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View fusion vis-à-vis a Bayesian interpretation of Black-Litterman
for portfolio allocation.

arXiv:2301.13594v1 [q-fin.PM] 31 Jan 2023

Trent Spears*†, Stefan Zohren∗ , Stephen Roberts∗
February 1, 2023

Summary
The Black-Litterman model extends the framework of the Markowitz Modern Portfolio Theory to incorporate investor views. We consider a case where multiple view estimates, including uncertainties,
are given for the same underlying subset of assets at a point in time. This motivates our consideration
of data fusion techniques for combining information from multiple sources. In particular, we consider
consistency-based methods that yield fused view and uncertainty pairs; such methods are not common to
the quantitative finance literature. We show a relevant, modern case of incorporating machine learning
model-derived view and uncertainty estimates, and the impact on portfolio allocation, with an example
subsuming Arbitrage Pricing Theory. Hence we show the value of the Black-Litterman model in combination with information fusion and artificial intelligence-grounded prediction methods.
Keywords: Black-Litterman portfolio allocation, information fusion, machine learning, financial timeseries analysis.

1

Introduction

The Black-Litterman model extends the Markowitz portfolio optimisation framework to incorporate investor
beliefs about asset returns [1]. Such beliefs are termed ‘views’. In its original formulation, the BlackLitterman framework is such that a given view induces an update of a prior assumption about the mean
parameter of a Gaussian return distribution. Making use of views in this way can result in non-trivial
differences to the estimated weights of a portfolio allocation strategy, and hence the relative investment
outcome.
Views can be obtained from diverse sources, including market pundit opinions, Reserve Bank statements,
and as output of statistical models. Consequently, in practice, a portfolio manager or trader may possess
multiple views about the same underlying asset, or subset of assets, over the next investment period. A
contribution of this work is to both consider and extend the Black-Litterman framework to incorporate
multiple such views.
It is natural to formalise elements of the Black-Litterman model with respect to ideas from Bayesian statistics. See [2] for a review summary, with [3, 4] intriguing supplements. We note that the modelling of this
paper presupposes the canonical Bayes-based Black-Litterman formalism. This approach is useful for practitioners in allowing for a strong theoretical grounding of the model prior in the Capital Asset Pricing Model
(CAPM) [5], and for the relative ease with which one can incorporate market views. Specifically, views are
* University of Oxford, Oxford-Man Institute of Quantitative Finance.
†

Corresponding author. E-mail: trent@robots.ox.ac.uk

1

given as a function of asset mean return estimates, and accompanied by an uncertainty estimate that encapsulates a notion of confidence about that mean. The specification of the view uncertainty is a domain for
debate in the literature. In contrast, the uncertainty specification is of critical importance to the mechanics
of the Bayesian estimation framework. The literature often tacitly assumes that uncertainty is to be specified
in excess of the given mean estimate. A prevailing method is to set the uncertainty term in proportion to a
quantity such as an estimate of the underlying return covariance. The proportionality constant can then be
tuned akin to a model hyperparameter. Such ad hoc specification can be unsatisfying; fortunately, scientific
modelling offers an alternative.
Machine learning methods are enjoying a resurgence as oracles for financial markets, particularly given the
advances underlying modern deep learning. Indeed, such methods have utility for forecasting time-series,
and predictions are often naturally partnered with uncertainty estimates. Such uncertainties are typically
categorised in the literature as aleatoric or epistemic. Aleatoric uncertainty relates to the modelled random error of the underlying data generating process, while epistemic uncertainty encompasses much more,
including error relating to the model specification, which also subsumes uncertainties about parameter estimates. These two categories of uncertainty are recognised across multiple research domains, and the models
contained therein. In any case, the estimation of views by statistical forecasting models, coupled with the
uncertainty estimate, allows for a more principled approach to setting view parameters when modelling with
Black-Litterman. We describe this further in Section 3 below. This interpretation stands in stark contrast to
existing methods – with one popular technique described above – that are more ad hoc. Our approach has
the further benefits of preserving the intent of the canonical model, and has utility for automating an aspect
of the investment decision-making process.
Further our extended modelling framework, we consider combining multiple views for expected returns
over the next time increment with respect to view correlation that may be non-zero. Although we consider the single-step case, we are inspired by the relationship between Black-Litterman and a state-space
modelling framework as recently described in [6]. Indeed, at the core of our approach are techniques from
the information fusion literature, that allow us to combine views in a principled way. In Section 3, we
present methods for fusing views assuming both zero and non-zero correlation between them, and consider
a conservative fusion method for sets of views that may be inconsistent with one other (according to some
distance metric between distributions). In their relative simplicity, the principles of these fusion methods
are understandable, and the resulting fused estimate are explainable. Some practitioners may find this advantageous relative to alternative black box methods. In our empirical work, we find fusion-based methods
that improve on of the mean and median of our 3 view-generating models in a statistically meaningful way.
This is interesting to the investment practitioner that cannot know ex-ante which of a set of views might
outperform; a fusion-based method may be selected instead.
To show the utility of data fusion, and in connection with recent work by [4], we offer an application of the
Black-Litterman model assuming a multi-factor model for asset returns. Factor models present in finance
with various specifications; see [7] for a recent review. We consider a conditional factor model, as inspired
by the Arbitrage Pricing Theory (APT) of Ross [8]. In Section 2, before we further develop view fusion in
Section 3, we expound on the Black-Litterman-APT as a Bayesian hierarchical model; a schematic for the
model is given in Figure 1 below. In Section 2.3 we carefully derive the solution for the optimal portfolio
weights under an unconstrained Markowitz portfolio allocation as a further contribution to the literature.
With a desire to extend our empirical work, we offer a preliminary model accounting for transaction costs,
with details specified in Section 2.4 This model permits a reasonable proof-of-concept, that could be built
upon for industrial application or as future academic work.
Notes for the data set construction are offered in Section 4. To the extent the raw data is available in the

2

Figure 1: LHS: Schematic of the Black-Litterman methodology as it applies to the APT model. Asset returns
are linearly projected onto factor returns, and the algorithm at the level of the factor model is such that at
each time-step views are collected, before the factor returns model is updated and predictions gleaned. RHS:
Hierarchical schematic suggestive of the relationship between terms in the BL-APT model, as described in
Section 2.
public domain (via the Wharton Research Data Service), these notes could lead to a straightforward replication. In Section 5, we use the ideas and results of the preceding sections to give empirical results that
showcase view fusion whereby estimates – including measures of uncertainty – are sourced from a collection of machine learning models. We discuss the ways in which fusion-based methods lead to investment
outperformance relative to methods based on a singular view. We also constrast our methods with the difficult benchmark of a buy-and-hold investment in the S&P 500. We conclude and summarise avenues for
future work in Section 6.

2

Black-Litterman

2.1

A recap

Assume there exist n investable assets within a market with excess return vector r whose distribution is
modelled as multivariate normal over discrete time increments, denoted r ∼ MVN(µ, Σ). Here µ denotes
the mean vector of returns, and Σ denotes the return covariance. Then in a Markowitz Modern Portfolio
Theory framework [9], assuming a single-period investment setting, the investor portfolio weights w =
(w1 , ..., wn ) are a solution to the mean-variance optimisation problem:
max
w∈C

E(w0 r) −

γ
var(w0 r).
2

(1)

Here C is a set of weight constraints, if any, and γ > 0 is the investor risk aversion parameter, denoting
preferences balancing return and risk. In the unconstrained case, differentiating eq. (1), setting it to 0 and
3

solving for the optimal portfolio weights w∗ in terms of (µ, Σ) yields
w∗ = (γΣ)−1 µ.

(2)

On analysing the second derivative one finds that this solution is indeed a maximum. In practice, these
weights could be solved for – and a portfolio subsequently implemented – upon estimation of the return
distribution parameters, for example, upon substitution of a computed sample mean vector and sample
covariance matrix, given historical data.
The Black-Litterman model [1] extends this framework to incorporate (subjective) investor beliefs about
asset returns, that are called ‘views’. A single view is assumed to be a function of real-valued vectors
of length n denoting portfolio weights p and expected asset returns µ, a real-valued scalar denoting the
portfolio value q, and a strictly positive real-valued scalar  denoting uncertainty in the portfolio value, that
can be written
p0 µ = q + .

(3)

To be explicit, let us call a collection of k views a ‘range’, for natural numbers k. Then similar to the original
presentation of Black and Litterman, a range of views can be arranged thus:
P µ = q + ,

 ∼ N (0, Ω).

(4)

Here P ∈ Rk×n is a real-valued matrix of portfolio weights, where the k th row is given by p0k as defined
above; q ∈ Rk is a real-valued vector collecting k portfolio values; and Ω ∈ Rk×k is assumed to be a
diagonal matrix of portfolio value estimate uncertainties, where we have further assumed independence
between views.
Given the above formulation, a view can express both ‘absolute’ and ‘relative’ relationships between mean
asset returns. The case of an absolute (normalised) view corresponds to p taking a unit value at the ith
entry, and zero otherwise, so that the left-hand side of eq. (3) equals µi . Any other (non-trival) view we call
relative; a common relative view, for example, is a belief on the return spread µi − µj = q. This could be
expressed by setting alternate positive and negative unit values at the i and j index of p, and zero elsewhere.
In the work that follows, we assume absolute views. We further assume that P is of full rank.
Given these preliminaries, the Black-Litterman model is such that the existing return distribution parameter
assumption is updated given views. We understand such updating in the context of Bayesian statistics, as
prior authors have noted [2–4, 6].
The Bayesian specification assumes a prior distribution p(θ), where θ denotes the target parameter of the
return distribution that we wish to express via a parametric probabilistic model. Most commonly, θ = µ.
The original authors – and many since – set the prior given an estimate of the CAPM model [5], and hence
underwriting the prior with a foundational theoretical result.
Next, the view is related to the target parameter via a likelihood function p(q|θ), such that, by the laws of
conditional probability and Bayes’ Theorem, an updated posterior distribution for the target parameter is
given by
p(θ|q) ∝ p(q|θ)p(θ).
Finally, we can write a posterior predictive distribution for the asset returns given the views as
Z
p(r|q) = p(r|θ)p(θ|q)dθ.

