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# Canonical Portfolios: Optimal Asset and Signal Combination

Nikan Firoozye Department of Computer Science, University College London.
E-mail<EMAIL_ADDRESS>Vincent Tan Oxford-Man Institute of Quantitative
Finance, University of Oxford. E-mail<EMAIL_ADDRESS>Stefan Zohren
Oxford-Man Institute of Quantitative Finance, University of Oxford. E-mail:
<EMAIL_ADDRESS>
(September 3, 2024)

# Canonical Portfolios: Optimal Asset and Signal Combination111 The authors
would like to thank Mihai Cucuringu, Brian Healy, Anthony Ledford, Ana Maria
Pires, Slavi Marinov, Attilio Meucci, and Yin Cheng Ng, as well as from
seminar participants at University of Oxford and Man AHL for the helpful
comments. This work was supported by the Oxford-Man Institute of Quantitative
Finance.

Nikan Firoozye Department of Computer Science, University College London.
E-mail<EMAIL_ADDRESS>Vincent Tan Oxford-Man Institute of Quantitative
Finance, University of Oxford. E-mail<EMAIL_ADDRESS>Stefan Zohren
Oxford-Man Institute of Quantitative Finance, University of Oxford. E-mail:
<EMAIL_ADDRESS>
(September 3, 2024)

We present a novel framework for analyzing the optimal asset and signal
combination problem, which builds upon the dynamic portfolio selection problem
of Brandt and Santa-Clara, (2006) in two phases. First, we reformulate their
original investment problem into a tractable vehicle, which admits a closed-
form solution that scales to large dimensions by imposing a joint Gaussian
structure on the asset returns and signals. Second, we recast the optimal
portfolio of correlated assets and signals into a set of uncorrelated managed
portfolios through the lens of Canonical Correlation Analysis of Hotelling,
(1936). The new investment environment of uncorrelated managed portfolios
offers unique economic insights into the joint correlation structure of our
optimal portfolio policy. We also operationalize our theoretical framework to
bridge the gap between theory and practice and showcase the improved
performance of our proposed method over natural competing benchmarks.

JEL Classification: G11, D81, C1.

Keywords: Canonical Correlation Analysis, Portfolio Selection, Statistical
Arbitrage

##  1 Introduction

The investment decisions of portfolio managers are typically guided by signals
that encompass their view on future returns.222The terminology “signals” is
also often referred to as predictors, characteristics, state variables, cross-
sectional anomalies, etc. in the finance literature. In order to harness the
predictive contents of these signals to drive those decisions, there is a
standard approach championed by Markowitz, (1952), which still to this day
remains the workhorse of modern portfolio theory and forms the bedrock of how
we think about diversification. Markowitz’s portfolio selection is
traditionally performed in two stages. In the first stage, the investor draws
inferences about the return-generating process either through plug-in
estimation or subjective belief formation. In the second stage, the optimal
portfolio weights are formed with estimates from the first stage.

In this paper, we borrow inspiration from Brandt and Santa-Clara, (2006) and
consider solving a dynamic portfolio selection problem with multiple return-
predictive signals in a single stage. Our first contribution is to reformulate
the authors’ original investment problem into a tractable one by modeling the
joint distribution of the returns and signals as a Gaussian, which yields a
solution that scales to large dimensions. The end result is a powerful
framework that (1) integrates both return forecasting and asset allocation
processes together, (2) exposes the multivariate contributions that are
embedded in both processes, which may impact the final allocation of capital,
and (3) is simple to implement. However, along with these benefits come
concomitant challenges. Unlike the standard mean-variance framework that only
requires one large-dimensional covariance as an input, our framework
necessitates two additional inputs; namely, the covariance of the signals and
the cross-covariances between both the returns and signals. These new inputs
not only warrant estimation but their interactions–between and within
variables–also introduce an additional layer of complexity that we have to
distill.

Our second contribution is to provide a simplification of this problem by
reorganizing the set of assets and accompanying signals into a set of
uncorrelated managed portfolios. As was hinted in Firoozye and Koshiyama,
(2020, Section 4), this can be achieved by a powerful tool from multivariate
analysis known as canonical correlation analysis (CCA) pioneered by Hotelling,
(1936). CCA is a generalization of principal component analysis (PCA) of
Hotelling, (1933) to two sets of random variables and a study of their joint
association. This is pertinent to our case where we have multiple correlated
assets and correlated signals that are mutually linked by a joint correlation
objective. CCA allows us to express any portfolio returns derived from our
framework in terms of its exposures to uncorrelated sources of returns by
reweighting the original set of financial variables such that their
correlation is maximally retained; we refer to these special weights as
canonical portfolios.333The idea of recasting the asset universe into their
orthogonal components through PCA can be traced back to Partovi and Caputo,
(2004) with further extensions and applications from Meucci, (2009) and
Avellaneda and Lee, (2010), as well as from Kozak et al., (2018, 2020) in the
context of risk pricing in a no near-arbitrage framework. Our work aims to
broaden that horizon to two sets of variables by also considering the signal
as an important input in the analysis of their joint association with returns.

The third contribution is to render our theoretical framework applicable to
practice, even in large dimensions. This is a challenging task given that all
three matrix inputs have degrees of freedom that scale quadratically in the
size of the problem. Thus, regularization of these large-dimensional objects
is paramount to encourage stability in the out-of-sample results. Our
prescription to this problem is by employing shrinkage to the covariances,
which is commonly used in portfolio selection problems; see, for example,
Frost and Savarino, (1986) and Ledoit and Wolf, 2004a ; Ledoit and Wolf, 2004b
; Ledoit and Wolf, (2017). Additionally, we retain the leading few canonical
portfolios that exhibit the greatest predictability while discarding noisy and
unstable ones. We show through a series of backtest simulations that our
proposed method responds positively to these innovations that we introduced
and outperforms competitive benchmarks.

Our paper builds upon a series of seminal works from Brandt, (1999), Ferson
and Siegel, (2001), Brandt and Santa-Clara, (2006), and Brandt et al., (2009)
that directly model the portfolio weights. Of relevance to our study is Brandt
and Santa-Clara, (2006), where the authors analyzed portfolio policies
conditioning on a small number of state variables through augmentation of the
asset space. However, we provide a counterpoint to the authors on two fronts.
First, we adopt a tractable approach to their problem that allows us to reveal
the correlation structure of our optimal portfolio through CCA. Second, since
our approach does not expand the dimension of the asset space, our method is
more computationally tractable in large dimensions, allowing it to accommodate
large-scale cross-sectional applications such as the equity universe. The
price that we have to pay for this tractability, however, is the need to
confront three large-dimensional objects.

Our study also extends the work of Firoozye and Koshiyama, (2019) on the use
of total least squares (TLS) of Golub and Van Loan, (1980) for optimally
combining signals on a univariate return. Their original study focused on a
novel objective function for return prediction. While most research analysts
will seek to find a good forecast for future returns via methods such as
ordinary least squares (OLS), or in the nonlinear context, via a large suite
of machine learning-based methods, Firoozye and Koshiyama, (2019, 2020) argued
that if the goal is to maximize the Sharpe ratio of trading strategy, then
this can be achieved through a linear combination of signals that maximizes
the correlation between their combination and return. In the linear context,
the solution to this problem comes out via TLS, which is an errors-in-
variables formulation of regression. TLS has been well-studied primarily among
numerical analysts and is used less formally on trading desks of investment
banks and hedge funds, typically under the moniker of PCA regression. Given
the close relation between TLS and CCA, our work extends this research towards
a multivariate context, which is more relevant for problems in portfolio
selection.

The remainder of the paper is organized as follows. Section 2 gives the
description of the dynamic portfolio selection problem. Section 3 provides the
financial interpretation of our optimizer with CCA and details our proposed
estimation approach. Section 4 describes the empirical methodology and
presents the results of the out-of-sample backtest exercise with Fama-French
equity sorted portfolios. Section 5 concludes. Appendix A–B contain all the
figures, tables, and additional mathematical derivations.

##  2 Setting the Stage

###  2.1 Notation

In this section, we introduce the notational convention for our analysis. Let
the subscript $i$ index the variables such that $i\in\\{1,\ldots,N\\}$, where
$N$ denotes the dimension of the asset universe. The subscript $t$ indexes the
trading dates such that $t\in\\{1,\ldots,T\\}$, where $T$ denotes the number
of observations. The notation $\mathsf{Cov}(\cdot,\cdot)$ represents the
covariance matrix between two random vectors, $\mathsf{Tr}(\cdot)$ represents
the trace of a square matrix, and $\mathsf{Diag}(\cdot)$ represents a diagonal
matrix of a given vector. Denote $\mathbb{I}_{N}$ to be an identity matrix of
dimensions $N\times N$. For any $N$-dimensional vector $z$, its
$\ell_{2}$-norm is given by $\|z\|_{2}:=\sqrt{z_{1}^{2}+\ldots z_{N}^{2}}$,
and for any real matrix $Z$, its Frobenius norm is
$\|Z\|_{\mathrm{F}}:=\sqrt{\mathsf{Tr}(Z^{\prime}Z)}$. Here, the symbol
$(^{\prime})$ denotes the transpose operator of a vector or matrix, the symbol
$:=$ is a definition sign, and the symbol $=$ is the equal sign.

Let $r_{t,i}$ be the return for a risky asset $i$ sampled at trading date $t$,
stacked into a vector $r_{t}:=(r_{t,1},\ldots,r_{t,N})^{\prime}$. Also, let
$x_{t,i}$ be an $M$-dimensional vector return-predictive signal at date $t$
for asset $i$, stacked into a vector
$x_{t}:=(x_{t,1},\ldots,x_{t,N})^{\prime}$. This means every asset is
accompanied by at least one signal and so $M$ is necessarily a positive
integer. For example, if there are two signals for one asset, then $M=2$. The
signals are understood to proxy the future expected returns conditional on the
investor’s information set at time $t$. We shall assume that both the asset
returns and the signals exhibit stationarity and ergodicity.

Their multivariate distributions are assumed to be Gaussian with zero
expectations and covariances
$\mathsf{Var}(r_{t+1}):=\mathsf{Cov}(r_{t+1},r_{t+1})=\Sigma_{r}$ and
$\mathsf{Var}(x_{t}):=\mathsf{Cov}(x_{t},x_{t})=\Sigma_{x}$, and a cross-
covariance $\mathsf{Cov}(r_{t+1},x_{t})=\Sigma_{rx}=\Sigma_{xr}^{\prime}$. The
column vector of stacked subsequent returns and signals
$(r_{t+1},x_{t})^{\prime}$ has a joint covariance matrix expressed in the
following partitioned form:

$\displaystyle\Sigma=\begin{pmatrix}\Sigma_{r}&\Sigma_{rx}\\\
\Sigma_{rx}^{\prime}&\Sigma_{x}\end{pmatrix}.$ (1)

In our notation, we denote estimated quantities of their population analogs
with a hat accent ($^$) on them. The sample covariances of the asset returns
and signals and the sample cross-covariance between the returns and signals
are denoted as $S_{r}:=T^{-1}\sum^{T}_{t=1}r_{t+1}r_{t+1}^{\prime}$,
$S_{x}:=T^{-1}\sum^{T}_{t=1}x_{t}x_{t}^{\prime}$, and
$S_{rx}:=T^{-1}\sum^{T}_{t=1}r_{t+1}x_{t}^{\prime}$, respectively.

Due to the symmetric positive semidefinite property of the covariance
matrices, they admit a spectral decomposition. For example,
$\Sigma_{r}=P_{r}\mathsf{Diag}(d_{r,1},\ldots,d_{r,N})P^{\prime}_{r}$, where
$P_{r}:=[p_{r,1},\ldots,p_{r,N}]$ is an orthogonal matrix
($P_{r}^{\prime}P_{r}=P_{r}P_{r}^{\prime}=\mathbb{I}_{N}$) whose columns
contain the eigenvectors and $\mathsf{Diag}(d_{r,1},\ldots,d_{r,N})$ is a
diagonal matrix containing the eigenvalues. The eigenvalues are assumed to be
sorted in ascending order. The matrix factors $\Sigma_{r}^{1/2}$ can be
obtained by retaining the eigenvectors of $\Sigma_{r}$, but reassembling them
with the square roots of the eigenvalues given by
$(\sqrt{d_{r,1}},\ldots,\sqrt{d_{r,N}})^{\prime}$. An analogous notation
applies for the decomposition of the covariance of signals $\Sigma_{x}$ and
its matrix of factors $\Sigma_{x}^{1/2}$.

###  2.2 Dynamic Strategies

We begin our analysis by considering the portfolio weights obtained at time
$t$ to be linear in the vector of signals, that is, $w_{t}:=A^{\prime}x_{t}$,
where $A$ is a $NM\times N$ matrix of coefficients.444 Note that the
assumption that the optimal portfolio weights are linear functions of the
signals is not entirely restrictive. Indeed, each of the signals in $x_{t}$
can be composed of non-linear functions of a more primitive set of underlying
signals. These weights are conditional on the strength of the signals but the
matrix $A$ is a static object where each column maps the signal vector into a
portfolio weight invested in each basis asset. From this parameterization, the
conditional portfolio selection problem that an investor faces at time $t$ can
be formulated in terms of a quadratic utility objective as follows

$\displaystyle\max_{A}\mathbb{E}_{t}[x_{t}^{\prime}Ar_{t+1}]-\frac{\gamma}{2}\mathsf{Var}_{t}[x_{t}^{\prime}Ar_{t+1}],$
(2)

where $\gamma$ is an investor’s risk aversion parameter.555Other
reformulations of this problem include the maximization of mean return or
minimization of risk. Regardless of the choice, all three problems result in
the same mean-variance trade-off. This objective function underscores that the
investor chooses to simultaneously allocate between the assets and signals in
order to yield a trading strategy that optimizes the returns of the portfolio
at time $t+1$, which we denote as $r_{t+1}^{w}$.

Brandt and Santa-Clara, (2006) showed that this problem formulation is
equivalent to an unconditional portfolio selection problem since the matrix of
coefficients is constant through time. Indeed, the matrix $A$ that optimizes
the investor’s conditional problem at any given date is the same for all dates
and hence, it also optimizes the investor’s unconditional problem. Hence, we
can drop the subscript $t$ in (2) and consider the following revised problem
instead

$\displaystyle\max_{A}\mathbb{E}[x_{t}^{\prime}Ar_{t+1}]-\frac{\gamma}{2}\mathsf{Var}[x_{t}^{\prime}Ar_{t+1}].$
(3)

Based on the assumption of joint Gaussianity between the returns and signals,
we can express this objective in terms of the unconditional second-moments
with the following proposition.

######  Proposition 1.

The objective function to the investor’s problem (3) is given by

$\displaystyle\max_{A}\mathsf{Tr}(A\Sigma_{rx})-\frac{\gamma}{2}\left(\mathsf{Tr}(\Sigma_{x}A\Sigma_{r}A^{\prime})+\mathsf{Tr}(\Sigma_{rx}A\Sigma_{rx}A)\right).$
(4)

A closed-form solution to this optimization problem is available in Theorem 2
from Appendix B.

