# Comprehensive search for buckled honeycomb binary compounds based on noble
metals (Cu, Ag, and Au)

Shota Ono<EMAIL_ADDRESS>Department of Electrical, Electronic and
Computer Engineering, Gifu University, Gifu 501-1193, Japan

###### Abstract

Honeycomb structure has been frequently observed in two-dimensional (2D)
materials. CuAu in the buckled honeycomb (BHC) structure has been synthesized
recently, which is the first case of 2D intermetallic compounds. Here, the
dynamical stability of 2D $AX$ in the BHC structure, where $A=$ Cu, Ag, and Au
and $X$ is a metallic element in the periodic table, is systematically studied
by calculating phonon dispersions from first-principles. Among 135 $AX$, more
than 50 $AX$ are identified to be dynamically stable. In addition, (i) a
relationship between the dynamical stability and the formation energy, (ii) a
correlation of dynamical stability between different constituents $A$, (iii) a
trend of lattice parameters, and (iv) electronic and magnetic properties are
discussed. Furthermore, a stable phase of B11-type AuZr is predicted based on
both the result (ii) and the stability relationship between 2D and three-
dimensional structures. The present findings stimulate future studies
exploring physics and chemistry of 2D intermetallic compounds.

##  I Introduction

Two-dimensional (2D) compound of CuAu has been synthesized experimentally
zagler , where 2D CuAu has a buckled honeycomb (BHC) structure, the same as
the structure of silicene and germanene. This discovery opens the way for
studying 2D compounds including only metallic elements. The fundamental
properties of 2D elemental metals, such as energetic and structural properties
nevalaita ; hwang , dynamical stability ono2020 ; ono2020_Po , and magnetic
properties ren , have been investigated systematically from first-principles,
also motivated by recent experimental synthesis of 2D metals wang2020 . It is
desirable to extend such investigations for 2D intermetallic compounds that
are difficult to be found (currently) in 2D materials databases ashton ;
choudhary ; haastrup ; feng ; fukuda .

Copper, silver, and gold belong to the noble metal column in the periodic
table, so that the effect of $d$ electrons on their electronic and optical
properties have been studied extensively under various conditions lin ; bauer
; xu ; giri . For the compounds including noble metals, it is not trivial to
understand their stability properties among Cu$X$, Ag$X$, and Au$X$, where $X$
is an element in the periodic table, as confirmed by, for example, Pettifor’s
structure map pettifor  or modern materials database MP . Recently, Alsalmi et
al. have studied the stability of CuZn, AgZn, and AuZn in the B2 (CsCl-type)
structure by using density functional theory (DFT) alsalmi  and explained
nonmonotonic behaviors of structural parameters, where AgZn takes the largest
lattice constant. This is never achieved by studying each compound
individually. A systematic investigation for the stability of 2D Cu$X$, Ag$X$,
and Au$X$ will thus be of fundamental importance.

In a previous paper, the author studied a stability relationship between 2D
and three-dimensional (3D) Cu-based compounds and found that if a Cu$X$ ($X$
is a metallic element) in the BHC structure is dynamically stable, that in the
Bh (WC-type) and L11 (CuPt-type) structures is also dynamically stable
ono2021SR . This is due to the geometric hierarchy: Bh (L11) structure can be
constructed from ABAB (ABC) stacking of hexagonal layers of Cu and $X$
alternately. In addition, it has been shown that an experimental synthesis of
Bh structure accounts for the dynamical stability of the BHC structure; for
example, LiRh, LiPd, LiIr, and LiPt ono_satomi . In this way, the BHC
structure serves as a building block for constructing 3D compounds.

In this paper, we show systematic investigations of the dynamical stability
for BHC Cu$X$, Ag$X$, and Au$X$, where $X$ is a metallic element from Li to Pb
in the periodic table, by using first-principles approach. We identify that
more than 50 compounds are dynamically stable by quantifying the profile of
the phonon spectra. We demonstrate that the quantity for the dynamical
stability of Ag$X$ is correlated with that of Au$X$. Based on this fact, we
predict that as a 3D compound AuZr in the B11 (CuTi-type) structure is
dynamically stable. Trends of structural parameters, formation energy,
electronic bands, and magnetic property are also discussed. We identify that
the BHC AuK is a wide-gap semiconductors and that the BHC AgCr, AuCr, CuMn,
AgMn, and AuMn have a magnetic moment about 4 $\mu_{\rm B}$ with a Bohr
magneton $\mu_{\rm B}$. The calculated results are listed in Table 1 including
the values of the modified formation energy defined in Eq. (1) and the degree
of the dynamical stability defined in Eq. (2) and Table 2 including the
lattice parameters of the BHC Cu$X$, Ag$X$, and Au$X$.

Table 1: The values of $E_{\rm form}$ in units of eV (the column 2, 3, and 4) and $S$ (the column 5, 6, and 7) for BHC $AX$. The value of $S$ is shown in bold if $S\geq 0.6$.  $X$ |  | $A$ |  |  | $A$ |
---|---|---|---|---|---|---
| Cu | Ag | Au | Cu | Ag | Au
Li | $-0.47$ | $-0.64$ | $-0.83$ | ${\bf 1.00}$ | ${\bf 0.82}$ | ${\bf 0.98}$
Na | $0.40$ | $-0.12$ | $-0.19$ | $-2.03$ | $-0.53$ | ${\bf 0.91}$
K | $0.71$ | $0.01$ | $-0.57$ | $-2.22$ | $-1.17$ | ${\bf 1.00}$
Rb | $0.70$ | $-0.00$ | $-0.61$ | $-2.75$ | $-1.05$ | $-1.11$
Cs | $0.65$ | $-0.11$ | $-0.71$ | $-1.25$ | $-2.54$ | $-0.67$
Be | $-0.42$ | $0.35$ | $0.25$ | ${\bf 0.97}$ | $-3.86$ | $-2.63$
Mg | $-0.24$ | $-0.56$ | $-0.66$ | $-0.21$ | ${\bf 0.89}$ | ${\bf 0.97}$
Ca | $0.02$ | $-0.40$ | $-0.97$ | $-3.01$ | $-2.03$ | $-1.51$
Sr | $0.19$ | $-0.25$ | $-0.90$ | $-4.53$ | $-3.28$ | $-2.01$
Ba | $0.12$ | $-0.28$ | $-1.00$ | $-3.57$ | $-4.00$ | $-1.64$
Sc | $-0.84$ | $-1.01$ | $-1.39$ | $-2.19$ | ${\bf 1.00}$ | ${\bf 0.66}$
Y | $-0.70$ | $-0.95$ | $-1.47$ | $-3.94$ | $-1.19$ | $-0.97$
Lu | $-0.89$ | $-1.08$ | $-1.66$ | $-4.08$ | $-0.86$ | $-0.98$
Ti | $-1.07$ | $-0.96$ | $-1.13$ | $0.01$ | ${\bf 1.00}$ | ${\bf 0.92}$
Zr | $-0.82$ | $-0.97$ | $-1.15$ | $-3.05$ | ${\bf 0.66}$ | ${\bf 0.83}$
Hf | $-1.06$ | $-1.11$ | $-1.35$ | $-3.13$ | ${\bf 1.00}$ | ${\bf 0.81}$
V | $-0.94$ | $-0.43$ | $-0.51$ | $-4.29$ | $-14.96$ | $-24.01$
Nb | $-0.90$ | $-0.79$ | $-0.89$ | $-1.60$ | $-1.70$ | $-1.22$
Ta | $-0.99$ | $-0.84$ | $-0.94$ | $-3.19$ | $-2.78$ | $-1.76$
Cr | $-1.07$ | $-0.67$ | $-0.70$ | $-3.47$ | ${\bf 0.85}$ | ${\bf 0.86}$
Mo | $-1.02$ | $-0.64$ | $-0.68$ | $-2.97$ | $-2.48$ | $-2.18$
W | $-1.08$ | $-0.70$ | $-0.70$ | $-2.31$ | $-2.51$ | $-3.08$
Mn | $-0.84$ | $-0.72$ | $-0.79$ | ${\bf 1.00}$ | ${\bf 1.00}$ | ${\bf 0.76}$
Tc | $-1.07$ | $-0.52$ | $-0.47$ | $-4.35$ | $-5.01$ | $-5.44$
Re | $-1.10$ | $-0.52$ | $-0.39$ | $-2.26$ | $-3.36$ | $-3.72$
Fe | $-0.75$ | $-0.22$ | $-0.23$ | $-2.63$ | $-2.42$ | $-3.12$
Ru | $-1.01$ | $-0.50$ | $-0.38$ | $-1.28$ | $0.12$ | $0.55$
Os | $-0.99$ | $-0.40$ | $-0.15$ | $-1.75$ | $-1.36$ | $-1.98$
Co | $-0.73$ | $-0.03$ | $0.03$ | ${\bf 1.00}$ | $-0.63$ | $-0.29$
Rh | $-0.96$ | $-0.60$ | $-0.43$ | ${\bf 0.75}$ | ${\bf 0.83}$ | ${\bf 0.85}$
Ir | $-0.90$ | $-0.44$ | $-0.13$ | $-0.25$ | $0.52$ | $-0.77$
Ni | $-0.76$ | $-0.12$ | $-0.02$ | ${\bf 0.91}$ | $-0.17$ | $-0.87$
Pd | $-0.89$ | $-0.78$ | $-0.56$ | ${\bf 0.86}$ | ${\bf 1.00}$ | ${\bf 0.76}$
Pt | $-0.75$ | $-0.54$ | $-0.22$ | ${\bf 0.79}$ | ${\bf 0.83}$ | ${\bf 0.97}$
Cu | $-0.51$ | $-0.14$ | $-0.09$ | ${\bf 1.00}$ | ${\bf 0.69}$ | ${\bf 0.88}$
Ag | $-0.14$ | $-0.33$ | $-0.25$ | ${\bf 0.69}$ | ${\bf 0.75}$ | ${\bf 0.88}$
Au | $-0.09$ | $-0.25$ | $-0.10$ | ${\bf 0.88}$ | ${\bf 0.88}$ | ${\bf 0.87}$
Zn | $-0.41$ | $-0.23$ | $-0.25$ | ${\bf 1.00}$ | $0.50$ | ${\bf 1.00}$
Cd | $0.02$ | $-0.31$ | $-0.30$ | $-0.85$ | ${\bf 0.82}$ | ${\bf 0.93}$
Hg | $0.45$ | $0.02$ | $0.10$ | $-3.29$ | $-2.90$ | $-2.18$
Al | $-0.72$ | $-0.57$ | $-0.64$ | ${\bf 1.00}$ | ${\bf 0.91}$ | ${\bf 0.87}$
Ga | $-0.42$ | $-0.44$ | $-0.44$ | ${\bf 0.65}$ | ${\bf 0.62}$ | ${\bf 0.83}$
In | $0.11$ | $-0.29$ | $-0.23$ | $-3.24$ | $-0.45$ | $-0.85$
Tl | $0.42$ | $-0.07$ | $0.05$ | $-5.67$ | $-1.67$ | $-3.17$
Sn | $0.07$ | $-0.27$ | $-0.19$ | $-4.98$ | $-0.62$ | $-1.16$
Pb | $0.33$ | $-0.11$ | $0.04$ | $-6.05$ | $-2.44$ | $-2.99$
Table 2: The values of $d(A\mathchar 45\relax A)$ (the column 2, 3, and 4) and $d(A\mathchar 45\relax X)$ (the column 5, 6, and 7) for BHC $AX$. The smaller value, $d(A\mathchar 45\relax A)$ or $d(A\mathchar 45\relax X)$, is underlined for each $AX$.  $X$ |  | $A$ |  |  | $A$ |
---|---|---|---|---|---|---
| Cu | Ag | Au | Cu | Ag | Au
Li | 2.63 | 2.92 | 2.91 | 2.51 | 2.68 | 2.60
Na | 2.84 | 3.06 | 2.98 | 2.93 | 3.08 | 2.99
K | 5.55 | 5.69 | 5.56 | 3.20 | 3.29 | 3.21
Rb | 5.75 | 5.95 | 5.78 | 3.32 | 3.44 | 3.34
Cs | 5.94 | 6.23 | 5.98 | 3.43 | 3.60 | 3.45
Be | 2.39 | 2.69 | 2.84 | 2.40 | 2.58 | 2.38
Mg | 2.84 | 2.99 | 2.98 | 2.72 | 2.92 | 2.83
Ca | 3.69 | 3.72 | 3.89 | 2.87 | 3.03 | 2.88
Sr | 4.07 | 4.13 | 4.27 | 3.03 | 3.19 | 3.04
Ba | 4.47 | 4.49 | 4.66 | 3.14 | 3.31 | 3.17
Sc | 3.10 | 3.14 | 3.29 | 2.67 | 2.90 | 2.74
Y | 3.44 | 3.43 | 3.64 | 2.81 | 3.01 | 2.86
Lu | 3.32 | 3.33 | 3.54 | 2.74 | 2.94 | 2.78
Ti | 2.71 | 2.85 | 2.89 | 2.66 | 2.90 | 2.80
Zr | 2.96 | 3.00 | 3.07 | 2.77 | 3.01 | 2.89
Hf | 2.97 | 3.01 | 3.09 | 2.72 | 2.96 | 2.83
V | 2.51 | 2.72 | 2.75 | 2.66 | 2.87 | 2.81
Nb | 2.71 | 2.80 | 2.82 | 2.77 | 3.00 | 2.95
Ta | 2.75 | 2.82 | 2.85 | 2.72 | 2.97 | 2.90
Cr | 2.44 | 2.88 | 2.87 | 2.60 | 2.86 | 2.82
Mo | 2.63 | 2.73 | 2.75 | 2.69 | 2.94 | 2.90
W | 2.67 | 2.74 | 2.76 | 2.67 | 2.93 | 2.90
Mn | 2.63 | 2.83 | 2.84 | 2.59 | 2.85 | 2.82
Tc | 2.59 | 2.69 | 2.71 | 2.64 | 2.89 | 2.87
Re | 2.61 | 2.69 | 2.71 | 2.64 | 2.90 | 2.88
Fe | 2.48 | 2.70 | 2.73 | 2.58 | 2.83 | 2.80
Ru | 2.58 | 2.70 | 2.72 | 2.62 | 2.88 | 2.89
Os | 2.60 | 2.68 | 2.70 | 2.63 | 2.91 | 2.91
Co | 2.44 | 2.67 | 2.70 | 2.55 | 2.77 | 2.78
Rh | 2.60 | 2.72 | 2.73 | 2.61 | 2.86 | 2.88
Ir | 2.61 | 2.69 | 2.71 | 2.63 | 2.91 | 2.95
Ni | 2.45 | 2.68 | 2.71 | 2.51 | 2.72 | 2.73
Pd | 2.65 | 2.78 | 2.77 | 2.64 | 2.84 | 2.89
Pt | 2.66 | 2.75 | 2.74 | 2.66 | 2.89 | 2.97
Cu | 2.50 | 2.73 | 2.74 | 2.57 | 2.76 | 2.78
Ag | 2.73 | 2.86 | 2.82 | 2.76 | 2.98 | 3.06
Au | 2.74 | 2.82 | 2.77 | 2.78 | 3.06 | 3.30
Zn | 2.53 | 2.76 | 2.75 | 2.74 | 2.96 | 2.99
Cd | 2.80 | 2.91 | 2.87 | 2.91 | 3.15 | 3.18
Hg | 2.88 | 2.99 | 2.89 | 2.93 | 3.19 | 3.34
Al | 2.65 | 2.81 | 2.83 | 2.61 | 2.86 | 2.82
Ga | 2.65 | 2.84 | 2.83 | 2.68 | 2.96 | 2.97
In | 2.98 | 3.06 | 3.01 | 2.83 | 3.09 | 3.06
Tl | 3.26 | 3.19 | 3.36 | 2.84 | 3.17 | 2.98
Sn | 3.13 | 3.13 | 3.12 | 2.76 | 3.05 | 2.98
Pb | 3.33 | 3.26 | 3.24 | 2.80 | 3.12 | 3.05