(5)

(6)

Hence, the expectation and covariance of the random quantity r|q can be computed analytically – or otherwise estimated if the integral is intractable – before substitution into, say, eq. (2) to yield a solution to the
portfolio allocation problem.
4

2.2

Black-Litterman in the context of Factor models

In line with recent work by [4], we offer an application of the Black-Litterman model assuming a factor
model for asset returns, and we conveniently conform with some of their notation.
For context, we note that in many investment studies of equity markets the empirical estimation of the return
covariance is challenging, especially when the universe of equities under consideration is large. The linear
factor model that we outline below, whereby k  n, is relatively practical and efficient. This has contributed
to its popularity within the literature, and bolsters the case for industrial use [10]. We consider a conditional
factor model, as inspired by the Arbitrage Pricing Theory (APT) of Ross [8]. In the single-period investment
setting, we model
r = Xf + r ,

r ∼ N (0, D).

(7)

Here r is the n-dimensional random vector containing the cross-section of excess returns for n equities over
the next time increment [t, t + 1]; X is a non-random n × k matrix of observable factor exposures estimated
at t, for k the number of explanatory asset factors; D is a covariance matrix for the returns, also known at t
and assumed to be diagonal; and f is a k-dimensional latent factor vector that we estimate at each time step
by cross-sectional regression. Within a Black-Litterman framework, consider that our parameter of interest
is now θ = µf , such that
f = µf + f ,

f ∼ N (0, F ).

(8)

We refer to the elements of µf as the factor risk premia.
Black-Litterman in the context of APT, henceforth labelled ‘BL-APT’, amounts to a Bayesian hierarchical
model. A schematic for the model is given in the right-hand panel of Figure 1. We note that in comparison
to the Black-Litterman model applied to the return mean, an analogue to the CAPM prior does not exist.
In the work that follows, we set the prior parameters as a simple function of recent data, per the discussion
in Section 5. On the other hand, view estimates for the factor risk premia at time t induce return forecasts
(with uncertainties) over the next time period. These parameter estimates can then be utilised to estimate
portfolio weights.
In the next section, we carefully show the derivation for the optimal portfolio weights of an unconstrained
‘BL-APT’ as a reference contribution to the literature.

2.3

Solving for optimal weights – Hierarchical Bayesian Black-Litterman for APT

We can express our model for asset returns, factor returns and views carefully as a hierarchical Bayesian
model, with a suggestive schematic shown in Figure 1 above. We derive the optimal portfolio allocation
after finding p(r|q), as defined below.
We begin by writing the joint distribution of random variables of interest as a conditional probability:
p(r, f , θ, q) = p(r, f |θ, q)p(θ, q).
Substituting p(θ, q) = p(θ|q)p(q) and dividing both sides by p(q) yields
p(r, f , θ|q) = p(r, f |θ, q)p(θ|q).
Integrating both sides over θ we find that
Z
p(r, f |q) =

p(r, f |θ, q) p(θ|q)dθ.
|
{z
}
=p(r,f |θ)

5

Again integrating both sides, this time with respect to f , we have that
ZZ
p(r|q) =
p(r, f |θ)p(θ|q)dθdf .

(9)

With one more simplification, writing p(r, f |θ) = p(r|f , θ)p(f |θ) = p(r|f )p(f |θ), we find
Z
Z
p(r|q) = p(r|f ) p(f |θ)p(θ|q)dθ df .
|
{z
}

(10)

=p(f |q)

Since we assume θ = µf , there holds the well-known facts that the posterior p(θ|q) is normal and that the
posterior predictive p(f |q) is normal. Hence it is clear that p(r|q) is also normal, with its expectation and
covariance given in closed-form by the following three-step process.
Firstly, choosing the prior p(θ) ∼ N (ξ, V ), by Bayes’ rule for linear Gaussian systems, as in [11], we have
that the posterior p(θ|q) is multivariate normal with covariance and mean
var(θ|q) = (V −1 + Ω−1 )−1 ,
E(θ|q) = var(θ|q) · (V −1 ξ + Ω−1 q).
i.i.d.

Secondly, recall the convolution integral for sums of random variables: if Xi ∼ N (µi , Σi ), i = 1, 2, then
Z
f (z) := fX1 (x)fX2 (z − x)dx ∼ N (µ1 + µ2 , Σ1 + Σ2 ).
(11)
Writing the inner integral of eq. (10) with the change of variable f 7→ f − θ, we find
Z
p(f |q) = p(f − θ|θ) p(θ|q)dθ
| {z }
∼N (0,F )

so that by eq. (11) we have that p(f |q) is normally distributed with expectation and covariance given by
E(f |q) = (V −1 + Ω−1 )−1 · (V −1 ξ + Ω−1 q) = E(θ|q),
var(f |q) = (V −1 + Ω−1 )−1 + F = var(θ|q) + F .
Thirdly, we solve the outer integral of eq. (10) (deferring some of the details of the algebraic manipulations
to the Appendix, for brevity). Writing
Z
p(r|q) = p(r|f )p(f |q)df ,
(12)
we collect and expand the terms in the exponent up to a factor of − 21 , complete the square in f , and integrate
what is recognisable as a Gaussian integral. By the recursive nature of the integrals, it is clear that we also
retrieve a Gaussian for r|q. Hence, as supported by further calculations within the Appendix below, we
have that the posterior predictive distribution p(r|q) is multivariate normal with covariance and mean


−1
−1
var(r|q) = D −1 − D −1 X X T D −1 X + var(f |q)−1 X T D −1 ,

−1
E(r|q) = var(r|q)D −1 X X T D −1 X + var(f |q)−1 var(f |q)−1 E(f |q).
By eq. (2), we write the optimal weights for a single-step Black-Litterman portfolio allocation as
h∗blb = γ −1 var(r|q)−1 E(r|q)

−1
= γ −1 D −1 X X T D −1 X + var(f |q)−1 var(f |q)−1 E(f |q).

(13)

This expression differs meaningfully with other optimal weights given in the literature. This is the case in
that we have carefully accounted for all of the specified terms within the hierarchy of the returns model.
6

2.4

An optimal weight model under transaction costs

In this section we consider a more realistic objective function for solving for portfolio weights, in that it
incorporates transaction costs. We do this by making a minimal adjustment to the Markowitz objective of
eq. (1):
γ
max
E(w0 r) − var(w0 r) − T C,
(14)
w∈C
2
with T C denoting transaction costs measured in percentage of invested wealth. Of the many ways to account
for transaction costs, we broadly follow the single-period model recently presented in [12]. Despite the
differences in our application domains, our analysis and results contrast somewhat consistently, and certainly
interestingly, with theirs.
Firstly, we choose a transaction cost model inspired by [13] so that transaction costs measured in dollars,
T C d , is given by
1
T Ctd = ∆0t Λt ∆t .
2
Here ∆t is the portfolio turnover vector for a time increment, defined as
∆t := Πt (wt − Rt−1 wt−1 ),
and the ‘market impact’ vector m is given by
1
mt = Λt ∆t .
2
We use the subscript t to explicitly account for time-dependence. The real scalar Πt denotes total wealth
investable at time t. Clearly, given the above specification dollar transaction-costs erode dollar returns
quadratically with increasing wealth. The term Rt−1 denotes a diagonal matrix whose diagonal elements
account for the realized returns for the assets held in the portfolio at t − 1, over the time step [t − 1, t), and
adjusted for wealth delta. We assume that the term Λt is a diagonal matrix whose elements depend on the
daily (dollar) volume traded in each underlying asset. Given the recent estimates of [14], we assume a 10
basis point market impact when trading 1% of the daily dollar volume Lt of US equity underlyings whether
long or short, so
1
mt = 10bps · 1 = Λt × 1%Lt
2
1
=⇒ Λt = L−1
.
5 t

(15)

In the work that follows, we set the values for daily dollar volumes at their 6 month rolling average. Further,
we assume that investor wealth grows proportionally with the market; specifically, we make the somewhat
arbitrary assumption that at each time step the total investable capital is one-tenth of the sum of the daily
dollar volumes of the assets in our trade universe, as known at the most recent time step.
Hence eq. (14) can be written, noting that T C = T C d /Π, as
max

wt ∈C

 Πt
 γ
0

∗
∗
wt − Rt−1 wt−1
Λt wt − Rt−1 wt−1
E wt0 rt − var wt0 rt −
2
2

and conveniently solved in closed-form for optimal weights:
−1

∗
wt∗ = γΣt + Πt Λt
µt + Πt Λt Rt−1 wt−1
.

(16)

(17)

Hence the optimal portfolio weights for the BL-APT model can be found by substituting estimates for
E(r|q) and var(r|q) into eq. (17), as an alternative to eq. (13), when taking transaction costs into account.
7

3

View fusion

3.1

Preliminaries

The Black-Litterman model – sans solving for the portfolio weights – can be expressed via a state-space
representation; this was recently presented in [6] in the context of modelling temporal dependency in asset
returns and parameter view estimates1 . In a simple case, consider ranges of views generated from a source
at discrete time increments k = 0, 1, ... . We can infer the underlying target mean µk in an online way, via
the linear time-varying system of equations:
µk+1 = µk + µ
k,

(18)

qk = Pk µk + qk .

