the black-litterman model in the light of bayesian portfolio analysis

26
Lecture Notes The Black-Litterman model in the light of Bayesian portfolio analysis Parameter Uncertainty and Learning in Dynamic Financial Decisions Daniel A. Bruggisser December 1, 2010

Upload: daniel-bruggisser

Post on 04-Jul-2015

1.797 views

Category:

Economy & Finance


0 download

DESCRIPTION

Financial Modelling; Risk Management; Bayesian Inference; Bayesian portfolio analysis; parameter uncertainty; asset allocation; Black-Litterman model; Bayesian learning; informative prior; return predictability; return forecasting.

TRANSCRIPT

Page 1: The Black-Litterman model in the light of Bayesian portfolio analysis

Lecture Notes

The Black-Litterman model in the light of Bayesianportfolio analysis

Parameter Uncertainty and Learning in Dynamic

Financial Decisions

Daniel A. Bruggisser

December 1, 2010

Page 2: The Black-Litterman model in the light of Bayesian portfolio analysis

Agenda

1. Introduction

2. Bayesian portfolio analysis

3. The mixed estimation model

4. The Black-Litterman model

5. Relating the Black-Litterman model to shrinkage estimation

6. Conclusion

2

Page 3: The Black-Litterman model in the light of Bayesian portfolio analysis

Introduction

• Parameter uncertainty is ubiquitous in finance.

• Given that the observation history is rarely a good predictor of the future and thatcertain parameters such as the mean of asset returns are difficult to estimate withprecision, it is evident that information other than the sample statistics of pastobservations may be very useful in a portfolio selection context.

• Such non-sample information may include fundamental analysis and the beliefs ina certain economic model such as market efficiency or equilibrium pricing.

• The Black-Litterman model is an application of Bayesian mixed estimation. It isdeeply rooted in the theory of Bayesian analysis.

• The Balck-Litterman model allows the investor to combine two sources of infor-mation: (1) The market equilibrium risk prima; (2) The investors subjective viewsabout some of the assets return forecasts.

3

Page 4: The Black-Litterman model in the light of Bayesian portfolio analysis

Bayesian Portfolio Analysis

• The classical portfolio selection problem:1

maxω

ET [U(WT+1)] = maxω

Ω

U(WT+1)p(rT+1|θ)drT+1 (1)

s.t. WT+1 = WT

(

ω′ exp(rT+1 + rfT ) + (1− ι′ω) exp(rfT )

)

, (2)

where Ω is the sample space, U(WT+1) is a utility function, WT+1 is the wealthat time T + 1, θ is a set parameters, ω are portfolio weights, p(rT+1|θ) is the

sample density of returns, rfT is the risk free rate, and ι is a vector of ones. Thecolumn vectors ι, ω and rT+1 are of the same dimension.

• The parameter vector θ is assumed to be known to the investor. However, someparameters are estimates and subject to parameter uncertainty.

4

Page 5: The Black-Litterman model in the light of Bayesian portfolio analysis

• Bayesian portfolio selection problem:2

maxω

ET [U(WT+1)] = maxω

Ω

U(WT+1)p(rT+1|ΦT )drT+1, (3)

s.t. WT+1 = WT

(

ω′ exp(rT+1 + rfT ) + (1− ι′ω) exp(rfT )

)

, (4)

where ΦT is the information available up to time T , and p(rT+1|ΦT ) is theBayesian predictive distribution (density) of asset returns.

• Conditioning takes place on ΦT instead of essentially uncertain parameters θ.

• There are many ways to derive the Bayesian predictive distribution depending onthe model at hand, the choice of the prior distribution of uncertain quantities, andthe information the investor assumes as known.

5

Page 6: The Black-Litterman model in the light of Bayesian portfolio analysis

• Bayesian decomposition, Bayes′ rule and Fubini′s theorem:3

ET [U(WT+1)] =

Ω

U(WT+1)p(rT+1|ΦT )drT+1 (5)

=

Ω×Θ

U(WT+1)p(rT+1,θ|ΦT )d(drT+1,θ)

=

Ω

Θ

U(WT+1)p(rT+1|θ)p(θ|ΦT )dθdrT+1

=

Ω

U(WT+1)

(∫

Θ

p(rT+1|θ)p(ΦT |θ)p(θ)dθ

)

drT+1,

where Θ is the parameter space, p(rT+1, θ|ΦT ) is the joint density of parametersand realizations, p(θ|ΦT ) is the posterior density, p(ΦT |θ) is the conditionallikelihood, and p(θ) is the prior density of the parameters.

