投资组合管理：广义布莱克-利特尔曼模型

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"这篇文章是关于一个通用化的Black-Litterman模型，由Shea D. Chen和Andrew E. B. Lim撰写，并发表在Operations Research期刊上。该模型扩展了经典的Black-Litterman模型，用于处理投资组合经理在设定市场预期时所面临的不确定性、噪声和有偏信息。" Black-Litterman模型是一种在投资组合优化中整合投资者观点和市场均衡的框架，由 Fischer Black 和 Robert Litterman 在1992年提出。传统的Black-Litterman模型将投资者的主观观点与资本资产定价模型（CAPM）或套利定价理论（APT）等市场均衡理论相结合，以更准确地估计资产预期回报。在这个通用化的版本中，作者考虑了投资者对资产收益率的间接和有偏见解，这些见解基于组合收益率而非资产本身的收益率，这意味着它们是线性变换的结果。这种变换可能反映了投资者的特定策略或者风险偏好。此外，由于未来的不确定性，这些观点还包含了噪声项，这表示即使有经验的投资者也无法完全准确预测市场动态。文章详细阐述了如何在模型中处理这些噪声和有偏信息，以提高投资决策的稳健性。通常，Black-Litterman模型包含以下几个关键步骤：首先，确定市场的无偏期望回报率；其次，投资者表达他们的观点，这些观点可能与市场预期有偏差；然后，通过信息矩阵（Information Matrix）和视窗函数（View Window）将这些观点纳入模型；最后，通过贝叶斯公式更新市场预期，得到调整后的预期回报率，用于构建优化的投资组合。在通用化的模型中，作者可能会探讨如何量化和处理噪声，以及如何调整信息矩阵来反映不同信息源的可靠性。他们可能还讨论了如何适应不同的市场环境和投资策略，以使得模型更加灵活和实用。这篇研究论文对于理解和应用Black-Litterman模型在实际投资中的复杂性和灵活性提供了深入的见解，对于投资组合管理和金融工程领域的专业人士来说具有很高的价值。通过这个通用化的模型，投资者可以更好地整合不确定和有偏的市场信息，从而制定更为明智的投资决策。

necessarily understood. One implication is that biases

for different views change randomly from period to

period and m ay be dependent. (For example, different

components may correspond to views of different

experts, who may (or may not) interact personally,

read research from the same sources, etc.) We model

this by assuming that the view bias b

at time t is an

independent draw of a normal random variable with

parameters (δ

, Θ

).Wenote,however,thatthebias

is not observed. Rather, the decision maker only sees

the view V

and the subsequent return r

,wherethe

view is normal with a mean Pr

+ b

. (In contrast, the

view in the classical model (2) is unbiased and has

mean Pr

.) The parameters of the bias distribution

(δ

, Θ

) are estimated using historical view and return

data and used to correct for the bias at time t.

The graphical model Figure 2 generalizes the clas-

sical model Figure 1 and illustrates the multiperiod

nature of the data being u sed in the generalized

model, whereas Figure 3 extendsthisfurtherbyin-

cluding t he model of view bias. Faced with the sit-

uation depicted in Figure 3, the plan is to use the

history of views and returns, (V

, r

), ..., (V

t−1

, r

t−1

) to

estimate the statistical proper ties (δ

, Θ

) of the bias,

and to use this information to correct for the bias b

in the view V

so as to improve the forecast of the future

return r

. As before, the history of views and returns

, r

), ..., (V

t−1

, r

t−1

) and the view V

at the start of

period t are observ ed and appear in boxes. The other

random variables, including the future return r

,are

not observed and are depicted in circles.

Consider now t he model for equilibrium returns. In

the classical model (1), returns are random variables

where the mean μ

is constant and assumed to equal

the estimate α of e xpected return s from the CAPM,

 α. In reality, however, the actual expected return

is likel y to differ from the CAPM forecast because

of breakdowns in the assumptions underlying the

CAPM or the noninclusion of relevan t factors and

unmodeled return dynamics (Fama and French 2004).

In particular, there is concern about the impact of using

a misspeciﬁed equilibrium model on the portfolio allo-

cations that come from the classical Black–Litterman

model, and various methods have been proposed to

account for this error (e.g., Black and Litterman (1991)

uses the scalar τ to scale the covariance matrix of the

expected returns vector).

We account for errors in the CAPM forecast of ex-

pected returns (model misspeciﬁcation, unmodeled

factors, etc.) by dropping the assum ption that μ

constant, observable, and equal to α, but allow μ

to change from period to period according to in-

dependent samples of a normal random variable with

ameanα +b

. Here, b

is a constant correction term

that models bias in the CAPM forecast of expected

returns, whereas randomness in μ

models devia-

tions f rom the mean α + b

due to unmodeled sto-

chastic factors that inﬂuence the expected returns

from on e period to the next. These modeling a s-

sumptions extend Figu re 2 to the graphical model

found in Figure 4.Weemphasize,however,thatthe

expected return μ

and the correction b

are not ob-

servable but need to be estimated using knowledge of

the CAPM forecast α and the history of returns.