(19)

q
Here µ
k and k model independent zero-mean white-noise processes with respective covariances Ψk and
Σk . We call equations (18) and (19) the ‘state’ and ‘measurement’ equations, respectively. Solving the statespace representation of Black-Litterman, assuming absolute views, we have the probability distribution for
µk |qk , analogous to eq. (5) and solved equivalently to solving for p(θ|q) per Section 2.1. The predictive
distribution for µk+1 |qk , analogous to eq. (6) can be solved equivalently to solving for p(f |q), also of
Section 2.1. On the other hand, within the Black-Litterman literature, as in our model from Section 2,
the prior is typically re-initialised to p(µ0 ) at each time-step, rather than dynamically updated given a
sequence of view estimates. In the following work, we follow this approach, and leave consideration of
prior updates based on a larger subset of recent data as future work. In any case, such state-space modelling
naturally inspires an extension of the Black-Litterman framework to handle the combination – or ‘fusion’ –
of multiple views held about a particular portfolio.

3.2

Information fusion

To this point we have described the Black-Litterman model assuming a singular source for a range of views.
Next, we consider the realistic – though in the context of Black-Litterman modelling, unexplored – case of
multiple sources generating views simultaneously for each time increment, and analyse the impact on the
predictive distribution. We extend eq. (19) thus (for brevity, keeping to the case of absolute views):
(q)

qk,s = µk,s + k,s ,

(20)
(q)

where qk,s denotes the view is from source s, 1 ≤ s ≤ S, and each k,s is a zero-mean white-noise process
with covariance Σk,s . Multiple views could realistically arise in the context of implementing a systematic
trading or asset management strategy, whereby multiple predictions across multiple (potentially, broadly
diverse) models need to be fused before capital allocation.
Information fusion has been common to the state-space literature since the work of [16]. However, fusion
is not limited to the state-space framework, and instead applies more generally to the problem of fusing
multiple estimates of an underlying random variable2 . Correspondingly, data fusion can take place at the
level of the state or measurement equations, though we assume the former for our empirical work below.
Typically, fusion techniques amount to solving for an optimal fused estimate ( b
µ, b
Σ) for the true underlying
b
b s , Σs ) : 1 ≤ s ≤ S}, with the mean
values (µ, Σ) given a collection of S underlying estimates {(µ
1

There exists an extensive literature on state-space modelling, with applications across many domains. See, for example, [15] for
an instructive review of filtering methods for mathematical finance.
2
Various approaches exist in the data fusion literature for incorporating information from multiple sources; introductory reviews
can be found in [17, 18].

8

Figure 2: Concentration ellipses for three example bivariate Gaussian measurements of a single state x =
(x1 , x2 ) (light orange), and the optimal fused state estimates for the four methods outlined in Section 3.
parameter a weighted linear combination of the underlying estimates:
X
b=
bs.
µ
Ws µ

(21)

s

b is minimal
[We have omitted the subscript k, for ease of notation.] ‘Optimal’ fusion is such that (i) Σ
in some sense – typically with respect to matrix trace or determinant; and also that (ii) b
Σ is consistent.
b
b
Within the fusion literature, a consistent estimate is such that Σ ≥ Σ, in the sense that Σ − Σ is positive
semi-definite. That is, the consistent estimate subsumes a covariance no less than the covariance Σ of the
distribution of the true underlying process; see, for example, [19]. We note that this definition is independent
of the well-known definition of a consistent estimator from the statistical literature, whereby a parameter is
called consistent if it converges in probability to the true value of the parameter. The consistency property is
crucial for many investors conscious of risk, in particular those that do not want to understate risk3 . Another
benefit of consistency is that probabilistic bounds for the state of the system can be deduced, as discussed in
[21].
Given the mean and covariance of a bivariate random variable, recall the ‘concentration ellipse’ [22] defined
as the locus of points {x : (x − µ)T Σ−1 (x − µ) = 1}. For visual intuition, we depict such ellipses,
estimated using the fusion methods described in the following subsections, in Figure 2 above. These results
are shown for the case of three underlying data sources, assumed to each yield a noisy estimate of the same
target.
3

In the case of (multi-period) Kelly investing, to which the fusion of this paper could directly apply, investment size is inversely
proportional to variance. But the investor faces almost sure ruin if they sufficiently over-invest (which can happen if uncertainty is
estimated to be too small) relative to the optimum strategy [20].

9

3.2.1

Precision-weighted fusion

We first consider precision-weighted (PW) fusion for the case that the cross-covariances between sources is
b s is a known error covariance
b s is an unbiased estimate, and each Σ
known to be 0. Suppose that for each s, µ
estimatePthat is consistent. It follows that the fused estimate given in eq. (21) is an unbiased estimate if and
only if s Ws = I, for weight matrices Ws , and I the identity matrix. To find the weight estimates we
solve
X
Ws = I.
min trace( b
Σ) subject to
W

s

Let b
Σij denote the cross-covariance estimate between the unique sources si and sj , and write
X
X
b sW T +
b ij W T .
b =
Ws Σ
Wsi Σ
Σ
s
sj
s

(22)

si 6=sj

Assuming the cross-covariances are everywhere 0, solving for the optimal weights one finds that, on weight
substitution,
b −1 )−1 ,
b = (Σ
b −1 + ... + Σ
Σ
1
S
b= b
b 1 + ... + b
b S ).
µ
Σ( b
Σ−1 µ
Σ−1 µ
1

S

(23)
(24)

Further details of this well-known result can be found in [16, 23]. The analogous result for fusing two
sources that have known, non-zero correlation is given in [24], though for our financial modelling framework
this is less useful since the cross-correlations are not estimated. On the other hand, the case of unknown,
potentially non-zero cross-correlation is the domain of the Covariance Intersection method, as presented in
the following subsection.
3.2.2

Covariance Intersection

One could reasonably expect that views on the same underlying asset returns, as functions of similar or
equal underlying data, exhibit non-zero cross-correlation. However, the cross-correlation is not necessarily
known, or easily estimable. Such a setting is the domain of application for the Covariance Intersection (CI)
b s}
fusion method of [25]. The insight of this approach is that, given eq. (22), the covariance ellipses for {Σ
b
contain Σ in their intersection, for any values of the cross-covariance terms. Hence the method yields
X
b −1 =
b −1 ,
Σ
ωs Σ
(25)
s
s

b
b=Σ
µ

X

b −1 µ
ωs Σ
s bs,

(26)

s

P
b encloses the
for scalars ωs ∈ [0, 1] with s ωs = 1, since a convex combination of covariances ensures Σ
intersection region. The ωs terms can be computed via a standard optimization routine subject to minimising
b Further, it has been shown that the CI method applied to a pair of consistent estimates yields
the trace of Σ.
a consistent estimate [19]. Finally, equations (25) and (26) as given above extend the canonical case of two
sources, as offered in [26].
CI is a straightforward method that holds for any cross-covariance, but a cost of its flexibility is that the estimate can be too conservative for many practical use cases. We next consider Inverse Covariance Intersection
that attempts to retain the consistency benefit of CI, while being less conservative.

10

3.2.3

Inverse Covariance Intersection

The method of inverse Covariance Intersection (ICI) is similar in approach to that of CI, but achieves a less
conservative estimate while still maintaining consistency. The algorithm was presented recently in [27]. A
central insight, inspired by [28], is to first consider estimates pairwise, such that for s = 1, 2:


b s (Σ
e s )−1 µ
e s + Γ−1 γ ,
bs = Σ
(27)
µ

−1
b
.
(28)
Σs = ( e
Σs )−1 + Γ−1
es, e
This is to say, that each estimate is itself the PW fusion of a sub-estimate from the collection {(µ
Σs ) :
1 ≤ s ≤ 2}, for sub-estimates uncorrelated between sources, and a shared set of information denoted
(γ, Γ), also uncorrelated with each sub-estimate; we proceed assuming such a decomposition exist. If the
shared information is known, then fusion can be solved optimally by subtracting the shared covariance from
the PW fusion covariance for the original pair of estimates [27]. But since it is not known, the second
insight is to instead solve for, and subtract, an upper bound for the shared covariance, valid for any true
underlying shared information. Further, regardless of the shared information, it is shown that a consistent,
tight covariance estimate – if it exists – is of the form
b
Σ−1 =

2
X

b
Σ−1
s −

2
X

ωs b
Σs

−1

,

(29)

s=1

i=1

for scalars ωs as in the case of CI, with the ICI covariance estimate achieving this form exactly [27]. With a
modicum of algebra, it follows that the fused mean is given by
b=
µ

2
X

bs,
Cs µ

(30)

s=1
2

X
−1 
−1
b
b
bs
where Cs = Σ · Σs − ωs
ωs Σ
.
s=1

It is important to regonise that this formula does not generalise to more than two information sources in an
obvious way – the ICI algorithm fuses information sources pairwise. However, ICI can be applied recursively. This follows from the assumption that any pairwise-fused estimate has a decomposition analogous
to equations (27) and (28) [27]. For further results and details we also recommend [29].
3.2.4

Covariance Union

The fusion methods outlined above assume that each source’s estimate is itself consistent with respect to
the target. We relax this assumption by requiring that at least one of the source estimates is consistent. A
given estimate could be inconsistent, as evidenced by, say, mean components sufficiently different between
estimates, given relatively smaller covariance estimates. This could be apparent, for example, when the
Mahalanobis distance between some pair of estimates exceeds a critical threshold. Further, we may not
know precisely which of the underlying estimates is consistent, or it may be (in some sense) costly to
resolve. The Covariance Union (CU) algorithm has been proposed as a fusion method for this setting [30].
CU constructs a fused estimate guaranteed to be consistent, since the fused estimate is consistent with
respect to each underlying estimate:
b
b−µ
b s )(µ
b−µ
b s )T ,
Σ≥ b
Σs + (µ
11

1 ≤ s ≤ S.