6

Page 7: The Black-Litterman model in the light of Bayesian portfolio analysis

The mixed estimation model

• Mixed estimation allows the investor to combine different sources of information.4

Let the sample density of returns be given a multivariate normal density

p(rt|µ,Σ) = N(

µ,Σ)

, (6)

and assume that the prior density for the m × 1 vector of means µ also hasmultivariate normal density with

p(µ) = N (m0,Λ0) . (7)

The investor expresses views about µ by imposing

p(v|µ) = N (Pµ,Ω) , (8)

where P is an n×m design matrix that selects and combines returns into portfoliosabout which the investor is able to express his views. v is a n× 1 vector of viewsand the n× n matrix Ω expresses the uncertainty of those views.

7

Page 8: The Black-Litterman model in the light of Bayesian portfolio analysis

Applying Bayes′ rule, the posterior of µ given the investors views v is then

p(µ|v) ∝ p(v|µ)p(µ). (9)

It emerges that the posterior of µ updated by the views v is (see Appendix 1 fora proof)

p(µ|v) = N (mv,Λv) (10)

mv =(

Λ−10 +P

′Ω

−1P)−1 (

Λ−10 m0 +P

′Ω

−1v)

Λv =(

Λ−10 +P

′Ω

−1P)−1

.

Then, the predictive density of one period ahead returns is obtained by integratingover the unknown parameter µ

p(rT+1|v,Σ) =

Θ

p(rT+1|µ,Σ)p(µ|v)dµ, (11)

8

Page 9: The Black-Litterman model in the light of Bayesian portfolio analysis

which can be shown to result in (see Appendix 1 for a proof)

p(rT+1|v,Σ) = N(

mv,Σ+Λv

)

. (12)

An interesting effect of parameter uncertainty is that in the long-run (buy-and-holdinvestor), assets are viewed riskier than at short-sight. It can be shown that thek-period predictive density is (see Appendix 1 for a proof)

p(rT+k|v,Σ) = N (kmv, kΣ+ k2Λv). (13)

This effect of parameter uncertainty has first been noted by Barberis (2000).

9

Page 10: The Black-Litterman model in the light of Bayesian portfolio analysis

Black-Litterman model

• The Black-Litterman model

Black & Litterman (1992) suggest using the market equilibrium model as a priorobtained by reverse optimization5

µequ = γΣω∗mkt, (14)

where γ is the risk aversion of a power utility investor and ω∗mkt are the market

portfolio weights (fractions of the market capitalization). Black & Littermanassume a natural conjugate prior for the vector of means such that

p(µ) = N(

µequ, λ0Σ)

. (15)

The investor expresses views about µ by imposing

p(v|µ) = N (Pµ,Ω) . (16)

10

Page 11: The Black-Litterman model in the light of Bayesian portfolio analysis

It follows that the posterior of µ updated by the views is

p(µ|v) = N (mv,Λv) (17)

mv =(

(λ0Σ)−1

+P′Ω

−1P

)−1 (

(λ0Σ)−1

µequ +P′Ω

−1v

)

(18)

Λv =(

(λ0Σ)−1

+P′Ω

−1P

)−1

. (19)

Then, the Bayesian predictive density of one period ahead returns is obtained bythe same argument as in mixed estimation (see Appendix 1 for a proof)6

p(rT+1|v,Σ) = N(

mv,Σ+Λv

)

(20)

and the k-period predictive density is again

p(rT+k|v,Σ) = N (kmv, kΣ+ k2Λv). (21)

• An example of the Black-Litterman model is given in Appendix 2.

11

Page 12: The Black-Litterman model in the light of Bayesian portfolio analysis

Relating the Black-Litterman model to shrinkage estimation

• The Black-Litterman model can be aligned to shrinkage estimation by matrixalgebra.7

If P and Ω are m × m with full rank (n = m), and v is an m × 1 vector, themean of the posterior in (18) can be written in shrinkage form:

µv = δµequ + (I− δ)(P′P)−1

P′v, (22)

where I is an m×m identity matrix with principal diagonal elements of one andzero elsewhere. δ is called the posterior shrinkage factor. It can be shown that8

δ =(

(λ0Σ)−1 +P′Ω

−1P

)−1

(λ0Σ)−1 (23)

=(

[prior covariance]−1 + [conditional covariance]−1)−1

[prior covariance]−1

= [posterior covariance][prior covariance]−1.

12

Page 13: The Black-Litterman model in the light of Bayesian portfolio analysis

Conclusion (1)

• Bayesian portfolio analysis has a long tradition in finance.