It is analytically convenient to assume that
$\mathsf{Tr}(\Sigma_{rx}A\Sigma_{rx}A)\approx 0$ in Equation (4) from
Proposition 1.666This approximation allows us to express the optimal portfolio
policy solely in terms of the unconditional moments without any recourse to
canonical correlation analysis, which is required to solve problem (4). It is
a reasonable approximation provided the squared expected returns of the
dollar-neutral managed portfolios–whose returns are of the form $x_{t}$ times
the return of each asset–as well as their weighted sum contributions are small
and can be approximated with zero. Notwithstanding this approximation, we have
also empirically tested both approaches in Section 4. The following auxiliary
problem is sufficient for us to study.

######  Proposition 2.

Consider the following objective function

$\displaystyle\max_{A}\mathsf{Tr}(A\Sigma_{rx})-\frac{\gamma}{2}\mathsf{Tr}(\Sigma_{x}A\Sigma_{r}A^{\prime}).$
(5)

The solution to this problem is given by

$\displaystyle
A=\frac{1}{\gamma}\Sigma^{-1}_{x}\Sigma_{rx}^{\prime}\Sigma^{-1}_{r},$ (6)

The weight $w_{t}$ allocated to each asset conditional on the signal at time
$t$ is then

$\displaystyle
w_{t}=\frac{1}{\gamma}\Sigma^{-1}_{r}\Sigma_{rx}\Sigma^{-1}_{x}x_{t}.$ (7)

In absence of any further constraints on $w_{t}$, this is the frictionless
portfolio policy that invests directly in each asset, and the size of the
portfolio is determined by the level of the investor’s risk aversion. The
dynamics of the portfolio is guided by the trajectory of the signal $x_{t}$ at
each period. We see that the portfolio takes into account the cross-sectional
information from the asset returns, the signals as well as their cross-
relationships. The cross-covariance matrix $\Sigma_{rx}$ can be interpreted as
a matrix consisting of managed portfolios that encapsulate the potential
earnings opportunities that are available to an investor.777In Kelly et al.,
(2022), the authors term the cross-covariance matrix as the “prediction
matrix”, which is also another valid interpretation since the correlation
between an asset return and some signal is one measure of signal-return
predictability. Given that a cross-covariance matrix is not symmetric in
general, the predictive strength of a signal $i$ on asset $j$ may be different
from that of signal $j$ on asset $i$. Each managed portfolio, formed in the
spirit of Hansen and Richard, (1987), Cochrane, (1996) and Brandt and Santa-
Clara, (2006) among others, invests in each asset a proportion determined by
the size of the conditioning variable. For this reason itself, we term this
matrix a “conditional portfolios” matrix.

######  Example 1 (Univariate Portfolio).

Let us consider one return-accompanying signal and postulate that all
covariances in Equation (7) are the identity matrix. In this case, the weight
is proportional to the conditioning variable. This portfolio policy is
interesting in its own right because it ignores the impact that the returns
and signals can have on the covariances. Hence, it is essentially a univariate
system in which the signals only forecast their associated asset. We shall
henceforth refer to this conditional portfolio as the univariate factor (UNI).

######  Example 2 (Two-Assets Portfolio).

Suppose that $N=2$, $M=1$, and $\gamma=1$. If the asset returns and signals
have unit variances, then the covariances are

$\Sigma_{r}=\left({\begin{array}[]{cc}1&\rho_{r}\\\ \rho_{r}&1\\\
\end{array}}\right),\quad\Sigma_{x}=\left({\begin{array}[]{cc}1&\rho_{x}\\\
\rho_{x}&1\\\
\end{array}}\right),\quad\text{and}\quad\Sigma_{rx}=\left({\begin{array}[]{cc}\xi_{11}&\xi_{12}\\\
\xi_{21}&\xi_{22}\\\ \end{array}}\right).$ (8)

For simplicity, also suppose that the signals are uncorrelated, that is,
$\rho_{x}=0$. Inserting these objects into the optimal matrix $A$, we can
write the weights invested in the two basis assets as

$w_{t}=\left(\frac{\xi_{11}-\rho_{r}\xi_{21}}{1-\rho^{2}_{r}}x_{1}+\frac{\xi_{12}-\rho_{r}\xi_{22}}{1-\rho_{r}^{2}}x_{2},\frac{\xi_{21}-\rho_{r}\xi_{11}}{1-\rho^{2}_{r}}x_{1}+\frac{\xi_{22}-\rho_{r}\xi_{12}}{1-\rho_{r}^{2}}x_{2}\right).$
(9)

Moreover, the expected portfolio return is

$\displaystyle\mathsf{Tr}(A\Sigma_{rx})=\mathsf{Tr}(\Sigma_{r}^{-1}\Sigma_{rx}\Sigma_{x}^{-1}\Sigma_{rx}^{\prime})$
$\displaystyle=\frac{\xi_{11}^{2}+\xi_{12}^{2}+\xi_{21}^{2}+\xi_{22}^{2}-2\rho_{r}(\xi_{11}\xi_{21}+\xi_{12}\xi_{22})}{1-\rho_{r}^{2}}.$
(10)

This expression is non-negative given $\rho_{r}\in[-1,1]$ since
$\xi_{11}^{2}+\xi_{21}^{2}\geq 2\rho_{r}\xi_{11}\xi_{21}$ and
$\xi_{12}^{2}+\xi_{22}^{2}\geq 2\rho_{r}\xi_{12}\xi_{22}$ by the inequality of
arithmetic and geometric means. Let us consider three cases. First, if
$\rho_{r}=0$, then the expected return is simply the squared expected returns
of all conditional portfolios that exploit own- and cross-predictabilities.
Second, if $\rho_{r}\in(-1,0)$, then there is an additional positive
contribution to the conditional portfolios that arises from diversifying into
imperfectly negatively correlated assets. The position is leveraged according
to a factor of $1/(1-\rho_{r}^{2})$, which increases as $|\rho_{r}|$
approaches one from below. Third, if $\rho_{r}\in(0,1)$, then the expected
portfolio return is now reduced since the gains to diversification are limited
by the extent to which the assets are positively correlated. Notwithstanding
the diminished returns, the optimal strategy is also leveraged with a factor
$1/(1-\rho_{r}^{2})$ in order to increase the portfolio returns to meet its
profit objective. A similar conclusion holds for the case $\rho_{r}=0$ and
$\rho_{x}\neq 0$.

######  Remark 1 (Conditional Portfolio Policies).

We can gain further intuition into the role of conditioning that our problem
posits. Consider the following conditional portfolio selection problem:

$\displaystyle\max_{w_{t}}\mathbb{E}_{t}[w_{t}^{\prime}r_{t+1}]-\frac{\gamma}{2}\mathsf{Var}_{t}[w_{t}^{\prime}r_{t+1}].$
(11)

Since the weights are known at time $t$, the problem can be solved to give

$\displaystyle
w_{t}=\frac{1}{\gamma}\mathsf{Var}_{t}[r_{t+1}]^{-1}\mathbb{E}_{t}[r_{t+1}].$
(12)

Observe that this optimal portfolio is determined by the mean and variance of
returns that incorporate all available information up to time $t$. Hence,
restricting the weights to be linear functions of a set of signals also
restricts the solution set, which necessarily leads to a suboptimal solution.
However, specifying the conditional means and variances of the returns is a
notoriously difficult task. Our weights restriction simplifies these dual
tasks by converting a dynamic problem into an equivalent static problem for
which a conditional portfolio policy is available solely in terms of the
unconditional moments of the assets and signals. That is, our conditional
policy (7) does not rely on how the conditional moments of returns depend on
the signals or how the signals vary over time.

######  Remark 2 (Regression-Based Policies).

Notice that the portfolio weights (7) can be seen as one that had been
obtained from a two-stage process of forecasting and asset allocation. To see
this, let us consider the following return-generating process:

$\displaystyle r_{t+1}=B^{\prime}x_{t}+\varepsilon_{t+1},$ (13)

where $B$ is a matrix of coefficients and $\varepsilon_{t}$ is an
idiosyncratic error that is independent and identically distributed (i.i.d.).
A simple forecasting approach that captures this relationship between both
variables would be to perform an ordinary least squares (OLS) regression. In
particular, one would first perform $N$ independent univariate OLS regressions
of the returns to assets on the signals in order to obtain a vector of cross-
sectional predictive signals, and then insert them into (12) to arrive at the
vector of allocations:

$\displaystyle
w_{t}=\frac{1}{\gamma}\Sigma_{\varepsilon}^{-1}\Sigma_{rx}\Sigma_{x}^{-1}x_{t},$
(14)

where
$\Sigma_{\varepsilon}=\mathsf{Cov}(\varepsilon_{t})=\Sigma_{r}-\Sigma_{rx}\Sigma_{x}^{-1}\Sigma_{rx}^{\prime}$
is the covariance matrix of the residuals. In fact, this portfolio is also
optimal under our assumption of joint Gaussianity between the asset returns
and signals. By invoking the standard properties of the multivariate Gaussian
distribution, the conditional distribution of $r_{t+1}$ given $x_{t}$ is also
multivariate Gaussian distributed with means and variances given by

$\displaystyle\mathbb{E}_{t}[r_{t+1}]$
$\displaystyle=\Sigma_{rx}\Sigma^{-1}_{x}x_{t},$ (15)
$\displaystyle\mathsf{Var}_{t}[r_{t+1}]$
$\displaystyle=\Sigma_{r}-\Sigma_{rx}\Sigma^{-1}_{x}\Sigma_{rx}.$ (16)

Inserting these expressions into (12) gives portfolio (14). In contrast to our
portfolio (7), this portfolio depends on the covariance structure of the
unexplained parts of the returns. Another key difference here is that our
portfolio policy respects symmetry; exchanging the roles of the assets and
signals yields identical returns to the portfolio. This symmetry property may
be warranted in situations where the signals have an influence on the asset
returns but at the same time, the reverse association is also economically
plausible. It also implies that our approach is more closely related to total
least squares (TLS) than it is to OLS since TLS is an errors-in-variables
regression model that optimizes for the correlation of both variables, which
is a symmetric metric.

###  2.3 Fully Invested Constraint

In quantitative equity investing, it is commonplace to impose economic
constraints on the portfolio weights. A prominent example would be the fully-
invested portfolio, which enforces a budget constraint so that the weights add
to one. This equality constraint can be included in our optimization problem
and given that it is linear in the matrix of coefficients $A$, the portfolio
weights can be fortunately solved in closed form.

######  Proposition 3.

We can formulate the optimal asset and signal combination problem with a
fully-invested constraint as follows

$\displaystyle\max_{A}\mathsf{Tr}(A\Sigma_{rx})-\frac{\gamma}{2}\mathsf{Tr}(\Sigma_{x}A\Sigma_{r}A^{\prime})$
(17) subject to $\displaystyle\mathbf{1}^{\prime}A^{\prime}x_{t}=1,$

where $\mathbf{1}$ denotes the conformable vector of ones of dimension $N$.
The problem has the following analytical solution

$\displaystyle w_{t}^{\mathrm{FI}}$
$\displaystyle=(1-\kappa)\frac{\Sigma_{r}^{-1}\mathbf{1}}{\mathbf{1}^{\prime}\Sigma_{r}^{-1}\mathbf{1}}+\kappa\frac{\Sigma_{r}^{-1}\Sigma_{rx}\Sigma_{x}^{-1}x_{t}}{\mathbf{1}^{\prime}\Sigma_{r}^{-1}\Sigma_{rx}\Sigma_{x}^{-1}x_{t}},$
(18)

where
$\kappa:=\gamma^{-1}\mathbf{1}^{\prime}\Sigma_{r}^{-1}\Sigma_{rx}\Sigma_{x}^{-1}x_{t}$
is a scalar value.

Proposition 3 shows that the portfolio weight is expressed as a convex linear
combination of the global minimum variance portfolio and the optimal
unconstrained portfolio policy that is renormalized to one. The latter
portfolio resembles the so-called ‘tangency portfolio’, which is the highest
Sharpe ratio of a portfolio of risky assets from the standard mean-variance
framework. The fully-invested portfolio $w_{t}^{\mathrm{FI}}$ has a similar
form to the one obtained from the standard Markowitz paradigm found in Huang
and Litzenberger, (1988, Sections 3.8 and 3.9).

######  Example 3.

Suppose there exists a vector of coefficients $\theta$ that maps the vector of
signals $x_{t,i}$ into a real-valued signal $y_{t,i}$ for asset $i$ according
to $y_{t,i}=\theta^{\prime}x_{t,i}$. Furthermore, suppose that all the
covariances in the fully-invested portfolio are identity matrices with respect
to this newly defined $N$-dimensional signal vector $y_{t}$. Then our fully-
invested portfolio (18) simplifies to

$\displaystyle
w_{t}^{\text{FI}}=\frac{1}{N}\mathbf{1}+\frac{1}{\gamma}\left(y_{t}-\frac{\mathbf{1^{\prime}}y_{t}}{N}\right).$
(19)

The resulting portfolio can now be seen as a combination of an equally-
weighted portfolio and a long-short portfolio that sums to zero. This
corresponds to the parametric portfolio policy of Brandt et al., (2009) but
specialized in an equally-weighted deterministic benchmark portfolio, and a
quadratic utility preference.

###  2.4 Relation to Existing Literature

At this juncture, it is important to highlight the differences between our
work from those of the existing literature. We first draw our attention to the
fact that the objective function (5) under our consideration is similar to
that of Brandt and Santa-Clara, (2006).888A subtle difference worth pointing
out is that the authors model the portfolio policy to be affine in the state
variables, and hence the allocations are also driven by the assets themselves.
Here, we work in a slightly restricted setting where the portfolio policy is
only driven by the conditional managed portfolios themselves. If we assume
joint Gaussianity of the returns and signals and the squared returns of
conditional managed portfolios are relatively small and can be approximately
zero.999For example, if a portfolio managed on the basis of a momentum signal
has an average return of 5%, then the square of its return is 0.25%.  Then we
can express the optimal matrix of coefficients $A$ from Equation (6) in
vectorized form as

$\displaystyle\mathsf{vec}(A)$
$\displaystyle=\frac{1}{\gamma}\mathsf{vec}(\Sigma^{-1}_{x}\Sigma_{rx}^{\prime}\Sigma^{-1}_{r})$
(20)
$\displaystyle=\frac{1}{\gamma}(\Sigma_{x}^{-1}\otimes\Sigma_{r}^{-1})\mathsf{vec}(\Sigma_{rx}^{\prime})$
(21)
$\displaystyle=\frac{1}{\gamma}(\Sigma_{x}\otimes\Sigma_{r})^{-1}\mathsf{vec}(\Sigma_{rx}^{\prime})$
(22)
$\displaystyle\approx\frac{1}{\gamma}\mathbb{E}[(x_{t}x_{t}^{\prime})\otimes(r_{t+1}r_{t+1}^{\prime})]^{-1}\mathbb{E}[x_{t}\otimes
r_{t+1}],$ (23)

where $\mathsf{vec}(\cdot)$ is an operator that stacks the columns of matrix
$A$ into a vector, and $\otimes$ is the Kronecker product of two matrices. The
fourth (approximate) equality invokes our sparsity assumptions so that the
second-moment matrix of the conditional product returns $x_{t}\otimes r_{t+1}$
is a Kronecker product of two covariances. This corresponds to the solution
that one obtains by rewriting the portfolio returns as
$x_{t}^{\prime}Ar_{t+1}=\mathsf{vec}(A)(x_{t}\otimes r_{t+1})$, and optimizing
over $\mathsf{vec}(A)$ with respect to $x_{t}\otimes r_{t+1}$.

