InfoBox

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

Trent Spears , Stefan Zohren∗, Stephen Roberts∗ University of Oxford, Oxford-
Man Institute of Quantitative Finance. Corresponding author. E-mail:
<EMAIL_ADDRESS>
###### 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 time-series 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 Black-Litterman
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 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

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 $\bm{r}$ whose distribution is modelled as multivariate normal over
discrete time increments, denoted
$\bm{r}\thicksim\text{MVN}(\bm{\mu},\bm{\Sigma})$. Here $\bm{\mu}$ denotes the
mean vector of returns, and $\bm{\Sigma}$ denotes the return covariance. Then
in a Markowitz Modern Portfolio Theory framework [9], assuming a single-period
investment setting, the investor portfolio weights $\bm{w}=(w_{1},...,w_{n})$
are a solution to the mean-variance optimisation problem:

$\displaystyle\max_{\bm{w}\in\mathcal{C}}\qquad\mathbb{E}(\bm{w}^{\prime}\bm{r})-\frac{\gamma}{2}\text{var}(\bm{w}^{\prime}\bm{r}).$
(1)

Here $\mathcal{C}$ is a set of weight constraints, if any, and $\gamma>0$ is
the investor risk aversion parameter, denoting preferences balancing return
and risk. In the unconstrained case, differentiating eq. (1), setting it to 0
and solving for the optimal portfolio weights $\bm{w}^{*}$ in terms of
$(\bm{\mu},\bm{\Sigma})$ yields

$\displaystyle\bm{w}^{*}=(\gamma\bm{\Sigma})^{-1}\bm{\mu}.$ (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 $\bm{p}$ and expected asset returns $\bm{\mu}$, a
real-valued scalar denoting the portfolio value $q$, and a strictly positive
real-valued scalar $\epsilon$ denoting uncertainty in the portfolio value,
that can be written

$\displaystyle\bm{p}^{\prime}\bm{\mu}=q+\epsilon.$ (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:

$\displaystyle\bm{P}\bm{\mu}=\bm{q}+\bm{\epsilon},\qquad\bm{\epsilon}\thicksim
N(\bm{0},\bm{\Omega}).$ (4)

Here $\bm{P}\in\mathbb{R}^{k\times n}$ is a real-valued matrix of portfolio
weights, where the $k^{th}$ row is given by $\bm{p}^{\prime}_{k}$ as defined
above; $\bm{q}\in\mathbb{R}^{k}$ is a real-valued vector collecting $k$
portfolio values; and $\bm{\Omega}\in\mathbb{R}^{k\times k}$ 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 $\bm{p}$ taking a unit value at the $i^{th}$ entry, and
zero otherwise, so that the left-hand side of eq. (3) equals $\mu_{i}$. Any
other (non-trival) view we call relative; a common relative view, for example,
is a belief on the return spread $\mu_{i}-\mu_{j}=q$. This could be expressed
by setting alternate positive and negative unit values at the $i$ and $j$
index of $\bm{p}$, and zero elsewhere. In the work that follows, we assume
absolute views. We further assume that $\bm{P}$ 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, 3, 4, 6].

The Bayesian specification assumes a prior distribution $p(\bm{\theta})$,
where $\bm{\theta}$ denotes the target parameter of the return distribution
that we wish to express via a parametric probabilistic model. Most commonly,
$\bm{\theta}=\bm{\mu}$. 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
$p(\bm{q}|\bm{\theta})$, such that, by the laws of conditional probability and
Bayes’ Theorem, an updated posterior distribution for the target parameter is
given by

$\displaystyle p(\bm{\theta}|\bm{q})\propto
p(\bm{q}|\bm{\theta})p(\bm{\theta}).$ (5)

Finally, we can write a posterior predictive distribution for the asset
returns given the views as

$\displaystyle p(\bm{r}|\bm{q})=\int
p(\bm{r}|\bm{\theta})p(\bm{\theta}|\bm{q})d\bm{\theta}.$ (6)

Hence, the expectation and covariance of the random quantity $\bm{r}|\bm{q}$
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.

###  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 $k\ll n$, 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

$\displaystyle\bm{r}=\bm{X}\bm{f}+\bm{\epsilon}_{r},\qquad\bm{\epsilon}_{r}\thicksim
N(\bm{0},\bm{D}).$ (7)

Here $\bm{r}$ is the $n$-dimensional random vector containing the cross-
section of excess returns for $n$ equities over the next time increment
$[t,t+1]$; $\bm{X}$ is a non-random $n\times k$ matrix of observable factor
exposures estimated at $t$, for $k$ the number of explanatory asset factors;
$\bm{D}$ is a covariance matrix for the returns, also known at $t$ and assumed
to be diagonal; and $\bm{f}$ 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
$\bm{\theta}=\bm{\mu_{f}}$, such that

$\displaystyle\bm{f}=\bm{\mu_{f}}+\bm{\epsilon}_{f},\qquad\bm{\epsilon}_{f}\thicksim
N(\bm{0},\bm{F}).$ (8)

We refer to the elements of $\bm{\mu_{f}}$ 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
$p(\bm{r}|\bm{q})$, as defined below.

We begin by writing the joint distribution of random variables of interest as
a conditional probability:

$\displaystyle
p(\bm{r},\bm{f},\bm{\theta},\bm{q})=p(\bm{r},\bm{f}|\bm{\theta},\bm{q})p(\bm{\theta},\bm{q}).$

Substituting $p(\bm{\theta},\bm{q})=p(\bm{\theta}|\bm{q})p(\bm{q})$ and
dividing both sides by $p(\bm{q})$ yields

$\displaystyle
p(\bm{r},\bm{f},\bm{\theta}|\bm{q})=p(\bm{r},\bm{f}|\bm{\theta},\bm{q})p(\bm{\theta}|\bm{q}).$

Integrating both sides over $\bm{\theta}$ we find that

$\displaystyle
p(\bm{r},\bm{f}|\bm{q})=\int\underbrace{p(\bm{r},\bm{f}|\bm{\theta},\bm{q})}_{=p(\bm{r},\bm{f}|\bm{\theta})}p(\bm{\theta}|\bm{q})d\bm{\theta}.$

Again integrating both sides, this time with respect to $\bm{f}$, we have that

$\displaystyle p(\bm{r}|\bm{q})=\iint
p(\bm{r},\bm{f}|\bm{\theta})p(\bm{\theta}|\bm{q})d\bm{\theta}d\bm{f}.$ (9)

With one more simplification, writing
$p(\bm{r},\bm{f}|\bm{\theta})=p(\bm{r}|\bm{f},\bm{\theta})p(\bm{f}|\bm{\theta})=p(\bm{r}|\bm{f})p(\bm{f}|\bm{\theta})$,
we find

$\displaystyle p(\bm{r}|\bm{q})=\int p(\bm{r}|\bm{f})\underbrace{\int
p(\bm{f}|\bm{\theta})p(\bm{\theta}|\bm{q})d\bm{\theta}}_{=p(\bm{f}|\bm{q})}d\bm{f}.$
(10)

Since we assume $\bm{\theta}=\bm{\mu_{f}}$, there holds the well-known facts
that the posterior $p(\bm{\theta}|\bm{q})$ is normal and that the posterior
predictive $p(\bm{f}|\bm{q})$ is normal. Hence it is clear that
$p(\bm{r}|\bm{q})$ is also normal, with its expectation and covariance given
in closed-form by the following three-step process.