##  II Basic concepts

The primitive lattice vectors of the BHC structure are $\bm{a}_{1}=a_{\rm
lat}(1,0,0)$, $\bm{a}_{2}=a_{\rm lat}(-1/2,\sqrt{3}/2,0)$, and
$\bm{a}_{3}=(0,0,c)$ with the lattice constant $a_{\rm lat}$. For the BHC $AX$
with $A=$ Cu, Ag, and Au and $X$ a metallic element, the basis vectors are
$A(0,0,\delta)$ and $X(0,a_{\rm lat}/\sqrt{3},-\delta)$ with the total
thickness $2\delta$. The interatomic distance $d(A\mathchar 45\relax A)$
between the same species is equal to $a_{\rm lat}$, while the interatomic
distance $d(A\mathchar 45\relax X)$ between atoms $A$ and $X$ is given by
$a_{\rm lat}\sqrt{1/3+4r^{2}}$ with $r=\delta/a_{\rm lat}$ (see Fig. 1).

Figure 1: Top and side views for the 2D $AX$ in the BHC structure with the
lattice constant $a_{\rm lat}$, the thickness $2\delta$, and the interatomic
distances $d(A\mathchar 45\relax A)$, $d(A\mathchar 45\relax X)$, and
$d(X\mathchar 45\relax X)$.

The formation energy is usually defined as the energy difference between the
total energy of the binary system considered (such as ordered compounds and/or
alloys) and that of the reference system (ground state structure of elements,
such as fcc, bcc, and hcp). Application of this standard formula to 2D binary
compounds yielded positive formation energies for most cases because 2D
structures are energetically unstable (but might be dynamically stable)
ono_satomi . To obtain a useful insight from the energy calculations, we
define a modified formation energy as

$\displaystyle E_{\rm form}(AX)=\varepsilon_{\rm
BHC}(AX)-\left[\varepsilon_{\rm HX}(A)+\varepsilon_{\rm HX}(X)\right],$ (1)

where $\varepsilon_{\rm BHC}(AX)$ is the total energy of $AX$ in the BHC
structure and $\varepsilon_{\rm HX}(A)$ is the total energy of $A$ in the
hexagonal (HX) structure. The physics behind Eq. (1) is that vertically
stacking the HX layers of $A$ and $X$ produces the BHC structure of $AX$ by
releasing or absorbing energies: When $E_{\rm form}(AX)<0$, BHC $AX$ is
energetically suitable to be formed from HX layers.

For the elemental metals of $X$, 46 elements were investigated in the present
work, so that the phonon dispersions for 135 compounds were calculated. To
systematically understand the dynamical stability property of $AX$, we define
the degree of dynamical stability $S$ as

$\displaystyle S=\frac{R-(1-p)}{p}$ (2)

with

$\displaystyle R=\frac{\omega_{\rm max}+{\rm sgn}(\omega_{\rm
min}^{2})|\omega_{\rm min}|}{\omega_{\rm max}},$ (3)

where $\omega_{\rm max}$ and $\omega_{\rm min}$ are the maximum and minimum
phonon frequencies, respectively, which are calculated by phonon density-of-
states (DOS). An imaginary frequency is represented as a negative frequency by
using the sign function of ${\rm sgn}$. If no imaginary frequency is observed
in the phonon DOS, $R=1$ exactly, otherwise $R<1$. Due to a finite size of $q$
grid in the Brillouin zone, small negative frequencies may appear around the
$\Gamma$ point. In order to identify an unstable structure clearly, we
subtract $1-p$ from $R$ and normalize the value by dividing $p$. In the
present work, we adapted the value of $p=0.1$. The negative (or small
positive) value of $S$ indicates an instability of $AX$ in the BHC structure.

##  III Results and Discussion

Figure 2: Plot of $S$ as a function of $E_{\rm form}$ for the BHC Cu$X$,
Ag$X$, and Au$X$. Dashed lines indicate $E_{\rm form}=0$ (vertical) and $S=0$
(horizontal). Figure 3: Relationship of $S$ between $AX$ and $BX$: (a)
$(A,B)=$ (Cu, Ag), (b) $(A,B)=$ (Cu, Au), and (c) $(A,B)=$ (Ag, Au). Diagonal
line (dashed) indicates $S(AX)=S(BX)$.  Figure 4: (a) The phonon dispersions
of AuZr in the B11 structure and (b) the atomic distribution of the
3$\times$2$\times$2 supercell after the first-principles MD simulation of 3.2
ps under 500, 1500, and 2000 K. The views are from the $c$ axis (top) and $a$
axis (bottom).

###  III.1 Dynamical stability

Table 1 lists the values of $E_{\rm form}(AX)$ in Eq. (1) and $S$ in Eq. (2)
for Cu$X$, Ag$X$, and Au$X$ in the BHC structure. The values of $S$ are
distributed from $-6$ (unstable) to $1$ (stable), except for AgV and AuV. The
BHC $AX$ having relatively high value of $S$ will be a cousin of 2D CuAu
zagler , which will expand the family of 2D intermetallic compounds. The
calculated phonon dispersions for Cu$X$, Ag$X$, and Au$X$ are provided in the
Supplemental Material.

We first study a correlation between $E_{\rm form}(AX)$ and $S$. As shown in
Fig. 2, the $AX$ having positive $S$ shows negative $E_{\rm form}(AX)$,
indicating that the formation energy modified to treat an energy gain for
stacking HX layers, defined as Eq. (1), may be a good quantity for studying
the dynamical stability of $AX$. This improves our previous results ono_satomi
, where the 2D compound that is dynamically stable can have a positive
formation energy (even in 2D CuAu) when the standard formation energy is
applied. However, negative $E_{\rm form}(AX)$ is not still a sufficient
condition for positive $S$, which is left for an open question.

As shown in Fig. 2, the distribution of $E_{\rm form}({\rm Cu}X)$ seems to be
different from that of $E_{\rm form}({\rm Ag}X)$ and $E_{\rm form}({\rm
Au}X)$. To understand the dynamical stability relationship among 2D compounds,
we plot $S$ for Ag$X$ and Au$X$ as a function of Cu$X$ in Fig. 3(a) and 3(b),
respectively. The trends of the dynamical stability for Cu$X$ is not similar
to those for Ag$X$ and Au$X$ because the plots of $S$ are scattered around
dashed lines $S({\rm Cu}X)=S({\rm Ag}X)$ and $S({\rm Cu}X)=S({\rm Au}X)$. In
addition, even if Cu$X$ in the BHC structure is dynamically stable, i.e.,
$S({\rm Cu}X)\simeq 1$, Ag$X$ and Au$X$ in the BHC structure are unstable,
i.e., $S({\rm Ag}X)\ll 1$ or $S({\rm Au}X)\ll 1$, and vice versa: For $X=$ Be,
Co, and Ni, Cu$X$ is dynamically stable, while Ag$X$ and Au$X$ are unstable.
On the other hand, for $X=$ Mg, Sc, Ti, Zr, Hf, Cr, and Cd, Ag$X$ and Au$X$
are dynamically stable, while Cu$X$ is unstable. These anomalies can be
explained by negatively large values of $E_{\rm form}(AX)$ for $X=$ Be, Co,
Ni, Mg, Sc, and Cd (see Table 1). An interesting finding is that the value of
$S$ for Ag$X$ is well correlated with that of Au$X$, as shown in Fig. 3(c).
The correlation coefficient between Ag$X$ and Au$X$ is 0.94, whereas that
between Cu$X$ and Ag$X$ (Au$X$) is 0.49 (0.42). The strong correlation of $S$
between Ag$X$ and Au$X$ is attributed to the similar distribution of $E_{\rm
form}$ in Fig. 2.

Combined with the correlation of $S$ between Ag$X$ and Au$X$ and the stability
relationship between 2D and 3D structures, we can predict stable compounds
that have not been synthesized yet experimentally. Let us consider the cases
of $X=$ Ti, Zr, and Hf, where Ag$X$ and Au$X$ are dynamically stable as listed
in Table 1. By using Atom Works database nims , we found that AgTi AgTi , AgZr
AgZr , AgHf AgHf , AuTi AuTiAuHf , and AuHf AuTiAuHf  in the B11 structure
have already been synthesized experimentally. We thus identify an anomaly that
no synthesis of AuZr in the B11 structure has been reported so far. We then
calculated the phonon dispersions of AuZr, as shown in Fig. 4(a). No imaginary
phonon frequencies are observed along the symmetry lines. To study the
temperature effect on the dynamical stability, we also performed first-
principles molecular dynamics (MD) simulation. The B11 AuZr is stable up to
1500 K (see Fig. 4(b)).

