# Modeling asset allocation strategies and a new portfolio performance score

Apostolos Chalkis  Department of Informatics & TelecommunicationsNational &
Kapodistrian University of Athens, Greece  ATHENA Research & Innovation
Center, Greece  Emmanouil Christoforou  Department of Informatics &
TelecommunicationsNational & Kapodistrian University of Athens, Greece  ATHENA
Research & Innovation Center, Greece  Ioannis Z. Emiris  ATHENA Research &
Innovation Center, Greece  Department of Informatics &
TelecommunicationsNational & Kapodistrian University of Athens, Greece
Theodore Dalamagas  ATHENA Research & Innovation Center, Greece

( )

###### Abstract

We discuss and extend a powerful, geometric framework to represent the set of
portfolios, which identifies the space of asset allocations with the points
lying in a convex polytope. Based on this viewpoint, we survey certain state-
of-the-art tools from geometric and statistical computing in order to handle
important and difficult problems in digital finance. Although our tools are
quite general, in this paper we focus on two specific questions.

The first concerns crisis detection, which is of prime interest for the public
in general and for policy makers in particular because of the significant
impact that crises have on the economy. Certain features in stock markets lead
to this type of anomaly detection: Given the assets’ returns, we describe the
relationship between portfolios’ return and volatility by means of a copula,
without making any assumption on investor strategies. We examine a recent
method relying on copulae to construct an appropriate indicator that allows us
to automate crisis detection. On real data, the indicator detects all past
crashes in the cryptocurrency market, whereas from the DJ600-Europe index,
from 1990 to 2008, the indicator identifies correctly 4 crises and issues one
false positive for which we offer an explanation.

Our second contribution is to introduce an original computational framework to
model asset allocation strategies, which is of independent interest for
digital finance and its applications. Our approach addresses the crucial
question of evaluating portfolio management, and is relevant to individual
managers as well as financial institutions. To evaluate portfolio performance,
we provide a new portfolio score, based on the aforementioned framework and
concepts. In particular, our score relies on statistical properties of
portfolios, and we show how they can be computed efficiently.

##  1 Introduction

Modern finance has been pioneered by Markowitz who set a framework to study
choice in portfolio allocation under uncertainty [28], and for which he earned
the Nobel Prize in economics, 1990. Within this framework, Markowitz
characterized portfolios by their return and their risk; the latter is
formally defined as the variance of the portfolios’ returns111Throughout this
paper we refer to the variance of a portfolio’s return as portfolio
volatility.. An investor builds a portfolio that would maximize its expected
return for a chosen level of risk; it has since become common for asset
managers to optimize their portfolio within this framework. This approach has
led a large part of the empirical finance research to focus on the so-called
efficient frontier which is defined as the set of portfolios presenting the
lowest risk for a given expected return. The efficient frontier is associated
with a well-known family of convex functions, studied by Markowitz [29].
Moreover, the distributional properties of the optimal portfolio weights have
been used for efficient portfolio selection [4, 5, 8, 19, 18].

It is known, from the relevant literature, that financial markets exhibit
three types of behavior. In normal times, stocks are characterized by slightly
positive returns and moderate volatility, in up-market times (typically
bubbles) by high returns and low volatility, and during financial crises by
strongly negative returns and high volatility, see [3] for details. So,
following Markowitz’ framework, in normal and up-market times, the stocks and
portfolios with the lowest volatility should present the lowest returns,
whereas during crises those with lowest volatility should present the highest
returns. The detection of normal and crisis periods is also crucial for
computing an efficient asset allocation [16, 15, 35].

However, these tools, when used to build a portfolio, do not always guarantee
a good performance in practice [27]. Thus, the analysis of investment
performance is of special interest in modern finance, especially given the
growth of the asset management industry in the last decades. Research in this
area is axed on Sharpe-like ratios proposed in the 1960’s [17, 39, 40]. In
practice, the performance of a portfolio manager, over a given period, is
usually measured as the ratio of his ”excess” return, with respect to a
benchmark portfolio, over a risk measure [12]. Managers are then ranked
according to these ratios, and the one achieving the highest and steadiest
returns receives the best score. The major drawback of these techniques is the
identification of benchmark portfolios, since the formation of such portfolios
remains controversial. Moreover, they suffer from non-negligible estimation
errors [25], which prevent any performance comparison to be significant. In
[36] -and independently in [13, 2]\- they use a geometric representation of a
stock market to define a cross-sectional score of a portfolio given a vector
of assets’ returns. The score of a portfolio is defined as the proportion of
all possible asset allocations that the portfolio outperforms in terms of
return. The aim is to measure the relative performance with respect to all
possible alternative allocations offered to the manager. The term cross
section is used to underline that the score takes into account portfolios that
are diversified over all sections of assets, without studying -separately- the
performance on specific sections of stocks. In [2], they follow the same
approach by defining what they call naive investor’s strategy. A naive
investor’s strategy selects uniformly a portfolio from the set of portfolios,
as it is agnostic of the asset returns generating process, and hence does not
use any such information.

###  1.1 Contributions

First, we briefly survey the computational framework in [9], which uses the
geometric representation of long-only portfolios in [36] and the copula
representation for the dependency between portfolios’ return and volatility. A
copula is a multivariate joint distribution where the marginal distributions
are uniform; for more details on copulae, we refer to [31]. We enhance this
framework significantly by employing clustering methods on copulae, and we use
it to detect all past crash events in the cryptocurrency market and all past
crises from 1990 to 2008, using real data from DJ600.

We extend the geometric framework in [36] to model additional asset
allocations to long-only portfolios, e.g. the ”150/50” or the ”130/30”
strategies, which recently have gained popularity [24]. In particular, we work
with the set of fully-invested portfolios, i.e., portfolios whose weights sum
up to 1, which is the default choice for the bulk of the asset management
industry. More precisely, we allow the weights be negative and we use the
norm-constraint in [43] to set a lower bound on the weights’ values. Then, we
introduce a transformation to represent the set of all possible fully-invested
portfolios by a convex polytope; i.e., each point in the interior of the
polytope corresponds to a single asset allocation.

We use this geometric representation to introduce a new mathematical model of
portfolio allocation strategies in a stock market. We consider the concept
where portfolio managers compute and propose portfolio allocations, which we
call formal allocation proposals. Then, an investor decides which asset
allocation proposal to select. Second, she decides how much to modify this
proposal to build her final portfolio. Thus, we expect the portfolios of the
investors who have chosen the proposal of a manager to be ”concentrated
around” that proposal. To model this procedure, we employ multivariate log-
concave distributions. The support of the Probability Density Function (i.e.
the subset of ${\mathbb{R}}^{n}$ which is not mapped to zero) of each
distribution is the set of all possible portfolios, i.e. a convex polytope. In
particular, we say that a portfolio allocation strategy $F_{\pi}$ is induced
from a log-concave distribution $\pi$ as follows: to build a portfolio with
strategy $F_{\pi}$, sample a point/portfolio from $\pi$. Then, we call the
mode of $\pi$ a formal allocation proposal of the allocation strategy
$F_{\pi}$.

We use Markowitz’s framework to parameterize the allocation strategies by the
level of risk that a certain group of investors selects. Similarly, for a
given level of risk, we use the variance to parameterize to what extent around
the formal allocation proposal a subgroup of investors may decide to stick.
Finally, as in any stock market, plenty of strategies may appear which are
chosen by groups of investors. Thus, we define the mixed strategy, induced by
a convex combination of log-concave distributions, i.e. a mixture
distribution.

We use this model of portfolio allocation strategies to define a new portfolio
score to evaluate the performance of an investment. Our new score considers
the set of truly invested portfolios in a stock market in a given time period.
We evaluate the performance of a portfolio, for a given time period, by
comparing the portfolio against a mixed strategy $F_{\pi}$. Thus, we define
the score of a portfolio as the expected number of truly invested portfolios
that the first outperforms –in terms of return– when the portfolios have been
invested according to the mixed strategy $F_{\pi}$. To estimate the new cross-
sectional score within an arbitrarily small error, we provide an efficient
algorithm, based on Markov Chain Monte Carlo integration. In extreme cases,
our new score becomes equal to that in [36, 13, 2]. Thus, it can also be seen
as a generalization of the latter cross-sectional score. Moreover, as one may
have limited knowledge about how the investors behave in a stock market, or
her knowledge may vary from a time period to another, we extend our framework
to handle these issues. We also provide different versions of our score. Each
version provides a piece of different information about the portfolio
allocation we would like to evaluate.

We also provide an open-source
implementation222https://github.com/TolisChal/portfolio_scoring.git to
simulate (mixed) allocation strategies and to compute our new score given a
portfolio. Our implementation scales up to a few hundred assets and allocation
strategies. We provide a pseudo-real time example in the cryptocurrency
market, using the 12 cryptocurrencies with longest history. We provide
extended arithmetic results to show that the informativeness of our new score
can be higher than that of existing and well-known performance measures (e.g.
Sharpe, Sortino ratios, and Jensen’s alpha). Moreover, we use our computations
of the distribution of a portfolio’s score –assuming a distribution on the
assets’ returns– to discuss how it could lead to useful insights about its
performance. We also compute copulae of portfolios’ return and volatility
under the assumption that the portfolios have been built according to a mixed
strategy. We show that a copula of a certain time period can be very different
from that in [9]. We believe that the last two simulations pave the way for
future work in the problems of crisis detection and portfolio allocation.

Finally, since the simulation of allocation strategies and the computation of
the score and copulae rely only on sampling from high dimensional log-concave
distributions supported on the set of portfolios, our framework works also for
a singular covariance matrix. That is, we can incorporate in our framework the
results in [14, 6, 30, 7, 33]. However, to keep the presentation simple, in
Sec. 4 we assume that the covariance matrix of the asset returns is positive
definite. More details about our computational method and its efficiency are
found in Appendix A.

##### Paper structure.

The next section presents our geometric representation of portfolios. Sec. 3
surveys our work on copulae and the ensuing crisis indicator; our approach is
corroborated by two applications on real data. Some elements of this section
are presented in [9], but here we present a broader class of methods (i.e.
clustering copulae) and a new result on the cryptocurrency market. Sec. 4
introduces a new framework for modeling allocation strategies and evaluating
portfolio performance by defining a new score of a portfolio. Sec. 6 presents
our pseudo-real time example on real data to illustrate our new framework and
the usefulness of our new score. Finally, in Sec. 7, we briefly discuss
conclusions and future work.

##  2 Geometric representation of the set of portfolios

In this section we formalize the geometric representation of sets of
portfolios with an arbitrary large number of assets $n$. First we handle the
case of long-only strategies and then, we extend this representation to fully-
invested portfolios. In both cases the set of portfolios is a convex polytope
in ${\mathbb{R}}^{n}$.

###  2.1 Long-only portfolios

In this case no short sales are allowed. Let a portfolio $x$ investing in $n$
assets, whose weights are $x=(x_{1},\dots,x_{n})\in{\mathbb{R}}^{n}$. The
portfolios in which a long-only asset manager can invest are subject to
$\sum\limits_{i=1}^{n}x_{i}=1$ and $x_{i}\geq 0,\forall i$. Thus, the set of
portfolios available to this asset manager is the unit $(n-1)-$dimensional
canonical simplex, denoted by $\Delta^{n-1}$ and defined as

$\Delta^{n-1}:=\left\\{x\in\mathbb{R}^{n}\ \left|\
\sum_{i=1}^{n}x_{i}=1,\mbox{ and }x_{i}\geq
0,i\in[n]\right.\right\\}\subset{\mathbb{R}}^{n}.$ (1)

The simplex $\Delta^{n-1}$ is the smallest convex polytope with nonzero volume
in a given dimension. For instance, in the plane any triangle is a simplex,
while a triangular pyramid, or tetrahedron, is the simplex in 3d space. The
vertices of $\Delta^{n-1}$ represent portfolios composed entirely of a single
asset.

###  2.2 Fully-invested portfolios

When short sales are allowed we write the set of all possible portfolios as,

$P:=\left\\{x\in\mathbb{R}^{n}\ \left|\ \sum_{i=1}^{n}x_{i}=1,\mbox{ and
}\|x\|_{1}\leq\gamma,i\in[n],\ \gamma\geq
1\right.\right\\}\subset{\mathbb{R}}^{n},$ (2)

where the $L_{1}$-norm $\|x\|_{1}=\sum_{i=1}^{n}|x_{i}|$. When $\gamma=1$ no
short sales are allowed and $P=\Delta^{n}$. When $\gamma=1.6$ then $P$
corresponds to fully-invested portfolios of the 130/30 type and $\gamma=2$ to
150/50. To show that $P$ is a convex polytope for any $\gamma\geq 1$, we
replace the norm-constraint $\|x\|_{1}\leq\gamma$ with a set of linear
inequalities. Since $|x_{i}|=\max\\{-x_{i},x_{i}\\}$, for each $x_{i}$, we add
an auxiliary variable $y_{i}$ such that,

$y_{i}\geq-x_{i},\ y_{i}\geq x_{i},\ y_{i}\geq 0.$ (3)

Then, the set of all possible portfolios is given by,

$\small\tilde{P}:=\left\\{(x,y)\in{\mathbb{R}}^{2n}\ \bigg{|}\
\sum_{i=1}^{n}x_{i}=1,\ -y_{i}\leq x_{i}\leq y_{i},\
\sum_{i=1}^{n}y_{i}\leq\gamma,\ i\in[n],\ \gamma\geq
1\right\\}\subset{\mathbb{R}}^{2n},$ (4)

which is a convex polytope as the feasible space is defined only by a set of
linear inequalities (half-spaces).