Firstly, choosing the prior $p(\bm{\theta})\thicksim N(\bm{\xi},\bm{V})$, by
Bayes’ rule for linear Gaussian systems, as in [11], we have that the
posterior $p(\bm{\theta}|\bm{q})$ is multivariate normal with covariance and
mean

$\displaystyle\text{var}(\bm{\theta}|\bm{q})$
$\displaystyle=(\bm{V}^{-1}+\bm{\Omega}^{-1})^{-1},$
$\displaystyle\mathbb{E}(\bm{\theta}|\bm{q})$
$\displaystyle=\text{var}(\bm{\theta}|\bm{q})\cdot(\bm{V}^{-1}\bm{\xi}+\bm{\Omega}^{-1}\bm{q}).$

Secondly, recall the convolution integral for sums of random variables: if
$\bm{X}_{i}\stackrel{{\scriptstyle
i.i.d.}}{{\thicksim}}N(\bm{\mu}_{i},\bm{\Sigma}_{i}),i=1,2$, then

$\displaystyle f(\bm{z}):=\int
f_{\bm{X}_{1}}(\bm{x})f_{\bm{X}_{2}}(\bm{z}-\bm{x})d\bm{x}\thicksim
N(\bm{\mu}_{1}+\bm{\mu}_{2},\bm{\Sigma}_{1}+\bm{\Sigma}_{2}).$ (11)

Writing the inner integral of eq. (10) with the change of variable
$\bm{f}\mapsto\bm{f}-\bm{\theta}$, we find

$p(\bm{f}|\bm{q})=\int\underbrace{p(\bm{f}-\bm{\theta}|\bm{\theta})}_{\thicksim
N(\bm{0},\bm{F})}p(\bm{\theta}|\bm{q})d\theta$

so that by eq. (11) we have that $p(\bm{f}|\bm{q})$ is normally distributed
with expectation and covariance given by

$\displaystyle\mathbb{E}(\bm{f}|\bm{q})$
$\displaystyle=(\bm{V}^{-1}+\bm{\Omega}^{-1})^{-1}\cdot(\bm{V}^{-1}\bm{\xi}+\bm{\Omega}^{-1}\bm{q})=\mathbb{E}(\bm{\theta}|\bm{q}),$
$\displaystyle\text{var}(\bm{f}|\bm{q})$
$\displaystyle=(\bm{V}^{-1}+\bm{\Omega}^{-1})^{-1}+\bm{F}=\text{var}(\bm{\theta}|\bm{q})+\bm{F}.$

Thirdly, we solve the outer integral of eq. (10) (deferring some of the
details of the algebraic manipulations to the Appendix, for brevity). Writing

$\displaystyle p(\bm{r}|\bm{q})=\int p(\bm{r}|\bm{f})p(\bm{f}|\bm{q})d\bm{f},$
(12)

we collect and expand the terms in the exponent up to a factor of
$-\frac{1}{2}$, complete the square in $\bm{f}$, 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 $\bm{r}|\bm{q}$. Hence, as
supported by further calculations within the Appendix below, we have that the
posterior predictive distribution $p(\bm{r}|\bm{q})$ is multivariate normal
with covariance and mean

$\displaystyle\text{var}(\bm{r}|\bm{q})$
$\displaystyle=\big{[}\bm{D}^{-1}-\bm{D}^{-1}\bm{X}\big{[}\bm{X}^{T}\bm{D}^{-1}\bm{X}+\text{var}(\bm{f}|\bm{q})^{-1}\big{]}^{-1}\bm{X}^{T}\bm{D}^{-1}\big{]}^{-1},$
$\displaystyle\mathbb{E}(\bm{r}|\bm{q})$
$\displaystyle=\text{var}(\bm{r}|\bm{q})\bm{D}^{-1}\bm{X}\big{[}\bm{X}^{T}\bm{D}^{-1}\bm{X}+\text{var}(\bm{f}|\bm{q})^{-1}\big{]}^{-1}\text{var}(\bm{f}|\bm{q})^{-1}\mathbb{E}(\bm{f}|\bm{q}).$

By eq. (2), we write the optimal weights for a single-step Black-Litterman
portfolio allocation as

$\displaystyle\bm{h}_{blb}^{*}$
$\displaystyle=\gamma^{-1}\text{var}(\bm{r}|\bm{q})^{-1}\mathbb{E}(\bm{r}|\bm{q})$
$\displaystyle=\gamma^{-1}\bm{D}^{-1}\bm{X}\big{[}\bm{X}^{T}\bm{D}^{-1}\bm{X}+\text{var}(\bm{f}|\bm{q})^{-1}\big{]}^{-1}\text{var}(\bm{f}|\bm{q})^{-1}\mathbb{E}(\bm{f}|\bm{q}).$
(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.

###  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):

$\displaystyle\max_{\bm{w}\in\mathcal{C}}\qquad\mathbb{E}(\bm{w}^{\prime}\bm{r})-\frac{\gamma}{2}\text{var}(\bm{w}^{\prime}\bm{r})-TC,$
(14)

with $TC$ 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, $TC^{d}$, is given by

$TC_{t}^{d}=\frac{1}{2}\bm{\Delta}_{t}^{\prime}\bm{\Lambda}_{t}\bm{\Delta}_{t}.$

Here $\bm{\Delta}_{t}$ is the portfolio turnover vector for a time increment,
defined as

$\bm{\Delta}_{t}:=\Pi_{t}(\bm{w}_{t}-\bm{R}_{t-1}\bm{w}_{t-1}),$

and the ‘market impact’ vector $\bm{m}$ is given by

$\bm{m}_{t}=\frac{1}{2}\bm{\Lambda}_{t}\bm{\Delta}_{t}.$

We use the subscript $t$ to explicitly account for time-dependence. The real
scalar $\Pi_{t}$ denotes total wealth investable at time $t$. Clearly, given
the above specification dollar transaction-costs erode dollar returns
quadratically with increasing wealth. The term $\bm{R}_{t-1}$ 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 $\bm{\Lambda}_{t}$ 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
$\bm{L}_{t}$ of US equity underlyings whether long _or short_ , so

