Topology of Complex Networks and Demand of Intraday Liquidity: Based on the Real-Time Gross Settlement System

The liquidity flow in the financial system will change rapidly in direction or/and volume in a very short time frame. This characteristic of liquidity flow could make risk management and supervision more difficult. A significant number of studies in the literature have attempted to research how the topology of complex financial networks contributes to the expanding and shrinking of liquidity in a system. These studies include work done on the topology of complex financial networks and their formation (e.g., Boss et al. 2004; Garratt et al. 2011; Silva et al. 2016), the robustness of systems and the importance of nodes under different network topologies (e.g., Iori et al. 2006; Soramäki et al. 2013; Liu et al. 2016), and the contagiousness of systemic risk in a financial network (e.g., Imakubo and Soejima 2010; Lenzu and Tedeschi 2012; Hoffman et al. 2015). In the field of intraday liquidity, recent papers focus mainly on settlement mechanisms and their involvement (e.g., Bech and Soramäki 2005; Martin and McAndrews 2010; Tsuchiya 2013), the behaviour of members (e.g., Bech and Garratt 2003; Mills and Nesmith 2008; Abbink et al. 2010; Galbiati and Soramäki 2013), systemic externality (e.g., Bedford et al. 2004; Bech and Garratt 2012) and policies of central banks (e.g., Ball et al. 2011; Zhang 2012; Munoz and Gonzalez 2013). However, there are few papers that attempt to investigate the implications of network topology on the liquidity demand of individual members or on whole systems by analysing how the liquidity circulates through a complex network.

This paper aims to contribute to this understanding by offering a model of intraday liquidity demand and circulation in a complex network, and then by studying how the topology of network affects the intraday liquidity demand in an RTGS system through simulations. The paper is organized as follows. Section 2 provides a model to show how the liquidity of a member circulates through the network to settle the payments in one day. Section 3 describes the design of the network and the details of the simulation. Section 4 presents the results and Section 5 concludes.

In an RTGS system, the members settle their obligations using their intraday liquidity every trading day. Let the matrix P_n×n denote this directed and weighted intraday payment network, and let us suppose that every member in the system will have nonzero inflows or outflows or both in a trading day. Let the p_ij of P_n×n denote the total value of payment from member i to j in day, and, namely, the directed and weighted linkage between node i and node j. This yields p_ij = 0,∀i ≠ j and p_ij ≥ 0,∀i ≠ j. Using the terms of a complex network, the out-strength and in-strength of i equals the total outflow and inflow if i in day, respectively, i.e., OS_i = Σ_jp_ij and IS_i = Σ_jp_ij. Thus, the total strength of the network is the total value of payments in the system, i.e., TS = Σ _i OS_i = Σ _i IS_i = Σ _i Σ_j p_ij.

The real-time gross settlement is driven by the intraday liquidity held by members. The initial liquidity held by members at the beginning of a day is denoted as the vector 𝕃_n×1 (l_i ≥ 0∀i). We suppose that no additional liquidity can be acquired by any members during the day for settlement. 𝕃_n×1 will be injected into the network through the settlement of payments by members, and will circulate in the network to settle yet more payments.

From P_n×n we can generate the transition matrix Z_n×n′ the element of which is

zij=pijOSi.$$\begin{array}{} \displaystyle z_{ij}=\frac{p_{ij}}{OS_i} . \end{array} $$

The transitional probability z_ij is the chance that a payment from i is paid to j exactly, such that z_i: also shows the distribution of the total out-payment of i among other members (or nodes). Let d(𝕃) denote the function that transforms a liquidity vector to a diagonal matrix, with that vector as the main diagonal element. Let 𝕀_n×n denote a n × n identity matrix. Thus, the process of circulation of liquidity in the payment network can be expressed as:

Xn×n0=d(L)$$\begin{array}{} \displaystyle {\mathbb X}_{n\times n}^0=d({\mathbb L}) \end{array} $$(1)

Xn×n1=d(L)Z$$\begin{array}{} \displaystyle {\mathbb X}_{n\times n}^1=d({\mathbb L}){\mathbb Z} \end{array} $$(2)

Xn×nr=d(L)Zr−<!-- - -->1$$\begin{array}{} \displaystyle {\mathbb X}_{n\times n}^r=d({\mathbb L}){\mathbb Z}^{r-1} \end{array} $$(3)