##  3 Crisis detection

In this section, we present our computational methods to address the problem
of crisis detection in stock markets. We focus on long-only portfolios, which
means that the set of portfolios in the following computations corresponds to
the canonical simplex $\Delta^{n-1}$.

It is difficult to capture the dependency between portfolios return and
volatility by the usual mean-variance representation. So we rely on the copula
representation. A copula is a joint probability distribution for which all the
marginal probability distributions are uniform. Fig. 1 illustrates such a
copula and shows a positive dependency between portfolios’ return and
volatility. Given a vector of assets’ returns $R\in{\mathbb{R}}^{n}$ and the
covariance matrix $\Sigma\in{\mathbb{R}}^{n\times n}$ of the assets’ returns
distribution, we say that any portfolio $x\in\Delta^{n-1}$ has return
$f_{ret}(x,R)=R^{T}x$ and variance (or volatility)
$f_{vol}(x,\Sigma)=x^{T}\Sigma x$.

Figure 1: Copula representation of the portfolios distribution, by return and
variance. The market considered is made of the 19 sectoral indices of DJSTOXX
600 Europe. The data is from Oct. 16, 2017 to Jan. 10, 2018. Each line and
column sum to 1% of the portfolios.

To estimate the copula between portfolios’ return and volatility we consider
the following discretization on the values of each quantity. We fix two
sequences $s_{0}<\dots<s_{m}$ and $u_{0}<\dots<u_{m}$ such that

$\frac{\mbox{vol}(S_{i})}{\mbox{vol}(\Delta^{n-1})}\approx p\quad\text{ and
}\quad\frac{\mbox{vol}(U_{i})}{\mbox{vol}(\Delta^{n-1})}\approx
p,\;i=0,\dots,m-1,$ (5)

where $S_{i}:=\\{x\in{\mathbb{R}}^{n}\ |\ s_{i}\leq f_{ret}(x,R)\leq
s_{i+1}\\}$ and $U_{i}:=\\{x\in{\mathbb{R}}^{n}\ |\ u_{i}\leq
f_{vol}(x,\Sigma)\leq u_{i+1}\\}$ and $p<1$ a small constant (e.g. $p=0.01$).
Equation (5) implies that a constant percentage $p$ of the portfolios have
return less than $s_{i+1}$ and higher than $s_{i}$. The same occurs for all
the sets $U_{i}$, which contain portfolios with bounded volatility.

Furthermore, the sets $S_{i},\ U_{i}$ define a grid of convex bodies, obtained
by a family of parallel hyperplanes and a family of concentric ellipsoids
–centered at the origin– intersecting $\Delta^{n-1}$. In fact, for given
integers $i,j\leq m-1$ the body

$Q_{ij}:=\\{x\in\Delta^{n-1}\ |\ s_{i}\leq f_{ret}(x,R)\leq s_{i+1}\text{ and
}u_{j}\leq f_{vol}(x,\Sigma)\leq u_{j+1}\\},$ (6)

contains the portfolios with return less than $s_{i+1}$ and higher than
$s_{i}$ and volatility less than $u_{j+1}$ and higher than $u_{j}$. Now, to
obtain the aforementioned copula one has to estimate the ratios
$\frac{\mbox{vol}(Q_{ij})}{\mbox{vol}(\Delta^{n-1})}$ for $i,j=0,\dots,m-1$.

We use Monte Carlo to estimate each volume ratio. We leverage direct,
efficient uniform sampling from $\Delta^{n-1}$ following [37] and then count
the number of points per body in the grid. In Sec. 3.2 this leads to an
indicator to decide the state of the stock market that the estimated copula
corresponds to.

Considering the computational efficiency of this method, it can be applied to
stock markets with a few thousands of assets, since the cost per uniformly
distributed sample in $\Delta^{n-1}$ using the exact sampler in [37] is
$O(n)$. For runtimes see Appendix A.

###  3.1 Computing copulae

In our computations, to define the family of parallel hyperplanes, we consider
compound returns over periods of $k$ observations. Let the asset returns
$r_{i}=(r_{i,1},\dots,r_{i,n})\in\mathbb{R}^{n}$, $i\in[k]$, then the
component $j$ of the compound return equals:

$R_{j}=(1+r_{i,j})(1+r_{i+1,j})\cdots(1+r_{i+k-1,j})-1,\quad j=1,\dots,n.$ (7)

This defines vector $R\in{\mathbb{R}}^{n}$ normal to a family of parallel
hyperplanes, whose equations are fully defined by selecting appropriate
constants.

The covariance matrix $\Sigma$ of the assets’ returns is computed using the
shrinkage estimator of [21],333Matlab code at
http://www.econ.uzh.ch/en/people/faculty/wolf/publications.html. as it
provides a robust estimate even when the sample size is short with respect to
the number of assets.

To compute the copulae, we determine constants defining hyperplanes and
ellipsoids so that the volume between two consecutive such objects is $p=1\%$
of the simplex volume. Let us refer to the method outlined at Equation (5)
using notation introduced just before this equation. The sequence of
$s_{0}<\dots<s_{m}$ are determined by bisection using Varsi’s algorithm. For
ellipsoids, we sample from the simplex and look for $u_{0}<\dots<u_{m}$ such
that there is an equal number of uniformly distributed points in each
intersection.

We set $m=100$, to estimate each copulae. We thus get $100\times 100$ copulae
representing the distribution of the portfolios with respect to the portfolio
returns and volatilities. Fig. 2 illustrates such copulae, and shows the
different relationship between returns and volatility in good (left) and bad
(right, Covid-19 shock event) times.

Figure 2: Copulae that correspond to cryptocurrencies’ states. Left, a normal
period (16/12/2017) and right, a shock event due to Covid-19 (15/03/2020). The
middle plot shows the mass of interest to characterize the market state.

We analyze real data consisting of regular interval (e.g. daily) returns from
two different asset sections: stocks from the Dow Jones Stoxx 600
Europe™(DJ600) and cryptocurrencies. We apply the methodology to a subset of
assets drawn from the DJ 600 constituents using daily data covering the period
from 01/01/1990 to 31/11/2017444Our data is from Bloomberg™.. Since not all
stocks are tracked for the full period of time, we select the 100 assets with
the longest history in the index, and juxtapose stock returns and stock
returns covariance matrix over the same period to detect crises. For the
cryptocurrency assets, we use the daily returns of 12 out of the top 100
cryptocurrencies, ranked by CoinMarketCap’s555https://coinmarketcap.com/
market cap (cmc_rank) on 22/11/2020, having the longest available history
(Table 6). We compute the daily return for each coin using the daily close
price obtained by CoinMarketCap, for several notable coins such as Bitcoin,
Litecoin and Ethereum.

Figure 3: Representation of the periods over which the indicator is greater
than one for 61-100 days (yellow) and over 100 days (red).

###  3.2 Indicator and crisis detection

When we work with real data in order to build the indicator, we wish to
compare the densities of portfolios along the two diagonals. In normal and up-
market times, the portfolios with the lowest volatility present the lowest
returns and the mass of portfolios should be on the up-diagonal. During
crises, the portfolios with the lowest volatility present the highest returns
and the mass of portfolios should be on the down-diagonal, see Fig. 2 as
illustration. Thus, setting up- and down-diagonal bands, we define the
indicator as the ratio of the down-diagonal band over the up-diagonal band,
discarding the intersection of the two. The construction of the indicator is
illustrated in Fig. 2 (middle) where the indicator is the ratio of the mass of
portfolios in the blue area over the mass of portfolios in the red one.

The indicator is estimated on copulae by drawing $500,000$ uniformly
distributed points. We compute the indicator per copula over a rolling window
of $k=60$ days and with a band of $\pm 10\%$ with respect to the diagonal. We
experimentally determined both values. The latter corresponds to roughly 3
months when observations are daily. When the indicator exceeds 1 for more than
60 days but less than 100 days, we report the time interval as a “warning”
(yellow color), while when exceeds 1 for more than 100 days, we report the
interval as a “crisis” (red); see Fig. 3, and Fig. 4. The periods are at least
60 days long to avoid detection of isolated events whose persistence is only
due to the auto-correlation implied by the rolling window.

We compare DJ 600 results with the database of financial crises in European
countries in [11]. The first crisis (May 1990 to Dec. 1990) corresponds to the
early 90’s recession, the second one (May 2000 to May 2001) to the dot-com
bubble burst, the third one (Oct. 2001 to Apr. 2002) to the stock market
downturn of 2002, the fourth one (Nov. 2005 to Apr. 2006) is not listed in the
European database and is either a false positive of our method or may be due
to a bias in the companies selected in the sample, and the fifth one (Dec.
2007 to Aug. 2008) can be associated with the sub-prime crisis.

Our cryptocurrencies indicator detects successfully the 2018 (great)
cryptocurrency crash; see Fig. 4. The first shock event detected in 2018 (mid-
January to late March) corresponds to the crash of nearly all
cryptocurrencies, following Bitcoin’s, whose price fell by about 65% from 6
January to 6 February 2018, after an unprecedented boom in 2017. Intermediate
warnings (mid-May to early August) should correspond to cryptocurrencies
collapses (80% from their peak in January) until September. The detected crash
at the end of 2018 (November 2018 until early January 2019) corresponds to the
fall of Bitcoin’s market capitalization (below $100 billion) and price by over
80% from its peak, almost one-third of its previous week value. Finally, the
detected event in early 2020 corresponds to the shock event due to covid-19.

Figure 4: Warning (yellow) and Crises (red) periods detected by the indicator
for cryptocurrencies (2014-2020).

####  3.2.1 Clustering of copulae agrees with indicator

To cluster the probability distributions distances of the copulae, we computed
a distance matrix ($D$) between all copulae using the earth mover’s distance
(EMD) [38]. The EMD between two distributions is the minimum amount of work
required to turn one distribution into the other. We use a fast and robust EMD
algorithm, which appears to improve both accuracy and speed [34]. Then, we
apply spectral clustering [32], a method to cluster points using the
eigenvectors of the affinity matrix ($A$) which we derive from the distance
matrix, computed by the radial basis function kernel, replacing the Euclidean
distance with EMD, where $A_{ij}=exp(-D_{ij}^{2}/2\sigma^{2})$, and for
$\sigma$ we chose the standard deviation of distances. Using the $k$ largest
eigenvectors of the laplacian matrix, we construct a new matrix and apply
k-medoids clustering by treating each row as a point, so as to obtain $k$
clusters. The results with $k=6$ and $k=8$ are shown on the indicators’ values
in Fig. 13, 14, and 15. Clusters appear to contain copulae with similar
indicator values. Crisis and normal periods are assigned to clusters with high
and low indicator values respectively. Therefore, the clustering of the
copulae is proportional to discretising the values of the indicator. We do not
use any data-driven techniques to select an optimal cluster size, since we
apply clustering only to demonstrate that the resulting clusters validate the
indicator and distinguish different market states according to the indicator.
Additional results on clustering copulae can be found in Appendix C.

Figure 5: Spectral clustering of copulae, with $k=6$ clusters, on the earth
mover’s distances (EMD) of the copulae. Results are shown on the values of the
indicator for every copula. There are 6 different plots, one for every
cluster. Red points indicate the copulae assigned to the specific cluster,
while the blue points are the copulae assigned to other clusters. Yellow and
red time intervals are the identified by the indicator warning and crises
periods respectively.

##  4 Modeling allocation strategies and a new portfolio score

We provide an original framework for modeling allocation strategies and a new
cross-sectional portfolio score. We define the score of a given asset
allocation as the expected value of the proportion of truly invested
portfolios in a stock market, that the first outperforms when the portfolios
have been built according to, what we call, a mixed strategy.

Here, we assume that in a stock market the portfolio managers make allocation
proposals. Then, the investors choose which proposal to select and how much to
modify it before they build their final portfolio. Thus, we model a portfolio
allocation strategy by a log-concave distribution supported on the portfolio
domain $P$, with its mode being at a benchmark portfolio. Then, an investor
builds a portfolio according to that strategy, by generating a point/portfolio
from the corresponding distribution.