$\displaystyle\bm{m}_{t}$
$\displaystyle=10\text{bps}\cdot\bm{1}=\frac{1}{2}\bm{\Lambda}_{t}\times
1\%\bm{L}_{t}$
$\displaystyle\implies\bm{\Lambda}_{t}=\frac{1}{5}\bm{L}_{t}^{-1}.$ (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 $TC=TC^{d}/\Pi$, as

$\displaystyle\max_{\bm{w}_{t}\in\mathcal{C}}\qquad\mathbb{E}\big{(}\bm{w}_{t}^{\prime}\bm{r}_{t}\big{)}-\frac{\gamma}{2}\text{var}\big{(}\bm{w}_{t}^{\prime}\bm{r}_{t}\big{)}-\frac{\Pi_{t}}{2}\big{(}\bm{w}_{t}-\bm{R}_{t-1}\bm{w}^{*}_{t-1}\big{)}^{\prime}\bm{\Lambda}_{t}\big{(}\bm{w}_{t}-\bm{R}_{t-1}\bm{w}^{*}_{t-1}\big{)}$
(16)

and conveniently solved in closed-form for optimal weights:

$\displaystyle\bm{w}_{t}^{*}=\big{(}\gamma\bm{\Sigma}_{t}+\Pi_{t}\bm{\Lambda}_{t}\big{)}^{-1}\big{(}\bm{\mu}_{t}+\Pi_{t}\bm{\Lambda}_{t}\bm{R}_{t-1}\bm{w}^{*}_{t-1}\big{)}.$
(17)

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

##  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 estimates111There 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.. 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
$\bm{\mu}_{k}$ in an online way, via the linear time-varying system of
equations:

$\displaystyle\bm{\mu}_{k+1}$
$\displaystyle=\bm{\mu}_{k}+\bm{\epsilon}^{\bm{\mu}}_{k},$ (18)
$\displaystyle\bm{q}_{k}$
$\displaystyle=\bm{P}_{k}\bm{\mu}_{k}+\bm{\epsilon}^{\bm{q}}_{k}.$ (19)

Here $\bm{\epsilon}^{\bm{\mu}}_{k}$ and $\bm{\epsilon}^{\bm{q}}_{k}$ model
independent zero-mean white-noise processes with respective covariances
$\bm{\Psi}_{k}$ and $\bm{\Sigma}_{k}$. We call equations (18) and (19) the
‘state’ and ‘measurement’ equations, respectively. Solving the state-space
representation of Black-Litterman, assuming absolute views, we have the
probability distribution for $\bm{\mu}_{k}|\bm{q}_{k}$, analogous to eq. (5)
and solved equivalently to solving for $p(\bm{\theta}|\bm{q})$ per Section
2.1. The predictive distribution for $\bm{\mu}_{k+1}|\bm{q}_{k}$, analogous to
eq. (6) can be solved equivalently to solving for $p(\bm{f}|\bm{q})$, 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
$p(\bm{\mu}_{0})$ 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):

$\displaystyle\bm{q}_{k,s}$
$\displaystyle=\bm{\mu}_{k,s}+\bm{\epsilon}^{(\bm{q})}_{k,s},$ (20)

where $\bm{q}_{k,s}$ denotes the view is from source $s$, $1\leq s\leq S$, and
each $\bm{\epsilon}^{(\bm{q})}_{k,s}$ is a zero-mean white-noise process with
covariance $\bm{\Sigma}_{k,s}$. 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 variable222Various approaches exist in the data fusion
literature for incorporating information from multiple sources; introductory
reviews can be found in [17, 18].. 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
$(\widehat{\bm{\mu}},\widehat{\bm{\Sigma}})$ for the true underlying values
$(\bm{\mu},\bm{\Sigma})$ given a collection of $S$ underlying estimates
$\\{(\widehat{\bm{\mu}}_{s},\widehat{\bm{\Sigma}}_{s}):1\leq s\leq S\\}$, with
the mean

Figure 2: Concentration ellipses for three example bivariate Gaussian
measurements of a single state $\bm{x}=(x_{1},x_{2})$ (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:

$\displaystyle\widehat{\bm{\mu}}=\sum_{s}\bm{W}_{s}\text{
}\widehat{\bm{\mu}}_{s}.$ (21)

[We have omitted the subscript $k$, for ease of notation.] ‘Optimal’ fusion is
such that (i) $\widehat{\bm{\Sigma}}$ is minimal in some sense – typically
with respect to matrix trace or determinant; and also that (ii)
$\widehat{\bm{\Sigma}}$ is _consistent_. Within the fusion literature, a
consistent estimate is such that $\widehat{\bm{\Sigma}}\geq\bm{\Sigma}$, in
the sense that $\widehat{\bm{\Sigma}}-\bm{\Sigma}$ is positive semi-definite.
That is, the consistent estimate subsumes a covariance no less than the
covariance $\bm{\Sigma}$ 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 risk333In 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].. 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
$\\{\bm{x}:(\bm{x}-\bm{\mu})^{T}\bm{\Sigma}^{-1}(\bm{x}-\bm{\mu})=1\\}$. 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.

####  3.2.1 Precision-weighted fusion

We first consider precision-weighted (PW) fusion for the case that the cross-
covariances between sources is known to be 0. Suppose that for each $s$,
$\widehat{\bm{\mu}}_{s}$ is an unbiased estimate, and each
$\widehat{\bm{\Sigma}}_{s}$ is a known error covariance estimate that is
consistent. It follows that the fused estimate given in eq. (21) is an
unbiased estimate if and only if $\sum_{s}\bm{W}_{s}=\bm{I}$, for weight
matrices $\bm{W}_{s}$, and $\bm{I}$ the identity matrix. To find the weight
estimates we solve

$\displaystyle\min_{\bm{W}}\text{ trace}(\widehat{\bm{\Sigma}})\text{ subject
to }\sum_{s}\bm{W}_{s}=\bm{I}.$

Let $\widehat{\bm{\Sigma}}^{ij}$ denote the cross-covariance estimate between
the unique sources $s_{i}$ and $s_{j}$, and write

$\displaystyle\widehat{\bm{\Sigma}}=\sum_{s}\bm{W}_{s}\widehat{\bm{\Sigma}}_{s}\bm{W}_{s}^{T}+\sum_{s_{i}\neq
s_{j}}\bm{W}_{s_{i}}\widehat{\bm{\Sigma}}^{ij}\bm{W}_{s_{j}}^{T}.$ (22)

Assuming the cross-covariances are everywhere 0, solving for the optimal
weights one finds that, on weight substitution,