It is quite interesting that the synthesis probability of B11 AuZr has been
predicted to be 91.1 %, according to the MaterialNet program aykol . This
synthesizability was derived from machine learning the time evolution of
materials stability network, where the network is constructed from the convex
free-energy surface of inorganic materials. In this way, different approaches
have predicted the stability of B11 AuZr, so that we expect that AuZr in the
B11 structure will be potentially synthesized in future experiments.

By considering the stability relationship between 2D and 3D structures again,
the positive values of $S$ for BHC AgPd, AgPt, and CuPt listed in Table 1 are
consistent with the prediction by Nelson et al. nelson , where AgPd, AgPt, and
CuPt have the L11 structure as its ground state. This is because if the BHC
structure is dynamically stable, then the L11 structure is also stable
ono2021SR . An implication in this section is that the 2D and 3D structures
must be interrelated via the dynamical stability. A similar interrelation can
hold for the Ag-based and Au-based compounds.

###  III.2 Structure relationship

We next study how the structural parameters of the BHC structure are related
to those of the 3D counterparts. To investigate it, we list the values of
$d(A\mathchar 45\relax A)$ and $d(A\mathchar 45\relax X)$ in Table 2. If the
value of $d(A\mathchar 45\relax A)$ is smaller than that of $d(A\mathchar
45\relax X)$, the coordination number is equal to six because of a strong
intralayer bonding ($A$-$A$ and $X$-$X$). In contrast, if an interlayer
bonding ($A$-$X$) is rather strong, an inequality $d(A\mathchar 45\relax
A)/d(A\mathchar 45\relax X)>1$ will hold, leading to smaller coordination
number of three.

We consider the cases of $X=$ Cu, Ag, and Au, where an inequality
$d(A\mathchar 45\relax A)/d(A\mathchar 45\relax X)<1$ holds. Let us just
remind that the noble metals (Cu, Ag, and Au) prefer to have the fcc structure
with the largest coordination number of twelve. Therefore, the inequality of
$d(A\mathchar 45\relax A)/d(A\mathchar 45\relax X)<1$ for the 2D Cu, Ag, and
Au in the BHC structure reflects the close-packed structure in the 3D Cu, Ag,
and Au. By using Atom Works database nims , we found that the mixing of noble
metals produces a solid solution in the fcc structure represented by
Cu0.5Ag0.5 linde , Cu0.5Au0.5 nahm , and Ag0.5Au0.5 venudhar . These support
the high coordination number in 2D CuAg, CuAu, and AgAu.

To study the structural trends for the $AX$ in the BHC structure, we plot the
values of $d(A\mathchar 45\relax A)/d(A\mathchar 45\relax X)$ versus the
column of $X$ in the periodic table, as shown in Fig. 5. When $X$ is an
element of the column 1, 2, 3, and 4, an inequality $d(A\mathchar 45\relax
A)>d(A\mathchar 45\relax X)$ holds. (The $AX$s including alkali metals of K,
Rb and Cs have $d(A\mathchar 45\relax A)/d(A\mathchar 45\relax X)=\sqrt{3}$,
i.e., no buckling structure.) Again, let us just remind that noble metals can
mix with some elements of the column 1, 2, and 3 to form the B2 structure,
such as AuRb and AuCs tinelli , AgMg AgMg , AuMg AuMgAuCd , and AgSc and AuSc
Sc , where the coordination number is eight that is smaller than twelve. We
can attribute the inequality of $d(A\mathchar 45\relax A)/d(A\mathchar
45\relax X)>1$ partially to the formation of the B2 structure of $AX$. When
the column of $X$ is from 5 to 10, the magnitude of $d(A\mathchar 45\relax A)$
is smaller than that of $d(A\mathchar 45\relax X)$, implying a strong
intralayer bonding as in the BHC Cu, Ag, and Au (the column 11). However, for
$X=$ the column 12, it is difficult to rationalize the inequality
$d(A\mathchar 45\relax A)<d(A\mathchar 45\relax X)$ because $A$Zn, AgCd AgCd ,
and AuCd AuMgAuCd  in the B2 structure have been synthesized, as for $X=$ the
column of 1, 2, and 3. For $X=$ the column 13 and 14 elements, no compounds of
the forms of $AX$ and $A_{\rm 0.5}X_{\rm 0.5}$ have been reported except for
$X=$ Al and Ga.

It will be useful to study trends of structural parameters with $X$ fixed.
Recently, Alsalmi et al. have studied the stability of CuZn, AgZn, and AuZn in
the B2 structure by using DFT alsalmi . They have shown that the bonding
strength between noble metal and Zn atom becomes stronger as the $d$ orbitals
are more extended. In contrast, the core-core repulsion enhances the lattice
constant, in turn, giving rise to a decrease in the overlap of wavefunctions.
These effects yield nonmonotonic stability properties as one goes CuZn, AgZn,
and AuZn and a large lattice parameter of AgZn. In the present calculations,
similar behaviors can be observed in BHC $AX$: an inequality $d({\rm
Cu}\mathchar 45\relax{\rm Cu})<d({\rm Au}\mathchar 45\relax{\rm Au})<d({\rm
Ag}\mathchar 45\relax{\rm Ag})$ or $d({\rm Cu}\mathchar 45\relax X)<d({\rm
Au}\mathchar 45\relax X)<d({\rm Ag}\mathchar 45\relax X)$ can hold. For the
cases of $X=$ Tl, Sn, and Pb, the relationship between $d({\rm Cu}\mathchar
45\relax{\rm Cu})$, $d({\rm Ag}\mathchar 45\relax{\rm Ag})$, and $d({\rm
Au}\mathchar 45\relax{\rm Au})$ is quite different.

Figure 5: Plot of $d(A\mathchar 45\relax A)/d(A\mathchar 45\relax X)$ versus
the column of $X$ in the periodic table for the BHC Cu$X$, Ag$X$, and Au$X$.

###  III.3 Electronic and magnetic properties

Toward practical applications, bandgap engineering of 2D materials has been
studied by introducing an electric field, strain, and foreign atoms chaves .
It is noteworthy that for 3D compounds, AuRb and AuCs in the B2 structure have
been known as semiconductors spicer , although both Rb and Cs are metallic
elements. Recently, Adeleke et al. have predicted that at high pressure K2Ni
is a semiconductor with an indirect bandgap of 0.65 eV, which is due to a
Peierls gap opening adeleke . It will be interesting to study which 2D
compounds including two metallic elements may exhibit a semiconducting
property.

The spin-dependent electron band structure for Cu$X$, Ag$X$, and Au$X$ in the
BHC structure is provided in the Supplemental Material. Basically, nearly flat
$d$-bands are located below the Fermi level by a few eV. As in the band
structure of elemental metals, the location of $d$-bands tend to be shallow
and deep for Cu$X$ and Ag$X$, respectively. The $s$ electron band is modified
by the mixing with other electrons of $X$ as well as the $s$-$d$ coupling and
crosses the Fermi level.

We identify that the compounds with alkali metals $X=$ K, Rb, and Au are
semiconductors. Figure 6(a) show the electron band structure of AuK, showing
an indirect semiconductor with a bandgap of 3.16 eV, where the valence band
maximum and conduction band minimum are located at K and M point,
respectively. It is well known that the DFT within the generalized gradient
approximation underestimates the magnitude of the bandgap. In this way, BHC
AuK can be included to one of the wide-gap semiconductors such as hexagonal BN
2DBN  and 2D GaN 2DGaN . Although AuK in the BHC structure is predicted to be
dynamically stable at zero temperature, first-principles MD simulation
predicts that it is unstable even at low temperature of 150 K, as shown in
Fig. 6(b). While the hexagonal symmetry is almost preserved (top view), the
out-of-plane displacement is significantly large (side view) at 1 ps, giving
rise to a failure of self-consistent field convergence. The instability
against the temperature is due to small phonon energies (a few meV) of the
flexural modes around the M and K points (see Fig. S1 in the Supplemental
Material). It is highly desirable to propose a substrate material for
stabilizing the BHC AuK.

Figure 6: (a) The electron band structure of AuK in the BHC structure and (b) the top and side views of the atomic distribution of the 4$\times$4$\times$1 supercell (32 atoms) after the first-principles MD simulation of 1 ps under 150 K, indicating an instability against out-of-plane vibrations.  Table 3: The magnetic moment (in units of $\mu_{\rm B}$ per a cell) of Cu$X$, Ag$X$, and Au$X$ in the BHC structure. The magnetic moments for the dynamically stable structures ($S\geq 0.6$) are shown in bold.  $X$ |  | $A$ |
---|---|---|---
| Cu | Ag | Au
Sr | 0.17 | 0.14 | 0.34
Ba | 0.66 | 0.61 | 0.65
Ti | 0.87 | 1.01 | 1.14
Zr | 0.70 | 0.64 | 0.82
Hf | 0.75 | 0.75 | 0.90
V | 0.02 | 1.13 | 0.88
Cr | 0.00 | 4.29 | 4.28
Mn | 3.85 | 4.20 | 4.31
Fe | 2.64 | 3.00 | 3.09
Ru | 0.01 | 1.19 | 1.42
Os | 0.00 | 0.01 | 0.38
Co | 1.74 | 1.86 | 1.95
Rh | 0.59 | 0.82 | 0.98
Ir | 0.01 | 0.17 | 0.71
Ni | 0.46 | 0.61 | 0.72

We move on to the study of magnetic property. In general, reduced
dimensionality can have a possibility for enhancing the magnitude of magnetic
moments vaz . It has been recently predicted that 18 elemental metals show a
finite magnetic moment in the 2D structures, while there are only 5 magnetic
elements in the 3D structures ren . Also, the Cu-based compounds in the BHC
structure exhibit a relatively high magnetic moment, compared to the 3D
counterparts ono2021SR . The replacement of Cu with Ag or Au in the 2D
compounds will change a bonding character between adjacent atoms and the
degree of $d$ electron localization as well, leading to an increase and/or
decrease in the magnetic moment.

Table 3 lists the magnetic moment of selected compounds $AX$ in the BHC
structure. Even when nonmagnetic elements ($X=$ Sr, Ba, Ti, Zr, Hf, V, Ru, Os,
Rh, and Ir) are included, finite magnetic moments appears due to the different
profile of electron bands between up and down spins (see the Supplemental
Material). This tendency is similar to the DFT calculations for 2D elemental
metals ren . The Cu$X$ tends to have smaller magnetic moment than Ag$X$ and
Au$X$. The following interpretation will be applied: Among Cu, Ag, and Au, the
lattice parameters of Cu$X$ is smallest, as listed in Table 2 and discussed in
Sec. III.2; the hybridization of the orbital between different atoms will be
strong in Cu$X$; and the delocalization of $d$ electron yields a decrease in
the band splitting.

The coupling between the magnetism and the dynamical stability has long been
investigated for elemental metals grimvall . An interesting feature in the
present calculations is that the 2D compounds including anti-ferromagnetic
elements of Cr and Mn have large magnetic moment about 4 $\mu_{\rm B}$ and are
also dynamically stable. On the other hand, the 2D compounds with
ferromagnetic elements of Fe, Co, and Ni have finite magnetic moments but are
unstable except for CuCo and CuNi. The latter is similar to the tendency
observed in our previous calculations ono2020 , where magnetic effects lead to
an instability of 2D Fe and Co in the buckled square structure.

In our previous work ono_satomi , we predicted that the BHC AuTi is unstable
against the excitation of phonons at M and K points, by performing spin-
unpolarized calculations. In the present work, BHC AuTi have finite magnetic
moment listed in Table 3 and is dynamically stable. The magnetic effect on the
dynamical stability of $AX$ seems to strongly depend on the column of $X$.