Xn×n=∑<!-- ? -->rXn×nr=d(L)(I+Z+Z2+⋯<!-- ? -->+Zr−<!-- - -->1+…)$$\begin{array}{} {\mathbb X}_{n\times n}=\sum_r {\mathbb X}_{n\times n}^r=d({\mathbb L})({\mathbb I+}{\mathbb Z}+{\mathbb Z}^2+\dots+{\mathbb Z}^{r-1}+\dots) \end{array} $$(4)

where the element xijr$\begin{array}{} x_{ij}^r \end{array} $ of matrix Xn×nr$\begin{array}{} {\mathbb X}_{n\times n}^r \end{array} $ is the amount of initial liquidity that has flowed from i to j in the r-th round of the liquidity circulation. By summing the flows of all rounds, matrix 𝕏 shows the final settlement situation.

The following insights or influencing factors can be obtained from the model:

Smooth Circulation. If there is no liquidity leakage or stopping mechanism, equation (4) will not converge, i.e., every nonzero initial liquidity will circulate in the network forever, and therefore, an infinitive value of payments can be settled in the system.

Let 𝒯 = {0, 1, ..., T} ∈N be the set of trading time in a trading day, Pit$\begin{array}{} {\mathcal P}_i^t \end{array} $ be the set of payments of member i at the specific time t and let t∈𝒯 and Pt=∪<!-- ? -->i=1nPit$\begin{array}{} {\mathcal P}^t=\cup_{i=1}^n{\mathcal P}_i^t \end{array} $ be the set of payments of the entire system. Similarly, let St=∪<!-- ? -->n=1nSit$\begin{array}{} {\mathcal S}^t=\cup_{n=1}^n{\mathcal S}_i^t \end{array} $ be the set of settled payments and Qt=∪<!-- ? -->i=1nQit$\begin{array}{} {\mathcal Q}^t=\cup_{i=1}^n{\mathcal Q}_i^t \end{array} $ be the queued payments of the system at time t. This gives us Qit=∪<!-- ? -->Sit=Pit.$\begin{array}{} {\mathcal Q}_i^t=\cup\,\,{\mathcal S}_i^t={\mathcal P}_i^t. \end{array} $

According to the general rules of RTGS, the settlement of payment pijt$\begin{array}{} p_{ij}^t \end{array} $ will be attempted at time t following the time order, and will be settled successfully if and only if lit≥<!-- = -->pijt$\begin{array}{} l_i^t\ge p_{ij}^t \end{array} $ and 𝓭^t ≠ Ø. Then the payment sijt=pijt$\begin{array}{} s_{ij}^t= p_{ij}^t \end{array} $ will be added to s^t. Otherwise, if lit<<!-- ? -->pijt$\begin{array}{} l_{i}^t\lt p_{ij}^t \end{array} $ or Qt=∅,pijt$\begin{array}{} {\mathcal Q}^t=\emptyset,\, p_{ij}^t \end{array} $ will be queued into 𝓭^t, denoted by qijt.$\begin{array}{} q_{ij}^t. \end{array} $ When lit><!-- > -->lit−<!-- - -->1,$\begin{array}{} l_i^t\gt l_i^{t-1}, \end{array} $ the orders in i′s queue will be settled according to FIFO. The condition for which payment qijτ<!-- t -->$\begin{array}{} q_{ij}^{\tau} \end{array} $ can be settled is lit≥<!-- = -->qijτ<!-- t -->,∃<!-- ? -->qijτ<!-- t -->∈<!-- ? -->Qitand∄qijv∈<!-- ? -->Qit,v<<!-- ? -->τ<!-- t -->≤<!-- = -->t∈<!-- ? -->T.$\begin{array}{} l_{i}^{t}\ge q_{ij}^{\tau},\exists\, q_{ij}^{\tau}\in{\mathcal Q}_i^t\,\,\text{and}\,\,\not \exists\,q_{ij}^v\in{\mathcal Q}_i^t,v\lt\tau\le t\in{\mathcal T}. \end{array} $

At the end of a day, T, member i has settled all his payments if Qit$\begin{array}{} {\mathcal Q}_i^t \end{array} $ ≠ Ø, and has settled in real time only if Qit$\begin{array}{} {\mathcal Q}_i^t \end{array} $ = Ø, ∀t ∈ 𝒯. The minimum liquidity requirement to settle all payments of a member in real time is called upper bound liquidity (UBL) and is given by:

UBLi=max(maxt(Pi:t−<!-- − -->S:it),0)$$\begin{array}{} UBL_i=\max\Big(\underset{t}{\max}(P_{i:}^t-S_{:i}^t\Big),0) \end{array} $$ (6)

where Pi:t$\begin{array}{} P_{i:}^t \end{array} $ is the entire required payment value of i for unit of time t, and S:it$\begin{array}{} S_{:i}^t \end{array} $ is the whole settled payment value of i until time t, respectively. Similarly, the UBL of a system is defined by the sum of UBLs of all members as:

UBL=∑<!-- ? -->iUBLi$$\begin{array}{} UBL=\sum_i UBL_i \end{array} $$(7)

According to (6), we also have:

max(OSi−<!-- - -->ISi,0)≤<!-- = -->UBLi≤<!-- = -->OSi$$\begin{array}{} \max(OS_i-IS_i,0)\le UBL_i\le OS_i \end{array} $$(8)

∑<!-- ? -->imax(OSi−<!-- - -->ISi,0)≤<!-- = -->UBL≤<!-- = -->TS.$$\begin{array}{} \sum_i\max(OS_i-IS_i,0)\le UBL\le TS. \end{array} $$(9)

The relative UBL of member or system (RUBL) is defined as:

0≤<!-- = -->RUBLi=UBLiOSi≤<!-- = -->1$$\begin{array}{} \displaystyle 0\le RUBL_i=\frac{UBL_i}{OS_i}\le1 \end{array} $$(10)

0≤<!-- = -->RUBL≤<!-- = -->UBLTS≤<!-- = -->1.$$\begin{array}{} \displaystyle 0\le RUBL\le\frac{UBL}{TS}\le1. \end{array} $$(11)

This paper intends to study how the structural effect, i.e., the effect of strength, strength distribution or leakage, will affect the intraday liquidity demand of the entire system or a member to settle the payments in real time.

Other effects, such as the effect of gridlock or timing, will be controlled during the simulation.

at first, a directed and weighted globally coupled network with nodes is constructed as a bench network, denoted by FE. In FE, there is a direct link between every pair of nodes with equal weight w. Thus, the total strength of FE is TS = n(n - 1)w. In FE, there is no effect of strength distribution nor of leakage, because all nodes are homologous and their out-strength is equal to their in-strength. All other types of networks mentioned in this paper are derived from this bench network with an unchanged total strength of the system, TS, to eliminate the effect of strength in the system.

Distribution of total strength among the nodes. There are many complex financial networks for which the degree of distribution follows a power law. According to the BA model, we construct a scale-free network with the same number of nodes and total strength as the FE network, denoted by BS. The BS network is symmetric, i.e., the out-strength of every node is equal to its in-strength. In BS, there are not only effects of strength at the node level but also effects of strength distribution at the system level. The coefficient of variation of out-strength of all nodes, VOS, is calculated to denote the effect of strength distribution at the level of the system. OS_i is used to denote the effect of strength at the level of nodes.

By setting j = 1,2, ... 10 and transposing the network, a total of 84 networks (2 FEs, 2 BSs, 40 ESs and 40 Bas) are constructed. The TL and VOS or VTS of these networks are depicted in Fig. 1, as a symmetric network with homogenous nodes, are located at origin. BSs without a leakage effect are located on the horizontal axis. ESs and BAs have effects of both strength distribution and leakage, but both effects are greater in the BAs. Furthermore, the ESs have nonzero VOSs but zero VTSs.

Fig. 1

At the level of nodes (Fig. 2), all nodes of FEs are located at the origin, as well. In ESs, the total strengths of all nodes are equal, and VOS_i is negatively correlated to TL_i. All nodes of BSs are located on the horizontal axis. BA nodes have greater differences in both VOS_i and TL_i and there is no linear relationship between them.

Fig. 2

For every payment network, data of payment flows over 500 trading days are generated. The value per payment is generalized to one to control the grid-lock effect. The timing of payments in one day is evenly distributed. The payment flows are the same among all days with the exception of timing. The simulations are based on the algorithm of unlimited real time gross settlement to get the UBL of every day for every node and for the entire system. The UBL of 500 days is averaged to eliminate the effect of timing.

In this paper, we analyse the different influencing factors of intraday liquidity demand by modelling the circulating mechanism of liquidity in a network, and by examining the different structural effects on the liquidity demand of real-time settlement of both a member and of the whole system using different simulation methods. We find the following robust results:

that there is a more efficient liquidity circulation in symmetric network compared to an asymmetric network;

In light of previous findings, our results are useful for the management and supervision of short-term liquidity demand in complex financial systems, and for liquidity risk management and liquidity rescue policymaking.