$\displaystyle\widehat{\bm{\Sigma}}$
$\displaystyle=(\widehat{\bm{\Sigma}}_{1}^{-1}+...+\widehat{\bm{\Sigma}}_{S}^{-1})^{-1},$
(23) $\displaystyle\widehat{\bm{\mu}}$
$\displaystyle=\widehat{\bm{\Sigma}}(\widehat{\bm{\Sigma}}_{1}^{-1}\widehat{\bm{\mu}}_{1}+...+\widehat{\bm{\Sigma}}_{S}^{-1}\widehat{\bm{\mu}}_{S}).$
(24)

Further details of this well-known result can be found in [23, 16]. 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) fusion method of [25]. The insight of this
approach is that, given eq. (22), the covariance ellipses for
$\\{\widehat{\bm{\Sigma}}_{s}\\}$ contain $\widehat{\bm{\Sigma}}$ in their
intersection, for any values of the cross-covariance terms. Hence the method
yields

$\displaystyle\widehat{\bm{\Sigma}}^{-1}$
$\displaystyle=\sum_{s}\omega_{s}\widehat{\bm{\Sigma}}_{s}^{-1},$ (25)
$\displaystyle\widehat{\bm{\mu}}$
$\displaystyle=\widehat{\bm{\Sigma}}\sum_{s}\omega_{s}\widehat{\bm{\Sigma}}_{s}^{-1}\widehat{\bm{\mu}}_{s},$
(26)

for scalars $\omega_{s}\in[0,1]$ with $\sum_{s}\omega_{s}=1$, since a convex
combination of covariances ensures $\widehat{\bm{\Sigma}}$ encloses the
intersection region. The $\omega_{s}$ terms can be computed via a standard
optimization routine subject to minimising the trace of
$\widehat{\bm{\Sigma}}$. Further, it has been shown that the CI method applied
to a pair of consistent estimates yields 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.

####  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
$s=1,2$:

$\displaystyle\widehat{\bm{\mu}}_{s}$
$\displaystyle=\widehat{\bm{\Sigma}}_{s}\Big{(}(\widetilde{\bm{\Sigma}}_{s})^{-1}\widetilde{\bm{\mu}}_{s}+\bm{\Gamma}^{-1}\bm{\gamma}\Big{)},$
(27) $\displaystyle\widehat{\bm{\Sigma}}_{s}$
$\displaystyle=\Big{(}(\widetilde{\bm{\Sigma}}_{s})^{-1}+\bm{\Gamma}^{-1}\Big{)}^{-1}.$
(28)

This is to say, that each estimate is itself the PW fusion of a sub-estimate
from the collection
$\\{(\widetilde{\bm{\mu}}_{s},\widetilde{\bm{\Sigma}}_{s}):1\leq s\leq 2\\}$,
for sub-estimates uncorrelated between sources, and a shared set of
information denoted $(\bm{\gamma},\bm{\Gamma})$, 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

$\displaystyle\widehat{\bm{\Sigma}}^{-1}=\sum_{i=1}^{2}\widehat{\bm{\Sigma}}_{s}^{-1}-\Big{(}\sum_{s=1}^{2}\omega_{s}\widehat{\bm{\Sigma}}_{s}\Big{)}^{-1},$
(29)

for scalars $\omega_{s}$ 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

$\displaystyle\widehat{\bm{\mu}}$
$\displaystyle=\sum_{s=1}^{2}C_{s}\widehat{\bm{\mu}}_{s},$ (30)
$\displaystyle\text{where }C_{s}$
$\displaystyle=\widehat{\bm{\Sigma}}\cdot\Big{(}\widehat{\bm{\Sigma}}_{s}^{-1}-\omega_{s}\Big{(}\sum_{s=1}^{2}\omega_{s}\widehat{\bm{\Sigma}}_{s}\Big{)}^{-1}\Big{)}.$

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:

$\displaystyle\widehat{\bm{\Sigma}}$
$\displaystyle\geq\widehat{\bm{\Sigma}}_{s}+(\widehat{\bm{\mu}}-\widehat{\bm{\mu}}_{s})(\widehat{\bm{\mu}}-\widehat{\bm{\mu}}_{s})^{T},\qquad
1\leq s\leq S.$

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, $\bm{x}$. The first $n$ elements of $\bm{x}$
are the elements of $\widehat{\bm{\mu}}$, and the remaining $\frac{n(n+1)}{2}$
elements of $\bm{x}$ are the elements of the upper triangle of
$\widehat{\bm{\Sigma}}$. Then $\bm{x}$ becomes the optimization target for
SolvOpt. We seek to minimise the determinant of $\widehat{\bm{\Sigma}}$
subject to the constraint that

$\displaystyle\forall
s,\qquad\widehat{\bm{\Sigma}}_{s}^{*}:=\widehat{\bm{\Sigma}}$
$\displaystyle-\widehat{\bm{\Sigma}}_{s}-(\widehat{\bm{\mu}}-\widehat{\bm{\mu}}_{s})(\widehat{\bm{\mu}}-\widehat{\bm{\mu}}_{s})^{T}\geq
0.$ (31)

A particular feature of the SolvOpt software is that it requires a constraint
$c(\bm{x})$ such that, for any input, $c(\bm{x})\leq 0$. Since
$\widehat{\bm{\Sigma}}_{s}^{*}$ is required to be positive semi-definite, it
must have non-negative eigenvalues for all choices of $s$, and the constraint
is satisfied for a choice of $c$ such that

$-c(\bm{x}):=\min\\{\\{\text{eigenvalues}(\widehat{\bm{\Sigma}}_{s}^{*}):\forall
s\\}\\}\geq 0.$

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 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 time-series, 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
$\bm{q},\bm{F},\bm{\Omega},\bm{\xi},\bm{V}$ and $\bm{D}$. We set $\bm{q}$ 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 $\bm{\Omega}$, and $\bm{F}$ is set equal to the aleatoric uncertainty
estimate. The prior hyperparameter $\bm{\xi}$ is estimated as an average of
$\bm{f}$ 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 $\bm{V}$ as this estimate.
Our estimate for $\bm{D}$ is a diagonal covariance matrix equal to
$\widehat{\sigma}^{2}\bm{I}$, where $\widehat{\sigma}^{2}$ is the mean square
error yielded from estimating eq. (7), and $\bm{I}$ 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

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 left-most 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 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 (one-sided)
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 Shaprio-Wilks 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.