The authors considered this new optimization scheme to be an augmented asset
space form of the mean-variance optimization. They had removed one step in the
process of forecasting and allocation and combined them into one step, thus
reducing noise in the allocation process. However, there are two issues with
this approach. First, augmenting the asset space results in a severe expansion
of the problem dimension. The unrestricted covariance matrix in (23) has
$N^{2}M(N^{2}M+1)/2$ parameters, while our Kronecker product factorization in
(22) has only $N(N+1)/2+NM(NM+1)/2$ parameters. Clearly, the former becomes
less computationally tractable when the dimensions of both the asset and
signal space are large. Second, there is less clarity on the structure of
their optimal solution due to the complicated interactions in the covariance
between the subsequent returns and signals. Notwithstanding these
shortcomings, our efforts in recasting the optimal strategy (23) into (6) (or
(46) in Theorem 2) help to amend their approach towards large-scale financial
applications. Although this requires us to model the joint distribution of
returns and signals, our solution lends itself nicely to the study of the
joint correlation structure of the optimal matrix of coefficients $A$; see
Section 3.3.

Second, our work is related to the classic portfolio selection problem of
Markowitz, (1952) in its static form. If we assume independent and identically
distributed (i.i.d.) returns with constant moments and time-invariant weights
$w_{t}\equiv w$, the investment problem (11) simplifies to an unconditional
one, which has the following well-known solution

$\displaystyle
w=\frac{1}{\gamma}\mathsf{Var}[r_{t+1}]^{-1}\mathbb{E}[r_{t+1}].$ (24)

This portfolio policy is typically implemented through plug-in estimation. In
this paper, we focus on the more realistic setting of non-i.i.d. returns,
which motivates our model parameterization and use of conditioning variables
that affect the joint distribution of returns.

Finally, our work is also related to that of Kelly et al., (2022) on cross-
predictability. In their work, the authors pursued a similar parameterization
for the portfolio weights but optimize an objective function that is subject
to a robust risk constraint that controls for the leverage of $A$. As a
result, their optimal portfolio policy depends only on the cross-covariance
matrix. When viewed in the context of our framework, the analysis is
simplified and avoids the challenges of inverting two additional large-
dimensional matrices. However, this departs from our paper on two fronts.
First, we assume that the investors have quadratic utility preferences in a
similar spirit to the standard mean-variance problem.101010Levy and Markowitz,
(1979) and Markowitz, (1991) argue that the quadratic utility preferences
serve as a reasonable approximation to other utility preferences in portfolio
selection problems. Second, our application of canonical correlation analysis
formally suggests that the cross-covariance between decorrelated asset returns
and signals is the more appropriate object for investment analysis as opposed
to the cross-covariance of the original asset returns and signals. This is
because the latter contains non-trivial variations that are embedded in both
financial variables that may obscure the inference.

##  3 Strategy Diversification

###  3.1 Canonical Correlation Analysis

The goal of canonical correlation analysis (CCA) is to perform dimension
reduction on two different data sets that comprise a large number of
interrelated variables, while simultaneously retaining as much of the
correlation present in the two data sets. This is made possible through a
transformation of the original set of variables into a new set of mutually
orthogonal paired variables, which are ranked so that the largest few retain
most of the correlation present in all of the original variables.

There are several standard approaches to solving the CCA problem as outlined
in Uurtio et al., (2017). Indeed, this may involve solving a standard
eigenvalue problem (Hotelling,, 1936), or a generalized eigenvalue problem
(Bach and Jordan,, 2002), or through singular value decomposition (SVD)
(Healy,, 1957). We adopt the latter approach in this brief exposition of the
classical subject by emphasizing a sequence of two change of basis operations
to arrive at a reduced problem whose financial interpretation we provide in
Section 3.3.

Given that scale invariance is a property of correlation, the first step is to
orthogonalize the random vectors $r$ and $x$ such that their covariances are
the identity matrices. This can be achieved through the following linear
transformations $\tilde{r}:=\Sigma^{-1/2}_{r}r$ and
$\tilde{x}:=\Sigma^{-1/2}_{x}x$. The transformed objects are now uncorrelated
and have a unit variance and their joint covariance matrix is given by

$\displaystyle\begin{pmatrix}\mathbb{I}_{N}&\Sigma_{r}^{-1/2}\Sigma_{rx}\Sigma_{x}^{-1/2}\\\
\Sigma_{x}^{-1/2}\Sigma_{rx}^{\prime}\Sigma_{r}^{-1/2}&\mathbb{I}_{NM}\end{pmatrix}.$
(25)

It is helpful to introduce the object

$\displaystyle\Sigma_{\tilde{r}\tilde{x}}:=\mathbb{E}[\tilde{r}\tilde{x}^{\prime}]=\Sigma_{r}^{-1/2}\Sigma_{rx}\Sigma_{x}^{-1/2},$
(26)

as the adjusted cross-correlation matrix between the transformed variables
$\tilde{r}$ and $\tilde{x}$.111111Note that the matrix (26) is different from
the unadjusted cross-covariance matrix $\Sigma_{rx}$. The adjusted cross-
covariance matrix is the central object in CCA as it internalizes the
variabilities from the asset returns and signals that might potentially
obscure the relationships between both of the variables. Moreover, since the
optimal matrix of coefficients for our problem (5) is indeed the adjusted
cross-covariance matrix under an appropriate change of basis (see Section
3.3), it makes sense for us to focus on this object for our analysis.

The second step is to perform an SVD operation on the adjusted cross-
covariance matrix $\Sigma_{\tilde{r}\tilde{x}}$. Let
$((s_{1},\ldots,s_{N});(u_{1},\ldots,u_{N});(v_{1},\ldots,v_{N}))$ denote a
system of singular values and singular vectors of the adjusted cross-
covariance matrix $\Sigma_{\tilde{r}\tilde{x}}$. We assume that the singular
values $s_{i}$ are sorted in increasing order. The canonical correlations
correspond to the singular values. The canonical variates of $r$ and $x$ are
defined as $u^{\prime}_{1}\tilde{r},\ldots,u^{\prime}_{N}\tilde{r}$ and
$v^{\prime}_{1}\tilde{x},\ldots,v^{\prime}_{N}\tilde{x}$, respectively. As the
singular values, $s_{i}$ are sorted in increasing order, the canonical variate
with the largest correlation is given by the pair
$(u^{\prime}_{N}\tilde{r},v^{\prime}_{N}\tilde{x})$ while the canonical
variate with the smallest correlation is
$(u^{\prime}_{1}\tilde{r},v^{\prime}_{1}\tilde{x})$.

Finally, since the solutions are expressed on a different basis than our
original problem, we have to translate them back to our original basis. By
inverting the change of variables we made, the $i$th canonical variates can be
written as
$(u^{\prime}_{i}\Sigma_{r}^{-1/2}r,v^{\prime}_{i}\Sigma_{x}^{-1/2}x)$. This
pair can be seen as a linear combination of the original assets and signals
with coefficients given by $q_{r,i}:=\Sigma^{-1/2}_{r}u_{i}$ and
$q_{x,i}:=\Sigma^{-1/2}_{x}v_{i}$, which are the so-called canonical
directions. The application of change of basis operations simplifies the
covariance structure considerably.

In order to implement the CCA in practice, a common approach is to replace the
population second-moments $\Sigma_{r},\Sigma_{x}$, and $\Sigma_{rx}$, with
their sample counterparts $S_{r},S_{x}$, and $S_{rx}$ computed from random
samples $r_{2},\ldots,r_{T+1}$ and $x_{1},\ldots,x_{T}$. Let
$((\hat{s}_{1},\ldots,\hat{s}_{N});(\hat{u}_{1},\ldots,\hat{u}_{N});(\hat{v}_{1},\ldots,\hat{v}_{N}))$
be a system of singular values (sorted in increasing order) and singular
vectors of the sample analog of (26) denoted by $S_{\tilde{r}\tilde{x}}$. Then
the sample canonical variates of $r$ and $x$ are given by
$\hat{u}^{\prime}_{1}\tilde{r},\ldots,\hat{u}^{\prime}_{N}\tilde{r}$ and
$\hat{v}^{\prime}_{1}\tilde{x},\ldots,\hat{v}^{\prime}_{N}\tilde{x}$,
respectively. However, one should be skeptical of sample canonical
correlations as they are not reflective of the population canonical
correlations. Indeed, the true canonical correlation of the $i$th canonical
variate pair $(\hat{u}^{\prime}_{i}\tilde{r},\hat{v}^{\prime}_{i}\tilde{x})$
is $\hat{v}_{i}^{\prime}\Sigma_{\tilde{r}\tilde{x}}^{\prime}\hat{u}_{i}$, as
opposed to
$\hat{s}_{i}=\hat{v}_{i}^{\prime}S_{\tilde{r}\tilde{x}}^{\prime}\hat{u}_{i}$.
This is because the sample canonical correlations are known to be inconsistent
estimates of their population counterparts when the dimensionality of the
problem for both variables is large relative to the number of observations.

To bring this point home, let us consider a specific case when the two
variable sets $r$ and $x$ are independent and each set consists of i.i.d.
Gaussian random variables. Figure 1 shows the distribution of sample canonical
correlations for various values of the limiting ratios
$N/T\in\\{0.003,0.03,0.3\\}$ and $M/T\in\\{0.006,0.06,0.6\\}$. We obtained
this figure by invoking the asymptotic formula derived by Wachter, (1980,
Theorem 3.1).121212Wachter’s result also holds under a less restrictive
setting of i.i.d. data with finite second-moments; see, for example, Yang and
Pan, (2012, Theorem 1). We observe that the sample canonical correlations are
biased upwards away from zero and the severity of the bias worsens as $N/T$ or
$M/T$ or both increases. This distributional shape is expected to hold when
$N$, $M$ and $T$ are approximately large and it only depends on the ratio
$N/T$ and $M/T$; this is true regardless of any particular realization of the
adjusted sample cross-covariance matrix. We attempt to address these problems
with regularization in Section 3.6.

###  3.2 Relation to PCA, PLS, and RRR

Having laid the groundwork for CCA, it is interesting to see how CCA is
related to a host of linear dimension-reduction techniques commonly used to
uncover underlying economic structures. We shall restrict our attention to the
principal component analysis (PCA), partial least squares (PLS), and reduced
rank regression (RRR) due to their tight connection with one another and
extensive applications in the finance and econometrics literature.131313For
example, PCA has been used to extract common factors from a covariance matrix
of individual stocks among others by Roll and Ross, (1980), Connor and
Korajczyk, (1986, 1988), and Lehmann and Modest, (1988). In the context of
describing risk compensation through estimation of the stochastic discount
factor, Kozak et al., (2018, 2020) invokes no near-arbitrage arguments for the
use of PCA to identify factor risk exposures that price the cross-section of
expected returns of characteristic-managed portfolios. For PLS, one of the
earliest applications to finance can be ascribed to Kelly and Pruitt, (2013),
where the authors used a three-pass regression filter (which is a special case
of PLS) to forecast market return and cash flow growth. The RRR, alongside
CCA, features prominently in the estimation of the celebrated vector-error-
correction model (VECM) of Engle and Granger, (1987) as a means to detect
cointegration, which are economic relationships that cannot be obtained
through standard regression approaches; see Johansen, (1988, 1991, 1995).

Starting with PCA developed by Hotelling, (1933), the method seeks to break
down the variation of the data into mutually orthogonal components whose
variances can be ranked from the smallest to the largest. In the context of a
paired dataset consisting of asset returns with their associated signals, PCA
focuses on searching for directional vectors that have the maximum variation
in each variable separately. In contrast to CCA, however, it does not take
into account any relationship between the two financial variables. In this
paper, we would like to focus on how financial variables are related, not how
much they vary. We summarize the key properties distinguishing PCA and CCA in
Table 1.

On the other hand, PLS was introduced by Wold, (1975) as an econometric
technique that aims to explain the relationship of a paired dataset by
studying the (cross-)covariance. This can be understood by relaxing the
variance constraints in the optimization problem of CCA, which results in a
maximum (cross-)covariance problem. The solution(s) can be obtained via
singular value decomposition of the cross-covariance matrix, $\Sigma_{rx}$,
and the obtained managed portfolios closely mimic that of Kelly et al.,
(2022). This differs from CCA where the cross-covariance matrix is normalized
with respect to the covariances from both the $r_{t+1}$ and the $x_{t}$
variables.

Finally, the RRR proposed by Anderson and Rubin, (1949, 1950) and Anderson,
(1951) can be understood through the regression framework (13) with the
following error minimization problem:

$\displaystyle\min_{B}\mathbb{E}\big{[}\big{\|}\Gamma^{1/2}(r_{t+1}-B^{\prime}x_{t})\big{\|}^{2}_{2}\big{]},$
(27)

where $\Gamma$ is a $N\times N$ weighting matrix, and the matrix $B$ is
subject to a rank constraint; see also, Velu and Reinsel, (2013). Expressing
the objective function in terms of the unconditional moments, we have

$\displaystyle\mathbb{E}\big{[}\big{\|}\Gamma^{1/2}(r_{t+1}-B^{\prime}x_{t})\big{\|}^{2}_{2}\big{]}$
$\displaystyle=\mathbb{E}\big{[}\big{\|}\Gamma^{1/2}r_{t+1}\big{\|}^{2}_{2}\big{]}-2\mathbb{E}\big{[}x_{t}^{\prime}B\Gamma
r_{t+1}\big{]}+\mathbb{E}\big{[}x_{t}^{\prime}B\Gamma B^{\prime}x_{t}\big{]}$
(28) $\displaystyle=N-2\mathsf{Tr}(B\Gamma\Sigma_{rx})+\mathsf{Tr}(B\Gamma
B^{\prime}\Sigma_{x})$ (29)
$\displaystyle=\big{\|}\Gamma^{1/2}B\Sigma^{1/2}_{x}-\Gamma^{1/2}\Sigma_{rx}\Sigma_{x}^{-1/2}\big{\|}^{2}_{\mathrm{F}}+\text{constant}.$
(30)

The solution to this rank-restricted minimization problem follows from the
theorem of Eckart and Young, (1936) and it relies on the singular value
decomposition of $\Gamma^{1/2}\Sigma_{rx}\Sigma_{x}^{-1/2}$. The typical
objective function of RRR occurs when $\Gamma=\mathbb{I}_{N}$. If
additionally, $\Sigma_{x}=\mathbb{I}_{M}$, then RRR coincides with PLS.
Finally, observe that if $\Gamma=\mathbb{I}_{N}$ and $r_{t+1}=x_{t}$, then the
objective function coincides with that of PCA.