######  Definition 1.

Let $\pi$ be a log-concave distribution supported on the portfolio domain
$P\subset{\mathbb{R}}^{n}$ with Probability Density Function (PDF) $\pi(x)$.
Then, a portfolio allocation strategy $F:\pi\rightarrow P$ is said to be
induced by the distribution $\pi$, and we write $F_{\pi}$. More precisely,
$F_{\pi}$ is induced by the following state:

“To build a portfolio with strategy $F_{\pi}$ sample a point/portfolio from
$\pi$”.

The mode of $\pi$ can be seen as the allocation proposal that a portfolio
manager has made. Then, we expect the portfolios of the investors, who have
chosen that proposal, to be concentrated around that proposal/mode.

######  Definition 2.

Let strategy $F_{\pi}$ induced by the unimodal distribution $\pi$. We call the
mode of $\pi$ formal allocation proposal or formal proposal of the portfolio
allocation strategy $F_{\pi}$.

In the sequel, we assume that in a stock market the set of truly invested
portfolios, are being built by a combination of different strategies used by
the investors (mixed strategy). First, we consider a sequence of log-concave
distributions $\pi_{1},\dots,\pi_{M}$ restricted to $P$. Each distribution
induces a portfolio allocation strategy, i.e. $F_{\pi_{1}},\dots,F_{\pi_{M}}$.
Then, the mixed strategy is induced by a convex combination of $\pi_{i}$, i.e.
by a mixture distribution, as the following definition states.

######  Definition 3.

Let $\pi_{1},\dots,\pi_{M}$ be a sequence of log-concave distributions
supported on the set of portfolios $P\subset{\mathbb{R}}^{n}$, and let the
mixture density be $\pi(x)=\sum_{i=1}^{M}w_{i}\pi_{i}(x)$, where $w_{i}\geq
0,\ \sum_{i=1}^{M}w_{i}=1$. We call $F_{\pi}$ the mixed strategy induced by
the mixture density $\pi$.

In Definition 3 each weight $w_{i}$ corresponds to the proportion of the
investors that build their portfolios according to the allocation strategy
$F_{\pi_{i}}$. Thus, the vector of weights $w\in{\mathbb{R}}^{M}$ implies how
the investors, in a certain stock market and time period, tend to behave. Now
we are ready to define the new cross-sectional score of an asset allocation
versus a mixed strategy.

######  Definition 4.

Let a stock market with $n$ assets and $F_{\pi}$ a mixed strategy induced by
the mixture density $\pi$. For given asset returns $R\in{\mathbb{R}}^{n}$ over
a single period of time, the score of a portfolio, providing a value of return
$R^{*}$, is

$s=\int_{P}g(x)\pi(x)dx,\quad g(x)=\left\\{\begin{array}[]{ll}1.&\mbox{ if
}R^{T}x\leq R^{*},\\\ 0,&\mbox{otherwise.}\\\ \end{array}\right.$ (8)

Clearly, the value of the integral in Equation (8) corresponds to the expected
proportion of portfolios that an allocation outperforms –in terms of return–
when the portfolios are invested according to the mixed strategy $F_{\pi}$.

###  4.1 Log-concave distributions in Markowitz’ framework

In this section, we model allocation strategies in Markowitz’s framework using
special multivariate log-concave distributions supported on the set of
portfolios $P$. A proper choice of log-concave distributions allows us to
parameterize a strategy by the level of risk and the level of dispersion
around the formal allocation proposal of the strategy.

In general, using Markowitz’ framework one can define, under certain
assumptions, the optimal portfolio $\bar{x}$ as the maximum of a concave
function $h(x),\ x\in P$. Then, the mode of the log-concave distribution with
PDF $\pi(x)\propto e^{\alpha h(x)}$ is $\bar{x}$, while the parameter
$\alpha>0$ controls the variance of the distribution. Large/small values of
$\alpha$ corresponds to small/large variance.

Notice that as the variance grows, $\pi$ converges to the uniform
distribution. Moreover, as the variance diminishes, the mass of $\pi$
concentrates around the mode of $\pi(x)$. Consequently, we use the variance to
parameterize the sequence $\pi_{i}\propto e^{\alpha_{i}h(x)}$. That is, small
variances correspond to allocation strategies used by investors who stick
around the formal allocation proposal. Large variances correspond to
allocation strategies used by investors who may modify the formal proposal a
lot. Thus, in the first case, the invested portfolios would be highly
concentrated around the formal allocation proposal of $F_{\pi}$ (or around the
mode of $\pi$). In the second case, the invested portfolios would be highly
dispersed around the mode of $\pi$.

In the extreme case of very large variance, $\pi$ is close to the uniform
distribution. Then, the induced allocation strategy becomes the naive strategy
as defined in [2]. We employ the $L_{2}$ norm of a log-concave distribution
$\pi$ with respect to (w.r.t.) the uniform distribution to characterize how
dispersed, around the formal proposal, the portfolios built according to
$F_{\pi}$ are. The $L_{2}$ norm of a distribution $f$ w.r.t a distribution
$g$, when both are supported on a set $P\subset{\mathbb{R}}^{n}$ is,

$\|f/g\|=\mathbb{E}_{f}\bigg{(}\frac{f(x)}{g(x)}\bigg{)}=\int_{P}\frac{f(x)}{g(x)}f(x)dx=\int_{P}\bigg{(}\frac{f(x)}{g(x)}\bigg{)}^{2}g(x)dx.$
(9)

We can now define what we call a $D$-dispersed allocation strategy.

######  Definition 5.

Let $\pi\propto e^{\alpha h(x)}$ be any log-concave distribution supported on
the set of portfolios $P$ and let $F_{\pi}$ be the induced portfolio
allocation strategy. We say that $F_{\pi}$ is $D$-dispersed, where $D$ is the
$L_{2}$ norm of $\pi$ w.r.t. the uniform distribution.

Figure 6: Left: illustration of PDFs $\pi_{q}\propto e^{-\alpha\phi_{q}(x)}$,
where $\alpha=1$ and from left to right $q_{1}=0.3,\ q_{2}=1,\ q_{3}=1.5$.
Right: $3$ illustrations of the mixture density of Equation (12), where
$M_{1}=3,\ M_{2}=2$. In both plots each black small star corresponds to a
formal allocation proposal of an allocation strategy. From yellow to blue:
high to low density regions.

Our main approach is to leverage the expected quadratic utility function,

$\phi_{q}(x)=x^{T}\Sigma x-q\mu^{T}x,\ x\in P\subset{\mathbb{R}}^{n},\quad
q\in[0,+\infty],$ (10)

where $\mu\in{\mathbb{R}}^{n}$ is the mean and $\Sigma\in{\mathbb{R}}^{n\times
n}$ is the covariance matrix of the assets’ returns and $n$ is the number of
assets. This parametric function delivers similar solutions to the original
Markowitz problem in [20, 23]. It is also used by the investors to compute the
efficient frontier and optimal portfolios. The $x^{T}\Sigma x$ is called risk
term, the $\mu^{T}x$ is called return term and the parameter $q$ controls the
trade-off between return and risk. Typically, in modern finance, a portfolio
manager builds an efficient asset allocation by selecting a value $q_{0}$
—which determines the level of risk of his allocation. Then, according to [29]
she/he solves the following optimization problem:

$\min\;\phi_{q_{0}}(x)=x^{T}\Sigma x-q_{0}\mu^{T}x,\;\text{ subject to }x\in
P.$

We call the portfolio $\bar{x}=\min\limits_{x\in P}\phi_{q_{0}}(x)$ as the
optimal mean-variance portfolio for the risk implied by $q_{0}$. Thus, the
efficient frontier can be seen as a parametric curve on $q$.

Let the log-concave distribution,

$\pi_{\alpha,q}\propto e^{-\alpha\phi_{q}(x)},$ (11)

supported on $P$. The left plot in Fig. 6 illustrates some examples of the
probability density function $\pi_{\alpha,q}$ where the mean $\mu$ and the
covariance matrix $\Sigma$ are randomly sampled once. Notice that for
different $q$, the mode (or the formal allocation proposal of the strategy
$F_{\pi_{\alpha,q}}$) is shifted.

We use the parameter $q$ to denote the level of risk of a portfolio allocation
strategy $F_{\pi_{\alpha,q}}$. Small values of $q$ correspond to low risk
strategies whereas large values of $q$ to high risk strategies. Thus, a
sequence of such densities can be parameterized by both $q$ (risk) and
$\alpha$ (dispersion). In particular, a mixed strategy $F_{\pi}$ can be
induced by the following mixture density:

$\pi(x)=\sum_{i=1}^{M_{1}}\sum_{j=1}^{M_{2}}w_{ij}e^{-a_{ij}\phi_{i}(x)},\
\text{where }\phi_{i}=x^{T}\Sigma x-q_{i}\mu^{T}x,\ x\in P,$ (12)

where each $q_{i}$ denotes the level of risk. For each $q_{i}$ the parameters
$\alpha_{ij}$ imply the level of dispersion of the strategy $F_{\pi_{ij}}$.
Notice that for each level of risk $q_{i}$ there are $M_{2}$ different levels
of dispersion that different groups of investors’ portfolios may appear around
the same formal allocation proposal. The right plot of Fig. 6 illustrates some
examples of this mixture density.

Since the portfolio score in Definition 4 is equal to the expectation of an
indicator function with respect to the measure induced by a mixture of log-
concave distributions, it can not be computed exactly (e.g. from a closed-
form). In the sequel, we discuss how we can estimate the value of the new
score by approximating the value of the corresponding multivariate integral.

###  4.2 Computation of the score

This section provides a Markov Chain Monte Carlo (MCMC) integration method to
guarantee fast and robust approximation within arbitrarily small error for the
score in Definition 4. Let the probability density function
$\pi(x)=\sum_{i=1}^{M}w_{i}\pi_{i}(x)$ to be a mixture of log-concave
densities (i.e. $\pi_{i}$ are log-concave distributions). Furthermore, let the
vector of assets’ returns $R\in{\mathbb{R}}^{n}$, the halfspace
$H(R^{*}):=\\{x\in{\mathbb{R}}^{n}\ |\ R^{T}x\leq R^{*}\\}$ and the indicator
function $g(x)=\left\\{\begin{array}[]{ll}1.&\mbox{ if }x\in H(R^{*}),\\\
0,&\mbox{otherwise.}\\\ \end{array}\right.$. Then the score in Equation (8)
can be written,

$\begin{split}s&=\int_{P}g(x)\sum_{i=1}^{M}w_{i}\pi_{i}(x)dx=\sum_{i=1}^{M}w_{i}\int_{P}g(x)\pi_{i}(x)dx\\\
&=\sum_{i=1}^{M}w_{i}\int_{P\cap
H(R^{*})}\pi_{i}(x)dx=\sum_{i=1}^{M}w_{i}\int_{S}\pi_{i}(x)dx,\end{split}$
(13)

where $S:=P\cap H(R^{*})$.

It is clear that the computation of the score $s$ is reduced to integrate $M$
log-concave functions over a convex set $S$, i.e. to compute each
$\int_{S}\pi_{i}(x)dx,\ i\in[M]$. For each one of these $M$ integrals, we use
the algorithm presented in [26] to approximate it within an arbitrarily small
error after a number of operations that grows polynomially with the dimension
(number of assets) $n$. First, we use an alternative representation of the
volume of $S$, employing a log-concave function $\pi(x)$,

$\begin{split}&\mbox{vol}(S)=\int_{S}\pi(x)dx\
\frac{\int_{K}\pi^{\beta_{1}}(x)dx}{\int_{S}\pi(x)dx}\
\frac{\int_{S}\pi^{\beta_{2}}(x)dx}{\int_{S}\pi(x)^{\beta_{1}}dx}\ \cdots\
\frac{\int_{S}1dx}{\int_{S}\pi(x)^{\beta_{k}}dx}\\\
&\Rightarrow\int_{S}\pi(x)dx=\mbox{vol}(S)\
\frac{\int_{S}\pi(x)^{\beta_{k}}dx}{\int_{S}1dx}\ \cdots\
\frac{\int_{S}\pi(x)dx}{\int_{S}\pi(x)^{\beta_{1}}dx},\end{split}$ (14)

where the sequence $\beta_{j},\ j\in[k]$ are factors applied on the variance
of $\pi(x)$.