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)
---|---|---|---|---|---|---
SPX | ICI | 0.14 | 0.21 | 0.06 | [ 0.15, 0.98 ] | [ -0.13, 0.90 ] | 0.07
| CI | 0.14 | 0.45 | 0.05 | [ 0.15, 0.99 ] | [ -0.07, 0.70 ] | 0.07
| CU | 0.12 | 0.75 | 0.05 | [ 0.19, 1.01 ] | [ 0.31, 1.03 ] | 0.06
| PW | 0.17 | 0.36 | 0.03 | [ 0.29, 1.22 ] | [ 0.07, 1.30 ] | 0.06
| ARIMA | 0.13 | 0.41 | 0.03 | [ 0.25, 1.09 ] | [ 0.17, 0.92 ] | 0.03
| GP | 0.17 | 0.51 | 0.03 | [ 0.20, 1.09 ] | [ 0.34, 1.10 ] | 0.03
| Boost | 0.16 | 0.32 | 0.05 | [ 0.18, 1.01 ] | [ -0.10, 0.83 ] | 0.08
ICI | CI | 0.80 | 0.03 | 0.47 | [ -0.08, 0.08 ] | [ -0.26, 0.28 ] | 0.31
| CU | 0.88 | 0.12 | 0.41 | [ -0.11, 0.14 ] | [ 0.01, 0.67 ] | 0.63
| PW | 0.94 | 0.90 | 0.15 | [ -0.03, 0.32 ] | [ 0.04, 1.00 ] | 0.19
| ARIMA | 0.70 | 0.75 | 0.08 | [ 0.02, 0.21 ] | [ -0.26, 0.36 ] | 0.10
| GP | 0.33 | 0.65 | 0.09 | [ 0.01, 0.19 ] | [ 0.25, 0.73 ] | 0.12
| Boost | 0.68 | 0.19 | 0.46 | [ -0.21, 0.25 ] | [ -0.51, 0.39 ] | 0.58
CI | CU | 0.56 | 0.85 | 0.43 | [ -0.12, 0.14 ] | [ 0.05, 0.51 ] | 0.55
| PW | 0.67 | 0.56 | 0.15 | [ -0.02, 0.32 ] | [ -0.29, 0.65 ] | 0.18
| ARIMA | 0.47 | 0.00 | 0.06 | [ 0.03, 0.19 ] | [ -0.17, 0.22 ] | 0.93
| GP | 0.11 | 0.74 | 0.13 | [ -0.01, 0.20 ] | [ 0.30, 0.67 ] | 0.16
| Boost | 0.35 | 0.91 | 0.47 | [ -0.17, 0.24 ] | [ -0.52, 0.10 ] | 0.50
CU | PW | 0.29 | 0.28 | 0.17 | [ -0.04, 0.28 ] | [ -0.57, 0.43 ] | 0.17
| ARIMA | 0.48 | 0.89 | 0.23 | [ -0.05, 0.22 ] | [ -0.46, -0.02 ] | 0.20
| GP | 0.75 | 0.22 | 0.23 | [ -0.06, 0.20 ] | [ -0.13, 0.37 ] | 0.21
| Boost | 0.73 | 0.98 | 0.49 | [ -0.21, 0.19 ] | [ -0.78, 0.01 ] | 0.52
PW | ARIMA | 0.95 | 0.64 | 0.36 | [ -0.17, 0.11 ] | [ -0.65, 0.24 ] | 0.65
| GP | 0.95 | 0.58 | 0.33 | [ -0.19, 0.08 ] | [ -0.31, 0.66 ] | 0.41
| Boost | 0.17 | 0.45 | 0.09 | [ -0.27, -0.03 ] | [ -1.12, -0.14 ] | 0.10
ARIMA | GP | 0.01 | 0.94 | 0.46 | [ -0.08, 0.07 ] | [ 0.22, 0.54 ] | 0.45
| Boost | 0.20 | 0.06 | 0.27 | [ -0.29, 0.09 ] | [ -0.51, 0.19 ] | 0.73
GP | Boost | 0.74 | 0.34 | 0.27 | [ -0.26, 0.07 ] | [ -0.96, -0.32 ] | 0.64

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.

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##  Appendix 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 $\bm{r}|\bm{q}$ 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 $-\frac{1}{2}$, complete the square in
$\bm{f}$, and integrate what is recognisable as a Gaussian integral:

$\displaystyle(\bm{r}-\bm{X}\bm{f})^{T}\bm{D}^{-1}(\bm{r}-\bm{X}\bm{f})+(\bm{f}-\bm{\mu})^{T}\bm{\Sigma}^{-1}(\bm{f}-\bm{\mu})$
$\displaystyle=\bm{r}^{T}\bm{D}^{-1}\bm{r}-2\bm{f}^{T}\bm{X}^{T}\bm{D}^{-1}\bm{r}$
$\displaystyle\qquad+(\bm{X}\bm{f})^{T}\bm{D}^{-1}\bm{X}\bm{f}+\bm{f}^{T}\bm{\Sigma}^{-1}\bm{f}-2\bm{f}^{T}\bm{\Sigma}^{-1}\bm{\mu}+\bm{\mu}^{T}\bm{\Sigma}^{-1}\bm{\mu}$
$\displaystyle=\bm{r}^{T}\bm{D}^{-1}\bm{r}+\bm{f}^{T}\bm{H}\bm{f}-2\bm{\eta}\bm{f}+\bm{\mu}^{T}\bm{\Sigma}^{-1}\bm{\mu}$

where $\bm{\mu}$ and $\bm{\Sigma}$ denote the mean and covariance for
$\bm{f}|\bm{q}$, $\bm{H}:=\bm{X}^{T}\bm{D}^{-1}\bm{X}+\bm{\Sigma}^{-1}$ and
$\bm{\eta}:=\bm{X}^{T}\bm{D}^{-1}\bm{r}+\bm{\Sigma}^{-1}\bm{\mu}$. Completing
the square,

$\bm{f}^{T}\bm{H}\bm{f}-2\bm{f}^{T}\bm{\eta}=(\bm{f}-\bm{\nu})^{T}\bm{H}(\bm{f}-\bm{\nu})-\bm{\nu}^{T}\bm{H}\bm{\nu},\qquad\bm{\nu}=\bm{H}^{-1}\bm{\eta},$

whereby

$\int\exp\Big{(}-\frac{1}{2}(\bm{f}-\bm{\nu})^{T}\bm{H}(\bm{f}-\bm{\nu})\Big{)}d\bm{f}=\sqrt{\frac{(2\pi)^{k}}{|\bm{H}|}}.$

Therefore

$\displaystyle
p(\bm{r}|\bm{q})\propto\exp\Big{(}-\frac{1}{2}\Big{[}\bm{r}^{T}\bm{D}^{-1}\bm{r}+\bm{\mu}^{T}\bm{\Sigma}^{-1}\bm{\mu}-\bm{\nu}^{T}\bm{H}\bm{\nu}\Big{]}\Big{)}=\exp\Big{(}-\frac{1}{2}\Big{[}\bm{r}^{T}\bm{D}^{-1}\bm{r}+\bm{\mu}^{T}\bm{\Sigma}^{-1}\bm{\mu}-\bm{\eta}^{T}\bm{H}^{-1}\bm{\eta}\Big{]}\Big{)},$
(32)

and we retrieve the exponent of the Gaussian for $\bm{r}|\bm{q}$. Inspired by
[4], we next make use of the following Lemma:

######  Lemma 1.

If a multivariate normal random variable $\bm{\theta}$ has density
$p(\bm{\theta})$ and $-2\log
p(\bm{\theta})=\bm{\theta}^{\prime}\bm{H}\bm{\theta}-2\bm{\nu}^{\prime}\bm{\theta}+(\text{terms
without }\bm{\theta})$ then $\text{var}(\bm{\theta})=\bm{H}^{-1}$ and
$\mathbb{E}\bm{\theta}=\bm{H}^{-1}\bm{\nu}$.