##  IV Summary

We systematically investigated the dynamical stability and electronic property
of BHC Cu$X$, Ag$X$, and Au$X$, where $X$ is 46 metallic element in the
periodic table, by using first-principles approach. More than 50 compounds
were identified to be dynamically stable by quantifying the profile of the
phonon spectra. We demonstrated that the dynamical stability property of Ag$X$
is correlated with that of Au$X$. We predicted that as a 3D compound AuZr in
the B11 structure is dynamically stable. This was speculated by a strong
correlation of the quantity for the dynamical stability between Ag$X$ and
Au$X$. Future work is to explore another interrelationship of structural and
dynamical properties between the 2D and 3D compounds through more
comprehensive search in the periodic table and/or other structures (including
solid solutions) and with the use of machine learning approach.

In addition, we calculated the electronic and magnetic properties of BHC $AX$.
The AuK was predicted to be a wide-gap semiconductor, while a substrate
material is needed to stabilize the out-of-plane vibrations. The AgCr, AuCr,
CuMn, AgMn, and AuMn were predicted to show a ferromagnetic phase with a large
magnetic moment. An interesting but complex relationship between the magnetism
and the dynamical stability was also identified.

##  Appendix A Methods

We optimized the parameters $a_{\rm lat}$ and $\delta$ for BHC $AX$ by using
the DFT implemented to Quantum ESPRESSO (QE) package qe . We used ultrasoft
pseudo-potential dalcorso  and exchange-correlation functional parametrized by
Perdew, Burke, and Ernzerhof within the generalized gradient approximation pbe
. The self-consistent field (scf) calculations were performed by setting the
energy cutoff to be 80 Ry and 800 Ry for wavefunction and charge density,
respectively, and the energy convergence to be $10^{-10}$ Ry. The $k$ grid was
assumed to be 30$\times$30$\times$1 MK  with the smearing parameter of 0.02 Ry
smearing . The value of $c$ in $\bm{a}_{3}$ was set to be 14 Å to avoid
spurious interactions between different cells along the $z$ axis. The
threshold values for geometry optimization were set to be $10^{-5}$ Ry for the
total energy and $10^{-4}$ a.u. for the total force. Spin-polarized
calculations were performed for all compounds, where a ferromagnetic phase was
assumed as an initial guess of the scf calculations. The parameters (the size
of the $k$ grid and the threshold values for geometry optimization) used in
the present work are equal to those used in the previous work for elemental
metals ono2020 .

We next calculated the phonon dispersions along the symmetry lines of
$\Gamma$-M-K-$\Gamma$ by using density-functional perturbation theory (DFPT)
dfpt  implemented to QE. The parameters for the scf calculations were the same
as the geometry optimization calculations above. The convergence parameter for
the DFPT calculations was set to be tr2_ph=10-14. The $q$ grid was assumed to
be 6$\times$6$\times$1 (seven $q$ points in the irreducible Brillouin zone).
By diagonalizing the dynamical matrix, we finally obtain the phonon
frequencies $\omega_{\alpha}(\bm{q})$ at the wavevector $\bm{q}$ in the
$\alpha$th band with $\alpha=1,\cdots,6$. If $\omega_{\alpha}(\bm{q})$ is
imaginary, the BHC structure is unstable against such a phonon mode.

By using the QEqe , we optimized the geometry of B11 AuZr with the use of the
cutoff energies of 60 and 600 Ry for wavefunction and charge density,
respectively, and 16$\times$16$\times$16 $k$ grid. The optimized lattice
parameters were $a=3.554$ Å and $c=6.391$ Å for the tetragonal unit cell. The
phonon dispersion calculation was done by assuming 3$\times$3$\times$3 $q$
grid. We also performed first-principles MD simulation implemented in QE qe .
We used a 3$\times$3$\times$2 supercell (72 atoms) by considering the $\Gamma$
point in the Brillouin zone and set the ionic temperature of 500, 1000, 1500,
and 2000 K by using the velocity scaling method. The time step of 20 a.u.
(0.96 fs) was adapted and 3300 MD steps were done, corresponding to 3.2 ps.

###### Acknowledgements.

This study was supported by the a Grant-in-Aid for Scientific Research (C)
(Grant No. 21K04628) from JSPS. The computation was carried out using the
facilities of the Supercomputer Center, the Institute for Solid State Physics,
the University of Tokyo.

## References

  * (1) G. Zagler, M. Reticcioli, C. Mangler, D. Scheinecker, C. Franchini, and J. Kotakoski, CuAu, a hexagonal two-dimensional metal, 2D Mater. 7, 04501 (2020).
  * (2) J. Nevalaita and P. Koskinen, Atlas for the properties of elemental two-dimensional metals, Phys. Rev. B 97, 035411 (2018).
  * (3) J. Hwang, Y. J. Oh, J. Kim, M. M. Sung, and K. Cho, Atomically thin transition metal layers: Atomic layer stabilization and metal-semiconductor transition, J. Appl. Phys. 123, 154301 (2018).
  * (4) S. Ono, Dynamical stability of two-dimensional metals in the periodic table, Phys. Rev. B 102, 165424 (2020).
  * (5) S. Ono, Two-dimensional square lattice polonium stabilized by the spin-orbit coupling, Sci. Rep. 10, 11810 (2020).
  * (6) Y. Ren, L. Hu, Y. Shao, Y. Hu, L. Huang, and X. Shi, Magnetism of elemental two-dimensional metals, J. Mater. Chem. C 9, 4554 (2021).
  * (7) T. Wang, M. Park, Q. Yu, J. Zhang, and Y. Yang, Stability and synthesis of 2D metals and alloys: a review, Mater. Today Adv. 8, 100092 (2020).
  * (8) M. Ashton, J. Paul, S. B. Sinnott, and R. G. Hennig, Topology-Scaling Identification of Layered Solids and Stable Exfoliated 2D Materials, Phys. Rev. Lett. 118, 106101 (2017).
  * (9) K. Choudhary, I. Kalish, R. Beams, and F. Tavazza, High-throughput Identification and Characterization of Two-dimensional Materials using Density functional theory, Sci. Rep. 7, 5179 (2017).
  * (10) S. Haastrup, M. Strange, M. Pandey, T. Deilmann, P. S. Schmidt, N. F. Hinsche, M. N. Gjerding, D. Torelli, P. M. Larsen, A. C. Riis-Jensen, J. Gath, K. W. Jacobsen, J. J. Mortensen, T. Olsen, and K. S. Thygesen, The Computational 2D Materials Database: high-throughput modeling and discovery of atomically thin crystals, 2D Materials 5, 042002 (2018).
  * (11) J. Zhou, L. Shen, M. D. Costa, K. A. Persson, S. P. Ong, P. Huck, Y. Lu, X. Ma, Y. Chen, H. Tang, and Y. P. Feng, 2DMatPedia, an open computational database of two-dimensional materials from top-down and bottom-up approaches, Sci. Data 6, 86 (2019).
  * (12) M. Fukuda, J. Zhang, Y. T. Lee, and T. Ozaki, A structure map for AB2 type 2D materials using high-throughput DFT calculations. Mater. Adv. 2, 4392 (2021).
  * (13) Z. Lin, L. V. Zhigilei, and V. Celli, Electron-phonon coupling and electron heat capacity of metals under conditions of strong electron-phonon nonequilibrium, Phys. Rev. B 77, 075133 (2008).
  * (14) M. Xu, J. Y. Yang, S. Zhang, and L. Liu, Role of electron-phonon coupling in finite-temperature dielectric functions of Au, Ag, and Cu, Phys. Rev. B 96, 115154 (2017).
  * (15) M. Bauer, A. Marienfeld, and M. Aeschlimann, Hot electron lifetimes in metals probed by time-resolved two-photon photoemission, Progress in Surface Science 90, 319 (2015).
  * (16) A. Giri, J. T. Gaskins, L. Li, Y. S. Wang, O. V. Prezhdo, and P. E. Hopkins, First-principles determination of the ultrahigh electrical and thermal conductivity in free-electron metals via pressure tuning the electron-phonon coupling factor, Phys. Rev. B 99, 165139 (2019).
  * (17) D. Pettifor, Bonding and Structure of Molecules and Solids (Oxford University Press, New York, 2002).
  * (18) A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, and K. A. Persson, The Materials Project: A materials genome approach to accelerating materials innovation, APL Mater. 1, 011002 (2013).
  * (19) O. Alsalmi, M. Sanati, R. C. Albers, T. Lookman, and A. Saxena, First-principles study of phase stability of bcc $X\mathrm{Zn}$ ($X=\mathrm{Cu}$, Ag, and Au) alloys, Phys. Rev. Materials 2, 113601 (2018).
  * (20) S. Ono, Metastability relationship between two- and three-dimensional crystal structures: a case study of the Cu-based compounds, Sci. Rep. 11, 14588 (2021).
  * (21) S. Ono and H. Satomi, High-throughput computational search for two-dimensional binary compounds: Energetic stability versus synthesizability of three-dimensional counterparts, Phys. Rev. B 103, L121403 (2021).
  * (22) Y. Xu, M. Yamazaki, and P. Villars, Inorganic Materials Database for Exploring the Nature of Material, Jpn. J. Appl. Phys. 50, 11RH02 (2011).
  * (23) V. N. Eremenko, Yu. I. Buyanov, and N. M. Panchenko, Constitution diagram of the system titanium-silver, Powder Metall. Met. Ceram. 8, 562 (1969).
  * (24) Z. Kanghou, Z. Huaizhi, and Z. Yuehua, An investigation of the Ag-Zr phase diagram, J. Less-Common Met. 138, 173 (1988).
  * (25) K. Schubert, H. G. Meissner, A. Raman, and W. Rossteutscher, Einige Strukturdaten metallischer Phasen (9), Naturwissenschaften 51, 287 (1964).
  * (26) K. Schubert, H. G. Meissner, and M. Pötzschke, W. Rossteutscher, and E. Stolz, Einige Strukturdaten metallischer Phasen (7), Naturwissenschaften 49, 57 (1962).
  * (27) M. Aykol, V. I. Hegde, L. Hung, S. Suram, P. Herring, C. Wolverton, and J. S. Hummelshøj, Network analysis of synthesizable materials discovery, Nat Commun. 10, 2018 (2019).
  * (28) L. J. Nelson, G. L. W. Hart, and S. Curtarolo, Ground-state characterizations of systems predicted to exhibit $L{1}_{1}$ or $L{1}_{3}$ crystal structures, Phys. Rev. B 85, 054203 (2012).
  * (29) R. K. Linde, Lattice Parameters of Metastable Silver-Copper Alloys, J. Appl. Phys. 37, 934 (1966).
  * (30) T. U. Nahm, K. H. Park, S. J. Oh, S. M. Chung, and G. K. Wertheim, Partial spectral weights of disordered Cu-Au alloys, Phys. Rev. B 52, 16466 (1995).
  * (31) Y. C. Venudhar, L. Iyengar, K.V. K. Rao, Temperature dependence of the lattice parameter and the thermal expansion of Ag-Au (50 AT. %) alloy by an X-ray method, J. Less-Common Met. 60, 41 (1978).
  * (32) G. A. Tinelli, D. F. Holcomb, NMR and structural properties of CsAu and RbAu, J. Solid State Chem. 25, 157 (1978).
  * (33) S. N. Sharma, A. Weiss, Thermal and X-ray investigations of the quasi-binary system Ag1-xPdxMg (if $0\leq x\leq 0.5$), J. Less-Common Met. 104, L5 (1984).
  * (34) Y. Matsuo, S. Minamigawa, K. Katada, Phase Relation of the AuCd-AuMg and AuCd-AuZn Quasibinary Alloys, Trans. Jpn. Inst. Met. 22, 367 (1981).
  * (35) A T. Aldred, Intermediate phases involving scandium, https://www.osti.gov/servlets/purl/4772375.
  * (36) C. B. Walker and M. Marezio, A Single-Crystal X-Ray Study of Order in $\beta^{\prime}$-AgCd, J. Appl. Phys. 34, 1443 (1963).
  * (37) A. Chaves, J. G. Azadani, H. Alsalman, D. R. da Costa, R. Frisenda, A. J. Chaves, S. H. Song, Y. D. Kim, D. He, J. Zhou, Bandgap engineering of two-dimensional semiconductor materials, npj 2D Mater. Appl. 4, 29 (2021).
  * (38) W. E. Spicer, Photoemission and Band Structure of the Semiconducting Compound CsAu, Phys. Rev. 125, 1297 (1962).
  * (39) A. A. Adeleke, E. Stavrou, A. O. Adeniyi, B. Wan, H. Gou, and Y. Yao, Two good metals make a semiconductor: A potassium-nickel compound under pressure, Phys. Rev. B 102, 134120 (2020).
  * (40) C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, and J. Hone, Boron nitride substrates for high-quality graphene electronics, Nature Nanotech 5, 722 (2010).
  * (41) Z. Al Balushi, K. Wang, R. K. Ghosh et al., Two-dimensional gallium nitride realized via graphene encapsulation. Nature Mater. 15, 1166 (2016).
  * (42) C. A. F. Vaz, J. A. C. Bland, and G. Lauhoff, Magnetism in ultrathin film structures, Rep. Prog. Phys. 71, 056501 (2008).
  * (43) G. Grimvall, B. Magyari-Köpe, V. Ozoliņš, and K. A. Persson, Lattice instabilities in metallic elements, Rev. Mod. Phys. 84, 945 (2012).
  * (44) P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B. Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, and M. Cococcioni et al., Advanced capabilities for materials modeling with Quantum ESPRESSO, J. Phys.: Condens. Matter 29, 465901 (2017).
  * (45) A. Dal Corso, Pseudopotentials periodic table: From H to Pu, Computational Material Science 95, 337 (2014).
  * (46) J. P. Perdew, K. Burke, and M. Ernzerhof, Generalized Gradient Approximation Made Simple, Phys. Rev. Lett. 77, 3865 (1996).
  * (47) H. J. Monkhorst and J. D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13, 5188 (1976).
  * (48) N. Marzari, D. Vanderbilt, A. De Vita, and M. C. Payne, Thermal Contraction and Disordering of the Al(110) Surface, Phys. Rev. Lett. 82, 3296 (1999).
  * (49) S. Baroni, S. Gironcoli, A. D. Corso, and P. Giannozzi, Phonons and related crystal properties from density-functional perturbation theory, Rev. Mod. Phys. 73, 515 (2001).