• Given that the observation history is rarely a good predictor of the future, it isevident that information other than the sample statistics of past observations maybe very useful in a portfolio selection context.

• Furthermore, portfolio choices are by nature subjective decisions and not objectiveinference problems as the mainstream literature on portfolio choice might suggest.Therefore, there is no need to facilitate comparison.9

• The mixed estimation model and the Black-Litterman model in particular allowthe investor to combine different sources of information.

• If the investor uses the market equilibrium risk prima as a prior, the mixedestimation model is the Black-Litterman model.

13

Page 14: The Black-Litterman model in the light of Bayesian portfolio analysis

• Portfolios constructed from Black-Litterman model exhibit overall more stabilityin the optimal allocation decision compared to the case where sample statisticsare used.

14

Page 15: The Black-Litterman model in the light of Bayesian portfolio analysis

Appendix 1: Proof of mixed estimation

• Derivation of the posterior

The proof follows Satchell & Scowcroft (2000), Scowcroft & Sefton (2003), and Theil & Goldberger

(1961). The prior on the m × 1 vector of means µ is multivariate normal such that

p(µ) = N (m0,Λ0) (24)

where m0 is a m × 1 vector and Λ0 is a m × m and matrix assumed non-singular. In explicit

form, the prior can be written as

p(µ) = (2π)−m/2 |Λ0|

−1/2exp

−1

2(µ − m0)

′Λ

−10 (µ − m0)

(25)

= (2π)−m/2 |Λ0|

−1/2exp

−1

′Λ

−10 µ + µ

′Λ

−10 m0 −

1

′0Λ

−10 µ0

(26)

∝ exp

−1

′Λ

−10 µ + µ

′Λ

−10 m0

. (27)

The probability density of the views is also multivariate normal

p(v|µ) = N (Pµ,Ω) (28)

15

Page 16: The Black-Litterman model in the light of Bayesian portfolio analysis

where P is a n × m design matrix and Ω is an n × n matrix. n is the number of views and m

the number of assets. The explicit form of the views probability is

p(v|µ) = (2π)−m/2 |Ω|−1/2exp

−1

2(v − Pµ)′Ω−1(v − Pµ)

(29)

∝ exp

−1

′(P

′Ω

−1P)µ + µ

′(P

′Ω

−1v)

. (30)

Combining (27) and (30) using Bayes’ rule

p(µ|v) ∝ p(v|µ)p(µ) (31)

gives

p(µ|v) ∝ exp

−1

′(P′Ω

−1P)µ + µ

′(P′Ω

−1v)

×

exp

−1

′Λ

−10 µ + µ

′Λ

−10 m0

, (32)

which implies that the distribution of µ conditional on the views v is also multivariate normal.

16

Page 17: The Black-Litterman model in the light of Bayesian portfolio analysis

Collecting terms in (32), it follows that

p(µ|v) = N (mv,Λv) (33)

mv =(

Λ−10 + P

′Ω

−1P

)−1 (

Λ−10 m0 + P

′Ω

−1v

)

(34)

Λv =(

Λ−10 + P

′Ω

−1P

)−1

. (35)

• Derivation of the shrinkage form

If P and Ω are m × m with full rank (n = m), and v is an m × 1 vector, the above posterior

can be brought into shrinkage form by expanding the last term in (34) by P(P′P)−1

P′. Then10

mv =(

Λ−10 + P

′Ω

−1P

)−1 (

Λ−10 m0 + P

′Ω

−1P(P

′P)

−1P

′v

)

(36)

has shrinkage form and can be written11

mv = δm0 + (I − δ)(P′P)−1

P′v (37)

δ =(

Λ−10 + P

′Ω

−1P

)−1

Λ−10 . (38)

17

Page 18: The Black-Litterman model in the light of Bayesian portfolio analysis

• Derivation of the Bayesian predictive density

The Bayesian predictive density of one period ahead returns is obtained by integrating over the

unknown parameter µ

p(rT+1|v,Σ) =

Θ

p(rT+1|µ,Σ)p(µ|v)dµ, (39)

which can be shown to result in

p(rT+1|v,Σ) = N(

mv,Σ + Λv

)

. (40)

We can avoid the tedious effort of integration by making use of the well known properties of

multivariate normal densities. Note that p(rT+1|µ,Σ) = N(

µ,Σ) and p(µ|v) = N (mv,Λv).