Interestingly, if $\Gamma=\Sigma_{r}^{-1}$, then the solution of RRR
generalizes to CCA. This speaks to the difference between optimizing a
regression-based objective and a correlation-based objective. Indeed,
regression attempts to explain as much of the variation in the returns
$r_{t+1}$ as possible. However, if $r_{t+1}$ is decorrelated and isotropic
along all variance directions, then minimizing a regression loss function
becomes equivalent to maximizing a correlation objective. Unlike conventional
regression-based approaches, there is no notion of independent and dependent
variables in CCA because the correlation metric is symmetric; that is, if we
exchange the roles of asset returns and signals, CCA yields identical outcomes
for the canonical directions and canonical correlations.

In summary, RRR, PLS, and CCA can all be viewed under the same framework,
which is solving a cross-covariance maximization problem. Each of these linear
dimensional reduction techniques is subject to different normalization schemes
with CCA serving as the most general one that nests these methods.

###  3.3 Reformulation as Canonical Portfolios

The portfolio returns $r_{t+1}^{w}$ at time $t+1$ obtained with (6) is not
particularly intuitive as the expression involves large-dimensional matrix
inversion and multiplication operations. However, we can make progress by
performing a CCA in order to decompose the portfolio selection problem into
one that we can provide financial interpretation.

We start by expressing the portfolio returns in terms of their transformed
objects as

$\displaystyle
x_{t}^{\prime}Ar_{t+1}=\frac{1}{\gamma}x_{t}^{\prime}\Sigma^{-1}_{x}\Sigma_{rx}^{\prime}\Sigma^{-1}_{r}r_{t+1}=\frac{1}{\gamma}\tilde{x}_{t}^{\prime}\Sigma_{\tilde{r}\tilde{x}}\tilde{r}_{t+1},$
(31)

Since the decorrelated objects $\tilde{r}_{t+1}$ and $\tilde{x}_{t}$ span the
same universe as the original assets and signals, we shall term them
“synthetic assets” and “synthetic signals”. These newly defined objects have
identity covariances, and so this is the cross-sectional version of risk
parity.

From expression (31), we can see that the portfolio returns depend intricately
on the synthetic asset returns and synthetic signals, which are coupled
through their adjusted cross-covariances. The portfolio returns do not depend
on whether we exchange the role of the asset returns of signals; what matters
is how they are related to each other.

Our next step is to choose basis vectors such that
$\Sigma_{\tilde{r}\tilde{x}}$ is diagonal. In particular, we perform an SVD
operation such that all cross-relationships are eliminated in the new basis in
order to arrive at the following simplified expression for the portfolio
returns:

$\displaystyle
x_{t}^{\prime}Ar_{t+1}=\frac{1}{\gamma}\sum^{N}_{i=1}s_{i}(v_{i}^{\prime}\tilde{x}_{t})(u_{i}^{\prime}\tilde{r}_{t+1}).$
(32)

Thus, we see that a generic strategy return can be viewed either as a
combination of the original assets and signals that have been optimally
blended with the matrix of coefficients $A$, or as a combination of $N$
uncorrelated managed long-short portfolios weighted by their canonical
correlations. The $N$ long-short managed portfolios span the same space of
investment opportunities as the original assets and signals.

Economically, we interpret the change of basis as an operation that
reorganizes the set of $N$ synthetic assets and $N$ synthetic signals into a
set of $N$ uncorrelated managed portfolios. Each of these managed portfolios
is expressed as a certain weighted combination of the assets and signals
themselves. These particular combinations are determined by the singular
vectors or canonical directions; in either case, we shall refer to them as
canonical portfolios henceforth to reinforce the idea that these vectors are
essentially managed portfolios that are loading onto their respective asset
and signal variables. The higher the $i$th canonical correlation, the higher
the return of the $i$th canonical portfolio. Said differently, the canonical
portfolios are ordered in such a way that the highest canonical correlation
corresponds to the most linearly predictable portfolio, and the second highest
canonical correlation, the second most linearly predictable portfolio, and so
on. This notion of predictability distinguishes canonical portfolios from
other concepts of orthogonal managed portfolios such as those constructed by
PCA.

Circling back to Equation (32), the source of the portfolio returns can be
seen to be distributed across the canonical portfolios of
$\Sigma_{\tilde{r}\tilde{x}}$. This reinforces the economic notion of
diversification: Having exposure to different uncorrelated canonical
portfolios is akin to putting each of your eggs into different baskets. We see
that the capital assigned to each canonical portfolio is proportional to its
original returns $r_{t+1}$ and signal $x_{t}$, and proportional to the
correlation $s_{i}$. Put differently, we want to assign capital to strategies
that have high correlations but are also orthogonal to each other.

###  3.4 Canonical Portfolio Analysis

With our CCA decomposition, we can also gain some insight into how the
canonical portfolios impact the returns of a portfolio. We start by working
out the expected returns and variance of these canonical portfolios with the
next proposition.

######  Proposition 4.

Let
$\pi_{ti}:=s_{i}(v_{i}^{\prime}\tilde{x}_{t})(u_{i}^{\prime}\tilde{r}_{t+1})$
be the return that the $i$th canonical portfolio generates. The expected value
and variance return of each canonical portfolio are

$\displaystyle\mathbb{E}[\pi_{ti}]$
$\displaystyle=s_{i}^{2},\quad\text{and}\quad\mathsf{Var}[\pi_{ti}]=s_{i}^{2}(1+s_{i}^{2}).$
(33)

Furthermore, the expected value and variance of the optimal portfolio is

$\displaystyle\mathbb{E}[x_{t}^{\prime}Ar_{t+1}]$
$\displaystyle=\frac{1}{\gamma}\sum_{i=1}^{N}s_{i}^{2},\quad\text{and}\quad\mathsf{Var}[x_{t}^{\prime}Ar_{t+1}]=\frac{1}{\gamma}\sum_{i=1}^{N}s_{i}^{2}.$
(34)

We see that the expected portfolio returns are positive, and hence, the
optimal portfolio is a profit-generating strategy that has economic value.
Moreover, the expected portfolio returns can be expressed as the sum of the
contributions from the squared canonical correlations; the larger the expected
return of the $i$th canonical portfolio, the larger its contribution towards
the overall returns.141414The formulas (33) and (34) have some close
similarities to a result in Kelly et al., (2022, Proposition 4) for the
expected returns. One notable difference, however, is that our singular values
are raised to the power of two. This is implicit in the fact that the expected
return of each canonical portfolio is scaled by a leverage factor of $s_{i}$.
On the other hand, this multiplicative factor is absent in their expression
since leverage is explicitly controlled for in their problem formulation. This
suggests that the squared canonical correlations can serve as a natural
measure to rank the performance of the canonical portfolios.

Interestingly, the canonical correlations are also intimately connected to the
Sharpe ratio of the returns associated with our portfolio policy.

######  Corollary 1.

The Sharpe ratio of the $i$th canonical portfolio and the optimal portfolio is
given by

$\displaystyle\mathsf{SR}_{i}:=\frac{\mathbb{E}[\pi_{ti}]}{\sqrt{\mathsf{Var}[\pi_{ti}]}}=\frac{s_{i}}{\sqrt{1+s_{i}^{2}}},\quad\text{and}\quad\mathsf{SR}:=\frac{\mathbb{E}[x_{t}^{\prime}Ar_{t+1}]}{\sqrt{\mathsf{Var}[x_{t}^{\prime}Ar_{t+1}]}}=\sqrt{\sum^{N}_{i=1}s_{i}^{2}}.$
(35)

From Corollary 1, we see the squared canonical correlations also influence the
portfolio’s Sharpe ratio. We end this section with two simple examples to give
further intuition into these results.

######  Example 4 (Equal canonical correlations).

If we suppose $s_{i}=s$ for all $i$, then Equation (35) simplifies to
$\mathsf{SR}=s\sqrt{N}$. In this special case, we arrive at the Fundamental
Law of Active Management of Grinold, (1989), where the so-called “information
coefficient” is $s$ and the effective number of statistically independent
investments (also referred to as ‘breadth’) is $N$. We see that the Sharpe
ratio improves with the average canonical correlation and the number of
canonical portfolios, with the latter exhibiting diminishing gains to returns
due to the square-root scaling in its exponent. In the general case where the
canonical correlations are not identical across all canonical portfolios, the
canonical correlations are thought to be the information coefficient over
clusters of signals.

######  Example 5 (Bias of In-Sample Returns).

Let $\hat{A}:=\gamma^{-1}S_{x}^{-1}S_{rx}^{\prime}S_{r}^{-1}$ is the optimal
matrix of coefficients that replaces the population moments in $A$ with sample
moment-based estimates. For simplicity, suppose that $S_{r}=\Sigma_{r}$ and
$S_{x}=\Sigma_{x}$. Since the Frobenius norm is a convex function, by Jensen’s
inequality, the expected portfolio return satisfies

$\displaystyle\mathbb{E}[\mathsf{Tr}(\hat{A}S_{rx})]=\frac{1}{\gamma}\mathbb{E}[\operatorname{\mathsf{Tr}}{(S_{\tilde{r}\tilde{x}}^{\prime}S_{\tilde{r}\tilde{x}}})]\geq\frac{1}{\gamma}\mathsf{Tr}(\mathbb{E}[S_{\tilde{r}\tilde{x}}]^{\prime}\mathbb{E}[S_{\tilde{r}\tilde{x}}])=\frac{1}{\gamma}\mathbb{E}[\operatorname{\mathsf{Tr}}{(S_{\tilde{r}\tilde{x}}^{\prime}\Sigma_{\tilde{r}\tilde{x}}})],$
(36)

where we used the fact that $\mathbb{E}[S_{rx}]=\Sigma_{rx}$ in the last
equality due to the unbiasedness of the sample cross-covariance. Hence,

$\displaystyle\sum^{N}_{i=1}\mathbb{E}[\hat{s}_{i}^{2}]\geq\sum^{N}_{i=1}s_{i}^{2}=\sum^{N}_{i=1}\mathbb{E}[\hat{s}_{i}s_{i}^{\circ}],$
(37)

where $s_{i}^{\circ}:=\hat{q}_{r,i}^{\prime}\Sigma_{rx}\hat{q}_{x,i}$ is the
out-of-sample ‘canonical correlation’ associated with the $i$th estimated
canonical portfolio $\hat{q}_{r,i}$ and $\hat{q}_{x,i}$.151515In fact, a more
precise relationship between the in-sample and out-of-sample portfolio returns
has been established by Benaych-Georges et al., (2019, Proposition 2.9).
Hence, on average, the in-sample returns are always optimistic but the out-of-
sample evaluation disappoints. This is because both the in-sample singular
values and singular vectors are estimated with a bias. Hence, in order to
ensure that the in-sample and out-of-sample returns are more in sync, we have
to shrink the in-sample singular values and align singular vectors closer to
the truth.

###  3.5 Static and Dynamic Returns Decomposition

In our standing assumptions, we supposed that both the asset returns and
signals are centered and have zero mean. However, when evaluating the out-of-
sample performance of the portfolio, the original non-demeaned returns and
signals tend to be used instead. Moreover, the means of both financial
variables may be non-negligible and contain useful information for describing
the cross-predictability of future returns. In general, we can write the
second-moment matrix of conditional portfolios using the following
decomposition:

$\Sigma_{rx}=\mathbb{E}[r_{t+1}x_{t}^{\prime}]=\mathbb{E}[r_{t+1}]\mathbb{E}[x_{t}]^{\prime}+\mathsf{Cov}(r_{t+1},x_{t}).$
(38)

The first component is a rank-one outer product matrix of the unconditional
means in the asset returns and signals while the second component is the
cross-covariance matrix of the demeaned returns and signals. In particular,
the first component places emphasis on the cross-sectional differences driven
by the unconditional levels since all time-series variabilities have been
averaged out, while the second component captures time-series variations in
the returns and signals that are expressed as the deviations from their
unconditional means. Kelly et al., (2022) describe the first component as
static bets and the second component as dynamic bets.

We can derive the expected returns of the optimal portfolio in this setting in
the next proposition.

######  Proposition 5.

If the expected value of the returns and signals are
$\mathbb{E}[r_{t+1}]=\mu_{r}$ and $\mathbb{E}[x_{t}]=\mu_{x}$, then the
expected portfolio return utilizing the second-moment matrix (38) is given by

$\displaystyle\mathbb{E}[x_{t}^{\prime}Ar_{t+1}]=\frac{1}{\gamma}(\mu_{r}^{\prime}\Sigma_{r}^{-1}\mu_{r})(\mu_{x}^{\prime}\Sigma_{x}^{-1}\mu_{x})+\frac{1}{\gamma}\sum^{N}_{i=1}s_{i}^{2}.$
(39)

Proposition 5 shows that there is an additional non-negative contribution to
the portfolio returns, which is due to the (squared) maximum Sharpe ratio
achievable from the assets. This observation aligns nicely with our
interpretation that the sum of the squared canonical correlations is related
to the squared Sharpe ratio of optimal portfolio returns from Corollary 1.
That is, the expected portfolio returns can be attributed to the Sharpe ratios
from both static and dynamic bets.

We shall let our portfolio exploit both investment opportunities that arise
from the second-moment matrix of managed portfolio returns. This is possible
within our framework since the assumption of zero-mean returns and signals
strictly applies to the variance of the portfolio returns, and so only the
covariances of the returns and signals have to be centered. However, we shall
apply CCA to the cross-covariance matrix to fulfill its modeling assumptions
since otherwise, it can result in a generic top canonical portfolio that is
mainly driven by the unconditional levels.161616CCA is sometimes applied to
non-demeaned variables. In such a case, this is more closely related to using
a cosine similarity objective instead of a correlation objective.

###  3.6 Estimation

Our optimal portfolio policy requires the knowledge of the population
covariances of the asset returns, the covariances of the signals, and cross-
covariance between both variables. These objects are generally unknown to us,
and so in order to render our framework to practice, we have to estimate them
with real data. Given the large-dimensional nature of our problem, it is
necessary to regularize the covariances of our portfolio to reduce the
estimation errors. Unfortunately, simply maintaining the top canonical
portfolio is not sufficient for empirical analysis. Indeed, if either the
assets or the signals has rank $T$ (which tends to be the case when $N\gg T$),
then the canonical portfolios can take on any arbitrary values. Moreover,
since the smallest amount of variability in each dataset gets rescaled to one,
CCA can produce spurious outcomes.

The challenge of estimating the covariance matrix of financial covariances is
well known amongst practitioners (Jobson and Korkie,, 1980). A standard
approach is to use the sample covariance matrix. However, when the
dimensionality of the problem is large relative to the number of observations,
estimation error of the sample covariance matrix can create issues for
portfolio optimizers; they tend to place extreme bets on low-risk sample
eigenvectors. In fact, this observation led Michaud, (1989) to refer to mean-
variance optimizers as “error maximization” schemes. There have been several
approaches from practice to address this problem using methods from
bootstrapping (Michaud and Michaud,, 2008) to Bayesian estimators (Black and
Litterman,, 1992; Lai et al.,, 2011).

