OPTIMAL CENTRALIZATION OF LIQUIDITY MANAGEMENT†

SEBASTIAN POKUTTA AND CHRISTIAN SCHMALTZ

ABSTRACT. Liquidity is a key resource that banks have to manage on a daily basis. Large banking groups face thequestion of how to optimally allocate and generate liquidity: in a central liquidity hub or in many decentralized branchesacross different time zones, jurisdictions, and FX zones. We rephrase the question as a facility-location problem underuncertainty. We show that volatility is a key driver of the degree of (de-)centralization. As expected, in a deterministicsetup liquidity should be managed centrally in the most profitable branch. However, under stochastic liquidity demandand different time zones, FX zones, and jurisdictions we find that liquidity is preferably managed and generated in adecentralized fashion to a certain extend. We provide an analytical solution for the 2-branch model. In our setup, aliquidity center is an option on immediate liquidity. Therefore, its value can be interpreted as the “price of information”,i.e. the price of knowing the demand. Furthermore, we derive the threshold to open a liquidity center and show thatit is a function of the volatility and the characteristic of the bank network. Finally, we discuss the n-branch model forreal-world banking groups (n = 10 to 60 branches) and show that it can be solved with high granularity (100 scenarios)within less than 30 seconds.

1. INTRODUCTION

We study how a large banking group should organize its liquidity management. Such a banking group hasmany branches in various time zones, jurisdictions and FX-zones. Each branch faces a stochastic liquidity demanddriven by customer behavior. We ask whether such a group should run one global liquidity center, a set of regionalones or many local ones?

Liquidity management in general and its organization in particular are of utmost importance since the financialcrisis 2007/-09 that started as liquidity crisis and rose doubts on the soundness of the liquidity management ofmany banks. During the crisis the transferability of liquidity across FX-zones and jurisdictions was reduced (see[5, p. 13]). An issue that particularly globally operating banking groups face and that our model addresses.

Banks and liquidity are two closely related concepts: Both exist because of information asymmetries betweenlenders and borrowers. Banks internalize these asymmetries by producing illiquid assets (loans) that they keepon their balance sheet. Often these assets are funded by liquid deposits. The arising liquidity mismatch exposesthe banks to liquidity risk. Hence managing liquidity is a core activity of banks.

Research on banks’ liquidity management is as old as banks. Nevertheless, we are not aware of a paper thathas already addressed the organizational setup of liquidity production in a banking group. Many papers modelthe bank as a nucleus that interacts with outsiders as customers, banks or regulators. By contrast, our paper drillsdown in the nucleus splitting it up into a set of branches.

To identify the related literature, we outline our model setup.In each of the n branches of a globally operating banking group, clients unexpectedly deposit and withdraw

funds leading to a stochastic liquidity demand (Assumption 1). Intraday, the demand can be satisfied via thecentral bank account. However, at the end of the day the account must be cleared. Liquidity demand can becovered by liquidity production. We consider the cases of producing liquidity locally in a given branch itselfor remotely in a different branch. In the latter case additional costs for the transfer are incurred (Assumption2). Each branch can run a specialized department (treasury/liquidity center) that produces liquidity at the localinterbank market. In that case, we say that the branch is a “liquidity center”. Without the specialized department,the branch can only raise funds at the central bank (Assumption 3). In that case, we say that the branch is a

Date: October 19, 2009/Draft.Key words and phrases. Liquidity management, liquidity center location problem, facility location problem, real options.† The views and opinions expressed in this article are those of the authors only, and do not necessarily represent the views and opinions

of KDB Krall Demmel Baumgarten, or any of their affiliates and employees.We would like to thank Roland Demmel, Natalie Packham and the seminar participants at the Frankfurt School for Finance and Manage-

ment for their helpful comments.

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“non liquidity center”. We assume that interbank funds are substantially cheaper than the central bank funds(Assumption 4). Establishing a liquidity center requires an investment in staff and infrastructure which incursa fixed cost. A further complication is added by a time component. We assume that the remotely producedliquidity is not immediately available, but only with a certain delay motivated by the fact that executing a remoteliquidity request takes time (Assumption 5). The locally produced liquidity though is immediately availableas no transfer and legal clearance is required. Hence, a liquidity center is an option on cheap and immediate(interbank) liquidity. We further assume that in a first step, the board can only decide once upon the liquiditycenters and this decision is then assumed to be non-revisable for the considered time horizon. The model hasone production/transportation period under stochastic demand.

Hence, our paper is naturally related to the literature on (i) how banks manage liquidity, (ii) how they transferit between branches and (iii) how facilities (liquidity centers) should be allocated (facility-location literature).

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Gap

FIGURE 1.1. Asset-liability mismatch and liquidity gap

Banks must produce liquidity because outgoing cash flows (liabilities) are usually of shorter maturity thanincoming cash flows as suggested by Figure 1.1. The gap between incoming and outgoing cash flows is the socalled liquidity gap. In reality, the liquidity gap appears as a negative balance on the central bank account. Theliquidity gap is the challenge of liquidity management as it is a stochastic quantity driven by customer behavior.The variable in our model that corresponds to the liquidity gap is the liquidity demand. It is the stochastic driverof our model (see Assumption 1).

The gap (central bank account) must be closed (cleared) at the end of each day by additionally producedliquidity. Liquidity can be produced in two ways:

(1) Liquidation of assets: Liquid assets are sold on the market in order to generate liquidity.(2) Raising external funds: Banks seek out to obtain liquidity from other banks or the central bank.