Since $S$ is the intersection of a halfspace with the convex polytope $P$ we
use the algorithm in [10] to approximate $\mbox{vol}(S)$ within error
$\epsilon$ after $O^{*}(n^{3})$, where $O^{*}(\cdot)$ suppresses
polylogarithmic factors and dependence on $\epsilon$. In the special case of
$P=\Delta^{n-1}$, we can compute the exact value of $\mbox{vol}(S)$ using
Varsi’s algorithm [41] after $n^{2}$ operations at most. Consequently, the
computation of $\int_{S}\pi(x)dx$ is reduced to compute $k$ ratios of
integrals. For each ratio we have,

$\begin{split}r_{j}=\frac{\int_{S}\pi(x)^{\beta_{j-1}}dx}{\int_{S}\pi(x)^{\beta_{j}}dx}&=\frac{1}{\int_{S}\pi(x)^{\beta_{j}}dx}\int_{S}\frac{\pi(x)^{\beta_{j-1}}}{\pi(x)^{\beta_{j}}(x)}\pi(x)^{\beta_{j}}(x)dx\\\
&=\int_{S}\frac{\pi(x)^{\beta_{j-1}}}{\pi(x)^{\beta_{j}}}\frac{\pi(x)^{\beta_{j}}}{\int_{S}\pi(x)^{\beta_{j}}dx}dx.\end{split}$
(15)

Thus, to estimate $r_{j}$ we just have to sample $N$ points from the
distribution proportional to $\pi(x)^{\beta_{j}}$ and restricted to $S$. Then,

$r_{j}\approx\frac{1}{N}\sum_{i=1}^{N}\frac{\pi(x_{i})^{\beta_{j-1}}}{\pi(x_{i})^{\beta_{j}}}$
(16)

as $N$ grows. The key for an efficient approximation of $r_{j}$ using Monte
Carlo integration is to set $\beta_{j},\ \beta_{j+1}$ such that the variance
of $r_{j}$ is as small as possible (ideally a constant) for $N$ as small as
possible. In particular, In [26], they prove that the sequence of
$\beta_{1},\dots,\beta_{k}$ can be fixed such that the variance of each
$r_{j},\ j\in[k]$ is bounded by a constant. Moreover, $N=O^{*}(\sqrt{n})$
points per integral ratio $r_{j}$ and $k=O^{*}(\sqrt{n})$ ratios in total
suffices to approximate each $\int_{S}\pi_{i}(x)dx,i\in[M]$ within error
$\epsilon$. Thus, $O^{*}(n)$ points suffices to estimate each
$\int_{S}\pi_{i}(x)dx$.

######  Lemma 6.

Let the probability density function $\pi(x)$ in the Definition 4 be a mixture
of $M$ log-concave densities. The integral ratio in Equation (16) can be
estimated with $O^{*}(n)$ samples from $\pi(x)^{\beta_{j}}$ within error
$\epsilon$. Thus, the portfolio score in Equation (8) can be estimated using
$O^{*}(Mn)$ samples.

To sample from each target distribution proportional to $\pi(x)^{\beta_{j}}$
and restricted to $S$, in [26] they use Hit-and-Run random walk [42]. This
implies a total number of $O^{*}(n^{4})$ arithmetic operations per generated
point. Thus the total number of arithmetic operations to estimate the score
$s$ is $O^{*}(Mn^{5})$. In our implementation, to sample from a log-concave
distribution supported on $P$, we use the reflective Hamiltonian Monte Carlo
in [1] which is more efficient in practice than Hit-and-Run. For an extended
introduction to geometric random walks we suggest [42].

##  5 Mixed strategies

An important question is how one could set the risk and dispersion parameters
$q_{i},\ \alpha_{ij}$ and the weight $w_{ij}$ of each allocation strategy
$F_{\pi_{q_{i},\alpha_{ij}}}$ in a certain stock market. The issue is that our
knowledge about the stock market and the behavior of the investors in it might
be weak or vary from a time period to another.

In this section, we provide practical methods to set the parameters of a
sequence of log-concave distributions. We also present different versions of
the score than those given in Definition 4. We detail the computational
methods of this section in Appendix A.

###  5.1 Set the levels of dispersion

Let the concave function $h(x):P\rightarrow{\mathbb{R}}$, where
$P\subset{\mathbb{R}}^{n}$ the set of portfolios. Also let the log-concave
probability density function,

$\pi_{\alpha}(x)\propto e^{\alpha h(x)},\ \alpha>0,$ (17)

supported also on $P$ and $\tilde{x}\in P$ the mode of $\pi_{\alpha}$. Recall
that small/large values of $\alpha$ correspond to large/small values of
variance of $\pi_{\alpha}$. Thus, first we compute a value $\alpha_{L}$ such
that $F_{\pi_{\alpha_{L}}}$ is a $e$-dispersed allocation strategy; that is
the distribution $\pi_{\alpha_{L}}$ is e-close to the uniform distribution
according to the $L_{2}$ norm. Second, we compute a value $\alpha_{U}$ such
that the mass of the distribution $\pi_{\alpha_{U}}$ is almost entirely
concentrated in a ball $B(\tilde{x},\delta)$, that is a ball centered at the
mode $\tilde{x}$ and with a small radius $\delta>0$. Then, we compute a
sequence $\alpha_{L}=\alpha_{1}<\dots<\alpha_{k}=a_{U}$ such that,

$\|\pi_{\alpha_{i+1}}/\pi_{\alpha_{i}}\|=\|\pi_{\alpha_{i}}/\pi_{\alpha_{i-1}}\|,\
i\in\\{2,\dots,k-1\\}.$ (18)

To compute $\alpha_{L}$ we start with $\alpha_{0}=1$ and we use the annealing
schedule in [10]. In particular we generate the sequence,

$\alpha_{i}=\alpha_{0}\bigg{(}1-\frac{1}{n}\bigg{)}^{i},\ i\in\mathbb{N_{+}}.$
(19)

This schedule guarantees that a sample from $\pi_{\alpha_{i}}$ is a warm start
to sample from $\pi_{\alpha_{i+1}}$ for several random walks [26, 10] and
moreover, the variance of the distribution which is proportional to
$e^{(\alpha_{i+1}-\alpha_{i})h(x)}$ is $O(1)$; that is, each jump to the next
distribution in the sequence is ”small”. Next, for each $\alpha_{i}$ we
estimate the $L_{2}$ norm of $\pi_{\alpha_{i}}$ w.r.t. the uniform
distribution, by sampling from $\pi_{\alpha_{i}}$. We stop when the norm is
smaller than a given threshold.

To compute $\alpha_{U}$ we use the same annealing schedule, but now we
generate an increasing sequence,

$\alpha_{i}=\alpha_{0}\bigg{(}1+\frac{1}{n}\bigg{)}^{i},\ i\in\mathbb{N_{+}}.$
(20)

We stop when we meet the smallest $i$ such that the $100(1-\epsilon)\%$ of the
mass of $\pi_{\alpha_{i}}$ is inside the ball $B(\tilde{x},\delta)$ with high
probability. We probabilistically guarantee this by sampling a sufficiently
large number of points from $\pi_{\alpha_{i}}$ and by splitting the sample to
$\nu$ sub-samples. For each sub-sample we compute the ratio of points that lie
in $B(\tilde{x},\delta)$; that is we obtain $\nu$ ratios. Then, we perform a
t-test using those ratios while the null hypothesis states that the overall
ratio is larger than $(1-\epsilon)$. We stop for an $\alpha_{i}$ that results
in rejecting the null hypothesis.

Finally, to compute a sequence of equidistant distributions as in Equation
(18), we estimate
$d=\max\limits_{i\in[k-1]}\\{\|\pi_{\alpha_{i+1}}/\pi_{\alpha_{i}}\|\\}$.
Then, we start from $\alpha_{1}=\alpha_{L}$. Given $\alpha_{i}$, to compute
the next value of parameter in the sequence, namely $\alpha_{i+1}$, we perform
bisection method in the interval $[\alpha_{i},\alpha_{U}]$ to compute a value
such that the $L_{2}$ norm of $\pi_{\alpha_{i+1}}$ w.r.t. $\pi_{\alpha_{i}}$
is $d\pm\epsilon$ with a high probability and a small $\epsilon>0$. We stop
when we compute an $\alpha_{i}>\alpha_{U}$ and we set $\alpha_{k}=\alpha_{i}$.
To select $M$ values of $\alpha$ we pick $\alpha_{1}$ and $\alpha_{k}$ and
then, we equidistantly pick $M-2$ values in between them.

###  5.2 Set the levels of risk

Our practical method computes a sequence $q_{1}<\dots<q_{M}$. The values
$q_{i}$ are equidistant concerning the portfolio volatility that each $q_{i}$
corresponds to. First, we compute the minimum and the maximum value of
portfolio volatility. The first one is also called Global Minimum Variance
portfolio [43]. In particular, we solve the following optimization problems,

$\min/\max\ x^{T}\tilde{\Sigma}x^{T},\ x\in P,\
\tilde{\Sigma}\in{\mathbb{R}}^{n\times n}\text{ pos. def.}$ (21)

where $P$ is the set of portfolios and $\tilde{\Sigma}$ is an estimation of
the covariance using the shrinkage estimate in [21]. Let the values of the
minimum and the maximum portfolio volatility $v_{\min}$ and $v_{\max}$
respectively. To compute $M$ values of the parameter $q$, we equidistantly
select $M$ values of portfolio volatility
$v_{\min}<v_{1}<\dots<v_{M}<v_{\max}$. Then, for each $v_{i}$ we perform a
bisection method in a proper interval $[q_{\min},q_{\max}]$ to compute a
$q_{i}$ such that,

$|\min\limits_{x\in P}\phi_{q_{i}}(x)-v_{i}|\leq\epsilon,\ i\in[M],$ (22)

for a sufficiently small value of $\epsilon>0$, while $\phi_{q}(x)$ is the
expected quadratic utility function in Equation (10). In particular, for each
$q_{i}$ we search in $[q_{i-1},q_{M}],\ i\in\\{2,\dots,M-1\\}$; for $q_{1}$ we
search in $[0,q_{M}]$. To compute $q_{M}$ we search for the smallest non-
negative integer $j$ such that $\min\limits_{x\in P}\phi_{2^{j}}(x)>v_{M}$.
Then, we perform a bisection method in $[0,2^{j}]$ to compute $q_{M}$.

###  5.3 Set the composition of the investors

The computation of both sequences of $q$ and $\alpha$ allow to specify the
sequence of log-concave distributions,

$\pi_{ij}=e^{-\alpha_{ij}\phi_{q_{i}}(x)},\ i\in[M_{1}],\ j\in[M_{2}],$ (23)

where we assume that for each level of risk $q_{i}$ we have $M_{2}$ levels of
dispersion. However, to determine a mixed strategy one has to determine the
weights $w_{ij}$ in the corresponding mixture distribution. We recall that
each $w_{ij}$ implies the proportion of investors that build their portfolios
according to the allocation strategy induced by $\pi_{ij}$. Setting $w_{ij}$
forms the mixed strategy $F_{\pi}$ while the score in Definition 4 becomes,

$s=\sum_{i=1}^{M_{1}}\sum_{j-1}^{M_{2}}w_{ij}\int_{S}\pi_{ij}(x)dx,\ S:=P\cap
H(R^{*}).$ (24)

First, we allow setting additional bounds on $w_{ij}$. For example, one would
provide an upper/lower bound on the proportion of the investors who chose a
specific allocation strategy. In particular, let us assume that we estimate
the $M=M_{1}M_{2}$ integrals of Equation (24) as described in Sec. 4.2. $M$ is
the total number of allocation strategies in a certain stock market. Then, let
the $M$ values to form a vector $c\in{\mathbb{R}}^{M}$. Also let the
corresponding weights $w_{ij}$ in Equation (24) to form a vector
$w\in{\mathbb{R}}^{M}$ such that the score,

$s=\langle c,w\rangle,$ (25)

where $\langle\cdot,\cdot\rangle$ denotes the inner product between two
vectors. Given a matrix $A\in{\mathbb{R}}^{N\times M}$ and a vector
$b\in{\mathbb{R}}^{N}$, let the following feasible region of weights,

$Q=\left\\{w\in{\mathbb{R}}^{M}\ \bigg{|}\ Aw\leq b,\ w_{i}\geq 0,\
\sum_{i}^{M}w_{i}=1\right\\}\subset{\mathbb{R}}^{M}$ (26)

The matrix $A$ and the vector $b$ used to express $N$ further constraints on
the weights (e.g. lower, upper bounds or any linear constraint on $w_{ij}$).
Notice that if no further constraints are given on the weights, then the
feasible region $Q$ is the canonical simplex $\Delta^{M-1}$.

Now let us define three new versions of the score $s$ in Equation (24).

Let the weights $w\in Q$, where $Q\subset{\mathbb{R}}^{M}$ the feasible region
in Equation (26). 1. min score, $s_{\min}:=\min\langle c,w\rangle,\
\text{subject to }Q$. 2. max score, $s_{\max}:=\max\langle c,w\rangle,\
\text{subject to }Q$. 3. mean score,
$\bar{s}:=\frac{1}{\mbox{vol}(Q)}\int_{Q}\langle c,w\rangle\ dw$.