We obtain regularized covariance matrices by applying the linear shrinkage
technology from Ledoit and Wolf, 2004b . The covariances of returns will be
estimated as
$\hat{\Sigma}_{r}=(1-\delta_{r})S_{r}+\delta_{r}N^{-1}\mathsf{Tr}(S_{r})\mathbb{I}_{N}$,
where the shrinkage intensity $\delta_{r}$ is determined based on an
asymptotic formula. We also apply the same linear shrinkage technology for the
covariances of signals $\hat{\Sigma}_{x}$ with a shrinkage intensity parameter
$\delta_{x}$. The choice of linear shrinkage of the covariances has a nice
interpretation in our context in that varying the shrinkage intensities allows
us to interpolate between the maximum covariance (PLS) and maximum correlation
(CCA) problems. Shrinking towards maximum covariance helps to break the
singularities by considering managed canonical portfolios that have better
out-of-sample properties. In the extreme regularization setting with both
$\hat{\Sigma}_{r}$ and $\hat{\Sigma}_{x}$ being identity matrices, our
estimated canonical portfolios mimics the principal portfolios approach of
Kelly et al., (2022).

With both of these estimated covariances at hand, we proceed to build a
regularized adjusted cross-covariance matrix by pre-conditioning the sample
covariances $S_{rx}$ on both sides with the estimated matrix factors
$\hat{\Sigma}_{r}^{1/2}$ and $\hat{\Sigma}_{x}^{1/2}$. One can then choose to
maintain the top few canonical correlations of the regularized adjusted sample
cross-covariance matrix defined as

$\displaystyle\hat{\Sigma}_{r}^{-1/2}S_{rx}\hat{\Sigma}_{x}^{-1/2},$ (40)

and set the bottom ones to zero.171717The regularized sample adjusted cross-
covariance matrix (40) has a close similarity to that of Vinod, (1976), where
ridge regression was proposed as a means of regularizing the sample adjusted
cross-covariance matrix. We resort to the class of linear shrinkage estimators
due to its ability to also reduce the influence of large variance directions.
This thresholding operation has the effect of regularizing the problem as it
reduces the effective number of parameters that we have to estimate by
maintaining the top few canonical portfolios that are the most predictable.

##  4 Empirical Analysis

###  4.1 Data and Portfolio Construction Rules

For our empirical analysis, we download six datasets from Kenneth French’s
data library, which are characteristic-sorted long-short stock
portfolios.181818The description of all portfolio construction can be found on
Kenneth French’s website:
https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html. At
the time of writing, these datasets were based on the 10-2022 CRSP database.
They are daily returns on portfolios of stocks sorted on the basis of size and
book-to-market (FF), size and operating profitability (ME/OP), and size and
investment (ME/INV), each of which is of universe size 25 and 100. In
comparison to individual stocks, each return of a portfolio is an average
return of a group of stocks sharing similar characteristics and so they are
less subject to large variabilities due to idiosyncratic risks. Hence, they
serve as useful test assets that may allow us to more easily harvest the
predictability in stock returns.

Although some of these portfolio returns have been available since 1926, we
conduct most of our analysis on the period from July 1963 to October 2022, for
which most of the returns are available. For simplicity, we suppose that 21
consecutive days constitute one trading ‘month’ and 252 consecutive days as
one trading ‘year’. We adopt a sequential updating scheme and rebalance the
portfolio every ’month’ on a rolling walk-forward basis. To obtain a well-
defined investment universe for which we can estimate the portfolios, we use
the following rule. For each rebalancing date, we select test assets that have
a complete return history over the most recent $T=120$ months as well as a
complete return ‘future’ over the subsequent trading month. This provides us
with 578 monthly out-of-sample returns, which covers the out-of-sample
investment period from 09-09-1974 through 10-21-2022.

###  4.2 Signal Construction

As a demonstration of our method to a dynamic portfolio selection problem
between equity sorted portfolios, we start with a single conditioning variable
as our signal. We consider the momentum signal as there is substantial
empirical evidence that documents this anomaly in the returns of individual
stocks (Jegadeesh,, 1990; Jegadeesh and Titman,, 1993), industries (Moskowitz
and Grinblatt,, 1999), and of size and value portfolios (Lewellen et al.,,
2010). This application is similar to Kelly et al., (2022) for
reproducibility.

To construct a momentum signal, we compute for each asset the lagged one-month
return defined as the simple average return over the previous 21 trading days.
Similar to Asness et al., (2019), Freyberger et al., (2020), Kozak et al.,
(2020) and Kelly et al., (2022), we rank the momentum signals across the
assets from 1 to $N$, dividing the ranks by the number of assets, and then
centering the normalized ranks to map the signals into the range $[-0.5,0.5]$.
This provides us with a set of dollar-neutral signals that are insensitive to
outliers for which we further divide by the sum of their absolute values. This
keeps the gross exposure (that is, the sum of the absolute amount of long and
short positions) fixed since otherwise doubling the number of assets at any
time $t$ will result in signals that are two times more aggressive even though
the investment opportunities remain the same. We assume $1 of capital is
invested to $1 of long and short positions.

We collect the individual momentum values of the $N$ assets to yield a
predictive signal $x_{t}$ for the subsequent monthly returns. Additionally, we
impose a one-day buffer between the constructed signals and the subsequent
returns to limit the effects of illiquidity from driving our results and to
bring our backtest simulations closer to being tradeable practice.191919The
challenge working with the equity portfolios from Kenneth French’s website is
that they contain small illiquid stocks. Moreover, given that we are using
daily returns, which are close-to-close returns, asynchronous trading at the
end of the day may arise; see, for example, Lo and MacKinlay, (1990).
Consequently, this may induce some lead-lag relationships among the stocks,
which can lead to autocorrelation in the portfolio returns and spurious
correlation estimates due to the Epps, (1979) effect. The latter problem can
be particularly acute given our use of multivariate techniques in the
estimation of portfolios. Therefore, we employ a one-day buffer along with
monthly returns to mitigate the effects that asynchronous or infrequent
trading can have on the portfolio returns.

###  4.3 Candidate Portfolios

Given the time series panels of asset returns and signals, we consider the
following portfolios in our study:

  * •

CP2: Our proposed portfolio contruction methodology based on Equation (7).

  * •

MVO: The mean-variance optimization portfolio of Markowitz, (1952).

  * •

PP2: The Principal Portfolios of Kelly et al., (2022).

  * •

UNI: The univariate factor where the weights are the signals.

The competing portfolios have been chosen because they can be subsumed in our
proposed method and hence, serve as natural benchmarks for us to determine
where the contribution to any improved performance comes from. The suffix
number attached to the portfolio labels CP and PP indicates the number of
managed portfolios that we retain; for example, CP2 means that we choose to
keep the leading two most predictable managed portfolios.202020We cross-
sectionally demean the returns in the construction of the cross-covariance
matrix for PP as suggested in Kelly et al., (2022) to focus on the cross-
sectional differences. However, we choose to ignore otherwise for the other
portfolios.

The covariance of returns in CP2 and MVO is estimated with the linear
shrinkage of Ledoit and Wolf, 2004b . Additionally, we also apply linear
shrinkage to the covariance of signals for CP2 but choose a high shrinkage
intensity with value $\delta_{x}=0.9$ without relying on the asymptotic
formula from Ledoit and Wolf, 2004b  for this purpose since it was developed
for financial returns that are assumed to be independent and identically
distributed.

Finally, we renormalize the estimated portfolios so that the sum of the
absolute value of their weights equals one. This allows all the portfolios to
be comparable in scale. It also implies that the gross exposure for all
portfolios is one dollar by construction, that is, we apply one dollar of
capital for one dollar of long and short positions. This is sensible for long-
short equity hedge fund managers who face institutional constraints such as
limits on gross exposure by their prime brokerage.212121 The insights are
similar if we renormalize the portfolios to achieve a target level of
volatility or return.

###  4.4 Evaluation Methodology

To evaluate the performance of the different portfolios, we report three main
out-of-sample performance measures: the average cumulative out-of-sample
returns, the standard deviation of the out-of-sample returns, and the Sharpe
ratio defined as the ratio of the average returns to the standard deviation of
returns. For ease of interpretability, all performance measures are annualized
with 12 trading ‘months’. The Sharpe ratio is computed with respect to the
actual returns (as opposed to returns in excess of the risk-free rate) since
we believe it is more relevant in our context where the portfolios are formed
solely on the basis of risky assets.

We also report three additional performance measures based on the out-of-
sample returns in excess of a 6-factor benchmark; that is, the 5 factors from
Fama and French, (2015) augmented with out-of-sample returns from UNI. We
compute the Jensen’s alpha, beta to the out-of-sample UNI returns, and
information ratio from a 6-factor regression model. This is done for all
portfolios except for UNI. The alpha and information ratio are annualized with
12 trading ‘months’. We also provide the t-statistics of the Sharpe ratio and
information ratio, which are computed with approximate standard errors from
Lo, (2002).

Additionally, we report the following portfolio weight statistics averaged
over the 578 trading months: turnover, proportional leverage (computed as the
fraction of negative weights), the sum of negative weights, and the minimum
and maximum weight. Note that these statistics are not our primary focus since
our proposed method is not optimized to account for these measures.
Nevertheless, they are provided to give a better overview of the different
methods.

###  4.5 Application of Canonical Portfolio Analysis

We can use the results of Proposition 4 to identify the sources of portfolio
returns by estimating the canonical portfolios that contribute most to its
profitability. To this end, we will consider the FF25, ME/OP25, and ME/INV25
test assets for this purpose. The FF25 is widely studied amongst academics and
it allows us to check if our method produces the expected results. Moreover,
we know from Lewellen et al., (2010) that the size and value portfolio returns
have a strong factor structure explained mostly by the three-factor model Fama
and French, (1993). Kozak et al., (2018) finds that retaining the first three
principal components extracted from FF25, closely reproduces the Fama and
French three-factor model. Thus, this observation could potentially be
exploited in our method.

The left panels of Figure 3(c) show the squared sample canonical correlations
of the demeaned conditional portfolios matrix for the three test assets. It is
also overlaid with the squared canonical correlations generated from randomly
permuting the signals observations for each asset and repeating a similar
exercise.222222Randomly shuffling the time series helps generate a null
distribution. It is a useful heuristic to determine the importance of each
canonical correlation relative to a random benchmark compared to a formal
statistical test of significance from Yang and Pan, (2015). The squared
canonical correlations are averaged over the rebalancing dates and ordered
from the smallest to the largest. From this in-sample analysis, we can see
that the leading two squared sample canonical correlations have values larger
than their pseudo-random generated counterparts and that the top one ‘sticks
out’ and extends beyond the value of one.232323The values of (squared)
regularized canonical correlations can exceed one unlike its sample-based
analog, which is constrained to the interval (0,1). Moreover, the spacing
between the sample canonical correlations is wider at the top end of the
spectrum and more uniform at the bottom end of the spectrum.

We contrast these findings with the right panels of Figure 3(c), which shows
the out-of-sample returns of each canonical portfolio computed as the product
of the in-sample and out-of-sample canonical correlation, accompanied by their
standard error. Not surprisingly, we find a performance deterioration in the
out-of-sample performance due to the bias in the in-sample predictions.
Nevertheless, there is some coherence between the in-sample predictions and
the out-of-sample returns in that the top canonical portfolio possesses the
most realized returns followed by the second one, while the bottom ones are
close to zero.

Given the prominence of the leading canonical correlation, we plot its
corresponding weights. The left panels of Figure 4(c) show the top canonical
portfolios (that is, the canonical directions) averaged over the rebalancing
dates. To ensure the signs of the canonical portfolios are consistent across
time, we flip the sign of the canonical portfolios at any given rebalancing
date if its cosine similarity with the canonical portfolios obtained from the
previous rebalancing date is negative. While there is not any clear pattern of
trades that we can immediately discern, it nonetheless differs from other
research findings vis-á-vis PCA; for example, Kelly et al., (2022) finds that
the top principal component of a symmetrized cross-covariance matrix goes long
(short) on big (small) equity portfolios, and long (short) on value (growth)
portfolios. The right panels of Figure 4(c) provide the final weights invested
in each asset.

###  4.6 Empirical Results

Table 2 summarizes the performance of the various portfolio methods for the
different test assets. Restricting our attention to the FF25 column, we see
that CP2 has an average return, which is lower than that of PP2 (2.20% versus
3.76%) but has much lower volatility (2.18% versus 5.85%).242424Note that the
magnitudes of the returns for all portfolios appear to be low in comparison to
those published in hedge fund return indices. This is expected since the gross
exposure of our portfolios are all constrained to one. In practice, one would
typically apply a leverage factor greater than one in order to magnify the
returns. Altogether, this translates into Sharpe ratio of $1.01$ which is a
50% improvement over the closest competitor, PP2, at $0.66$. The alpha of the
CP2 is 1.84% and has a low beta to the univariate factor return of 0.15.
Adjusting the alpha by the idiosyncratic volatility of 2.04% gives an
information ratio of 0.9 for CP2. A similar conclusion holds for the ME/OP25
and ME/INV25 test assets, although we observe a deterioration in the Sharpe
ratio for all portfolios in larger-sized test asset possibly due to estimation
errors. CP2 also underperforms PP2 in the ME/INV100 dataset.

Table 3 describes the distribution of the portfolio weights of the estimated
portfolios. In the FF25 column, we see that CP2 has the lowest turnover. This
is interesting given that we make no effort to control the trajectory of the
weights. The average sum of negative weights in the CP2 is $-0.496$ and the
average proportional leverage in CP2 is less than 0.5 indicating a slight tilt
towards long positions. The weights are CP2 appear to be the most dispersed
but do but are relatively not extreme. We draw a similar conclusion for the
other test assets.

It would be interesting to investigate where the performance improvement of
CP2 comes from: Is better at selecting assets that have historically performed
well on average or at varying the positions in the assets dynamically? One way
to discern between the two possible explanations is to decompose the portfolio
returns into the following components:

$\displaystyle\mathbb{E}[w_{t}^{\prime}r_{t+1}]=\underbrace{\mathbb{E}[w_{t}]^{\prime}\mathbb{E}[r_{t+1}]}_{\text{static}}+\underbrace{\mathsf{Cov}(r_{t+1},w_{t})}_{\text{dynamic}}.$
(41)

Intuitively, static bets refer to the investor’s ability to get the long-run
allocations right. On the other hand, the covariance between the signal and
subsequent returns refers to an investor’s ability to tactically ‘time’ the
market, this is, to accurately identify movements in the assets and gain
exposure to those assets accordingly.

Table 4 reports the results of this decomposition, where we estimate the
static and dynamic components of returns according to Equation (41) with their
corresponding sample analogs. Panels A and B show that both the static and
dynamic components contribute positively to the overall performance of the
portfolio. Panel C of Table 4 breaks down the share of total returns due to
taking dynamic bets for CP2 and P2. It shows that on average the returns of
CP2 and PP2 come from taking dynamic bets. Therefore, the performance of our
proposed method does not come from its ability to long (short) more highly
performing (underperforming) assets; it must emanate from its ability to
tactically time the market.

Finally, we also isolate the portfolio returns due to the long and short legs
of the trade. We write $r^{w}_{t}=l((r^{w}_{t})_{+}-(r^{w}_{t})_{-})$ where
$(r^{w}_{t})_{+}$ is the return on the long leg and $(r^{w}_{t})_{-}$ is the
return on the short leg, with weights of both trade legs normalized to sum to
one. Here, $l$ reflects the leverage of the long-short portfolio, but since
our estimated portfolios have a unit gross exposure by construction, $l$ is
close to one. Panel D and E of Table 4 present the statistics of the long and
short legs, respectively. For CP2, the average return on the long leg is 8.01%
and the short leg is 5.71%. This indicates that the profits of our portfolio
come from the long side of the trades.

