market equilibrium estimate of the expected re turn
(see Walters 2013). One limitation of this approach is
that it downweights the forecasting information that
is available in the CA PM estimate and ignores the
possibility that systematic errors such as forecast bias
can potentially be estimated a nd removed. A nat ural
approach to address the acknowledged limitations of
theCAPMisexploredinKrishnanandMains(2005),
which includes additional factors in the equilibrium
model. Similar c oncerns about misspecification arise
in the context of the expert model, where much of the
focusiscenteredaroundhowconfidence levels for
experts can be chosen (see, e.g., Walters 2011,where
several methods are discussed).
In this paper, we propose a substantial general-
ization of the traditional model that accounts for
its shortcomings directly. Specifically, we propose a
Bayesian graph ical model t hat acco unts for syst em-
atic biases and unmodeled factors/dynamics in both
the expert and e quilibrium models and show how it
can be estimated in a Bayesian framework using a
history of expert forecasts and realized returns. We
characterize the optimal mean-variance portfolio in
terms of the mean and variance of returns under the
posterior and show how it can be estimated using
Gibbs sampling. We also show that this estimate is
consistent, asymptotically normal, and converges at
rate 1/
m
√
(where m is the number of Gibbs samples),
generalizing analogous results for sample average
approximation ( Shapiro et al. 2014)tothecasewhen
samples are dependent and there is a risk function
in the objective. Empi rical tests with simulated and
historical data are also provided.
As we have noted, much of the literature on the
Black–Litterman m odel stays close to the original
framework, and relatively little has been done to
address its shortcomings by generalizing the model,
which is one contribution of this paper. One nota-
bleexceptionisthepaperbyBertsimasetal.(2012),
which combines forecasts from the equilibrium model
and experts using ideas from inverse optimization.
The advantage of their framework is that it allows for
nonstandard expert opinions (e.g., on volatility) and
the use of risk measures such as conditional value at
risk (CVaR). It can also handle uncertainty about the
covariance matrix using ideas from robust optimi-
zation. W e note, however, that it does not address
concerns about misspecification in either the equi-
librium or expert models (and hence the forecast of
asset returns that is generated by combining these
models). Our paper is different in that we develop
equilibrium and view m odels with both learnable
bias and unresolvable estimation uncertainty using
probabilistic and Bayesian techniques and s how how
sampling methods can be used to compute optimal port-
folios. Although we focus on mean-variance optimization,
we believe that mean-CVaR problems can be addressed
by combining the posterior sampling methods used i n
this paper with the data-driven method from Rockafellar
and Urya sev (2000). In summary, both papers address
different limitations of the classical Black–Litterman
model with different tools.
The remainder of this paper is as follows: Section 2
covers the bac kground of the Black –Litterman model
and its graphical representation. Section 3 introduces
the generalized Black–Litterman model (GBL) and the
associated Bayesian graphical model. Section 4 dis-
cusses the problem of estimation and shows how
Gibbs sampling can be used to calibrate the model
and generate samples from the updated posterior
distribution of re turn s. The op timal por tfolio is char-
acterized in terms of the mean and variance of returns
of the posterior distribution in S e c ti o n 5 and can be
estimated using Gibbs sampling. We show that this
estimate is consistent and asymptotically normal and
converges a t rate 1/
m
√
(the number of Gibbs sam-
ples). Empirical tests on simulated and real data are
provided in Section 6. W e conclude in Section 7.
2. The Black–Litterman Model
The Black –Litterman model estimates future returns
by combining a backward-looking equilibrium model
with forward-looking expert views. We brieflysum-
marize the Bayesian perspective of the classical model
(see also Qian and Gorman 2001)andprovideagraph-
ical representation of this model.
2.1. Description
A central concern in a portfolio choice problem is
modeling the distribution of asset returns. The clas-
sical Black–Litterman model constructs this distribu-
tion by combining a backward-looki ng equilibrium
model with forward-looking expert views where the
following sequence of events is assumed: (i) at the
start of the investment period, the investor specifies a
“prior” distribution for the vector of asset returns that
is calibrated using a backward-looking equilibrium
model, (ii) the investor receives expert views which
are modeled as noisy observations of the future asset
returns, (iii) expert views are combined with the
equilibrium model using Bayes’ rule to generate an
updated return distribution, (iv) a portfolio allocation
is made on the basis of the updated distribution, a nd
(v) asset returns are realize d.
2.1.1. Equilibrium Model and Prior Distribution for
Returns.
Our financial market c onsists of N risky
assets and a risk-free a sset. Let r denote t he N-
dimensional vector of returns for the risky assets. We
assume without loss of generality that t he risk-free
rate is zero. (If the risk-free rate is nonzero, simply
replace r with the vector of excess returns and our
Chen and Lim: A Generalized Black– Litterman Model
2 Operations Research, Articles in Advance, pp. 1–30, © 2020 The Author(s)
