal: fat tails, skewness, tail risk codependence,…

Optimization

Views/Scenarios

+ uncertainty

Market distribution

Market is not only normalMarket is not only normal

Market is not only equilibrium: historical estimates, implied values, …

Optimization

Views/Scenarios

+ uncertainty

Market distribution

Market is not only equilibrium

Views are not only portfolios: generic non-linear functions

Optimization

Views/Scenarios

+ uncertainty

Market distribution

Views are not only portfolios

Views are not only on expectations: correlations, volatilities,tail behavior, copulas, …

Optimization

Views/Scenarios

+ uncertainty

Market distribution

Views are not only on expectations

Views are not only equalities: stock ranking, qualitative views

Optimization

Views/Scenarios

+ uncertainty

Market distribution

Views are not only on expectations

Views are not only equalities

Optimization is not only mean variance: mean-CVaR, mean-VaR, …

Optimization

ESTIMATION RISKESTIMATION RISK

SCENARIO ANALYSIS

ENTROPY POOLING - Theory

- Implementation- Implementation

CASE STUDIES

Market distr. not returns, not normal, not equilibrium 1Xe.g.

2X2-yr swap rate

5-yr swap rate2X 5-yr swap rate5y

22yr swap

not returns, not normal, not equilibriumMarket distr. 1Xe.g.

2X2-yr swap rate

22yr swap

P i i d l / / f ll i ie.g.

d ti itPricing delta/gamma/vega, full pricing, … duration + convexity

2X2-yr swap rate

22yr swap

P i i d l / / f ll i ie.g.

d ti itPricing

Optimization

delta/gamma/vega, full pricing, …

utility, mean-CVaR, … mean-variance

duration + convexity

2X2-yr swap rate

5-yr swap rate

Focus non-linear functions and external factors2X 5-yr swap rate

2 21 2V X X

convexity factor

2X2-yr swap rate

5-yr swap rate

Focus non-linear functions and external factors2X 5-yr swap rate

22yr swap

2X2-yr swap rate

5-yr swap rate

non-linear functions and external factors

full distribution specification

2X 5-yr swap rate

22yr swap

2X2-yr swap rate

5-yr swap rate

2X 5-yr swap rate

view on expectations (BL), medians

22yr swap

2X2-yr swap rate

5-yr swap rate

2X 5-yr swap rate

ranking

2X2-yr swap rate

5-yr swap rate

2X 5-yr swap rate

ranking

views on volatilities

22yr swap

2X2-yr swap rate

5-yr swap rate

2X 5-yr swap rate

ranking

correlation stress-testing

2X2-yr swap rate

5-yr swap rate

2X 5-yr swap rate

ranking

correlation stress-testing

i t il b h iview on tail behavior

2X2-yr swap rate

5-yr swap rate

2X 5-yr swap rate

partial distribution specification

2X2-yr swap rate

5-yr swap rate

2X 5-yr swap rate

2yr swap

2X2-yr swap rate

5-yr swap rate

non-linear functions and external factors2X 5-yr swap rate

2yr swap

2X2-yr swap rate

5-yr swap rate

non-linear functions and external factors2X 5-yr swap rate

Posterior ?2yr swap

2X2-yr swap rate

5-yr swap rate

2X 5-yr swap rate

Posterior ?

“distance” btw. distributionsrelative entropy

2X2-yr swap rate

5-yr swap rate

2X 5-yr swap rate

Posterior least distance from prior

not returns, not normal, not equilibriumMarket distr.