# Towards Evology: a Market Ecology Agent-Based Model of US Equity Mutual
Funds

Aymeric Vie<EMAIL_ADDRESS>0000-0002-7178-1380 Mathematical Institute,
University of OxfordUK Institute for New Economic Thinking at the Oxford
Martin School, University of OxfordUK ,  Maarten Scholl  Department of
Computer Science, University of OxfordUK Institute for New Economic Thinking
at the Oxford Martin School, University of OxfordUK ,  Alissa M.
Kleinnijenhuis  Stanford UniversityUSA Institute for New Economic Thinking at
the Oxford Martin School, University of OxfordUK and  James D. Farmer
Mathematical Institute, University of OxfordUK Institute for New Economic
Thinking at the Oxford Martin School, University of OxfordUK Santa Fe
InstituteUSA

(2022)

###### Abstract.

The profitability of various investment styles in investment funds depends on
macroeconomic conditions. Market ecology, which views financial markets as
ecosystems of diverse, interacting and evolving trading strategies, has shown
that endogenous interactions between strategies determine market behaviour and
styles’ performance. We present Evology: a heterogeneous, empirically
calibrated multi-agent market ecology agent-based model to quantify endogenous
interactions between US equity mutual funds, particularly Value and Growth
investment styles. We outline the model design, validation and calibration
approach and its potential for optimising investment strategies using machine
learning algorithms.

agent-based model, calibration, finance, investment, market ecology, mutual
funds, validation

††copyright: acmcopyright††journalyear: 2022††conference: 2022 ACM
International Conference on Artificial Intelligence in Finance; November 2022;
New York, USA††booktitle: Benchmarks for AI in Finance Workshop††ccs: Applied
computing Economics††ccs: Computer systems organization Heterogeneous (hybrid)
systems††ccs: Computer systems organization Self-organizing autonomic
computing

##  1\. Introduction

###  1.1. Motivation

##### Research question

One key financial topic of discussion is the Value vs Growth debate. Does
Value, focusing on stocks trading for less than their intrinsic or book value,
beat Growth, investing in fast-growing companies, over the long term? What
investment styles can profit-driven machine learning search find in a multi-
fund simulation?

##### Motivation

A first piece of the answer is undoubtedly macroeconomic and monetary
conditions. The last ten years in financial markets have seen the Growth
investment style significantly outperform Value. One common reason to justify
this performance is the profitability of Growth investing in low-interest
rates environments. Growing companies rely on borrowing to fuel their
expansion. Hence low-interest rates facilitate this growth. With the recent
rising interest rates, and the 2022 bear market particularly affecting tech
stocks, those growth prospects appear less favourable and suggest a comeback
of the Value style. Outside of those external factors, does Value and Growth
performance depend on more endogenous factors such as their respective shares
of invested wealth and market composition? Can Value/Growth returns cycles
emerge from the endogenous interaction of those different investment styles?
Over the last 30 years, this rotation of winners visible in Figure 2 in the
appendix resembles the oscillations and cycles typically observed in
population dynamics. The theory of market ecology (Farmer, 2002; LeBaron,
2002; Musciotto et al., 2018; Lo, 2019; Levin and Lo, 2021; Scholl et al.,
2021) borrows concepts from ecology and biology to study financial markets.
Trading strategies are analogous to biological species: they exploit market
inefficiencies and compete for survival or profit. (Scholl et al., 2021) has
highlighted the nature of interactions between common trading strategies and
the strong density dependence of their returns for stylised trading styles. We
here propose to expand this agent-based approach more quantitatively to
investigate the Value/Growth interactions in investment funds. We focus on
mutual funds: investment companies that pool money from shareholders and
invest in securities portfolios.

##### Related work

This research is in the continuity of the rich area of financial agent-based
models and market selection with heterogeneous beliefs (Blume and Easley,
1992, 2006). For example, several ABMs have recently been introduced for
market-making optimisation (Spooner et al., 2018), understanding flash crashes
(Paulin et al., 2019) and providing sophisticated financial architectures for
trading training (Byrd et al., 2019). We attempt to develop the complementary
approach of market ecology (Farmer, 2002; Scholl et al., 2021) by focusing on
the ecological interactions between the different types of agents and
strategies.

##### Significance

This research topic participates in an active area of debate with a novel
approach. We describe how some particular results of the market ecology model
provide a new, exciting challenge for optimising investment styles using
machine learning algorithms. Such simulation-based training can account for
interactions and density dependence effects that could be significant and
overlooked by traditional time-series training. However, its importance is not
limited to the world of financial investment professionals. In the US alone,
according to the Investment Company Institute, more than 102 million
individuals and an estimated 48% of households own mutual fund shares
(Institute, 2022). The total retirement market assets in the US represent 39
trillion dollars, of which more than 12 trillion are invested in mutual funds.
Net sales of regulated open-end funds surged in 2021, with investors placing
more than 3 billion dollars in the sector, which holds an increasing share of
worldwide equity and debt securities (27%). Any currently unknown endogenous
dynamics at play within the market fund ecosystem thus carry actual, high-
magnitude economic impact.

###  1.2. Stylised facts of US equity mutual funds

Developing a more quantitative agent-based model of the mutual fund industry
requires laying down the characteristics of the system we are trying to model.
One of the critical elements of validation of the model is its correspondence
to the key attributes of real regulated funds. At year-end 2021, more than 34
trillion US$ were invested in open-end funds, which issue new shares and
redeem existing shares on demand. This broad category with total net assets of
34 trillion dollars includes mutual funds (27 trillion) and exchange-traded
funds (ETFs, 7.2 trillion) but also unit investment trusts (95 billion) and
closed-end funds (309 billion). The US contain more than 8,800 mutual funds
and 2,800 ETFs. The Investment Company Institute 2022 Fact Book (Institute,
2022) describes the total aggregate assets under management and the number of
funds of each type. We discuss modelling approaches (few aggregate agents vs
many agents) and present some mutual fund data in the appendix.

##  2\. Model

We consider a population of $n$ investment funds, which trade shares of a
single representative asset and cash. Every period-day $t$, the funds can buy,
sell and short-sell shares of the asset in constant supply. Asset shares pay
daily dividends $\delta(t)$ following an autocorrelated Geometric Brownian
Motion. Cash yields an interest rate $r$ paid daily. We build over the model
of (Scholl et al., 2021), with notable improvements in empirical calibration.
We show an example simulation run in the appendix.

InitialiseTrading signalDemandMarket
clearingDividends/interestSolvencyInvestmentTerminateFor $T_{\text{max}}$
periodsAfter $T_{\text{max}}$ periods Figure 1. Visual summary of the model
components.

###  2.1. Trading strategies and signals

Like real markets, our financial market model features a diverse sample of
stylised versions of the most common funds’ trading strategies (Scholl et al.,
2021). The previous section outlined the main results from the ICI data
(Institute, 2022), which suggests including Value and Growth funds, possibly
divided by their cap, in the model. For the current version of the model in
development, we include three styles: Value, Noise trading and Momentum (trend
following). Value investors (VI) form heterogeneous subjective valuations
$V_{i}$ of the asset based on discounted sums of dividends. Noise traders (NT)
trade on a similar valuation perturbed by a mean-reverting Ornstein Uhlenbeck
process $X(t)$, mimicking exogenous sentiment dynamics. We calibrate the
parameters of the Ornstein Uhlenbeck process to match empirical excess
volatility (Scholl et al., 2021). Trend followers (TF) trade on the existence
of trends in the asset price over various time horizons. Agents’ trading
strategies are represented by their trading signals $\phi(t)$.

(1)
$\displaystyle\phi_{i}^{\text{NT}}(t)=\log_{2}\left(X_{i}(t)V_{i}(t)/p(t)\right)$
(2) $\displaystyle\phi_{i}^{\text{VI}}(t)=\log_{2}\left(V_{i}(t)/p(t)\right)$
(3)
$\displaystyle\phi_{i}^{\text{TF}}(t)=\log_{2}\left(p(t-1)/p(t-\theta_{i})\right)$

###  2.2. Asset demand

The funds’ daily trading signals are inputs of the excess demand function for
the asset (Scholl et al., 2021). The excess demand function expresses the
demand of the fund for the asset as a function of the unknown price $p(t)$.
Fund wealth $W$ is the sum of agent cash, present value of asset shares and
liabilities. Our demand function features maximum leverage ${\lambda}$ and
strategy aggression $\beta$. Our demand function simply represents an investor
with asset position $S(t)$ and budget $\lambda W(t)$ spending a share
$\tilde{\phi}(t)$ of her budget on the asset, and $1-\tilde{\phi}(t)$ on the
cash (Poledna et al., 2014; Scholl et al., 2021).

(4) $D(t,p(t))=\tilde{\phi}(t)\frac{\lambda W(t)}{p(t)}-S(t)$

For $\tilde{\phi}(t)=\tanh(\beta\phi(t))$: the $\tanh$ function smooths and
bounds the trading signal in the range $[-1,1]$, so that the demand never
exceeds the agent budget including leverage. This demand function is
continuous, allows short-selling, and enforces deleveraging & margin
calls111Previous designs of the demand function in our model tended to
generate huge short positions. Because of the embedded deleveraging, i.e.
reduction of short positions in case of price increase, this demand function
over the simulation test run gives an average short position size equivalent
to $1.03\%$ of asset supply. This level is in line with the top10 NASDAQ
stocks ($1.17\%$ of float on average). If leverage increases, the short ratio
increases to riskier stock levels., and always deliver orders that respect the
budget constraint.

###  2.3. Market-clearing

The market-clearing process finds the price for which the sum of the funds’
demands equals the fixed asset supply $Q$, demand matching supply (Poledna et
al., 2014). This is equivalent to the market-clearing procedure of finding the
root of the aggregate excess demand function (Scholl et al., 2021). The
market-clearing condition here is thus: $\sum_{i}D_{i}(t,p(t))=Q$. While many
financial agent-based models use limit-order books (LOBs), our focus here is
on long timescales from decades to centuries, while those models focus on
intraday dynamics. We do not exclude using LOBs in this model’s future but
believe that market clearing is sufficient for our time horizon.