Figure 3: Cumulative factor returns to risk premia as described in Section 2.2; replication of Figure 1 of [4].
A single year is shown for ease of visualisation. The year 2011 is arbitrarily chosen.
While the idea behind CU is relatively simple, the numerical implementation for estimating the fused estimate is more challenging. In an attempt to simplify the computation, we offer the following details. Firstly,
CU can be implemented using non-convex optimisation software such as SolvOpt [31]. We follow [32] for
implementing a SolvOpt-based solution for calculating the results of Section 5 below. Secondly, a key step
is to define the coefficient vector that we wish to solve for, x. The first n elements of x are the elements of
b Then x becomes the
b
elements of x are the elements of the upper triangle of Σ.
µ, and the remaining n(n+1)
2
b subject to the constraint that
optimization target for SolvOpt. We seek to minimise the determinant of Σ
∀s,

b ∗ := Σ
b −Σ
b s − (µ
b−µ
b s )(µ
b−µ
b s )T ≥ 0.
Σ
s

(31)

A particular feature of the SolvOpt software is that it requires a constraint c(x) such that, for any input,
b ∗ is required to be positive semi-definite, it must have non-negative eigenvalues for all
c(x) ≤ 0. Since Σ
s
choices of s, and the constraint is satisfied for a choice of c such that
b ∗ ) : ∀s}} ≥ 0.
−c(x) := min{{eigenvalues(Σ
s
Finally, for the interested reader, the works of [32, 33] offer further efficient approximation methods for
implementing CU, as required for particular practical use cases.

4

Data

We describe the data set construction for replicating the empirical example for BL-APT of [4]. This construction matches the original descriptions as closely as possible, and we make note of any additional required assumptions made. While we have attempted to carefully reproduce the database, our results need
12

not match exactly. Not in the least, we have no guarantee that the underlying database that we call is the
same 5 years later, even if our data replication process is perfectly faithful. A visual comparison can be
made between our factor values against the source given Figure 3 above, which demonstrates a reasonable
replication.
The data is sourced fom CRSP and IBES databases with access granted via the Wharton Research Data
Service. The original work sourced daily US equity market data from 1992-2015; we extend our data set till
2022 to include the market crash of March 2020, and subsequent market rebound, for interest’s sake.
For each day, set n = 2000 and select the top n stocks within the US market sorted by market capitalisation.
The data set is filtered for USD denominated common stock only – there are no closed end funds, REITs,
ETFs, unit trusts, depository receipts, warrants etc. We also collected the S&P500 (excess) return timeseries, stock market capitalisations, and earnings per share, adjusted for splits.
The data set requires the creation of 5 risk factors: market beta, size, volatility, momentum and value. The
reasonableness of our data set construction is per the output of Figure 3, which accords with expectations.
We also create 70 industry factors; these are one-hot encoded based on the industry given by SIC. Calculating
size is relatively straightforward in CRSP as the product of shares outstanding and the daily close price.
Value was determined by analysing IBES data, and could be merged back to the CRSP database based on
the CUSIP identifier. We note that risk factor construction is often non-trivial. Momentum was codable
using CRSP data and the definitions of [34]; we calculated the daily compounded 12 month returns sans the
most recent 1 month of data. We created the market Beta and volatility factors using the relevant CRSP data
and the details of [4]. Indeed, over a two year rolling window we estimate a collection of linear functions
by regressing each asset’s daily excess returns against the S&P500 excess returns. The market Beta is the
regression gradient, while the volatility factor is set to be the regression mean-square error.
The data is constructed for bi-monthly rebalancing to facilitate the experiments of the following section.

5

Results

View models. To proxy multiple sources of view generation, we implemented various models with utility
for forecasting financial time-series. Indeed, machine learning models have proven useful for not only
prediction, but also for estimating prediction uncertainty. We implement an ARIMA, boosted regression and
Gaussian Process model for generating views, and calculate model-derived uncertainty estimates. Relevant
details are included in the Appendix, since time-series prediction modelling is not the primary focus of this
paper. We denote the three view models ‘ARIMA’, ‘Boost’ and ‘GP’, respectively. We also implement the
4 fusion methods outlined in Section 3.
Parameter choices. Per Figure 1 and the outline in Section 2, the BL-APT model requires the specification
of q, F , Ω, ξ, V and D. We set q as the mean prediction estimate given by a view model, with the view
models as previously described. The accompanying epistemic uncertainty estimate is set to Ω, and F is
set equal to the aleatoric uncertainty estimate. The prior hyperparameter ξ is estimated as an average of f
over a recent rolling lookback window of the most recent 20 observations. This window size was arbitraily
decided, and is the same for all rolling windows introduced hereafter. The total variance of each underlying
factor is calculated relative to out-of-sample data, and we yield an approximation to epistemic uncertainty
by the estimation method as described for the ARIMA model in the Appendix below. We set V as this
estimate. Our estimate for D is a diagonal covariance matrix equal to σ
b2 I, where σ
b2 is the mean square
error yielded from estimating eq. (7), and I is the identity matrix. Finally, we set the relative risk aversion
parameter to a constant value of 10.
Transaction costs. An effect of accounting for transaction costs, and carefully setting the transaction cost
model parameters, is to greatly reduce portfolio turnover relative to the unconstrained Markowitz approach
13

Figure 4: View-based portfolio excess return performance net of trading costs. Benchmark SPX strategy is
gross of (negligible) transaction costs, but total returns are calculated in excess of the risk-free rate.
with costs assumed nil. Indeed, without transaction costs, it is well-known that this latter approach usually
result in unrealistic and unreasonably large leverage suggested for the optimal portfolio, and high portfolio
turnover. Recent results of [12] show that incorporating an appropriate transaction cost model goes some
way to mitigate these limitations in empirical studies. We implement the model outlined in Section 2.4 above
to the same ends. We set our portfolio rebalance period to be bi-monthly, which is sufficiently long – given
our assumed wealth process, and the potential return being signficantly larger than the cost of rebalancing –
such that the reward of updating weights is not dominated by the cost of trading.
Results – benchmark. We have tabulated performance results by year and model over a 29 year period, and
include them in the Appendix. The method ‘S&P500 (TR)’ denotes the total return of the S&P 500 for the
year. The annual returns may appear slightly less than commonly reported since they are net of the risk-free
rate. The S&P500 (TR) represents a useful benchmark in that it represents the market buy-and-hold strategy,
accessible to the modern day investor and with a potential for exceptionally competitive fees, especially in
the more recent years. For comparisons sake, all results dependent on an underlying view model are reported
based on normalised underlying data, such that ex-post annualised return volatility is equal to that of the
benchmark portfolio. We make a note that, on perusing the tables, there is obvious decay in the performance
of the view-based models over the 30 year period, as evidenced, for example, by the significant decline in
Sharpe ratios over time.
Results – statistics. We depict the Sharpe ratio statistics from the tabulated data in Figure 4 above. The leftmost chart shows box & whisker plots for the annual Sharpe ratio for the collection of portfolio’s over the 29
year period. Investment in the S&P500 (TR) is clearly superior to our underlying modeling approach. On the
other hand, there are improvements that could increase the view model performance that we deem beyond
the scope of this work. For an easier comparison of centrality for the various method’s Sharpe distributions,
we show each portfolio’s median and mean Sharpe ratio over the 29 year period in the right-most chart.
This visual depiction summarises how fusion and non-fusion based methods performed relative to each
other through time (and relative to the market total return portfolio). We observe that fusion-based medians
14

are increasing through PW (0.11), CU (0.16), CI (0.38) and ICI (0.48), with the means showing a similar but
less pronounced trend. The set of mean Sharpe ratios for the single-view based methods have a smaller range
than the medians (0.21-0.30 versus 0.01–0.53, respectively); similarly for the fusion-based methods (0.17–
0.31 versus 0.11–0.48). When considering the mean Sharpe ratio over the full data sample, ICI outperforms
amongst the set of fusion-based methods and the single-view-based models. Similarly to the median, ICI
outperforms amongst the set of fusion-based methods and sits closely to the best of the single-view-based
models. We test for statistical significance of differences in these measures of distribution centrality per the
results in Figure 5 below. Despite the small sample sizes there is some interesting supporting evidence for
the outperformance of fusion-based methods.
We test for the difference of means between methods with a collection of pairwise 1-sided t-tests. Such
tests make a critical assumption of independence in the pairwise Sharpe ratio differences as indexed by year.
We note that despite individual time-series of model performance statistics potentially displaying serial
correlation, it seems reasonable to suggest that the the performance between models over the years does
not. This is clearly a critical claim. We examine the autocorrelation function for the 28 possible pairwise
comparisons and find 1 case of rejecting the null hypothesis of no autocorrelation in a Ljung-Box test at
the 10% level of significance. We further assume normality of the differences and test this with a (onesided) Shapiro-Wilk test for normality, also at the 10% level of significance. We find that this assumption
is rejected for 3 cases. Finally, we assume the non-existence of outliers in the difference data, and validate
this claim on visual inspection of box and whisker plots.
On testing for the difference of means with a paired-t test, we find, of particular interest, that ICI outperforms
the ARIMA and GP models, while CI outperforms the ARIMA only (though in this case the ShaprioWilks test leads us not to rely on some of the t-test p-values). To support testing for significance in the
difference of means between groups, we bootstrap 2-sided bias-corrected and accelerated (BCa) bootstrap
90% confidence intervals for paired data [35]. We find evidence that ICI outperforms the ARIMA and GP
models, and again the CI outperforms the ARIMA model. We also test for the difference in medians between
groups with BCa confidence intervals, but they are broadly less conclusive. This is potentially owing to the
small sample size. On the other hand, we have evidence that both ICI and CI outperforms GP on this metric.
Alternatively to testing for the difference of medians across all samples, we can test for the median of the
differences between pairwise annual Sharpes. To this end, we utilise the Wilcoxon signed-rank test for
paired samples [36]. In addition to assuming independence, a critical assumption is that of symmetry about
the distributions of paired differences. Given the results of the Shapiro-Wilk tests, and visual inspection of
what appear to be reasonable QQ plots, we continue assuming symmetry is satisfied. We perform 1-sided
tests at the 10% level of signficance and find evidence that ICI outperforms the ARIMA model for this
comparison.
It is interesting that the CI and ICI methods achieve more compelling outperformance than the CU and PW
fusion methods, relative to the underlying view models. The assumption of PW that the underlying views
have 0 cross-covariance may be suboptimal, such that the method is yielding inconsistent estimates and in
turn impacting trading performance. On the other hand, perhaps CU could be utilised more sparingly and
strategically, such as when the underlying views have relatively high dispersion and so appear sufficiently
contradictory. An alternative fusion method such as ICI could be employed otherwise.
Corroborating our intuition about the relative strength of the S&P500 (TR) strategy relative to all others, we
see that SPX improves on all models – both single-view- and fusion-based – for almost all tests. Sufficiently
strong view models are required to compete with the market portfolio, but beyond the scope of this work. It
is perhaps surprising that view- and fusion-based models net of transaction costs have net positive Sharpe
ratios at all. Indeed, we estimate global Sharpe ratios between 0.10 (PW) and 0.34 (ICI) for the entirety of
the 29 year period.