Therefore, the partitioned matrix for the joint movement of rT+1 and µ is

[

µ

rT+1

∣v,Σ

]

∼ N

([

mv

mv

]

,

[

Λv H12

H21 H22

])

. (41)

The following equality must hold for the mean of the conditional density p(rT+1|µ,Σ)

µ ≡ mv + H21Λ−1v (µ − mv) (42)

18

Page 19: The Black-Litterman model in the light of Bayesian portfolio analysis

and therefore H21 = Λv. By symmetry of the covariance H12 = Λv. Furthermore, because

Σ ≡ H22 − ΛvΛ−1v Λv (43)

it is clear that H22 = Σ + Λv

The complete partitioned matrix is then (Bauwens, Lubrano & Richard, 1999, p. 300)

[

µ

rT+1

∣v,Σ

]

∼ N

([

mv

mv

]

,

[

Λv Λv

Λv Σ + Λv

])

. (44)

and therefore

p(rT+1|v,Σ) = N(

mv,Σ + Λv

)

. (45)

• k-period predictiv density. The argument is that the k-period sample density is p(rT+k|µ,Σ) =

N (kµ, kΣ) and the posterior density of µ is given by p(µ|v) = N (mv,Λv), then, the Bayesian

predictive density is obtained from solving the integral

p(rT+k|v,Σ) =

Θ

p(rT+k|µ,Σ)p(µ|v)dµ. (46)

It can be shown by the same argument as for the one period case, that the k-period predictive

density

p(rT+k|v,Σ) = N (kmv, kΣ + k2Λv). (47)

19

Page 20: The Black-Litterman model in the light of Bayesian portfolio analysis

Appendix 2: Example

• The Black-Litterman model and the idea of implied equilibrium risk premia is best illustrated

through an example.

• The investor is given the following descriptive statistics of six portfolios of all AMEX, NASDAQ

and NYSE stocks sorted by their market capitalization and book-to-market ratio.12

Table 1: Descriptive statistics

Size Book to Historical Volatility Correlations

Market risk premia

Small Low 5.61% 24.56% 1

Small Medium 12.75% 17.01% 0.926 1

Small High 14.36% 16.46% 0.859 0.966 1

Big Low 9.72% 17.07% 0.784 0.763 0.711 1

Big Medium 10.59% 15.05% 0.643 0.768 0.763 0.847 1

Big High 10.44% 13.89% 0.555 0.698 0.735 0.753 0.913

• The investor calculates equilibrium risk premia implied by market capitalization weights for

20

Page 21: The Black-Litterman model in the light of Bayesian portfolio analysis

preferences with different levels of risk aversion γ.13

Table 2: Equilibrium risk prima

Size Book to Market Equilibrium risk prima Historical

Market weight γ = 1 γ = 2.5 γ = 5 γ = 7.5 risk premia

Small Low 2.89% 3.07% 7.69% 15.37% 23.06% 5.61%

Small Medium 3.89% 2.21% 5.52% 11.03% 16.55% 12.75%

Small High 2.21% 2.04% 5.11% 10.22% 15.33% 14.36%

Big Low 59.07% 2.62% 6.55% 13.10% 19.64% 9.72%

Big Medium 23.26% 2.18% 5.44% 10.88% 16.32% 10.59%

Big High 8.60% 1.97% 4.91% 9.83% 14.74% 10.44%

• A striking result of Table 2 is the differences that exist between market equilibrium risk prima and

historical risk prima.

• For some reasons, the investor beliefs that the market has γ = 2.5 and expresses his personal

views identical to historical evidence, that is, the historical risk prima. Furthermore, his uncertainty

about these views is the historical variance. His confidence in market equilibrium is quite strong,

so he chooses to set λ0 = 1/T , with T = 20 years, the length of the observation history.14

21

Page 22: The Black-Litterman model in the light of Bayesian portfolio analysis

• The views of the investor translate into the following matrices:

P =

1 0 0 0 0 0

0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

, v =

0.0561

0.1275

0.1436

0.0972

0.1059

0.1044

(48)

Ω =

0.0603 0 0 0 0 0

0 0.0289 0 0 0 0

0 0 0.0271 0 0 0

0 0 0 0.0291 0 0

0 0 0 0 0.0227 0

0 0 0 0 0 0.0193

(49)

Using equations (17)-(19), the investor calculates the posterior of µ given the views: With