###  2.4. Dividends, interest and investment flows

After computing the clearing price, funds execute the resulting demand orders.
Agents receive capital gains: the dividends $\delta(t)$ and interest $r$
corresponding to their new positions. The actions of external investors play
an essential role in the wealth dynamics of mutual funds. Depending on the
performance of the funds, external investors can choose to redeem their shares
or buy new fund shares. Our model models those inflows and outflows in the
investment module according to empirical data on fund flows.

###  2.5. Solvency

Funds with negative wealth enter bankruptcy and exit the market. An
administrator slowly liquidates their shares. The wealthiest fund will split
into several identical, equal-sized entities to fill the vacant spot. This
mechanism keeps the number of funds and asset shares constant and limits
market perturbations due to insolvencies.

##  3\. Calibration and Validation

Calibration and validation of agent-based models (ABMs) are crucial (Paulin et
al., 2018). A common criticism of ABMs is that they often have too many
parameters and risk being unrealistic. Our validation includes three main
targets. The first is for the model to reproduce the stylised financial
properties of asset returns (Cont, 2001, 2007). The second is to model
realistic fund flows. The third is for the fund agents to be consistent in
various properties with the empirical data on mutual funds (Institute, 2022):
their number/size, returns, and investment styles. While we achieve the first
two, more work is necessary to satisfy the latter, as we detail in the
appendix.

###  3.1. Reproducing stylised facts of financial markets

Generating the so-called financial “stylised facts” is a popular requirement
for validating financial ABMs. Our model reproduces the main stylised facts of
asset prices (Cont, 2001). Our log prices display intermittency. The log price
returns do not show significant linear autocorrelations past trivial
frequencies. Returns show heavy tail distributions with excess kurtosis
compared to a normal distribution. We can also reproduce the leverage effect
-negative correlation between price returns and volatility- a positive volume-
volatility correlation and slow decay of autocorrelation in absolute returns.
We provide more details in the appendix.

###  3.2. Calibrating Investment flows

We calibrate model investment flows to ensure that funds’ returns lead to
realistic sales and redemption flows, as these flows represent a significant
share of funds’ net value. Investment companies are subject to reporting
requirements of their assets, redemptions, sales, and other indicators through
various SEC forms. (Ha and Ko, 2019) analysed N-SAR reports and fund returns
and identified a linear, positive relationship between funds’ excess return
and investment flow, suggesting that external investors are chasing returns,
confirming earlier results (Chevalier and Ellison, 1997). We estimated linear
regression models to predict net fund flows from fund excess returns at
various lags. Indeed, investors may look at fund returns on a monthly, yearly
or even multi-year basis. We identified a few significant predictors of net
flows. The constant regression term is negative, illustrating investors
redeeming their shares for liquidity purposes. 10-year excess returns have a
positive coefficient, and the constant is negative.

##  4\. Results

We present some vanilla dynamics of the Evology ABM without strategy
evolution: the asymptotic distribution of wealth between strategies. We are
interested in their dependency on the initial wealth distribution. On each
point of a uniform sample of $500$ points in the three-dimensional simplex, we
run the simulation $10$ times for $150$ years of trading. We measure the
average wealth share of the last $10,000$ days of trading. This sampling
ensures convergence in wealth distributions and accounts for stochasticity.
The returns and wealth shares of the strategies significantly vary with their
position in the simplex, showing density-dependence (Farmer, 2002; Scholl et
al., 2021). Experiment parameters ensure that we observe the asymptotic wealth
shares after convergence. Figure 2 provides an example of the price and
dividend series generated during a single simulation run, and appendix Table 2
presents the performance of the base strategies in the example run.

Figures 3-5 show the final wealth shares of each strategy after 150 years,
depending on the initial condition. Outside a specific corner of the simplex,
the wealth percentage of noise traders goes down to negligible levels of 5 to
20%. Value investors are almost absent from the left boundary but dominate
most of the simplex configurations with a majority of 70% of the wealth. Trend
followers dominate the unstable top region and are absent around the bottom
axis. Early extinction of the other strategies characterises this unstable top
region: if initialised in high proportion to other strategies, TFs are
detrimental to the other species.

##  5\. Trading Strategy Optimisation in Evology

Trading strategy search is a popular topic for applying machine learning.
Quantitative trading systems driven by linear & logistic regression, support
vector machines, reinforcement learning, deep neural networks, random forests,
genetic algorithms, and genetic programming have successfully created
profitable strategies (Allen and Karjalainen, 1999; Dempster and Jones, 2001;
Zhang and Maringer, 2016; Ta et al., 2018; Hasan et al., 2020). Derivative-
free methods are relevant since the profit objective usually cannot be reduced
to an explicit, derivable objective function. Evolution strategies and program
synthesis can also be particularly performant for this task.

The returns of a trading strategy depend on the wealth distribution of the
market, i.e. the wealth shares owned by the different strategies (Scholl et
al., 2021), challenging optimisation. Strategies experience crowding: their
returns decrease as their size grows and exceeds the carrying capacity of
their niche (Farmer, 2002; Scholl et al., 2021). Offline tuning, which assumes
no interactions between the strategy and the training data, can overlook this
effect. These additional difficulties invite reassessing the performance of
those popular machine learning approaches for trading strategy search in a
simulation environment. In addition, provided that the simulation environment
is realistic enough, training on the data-generating process behind market
dynamics can lead to more robust strategies than training on single sample
paths of this market process.

###  5.1. Benchmark learning tasks

##### General challenges and environments

Beyond the tasks described below, the strategies evolved in Evology should
satisfy some higher-end goals. We do not desire the machine learning
algorithms to result in incomprehensible, over-fitting strategies to maximise
profits. For the trained strategies to be interesting, they need to be
interpretable -display some level of economic insight- and robust
-successfully operate under various market conditions-. We develop in more
detail those general challenges and provide more details on the tasks
mentioned below in the appendix. We can consider two different environments:
an environment of stable strategies where the strategies of all the agents
(except the optimising agent) stay fixed over time. The alternative is a
coevolution setting where a small fraction of the population is adaptive,
imitating the activity of actively managed/hedge funds. This adaptation could
use an extensive range of models from the intensity of choice and imitation to
more sophisticated evolutionary or machine learning approaches. This
coevolution setting brings an additional challenge to strategy optimisation.

##### Task 1: Trading strategy optimisation

We are interested in an individual fund optimising its trading strategy
$\phi(t)$ to maximise profits during a $T$-period market run, using a profit
measure such as the cumulative return or the Sharpe ratio. The baseline levels
to achieve would be i) become more profitable than the base strategies and ii)
become more profitable than empirical strategies.

##### Task 2: Investment strategy optimisation

We are interested in an individual investor optimising its investment
strategy. The investor is learning how to invest, i.e. a function $\nu$
mapping the various fund characteristics (return, size…) to positive or
negative investment amounts, intending to maximise the profitability of their
investments.

##  6\. Conclusion

We present Evology222Available open-source
https://github.com/aymericvie/evology, an empirically calibrated financial
agent-based model of US equity mutual funds grounded on the market ecology
perspective. The complexity of strategies’ interactions and the density-
dependence of returns make this specific optimisation problem challenging for
search algorithms: dynamic, deceptive, and no clear optimal solution. This
multi-agent model has the potential to become a new training ground for
investment strategy search with machine learning methods. We further discuss
the model limitations and next steps in the appendix.

###### Acknowledgements.

This publication is based on work supported by the EPSRC Centre for Doctoral
Training in Mathematics of Random Systems: Analysis, Modelling and Simulation
(EP/S023925/1) and by Baillie Gifford. The authors thank Christian Schroeder
de Witt for helpful advice, Yeonjeong Ha and Kwangsoo Ko for sharing their
N-SAR and CRSP data (Ha and Ko, 2019), the Mathematical Institute of the
University of Oxford for the use of computing servers, the ICML22 AI4ABM
workshop where a previous version of this research was discussed (Vie et al.,
2022), members of Oxford INET and Fidelity Investments for advice and
anonymous referees for their valuable suggestions.

## References

  * (1)
  * Allen and Karjalainen (1999) Franklin Allen and Risto Karjalainen. 1999.  Using genetic algorithms to find technical trading rules.  _Journal of financial Economics_ 51, 2 (1999), 245–271.
  * Blume and Easley (1992) Lawrence Blume and David Easley. 1992.  Evolution and market behavior.  _Journal of Economic theory_ 58, 1 (1992), 9–40.
  * Blume and Easley (2006) Lawrence Blume and David Easley. 2006.  If you’re so smart, why aren’t you rich? Belief selection in complete and incomplete markets.  _Econometrica_ 74, 4 (2006), 929–966.
  * Brockman et al. (2016) Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. 2016.  OpenAI gym.  _arXiv preprint arXiv:1606.01540_ (2016).
  * Byrd et al. (2019) David Byrd, Maria Hybinette, and Tucker Hybinette Balch. 2019\.  ABIDES: Towards high-fidelity market simulation for AI research.  _arXiv preprint arXiv:1904.12066_ (2019).
  * Chevalier and Ellison (1997) Judith Chevalier and Glenn Ellison. 1997.  Risk taking by mutual funds as a response to incentives.  _Journal of political economy_ 105, 6 (1997), 1167–1200.
  * Cont (2001) Rama Cont. 2001\.  Empirical properties of asset returns: stylized facts and statistical issues.  _Quantitative finance_ 1, 2 (2001), 223.
  * Cont (2007) Rama Cont. 2007\.  Volatility clustering in financial markets: empirical facts and agent-based models.  In _Long memory in economics_. Springer, 289–309.
  * Dempster and Jones (2001) Michael Alan Howarth Dempster and Chris M Jones. 2001.  A real-time adaptive trading system using genetic programming.  _Quantitative Finance_ 1, 4 (2001), 397.
  * Dyer et al. (2022) Joel Dyer, Patrick Cannon, J Doyne Farmer, and Sebastian M Schmon. 2022.  Calibrating Agent-based Models to Microdata with Graph Neural Networks.  _arXiv preprint arXiv:2206.07570_ (2022).
  * Farmer (2002) J Doyne Farmer. 2002\.  Market force, ecology and evolution.  _Industrial and Corporate Change_ 11, 5 (2002), 895–953.
  * Ha and Ko (2019) Yeonjeong Ha and Kwangsoo Ko. 2019.  Misspecifications in the fund flow-performance relationship.  _Journal of Financial Intermediation_ 38 (2019), 69–81.
  * Hasan et al. (2020) Afan Hasan, Oya Kalıpsız, and Selim Akyokuş. 2020\.  Modeling traders’ behavior with deep learning and machine learning methods: evidence from BIST 100 index.  _Complexity_ 2020 (2020).
  * Institute (2022) Investment Company Institute. 2022\.  .  ICI. 262 pages.
  * LeBaron (2002) Blake LeBaron. 2002\.  Building the Santa Fe artificial stock market.  _Physica A_ (2002), 1–20.
  * Lehman and Stanley (2011) Joel Lehman and Kenneth O Stanley. 2011.  Novelty search and the problem with objectives.  In _Genetic programming theory and practice IX_. Springer, 37–56.
  * Levin and Lo (2021) Simon A Levin and Andrew W Lo. 2021.  Introduction to PNAS special issue on evolutionary models of financial markets.  _Proceedings of the National Academy of Sciences_ 118, 26 (2021).
  * Lo (2019) Andrew W Lo. 2019\.  _The adaptive markets hypothesis_.  Princeton University Press.
  * Musciotto et al. (2018) Federico Musciotto, Luca Marotta, Jyrki Piilo, and Rosario N Mantegna. 2018.  Long-term ecology of investors in a financial market.  _Palgrave Communications_ 4, 1 (2018), 1–12.
  * Paulin et al. (2018) James Paulin, Anisoara Calinescu, and Michael Wooldridge. 2018\.  Agent-based modeling for complex financial systems.  _IEEE Intelligent Systems_ 33, 2 (2018), 74–82.
  * Paulin et al. (2019) James Paulin, Anisoara Calinescu, and Michael Wooldridge. 2019\.  Understanding flash crash contagion and systemic risk: A micro–macro agent-based approach.  _Journal of Economic Dynamics and Control_ 100 (2019), 200–229.
  * Poledna et al. (2014) Sebastian Poledna, Stefan Thurner, J Doyne Farmer, and John Geanakoplos. 2014.  Leverage-induced systemic risk under Basle II and other credit risk policies.  _Journal of Banking & Finance_ 42 (2014), 199–212.
  * Scholl et al. (2021) Maarten P Scholl, Anisoara Calinescu, and J Doyne Farmer. 2021\.  How market ecology explains market malfunction.  _Proceedings of the National Academy of Sciences_ 118, 26 (2021).
  * Spooner et al. (2018) Thomas Spooner, John Fearnley, Rahul Savani, and Andreas Koukorinis. 2018.  Market making via reinforcement learning.  _arXiv preprint arXiv:1804.04216_ (2018).
  * Ta et al. (2018) Van-Dai Ta, Chuan-Ming Liu, and Direselign Addis. 2018\.  Prediction and portfolio optimization in quantitative trading using machine learning techniques. In _Proceedings of the Ninth International Symposium on Information and Communication Technology_. 98–105.
  * Vie et al. (2022) Aymeric Vie, Maarten Peter Scholl, and Doyne James Farmer. 2022\.  Evology: an Empirically-Calibrated Market Ecology Agent-Based Model for Trading Strategy Search. In _ICML 2022 Workshop AI for Agent-Based Modelling_.
  * Zhang and Maringer (2016) Jin Zhang and Dietmar Maringer. 2016.  Using a genetic algorithm to improve recurrent reinforcement learning for equity trading.  _Computational Economics_ 47, 4 (2016), 551–567.