The generalized Black–Litterman model combines

both the expert (Figure 3) and equilibrium (Figure 4)

models and is shown in Figure 5. With t his blueprint

in mind, we can now discuss speciﬁcs.

Figure 2. Graphical Representation of Black–Litterman

After Multiple Periods

Figure 3. Graphical Representation of Generalized

Black–Litterman Accounting for Expert View Bias

Figure 4. Graphical Representation of Generalized

Black–Litterman Model Accounting for Equilibrium Errors

Chen and Lim: A Generalized Black– Litterman Model

3.2. Graphical Model

Our market consists of N risky assets and a risk-free

asset. The vector of expert views observed at the

beginning of period t  1, 2, ...is denoted by V

∈ R

and the vector of returns observed at the end of time

period is denoted by r

∈ R

.Wedenotethehistoryof

observed returns and expert views up to and including

period t by 5

≡ r

, r

, ..., r

{}

and 9

≡ V

, ..., V

{}

respectively.Atthestartofperiodt,theinvestorhas

a history of forecasts 9

t−1

{V

, ..., V

t−1

}and returns

t−1

{r

, ..., r

t−1

} an d a forecast V

of the return r

(which will be observed only at the end of the period).

The objective is to use these data, {9

t−1

, V

} and 5

t−1

to estimate the parameters and biases in the equi-

librium and forecast model and to generate a forecast

of the return r

, which is to be used to optimize the time

t asset allocation. We now provide details of the compo-

nents of the equilibrium return and forecast model.

3.2.1. Equilibrium Model. We assume throughout that

asset returns are normal with mean μ

and covariance

matrix Σ:

|μ

∼ N μ

, Σ



. (6)

The classical Black–Litterman model assumes that the

mean return μ

is dete rministic and equals the forecast

of exp ected returns α fr om the CAPM (i.e., μ

 α for

every t in (6)). In the generalized Black–Litterman

model, we assume that μ

is randomly drawn each

period from a normal distribution w ith mean α +b

and covariance mat rix β:

∼ N α + b

,β



. (7)

Here α is the CAPM forecast of e xpected return and is

known to the investor, whereas b

is a constant but is

unknown to the investor. The term b

models the bias

in the CAPM estimate of expected returns, whereas

the r andomness in μ

accounts for unmodele d sto-

chastic factors that change μ

over time. The term b

the average impact of these unmodeled stochastic

factors on the expected return, which can be learned

over time , whereas ﬂuctuations in these unmodeled

factors is captured by the assumption that μ

is latent

and generated as independent and identically dis-

tributed (i.i.d.) at the start of each p eriod with co-

variance β (and, hence, there is uncert ainty in μ

that is

impossible to r esolve).

Empirical failings of the CAPM are discussed in

Fama and French (2004) and provide some justiﬁca -

tion of our model. (Figures 2 and 3 in Fama and French

2004 are examples of annualize d expected returns

deviating signiﬁcantly from the predictions of the

CAPM.) Although many of th ese limitations of the

CAPM have been partly addre ssed by the Fama and

French (1993, 1996) three-factor model, it still re-

mains the ca se that these improved models are still

misspeciﬁed (e.g., the three-factor model is unable to

account for momentum effects; Fama and Fre nch 2004).

Thecovariancematrixβ is a par ameter of t he

model chosen by the investor that describes the un-

certainty in the latent expected return μ

.Byvirtue

of (7), one might choose β so that a reasonable range

of values for the expected return μ

is covered by the

spread associated with this normal distribution. For

example, β  0.2

I means that the 95% conﬁdence

region of μ

is ±1.96 ×0.2 ±0.392 (≈±40%). Alter-

natively, one can choose Σ and β so that the covariance

of returns under GBL is consistent with historical

returns. Both approaches, together with experimental

results, are discussed in Section 6.3.

We model uncertainty in the constant b

by spec-

ifying a prior N(δ

, Θ

), where the hyperparameters

(δ

, Θ

) are user speciﬁed and are updated using

Bayes’ rule conditional on the history of forecasts 9

t−1

and return s 5

t−1

availableatthestartofperiodt.

3.2.2. Expert Views. Views in the classical Black–

Litterman model are unbiased noisy observations

of (linear functions of) returns. In the generalized

Black–Litterman model, we allow for the possibility

that views a re biased by assuming

, b



∼ NPr

+ b

, Ω



, (8)

Figure 5. Graphical Representation of the Generalized

Black–Litterman Model that Accounts for Violations in

Black and Litterman’s(1991, 1992) Assumptions Regarding

the Market Equilibrium and Expert Views

Note. The terms in circles are unobserved factors at the start of time t

that need to be estimated, whereas the terms in the rectangles are data.

Chen and Lim: A Generalized Black– Litterman Model

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