22

Page 23: The Black-Litterman model in the light of Bayesian portfolio analysis

equilibrium risk premium µequ and Σ given by

µequ =

0.0769

0.0552

0.0511

0.0655

0.0544

0.0491

, Σ =

0.0603 0.0387 0.0347 0.0329 0.0238 0.0189

0.0387 0.0289 0.0270 0.0222 0.0197 0.0165

0.0347 0.0270 0.0271 0.0200 0.0189 0.0168

0.0329 0.0222 0.0200 0.0291 0.0218 0.0179

0.0238 0.0197 0.0189 0.0218 0.0227 0.0191

0.0189 0.0165 0.0168 0.0179 0.0191 0.0193

,

the posterior is p(µ|v) = N (mv,Λv) with

µv =

0.0901

0.0657

0.0614

0.0751

0.0638

0.0542

, Λv =

0.0025 0.0016 0.0014 0.0013 0.0009 0.0007

0.0016 0.0012 0.0011 0.0009 0.0008 0.0006

0.0014 0.0011 0.0011 0.0008 0.0007 0.0007

0.0013 0.0009 0.0008 0.0012 0.0009 0.0007

0.0009 0.0008 0.0007 0.0009 0.0009 0.0008

0.0007 0.0006 0.0007 0.0007 0.0008 0.0008

.

The investor then calculates his optimal portfolio holdings implied by the Bayesian predictive

distribution

p(rT+1|v,Σ) = N (mv,Σ + Λv) (50)

23

Page 24: The Black-Litterman model in the light of Bayesian portfolio analysis

• Table 3 presents the portfolios held by the investor for different assumed models: (1) ωmkt if he

holds the market portfolio, (2) ωBL if he uses the Black-Litterman model with views as described

above, (3) ωhist if he uses historical risk prima.

Table 3: Optimal portfolios

Size Book to Optimal portfolio holdings

Market ωmkt ωBL ωhist

Small Low 2.89% 1.55% −206.46%

Small Medium 3.89% 7.57% 246.60%

Small High 2.21% 6.23% 65.79%

Big Low 59.07% 50.41% 133.02%

Big Medium 23.26% 22.68% −120.83%

Big High 8.60% 11.56% −18.12%

• Portfolio weights obtained with historical means take extreme positions, either short-selling or

excessive buying of only a few stocks. The portfolio weights obtained by the Black-Litterman

model are more stable and can be better matched with market equilibrium holdings.

24

Page 25: The Black-Litterman model in the light of Bayesian portfolio analysis

Footnotes1See, e.g., Campbell & Viceira (2003, p. 22), Barberis (2000).

2See, e.g., Barberis (2000), Kandel & Stambaugh (1996, p. 388), Rachev et al. (2008, p. 96), Wachter (2007, p. 14).

3See Barberis (2000), Brandt (2010, p. 308), Brown (1976, 1978), Kandel & Stambaugh (1996, p. 388), Klein & Bawa

(1976), Pstor (2000), Skoulakis (2007, p. 7), and Zellner & Chetty (1965).

4Mixed estimation is attributed to the work of Theil & Goldberg (1961). It is also presented in Brandt (2010, p. 313),

Satchell & Scowcroft (2000), Scowcroft & Sefton (2003). The Black & Litterman (1992) model is a special case of mixedestimation.

5Note that Black & Litterman (1992) assume simple returns. The exact formula for power utility and continuouslycompounded excess returns is µequ = γΣω∗

mkt − σ2/2 (see Campbell & Viceira, 2003, p. 30). The optimal

allocation for a power utility investor with U(WT+1) = W1−γT+1

/(1 − γ) and continuously compounded excess returns

is ω = 1γΣ

(

µ + σ2/2)

, where σ2 is the vector of the diagonal elements of Σ. However, for short investment horizon,

the optimal allocation will not be significantly different for continuously compounded excess returns.

6See also Rachev et al. (2008, p. 148).

7Posterior shrinkage is a generalization of the Bayes-Stein estimator (Jorion, 1986) and is a direct result from

reformulating the posterior obtained by an informative prior in shrinkage form.

8See, e.g., Greene (2008, p. 607), Hoff (2009, p. 108), Koop, Poirier & Tobias (2007, p. 26).

25

Page 26: The Black-Litterman model in the light of Bayesian portfolio analysis

9See Brandt (2010, p. 311).

10See Rachev et al. (2008, p. 146).

11See, e.g., Greene (2008, p. 607), Hoff (2009, p. 108), Koop, Poirier & Tobias (2007, p. 26).

12The values of the example are taken from Brandt (2010) who uses monthly data from January 1983 through December

2003.

13The values of the example are taken from Brandt (2010) who uses monthly data from January 1983 through December2003.

14See Rachev et al. (2008, p. 147), who also use λ0 = 1/T . Note that the view forecasts are assumed to beindependent. Therefore, Ω is a diagonal matrix.

26