15

Pairwise
comparison

Ljung-Box

Shapiro-Wilk

paired-t p-vals
(diff of means)

BCa CIs
(diff of means)

BCa CIs
(diff of med.)

Wilcoxon p-vals
(med. of diffs)

ICI
CI
CU
PW
ARIMA
GP
Boost

0.14
0.14
0.12
0.17
0.13
0.17
0.16

0.21
0.45
0.75
0.36
0.41
0.51
0.32

0.06
0.05
0.05
0.03
0.03
0.03
0.05

[ 0.15, 0.98 ]
[ 0.15, 0.99 ]
[ 0.19, 1.01 ]
[ 0.29, 1.22 ]
[ 0.25, 1.09 ]
[ 0.20, 1.09 ]
[ 0.18, 1.01 ]

[ -0.13, 0.90 ]
[ -0.07, 0.70 ]
[ 0.31, 1.03 ]
[ 0.07, 1.30 ]
[ 0.17, 0.92 ]
[ 0.34, 1.10 ]
[ -0.10, 0.83 ]

0.07
0.07
0.06
0.06
0.03
0.03
0.08

ICI

CI
CU
PW
ARIMA
GP
Boost

0.80
0.88
0.94
0.70
0.33
0.68

0.03
0.12
0.90
0.75
0.65
0.19

0.47
0.41
0.15
0.08
0.09
0.46

[ -0.08, 0.08 ]
[ -0.11, 0.14 ]
[ -0.03, 0.32 ]
[ 0.02, 0.21 ]
[ 0.01, 0.19 ]
[ -0.21, 0.25 ]

[ -0.26, 0.28 ]
[ 0.01, 0.67 ]
[ 0.04, 1.00 ]
[ -0.26, 0.36 ]
[ 0.25, 0.73 ]
[ -0.51, 0.39 ]

0.31
0.63
0.19
0.10
0.12
0.58

CI

CU
PW
ARIMA
GP
Boost

0.56
0.67
0.47
0.11
0.35

0.85
0.56
0.00
0.74
0.91

0.43
0.15
0.06
0.13
0.47

[ -0.12, 0.14 ]
[ -0.02, 0.32 ]
[ 0.03, 0.19 ]
[ -0.01, 0.20 ]
[ -0.17, 0.24 ]

[ 0.05, 0.51 ]
[ -0.29, 0.65 ]
[ -0.17, 0.22 ]
[ 0.30, 0.67 ]
[ -0.52, 0.10 ]

0.55
0.18
0.93
0.16
0.50

CU

PW
ARIMA
GP
Boost

0.29
0.48
0.75
0.73

0.28
0.89
0.22
0.98

0.17
0.23
0.23
0.49

[ -0.04, 0.28 ]
[ -0.05, 0.22 ]
[ -0.06, 0.20 ]
[ -0.21, 0.19 ]

[ -0.57, 0.43 ]
[ -0.46, -0.02 ]
[ -0.13, 0.37 ]
[ -0.78, 0.01 ]

0.17
0.20
0.21
0.52

PW

ARIMA
GP
Boost

0.95
0.95
0.17

0.64
0.58
0.45

0.36
0.33
0.09

[ -0.17, 0.11 ]
[ -0.19, 0.08 ]
[ -0.27, -0.03 ]

[ -0.65, 0.24 ]
[ -0.31, 0.66 ]
[ -1.12, -0.14 ]

0.65
0.41
0.10

ARIMA

GP
Boost

0.01
0.20

0.94
0.06

0.46
0.27

[ -0.08, 0.07 ]
[ -0.29, 0.09 ]

[ 0.22, 0.54 ]
[ -0.51, 0.19 ]

0.45
0.73

GP

Boost

0.74

0.34

0.27

[ -0.26, 0.07 ]

[ -0.96, -0.32 ]

0.64

16

SPX

Figure 5: Tabulated p-values and confidence intervals for a battery of statistical tests for comparing annual 29 years of annual Sharpe ratios pairwise
between strategies. Yellow highlighting is applied for significance at the 10% level. The rightmost four columns represent one-sided tests.

Closing remark. The results presented offer preliminary evidence supporting fusion-based methods within
a Black-Litterman framework subsuming a relatively complex investment strategy net of transaction costs.
Fusion-based methods result in a weighted-average performance respectful of uncertainty estimates. These
methods could certainly be ex-ante preferred when all views are to be incorporated into an investment decision, in the case that the best view model is unknown (and/or as in our presented work, when the method
constituting the best view model is both largely unpredictable, and changing throught time). We have supporting evidence for a rare ‘free lunch’ in financial investing by way of model fusion, essentially amounting
to a benefit we could call ‘model diversification’. This could be of further application in alternative trading domains, perhaps with more sophisticated underlying view models, and we hope to inspire readers to
consider their practical use.

6

Conclusion and next steps

In this work, we further characterise view uncertainty within a Bayesian Black-Litterman framework. In
doing so, we offer optimal portfolio weights for a case of Black-Litterman in an Arbitrage Pricing Theory
setting. Considering the Bayesian Black-Litterman model via an equivalent state-space representation, we
are inspired by the information fusion literature for combining correlated prediction mean and uncertainty
estimates streamed from diverse sources. Hence, we demonstrate prediction and uncertainty fusion for
multiple, simultaneous views within the Black-Litterman framework, and describe consistent uncertainty
estimation within financial investing contexts. In our empirical work, we demonstrate the utility of our
approach for BL-APT. Future practical work could be based on more realistic investment strategies, not the
least incorporating methods of drawdown control.

Acknowledgements
The authors would like to thank the Oxford-Man Institute of Quantitative Finance for its generous support.
Professor Stephen Roberts would like to further thank the Royal Academy of Engineering. Trent Spears
would like to further thank Professor Nir Vulkan and Dr Jan-Peter Calliess for their helpful comments and
suggestions.

References
[1] Fischer Black and Robert B. Litterman. Asset Allocation: Combining Investor Views with Market Equilibrium. The Journal
of Fixed Income, 1(2):7–18, 1991.
[2] Jay Walters. The black-litterman model in detail. https://ssrn.com/abstract=1314585, 2014. [Online; last
accessed 10-Aug-2022].
[3] Stephen Satchell and Alan Scowcroft. A demystification of the black–litterman model: Managing quantitative and traditional
portfolio construction. Journal of Asset Management, 1:138–150, 2000.
[4] Petter N. Kolm and Gordon Ritter. On the Bayesian interpretation of Black–Litterman. European Journal of Operational
Research, 258(2):564–572, 2017.
[5] William F. Sharpe. Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance, 19
(3):425–442, 1964.
[6] Hens Steehouwer Martin van der Schans. Time-dependent black–litterman. Journal of Asset Management, 18:371–387, 2017.
[7] Stefano Giglio, Bryan Kelly, and Dacheng Xiu. Factor models, machine learning, and asset pricing. Annual Review of
Financial Economics, 14(1):337–368, 2022.
[8] Stephen A. Ross. The arbitrage theory of capital asset pricing. Journal of Economic Theory, 13(3):341–360, 1976.
[9] Harry Markowitz. Portfolio Selection. The Journal of Finance, 7(1):77–91, 1952.
[10] Frank J. Fabozzi, Sergio M. Focardi, and Petter N. Kolm. Incorporating Trading Strategies in the Black-Litterman Framework.
The Journal of Trading, 1(2):28–37, 2006.