For the scores $s_{\min}$ and $s_{\max}$ one has to solve a linear program for
each one of them. The score $\bar{s}$ requires the computation of an integral
which can be computed with MCMC integration employing uniform sampling from
$Q$; otherwise, it can be reduced to the computation of the volume of a convex
polytope since $\langle c,w\rangle$ is a linear function of $w$ with the
domain being the set $Q$.

Let $w_{1}\in Q$ such that the min score $s_{\min}=\langle c,w_{1}\rangle$.
The weights denoted by the vector $w_{1}$ imply the proportions of the
investors that select each allocation strategy such that the portfolio score
$s$ takes its possibly minimum value. Similarly, the vector of weights
$w_{2}\in Q$ such that the max score $s_{\max}=\langle c,w_{2}\rangle$,
implies the proportions of the investors that select each allocation strategy
such that the portfolio score $s$ takes its possibly maximum value. Moreover,
it is easy to notice that the mean score $\bar{s}=\langle c,\bar{w}\rangle$,
while the vector of weights $\bar{w}$ is the center of mass of $Q$. For
example, if $Q=\Delta^{M-1}$ (i.e. the case where no further constraints are
given on the weights) the vector $\bar{w}$ is the equally weighted vector.

However, one may have additional knowledge on how the investors tend to behave
in a certain stock market, i.e. which allocation strategies they tend to
select. We also allow for these degrees of freedom by providing the notion of
behavioral functions in our context.

####  5.3.1 Behavioral functions

In this section, we assume that we are given a set of functions that
represents the knowledge, that one may have, related to which allocation
strategies the investors tend to select in a certain stock market and time
period. We assume that we are given $M_{1}+1$ functions $f_{q},f_{\alpha,i},\
i\in[M_{1}]$ with the domain being $[0,1]$ for all of them. We call these
functions behavioral functions and we use them to create a vector of weights
$w\in{\mathbb{R}}^{M}$, that emphasizes specific strategies, where
$M=M_{1}M_{2}$ is the total number of allocation strategies that take place in
the stock market. More specifically, $f_{q}$ declares the level of risks that
the investors tend to select, while $f_{\alpha,i}$ declares the level of
dispersion that the investors’ portfolios –who select risk $q_{i}$– tend to
have around the formal allocation proposal.

Figure 7: Examples of behavioral functions.

The plots in Fig. 7 demonstrate $4$ possible choices of such functions. For
example, if plot $C$ is $f_{q}$ then the investors tend to select low-risk
investments; the value of $f_{q}$ is high for small values of $q$ (low risk)
and low for high values of $q$ (high risk). In addition, if the plot $D$ is
$f_{\alpha,i}$ then, the portfolios of the investors who select risk $q_{i}$
tend to be highly stuck around the formal allocation proposal that corresponds
to $q_{i}$; the value of $f_{\alpha,i}$ is large for large values of $\alpha$
(low dispersion) and small for small values of $\alpha$ (high dispersion).

To compute a weight vector $w$ we map the intervals $[a_{i1},\alpha_{iM2}]$
and $[v_{\min},v_{\max}]$ we compute in Sec. 5.1 and Sec. 5.2 onto $[0,1]$, by
using the following transformation:

$z(t)=\frac{1}{d-c}(t-c),\ t\in[c,d].$ (27)

Throughout this paper, when we write $z(\cdot)$ we assume that the interval
$[c,d]$ is defined properly according to the input.

The following pseudocode describes how we compute such a weight vector when
$M_{1}+1$ behavioral functions are given.

Compute the weight vector $w$
Input: risk and dispersion parameters $q_{i}$ and $\alpha_{ij}$,
$i\in[M_{1}],\ j\in[M_{2}]$
computed as in Sec. 5.1 and 5.2 and $M_{1}+1$ behavioral
functions$f_{q},\ f_{\alpha,i}$. 1. Let the portfolio volatilities
$v_{i}=\min\limits_{x\in P}\phi_{q_{i}}(x),\ i\in[M_{1}]$. 2. For each pair of
$(i,j)$ set $r_{(i-1)M_{1}+j}\leftarrow
f_{q}(z(v_{i}))f_{\alpha,i}(z(\alpha_{ij}))$. 3. Normalize the vector
$r_{j}\leftarrow r_{j}/\sum_{i=1}^{M}r_{i},\ j\in[M]$ and $M=M_{1}M_{2}$. 4.
Set the weight vector $w\leftarrow r$.

Given the behavioral functions, one could use the vector of weights
—determined as in the above pseudo-code— and then, the portfolio score is
$s=\langle c,w\rangle$, while $c\in{\mathbb{R}}^{M}$ is the vector that
contains the values of the integrals in Equation (24).

####  5.3.2 Parametric score

In this section, we assume a weaker knowledge of how the investors tend to
behave than that in Sec. 5.3.1. Thus, we do not explicitly determine the
vector of weights $w\in{\mathbb{R}}^{M}$, where $M$ is the number of
allocation strategies in a certain stock market. In particular, let the
coordinates of the vector $r\in{\mathbb{R}}^{M}$ as in Sec. 5.3.1,

$r_{(i-1)M_{1}+j}\leftarrow f_{q}(z(v_{i}))f_{\alpha,i}(z(\alpha_{ij})),\
i\in[M_{1}],\ j\in[M_{2}]$ (28)

where $f_{q},\ f_{\alpha,i}$ the $M_{1}+1$ behavioral functions. Then, we use
the vector $r$ to denote a bias on the investors’ behavior. First, we again
allow further bounds and linear constraints on the weights. That is, we let
the feasible region of the weights to be the set $Q$ in Equation (26). To
denote the bias on the investors’s behavior, we employ the exponential
distribution,

$p_{T}(w)\propto e^{\langle r,w\rangle/T},\ T>0,$ (29)

with the support of $p_{T}(w)$ being the set $Q\subset{\mathbb{R}}^{M}$.

The distribution $p_{T}(w)\propto e^{\langle r,w\rangle/T}$ is usually called
Boltzmann distribution and the vector $r$ bias vector. In general, the
Boltzmann distribution gives the probability that a system will be in a
certain state as a function of that state’s energy and the temperature $T$ of
the system. The bias vector $r$ determines how the mass tends to distribute in
$Q$ and the temperature parameter $T$ how strong the bias denoted by $r$ is.

For example, when $Q$ is the canonical simplex $\Delta^{M-1}$, the mass of
$p_{T}$ tends to concentrate around the vertices which correspond to the
coordinates of $r$ with larger values than the other coordinates. Moreover, as
the temperature $T\rightarrow 0$ this tendency becomes stronger until almost
all the mass concentrates around the vertex which corresponds to the
coordinate of the largest value of $r$. As $T\rightarrow\infty$, $p_{T}$
converges to the uniform distribution and the bias denoted by $r$ disappears.

We intend to use the temperature $T$ to parameterize how strong the tendency
on the investors’ behavior, that the bias vector $r$ implies, is. Then the
parametric score is given as,

$\begin{split}&s(T):=\int_{S}\langle c,w\rangle\ p_{T}(w)dw,\ \text{where
}p_{T}(w)\propto e^{\langle r,w\rangle/T},\ T>0\\\ &\text{ and each coordinate
}\quad r_{(i-1)M_{1}+j}=f_{q}(z(v_{i}))f_{\alpha,i}(z(\alpha_{ij})),\\\
&\text{ and }v_{i}=\min\limits_{x\in P}\phi_{q_{i}}(x),\ i\in[M_{1}],\
j\in[M_{2}]\end{split}$ (30)

Let the center of mass $\bar{w_{T}}$ in $Q$ when the mass is distributed
according to $p_{T}(w)$. Notice that $\bar{w}_{T}$ can be seen as a parametric
curve on $T$. Furthermore, it is easy to notice that, for fixed $T$, the
parametric score $s(T)=\langle c,\bar{w}_{T}\rangle$. Thus, the score $s(T)$
is evaluated on that parametric curve. Following these observations, we are
ready to state the following Lemma.

######  Lemma 7.

Let a stock market with $M$ allocation strategies. Assume that we are given
the parameters $q_{i},\ \alpha_{ij}$ of Sec. 5.1 and 5.2 and any behavioral
functions $f_{q},\ f_{\alpha_{i}},\ i\in[M_{1}],\ j\in[M_{2}]$ and
$M=M_{1}M_{2}$ the number of allocation strategies that take place in the
stock market. Let the feasible set $Q\subset{\mathbb{R}}^{M}$ of the weights
as in Equation (26), the min score $s_{\min}$, the max score $s_{\max}$ and
the mean score $\bar{s}$ in Sec. 5.3 and the parametric score in Equation
(30). Then, the followings hold,

$\begin{split}&s_{\min}\leq s(T)\leq s_{\max},\ \forall T>0,\\\
&\bar{s}=\lim_{T\rightarrow\infty}s(T)\end{split}$ (31)

Notice that the Equation (31) holds for any set of behavioral functions as the
scores $s_{\min},\ s_{\max}$ always bound the parametric score. Furthermore,
when $T\rightarrow\infty$ the distribution $p_{T}(w)$ converges to the uniform
distribution over the feasible region of the weights $Q$ and thus the
parametric score is equal to the mean score $\bar{s}$.

To obtain the parametric score we compute a sequence of temperatures $T_{i}$
that correspond to a sequence of exponential distributions $p_{T_{i}}$.
Similarly to Sec. 5.1, we compute two temperatures $T_{\max}$ and $T_{\min}$.
The $L_{2}$ norm of $p_{T_{\max}}$ w.r.t. the uniform distribution over $Q$ is
smaller than a given threshold and the $100(1-\epsilon)\%$ of the mass of
$p_{T_{\min}}$ is inside a ball of a small radius $\delta>0$, centered at the
mode of $p_{T_{\min}}$. Then, we use the sequence,
$T_{i}=T_{\max}(1-\frac{1}{M})^{i}$, $i\in\mathbb{N}_{+}$, $T_{i}\geq
T_{\min}$ and the method in Sec. 5.1 to compute an equidistant –with respect
to $L_{2}$ norm– sequence of exponential distributions.

##  6 Simulations on allocation strategies

In this section, we take the set of portfolios $P\subset{\mathbb{R}}^{n}$ to
be the canonical simplex, $\Delta^{n-1}$, which means that we consider long-
only portfolios. We illustrate the usefulness of the new score in analyzing
the performance of a portfolio allocation given the asset returns. We also
compare it to several well-known portfolio scores. Moreover, we consider the
score of a given portfolio as a random variable in a stock market, where the
asset returns follow a multivariate distribution. We also illustrate how our
modeling of allocation strategies could be used to study the state of a stock
market, computing the copula between portfolios’ return and volatility while
the portfolios have been build according to a mixed strategy.

###  6.1 Portfolio score

In our simulation, we consider the daily returns of the $12$ cryptocurrencies
with the longest history, reported in Table 6. To illustrate our new score, we
consider a pseudo-real time example, where we take $100$ consecutive asset
returns from 22/10/2016 until 29/01/2017. We compute the optimal Mean-Variance
(MV) portfolio using the shrinkage estimate of the covariance matrix of [21],
while we fix its volatility equal to the average in-sample volatility of the
long-only portfolios. We also compute the Equally-weighted Risk Contributions
(ERC) portfolio by [27] which also uses the shrinkage estimator and the
Bitcoin (BTC) portfolio. For the sake of completeness we report the estimated
covariance matrix and the average assets’ return for the period of $100$ days,
that we used to compute the MV portfolio, in Appendix B. We also report the
three portfolios in Table 1.

Portfolios | BTC | LTC | ETH | XRP | XMR | USDT
---|---|---|---|---|---|---
MV | 0% | 58.5% | 0% | 0% | 0% | 0%
ERC | 8.94% | 5.81% | 6.81% | 16.18% | 7.02% | 7.70%
BTC | 100% | 0% | 0% | 0% | 0% | 0%
Portfolios | DASH | XLM | DOGE | DGB | XEM | SC
MV | 0% | 9.98% | 0% | 1.15% | 30.39% | 0%
ERC | 6.38% | 5.80% | 8.82% | 3.88% | 5.89% | 16.77%
BTC | 0% | 0% | 0% | 0% | 0% | 0%
Table 1: The Mean-variance (MV) optimal portfolio, the Equally-weighted Risk
Contributions (ERC) portfolio and the Bitcoin portfolio (BTC).

To evaluate those portfolios we take the average of the $10$ vectors of
assets’ returns after the $100$ daily asset returns. We report this vector of
assets’ returns in B. Together with our score, we report Jensen’s alpha,
Sharpe ratio, Sortino ratio, and the cross-sectional score in [13]. The latter
is equal to the proportion of all possible allocations that our portfolio
outperforms. To compute the Jensen’s alpha and the Sortino ratio, we consider
the return of the Global Minimum Variance portfolio as the risk-free rate and
for the market portfolio, we set the equally weighted portfolio. To compute
the Sharpe ratio we also set the equally weighted portfolio as the benchmark
portfolio.