Posterior least distance from prior, views satisfied

LIVELIVE

Market distr. 1X2X

2-yr swap rate

5-yr swap rate2X 5-yr swap rate

prior USD SWAP 2Y rate prior USD SWAP 5Y rate

2yr swap

prior USD SWAP 2Y rate p

normal fitestimated

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

2yr swap 5yr swap

Market distr. 1X2X

2-yr swap rate

prior USD SWAP 2Y rate prior USD SWAP 5Y rateprior USD SWAP 2Y rate p

normal fitestimated

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

2yr swap 5yr swap

Market distr. 1X2X

2-yr swap rate

Focus1V X

prior USD SWAP 2Y rate prior USD SWAP 5Y rateprior USD SWAP 2Y rate p

normal fitestimated

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

2yr swap 5yr swap

Market distr. 1X2X

2-yr swap rate

m V m V bp

3.26 2

m V m V bp

normal fitestimated

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

2yr swap 5yr swap

Market distr. 1X2X

2-yr swap rate

1V XFocus

Vm V m V bp

3.26 2

m V m V bp

normal fitestimated

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

2yr swap 5yr swap

Posterior

0posterior USD SWAP 2Y rate

correctnormal fit

-0.3 -0.2 -0.1 0 0.1 0.2 0.30

2yr swap

Market distr. 1X2X

2-yr swap rate

1V XFocus

Vm V m V bp

3.26 2

m V m V bp

normal fitestimated

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

2yr swap 5yr swap

Posterior

0posterior USD SWAP 2Y rate posterior USD SWAP 5Y rate

correctnormal fit

-0.3 -0.2 -0.1 0 0.1 0.2 0.30

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

2yr swap 5yr swap

Confidence multi-user, multi-confidence

100(1-c) % of times: PRIOR

100c % of times: POSTERIOR

Confidence

multi-user, multi-confidence

d l / / f ll i iPricing delta/gamma/vega, full pricing, …

Confidence

multi-user, multi-confidence

d l / / f ll i iPricing

Optimization mean-variance, mean-CVaR, …

delta/gamma/vega, full pricing, …

ESTIMATION RISKESTIMATION RISK

SCENARIO ANALYSIS

ENTROPY POOLING - Theory

- Implementation- Implementation

CASE STUDIES

Entropy PoolingParametric Fully Flexible

Implementation

DistributionsFully Flexible Probabilities

Entropy PoolingParametric

Implementation

Parametric

Distributions

Parametric Entropy Pooling

Implementation

Parametric

Distributions

Normal (+dimension reduction)

Implementation

Parametric

Distributions

NormalLinear-Quadratic

Market distr.

Analytical implementation

Posterior

Confidence

P i iPricing

Optimization

Market distr.

Posterior

Confidence

P i iPricing

Optimization

Market distr.

Posterior

Confidence

P i iPricing

Optimization

Market distr.

Posterior

Confidence

P i iPricing

Optimization

Market distr.

Posterior

Confidence

P i iPricing

Optimization

Market distr.

Posterior

Confidence

P i i …

Pricing

Optimization

Fully FlexibleEntropy Pooling

Parametric

Non-parametric implementation

Fully Flexible ProbabilitiesDistributions

joint scenario of N risk drivers probability of joint scenario

Parametric

Fully Flexible Probabilities

Parametric N P i

Distributions

Non-Parametric Entropy Pooling

Parametric

Fully Flexible Probabilities

Parametric N P i

Distributions

Fuzzy Membership

Bayesian networks

State Conditioning

KernelsCrisp

ConditioningNormalLinear-Quadratic

Time Conditioning

A. Meucci - Fully Flexible Views (Entropy Pooling) Non-parametric implementation

probability = 1 / num scenarios

“prior”

probability < 1 / num scenarios

probability > 1 / num scenarios

“prior”

bearish bullish

regular market

probability < 1 / num scenarios

probability > 1 / num scenarios

regular market

low volatility high volatility

market distribution

probability = 0

probability = 1

scenario analysis

market distribution

scenario analysis

Market distr.

joint scenario of N risk driversj

Probability of joint scenariojoint scenario

Market distr.

joint scenario of securities prices

P i i Pricing Probability of joint scenario

Market distr.

P i i Pricing

Optimization …

Market distr.

Posterior

Confidence

P i iPricing

Optimization

Market distr.

Views scenario index

Posterior

Confidence

P i iPricing

Optimization

Market distr.

2-yr swap rate

2 yr swap rate

5-yr swap rate

j jp V

Posterior

Confidence

P i iPricing

Optimization

Market distr.

Posterior

Market distr.

Posterior

Market distr.

Posterior

Dual formulation: li l i dlinearly constrained

convex optimization in # variables = # views

Market distr.

Posterior Confidence

P i iPricing

Optimization

Market distr.

Posterior Confidence P i iPricing

Optimization

Market distr.

Pricing

Optimization

Market distr.

Pricing

Optimization

ESTIMATION RISK

SCENARIO ANALYSIS

ENTROPY POOLING

CASE STUDIES - Portfolio from sorts

- Implied expected returns

Distributional stress testing- Distributional stress-testing

- Option trading

expected returns no views1

frontier no viewsexpected returns efficient frontier

Case study: portfolios from sorts

ghts S7

S8S9S10

DellS3

IBMDell

0 2 4 6 8 10 12

8 9 10 11 12 13 14 150

expected return volatility

IBMIBM

ghts S7

S8S9S10

DellS3

IBMDell

0 2 4 6 8 10 12

8 9 10 11 12 13 14 150

view: 3 4E R E Rexpected return volatility

IBMIBM

ghts S7

S8S9S10

DellS3

IBMDell

0 2 4 6 8 10 12

8 9 10 11 12 13 14 150

view: 3 4E R E Rexpected return volatility

IBMIBM

expected returns after views

1frontier after viewsexpected returns efficient frontier

ts S7S10

S8S9S10

Dell S3S6

S9S8S7S6S5

0 2 4 6 8 10

8 9 10 11 12 13 14 150

0.2por

volatilityexpected return

DellS5

IBMDellS3

A. Meucci - Fully Flexible Views (Entropy Pooling) Case study: portfolios from sorts

Views:

Sample means Standard ranking Centroid Entropy Pooling

UTXPGHDDISKOBA

MMMKFTMCD

XOMCSCOWMT

JNJAXPIBM

PFEHPQCVXMRK

MSFTINTCBACXOM

-20 0 20 40

JPMTRVPFE

Prior [% p.a.]-20 0 20 40

Common [% p.a.]-20 0 20 40

AC [% p.a.]-20 0 20 40

EP [% p.a.]