17

[11] Kevin P. Murphy. Machine Learning: A Probabilistic Perspective. The MIT Press, 2012.
[12] Theis I. Jensen, Bryan T. Kelly, Semyon Malamud, and Lasse H. Pedersen. Machine learning and the implementable efficient
frontier, swiss finance institute research paper no. 22-63. https://ssrn.com/abstract=4187217, 2022. [Online;
last accessed 10-Aug-2022].
[13] Nicolae Gârleanu and Lasse H. Pedersen. Dynamic trading with predictable returns and transaction costs. The Journal of
Finance, 68(6):2309–2340, 2013.
[14] Andrea Frazzini, Ronen Israel, and Tobias J. Moskowitz. Trading costs. https://ssrn.com/abstract=3229719,
2018. [Online; last accessed 10-Aug-2022].
[15] Paresh Date and Ksenia Ponomareva. Linear and non-linear filtering in mathematical finance: A review. IMA Journal of
Management Mathematics, 21, 06 2010.
[16] Dieter Willner, Chaw-Bing Chang, and Keh-Ping Dunn. Kalman filter algorithms for a multi-sensor system. 1976 IEEE
Conference on Decision and Control including the 15th Symposium on Adaptive Processes, pages 570–574, 1976.
[17] Bahador Khaleghi, Alaa Khamis, Fakhreddine O. Karray, and Saiedeh N. Razavi. Multisensor data fusion: A review of the
state-of-the-art. Information Fusion, 14(1):28–44, 2013.
[18] Federico Castanedo. A review of data fusion techniques. The Scientific World Journal, 2013.
[19] Jeffrey K. Uhlmann. Dynamic Map Building and Localization : New Theoretical Foundations. PhD thesis, University of
Oxford, 1995.
[20] Edward O. Thorp. The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market. Handbook of Asset and Liability
Management, 1:385–428, 2006.
[21] Steven Reece and Stephen Roberts. Generalised Covariance Union: A Unified Approach to Hypothesis Merging in Tracking.
IEEE Transactions on Aerospace and Electronic Systems, 46, 2010.
[22] Arthur P. Dempster. Elements of continuous multivariate analysis. Addison-Wesley Pub. Co., 1969.
[23] Peter S. Maybeck. Stochastic models, estimation, and control, Vol. 1. Academic Press, New York, 1979.
[24] Yaakov Bar-Shalom. On the track-to-track correlation problem. IEEE Transactions on Automatic Control, 26(2):571–572,
1981.
[25] Simon J. Julier and Jeffrey K. Uhlmann. A non-divergent estimation algorithm in the presence of unknown correlations. In
Proceedings of the 1997 American Control Conference, volume 4, pages 2369–2373. IEEE, 1997.
[26] Lingji Chen, P.O. Arambel, and R.K. Mehra. Fusion under unknown correlation - covariance intersection as a special case. In
Proceedings of the Fifth International Conference on Information Fusion. FUSION 2002., volume 2, pages 905–912, 2002.
[27] Benjamin Noack, Joris Sijs, Marc Reinhardt, and Uwe D. Hanebeck. Decentralized data fusion with inverse covariance
intersection. Automatica, 79:35–41, 2017.
[28] Joris Sijs and Mircea Lazar. State fusion with unknown correlation: Ellipsoidal intersection. Automatica, 48(8):1874–1878,
2012.
[29] Benjamin Noack, Joris Sijs, and Uwe Hanebeck. Inverse covariance intersection: New insights and properties. In Proceedings
of the 20th International Conference on Information Fusion, 2017.
[30] Jeffrey K. Uhlmann. Covariance consistency methods for fault-tolerant distributed data fusion. Information Fusion, 4:201–
215, 2003.
[31] Alexei Kuntsevich and Franz Kappel. Solvopt: The solver for local nonlinear optimization problems. Institute for Mathematics, University of Graz, 1997.
[32] Ottmar Bochardt, Ryan Calhoun, Jeffrey K. Uhlmann, and Simon J. Julier. Generalized information representation and
compression using covariance union. 2006 9th International Conference on Information Fusion, pages 1–7, 2006.
[33] Simon J. Julier, Jeffrey K. Uhlmann, and David Nicholson. A method for dealing with assignment ambiguity. Proceedings of
the 2004 American Control Conference, 5:4102–4107 vol.5, 2004.
[34] Clifford S. Asness, Tobias J. Moskowitz, and Lasse Heje Pedersen. Value and Momentum Everywhere. The Journal of
Finance, 68(3):929–985, 2013.
[35] Bradley Efron. Better bootstrap confidence intervals. Journal of the American Statistical Association, 82(397):171–185,
1987.
[36] Frank Wilcoxon. Individual comparisons by ranking methods. Biometrics Bulletin, 1(6):80–83, 1945.
[37] Eyke Hüllermeier and Willem Waegeman. Aleatoric and epistemic uncertainty in machine learning: an introduction to
concepts and methods. Machine Learning, 110:457–506, 2021.
[38] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg,
J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine Learning in
Python. Journal of Machine Learning Research, 12:2825–2830, 2011.
[39] Stephen Roberts, Michael Osborne, Mark Ebden, Steven Reece, N Gibson, and Suzanne Aigrain. Gaussian processes for timeseries modelling. Philosophical transactions. Series A, Mathematical, physical, and engineering sciences, 371:20110550, 02
2013.

18

[40] Liudmila Prokhorenkova, Gleb Gusev, Aleksandr Vorobev, Anna Veronika Dorogush, and Andrey Gulin. Catboost: unbiased
boosting with categorical features. In Advances in Neural Information Processing Systems, volume 31, 2018.
[41] Aleksei Ustimenko, Liudmila Ostroumova Prokhorenkova, and Andrey Malinin. Uncertainty in gradient boosting via ensembles. 06 2020.
[42] Salem Lahlou, Moksh Jain, Hadi Nekoei, Victor Butoi, Paul Bertin, Jarrid Rector-Brooks, Maksym Korablyov, and Yoshua
Bengio. Deup: Direct epistemic uncertainty prediction. https://arxiv.org/abs/2102.08501, 2022. [Online; last
accessed 04-Dec-2022].

19

A

Appendix

A.1

Further APT calculations

This section adds further detail to the calculations of Section 2.3. In determining the expectation and covariance for
r|q in the case of unknown mean but known covariance, given eq. (12) we collect and expand the terms in the exponent
up to a factor of − 12 , complete the square in f , and integrate what is recognisable as a Gaussian integral:
(r − Xf )T D −1 (r − Xf ) + (f − µ)T Σ−1 (f − µ) = r T D −1 r − 2f T X T D −1 r
+ (Xf )T D −1 Xf + f T Σ−1 f − 2f T Σ−1 µ + µT Σ−1 µ
= r T D −1 r + f T Hf − 2ηf + µT Σ−1 µ
where µ and Σ denote the mean and covariance for f |q, H := X T D −1 X + Σ−1 and η := X T D −1 r + Σ−1 µ.
Completing the square,
f T Hf − 2f T η = (f − ν)T H(f − ν) − ν T Hν,

ν = H −1 η,

whereby
Z


1
exp − (f − ν)T H(f − ν) df =
2

s



(2π)k
.
|H|

Therefore
 1h
i
 1h
i
p(r|q) ∝ exp − r T D −1 r + µT Σ−1 µ − ν T Hν = exp − r T D −1 r + µT Σ−1 µ − η T H −1 η ,
2
2
(32)
and we retrieve the exponent of the Gaussian for r|q. Inspired by [4], we next make use of the following Lemma:
Lemma 1. If a multivariate normal random variable θ has density p(θ) and −2 log p(θ) = θ 0 Hθ − 2ν 0 θ +
(terms without θ) then var(θ) = H −1 and Eθ = H −1 ν.
Collecting quadratic terms in r from the exponent of eq. (32):
−η T H −1 η + r T D −1 r = −[X T D −1 r + Σ−1 µ]T H −1 [X T D −1 r + Σ−1 µ] + r T D −1 r
= −r T D −1 XH −1 X T D −1 r + r T D −1 r + lower-order terms in r
T

= r [−D

−1

XH

−1

T

X D

−1

+D

−1

(33)

]r + lower-order terms in r

Making use of the lemma, we have that the posterior predictive distribution p(r|q) is multivariate normal with covariance
var(r|q) = [−D −1 XH −1 X T D −1 + D −1 ]−1 ,
= [D −1 − D −1 X[X T D −1 X + [(V −1 + Ω−1 )−1 + F ]−1 ]−1 X T D −1 ]−1 .
Futher, we collect the linear terms in r from eq. (33):
−r T D −1 X T H −1 Σ−1 µ − µT Σ−1 H −1 X T D −1 r = −2µT Σ−1 H −1 X T D −1 r,
so that again by the lemma
E(r|q) = var(r|q)D −1 XH −1 Σ−1 µ
= var(r|q)D −1 X[X T D −1 X + [(V −1 + Ω−1 )−1 + F ]−1 ]−1
· [(V −1 + Ω−1 )−1 + F ]−1 (V −1 + Ω−1 )−1 (V −1 ξ + Ω−1 q).

20

A.2

View fusion under model-based forecasts

Suppose we model y = f (x) +  for some function f parameterised by θ, with  ∼ N (0, σ 2 ). Then we can write
y|x, θ ∼ N (f (x), σ 2 ). Assuming θ has prior distribution p(θ), then for a collection of data D we can update the prior
via the likelihood analogous to eq. (5) above: p(θ|D) ∝ p(D|θ)p(θ). Then for a new data point x∗ , the predictive
distribution for the corresponding y ∗ can be written
Z
∗ ∗
p(y |x , D) = p(y ∗ |x∗ , θ)p(θ|D)dθ.
Further, the variance of the prediction, var(y ∗ |x∗ , D), can be decomposed making use of the law of total variance, that
is:


var(y ∗ |x∗ , D) = varθ|D E(y ∗ |x∗ , θ) + Eθ|D var(y ∗ |x∗ , θ) .
(34)
Here the subscript on the outer variance and expectation terms on the right-hand side denotes that the integral is
calculated with respect to the conditional density for θ|D. The first term on the right-hand side we call the epistemic
uncertainty, sometimes described as a ‘model’ uncertainty associated with estimating the model parameters, and that
is reducible with increasing data. The second term we call the aleatoric uncertainty; this models the underlying
stochasticity of the data distribution, and is irreducible.
Estimation of predictive distributions or the components of model uncertainty is not always tractable; consequently,
various approximation methods exists for estimating these quantities. A review is beyond the scope of this paper; we
instead refer interested readers to the recent work of [37] for further details.