Portfolios | Jen. alpha | Sharpe r. | Sortino r. | UnBiasedSc
---|---|---|---|---
MV | -0.0026 | 0.070 | 1.38 | 70.2%
ERC | -0.0014 | -0.16 | 0.86 | 34.8%
BTC | -0.014 | -0.41 | 0.015 | 1.8%
Table 2: Four well known scores of the Mean-variance (MV) optimal portfolio,
the Equally-weighted Risk Contributions (ERC) portfolio and the Bitcoin
portfolio (BTC). From left to right: Jensen’s alpha, Sharpe ratio, Sortino
ratio, the cross sectional score in [13].

Regarding our new score, we study two scenarios that differ based on the
strategies that take place in the stock market. First, we take $3$ levels of
risk with $4$ levels of dispersion for each risk, that is $M=12$ strategies in
total. Second, we take $6$ levels of risk with $10$ level of dispersion for
each risk. In addition, we select $10$ levels of dispersion around the Bitcoin
portfolio. More precisely, consider the family of distributions,

$\pi(x)\propto e^{-\alpha(x-\bar{x})^{T}\Sigma(x-\bar{x})},\
x\in{\mathbb{R}}^{n},$ (32)

where $\bar{x}\in{\mathbb{R}}^{n}$ is the Bitcoin portfolio. Then, we compute
the sequence of dispersion as in Sec. 5.1. That is $M=70$ strategies in total.
In both cases, we do not impose any additional constraint for the proportion
of the investors that select a specific strategy. This means that the set of
weights $Q\subset{\mathbb{R}}^{M}$, which determines the composition of the
investors in the stock market, is the canonical simplex $\Delta^{M-1}$.
Considering the behavioral functions, for the risk we consider three cases:
(i) plot B with $x_{0}=1/2$, (ii) plot C and (iii) plot D in Fig. 7. The
function in case (i) favors strategies with medium level of risk, the function
in case (ii) favors strategies with low level of risk, and (iii) favors
strategies with high level of risk. Throughout this section for the dispersion
of each risk, we consider only the case of plot D in Fig. 7, which favors
strategies of low dispersion around the formal allocation proposal. For all
the behavioral functions, we set the ratio between its maximum over its
minimum value equal to $10$.

| 12 Allocation Strategies | 70 Allocation Strategies
---|---|---
Portfolios | $\bar{s}$ |  $s$ (HR) |  $s$ (MR) |  $s$ (LR) | $\bar{s}$ |  $s$ (HR) |  $s$ (MR) |  $s$ (LR)
MV | 67.6% | 57.4% | 72.9% | 82.1% | 71.2% | 53.7% | 71.6% | 81.0%
ERC | 22.1% | 13.7% | 19.8% | 29.9% | 33.7% | 22.7% | 32.2% | 41.8%
BTC | 0.20% | 0.07% | 0.06% | 0.08% | 0.42% | 0.14% | 0.21% | 0.25%
Table 3: The scores of the Mean-variance (MV) optimal portfolio, the Equally-
weighted Risk Contributions (ERC) portfolio, and the Bitcoin portfolio (BTC)
when $12$ and $70$ strategies take place in the stock market. $\bar{s}$ stands
for the mean score; $s$ (HR) stands for the score when the behavioral function
of the risk is given by plot D in Fig. 7 (favors high risk strategies); $s$
(MR) stands for the score when the behavioral function of the risk is given by
plot B in Fig. 7 (favors medium risk strategies); $s$ (LR) stands for the
score when the behavioral function of the risk is given by plot C in Fig. 7
(favors low risk strategies). For all cases, the behavioral function of the
dispersion is given by plot D in Fig. 7, i.e. it favors strategies of low
dispersion around the formal allocation proposal.

In Table 2 we report the values of the existing portfolio scores. All scores
agree that the performance of the MV portfolio is better than both ERC’s and
BTC’s. They all also agree that ERC’s performance is better than BTC’s.
Moreover, MV is the only portfolio that outperforms the equally weighted
portfolio. The cross-sectional score in [13] informs us that MV outperforms
the $70.2\%$ of all possible portfolios, ERC the $34.8\%$ and BTC outperforms
only the $1.8\%$ of all possible portfolios. In Table 3 we report the new
score when $M=12$ or $M=70$ allocation strategies take place in the stock
market. We report the mean score $\bar{s}$, which is the score when the
investors are equally divided among the allocation strategies and the scores
for the three different choices of the risk’s behavioral function, using the
weight vector $w$ of Sec. 5.3.1.

|  | Risk’s behavioral function
---|---|---
Strategy risk | High risk (Plot D) | Medium risk (Plot B) | Low risk (Plot C)
(proposal vol.) | $w_{1}$ | $w_{2}$ | $w_{3}$ | $w_{1}$ | $w_{2}$ | $w_{3}$ | $w_{1}$ | $w_{2}$ | $w_{3}$
1.1% | $\alpha_{\max}$ | 8.41% | 6.14% | 1.56% | 10.17% | 11.47% | 3.07% | 12.27% | 45.94% | 85.47%
$\dots$ | 7.66% | 4.66% | 1.14% | 7.93% | 4.36% | 1.11% | 8.60% | 6.08% | 1.64%
$\dots$ | 7.45% | 4.09% | 1.05% | 7.33% | 3.48% | 0.84% | 7.80% | 4.35% | 1.16%
$\alpha_{\min}$ | 7.46% | 4.11% | 1.02% | 7.14% | 3.31% | 0.81% | 7.60% | 3.94% | 1.08%
1.9% | $\alpha_{\max}$ | 9.78% | 10.33% | 2.65% | 12.29% | 42.03% | 85.31% | 9.86% | 9.47% | 2.68%
$\dots$ | 8.48% | 5.57% | 1.43% | 8.85% | 6.20% | 1.46% | 8.54% | 5.31% | 1.42%
$\dots$ | 7.87% | 4.42% | 1.15% | 7.62% | 3.93% | 0.95% | 7.66% | 4.11% | 1.13%
$\alpha_{\min}$ | 7.54% | 4.18% | 1.03% | 7.09% | 3.41% | 0.85% | 7.40% | 3.73% | 0.99%
2.7% | $\alpha_{\max}$ | 12.34% | 43.33% | 85.63% | 10.26% | 11.70% | 3.11% | 8.47% | 5.75% | 1.51%
$\dots$ | 7.70% | 4.45% | 1.14% | 7.18% | 3.46% | 0.85% | 7.28% | 3.84% | 0.98%
$\dots$ | 7.69% | 4.40% | 1.12% | 7.07% | 3.34% | 0.84% | 7.29% | 3.77% | 0.98%
$\alpha_{\min}$ | 7.64% | 4.33% | 1.09% | 7.08% | 3.30% | 0.80% | 7.23% | 3.72% | 0.96%
MV score | 65.4% | 43.9% | 11.1% | 68.8% | 78.0% | 94.4% | 70.5% | 86.2% | 96.6%
ERC score | 20.9% | 12.8% | 3.2% | 21.5% | 15.1% | 3.8% | 23.8% | 40.6% | 61.7%
BTC score | 0.18% | 0.13% | 0.02% | 0.19% | 0.01% | 0.02% | 0.18% | 0.11% | 0.02%
Table 4: The scores of the Mean-variance (MV) optimal portfolio, the Equally-
weighted Risk Contributions (ERC) portfolio, and the Bitcoin portfolio (BTC)
for various weight vectors $w_{i}$, when 12 strategies take place in the
market. Each subset of rows corresponds to a set of allocation strategies with
a certain risk level and each row to an allocation strategy with a certain
dispersion level. Each weight in $w_{i}$ gives the percentage of investors who
select the corresponding allocation strategy. Each column block corresponds to
a different risk behavioral function and a bias vector $r$ according to Eq.
28, each plot refers to Fig. 7; each weight vector in a column block
corresponds to the center of mass of the distribution $p_{T}(w)\propto
e^{\langle r,w\rangle/T}$ supported on $Q$ for different values of temperature
$T$. For each weight vector, we report the score of the portfolios.

For all portfolios, the score increases while the investors tend to select
allocation strategies with s lower level of risk. The performance of the MV
portfolio is very similar in both cases of $M=12$ and $M=70$. ERC portfolio
performs better when $M=70$ as each score is about $10\%$ larger than the case
of $M=12$. BTC also performs better when $M=70$. However, the score is quite
small in both cases. Comparing to the unbiased case of the score in [13], it
is clear that the value of our score is affected by the investors’ composition
and can be higher or smaller than the score in [13].

Figure 8: The parametric scores of the Mean-variance (MV) optimal portfolio,
the Equally-weighted Risk Contributions (ERC) portfolio, and the Bitcoin
portfolio (BTC) for different risk’s behavioral functions (high, medium, low
risk) and $M=12$ strategies in the stock market. We report the score for the
$i$-th temperature in the sequence we compute according to Sec. 5.3.2. With
different colors, we mark different choices for the risk’s behavioral
function. Plot B, C, D refer to Fig. 7.

In Table 4, we further illustrate how the performance of each portfolio is
related to the investors’ composition in the stock market. We consider the
case of $M=12$ and for each risk’s behavioral function, we compute the bias
vector $r$ of Sec. 5.3.2. For each bias vector, we report three different
weight vectors and the corresponding scores. Recall that a weight vector
represents the investor’s composition in a stock market. In particular, we set
three different temperatures $T$ in the distribution $p_{T}(w)\propto
e^{\langle r,w\rangle/T}$. Then, for each temperature, we estimate the center
of mass of $p_{T}$. In each column block and from left to right, we decrease
the temperature, and thus, we strengthen the tendency implied by the bias
vector $r$. For all portfolios, as the percentage of investors who select
strategies with a high level of risk increases, their score drastically
decreases. When the percentage of the investors who select strategies with a
medium level of risk increases, the performance of the MV portfolio improves
while both ERC’s and BTC’s decrease. When the percentage of investors who
select strategies with a low level of risk increases both the MV’s and ERC’s
scores increase while BTC’s score decreases. Moreover, BTC’s score is always
smaller than 0.2% which implies a quite poor performance.

The plots in Fig. 8 and Fig. 9 illustrate a comparison between the three
portfolios using their parametric scores. For both sets of allocation
strategies, the score of MV is always higher than the scores of ERC and BTC.
When the percentage of investors who select strategies with a high level of
risk increases the three parametric scores converge as they all go to zero.
When the investors tend to select strategies with a medium or a low level of
risk we have a major change in the performance of ERC when $M=70$. For
example, for medium risk and $M=12$ the parametric score of ERC converges to
0, while for $M=70$ it converges to 1. This is an example of how the
(parametric) score can change as the number of allocation strategies in the
stock market also changes.

Figure 9: The parametric scores of the Mean-variance (MV) optimal portfolio,
the Equally-weighted Risk Contributions (ERC) portfolio, and the Bitcoin
portfolio (BTC) for different risk’s behavioral functions (high, medium, low
risk) and $M=70$ strategies in the stock market. We report the score for the
$i$-th temperature in the sequence we compute according to Sec. 5.3.2. With
different colors, we mark different choices for the risk’s behavioral
function. Plot B, C, D refer to Fig. 7.

###  6.2 The score as a random variable

Once a portfolio is chosen and assuming a distribution for the asset returns,
one can estimate the distribution of the scores of this portfolio. This
distribution allows us to understand the risk for this portfolio to perform
worse, or better, than a mixed strategy. This estimation is obtained as
follows. First, we draw randomly $10^{4}$ vectors of asset returns. Then we
compute the corresponding scores using our implementation. Finally, we
estimate the distributions of the score by a normal kernel function bounded in
$[0,1]$. Moreover, considering the parametric score we compute a sequence of
distributions of scores. That is one distribution for each temperature in
Equation (30).

We show how the mixed strategy in the stock market affects the distribution of
the score of a given portfolio. We work under the assumption that the asset
returns follow a multivariate Gaussian distribution $\mathcal{N}(\mu,\Sigma)$.
We use the same covariance matrix and mean vector as in Sec. 6.1.

Figure 10: The distributions of scores of the Mean-variance (MV) optimal
portfolio for different risk’s behavioral functions (high, medium, low risk)
and a sequence of temperatures $T_{i}$ such that the distributions in the
sequence $p_{T_{i}}{w}\propto e^{\langle r,w\rangle/T_{i}}$ are equidistant
w.r.t. $L_{2}$ norm. Plot B, C, D refer to Fig. 7.

In our example, we focus on the MV portfolio in Table 1 considering the case
of the $M=70$ allocation strategies of Sec. 6.1. For each risk’s behavioral
function we use the same temperatures as in Fig. 9, where we compute the
parametric scores. Thus, the plots in Fig. 10 illustrate how the distribution
of score of the MV portfolio changes as the tendency of the investors’
behavior –induced by the behavioral functions– increases.