## Appendix

###  .1. Value and Growth performance

Figure 2. Value and Growth relative performance (1994-2021). Source:
Canterbury Investment Management, ”Value versus growth: a brief historical
view”

###  .2. Mutual Funds: overview of ICI data

Table 1. Key elements on Equity mutual funds’ investment styles at year-end 2021 (Institute, 2022) Investment styles | TNA (million US$) | Number of funds
---|---|---
Growth small-cap | 315,759 | 168
Growth mid-cap | 354,192 | 164
Growth large-cap | 1,131,329 | 247
Growth multi-cap | 991,526 | 140
Value small cap | 197,397 | 194
Value mid cap | 269,347 | 172
Value large cap | 811,646 | 279
Value multi-cap | 377,509 | 167
Blend small-cap | 338,924 | 179
Blend mid-cap | 473,702 | 133
Blend large-cap | 3,741,454 | 380
Blend multi-cap | 1,679,108 | 221

###  .3. Model validation - Empirical stylised facts of asset prices (Cont,
2001)

The following figures related to stylised facts of asset prices use data from
a single simulation run, starting from the $[1/3,1/3,1/3]$ initial condition
on wealth shares. The interest rate equals $0.01$, external investment is
inactive, and the random seed is equal to $0$ for reproducibility of the
dividend and noise processes.

##### Absence of autocorrelations

Autocorrelations of asset returns should be insignificant except for very
small time scales, in which the microstructure has some impact.

Figure 3. Daily log returns of the asset price. For a price $p(t)$, the log
return at the daily timescale is $r(t)=\ln{p(t)}-\ln{p(t-1)}$. Figure 4.
Autocorrelation function of log returns of the asset price for up to 21
periods. After a short timescale of 5 periods, autocorrelations are not
significant from zero, which confirms the absence of autocorrelations.

##### Heavy tails

The unconditional distribution of returns should display a power-law or a
Pareto tail, with finite variance, excluding the normal distribution.

Figure 5. Histogram of the daily log price returns shows heavy tails. The
Fisher’s (or excess) kurtosis ($\kappa$) value for the series is $1.3$, which
is superior to Fisher’s kurtosis of the normal distribution, which is equal to
$0$. $\kappa=E\left[\left(\frac{X-\mu}{\sigma}\right)^{4}\right]-3$.

##### Gain/loss asymmetry

One should observe large drawdowns in stock prices without equally large
upward movements.

Figure 6. Asset price series of the simulation run. Graphical analysis
highlights faster downward than upward price movements: it usually takes more
time periods to recover from a drawdown than the drawdown duration.

##### Aggregational Gaussianity

As we increase the timescale for calculating the returns, their distribution
should look more and more like a normal distribution.

Figure 7. Distribution of yearly log price returns is closer to a normal
distribution than the distribution of daily log price returns in Figure 5.
Quantitatively, the excess kurtosis for monthly returns is lower, equal to
$0.85$, and the excess kurtosis of yearly returns is even lower at $0.32$.
Hence, we establish aggregational Gaussianity.

##### Intermittency

Returns should display high variability, visible by the presence of irregular
bursts in time series of volatility estimators.

Figure 8. 5-year moving average volatility over time show intermittency. We
attribute the relatively higher level of volatility in the early steps from
the runs to the transient period due to the agent-based model initialisation.
However, the irregular bursts in the log-returns of Figure 3 are not obvious,
perhaps given the large number of periods represented.

##### Volatility clustering and slow decay of autocorrelation in absolute
returns

Different volatility measures should display a positive autocorrelation over
several days, showing that high-volatility events tend to cluster over time.
We can observe this volatility clustering from the slow decay of the
autocorrelation function of absolute returns.

Figure 9. Autocorrelation function of absolute log price returns show long
range dependence and slow decay as a function of time. The autocorrelation
remains significantly positive over many time lags, showing volatility
clustering (Cont, 2007).

##### Leverage effect

Measures of asset volatility should negatively correlate with the asset
returns. Over our simulation run, this Pearson correlation is negative and
equal to $-0.005$.

##### Volume/volatility correlation

Trading volume should correlate with volatility measures. Over our simulation
run, this Pearson correlation is positive and equal to $0.48$. In addition,
volume positively correlates with price returns ($0.001$).

###  .4. Simulation run example

Figure 10. Price, fundamental value, volume and strategy types wealth shares
during a 50,000-day (around 200 years) simulation run, starting from the
initial coordinates $[1/3,1/3,1/3]$. Figure 11. Noise traders’ final wealth
share Figure 12. Value investors’ final wealth share Figure 13. Trend
followers’ final wealth share Table 2. Returns (daily and yearly geometric
mean, in %) and Sharpe ratios (daily and yearly for 252 trading days) of base
strategies at the initial coordinates $[1/3,1/3,1/3]$. Base strategies are:
NT, VI, TF, BH (buy and hold), IR (interest rate). The IR strategy has
constant returns hence no standard deviation.
| NT | VI | TF | BH | IR
---|---|---|---|---|---
Returns (d) | 0.007 | 0.007 | 0.006 | 0.007 | 0.004
Returns (y) | 1.668 | 1.806 | 1.424 | 1.687 | 1.005
Sharpe (d) | 0.029 | 0.026 | 0.02 | 0.14 | NA
Sharpe (y) | 0.463 | 0.41 | 0.317 | 0.222 | NA

###  .5. Estimating fund performance-flow relationship

We employ two separate procedures to estimate how the fund performance impacts
net cash flows, i.e. inflows minus outflows expressed as a function of the
fund TNA. The first uses fund-level data with the SEC Form N-SAR filings (Ha
and Ko, 2019), and the second aggregate data from the Investment Company
Institute (Institute, 2022). Both sources confirm that 10-year returns
positively influence net flows, with similar results on both data sources. Net
flows follow a constant decrease of a few percentage points of TNA,
corresponding to the higher-level wealth movements from active to index funds
and investor portfolio rebalancing during equity gains.

We define the fund return $r_{k}(t)$ between period $t$ and period $t-k$, with
$f(t)$ estimated net flows at period $t$. We can thus interpret $r_{k}(t)$ as
the fund performance before cash flows. For example, a fund may show a TNA
gain of $10\%$ if its portfolio gained $5\%$ and sold $5\%$ of TNA in new
sales. Taking $f(t)$ into account solves this bias and retrieves the fund
performance.

(5) $r_{k}(t)=\frac{TNA(t)-TNA(t-k)}{TNA(t-k)}-f(t)$

Excess returns are the return of a fund minus the average return of the funds
at this period, weighted by their size.

Net flows $f(t)$ are the difference between new sales (inflows) and cash
redemptions (outflows), divided by the fund TNA.

(6) $f(t)=\frac{I(t)-O(t)}{TNA(t)}$

####  .5.1. N-SAR SEC data

##### Data

(Ha and Ko, 2019) evaluated return discrimination between funds and the
investor substitution effect in this flow-performance relationship: the
tendency of some investors to redeem the worst performing funds to move to
better-performing funds. For this task, they examine actual net flows, inflows
and outflows using the Form N-SAR filings from the Securities Exchange
Commission EDGAR database and returns from the Center for Research in Security
Prices (CRSP) mutual fund database.

Form N-SAR A/B is a semi-annual report that registered investment companies
must file. The SEC EDGAR database started in 1994; the sample period is from
January 1994 to June 2015. The data focuses on US domestic equity mutual funds
in growth, growth&income and mid-small-cap fund styles, excluding exchange-
traded funds (ETFs) and index funds. The data also excludes funds with less
than $15M in total assets under management (TNA) and aged less than three
years. The data after this screening process contains 3562 domestic equity
funds.

##### Results

We regress monthly net flows as a function of past returns above the benchmark
(the weighted average return of funds), using various time lags (1M, 6M, 1Y,
2Y, 3Y, 4Y, 5Y, 10Y). We include some controls: fund age, fund TNA, investment
style, fund expense ratio and time (months). Table 3 shows the results.

We observe a negative constant of $-0.0164$ ($p=0.002$), meaning that, on
average, funds lose 1.6% of their TNA regardless of performance. This negative
constant can correspond to redemptions because of liquidity needs. We can also
interpret it as the ongoing shift from actively managed funds to index funds
or bonds. Finally, portfolio rebalancing may affect the constant since equity
gains are usually higher than bonds. If equity gains have been substantial,
investors following a 60-40 equity-bond allocation end up overweight in equity
and must redeem some fund shares to return to their target. Only the 1-month
and 10-year lags are significant among all lags, and both positively impact
net flows. One percentage point of a 10-year return increases the fund TNA by
0.43%.

However, R-squared is very low, and the data present excess skewness and
kurtosis, which suggests that the above model is not well specified or that
the data is very noisy due to high-frequency self-reporting. We move to the
aggregate ICI data to confirm our findings on monthly flows.

Dep. Variable: | Net_flows | R-squared: | 0.000
---|---|---|---
Model: | OLS | Adj. R-squared: | 0.000
Method: | Least Squares | F-statistic: | 1.056
Date: | Wed, 14 Sep 2022 | Prob (F-statistic): | 0.391
Time: | 14:30:06 | Log-Likelihood: | -1.2203e+05
No. Observations: | 73380 | AIC: | 2.441e+05
Df Residuals: | 73371 | BIC: | 2.442e+05
Df Model: | 8 |  |
Covariance Type: | nonrobust |  |

| coef | std err | t | P$>|$t$|$ | [0.025 | 0.975]
---|---|---|---|---|---|---
const | -0.0164 | 0.005 | -3.039 | 0.002 | -0.027 | -0.006
exc_return1 | 0.1630 | 0.095 | 1.721 | 0.085 | -0.023 | 0.349
exc_return6 | -0.0006 | 0.019 | -0.033 | 0.974 | -0.038 | 0.037
exc_return12 | -0.0021 | 0.023 | -0.093 | 0.926 | -0.047 | 0.042
exc_return24 | -0.0014 | 0.016 | -0.086 | 0.932 | -0.033 | 0.030
exc_return36 | 0.0089 | 0.014 | 0.642 | 0.521 | -0.018 | 0.036
exc_return48 | -0.0042 | 0.012 | -0.336 | 0.737 | -0.029 | 0.020
exc_return60 | -0.0012 | 0.008 | -0.157 | 0.875 | -0.017 | 0.014
exc_return120 | 0.0043 | 0.002 | 1.961 | 0.050 | 2.11e-06 | 0.009

Omnibus: | 302458.360 | Durbin-Watson: | 1.717
---|---|---|---
Prob(Omnibus): | 0.000 | Jarque-Bera (JB): | 2852915030912.781
Skew: | -111.696 | Prob(JB): | 0.00
Kurtosis: | 30548.664 | Cond. No. | 67.6

Table 3. OLS Regression Results of monthly net flows with SEC-N-SAR form data

####  .5.2. ICI data

##### Data

Another way of estimating the fund flow relationship is to use aggregated data
(Institute, 2022). Table 17 of the 2022 ICI FactBook, ”Long-Term Mutual Funds:
Net New Cash Flow, Total Net Assets, and Flows as a Percentage of Previous
Year’s Total Net Assets”, allows to aggregate flows as a percentage of
previous year TNA and regress those on the TNA change, from 1995 to 2021,
distinguishing between equity, hybrid and bond funds. Unlike our last results,
the data contain annual net flows.