A.2.1

Brief overview of view models

To yield the results of Section 5 above we first implement three diverse view models. For completeness, we provide
the implementation details below.
We implement a Gaussian Process model using scikit-learn in Python [38]. The kernel function is given by the sum
of a radial basis function and a white-noise kernel; see, for example, [39] for context on Gaussian Processes for timeseries modelling. We re-fit the model on a bi-monthly basis on a rolling data window of 1 year. The predict method
is used to generate prediction estimates and an accompanying (total) uncertainty estimate. The uncertainty can be
decomposed into epistemic and aleatoric components by setting the latter as the noise-level parameter estimated for
the white-noise kernel. By eq. (34) above, the epistemic uncertainty is then the difference between the total uncertainty
and the aleatoric uncertainty estimates.
We implement a boosted regression model making use of CatBoost [40]. We re-fit the model on a bi-monthly basis
on a rolling data window of 2 years. The model yields an aleatoric uncertainty estimate owing to modelling this
uncertainty directly via the loss function. An estimate for epistemic uncertainty is approximated by the variance of the
underlying ensemble of predictions. Further details of this method can be found in [41].
We estimate an ARIMA model for view generation as a well-known baseline model. Further, this model naturally
yields an aleatoric uncertainty estimate. To approximate epistemic uncertainty for this model, we are inspired by
Algorithm 3 of [42]. We note that our method is somewhat ad hoc, and has subtle differences: we estimate prediction
error variance on a recent out-of-sample lookback window of size 6 months, estimating epistemic uncertainty as the
difference between this term and today’s aleatoric uncertainty estimate, to a minimum of 1e-8. We re-fit the model on
a bi-monthly basis on a rolling data window of 1 year.

21

A.3

Performance results by year

Performance metrics for market total return (in excess of the risk-free rate), against a collection of Black-Litterman
models with varying underlying prediction methods to generate and/or fuse views. Methods sans the index allocation
are normalised by return volatility to equal the ex-post annual return volatility of the market index.
1993

1994

Portfolio method

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

S&P 500 (TR)

7.11

8.59

0.82

-1.94

1.21

-8.84

-2.03

9.80

-0.21

-1.42

-0.28

-8.91

ARIMA
GP
Boost

-17.50
-15.42
-9.14

8.59
8.59
8.59

-2.04
-1.79
-1.06

-2.68
-2.60
-1.84

-2.01
-1.78
-1.18

-16.43
-15.90
-10.40

20.00
21.39
12.53

9.80
9.80
9.80

2.04
2.18
1.28

1.73
1.78
1.04

5.65
5.91
2.73

-18.76
-19.92
-13.63

PW
CU
CI
ICI

-12.04
-11.11
-20.11
-16.51

8.59
8.59
8.59
8.59

-1.40
-1.29
-2.34
-1.92

-2.20
-1.98
-2.81
-2.46

-1.44
-1.47
-2.22
-1.82

-12.37
-13.29
-18.60
-16.14

13.90
20.65
18.24
19.94

9.80
9.80
9.80
9.80

1.42
2.11
1.86
2.03

1.16
1.74
1.59
1.70

3.07
5.76
6.13
8.08

-14.73
-19.13
-17.71
-18.73

1995

1996

Portfolio method

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

S&P 500 (TR)

26.80

7.81

3.43

-2.57

5.72

-24.15

16.31

11.75

1.38

-1.60

1.96

-18.96

ARIMA
GP
Boost

-12.74
-11.30
-15.45

7.81
7.81
7.81

-1.63
-1.45
-1.98

-4.48
-4.33
-4.56

-1.67
-1.52
-2.06

-14.11
-13.81
-14.93

14.18
9.49
10.90

11.75
11.75
11.75

1.21
0.81
0.93

-0.33
-0.59
-0.49

2.78
1.65
1.62

-19.34
-16.73
-18.25

PW
CU
CI
ICI

-21.30
-11.64
-13.29
-12.45

7.81
7.81
7.81
7.81

-2.73
-1.49
-1.70
-1.59

-5.36
-4.53
-4.87
-4.78

-2.46
-1.57
-1.77
-1.67

-20.11
-14.14
-14.72
-14.07

9.34
20.40
25.06
21.15

11.75
11.75
11.75
11.75

0.79
1.74
2.13
1.80

-0.60
0.02
0.29
0.06

1.34
4.69
6.07
4.78

-19.69
-22.35
-25.21
-23.72

1997

1998

Portfolio method

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

S&P 500 (TR)

25.34

18.09

1.40

-1.04

2.04

-23.74

22.47

20.25

1.11

-0.51

1.57

-22.93

ARIMA
GP
Boost

26.74
31.28
32.70

18.09
18.09
18.09

1.48
1.73
1.81

-0.05
0.17
0.22

3.39
3.71
4.57

-21.90
-25.04
-26.85

12.62
10.57
5.64

20.25
20.25
20.25

0.62
0.52
0.28

-0.34
-0.38
-0.58

0.91
0.73
0.41

-16.82
-16.72
-19.44

PW
CU
CI
ICI

30.50
28.95
23.15
28.57

18.09
18.09
18.09
18.09

1.69
1.60
1.28
1.58

0.13
0.05
-0.22
0.04

3.69
3.75
2.61
3.58

-26.26
-23.06
-18.85
-22.90

15.04
29.10
17.15
26.14

20.25
20.25
20.25
20.25

0.74
1.44
0.85
1.29

-0.24
0.22
-0.18
0.12

1.06
2.41
1.24
2.20

-15.90
-27.24
-16.74
-24.05

1999

2000

Portfolio method

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

S&P 500 (TR)

16.15

18.04

0.90

-3.27

1.34

-14.82

-12.80

22.18

-0.58

-0.50

-0.81

-20.07

ARIMA
GP
Boost

-28.79
-21.79
19.04

18.05
18.05
18.05

-1.60
-1.21
1.06

-1.72
-1.58
-0.12

-1.71
-1.40
1.64

-27.22
-22.85
-25.24

-5.65
-8.89
-29.64

22.17
22.17
22.17

-0.25
-0.40
-1.34

-0.10
-0.23
-0.83

-0.33
-0.51
-1.55

-26.38
-26.27
-38.33

PW
CU
CI
ICI

-9.48
-3.66
-19.29
-32.13

18.05
18.05
18.05
18.05

-0.53
-0.20
-1.07
-1.78

-1.13
-0.94
-1.39
-1.89

-0.68
-0.27
-1.27
-1.89

-20.84
-21.42
-20.96
-30.89

-10.81
1.46
6.15
-6.63

22.17
22.17
22.17
22.17

-0.49
0.07
0.28
-0.30

-0.32
0.15
0.34
-0.12

-0.64
0.09
0.39
-0.38

-27.58
-24.08
-23.62
-27.23

Figure 6: Performance metrics, 1993-2000.

22

2001

2002

Portfolio method

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

S&P 500 (TR)

-14.12

21.51

-0.67

-2.25

-0.92

-30.91

-23.24

25.99

-0.89

-1.43

-1.28

-33.81

ARIMA
GP
Boost

36.09
36.18
25.65

21.51
21.51
21.51

1.68
1.68
1.19

1.72
1.67
1.34

2.57
2.60
1.64

-35.22
-34.88
-30.70

26.07
33.51
31.67

25.99
25.99
25.99

1.00
1.29
1.22

1.49
1.73
1.65

1.53
2.10
1.99

-34.82
-37.44
-36.37

PW
CU
CI
ICI

23.71
21.26
35.16
34.26

21.51
21.51
21.51
21.51

1.10
0.99
1.63
1.59

1.28
1.24
1.66
1.64

1.48
1.39
2.62
2.52

-29.85
-26.00
-33.08
-33.28

33.17
4.75
22.01
26.35

25.99
25.99
25.99
25.99

1.28
0.18
0.85
1.01

1.68
0.83
1.37
1.48

2.09
0.24
1.28
1.57

-37.85
-30.08
-31.67
-35.08

2003

2004

Portfolio method

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

S&P 500 (TR)

25.64

17.04

1.50

-2.42

2.32

-28.43

9.78

11.07

0.88

-1.37

1.28

-12.49

ARIMA
GP
Boost

-23.61
-27.81
-31.93

17.03
17.03
17.03

-1.39
-1.63
-1.87

-2.48
-2.70
-2.76

-1.53
-1.73
-1.95

-23.77
-26.21
-30.85

14.26
15.63
12.73

11.07
11.07
11.07

1.29
1.41
1.15

0.26
0.35
0.16

2.62
3.08
2.41

-15.21
-16.01
-14.50

PW
CU
CI
ICI

-31.18
-20.25
-20.35
-21.49

17.03
17.03
17.03
17.03

-1.83
-1.19
-1.19
-1.26

-2.80
-2.60
-2.32
-2.39

-1.86
-1.28
-1.36
-1.40

-29.23
-22.22
-21.51
-22.16

14.71
12.98
12.22
16.73

11.07
11.07
11.07
11.07

1.33
1.17
1.10
1.51

0.30
0.17
0.13
0.43

3.02
2.22
2.18
3.21

-14.74
-15.10
-14.11
-15.57

2005

2006

Portfolio method

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

S&P 500 (TR)

2.37

10.26

0.23

-3.66

0.33

-9.87

10.49

10.02

1.05

-3.14

1.59

-12.93

ARIMA
GP
Boost

3.65
-0.49
1.49

10.27
10.27
10.27

0.36
-0.05
0.15

-0.24
-0.57
-0.42

0.52
-0.07
0.22

-9.98
-11.40
-11.54

-3.92
-2.43
-4.06

10.02
10.02
10.02

-0.39
-0.24
-0.41

-1.17
-1.03
-1.14

-0.58
-0.36
-0.58

-15.33
-14.76
-15.78

PW
CU
CI
ICI

0.10
1.58
1.62
1.23

10.27
10.27
10.27
10.27

0.01
0.15
0.16
0.12

-0.53
-0.44
-0.41
-0.45

0.01
0.23
0.22
0.17

-12.06
-10.15
-10.65
-9.78

-6.36
3.40
-2.07
1.52

10.02
10.02
10.02
10.02

-0.63
0.34
-0.21
0.15

-1.33
-0.59
-1.02
-0.74

-0.87
0.61
-0.32
0.27

-16.78
-13.37
-14.31
-12.77

2007

2008

Portfolio method

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

S&P 500 (TR)