###  4.7 Robustness Checks

In this section, we inspect whether the outperformance of our proposed
portfolio construction methodology is robust to different revisions in the
current empirical set-up. In particular, we will be interested in results
based on (1) subsample period, (2) forecast horizon (3) shrinkage in the
signal covariance, (4) momentum lookback window, and (5) ‘approximate’ versus
‘actual’ portfolio policy.

####  4.7.1 Sub-Period Analysis

In this section, we check if there are any peculiar subsample effects that may
drive the performances of our proposed scheme. We divide the out-of-sample
period into four roughly equally-sized subsamples of 144 months (that is, 12
trading years) each: (1) 1986-09-25 to 1986, (2) 1986 to 1998, (3) 1998 to
2010, and (4) 2010 to 2022. Then we perform the same procedure in each
subsample. The results are provided in Table 5.

Generally, the performance of all portfolios appears stronger in the earlier
periods of the sample but poorer in the recent decade. We see that the
outperformance of CP2 over the competing portfolios is consistent over time
for FF25 and ME/INV25. However, CP2 underperforms PP2 in (1) ME/INV100 for
most of the subsamples, and (2) FF100, ME/OP25, and ME/OP100 in the earlier
subsample.

####  4.7.2 Forecast Horizon

To make a forecast of subsequent returns, we have used a horizon length of 21
trading days, which roughly corresponds to one month. Given our use of non-
overlapping observations between the signals and subsequent returns, this
implies that the data are sampled on a monthly basis. which also corresponds
to the frequency with which we rebalance our portfolios. We now change the
forecast horizon from 21 days to 1, 5, and 10 days. Each of these forecast
horizons covers an investment period from (1) 01-06-1965 to 10-21-2022, (2)
12-06-1966 to 10-21-2022, and (3) 06-12-1969 to 10-21-2022; this provides us
with 14550 daily, 2814 weekly, and 1347 fortnightly out-of-sample returns,
respectively.

Table 6 shows that the annualized Sharpe ratio of all portfolios tends to be
better at shorter holding periods. Barring the potential side effects of
illiquidity in daily returns, this suggests that the conditioning information
becomes more relevant as the holding period decreases. This makes sense since
the conditional portfolios become more reactive to changes in the states of
the market. Overall, the ranking of the methods remains similar relative to
Table 2 with the exception of the 1-day horizon, where CP2 is the best
performer for all test assets.

####  4.7.3 Shrinkage Intensity

We examine the effect of shrinkage on the covariance of signals in our
proposed method. Our default choice in the analysis was $\delta_{x}=0.9$. The
annualized Sharpe ratios for CP2 corresponding to shrinkage values of
$\delta_{x}\in\\{0,0.1,0.5,0.8,1\\}$ are presented in Table 7 over different
subsamples.

We see that there is no specific level of shrinkage that provides consistent
outperformance for all test assets. This indicates that the shrinkage
intensity $\delta_{x}$ is time-varying in nature. Shrinkage values greater
than or equal to 0.5 appears to work well across different subsamples for
larger-sized equity portfolios up until the recent decade, where there is some
benefit of using more sample information from the signal correlations in
datasets ME/OP100 and ME/INV100.

####  4.7.4 Momentum Lookback Widnow

The 21-day momentum signal that we used as a base case, which is perceived to
be a relatively ‘fast’ signal in that it reacts quickly to changes in market
conditions but is also one that is susceptible to noise. We now consider using
momentum signals computed with a longer lookback window of sizes 42 days, 63
days, 84 days, 126 days, and 252 days. The remaining details remain similar.

Table 8 demonstrates that CP2 generally outperforms the other portfolios in
terms of annualized Sharpe ratio for different momentum signals and different
test assets. Although we observe some performance degradation for all
portfolios as we increase the lookback size from 21 days to 126 days, the
ranking of the methods remains similar to Table 2. There is, however, a
significant performance gain in CP2 not seen in the other portfolios as we
extend the momentum lookback size from 126 days to 252 days. This is
comforting since momentum signals with longer lookback horizons tend to have
lower turnover and allow portfolio managers to target higher overall dollar
capacity to absorb large trades; for example, Novy-Marx and Velikov, (2016)
finds that a fund’s capacity decreases as turnover increases.

####  4.7.5 Approximate Versus Actual Solution

We have worked with the ‘approximated’ mean-variance problem from Proposition
2 throughout this paper since it helped us to simplify the analysis. Given
that we also have a closed-form expression to the ‘full’ mean-variance problem
from Theorem 2 in Proposition 1, we now check if it offers any practical
benefits over our approximate solution. The only difference between both
solutions is in terms of how the canonical correlations enter into the
reconstruction of the optimal portfolio policy. The full solution essentially
nonlinearly adjusts the canonical correlations, while no adjustment takes
place in the approximated solution. Figure 2 shows that both solutions are
similar for small canonical correlation values but for large canonical
correlation values, the conservative behavior of the optimal strategy is
reinforced by downweighting its influence. This makes sense because we are
taking into account more terms that affect the risk profile of the strategy.

From Table 9, we see that our approximated formula generally performs better
than the full formula across different test assets. One reason for this
observation is that the nonlinear adjustment of the canonical correlations may
be too conservative relative to the unadjusted one. For example, a canonical
correlation value of one gets reduced by half through the nonlinear
adjustment. This holds mechanically irrespective of the data. Consequently,
this behavior may inadvertently under-leverage the leading two canonical
portfolios, which are the most profitable streams.

###  4.8 Extension: Two-Signal Case

Our canonical portfolios modeling framework is flexible enough to accommodate
multiple signals. In this empirical exercise, we expand our signal vector to
include two momentum signals of different lookback windows; one with a 21-day
lookback window, and another with a 252-days lookback window. This expanded
$2N$-dimensional signal vector will serve as input for both CP and PP. On the
other hand, we assume that both MVO and UNI take in an equal-weighted average
of the two momentum signals as inputs since these methods do not have an
‘optimal’ way of blending different signals together in a single stage. The
rest of the empirical setup remains unchanged.

From Table 10, we see that the Sharpe ratio of all portfolios, with the
exception of MVO, generally improves in comparison to the one signal case.
More importantly, the outperformance of CP2 over competing methods continues
to hold up for different test assets.

Turning our attention to Table 11, we see that the turnover is reduced for all
portfolios compared to the base case. This observation can be attributed to
the inclusion of the signal with a long lookback window, which tends to have a
lower turnover than one with a short lookback window. This is appealing since
we can expect the performance after factoring in transaction costs to be
better than using a single conditioning variable. We also observe lower
proportional leverage throughout the portfolios indicating a tilt toward long
positions. The weights also appear to be less dispersed than in the base case.

##  5 Conclusion

In this paper, we provide a novel framework for portfolio managers and
academics alike to conceptualize the optimal asset and signal combination
problem with canonical correlation analysis (CCA). Our contribution can be
summarized as follows. First, we recast the original investment problem of
Brandt and Santa-Clara, (2006) into a tractable one that scales to large
dimensions. Our solution can be seen to generalize the classic Markowitz,
(1952) mean-variance optimization scheme, such that it is able to ingest
multiple return-predictive signals, and account for cross-predictability and
uncertainty in the signals. All of these properties are achieved by solving
the portfolio selection problem in a single stage and assuming Gaussianity on
the joint distribution of the assets and signals. Second, we lift the veil of
complexity from our large-dimensional problem through a novel application of
CCA. In particular, CCA breaks down the correlations of all of the asset
returns and signals into independent long–short managed portfolios, which we
term as canonical portfolios. Each of the canonical portfolios can be ranked
from the one with the smallest correlation to the one with the highest; the
canonical portfolios with the highest predictable returns get scaled up the
most.

Having established the theoretical results of our method, we bring it to the
empirical test. We ran backtest simulations on Fama-French sorted equity
portfolios with a momentum signal, and we have found that our proposed method
generally outperforms natural benchmarks. The performance of our method
further improves when the analysis is extended to two momentum signals of
different lookback windows. These results are made possible by introducing
regularization techniques to overcome the instabilities in the estimation and
exploiting the most predictable dimensions of the data.

In terms of future work, our proposed modeling framework is not set in stone
and is flexible enough to accommodate further improvements. Some research
ideas include introducing nonlinearities into the modeling process, relaxing
the Gaussianity assumption on the joint return distribution, as well as
designing signals and portfolios with maximal mutual information are promising
avenues. Additionally, one may consider incorporating constraints on the
portfolio weights such as transaction costs which will be pursued in a
forthcoming work. Another interesting avenue to explore would be to apply
these ideas to develop a test for asset pricing models or to form portfolios
from different asset classes such as individual stocks portfolios, fixed-
income portfolios, currency portfolios, and so forth. Last but not least,
recasting the portfolio selection problem into a CCA framework enables
researchers to leverage the insights and techniques from the rich literature
of CCA that has been expanded by developments in machine learning.

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##  Appendix A Figures and Tables

Figure 1: Wachter Law This figure displays the limiting spectral density of
the sample canonical correlations of Wachter, (1980) for different ratios
$N/T$ and $M/T$. The population cross-covariance matrix is the zero matrix,
and so all the population canonical correlation coefficients are equal to
zero.  Figure 2: Optimal singular value adjustment versus linear approximation
This figure displays the optimal adjustment of the $i$th singular value versus
an unadjusted $i$th singular value.

Figure 3: In-Sample and Out-of-Sample Canonical Portfolio Returns This figure
displays the in-sample and out-of-sample returns for each canonical portfolio.
The left panel shows the squared in-sample regularized canonical correlations
coefficients (black dots) overlaid with their pseudo-random generated
counterparts (maroon dots). The right panel shows the out-of-sample returns of
each canonical portfolio overlaid with a $\pm 2$ standard error band. Each dot
corresponds to values that are averaged over the rebalancing dates. The
estimates are obtained by using a 21-day momentum signal to predict subsequent
monthly returns and 120 non-overlapping monthly returns. The test assets are
the FF25 (top panel), ME/OP25 (middle panel), and MEINV25 (bottom panel). The
out-of-sample period is covered from 09-09-1974 until 10-21-2022.

(a) 25 Size and Book-to-Market

(b) 25 Size and Operating Profitability

(c) 25 Size and Investment

Figure 4: Weights for the Top Canonical Portfolio This figure displays the
weights of the leading canonical portfolio. The left panel shows the canonical
‘weights’ on the returns (blue) and signals (red). The right panel shows the
final portfolio weights invested in each obtained from combining the returns
and signals together through their canonical weights and scaling them up with
the corresponding canonical correlation. The height of each bar corresponds to
values that are averaged over the rebalancing dates, and they are overlaid
with a $\pm 2$ standard error band. The estimates are obtained by using a
21-day momentum signal to predict subsequent monthly returns and 120 non-
overlapping monthly returns. The test assets are the FF25 (top panel), ME/OP25
(middle panel), and MEINV25 (bottom panel). The out-of-sample period is
covered from 09-09-1974 until 10-21-2022.

(a) 25 Size and Book-to-Market

(b) 25 Size and Operating Profitability

(c) 25 Size and Investment

Table 1: Summary of PCA and CCA This table summarizes the key distinguishing features of PCA and CCA applied to a paired dataset of asset returns $r_{t+1}$ and signals $x_{t}$. For CCA (PCA), it displays (1) the correlation (variance) maximization objective, (2) the equivalent error minimization objective, (3) the $i$th canonical (principal) direction, and (4) the $i$th correlation (variance) of the corresponding canonical (principal) direction.  PCA |  (1) Maximize Variance |  $\displaystyle\max_{\begin{subarray}{c}q_{r},q_{x}\\\ \|q_{r}\|_{2}=\|q_{x}\|_{2}=1\end{subarray}}\mathsf{Var}(q_{r}^{\prime}r_{t+1})+\mathsf{Var}(q_{x}^{\prime}x_{t})$
---|---|---
(2) Minimize Error |  $\displaystyle\min_{\|q_{r}\|_{2}=\|q_{x}\|_{2}=1}\mathbb{E}\big{[}\big{\|}r_{t+1}-q_{r}q^{\prime}_{r}r_{t+1}\big{\|}^{2}_{2}\big{]}+\mathbb{E}\big{[}\big{\|}x_{t}-q_{x}q^{\prime}_{x}x_{t}\big{\|}^{2}_{2}\big{]}$
(3) Principal Direction |  $\displaystyle q_{r,i},q_{x,i}=\operatorname*{argmax}_{\begin{subarray}{c}\|q_{r}\|_{2}=\|q_{x}\|_{2}=1\\\ q_{r}^{\prime}q_{r,j}=0,\text{ }j=1,\ldots,i-1\\\ q_{x}^{\prime}q_{x,j}=0,\text{ }j=1,\ldots,i-1\end{subarray}}\mathsf{Var}(q_{r}^{\prime}r_{t+1})+\mathsf{Var}(q_{x}^{\prime}x_{t})$
(4) Principal Variance |  $\displaystyle d_{r,i}=\mathsf{Var}(q_{r,i}^{\prime}r_{t+1}),\quad d_{x,i}=\mathsf{Var}(q_{x,i}^{\prime}x_{t})$
CCA | (1) Maximize Correlation |  $\displaystyle\max_{\mathsf{Var}(q_{r}^{\prime}r_{t+1})=\mathsf{Var}(q_{x}^{\prime}x_{t})=1}\mathsf{Cov}(q_{r}^{\prime}r_{t+1},q_{x}^{\prime}x_{t})$
(2) Minimize Error |  $\displaystyle\min_{\mathsf{Var}(q_{r}^{\prime}r_{t+1})=\mathsf{Var}(q_{x}^{\prime}x_{t})=1}\mathbb{E}\big{[}\big{\|}q_{r}^{\prime}r_{t+1}-q^{\prime}_{x}x_{t}\big{\|}^{2}_{2}\big{]}$
(3) Canonical Direction |  $\displaystyle q_{r,i},q_{x,i}=\operatorname*{argmax}_{\begin{subarray}{c}q_{r}^{\prime}\Sigma_{r}q_{r}=q_{x}^{\prime}\Sigma_{x}q_{x}=1\\\ q_{r}^{\prime}q_{r,j}=0,\text{ }j=1,\ldots,i-1\\\ q_{x}^{\prime}q_{x,j}=0,\text{ }j=1,\ldots,i-1\end{subarray}}\mathsf{Cov}(q_{r}^{\prime}r_{t+1},q_{x}^{\prime}x_{t})$
(4) Canonical Correlation |  $\displaystyle s_{i}=\frac{\mathsf{Cov}(q_{r,i}^{\prime}r_{t+1},q_{x,i}^{\prime}x_{t})}{\sqrt{\mathsf{Var}(q_{r,i}^{\prime}r_{t+1})}\sqrt{\mathsf{Var}(q_{x,i}^{\prime}x_{t})}}$