##### Results

Using this data and computing annual lagged returns at 2, 5 and 10 years
horizons, the 10-year returns significantly positively impact net flows, as
shown in Table 4. One percent return at a 10-year horizon increases fund TNA
by 53% ($p=0.06$). Other lags do not appear significant. The constant is
significant ($p=0.015$) and negative with a high magnitude: mutual funds
annually appear to lose more than 2.4% of their TNA regardless of performance.
This negative constant can correspond in the data to the long-term outflows
from actively managed mutual funds to index funds, investor portfolio
rebalancing and liquidity shocks.

Dep. Variable: | Net Flows | R-squared: | 0.665
---|---|---|---
Model: | OLS | Adj. R-squared: | 0.601
Method: | Least Squares | F-statistic: | 10.42
Date: | Wed, 14 Sep 2022 | Prob (F-statistic): | 8.23e-05
Time: | 14:19:07 | Log-Likelihood: | -66.244
No. Observations: | 26 | AIC: | 142.5
Df Residuals: | 21 | BIC: | 148.8
Df Model: | 4 |  |
Covariance Type: | nonrobust |  |

| coef | std err | t | P$>|$t$|$ | [0.025 | 0.975]
---|---|---|---|---|---|---
const | -2.4610 | 0.933 | -2.638 | 0.015 | -4.401 | -0.521
return_1Y | 0.0175 | 0.043 | 0.403 | 0.691 | -0.073 | 0.107
return_2Y | 0.0420 | 0.031 | 1.361 | 0.188 | -0.022 | 0.106
return_5Y | 0.0080 | 0.014 | 0.577 | 0.570 | -0.021 | 0.037
return_10Y | 0.0053 | 0.003 | 1.961 | 0.063 | -0.000 | 0.011

Omnibus: | 1.195 | Durbin-Watson: | 0.686
---|---|---|---
Prob(Omnibus): | 0.550 | Jarque-Bera (JB): | 0.323
Skew: | 0.202 | Prob(JB): | 0.851
Kurtosis: | 3.367 | Cond. No. | 907.

Table 4. OLS Regression Results of annual flows using ICI data

###  .6. Supplementary details on the benchmark learning tasks

###  .7. Interpretability and robustness

Although interpretability in this context requires further definition, we can
start by observing that real-world strategies usually rest on some economic
reasoning. For example, trend followers assume that price trends are
persistent. Interpretable strategies should exhibit a generalisable “model” of
the financial market. As for robustness, the strategies can readily train on
various market initial conditions or face unseen policy interventions and
structural model changes, which are easy to implement in the ABM. For example,
in the current context of rising interest rates, we could train a strategy on
a probability distribution of interest rates rather than on a single interest
rate value. Finally, the Evology environment would be more useful if its
insights could generate useful signals for trading in real markets, an
additional challenge to robustness and ABM realism. Since the Lucas critique,
we know that changes in economic policies can lead to structural changes in
modelling and optimal behaviour. Since we are in a simulation environment, we
can alter market conditions, and training can include those changes. It is
easy to implement changes in the interest rates, dividend policies and other
interventions in an ABM. For example, Central Banks’ quantitative easing can
involve a fictitious agent with a constant positive excess demand for the
asset.

###  .8. Task 1: trading strategy

The trading is simplified to arbitrage between the asset and cash. Given a set
of market conditions $M$, the action space available to our adaptive fund is a
$T$-dimensional vector $\Phi$ of trading signal values. On each day $t$, the
fund decides if it is buying or selling asset shares.

(7) $\Phi=[\phi(0),\phi(1),\dots,\phi(T)],\phi(t)\in[-1,1]\ \forall t$

The optimisation problem faced by the adaptive fund consists in finding the
sequence of trading signals $\Phi$ that maximises some measure of performance
$\mathcal{P}$, typically a measure of profits such as the Sharpe ratio
(geometric mean of returns divided by the standard deviation of returns) or
wealth multipliers (ratio of increase of the funds’ wealth after $T$
simulation steps).

(8) $\max_{\Phi}\mathcal{P}$

The baselines to beat are first the performance of the base strategies
introduced in the model. The first definition of success for the adaptive fund
optimising a trading strategy is to achieve better performance than the NT, VI
and TF strategies present in the model, under the same bounded rationality
limitations. This environment requires the agent to understand its market
impact and the returns landscape and to learn to mitigate the effect of its
size on its returns. Once we achieve this target, setting a benchmark for
simulation performance is more difficult as this benchmark is new. However,
obtaining Sharpe ratios superior to empirical hedge fund levels, i.e. 10 or
20, would be a good starting point.

There are several interesting extensions of this task. The first is to
evaluate the performance of different optimisation goals $\mathcal{P}$. Does
profit maximisation lead to the highest profits, or is the problem so
deceptive that other metrics, multi-objective fitness, or even novelty search
(Lehman and Stanley, 2011) can lead to better strategies? We can also consider
two sub-tasks for Task 1. The optimised trading strategy could be a single
trading signal function $\phi(t)$ kept fixed during each evaluation: this
would be a static task. In a more dynamic approach, the agent could instead
evolve a sequence of different trading signal functions. Learning thus would
focus on the meta-strategy that governs the adoption at any period of a
specific trading signal.

##### Task 2: investment strategy optimisation

The investor is learning what characteristics of the funds should determine
their inflows and outflows so that the investor achieves the highest return on
their investment. Using a similar choice of the performance measure
$\mathcal{P}$, the investor maximisation problem is:

(9) $\max_{\nu}\mathcal{P}$

Where $\nu$ is a function that maps an investment decision (the quantity of
money to invest in or out of a fund) based on some fund characteristics x,
including measures of fund return at various timescales, the statistical
significance of those returns, fund size, fund strategy… It will be
fascinating to consider how this evolved investor behaves concerning
timescales (Scholl et al., 2021), compare its performance to typical
investment firms and how statistically significant their actions are. The
first baseline will be to achieve a higher success (e.g. profitability or
rationality) than the average investor behaviour implemented in the model and
derived from empirical data (Ha and Ko, 2019).

###  .9. Supplementary conclusions and limitations

##### Modelling approach

Modelling implies identifying the critical characteristics of the industry to
obtain a model that is simple enough to allow analysis but realistic enough to
be interesting. Two main approaches are possible to model the ecology of
mutual funds. The first is to focus on aggregate quantities of each investment
style and deal with a few agents, each representing Small-cap Value, Large-cap
Growth, and so on. This representation is possible with the total net assets
(TNA) data provided by (Institute, 2022). For instance, the 168 small-cap
growth mutual funds reduce to a single representative agent with a TNA of
315,759 million dollars. A second approach would be to model each of those 168
funds and do the same for all investment styles. This second approach, more
realistic, is much more data intensive. It is likely that no pair of funds in
the same style have the same strategy, expense ratios, and performance
history. Morningstar, Lipper, SEC’s EDGAR and the Center for Research on
Securities Prices (CRSP) can provide such detailed fund-level data at some
cost. In particular, this second approach needs fund-level data on the
portfolio strategy. If the data does not distinguish those strategies, those
168 identical agents are equivalent to a single agent aggregating the 168
funds TNAs. For data availability reasons, we are thus currently taking the
first approach. We consider the diversity of equity investment styles covered
by the ICI data and use representative, stylised agents for the different
investment styles: growth, value and blend. One additional common distinction
is also the market capitalisation of the stocks in the funds’ portfolio:
small-cap (company market capitalisation between 300 million and 2 billion),
mid-cap (2-10 billion), large-cap (¿10 billion, e.g. Apple) and multi-cap.

##### Possible developments of the model to other participants

Table 1 focuses on actively-managed Equity mutual funds. The three large
categories in the ICI classification are Equity, Hybrid and Bonds. Equity
(domestic and the world) represented more than 14.7 trillion dollars, hybrid
1.8 and bonds 5.5. Money-market funds, which invest in high-quality, short-
term debt, represent 4.7 trillion dollars. It is worth noting that the current
model does not consider the bond markets. Other participants may also reveal
themselves as necessary to include in the model. In particular, index funds
(index mutual funds and index ETFs) have recently gained in size and market
share, notably for their lower expense ratios. In the US alone, ETFs represent
7.2 trillion dollars in market share. Actively-managed mutual funds have been
experiencing continuous outflows while index funds have received inflows of
equal size. Index funds likely impact more and more the interactions between
mutual funds. Active funds represented in 2021 year-end more than 16 trillion
US$ (6680 entities) and index funds 5.7 trillion (496 entities). (Institute,
2022).

##### Improving the mutual fund and securities’ population in the ABM

We outlined in the Introduction some main characteristics of the mutual funds’
population and acknowledge that the current model partially meets those
characteristics. We identify key steps for our future work to improve
validation in this aspect:

  * •

Model entry and exit of funds (in 2021, 748 funds opened and 457 were merged
or liquidated),

  * •

Fit the model investment styles to the data with Value, Growth and Blend,

  * •

Multiply and diversify model assets to obtain small, mid and large-cap stocks
attached to a realistic set of companies at various stages of their expansion.
We would like a diverse set of companies that feature value and growth,

  * •

Ensure that the distribution of fund returns in the model is consistent with
empirical returns at various time horizons,

  * •

Adjust the initial conditions, i.e. the market composition in terms of TNA, to
the ICI data,

  * •

Consider the necessity of funds dividends, reinvestment and expense ratios,

  * •

Complement the ICI data with other classification methods to include more
diverse strategies, particularly ETFs, index funds and the bond market.

##### Next steps

A potential goal for this simulation could be to become an easy-to-use toolkit
for developing machine learning algorithms, just like the Open AI gym
(Brockman et al., 2016). It is thus essential that the environment offers
sufficient complexity to be of interest to the machine learning community and
is realistic enough for its insights to be potentially transferable to the
real world. The simulation should be efficient enough to allow training, e.g.
through developing a GPU implementation. Calibration and validation efforts
should continue to emulate market environments, mainly populating the ecology
with realistic strategies reconstructed from actual portfolios (Scholl et al.,
2021). Thanks to publicly available SEC data, model calibration constitutes a
benchmark task of interest. One task of potential interest that is not
mentioned yet in the model is the optimisation of financial regulation: we
could easily add to the model described in Figure 2 an additional regulation
component in which a policy-maker observes the market and attempts to limit
volatility, inflation or other measures of concern, by restricting agent
behaviour through maximum leverage, price interventions such as quantitative
easing or tightening. This addition would open a new task in which a machine
learning algorithm can attempt to optimise financial regulation to achieve
those goals.

##### Next-level calibration with simulation-based inference

Although we have reduced the number of free parameters in the model, we are
left with several free parameters of importance. We assumed that the various
heterogeneous sub-strategies (e.g. time horizon $\theta_{i}$, required rates
of return $\tilde{r}_{i}$) are uniform within arbitrary ranges. The leverage
of the various strategies is also significant to the ecology dynamics as they
impact the system’s position in the profit landscape. Several of the
abovementioned updates may also bring more free parameters to tackle. A poor
choice of summary statistics can lead to losing information from the empirical
data and reduce the quality of calibration (Dyer et al., 2022), which
motivates alternatives such as simulation-based inference using graph neural
networks. These procedures will likely require further exploration of
empirical data, such as the N-PORT forms and 13F filings.