2.07

15.96

0.13

-4.06

0.17

-10.66

-39.34

40.88

-0.96

-0.77

-1.31

-47.78

ARIMA
GP
Boost

-1.06
-1.52
-3.80

15.96
15.96
15.96

-0.07
-0.10
-0.24

-0.33
-0.35
-0.48

-0.10
-0.14
-0.35

-12.74
-13.27
-15.23

17.00
46.05
39.48

40.88
40.88
40.88

0.42
1.13
0.97

1.12
1.69
1.34

0.60
1.83
1.56

-42.30
-40.38
-43.34

PW
CU
CI
ICI

1.83
-13.65
-3.35
-0.99

15.96
15.96
15.96
15.96

0.11
-0.86
-0.21
-0.06

-0.21
-0.87
-0.44
-0.31

0.18
-1.09
-0.31
-0.08

-14.36
-17.35
-11.66
-9.66

56.22
67.57
47.33
60.76

40.88
40.88
40.88
40.88

1.38
1.65
1.16
1.49

1.81
2.27
1.79
2.33

2.61
3.13
2.16
2.91

-46.29
-51.49
-42.41
-49.16

Figure 7: Performance metrics, 2001-2008.

23

2009

2010

Portfolio method

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

S&P 500 (TR)

27.09

27.22

1.00

-0.83

1.45

-41.01

15.47

18.02

0.86

-1.81

1.23

-19.54

ARIMA
GP
Boost

17.54
7.78
-33.40

27.22
27.22
27.22

0.64
0.29
-1.23

-0.13
-0.43
-1.76

1.13
0.48
-1.51

-21.65
-18.56
-35.25

-41.11
-42.99
-33.99

18.02
18.02
18.02

-2.28
-2.38
-1.89

-2.30
-2.42
-2.07

-2.19
-2.33
-1.97

-37.28
-38.98
-33.55

PW
CU
CI
ICI

-9.72
19.15
25.99
27.58

27.22
27.22
27.22
27.22

-0.36
0.70
0.95
1.01

-1.00
-0.08
0.12
0.17

-0.55
1.30
1.52
1.74

-19.77
-21.64
-31.85
-27.00

-36.86
-36.30
-37.32
-38.24

18.02
18.02
18.02
18.02

-2.04
-2.01
-2.07
-2.12

-2.17
-2.14
-2.14
-2.16

-2.06
-2.03
-1.98
-1.99

-34.57
-35.08
-34.19
-33.98

2011

2012

Portfolio method

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

S&P 500 (TR)

4.78

23.24

0.21

0.83

0.28

-18.64

15.63

12.73

1.24

-0.23

1.89

-14.23

ARIMA
GP
Boost

-29.76
-29.17
-8.30

23.24
23.24
23.24

-1.28
-1.26
-0.36

-1.08
-1.03
-0.41

-1.35
-1.33
-0.45

-34.76
-33.83
-18.13

22.54
29.53
28.89

12.73
12.73
12.73

1.77
2.32
2.27

0.46
0.86
0.83

3.23
5.28
5.22

-19.67
-25.51
-25.12

PW
CU
CI
ICI

-8.41
-15.96
-34.31
-26.26

23.24
23.24
23.24
23.24

-0.36
-0.69
-1.48
-1.13

-0.42
-0.56
-1.19
-0.92

-0.48
-0.80
-1.53
-1.18

-16.08
-29.10
-32.76
-33.43

25.43
15.64
19.15
18.62

12.73
12.73
12.73
12.73

2.00
1.23
1.50
1.46

0.65
0.09
0.28
0.25

4.23
1.79
2.45
2.45

-24.06
-15.94
-18.43
-16.55

2013

2014

Portfolio method

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

S&P 500 (TR)

28.68

11.05

2.60

-1.62

3.96

-22.78

13.48

11.33

1.19

1.09

1.67

-18.27

ARIMA
GP
Boost

2.61
-4.37
7.30

11.05
11.05
11.05

0.24
-0.40
0.66

-1.74
-2.45
-1.47

0.34
-0.51
0.96

-11.82
-16.53
-8.04

8.11
7.28
16.40

11.33
11.33
11.33

0.72
0.64
1.45

-0.35
-0.42
0.18

1.74
1.33
3.85

-10.11
-8.94
-15.84

PW
CU
CI
ICI

-1.12
-11.24
4.64
-3.97

11.05
11.05
11.05
11.05

-0.10
-1.02
0.42
-0.36

-2.36
-2.96
-1.57
-2.40

-0.14
-1.17
0.63
-0.47

-14.72
-19.21
-9.25
-15.92

14.34
1.81
6.04
7.34

11.33
11.33
11.33
11.33

1.27
0.16
0.53
0.65

0.05
-0.79
-0.47
-0.39

3.14
0.29
1.26
1.43

-14.43
-7.87
-9.70
-9.50

2015

2016

Portfolio method

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

S&P 500 (TR)

2.57

15.46

0.17

0.91

0.23

-12.04

11.94

13.07

0.91

-1.32

1.30

-20.80

ARIMA
GP
Boost

29.50
25.13
28.98

15.46
15.46
15.46

1.91
1.62
1.87

1.56
1.34
1.47

2.82
2.52
3.17

-24.32
-24.29
-26.31

-0.11
0.75
-3.25

13.07
13.07
13.07

-0.01
0.06
-0.25

-0.90
-0.84
-1.08

-0.01
0.11
-0.39

-11.81
-12.31
-10.62

PW
CU
CI
ICI

28.45
31.35
27.01
22.50

15.46
15.46
15.46
15.46

1.84
2.03
1.75
1.46

1.56
1.83
1.45
1.12

2.77
2.96
2.56
2.42

-26.20
-26.59
-22.93
-23.22

-11.84
-1.96
2.62
6.23

13.07
13.07
13.07
13.07

-0.91
-0.15
0.20
0.48

-1.60
-1.01
-0.73
-0.52

-1.22
-0.25
0.38
1.03

-12.92
-12.55
-13.80
-13.95

Figure 8: Performance metrics, 2009-2016.

24

2017

2018

Portfolio method

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

S&P 500 (TR)

19.16

6.67

2.89

-1.04

4.55

-17.04

-4.81

17.03

-0.28

-0.77

-0.37

-19.81

ARIMA
GP
Boost

-1.08
0.06
-3.86

6.67
6.67
6.67

-0.16
0.01
-0.58

-2.14
-2.03
-2.50

-0.24
0.02
-0.79

-7.76
-5.84
-10.74

4.96
-0.59
6.59

17.03
17.03
17.03

0.29
-0.03
0.39

0.50
0.26
0.61

0.40
-0.04
0.56

-14.97
-14.22
-16.52

PW
CU
CI
ICI

-7.71
-0.88
2.51
4.71

6.67
6.67
6.67
6.67

-1.16
-0.13
0.38
0.71

-2.99
-2.16
-1.75
-1.48

-1.41
-0.20
0.61
1.32

-12.44
-6.45
-6.02
-4.78

13.39
7.20
2.18
-5.87

17.03
17.03
17.03
17.03

0.79
0.42
0.13
-0.34

0.87
0.57
0.37
0.05

1.17
0.57
0.17
-0.44

-17.03
-12.90
-12.58
-14.05

2019

2020

Portfolio method

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

S&P 500 (TR)

26.03

12.45

2.09

-0.79

3.02

-24.34

22.51

34.37

0.65

-2.26

0.90

-41.22

ARIMA
GP
Boost

22.69
20.72
23.24

12.45
12.45
12.45

1.37
1.66
1.87

-0.16
-0.30
-0.15

3.17
3.69
4.15

-22.74
-21.68
-20.27

17.36
-1.95
26.22

34.37
34.37
34.37

0.51
-0.06
0.76

-0.17
-0.59
0.03

0.61
-0.07
1.01

-38.09
-36.61
-39.32

PW
CU
CI
ICI

16.83
17.75
23.50
19.15

12.45
12.45
12.45
12.45

1.35
1.43
1.89
1.54

-0.50
-0.50
-0.15
-0.42

3.33
2.73
3.68
2.47

-19.62
-19.82
-21.02
-19.28

22.99
3.84
8.55
8.74

34.37
34.37
34.37
34.37

0.67
0.11
0.25
0.25

-0.04
-0.47
-0.36
-0.36

0.86
0.13
0.29
0.30

-37.70
-36.68
-36.89
-36.94

2021
Portfolio method

Cuml Ret

Ret Vol

Sharpe

IR

Sortino

Max DD

S&P 500 (TR)

26.10

13.07

2.00

1.55

2.97

-23.86

ARIMA
GP
Boost

-5.78
-2.48
6.64

13.07
13.07
13.07

-0.46
-0.20
0.53

-1.65
-1.47
-0.81

-0.60
-0.27
0.84

-12.93
-10.21
-12.56

PW
CU
CI
ICI

-5.24
-0.34
-1.12
-2.15

13.07
13.07
13.07
13.07

-0.42
-0.03
-0.09
-0.17

-1.55
-1.37
-1.34
-1.40

-0.53
-0.04
-0.13
-0.24

-12.61
-9.50
-9.80
-10.50

Figure 9: Performance metrics, 2017-2021.

25
</reference>

<statements>
1. View fusion / Bayesian BL (Spears, Zohren & Roberts 2023): information-fusion of multiple ML view+uncertainty estimates into BL, subsuming APT.
</statements>

Begin the assessment now. Output only the JSON list, without any conversational text or explanations.