Table 2: Performance Summary This table shows the out-of-sample performances (in percentages) of various portfolio selection methods using a 21-day momentum signal. The test assets are the 25 and 100 equity portfolios independently sorted on size and three other factors (book-to-market, operating profitability, and investment). Panels A-G display the average returns, the Sharpe ratio of returns, the alpha with respect to the Fama-French 5-factor regression augmented with the univariate factor (UNI), the beta of the simple factor, the idiosyncratic volatility, and the information ratio. The numbers in parenthesis are the t-statistics for the portfolio’s Sharpe ratio and information ratio using the standard error of Lo, (2002). The covariances are estimated on a rolling walk-forward basis with $T=120$ non-overlapping monthly returns. All measures are computed with 579 monthly portfolio weights from 09-09-1974 until 10-21-2022. | FF25 | FF100 |  ME/OP 25 |  ME/OP 100 |  ME/INV 25 |  ME/INV 100
---|---|---|---|---|---|---
Panel A. Average Returns (%)
CP2 | $2.196$ | $1.334$ | $1.741$ | $0.953$ | $1.809$ | $0.782$
PP2 | $3.762$ | $3.462$ | $3.806$ | $3.490$ | $3.377$ | $3.432$
MVO | $0.381$ | $0.188$ | $0.358$ | $-0.097$ | $0.365$ | $0.097$
UNI | $2.268$ | $1.787$ | $2.261$ | $1.477$ | $2.141$ | $1.473$
Panel B. Standard Deviation of Returns (%)
CP2 | $2.175$ | $1.366$ | $2.048$ | $1.260$ | $2.057$ | $1.248$
PP2 | $5.853$ | $5.690$ | $4.912$ | $4.931$ | $4.642$ | $4.433$
MVO | $1.857$ | $1.366$ | $1.685$ | $1.301$ | $1.595$ | $1.261$
UNI | $5.280$ | $4.814$ | $4.173$ | $3.663$ | $4.384$ | $3.970$
Panel C. Sharpe Ratio
CP2 | $\underset{(6.872)}{1.010}$ | $\underset{(6.655)}{0.977}$ | $\underset{(5.837)}{0.854}$ | $\underset{(5.208)}{0.759}$ | $\underset{(6.028)}{0.883}$ | $\underset{(4.341)}{0.631}$
PP2 | $\underset{(4.554)}{0.662}$ | $\underset{(4.325)}{0.628}$ | $\underset{(5.389)}{0.786}$ | $\underset{(4.954)}{0.722}$ | $\underset{(5.077)}{0.740}$ | $\underset{(5.377)}{0.785}$
MVO | $\underset{(1.485)}{0.214}$ | $\underset{(1.004)}{0.145}$ | $\underset{(1.528)}{0.220}$ | $\underset{(-0.473)}{-0.068}$ | $\underset{(1.639)}{0.236}$ | $\underset{(0.577)}{0.083}$
UNI | $\underset{(3.125)}{0.452}$ | $\underset{(2.719)}{0.393}$ | $\underset{(3.843)}{0.557}$ | $\underset{(2.899)}{0.419}$ | $\underset{(3.493)}{0.506}$ | $\underset{(2.690)}{0.389}$
Panel D. Alpha (%)
CP2 | $1.844$ | $1.189$ | $1.116$ | $0.823$ | $1.340$ | $0.683$
PP2 | $2.252$ | $2.240$ | $2.183$ | $2.347$ | $2.069$ | $2.565$
MVO | $-0.210$ | $-0.097$ | $-0.342$ | $-0.391$ | $-0.208$ | $-0.214$
Panel E. Beta to Simple Factor
CP2 | $0.146$ | $0.072$ | $0.259$ | $0.086$ | $0.172$ | $0.054$
PP2 | $0.684$ | $0.695$ | $0.703$ | $0.734$ | $0.583$ | $0.583$
MVO | $0.256$ | $0.168$ | $0.300$ | $0.201$ | $0.260$ | $0.188$
Panel F. Idiosyncratic Volatility (%)
CP2 | $2.041$ | $1.321$ | $1.749$ | $1.219$ | $1.910$ | $1.228$
PP2 | $4.583$ | $4.583$ | $3.951$ | $4.157$ | $3.889$ | $3.778$
MVO | $1.281$ | $1.102$ | $1.136$ | $1.077$ | $1.119$ | $1.018$
Panel G. Information Ratio
CP2 | $\underset{(6.170)}{0.903}$ | $\underset{(6.149)}{0.900}$ | $\underset{(4.393)}{0.638}$ | $\underset{(4.642)}{0.675}$ | $\underset{(4.819)}{0.701}$ | $\underset{(3.835)}{0.556}$
PP2 | $\underset{(3.396)}{0.491}$ | $\underset{(3.379)}{0.489}$ | $\underset{(3.810)}{0.553}$ | $\underset{(3.892)}{0.565}$ | $\underset{(3.671)}{0.532}$ | $\underset{(4.668)}{0.679}$
MVO | $\underset{(-1.137)}{-0.164}$ | $\underset{(-0.609)}{-0.088}$ | $\underset{(-2.087)}{-0.301}$ | $\underset{(-2.512)}{-0.363}$ | $\underset{(-1.292)}{-0.186}$ | $\underset{(-1.456)}{-0.210}$

Table 3: Weight Statistics This table shows the average monthly weight statistics of various portfolio selection methods estimated using a 21-day momentum signal. The test assets are the 25 and 100 equity portfolios independently sorted on size and three other factors (book-to-market, operating profitability, and investment). Panels A-F display the turnover, the proportional leverage, the sum of negative weights in the portfolio, and the minimum and maximum portfolio weights. The covariances are estimated on a rolling walk-forward basis with $T=120$ non-overlapping monthly returns. All measures are computed with 579 monthly portfolio weights from 09-09-1974 until 10-21-2022.  | FF25 | FF100 |  ME/OP 25 |  ME/OP 100 |  ME/INV 25 |  ME/INV 100
---|---|---|---|---|---|---
Panel A. Turnover
CP2 | $1.158$ | $1.202$ | $1.179$ | $1.225$ | $1.165$ | $1.217$
PP2 | $1.149$ | $1.148$ | $1.167$ | $1.154$ | $1.173$ | $1.162$
MVO | $1.377$ | $1.424$ | $1.378$ | $1.416$ | $1.381$ | $1.428$
UNI | $1.252$ | $1.282$ | $1.261$ | $1.295$ | $1.263$ | $1.298$
Panel B. Proportional Leverage
CP2 | $0.493$ | $0.497$ | $0.499$ | $0.494$ | $0.498$ | $0.495$
PP2 | $0.496$ | $0.497$ | $0.493$ | $0.497$ | $0.500$ | $0.502$
MVO | $0.499$ | $0.499$ | $0.504$ | $0.497$ | $0.502$ | $0.500$
UNI | $0.489$ | $0.500$ | $0.489$ | $0.500$ | $0.489$ | $0.500$
Panel C. Sum of Negative Weights
CP2 | $-0.496$ | $-0.497$ | $-0.498$ | $-0.498$ | $-0.497$ | $-0.498$
PP2 | $-0.500$ | $-0.500$ | $-0.500$ | $-0.500$ | $-0.500$ | $-0.500$
MVO | $-0.500$ | $-0.500$ | $-0.501$ | $-0.501$ | $-0.500$ | $-0.500$
UNI | $-0.500$ | $-0.500$ | $-0.500$ | $-0.500$ | $-0.500$ | $-0.500$
Panel D. Minimum Weight
CP2 | $-0.106$ | $-0.033$ | $-0.106$ | $-0.033$ | $-0.103$ | $-0.032$
PP2 | $-0.102$ | $-0.034$ | $-0.105$ | $-0.037$ | $-0.104$ | $-0.033$
MVO | $-0.097$ | $-0.030$ | $-0.094$ | $-0.030$ | $-0.094$ | $-0.029$
UNI | $-0.077$ | $-0.020$ | $-0.077$ | $-0.020$ | $-0.077$ | $-0.020$
Panel E. Maximum Weight
CP2 | $0.104$ | $0.033$ | $0.103$ | $0.032$ | $0.103$ | $0.032$
PP2 | $0.099$ | $0.033$ | $0.100$ | $0.036$ | $0.103$ | $0.033$
MVO | $0.098$ | $0.030$ | $0.095$ | $0.029$ | $0.096$ | $0.030$
UNI | $0.077$ | $0.020$ | $0.077$ | $0.020$ | $0.077$ | $0.020$
Panel F. Standard Deviation of Weights
CP2 | $0.051$ | $0.013$ | $0.051$ | $0.013$ | $0.051$ | $0.013$
PP2 | $0.051$ | $0.013$ | $0.052$ | $0.013$ | $0.052$ | $0.013$
MVO | $0.050$ | $0.012$ | $0.050$ | $0.012$ | $0.050$ | $0.012$
UNI | $0.047$ | $0.012$ | $0.047$ | $0.012$ | $0.047$ | $0.012$

Table 4: Portfolio Returns Decomposition This table shows the average returns (in percentages) of various portfolio selection methods due to static/dynamic trades and long/short trades. Panels A-C display the results for the dynamic and static trades as well as the share of returns due to dynamic trades. Panels D-E display the results for the long and short trades. The signal is a 21-day momentum. The test assets are the 25 and 100 equity portfolios independently sorted on size and three other factors (book-to-market, operating profitability, and investment). The covariances are estimated on a rolling walk-forward basis with $T=120$ non-overlapping monthly returns. All measures are computed with 579 monthly portfolio weights from 09-09-1974 until 10-21-2022.  | FF25 | FF100 |  ME/OP 25 |  ME/OP 100 |  ME/INV 25 |  ME/INV 100
---|---|---|---|---|---|---
Panel A. Static Returns (%)
CP2 | $0.184$ | $0.204$ | $0.182$ | $0.159$ | $0.178$ | $0.138$
PP2 | $0.043$ | $0.032$ | $0.209$ | $0.165$ | $0.081$ | $0.026$
MVO | $0.118$ | $0.159$ | $0.084$ | $0.103$ | $0.142$ | $0.153$
UNI | $0.168$ | $0.170$ | $0.190$ | $0.166$ | $0.157$ | $0.158$
Panel B. Dynamic Returns (%)
CP2 | $2.008$ | $1.127$ | $1.561$ | $0.791$ | $1.640$ | $0.657$
PP2 | $3.718$ | $3.429$ | $3.614$ | $3.349$ | $3.302$ | $3.426$
MVO | $0.263$ | $0.029$ | $0.268$ | $-0.191$ | $0.217$ | $-0.067$
UNI | $2.097$ | $1.614$ | $2.059$ | $1.303$ | $1.988$ | $1.316$
Panel C. Share of Dynamic Returns (%)
CP2 | $91.466$ | $84.510$ | $89.383$ | $83.179$ | $90.064$ | $82.603$
PP2 | $98.802$ | $99.069$ | $94.401$ | $95.229$ | $97.558$ | $99.237$
MVO | $68.917$ | $15.496$ | $75.939$ | $217.015$ | $60.439$ | $-77.498$
UNI | $92.467$ | $90.326$ | $91.399$ | $88.562$ | $92.652$ | $89.249$
Panel D. Long Leg Returns (%)
CP2 | $8.012$ | $7.621$ | $7.666$ | $7.505$ | $8.021$ | $7.725$
PP2 | $8.893$ | $8.532$ | $8.664$ | $8.506$ | $8.716$ | $8.924$
MVO | $7.506$ | $7.306$ | $7.248$ | $7.197$ | $7.570$ | $7.457$
UNI | $8.057$ | $7.916$ | $7.940$ | $7.822$ | $8.255$ | $8.015$
Panel E. Short Leg Returns (%)
CP2 | $5.714$ | $6.297$ | $5.785$ | $6.522$ | $6.011$ | $6.848$
PP2 | $4.764$ | $4.747$ | $4.543$ | $4.697$ | $5.081$ | $5.240$
MVO | $6.939$ | $6.979$ | $6.637$ | $7.152$ | $7.001$ | $7.224$
UNI | $5.476$ | $5.861$ | $5.442$ | $6.163$ | $5.866$ | $6.347$

Table 5: Subsample Analysis This table shows the annualized out-of-sample Sharpe ratio of various portfolio selection methods over different subsamples. Panels A-D display the results for Sample 1 (1974 through 1986), Sample 2 (1986 through 1998), Sample 3 (1998 through 2010), and Sample 4 (2010 through 2022). The signal is a 21-day momentum. The test assets are the 25 and 100 equity portfolios independently sorted on size and three other factors (book-to-market, operating profitability, and investment). The numbers in parenthesis are the t-statistics for the portfolio’s Sharpe ratio and information ratio using the standard error of Lo, (2002). The covariances are estimated on a rolling walk-forward basis with $T=120$ non-overlapping monthly returns.  | FF25 | FF100 |  ME/OP 25 |  ME/OP 100 |  ME/INV 25 |  ME/INV 100
---|---|---|---|---|---|---
Panel A. Sharpe Ratio, Sample Period 1974 to 1986
CP2 | $\underset{(4.208)}{1.254}$ | $\underset{(2.874)}{0.842}$ | $\underset{(3.685)}{1.094}$ | $\underset{(1.378)}{0.401}$ | $\underset{(4.411)}{1.324}$ | $\underset{(2.163)}{0.632}$
PP2 | $\underset{(3.741)}{1.107}$ | $\underset{(3.979)}{1.182}$ | $\underset{(4.224)}{1.264}$ | $\underset{(3.660)}{1.086}$ | $\underset{(3.973)}{1.184}$ | $\underset{(4.093)}{1.222}$
MVO | $\underset{(0.056)}{0.016}$ | $\underset{(-0.527)}{-0.152}$ | $\underset{(1.232)}{0.358}$ | $\underset{(-0.580)}{-0.168}$ | $\underset{(1.419)}{0.412}$ | $\underset{(0.707)}{0.205}$
UNI | $\underset{(3.181)}{0.935}$ | $\underset{(2.607)}{0.762}$ | $\underset{(3.802)}{1.130}$ | $\underset{(2.773)}{0.814}$ | $\underset{(3.446)}{1.020}$ | $\underset{(2.850)}{0.838}$
Panel B. Sharpe Ratio, Sample Period 1986 to 1998
CP2 | $\underset{(5.117)}{1.549}$ | $\underset{(5.199)}{1.577}$ | $\underset{(4.124)}{1.232}$ | $\underset{(3.854)}{1.146}$ | $\underset{(4.100)}{1.224}$ | $\underset{(3.604)}{1.069}$
PP2 | $\underset{(3.560)}{1.051}$ | $\underset{(3.989)}{1.185}$ | $\underset{(4.239)}{1.269}$ | $\underset{(3.892)}{1.159}$ | $\underset{(4.094)}{1.222}$ | $\underset{(4.074)}{1.216}$
MVO | $\underset{(2.025)}{0.589}$ | $\underset{(0.977)}{0.283}$ | $\underset{(1.185)}{0.344}$ | $\underset{(-0.431)}{-0.125}$ | $\underset{(1.616)}{0.470}$ | $\underset{(0.134)}{0.039}$
UNI | $\underset{(2.755)}{0.806}$ | $\underset{(2.466)}{0.720}$ | $\underset{(2.230)}{0.652}$ | $\underset{(1.393)}{0.405}$ | $\underset{(2.134)}{0.623}$ | $\underset{(1.307)}{0.380}$
Panel C. Sharpe Ratio, Sample Period 1998 to 2010
CP2 | $\underset{(3.010)}{0.883}$ | $\underset{(4.237)}{1.263}$ | $\underset{(3.086)}{0.909}$ | $\underset{(3.788)}{1.126}$ | $\underset{(3.390)}{1.002}$ | $\underset{(3.034)}{0.894}$
PP2 | $\underset{(1.407)}{0.408}$ | $\underset{(1.342)}{0.389}$ | $\underset{(2.747)}{0.807}$ | $\underset{(3.418)}{1.011}$ | $\underset{(2.369)}{0.693}$ | $\underset{(2.692)}{0.790}$
MVO | $\underset{(0.977)}{0.283}$ | $\underset{(1.214)}{0.351}$ | $\underset{(0.693)}{0.201}$ | $\underset{(-0.378)}{-0.109}$ | $\underset{(0.557)}{0.161}$ | $\underset{(0.416)}{0.121}$
UNI | $\underset{(1.140)}{0.330}$ | $\underset{(1.137)}{0.329}$ | $\underset{(1.606)}{0.467}$ | $\underset{(1.483)}{0.431}$ | $\underset{(1.373)}{0.399}$ | $\underset{(1.151)}{0.334}$
Panel D. Sharpe Ratio, Sample Period 2010 to 2022
CP2 | $\underset{(1.539)}{0.448}$ | $\underset{(1.339)}{0.389}$ | $\underset{(0.819)}{0.238}$ | $\underset{(1.105)}{0.322}$ | $\underset{(0.401)}{0.117}$ | $\underset{(-0.599)}{-0.174}$
PP2 | $\underset{(1.205)}{0.350}$ | $\underset{(0.515)}{0.149}$ | $\underset{(0.034)}{0.010}$ | $\underset{(-0.731)}{-0.213}$ | $\underset{(-0.468)}{-0.136}$ | $\underset{(-0.185)}{-0.054}$
MVO | $\underset{(-0.212)}{-0.062}$ | $\underset{(-0.021)}{-0.006}$ | $\underset{(0.186)}{0.054}$ | $\underset{(0.132)}{0.038}$ | $\underset{(-0.140)}{-0.041}$ | $\underset{(-0.552)}{-0.160}$
UNI | $\underset{(-0.037)}{-0.011}$ | $\underset{(-0.108)}{-0.031}$ | $\underset{(0.505)}{0.147}$ | $\underset{(0.412)}{0.120}$ | $\underset{(0.358)}{0.104}$ | $\underset{(0.327)}{0.095}$