When the investors are equally divided among allocation strategies, the
distribution of score is bimodal with the modes being around $0.2$ and $0.8$.
The latter implies that –for this investors’ composition– the score of MV has
a high probability to be around 0.8 or 0.2. As the percentage of investors who
select strategies with a medium level of risk increases the modes are shifting
to the extreme values, e.g. around 0 and 1. Moreover, the total mass over the
modes increases. Also, the mass over the mode of the largest value gets larger
than the mass over the mode of the smallest value. Thus, the probability of MV
to achieve a good score increases as the temperature $T$ decreases. On the
other side, the MV has a high probability to be either among the best or worst
performers, which implies that it is also quite risky w.r.t. to that score.
The same occurs as the percentage of investors who select strategies with a
low level of risk increases.

However, as the percentage of investors who select strategies with a high
level of risk increases, the modes are shifting towards the opposite
directions than in the previous cases. Moreover, for the median temperature,
the distribution of the score becomes unimodal and it is centered at 0.5. As
the temperature further increases the modes are shifting towards the extreme
scores, e.g. 0 and 1. In this case, the mass over the mode of the largest
value gets smaller than the mass over the mode of the smallest value. Thus, as
the percentage of investors who select high-risk strategies increases the
probability of MV achieving a bad score increases. However, when the
investors’ composition is implied by the median temperature, MV has a very
small probability to be among the best or worst performers, as the mass is
concentrated around the score of 0.5. The latter make the MV portfolio a safe
(stable) choice w.r.t. that score.

###  6.3 Alternative Copulae

Notice that the analysis in Sec. 3 is agnostic on allocation strategies by
working directly with the set of portfolios. Thus, it uses uniform sampling
from the set of portfolios to estimate the copula between portfolios’ return
and volatility. In this section we compute copulae when the portfolios have
been build according to a mixed strategy. To be more precise, we consider the
case of $M=70$ allocation strategies as in Sec. 6.1, which are computed using
the covariance matrix and the mean asset returns estimated on the period of
$100$ days of Sec. 6.1. Then, we consider the next $60$ days after the set of
$100$ days, and –in that time period– we compute one copula per mixed
allocation strategy. To compute each copula we use sampling from the
corresponding mixture distributions –for more details see Appendix A.

The upper left plot in Fig. 11 illustrates the copula computed with uniform
sampling from the set of long-only portfolios as in Sec. 3. The indicator of
this copula is equal to $0.32$, which implies a normal relation between
portfolios’ return and volatility. We also obtain similar copulae when the
portfolios are built according to three different mixed strategies which
differ based on the risk’s behavioral function (high-medium-low level of
risk), as in the two previous sections and the weight vectors $w$ have been
computed as in Sec. 5.3.1. The indicators for high, medium, and low risk are
$0.002,0.004,0.01$ respectively.

Figure 11: The copulae of portfolios’ return and volatility. The copula in the
upper left plot (Unbiased case) is computed according to Sec. 3. We compute
each one of the orthe three copulae when the portfolios have been built
according to three different mixed allocation strategies, while $M=70$
strategies take place in the stock market. The mixed strategies differ based
on the risk’s behavioral function (high, medium, and low risk). Plot B, C, D
refer to Fig. 7.

On the other hand, in Fig. 12 we compute the bias vector for the risk’s
behavioral function given by plot B in Fig. 7, which favors allocation
strategies with medium risk. The temperature $T_{1}$ corresponds to the mixed
strategy with equally divided investors over the $M=70$ allocation strategies.
We notice that as the temperature decreases, i.e. the percentage of the
investors who select allocation strategies with a medium level of risk
increases, a large percentage of the mass of the copula is shifting towards
the corners on the down diagonal. More precisely, the indicators for
$T_{1}>\dots>T_{4}$ are $0.03,0.03,0.04,2.19$ respectively. The latter implies
that the percentage of portfolios with either low return and high volatility
or high return and low volatility increases as the tendency of the investors
to select medium-risk allocation strategies increases in the stock market.

Figure 12: The copulae of portfolios’ return and volatility. We compute each
copula for a certain mixed allocation strategy, while $M=70$ strategies take
place in the stock market. We compute the bias vector using the risk’s
behavioral function given by Plot B in Fig. 7, which favors allocation
strategies with a medium level of risk. We compute a copula per temperature,
where $T_{1}>\dots>T_{4}$ correspond to equidistant exponential distributions
$p_{T_{i}}$ w.r.t. $L_{2}$ norm and the temperature $T_{1}$ correspond to a
mixed strategy with equally divided investors over the allocation strategies.

##  7 Conclusions and future work

We briefly survey existing work on crises detection and we strengthen its
results employing clustering algorithms for bivariate distributions. We also
use it to detect all the past crash events in the cryptocurrency markets. This
problem motivates us to develop a new computational framework to model asset
allocation strategies in a stock market and to define a new portfolio score
based on that framework. Our simulations show that the informativeness of our
new score can be higher than that of existing portfolio performance scores. To
provide efficient computations we develop high dimensional MCMC samplers for
log-concave distributions supported on a convex polytope. Our sampler scales
up to a few hundred dimensions/assets. We simulate mixed strategies to
estimate the distribution of the portfolio’s score –assuming a distribution
for the asset returns– and to compute alternative copulae of portfolios’
return and volatility.

A possible future work is to use nonlinear shrinkage, e.g. that in [22]. An
additional direction would be to further study the state of a stock market
using the alternative copula computations in Sec. 6.3. Furthermore, we believe
that it would be of special interest to use the distribution of the new score
to define new performance measures and thus, compute the optimal portfolios
with respect to those measures. In particular, the problem reduces to compute
a portfolio with a “good” distribution of score. Considering the copula
computation in Sec. 3 when the set of portfolios is the fully-invested
portfolios we can not use the exact sampler in [37] because the portfolio
domain $P\subset{\mathbb{R}}^{n}$ is a generic convex polytope. Thus, MCMC
sampling is the sole option. The latter is computationally more expensive than
the method in [37]. An interesting piece of future work is to develop
specialized MCMC uniform samplers for fully-invested portfolios. Last but not
least, one could use the clustering methods of Sec. 3 to detect intermediate
states of a market.

### Acknowledgment

Sec. 3 of this paper surveys work in collaboration with Ludovic Calès (JRC
Ispra, Italy).

## References

  * [1] H.M. Afshar and J. Domke.  Reflection, refraction, and Hamiltonian Monte Carlo.  In Advances in neural information processing systems, pages 3007–3015, 2015.
  * [2] A. Banerjee and C.-H. Hung.  Informed momentum trading versus uninformed “naive” investors strategies.  J. Banking & Finance, 35(11):3077–3089, 2011.
  * [3] M. Billio, M. Getmansky, and L. Pelizzon.  Dynamic risk exposures in hedge funds.  Comput. Stat. & Data Analysis, 56(11):3517–3532, 2012.
  * [4] T. Bodnar, S. Dmytriv, N. Parolya, and W. Schmid.  Tests for the weights of the global minimum variance portfolio in a high-dimensional setting.  IEEE Trans. Signal Processing, PP, 10 2017.
  * [5] T. Bodnar, S. Mazur, and Y. Okhrin.  Bayesian estimation of the global minimum variance portfolio.  European J. Operational Research, 256, 06 2016.
  * [6] T. Bodnar, S. Mazur, K. Podgorski, and J. Tyrcha.  Tangency portfolio weights for singular covariance matrix in small and large dimensions: Estimation and test theory.  J. Statistical Planning and Inference, 201, 12 2018.
  * [7] T. Bodnar, S. Mazur, and K. Podgórski.  Singular inverse wishart distribution and its application to portfolio theory.  J. Multivariate Analysis, 143:314–326, 2016.
  * [8] T. Bodnar and W. Schmid.  Econometrical analysis of the sample efficient frontier.  European J. Finance, 15:317–335, 04 2009.
  * [9] L. Calès, A. Chalkis, I.Z. Emiris, and V. Fisikopoulos.  Practical volume computation of structured convex bodies, and an application to modeling portfolio dependencies and financial crises.  In B. Speckmann and C.D. Tóth, editors, Proc. Intern. Symp. Computational Geometry (SoCG), volume 99 of Leibniz Intern. Proc. Informatics (LIPIcs), pages 19:1–19:15, Dagstuhl, Germany, 2018.
  * [10] B. Cousins and S. Vempala.  Bypassing KLS: Gaussian cooling and an ${O}^{*}(n^{3})$ volume algorithm.  In Proc. ACM STOC, pages 539–548, 2015.
  * [11] M. Lo Duca, A. Koban, M. Basten, E. Bengtsson, B. Klaus, P. Kusmierczyk, J.H. Lang, C. Detken, and T. Peltonen.  A new database for financial crises in european countries.  Technical Report 13, European Central Bank & European Systemic Risk Board, Frankfurt, Germany, 2017.
  * [12] M. Grinblatt and S. Titman.  A Study of Monthly Mutual Fund Returns and Performance Evaluation Techniques.  J. Financial and Quantitative Analysis, 29(3):419–444, 1994.
  * [13] D. Guegan, L. Calès, and M. Billio.  A Cross-Sectional Score for the Relative Performance of an Allocation.  Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00646070, HAL, 2011.
  * [14] Mårten Gulliksson and Stepan Mazur.  An iterative approach to ill-conditioned optimal portfolio selection.  Computational Economics, 56:773–794, 12 2020.
  * [15] A.A. Harzallah and M.B. Abbes.  The impact of financial crises on the asset allocation: Classical theory versus behavioral theory.  J. Interdisciplinary Economics, 32(2):218–236, 2020.
  * [16] V. Ivanyuk.  Formulating the concept of an investment strategy adaptable to changes in the market situation.  Economies, 9(3), 2021.
  * [17] M.C. Jensen.  The Performance of Mutual Funds in the Period 1945-1964.  SSRN Scholarly Paper ID 244153, Social Science Research Network, Rochester, NY, May 1967.
  * [18] J. D. Jobson and B. Korkie.  Estimation for markowitz efficient portfolios.  Journal of the American Statistical Association, 75(371):544–554, 1980.
  * [19] Raymond Kan and Daniel R. Smith.  The distribution of the sample minimum-variance frontier.  Management Science, 54(7):1364–1380, 2008.
  * [20] Yoram Kroll, H. Levy, and H. Markowitz.  Mean-variance versus direct utility maximization.  Journal of Finance, 39:47–61, 1984.
  * [21] O. Ledoit and M. Wolf.  Honey, I shrunk the sample covariance matrix.  J. Portfolio Management, 30(4):110–119, 2004.
  * [22] Olivier Ledoit and Michael Wolf.  Analytical nonlinear shrinkage of large-dimensional covariance matrices.  The Annals of Statistics, 48(5):3043 – 3065, 2020.
  * [23] H. Levy and H. M. Markowitz.  Approximating expected utility by a function of mean and variance.  The American Economic Review, 69(3):308–317, 1979.
  * [24] A. Lo and P. Patel.  130/30: The new long-only.  J. Portfolio Management, 34, 12 2007.
  * [25] A.W. Lo.  The Statistics of Sharpe Ratios.  Financial Analysts Journal, 58:36–52, 2002.
  * [26] L. Lovasz and S. Vempala.  Fast algorithms for logconcave functions: Sampling, rounding, integration and optimization.  In Proc. IEEE Symposium Foundations of Computer Science (FOCS’06), pages 57–68, 2006.
  * [27] Sébastien Maillard, Thierry Roncalli, and Jérôme Teïletche.  The properties of equally weighted risk contribution portfolios.  The Journal of Portfolio Management, 36(4):60–70, 2010.
  * [28] H. Markowitz.  Portfolio selection.  J. Finance, 7(1):77–91, 1952.
  * [29] H. Markowitz.  The optimization of a quadratic function subject to linear constraints.  Naval Research Logistics Quarterly, 3(1‐2):111–133, 1956.
  * [30] S. Mazur, T. Bodnar, and K. Podgorski.  A test for the global minimum variance portfolio for small sample and singular covariance.  AStA Advances in Statistical Analysis, 101, 07 2017.
  * [31] R.B. Nelsen.  An Introduction to Copulas (Springer Series in Statistics).  Springer, Berlin, 2006.
  * [32] A.Y. Ng, M.I. Jordan, and Y. Weiss.  On spectral clustering: Analysis and an algorithm.  In Proc. 14th Intern. Conf. Neural Information Processing Systems: Natural and Synthetic, NIPS’01, pages 849–856, Cambridge, MA, USA, 2001\. MIT Press.
  * [33] D. Pappas and G. Kaimakamis.  Optimal portfolio selection with singular covariance matrix.  International Mathematical Forum, 5:2305–2318, 01 2010.
  * [34] Ofir Pele and Michael Werman.  Fast and robust earth mover’s distances.  In IEEE 12th Intern. Conf. Computer Vision, pages 460–467. IEEE, Sep. 2009.
  * [35] Carlos Pinho and Augusto Melo.  The financial crisis impact on the composition of an optimal portfolio in the stock market-study applied to portuguese index psi 20.  International Journal of Business and Social Science, 8:38–47, 10 2017.
  * [36] I. Pouchkarev.  Performance evaluation of constrained portfolios.  PhD thesis, Erasmus Research Institute of Management, The Netherlands, 2005.
  * [37] R.Y. Rubinstein and B. Melamed.  Modern simulation and modeling.  Wiley, New York, 1998.
  * [38] Yossi Rubner, Carlo Tomasi, and Leonidas Guibas.  The earth mover’s distance as a metric for image retrieval.  International Journal of Computer Vision, 40:99–121, 01 2000.
  * [39] W.F. Sharpe.  Mutual Fund Performance.  J. Business, 39(1):119–138, 1966.
  * [40] J. L. Treynor.  How to Rate Management of Investment Funds.  In Treynor on Institutional Investing, pages 69–87. John Wiley & Sons, Ltd, 2015.
  * [41] G. Varsi.  The multidimensional content of the frustum of the simplex.  Pacific J. Math., 46:303–314, 1973.
  * [42] S.S. Vempala.  Geometric random walks: a survey.  In Combinatorial and Computational Geometry, volume 52 of MSRI Publications, Berkeley, 2005. MSRI.
  * [43] Z. Zhao, O. Ledoit, and H. Jiang.  Risk reduction and efficiency increase in large portfolios: leverage and shrinkage.  Technical Report 328, January 2020.  Revised version.