Collecting quadratic terms in $\bm{r}$ from the exponent of eq. (32):

$\displaystyle-\bm{\eta}^{T}\bm{H}^{-1}\bm{\eta}+\bm{r}^{T}\bm{D}^{-1}\bm{r}$
$\displaystyle=-[\bm{X}^{T}\bm{D}^{-1}\bm{r}+\bm{\Sigma}^{-1}\bm{\mu}]^{T}\bm{H}^{-1}[\bm{X}^{T}\bm{D}^{-1}\bm{r}+\bm{\Sigma}^{-1}\bm{\mu}]+\bm{r}^{T}\bm{D}^{-1}\bm{r}$
$\displaystyle=-\bm{r}^{T}\bm{D}^{-1}\bm{X}\bm{H}^{-1}\bm{X}^{T}\bm{D}^{-1}\bm{r}+\bm{r}^{T}\bm{D}^{-1}\bm{r}+\text{lower-
order terms in $\bm{r}$}$ (33)
$\displaystyle=\bm{r}^{T}[-\bm{D}^{-1}\bm{X}\bm{H}^{-1}\bm{X}^{T}\bm{D}^{-1}+\bm{D}^{-1}]\bm{r}+\text{lower-
order terms in $\bm{r}$}$

Making use of the lemma, we have that the posterior predictive distribution
$p(\bm{r}|\bm{q})$ is multivariate normal with covariance

$\displaystyle\text{var}(\bm{r}|\bm{q})$
$\displaystyle=[-\bm{D}^{-1}\bm{X}\bm{H}^{-1}\bm{X}^{T}\bm{D}^{-1}+\bm{D}^{-1}]^{-1},$
$\displaystyle=[\bm{D}^{-1}-\bm{D}^{-1}\bm{X}[\bm{X}^{T}\bm{D}^{-1}\bm{X}+[(\bm{V}^{-1}+\bm{\Omega}^{-1})^{-1}+\bm{F}]^{-1}]^{-1}\bm{X}^{T}\bm{D}^{-1}]^{-1}.$

Futher, we collect the linear terms in $\bm{r}$ from eq. (33):

$\displaystyle-\bm{r}^{T}\bm{D}^{-1}\bm{X}^{T}\bm{H}^{-1}\bm{\Sigma}^{-1}\bm{\mu}-\bm{\mu}^{T}\bm{\Sigma}^{-1}\bm{H}^{-1}\bm{X}^{T}\bm{D}^{-1}\bm{r}$
$\displaystyle=-2\bm{\mu}^{T}\bm{\Sigma}^{-1}\bm{H}^{-1}\bm{X}^{T}\bm{D}^{-1}\bm{r},$

so that again by the lemma

$\displaystyle\mathbb{E}(\bm{r}|\bm{q})$
$\displaystyle=\text{var}(\bm{r}|\bm{q})\bm{D}^{-1}\bm{X}\bm{H}^{-1}\bm{\Sigma}^{-1}\bm{\mu}$
$\displaystyle=\text{var}(\bm{r}|\bm{q})\bm{D}^{-1}\bm{X}[\bm{X}^{T}\bm{D}^{-1}\bm{X}+[(\bm{V}^{-1}+\bm{\Omega}^{-1})^{-1}+\bm{F}]^{-1}]^{-1}$
$\displaystyle\qquad\cdot[(\bm{V}^{-1}+\bm{\Omega}^{-1})^{-1}+\bm{F}]^{-1}(\bm{V}^{-1}+\bm{\Omega}^{-1})^{-1}(\bm{V}^{-1}\bm{\xi}+\bm{\Omega}^{-1}\bm{q}).$

###  A.2 View fusion under model-based forecasts

Suppose we model $y=f(\bm{x})+\epsilon$ for some function $f$ parameterised by
$\bm{\theta}$, with $\bm{\epsilon}\thicksim N(0,\sigma^{2})$. Then we can
write $y|\bm{x},\bm{\theta}\thicksim N(f(\bm{x}),\sigma^{2})$. Assuming
$\bm{\theta}$ has prior distribution $p(\bm{\theta})$, then for a collection
of data $\mathcal{D}$ we can update the prior via the likelihood analogous to
eq. (5) above: $p(\bm{\theta}|\mathcal{D})\propto
p(\mathcal{D}|\bm{\theta})p(\bm{\theta})$. Then for a new data point
$\bm{x}^{*}$, the predictive distribution for the corresponding $y^{*}$ can be
written

$p(y^{*}|\bm{x}^{*},\mathcal{D})=\int
p(y^{*}|\bm{x}^{*},\bm{\theta})p(\bm{\theta}|\mathcal{D})d\bm{\theta}.$

Further, the variance of the prediction,
$\text{var}(y^{*}|\bm{x}^{*},\mathcal{D})$, can be decomposed making use of
the law of total variance, that is:

$\displaystyle\text{var}(y^{*}|\bm{x}^{*},\mathcal{D})=\text{var}_{\bm{\theta}|\mathcal{D}}\big{(}\mathbb{E}(y^{*}|\bm{x}^{*},\bm{\theta})\big{)}+\mathbb{E}_{\bm{\theta}|\mathcal{D}}\big{(}\text{var}(y^{*}|\bm{x}^{*},\bm{\theta})\big{)}.$
(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 $\bm{\theta}|\mathcal{D}$. 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
time-series 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.