Table 6: Forecast Horizon This table shows the annualized out-of-sample Sharpe ratio of various portfolio selection methods for different rebalancing frequencies. Panels A-C display the results for 1 day, 5 days, and 10 days. The signal is a 21-day momentum. The test assets are the 25 and 100 equity portfolios independently sorted on size and three other factors (book-to-market, operating profitability, and investment). The numbers in parenthesis are the t-statistics for the portfolio’s Sharpe ratio and information ratio using the standard error of Lo, (2002). The covariances are estimated on a rolling walk-forward basis with $T=120$ non-overlapping monthly returns. Panel A is based on 14550 daily out-of-sample returns from 01-06-1965 until 10-21-2022, Panel B is based on 2814 weekly out-of-sample returns from 12-06-1966 until 10-21-2022, Panel C is based on 1347 fortnightly out-of-sample returns from 06-12-1969 until 10-21-2022.  | FF25 | FF100 |  ME/OP 25 |  ME/OP 100 |  ME/INV 25 |  ME/INV 100
---|---|---|---|---|---|---
Panel A. Sharpe Ratio, Horizon: 1 day
CP2 | $\underset{(15.920)}{2.104}$ | $\underset{(9.869)}{1.301}$ | $\underset{(14.031)}{1.853}$ | $\underset{(7.535)}{0.993}$ | $\underset{(13.717)}{1.811}$ | $\underset{(8.280)}{1.091}$
PP2 | $\underset{(9.299)}{1.226}$ | $\underset{(8.119)}{1.070}$ | $\underset{(8.767)}{1.155}$ | $\underset{(6.042)}{0.796}$ | $\underset{(10.133)}{1.336}$ | $\underset{(7.266)}{0.957}$
MVO | $\underset{(8.264)}{1.089}$ | $\underset{(-2.896)}{-0.381}$ | $\underset{(5.971)}{0.786}$ | $\underset{(-5.518)}{-0.727}$ | $\underset{(6.195)}{0.816}$ | $\underset{(-5.201)}{-0.685}$
UNI | $\underset{(8.702)}{1.147}$ | $\underset{(5.016)}{0.660}$ | $\underset{(7.444)}{0.981}$ | $\underset{(2.434)}{0.320}$ | $\underset{(7.679)}{1.012}$ | $\underset{(2.762)}{0.364}$
Panel B. Sharpe Ratio, Horizon: 5 days
CP2 | $\underset{(11.279)}{1.527}$ | $\underset{(8.810)}{1.187}$ | $\underset{(10.156)}{1.372}$ | $\underset{(6.313)}{0.848}$ | $\underset{(10.233)}{1.383}$ | $\underset{(7.980)}{1.074}$
PP2 | $\underset{(7.448)}{1.002}$ | $\underset{(7.676)}{1.033}$ | $\underset{(7.728)}{1.040}$ | $\underset{(6.946)}{0.934}$ | $\underset{(8.953)}{1.207}$ | $\underset{(8.004)}{1.078}$
MVO | $\underset{(4.140)}{0.555}$ | $\underset{(-1.096)}{-0.147}$ | $\underset{(3.097)}{0.415}$ | $\underset{(-3.351)}{-0.449}$ | $\underset{(3.627)}{0.486}$ | $\underset{(-1.336)}{-0.179}$
UNI | $\underset{(7.222)}{0.971}$ | $\underset{(5.075)}{0.681}$ | $\underset{(6.410)}{0.861}$ | $\underset{(3.378)}{0.453}$ | $\underset{(6.350)}{0.853}$ | $\underset{(3.469)}{0.465}$
Panel C. Sharpe Ratio, Horizon: 10 days
CP2 | $\underset{(9.356)}{1.302}$ | $\underset{(7.416)}{1.025}$ | $\underset{(9.917)}{1.382}$ | $\underset{(7.469)}{1.033}$ | $\underset{(9.255)}{1.287}$ | $\underset{(7.160)}{0.989}$
PP2 | $\underset{(6.702)}{0.925}$ | $\underset{(6.892)}{0.952}$ | $\underset{(6.882)}{0.950}$ | $\underset{(6.005)}{0.827}$ | $\underset{(6.266)}{0.864}$ | $\underset{(6.627)}{0.914}$
MVO | $\underset{(3.078)}{0.422}$ | $\underset{(1.123)}{0.154}$ | $\underset{(2.666)}{0.365}$ | $\underset{(-1.125)}{-0.154}$ | $\underset{(3.351)}{0.459}$ | $\underset{(0.202)}{0.028}$
UNI | $\underset{(5.727)}{0.789}$ | $\underset{(4.993)}{0.686}$ | $\underset{(6.245)}{0.861}$ | $\underset{(4.555)}{0.626}$ | $\underset{(5.650)}{0.778}$ | $\underset{(4.142)}{0.569}$

Table 7: Shrinkage Intensity This table shows the annualized out-of-sample Sharpe Ratio of Canonical Portfolio with the leading two components for different shrinkage intensities in the estimation of the covariance of the signals. Panels A-E display the results for the full sample period, Sample 1 (1974 through 1986), Sample 2 (1986 through 1998), Sample 3 (1998 through 2010), and Sample 4 (2010 through 2022). The rows within each panel correspond to results with shrinkage parameter $\delta_{x}\in\\{0,0.2,0.5,1\\}$. The signal is a 21-day momentum. The test assets are the 25 and 100 equity portfolios independently sorted on size and three other factors (book-to-market, operating profitability, and investment). The numbers in parenthesis are the t-statistics for the portfolio’s Sharpe ratio and information ratio using the standard error of Lo, (2002). The covariances are estimated on a rolling walk-forward basis with $T=120$ non-overlapping monthly returns. All measures are computed with 579 monthly portfolio weights from 09-09-1974 until 10-21-2022.  | FF25 | FF100 |  ME/OP 25 |  ME/OP 100 |  ME/INV 25 |  ME/INV 100
---|---|---|---|---|---|---
Panel A. Sharpe Ratio, Full Sample Period
CP2 ($\delta_{x}=1)$ | $\underset{(6.707)}{0.986}$ | $\underset{(6.400)}{0.939}$ | $\underset{(5.752)}{0.842}$ | $\underset{(4.359)}{0.634}$ | $\underset{(5.798)}{0.849}$ | $\underset{(4.047)}{0.588}$
CP2 ($\delta_{x}=0.8)$ | $\underset{(6.764)}{0.994}$ | $\underset{(6.766)}{0.995}$ | $\underset{(5.791)}{0.848}$ | $\underset{(5.817)}{0.851}$ | $\underset{(5.970)}{0.875}$ | $\underset{(4.545)}{0.661}$
CP2 ($\delta_{x}=0.5)$ | $\underset{(6.624)}{0.973}$ | $\underset{(5.805)}{0.849}$ | $\underset{(5.712)}{0.836}$ | $\underset{(6.232)}{0.914}$ | $\underset{(5.935)}{0.869}$ | $\underset{(4.994)}{0.728}$
CP2 ($\delta_{x}=0.2)$ | $\underset{(6.679)}{0.981}$ | $\underset{(3.393)}{0.491}$ | $\underset{(4.949)}{0.721}$ | $\underset{(3.778)}{0.548}$ | $\underset{(6.125)}{0.898}$ | $\underset{(3.694)}{0.536}$
Panel B. Sharpe Ratio, Subsample Period 1974 to 1986
CP2 ($\delta_{x}=1)$ | $\underset{(3.969)}{1.178}$ | $\underset{(3.284)}{0.966}$ | $\underset{(3.558)}{1.054}$ | $\underset{(1.364)}{0.396}$ | $\underset{(4.092)}{1.222}$ | $\underset{(1.482)}{0.431}$
CP2 ($\delta_{x}=0.8)$ | $\underset{(4.401)}{1.315}$ | $\underset{(2.593)}{0.757}$ | $\underset{(3.714)}{1.103}$ | $\underset{(1.701)}{0.495}$ | $\underset{(4.598)}{1.384}$ | $\underset{(2.566)}{0.752}$
CP2 ($\delta_{x}=0.5)$ | $\underset{(4.278)}{1.276}$ | $\underset{(1.672)}{0.485}$ | $\underset{(4.040)}{1.205}$ | $\underset{(1.645)}{0.479}$ | $\underset{(4.990)}{1.513}$ | $\underset{(2.896)}{0.852}$
CP2 ($\delta_{x}=0.2)$ | $\underset{(4.291)}{1.280}$ | $\underset{(0.203)}{0.059}$ | $\underset{(3.179)}{0.937}$ | $\underset{(-0.123)}{-0.036}$ | $\underset{(5.192)}{1.580}$ | $\underset{(1.670)}{0.486}$
Panel C. Sharpe Ratio, Subsample Period 1986 to 1998
CP2 ($\delta_{x}=1)$ | $\underset{(4.863)}{1.465}$ | $\underset{(4.680)}{1.406}$ | $\underset{(3.814)}{1.134}$ | $\underset{(2.485)}{0.728}$ | $\underset{(3.892)}{1.159}$ | $\underset{(3.521)}{1.043}$
CP2 ($\delta_{x}=0.8)$ | $\underset{(5.267)}{1.599}$ | $\underset{(5.493)}{1.676}$ | $\underset{(4.348)}{1.303}$ | $\underset{(4.518)}{1.358}$ | $\underset{(4.158)}{1.243}$ | $\underset{(3.644)}{1.081}$
CP2 ($\delta_{x}=0.5)$ | $\underset{(5.571)}{1.703}$ | $\underset{(5.113)}{1.548}$ | $\underset{(4.300)}{1.288}$ | $\underset{(4.917)}{1.489}$ | $\underset{(3.580)}{1.061}$ | $\underset{(3.854)}{1.147}$
CP2 ($\delta_{x}=0.2)$ | $\underset{(6.245)}{1.939}$ | $\underset{(4.176)}{1.244}$ | $\underset{(4.023)}{1.200}$ | $\underset{(3.500)}{1.036}$ | $\underset{(3.336)}{0.986}$ | $\underset{(2.669)}{0.783}$
Panel D. Sharpe Ratio, Subsample Period 1998 to 2010
CP2 ($\delta_{x}=1)$ | $\underset{(3.009)}{0.883}$ | $\underset{(3.896)}{1.156}$ | $\underset{(3.244)}{0.958}$ | $\underset{(4.052)}{1.209}$ | $\underset{(3.594)}{1.065}$ | $\underset{(3.599)}{1.067}$
CP2 ($\delta_{x}=0.8)$ | $\underset{(2.907)}{0.852}$ | $\underset{(4.338)}{1.295}$ | $\underset{(2.954)}{0.869}$ | $\underset{(3.463)}{1.025}$ | $\underset{(3.096)}{0.912}$ | $\underset{(2.720)}{0.798}$
CP2 ($\delta_{x}=0.5)$ | $\underset{(2.322)}{0.677}$ | $\underset{(3.819)}{1.132}$ | $\underset{(2.441)}{0.715}$ | $\underset{(2.556)}{0.749}$ | $\underset{(2.473)}{0.724}$ | $\underset{(2.193)}{0.641}$
CP2 ($\delta_{x}=0.2)$ | $\underset{(1.687)}{0.489}$ | $\underset{(1.547)}{0.448}$ | $\underset{(2.621)}{0.769}$ | $\underset{(0.980)}{0.285}$ | $\underset{(2.765)}{0.812}$ | $\underset{(2.025)}{0.591}$
Panel E. Sharpe Ratio, Subsample Period 2010 to 2022
CP2 ($\delta_{x}=1)$ | $\underset{(1.671)}{0.487}$ | $\underset{(1.295)}{0.376}$ | $\underset{(0.782)}{0.228}$ | $\underset{(0.416)}{0.121}$ | $\underset{(0.298)}{0.087}$ | $\underset{(-1.059)}{-0.308}$
CP2 ($\delta_{x}=0.8)$ | $\underset{(1.240)}{0.360}$ | $\underset{(1.348)}{0.392}$ | $\underset{(0.802)}{0.233}$ | $\underset{(1.764)}{0.516}$ | $\underset{(0.390)}{0.113}$ | $\underset{(-0.078)}{-0.023}$
CP2 ($\delta_{x}=0.5)$ | $\underset{(1.501)}{0.437}$ | $\underset{(0.937)}{0.272}$ | $\underset{(1.288)}{0.376}$ | $\underset{(3.318)}{0.984}$ | $\underset{(1.133)}{0.330}$ | $\underset{(1.160)}{0.338}$
CP2 ($\delta_{x}=0.2)$ | $\underset{(1.799)}{0.524}$ | $\underset{(0.762)}{0.221}$ | $\underset{(0.490)}{0.143}$ | $\underset{(3.083)}{0.912}$ | $\underset{(1.442)}{0.421}$ | $\underset{(0.898)}{0.261}$