##  Appendix A Computational methods

###  A.1 $L_{2}$ norm estimation

To estimate the $L_{2}$ norm of $\pi_{\alpha_{i+1}}\propto
e^{\alpha_{i+1}h(x)}$ w.r.t. the distribution $\pi_{\alpha_{i}}\propto
e^{\alpha_{i}h(x)}$, with $\alpha_{1}\neq\alpha_{2}$ and $h(x)$ being a convex
function over the set of portfolios $P$, we have,

$\begin{split}\|\pi_{\alpha_{i+1}}/\pi_{\alpha_{i}}\|&=\int_{P}\frac{e^{\alpha_{i+1}h(x)}}{\int_{P}e^{\alpha_{i+1}h(x)}dx}\frac{\int_{P}e^{\alpha_{i}h(x)}dx}{e^{\alpha_{i}h(x)}}\frac{e^{\alpha_{i+1}h(x)}}{\int_{P}e^{\alpha_{i+1}h(x)}dx}dx\\\
&=\frac{\int_{P}e^{\alpha_{i}h(x)}dx}{\int_{P}e^{\alpha_{i+1}h(x)}dx}\int_{P}e^{(\alpha_{i+1}-\alpha_{i})h(x)}\frac{e^{\alpha_{i+1}h(x)}}{\int_{P}e^{\alpha_{i+1}h(x)}dx}dx\end{split}$
(33)

While,

$\begin{split}\frac{\int_{P}e^{\alpha_{i}h(x)}dx}{\int_{P}e^{\alpha_{i+1}h(x)}dx}&=\int_{P}\frac{e^{\alpha_{i}h(x)}}{e^{\alpha_{i+1}h(x)}}\frac{e^{\alpha_{i+1}h(x)}}{\int_{P}e^{\alpha_{i+1}h(x)}dx}dx\\\
&=\int_{P}e^{(\alpha_{i}-\alpha_{i+1})h(x)}\frac{e^{\alpha_{i+1}h(x)}}{\int_{P}e^{\alpha_{i+1}h(x)}dx}dx\end{split}$
(34)

Thus, to estimate $\|\pi_{\alpha_{i+1}}/\pi_{\alpha_{i}}\|$ we sample
$x_{1},\dots,x_{k}$ points from the distribution which is proportional to
$e^{\alpha_{i+1}h(x)}$ and then, for sufficiently large $k$,

$\|\pi_{\alpha_{i+1}}/\pi_{\alpha_{i}}\|\approx\frac{\sum_{i=1}^{k}e^{(\alpha_{i}-\alpha_{i+1})h(x_{i})}}{k}\frac{\sum_{i=1}^{k}e^{(\alpha_{i+1}-\alpha_{i})h(x_{i})}}{k}$
(35)

###  A.2 Integral estimation

We implement the method implied by Equation (14). When, $\pi_{i}(x)\propto
e^{\alpha h_{i}(x)}$, where $h_{i}(x)$ a concave function supported on
$P\subset{\mathbb{R}}^{n}$, we re-write the Equation (13) as follows,

$s=\sum_{i=1}^{M}w_{i}\frac{\int_{S}e^{\alpha h_{i}(x)}dx}{\int_{P}e^{\alpha
h_{i}(x)}dx}.$ (36)

Thus, we estimate both integrals using the same method and we exactly compute
$\frac{\mbox{vol}(S)}{\mbox{vol}(P)}$ with Varsi’s algorithm [41].

###  A.3 Sampling from the set of portfolios

Let a low-dimensional convex polytope,

$LP=\\{x\in{\mathbb{R}}^{n}\ |\ Ax\leq b,A_{eq}x=b_{eq}\\},\
A\in{\mathbb{R}}^{m\times
n},b\in{\mathbb{R}}^{m},A_{eq}\in{\mathbb{R}}^{l\times
n},b_{eq}\in{\mathbb{R}}^{l}.$ (37)

To compute a full dimensional polytope, we compute the matrix
$N\in{\mathbb{R}}^{n\times d}$ of the right null space of $A_{eq}$. Then, we
obtain the full dimensional polytope,

$FP=\\{y\in{\mathbb{R}}^{d}\ |\ By\leq z\\},\ B\in{\mathbb{R}}^{m\times
d},z\in{\mathbb{R}}^{m},$ (38)

while $B=AN$ and $z=b-Ax^{*}$ and $x^{*}$ is a solution of the linear system
$A_{eq}x=b_{eq}$. Moreover, the matrix $N$ defines an isometric linear
transformation,

$f(y)=Ny+x^{*}.$ (39)

Thus, to sample from a log-concave distribution $\pi$ restricted to LP we
transform $\pi$ according to $N$ to obtain $\pi^{\prime}$. Next, we sample
from $\pi^{\prime}$ restricted to FP and map the generated points back to LP
using the inverse linear transformation.

To sample from a log-concave distribution truncated to a polytope we use the
reflective Hamiltonian Monte Carlo given by [1]. To sample from a mixture of
log-concave distribution

$\pi(x)=\sum_{i}w_{i}\pi_{i}(x),$ (40)

we implement the following method,

  1. 1.

Generate $u\sim\mathcal{U}(0,1)$.

  2. 2.

If $u\in[\sum_{i=1}^{k}w_{i},\sum_{i=1}^{k+1}w_{i}]$, generate a sample from
$\pi_{k}$.

  3. 3.

Repeat steps 1. and 2. until you have the desired amount of samples from the
mixture distribution.

###  A.4 Run-times

In Table 5 we report the run-times of the exact uniform sampler in [37] and
the Reflective Hamiltonian Monte Carlo we use to sample from a log-concave
distribution supported on the set of portfolios. All computations were
performed on a PC with Intel® Pentium(R) CPU G4400 @ 3.30GHz $\times$ 2 CPU
and 16GB RAM.

| Number of assets / Dimension
---|---
Method | 200 | 400 | 600 | 800 | 1000
Simpex sampler [37] | 2.42 | 5.19 | 6.93 | 9.37 | 11.69
ReHMC [1] | 8.39 | 64.12 | 620.2 | – | –
Table 5: The average run-times in seconds over $10$ runs for the exact uniform
sampler in [37] and our MCMC method (ReHMC) to sample log-concave
distributions supported on the set of portfolios $P\subset{\mathbb{R}}^{n}$.
For the ReHMC we sample from the spherical Gaussian centered at a uniformly
distributed point in $\Delta^{n-1}$. We set an upper bound on the run-time; if
the run-time exceeds $1$ hour we stop the execution.

##  Appendix B Crypto data

Coin | Symbol | Dates
---|---|---
Bitcoin | BTC | 28/04/2013 - 21/11/2020
Litecoin | LTC | 28/04/2013 - 21/11/2020
Ethereum | ETH | 07/08/2015 - 21/11/2020
XRP | XRP | 04/08/2013 - 21/11/2020
Monero | XMR | 21/05/2014 - 21/11/2020
Tether | USDT | 25/02/2015 - 21/11/2020
Dash | DASH | 14/02/2014 - 21/11/2020
Stellar | XLM | 05/08/2014 - 21/11/2020
Dogecoin | DOGE | 15/12/2013 - 21/11/2020
DigiByte | DGB | 06/02/2014 - 21/11/2020
NEM | XEM | 01/04/2015 - 21/11/2020
Siacoin | SC | 26/08/2015 - 21/11/2020
Table 6: Cryptocurrencies used to detect shock events in market.

The estimation of the covariance matrix $\Sigma$ we use in Section 6 is:

$\tiny\Sigma=10^{-3}\left[\begin{array}[]{rrrrrrrrrrrr}1.4290&0.9808&0.5099&0.3778&0.6328&0.8382&0.5957&0.7565&0.4754&0.9317&1.0839&0.2413\\\
0.9808&3.4804&0.7195&0.4652&1.1354&0.8545&1.1496&1.3633&0.8549&1.6933&1.3762&0.4291\\\
0.5099&0.7195&4.3024&0.3100&0.9297&0.6226&0.9011&1.3825&0.4979&1.0408&0.7013&0.3027\\\
0.3778&0.4652&0.3100&0.5014&0.3156&0.4654&0.3230&0.4687&0.2862&0.4977&0.4366&0.1317\\\
0.6328&1.1354&0.9297&0.3156&2.5198&0.6002&1.1857&0.9297&0.5557&1.9265&0.9451&0.3790\\\
0.8382&0.8545&0.6226&0.4654&0.6002&2.4615&0.5202&0.7768&0.5866&1.0493&1.1255&0.2875\\\
0.5957&1.1496&0.9011&0.3230&1.1857&0.5202&3.7467&1.0624&0.5390&2.0051&1.1436&0.3894\\\
0.7565&1.3633&1.3825&0.4687&0.9297&0.7768&1.0624&4.5856&0.8172&1.0103&1.0801&0.3444\\\
0.4754&0.8549&0.4979&0.2862&0.5557&0.5866&0.5390&0.8172&2.2931&0.9156&0.5850&0.3067\\\
0.9317&1.6933&1.0408&0.4977&1.9265&1.0493&2.0051&1.0103&0.9156&12.7352&1.2763&0.7029\\\
1.0839&1.3762&0.7013&0.4366&0.9451&1.1255&1.1436&1.0801&0.5850&1.2763&4.4734&0.2613\\\
0.2413&0.4291&0.3027&0.1317&0.3790&0.2875&0.3894&0.3444&0.3067&0.7029&0.2613&0.5760\\\
\end{array}\right]$

BTC | LTC | ETH | XRP | XMR | USDT
---|---|---|---|---|---
0.44% | 1.10% | 0.45% -0.07% | 0.53 | 0.11 |
DASH | XLM | DOGE | DGB | XEM | SC
0.48% | 0.96% | 0.64% | 0.96% | 1.05% | -0.07%
Table 7: The average assets’ returns from 22/10/2016 until 29/01/2017.  BTC | LTC | ETH | XRP | XMR | USDT
---|---|---|---|---|---
0.10% | 2.93% | -0.02% | 1.80% | 0.83% | 4.39%
DASH | XLM | DOGE | DGB | XEM | SC
-2.89% | 0.05% | 1.70& | 10.50% | 0.41% | -0.03%
Table 8: The average assets’ returns from 30/01/2017 until 08/02/2017.

##  Appendix C Clustering of copulae

Also, we apply clustering on features generated form the copulae, based on the
indicator. We generate vector representations for each copula using the rates
between all the possible combinations of the indicators’ corners: for $U_{L}$,
$U_{R}$ being the upper left and right corner of a copula respectively, and
for $L_{L}$, $L_{R}$ the lower left and right corners, the vector
representation is
$[\frac{U_{L}}{U_{R}}\frac{U_{L}}{L_{L}}\frac{U_{L}}{L_{R}}\frac{U_{R}}{L_{L}}\frac{U_{R}}{L_{R}}\frac{L_{L}}{L_{R}}]$.
These representations allow us to use clustering, such as k-medoids. Results
of the clustering also follow the values of the indicator as expected (Fig.
13, 16).

Figure 13: Left, spectral clustering ($k=6$) on EMD matrix. Right, k-medoids
($k=6$) on copulae features. Clusters appear to contain similar indicator
values.

Figure 14: Clustering of copulae using spectral clustering on EMD distances
with $k=6$.

Figure 15: Clustering of copulae using spectral clustering on EMD distances
with $k=8$.

Figure 16: Clustering using k-medoids on copulae features.