###  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 -17.50 8.59 -2.04 -2.68 -2.01 -16.43
20.00 9.80 2.04 1.73 5.65 -18.76 GP -15.42 8.59 -1.79 -2.60 -1.78 -15.90 21.39
9.80 2.18 1.78 5.91 -19.92 Boost -9.14 8.59 -1.06 -1.84 -1.18 -10.40 12.53
9.80 1.28 1.04 2.73 -13.63 PW -12.04 8.59 -1.40 -2.20 -1.44 -12.37 13.90 9.80
1.42 1.16 3.07 -14.73 CU -11.11 8.59 -1.29 -1.98 -1.47 -13.29 20.65 9.80 2.11
1.74 5.76 -19.13 CI -20.11 8.59 -2.34 -2.81 -2.22 -18.60 18.24 9.80 1.86 1.59
6.13 -17.71 ICI -16.51 8.59 -1.92 -2.46 -1.82 -16.14 19.94 9.80 2.03 1.70 8.08
-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 -12.74 7.81 -1.63 -4.48 -1.67
-14.11 14.18 11.75 1.21 -0.33 2.78 -19.34 GP -11.30 7.81 -1.45 -4.33 -1.52
-13.81 9.49 11.75 0.81 -0.59 1.65 -16.73 Boost -15.45 7.81 -1.98 -4.56 -2.06
-14.93 10.90 11.75 0.93 -0.49 1.62 -18.25 PW -21.30 7.81 -2.73 -5.36 -2.46
-20.11 9.34 11.75 0.79 -0.60 1.34 -19.69 CU -11.64 7.81 -1.49 -4.53 -1.57
-14.14 20.40 11.75 1.74 0.02 4.69 -22.35 CI -13.29 7.81 -1.70 -4.87 -1.77
-14.72 25.06 11.75 2.13 0.29 6.07 -25.21 ICI -12.45 7.81 -1.59 -4.78 -1.67
-14.07 21.15 11.75 1.80 0.06 4.78 -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 26.74 18.09 1.48 -0.05 3.39
-21.90 12.62 20.25 0.62 -0.34 0.91 -16.82 GP 31.28 18.09 1.73 0.17 3.71 -25.04
10.57 20.25 0.52 -0.38 0.73 -16.72 Boost 32.70 18.09 1.81 0.22 4.57 -26.85
5.64 20.25 0.28 -0.58 0.41 -19.44 PW 30.50 18.09 1.69 0.13 3.69 -26.26 15.04
20.25 0.74 -0.24 1.06 -15.90 CU 28.95 18.09 1.60 0.05 3.75 -23.06 29.10 20.25
1.44 0.22 2.41 -27.24 CI 23.15 18.09 1.28 -0.22 2.61 -18.85 17.15 20.25 0.85
-0.18 1.24 -16.74 ICI 28.57 18.09 1.58 0.04 3.58 -22.90 26.14 20.25 1.29 0.12
2.20 -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 -28.79 18.05 -1.60 -1.72
-1.71 -27.22 -5.65 22.17 -0.25 -0.10 -0.33 -26.38 GP -21.79 18.05 -1.21 -1.58
-1.40 -22.85 -8.89 22.17 -0.40 -0.23 -0.51 -26.27 Boost 19.04 18.05 1.06 -0.12
1.64 -25.24 -29.64 22.17 -1.34 -0.83 -1.55 -38.33 PW -9.48 18.05 -0.53 -1.13
-0.68 -20.84 -10.81 22.17 -0.49 -0.32 -0.64 -27.58 CU -3.66 18.05 -0.20 -0.94
-0.27 -21.42 1.46 22.17 0.07 0.15 0.09 -24.08 CI -19.29 18.05 -1.07 -1.39
-1.27 -20.96 6.15 22.17 0.28 0.34 0.39 -23.62 ICI -32.13 18.05 -1.78 -1.89
-1.89 -30.89 -6.63 22.17 -0.30 -0.12 -0.38 -27.23

Figure 6: Performance metrics, 1993-2000.

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 36.09 21.51 1.68 1.72 2.57
-35.22 26.07 25.99 1.00 1.49 1.53 -34.82 GP 36.18 21.51 1.68 1.67 2.60 -34.88
33.51 25.99 1.29 1.73 2.10 -37.44 Boost 25.65 21.51 1.19 1.34 1.64 -30.70
31.67 25.99 1.22 1.65 1.99 -36.37 PW 23.71 21.51 1.10 1.28 1.48 -29.85 33.17
25.99 1.28 1.68 2.09 -37.85 CU 21.26 21.51 0.99 1.24 1.39 -26.00 4.75 25.99
0.18 0.83 0.24 -30.08 CI 35.16 21.51 1.63 1.66 2.62 -33.08 22.01 25.99 0.85
1.37 1.28 -31.67 ICI 34.26 21.51 1.59 1.64 2.52 -33.28 26.35 25.99 1.01 1.48
1.57 -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 -23.61 17.03 -1.39 -2.48 -1.53
-23.77 14.26 11.07 1.29 0.26 2.62 -15.21 GP -27.81 17.03 -1.63 -2.70 -1.73
-26.21 15.63 11.07 1.41 0.35 3.08 -16.01 Boost -31.93 17.03 -1.87 -2.76 -1.95
-30.85 12.73 11.07 1.15 0.16 2.41 -14.50 PW -31.18 17.03 -1.83 -2.80 -1.86
-29.23 14.71 11.07 1.33 0.30 3.02 -14.74 CU -20.25 17.03 -1.19 -2.60 -1.28
-22.22 12.98 11.07 1.17 0.17 2.22 -15.10 CI -20.35 17.03 -1.19 -2.32 -1.36
-21.51 12.22 11.07 1.10 0.13 2.18 -14.11 ICI -21.49 17.03 -1.26 -2.39 -1.40
-22.16 16.73 11.07 1.51 0.43 3.21 -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 3.65 10.27 0.36 -0.24 0.52 -9.98
-3.92 10.02 -0.39 -1.17 -0.58 -15.33 GP -0.49 10.27 -0.05 -0.57 -0.07 -11.40
-2.43 10.02 -0.24 -1.03 -0.36 -14.76 Boost 1.49 10.27 0.15 -0.42 0.22 -11.54
-4.06 10.02 -0.41 -1.14 -0.58 -15.78 PW 0.10 10.27 0.01 -0.53 0.01 -12.06
-6.36 10.02 -0.63 -1.33 -0.87 -16.78 CU 1.58 10.27 0.15 -0.44 0.23 -10.15 3.40
10.02 0.34 -0.59 0.61 -13.37 CI 1.62 10.27 0.16 -0.41 0.22 -10.65 -2.07 10.02
-0.21 -1.02 -0.32 -14.31 ICI 1.23 10.27 0.12 -0.45 0.17 -9.78 1.52 10.02 0.15
-0.74 0.27 -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 -1.06 15.96 -0.07 -0.33
-0.10 -12.74 17.00 40.88 0.42 1.12 0.60 -42.30 GP -1.52 15.96 -0.10 -0.35
-0.14 -13.27 46.05 40.88 1.13 1.69 1.83 -40.38 Boost -3.80 15.96 -0.24 -0.48
-0.35 -15.23 39.48 40.88 0.97 1.34 1.56 -43.34 PW 1.83 15.96 0.11 -0.21 0.18
-14.36 56.22 40.88 1.38 1.81 2.61 -46.29 CU -13.65 15.96 -0.86 -0.87 -1.09
-17.35 67.57 40.88 1.65 2.27 3.13 -51.49 CI -3.35 15.96 -0.21 -0.44 -0.31
-11.66 47.33 40.88 1.16 1.79 2.16 -42.41 ICI -0.99 15.96 -0.06 -0.31 -0.08
-9.66 60.76 40.88 1.49 2.33 2.91 -49.16

Figure 7: Performance metrics, 2001-2008.