ESTIMATION RISK

SCENARIO ANALYSIS

ENTROPY POOLING

- Option trading

A. Meucci - Fully Flexible Views (Entropy Pooling) Case study: implied expected returns

market capital. weights

sample means Black-Litterman implied exp. returns

Entropy Pooling implied exp. return

TRVUTXVZ

WMTXOM

MCDMMMMRK

MSFTPFE

HPQIBM

INTCJNJJPM

BABACCAT

CSCOCVX

DDDISGE

0 5 10

AAAXPBA

Weights [%]0 2 4P i [% ]

0 2 4BL [% ]

0 2 4EP [% ]

TRVUTXVZ

WMTXOM

MCDMMMMRK

MSFTPFE

HPQIBM

INTCJNJJPM

BABACCAT

CSCOCVX

DDDISGE

0 5 10

AAAXPBA

Weights [%]0 2 4P i [% ]

0 2 4BL [% ]

0 2 4EP [% ]

TRVUTXVZ

WMTXOM

MCDMMMMRK

MSFTPFE

HPQIBM

INTCJNJJPM

BABACCAT

CSCOCVX

DDDISGE

0 5 10

AAAXPBA

Weights [%]0 2 4P i [% ]

0 2 4BL [% ]

0 2 4EP [% ]

Expectation -covariance ellipsoids

Sample

-10 -5 0 5 10

AXP-10 -5 0 5 10

Sample

Black-Litterman

AXP AA

10Entropy-Pooling

-10 -5 0 5 10

BAC-10 0 10

ESTIMATION RISK

SCENARIO ANALYSIS

ENTROPY POOLING

- Option trading

A. Meucci - Fully Flexible Views (Entropy Pooling) Case study: distributional stress-testing

Prior (exponential time decay)

State indicator (2yr swap rate - current level)State indicator (2yr swap rate current level)

State indicator (1->5yr ATM implied swaption vol - current level)

Membership (pseudo Gaussian kernel)

Posterior (Entropy Pooling mixture)

A. Meucci - Fully Flexible Views (Entropy Pooling) Case study: time decay + membership kernel

P&L scenarios

P&L distributionP&L distribution

P&L ex ante performance statisticsP&L ex-ante performance statistics

ESTIMATION RISK

SCENARIO ANALYSIS

ENTROPY POOLING

- Option trading

Black-Scholes formula: deterministic function of risk into price

Case study: option trading

deterministic function of risk into price

Black-Scholes formula: deterministic function of risk into price

deterministic function of risk into price

A. Meucci - Fully Flexible Views (Entropy Pooling) Case study: option trading

empirical smirk and smile

call option price at horizon

Portfolio: Microsoft 1 month Microsoft 2 months Microsoft 6 months

curve change (growth/inflation) not directly in pricing

Yahoo 1 month Yahoo 2 monthsYahoo 6 months Google 1 month G l 2 thGoogle 2 months Google 6 months

profit and loss is highly non-linear, highly non-normal?

?Mean-CVaR optimization ?

- long-short delta-neutral - no cash upfront - limit on leverage

Market distr.

P i i ?Pricing

Optimization

simulations

linear programming

Market distr.

P i i Pricing

Optimization linear programming

Market distr.

view: G 6m G 2m

Market distr.

Pricing

Optimization

view: G 6m G 2m

ESTIMATION RISK

SCENARIO ANALYSIS

CASE STUDIES

A. Meucci - Fully Flexible Views (Entropy Pooling) References

Entropy Pooling

Meucci (2008) http://symmys.com/node/158

References

Entropy Pooling

References

Parametric N P iParametric Entropy Pooling

Meucci (2008)NormalLinear-Quadratic

Entropy Pooling

References

Bayesian networks

Meucci (2008)NormalLinear-Quadratic

Meucci (2008) http://symmys.com/node/158 Kernels

Entropy Pooling

ReferencesMeucci (2008) http://symmys.com/node/158

Meucci, Ardia, Keel (2011) http://symmys.com/node/160

Bayesian networks

Meucci (2011)Normal (+dimension reduction)

Meucci (2008)

Fuzzy Membership State Conditioning

Meucci (2008) http://symmys.com/node/158 Kernels

Crisp Conditioning

Time ConditioningMeucci (2010)

http://symmys.com/node/150

A. Meucci - Fully Flexible Views (Entropy Pooling) conclusions

Entropy Pooling:

Market represented by generic non-linear risk factors, not only returns

Market distribution fully general, not only normal

Views/stress-testing on any function of the market, not only linear portfolios

Views on any feature, not only on expectations: median, volatility, correlations, tails

Views are equalities and inequalities: ranking is possible

Applications span several areas of portfolio management and risk management

Meucci Fully Flexible Views - Theory and Practice.ppt · A. Meucci - Fully Flexible Views (Entropy...

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