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 17.54 27.22 0.64 -0.13 1.13
-21.65 -41.11 18.02 -2.28 -2.30 -2.19 -37.28 GP 7.78 27.22 0.29 -0.43 0.48
-18.56 -42.99 18.02 -2.38 -2.42 -2.33 -38.98 Boost -33.40 27.22 -1.23 -1.76
-1.51 -35.25 -33.99 18.02 -1.89 -2.07 -1.97 -33.55 PW -9.72 27.22 -0.36 -1.00
-0.55 -19.77 -36.86 18.02 -2.04 -2.17 -2.06 -34.57 CU 19.15 27.22 0.70 -0.08
1.30 -21.64 -36.30 18.02 -2.01 -2.14 -2.03 -35.08 CI 25.99 27.22 0.95 0.12
1.52 -31.85 -37.32 18.02 -2.07 -2.14 -1.98 -34.19 ICI 27.58 27.22 1.01 0.17
1.74 -27.00 -38.24 18.02 -2.12 -2.16 -1.99 -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 -29.76 23.24 -1.28 -1.08 -1.35 -34.76
22.54 12.73 1.77 0.46 3.23 -19.67 GP -29.17 23.24 -1.26 -1.03 -1.33 -33.83
29.53 12.73 2.32 0.86 5.28 -25.51 Boost -8.30 23.24 -0.36 -0.41 -0.45 -18.13
28.89 12.73 2.27 0.83 5.22 -25.12 PW -8.41 23.24 -0.36 -0.42 -0.48 -16.08
25.43 12.73 2.00 0.65 4.23 -24.06 CU -15.96 23.24 -0.69 -0.56 -0.80 -29.10
15.64 12.73 1.23 0.09 1.79 -15.94 CI -34.31 23.24 -1.48 -1.19 -1.53 -32.76
19.15 12.73 1.50 0.28 2.45 -18.43 ICI -26.26 23.24 -1.13 -0.92 -1.18 -33.43
18.62 12.73 1.46 0.25 2.45 -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 2.61 11.05 0.24 -1.74 0.34
-11.82 8.11 11.33 0.72 -0.35 1.74 -10.11 GP -4.37 11.05 -0.40 -2.45 -0.51
-16.53 7.28 11.33 0.64 -0.42 1.33 -8.94 Boost 7.30 11.05 0.66 -1.47 0.96 -8.04
16.40 11.33 1.45 0.18 3.85 -15.84 PW -1.12 11.05 -0.10 -2.36 -0.14 -14.72
14.34 11.33 1.27 0.05 3.14 -14.43 CU -11.24 11.05 -1.02 -2.96 -1.17 -19.21
1.81 11.33 0.16 -0.79 0.29 -7.87 CI 4.64 11.05 0.42 -1.57 0.63 -9.25 6.04
11.33 0.53 -0.47 1.26 -9.70 ICI -3.97 11.05 -0.36 -2.40 -0.47 -15.92 7.34
11.33 0.65 -0.39 1.43 -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 29.50 15.46 1.91 1.56 2.82 -24.32
-0.11 13.07 -0.01 -0.90 -0.01 -11.81 GP 25.13 15.46 1.62 1.34 2.52 -24.29 0.75
13.07 0.06 -0.84 0.11 -12.31 Boost 28.98 15.46 1.87 1.47 3.17 -26.31 -3.25
13.07 -0.25 -1.08 -0.39 -10.62 PW 28.45 15.46 1.84 1.56 2.77 -26.20 -11.84
13.07 -0.91 -1.60 -1.22 -12.92 CU 31.35 15.46 2.03 1.83 2.96 -26.59 -1.96
13.07 -0.15 -1.01 -0.25 -12.55 CI 27.01 15.46 1.75 1.45 2.56 -22.93 2.62 13.07
0.20 -0.73 0.38 -13.80 ICI 22.50 15.46 1.46 1.12 2.42 -23.22 6.23 13.07 0.48
-0.52 1.03 -13.95

Figure 8: Performance metrics, 2009-2016.

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 -1.08 6.67 -0.16 -2.14 -0.24
-7.76 4.96 17.03 0.29 0.50 0.40 -14.97 GP 0.06 6.67 0.01 -2.03 0.02 -5.84
-0.59 17.03 -0.03 0.26 -0.04 -14.22 Boost -3.86 6.67 -0.58 -2.50 -0.79 -10.74
6.59 17.03 0.39 0.61 0.56 -16.52 PW -7.71 6.67 -1.16 -2.99 -1.41 -12.44 13.39
17.03 0.79 0.87 1.17 -17.03 CU -0.88 6.67 -0.13 -2.16 -0.20 -6.45 7.20 17.03
0.42 0.57 0.57 -12.90 CI 2.51 6.67 0.38 -1.75 0.61 -6.02 2.18 17.03 0.13 0.37
0.17 -12.58 ICI 4.71 6.67 0.71 -1.48 1.32 -4.78 -5.87 17.03 -0.34 0.05 -0.44
-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 22.69 12.45 1.37 -0.16 3.17
-22.74 17.36 34.37 0.51 -0.17 0.61 -38.09 GP 20.72 12.45 1.66 -0.30 3.69
-21.68 -1.95 34.37 -0.06 -0.59 -0.07 -36.61 Boost 23.24 12.45 1.87 -0.15 4.15
-20.27 26.22 34.37 0.76 0.03 1.01 -39.32 PW 16.83 12.45 1.35 -0.50 3.33 -19.62
22.99 34.37 0.67 -0.04 0.86 -37.70 CU 17.75 12.45 1.43 -0.50 2.73 -19.82 3.84
34.37 0.11 -0.47 0.13 -36.68 CI 23.50 12.45 1.89 -0.15 3.68 -21.02 8.55 34.37
0.25 -0.36 0.29 -36.89 ICI 19.15 12.45 1.54 -0.42 2.47 -19.28 8.74 34.37 0.25
-0.36 0.30 -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 -5.78 13.07 -0.46 -1.65 -0.60 -12.93
GP -2.48 13.07 -0.20 -1.47 -0.27 -10.21 Boost 6.64 13.07 0.53 -0.81 0.84
-12.56 PW -5.24 13.07 -0.42 -1.55 -0.53 -12.61 CU -0.34 13.07 -0.03 -1.37
-0.04 -9.50 CI -1.12 13.07 -0.09 -1.34 -0.13 -9.80 ICI -2.15 13.07 -0.17 -1.40
-0.24 -10.50

Figure 9: Performance metrics, 2017-2021.