Producing liquidity through any one of these channels involves cost — the haircut in case of liquidation and creditspread in case of raising external funds. Our model does not differentiate between the two channels as we do notwant to optimize across instruments. The literature on liquidity management gravitates around the stochasticliquidity gap. The distribution of the gap discusses [38]. Based on a data set of German Schmidt-Bank, thenormality assumption is rejected and a tailed distribution is proposed instead. Our real-world model (Section 3)uses a scenario approach, i.e., any empirical distribution can be implemented. In a simplified 2-branch model, weassume a normally distributed liquidity demand for analytical tractability (Section 2). Regulatory publications(see [9, 6]) discuss approaches to identify and measure the risk of the liquidity gap and the adequate size

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of risk buffers. In our model, liquidity buffers are implicitly considered if liquidity production is carried outremotely before the liquidity demand realizes. Papers that study the optimal instrument mix to manage theliquidity gaps are [23] and [18]. They present deterministic programming models that determine a mix onmoney market instruments that maximizes expected profit. Stochastic programming is also used in the contextof liquidity management. Such an approach for the optimal management of (independent) cash flow shocksusing short-term funds and investments has been presented in [33] and [17]. With these papers we sharethe assumption of a stochastic liquidity demand. Empirical studies on bank’s liquidity management are rare asmany liquidity variables are not publicly available. How banks react (using liquidity instruments) to externallychanged conditions is studied in [36]. The (empirical) role of loan sales and purchases as a new liquiditymanagement instrument is analyzed by [10]. Liquidity is produced on the money market. As the money marketis of systemic importance for the banking sector the study of its robustness is an active research field. The authorsof [28] provide an extensive survey of the money market literature and compile sources of market failures andthe appropriate regulatory preventive actions. However, this literature is only of secondary importance for ourpaper as we assume a smooth functioning of the money market. The money market can be further split up intointerbank and central bank market. We assume that the liquidity production on the central bank market is moreexpensive than on the interbank market (see Assumption 4). In general, the deposit rates of the central bankfloor, the lending rates cap the interbank market. Furthermore, we make the assumption that only specializeddepartments, i.e., liquidity centers, can trade on the interbank market. Standard branches have only access toexpensive central bank liquidity (see Assumption 3).

Liquidity is transferred between branches using payment systems. The payment systems in place are real-timegross settlement systems (RTGS). Their name-providing features are that the payments are irrevocably settledreal-time and on an order-by-order basis. In fact, Target2, the payment system of the Eurosystem settles 99,81%of the payments within 5 minutes (see [15, p.21,60]). A review of the academic literature on payment systemscan be found in [11]. A discussion from a practitioner’s point of view on the evolution, current situation andchallenges of payment systems can be found in [13]. There is a rich literature on the robustness of paymentsystems (see [22], [24], [26] , [32] , [21] and [25]). However, this literature is secondary for our paper as weassume the smooth functioning of payment systems. With respect to payment systems we make two assumptions:first, we assume that the transfer incurs (transportation) cost (see Assumption 2). Second, we assume that thetransfer takes time (see Assumption 5). As mentioned, the bulk of payments is settled within (negligible) 5minutes. However, our transfer time comprises all the time between observing a liquidity need, ordering liquidityin a remote branch and producing, transferring, settling the liquidity in the initial branch. Transfer cost compriseFX-conversion cost and any losses due to cross-border transactions (tax, corruption). In a recent paper [3] studythe FX-conversion cost in the light of the financial crisis. The assumption on the delay is an expert judgement aswe are not aware of an empirical paper that discusses time aspects for liquidity production in banks.

The facility location literature is the third stream that is relevant to our paper as our model is a facility loca-tion model. Facility location models optimize the locations across a network considering future production andtransportation. In our case, we optimize the allocation of liquidity centers across a branch network consideringthe future production and transfer of liquidity. The literature on this field is broad1. To identify the streams thatare relevant for our work, it is necessary to situate our model within the classification used in the literature.Following [12], our model is a discrete static stochastic facility location model. It is discrete, because liquiditycenters cannot be opened everywhere but only in (given) branches. It is static as the board can only decide onceon liquidity centers. In particular, it cannot revise its decision. It is stochastic as the distribution of our stochasticvariable (liquidity demand) is known. Extensive literature reviews of stochastic facility location models can befound in [35] and [30]. So far we specified the macro structure of the model. However, the solving methoddepends on the micro structure of the problem, i.e., the structure of the objective function and constraints. Weminimize expected cost although risk measures could be used instead or the objective function could be aug-mented by a (weighted) risk measure to include risk aversion. Our constraints are linear. The uncertainty liesin the state variable liquidity demand. The solving of stochastic models requires two steps: first, the conversionof the stochastic optimization problem into a deterministic one and second, the application of well-known algo-rithms (depending on objective, parameter and variable structure) to solve the deterministic model. For the firststep we use a scenario planning approach as discussed in [12] and [27, p. 75]. We then solve this problem using

1One might be tempted to write a survey for the surveys. In [31] it is mentioned that a survey from 1985 already lists around 1500references dealing with location and layout problems.

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a mixed integer linear programming approach in a second step. Despite the numerous applications of facilitylocation models to, e.g., Airline hubs, blood banks, day-care centers, fire stations (cf. [31], [8], [12] and refer-ences contained therein), this article is the first one, to the authors’ best knowledge, to apply the theory to thelocation of liquidity centers within a banking organization. The spirit of our application is similar to [4] that setup a location model with random demand and penalty cost for under- and overproduction. However, our modelis motivated by the principle of valuing liquidity centers as options. A model for the location of internationalfacilities incorporating exchange risk has been presented by [37].

A word on regulation. Banking is a highly regulated industry. In particular, we assume that regulators do notprescribe the treasury setup of a large banking group. However, recent publications indicate that regulators mightwant to have a tighter control over the organizational treasury setup of banks and thus in the future requirementson the liquidity setup might be postulated by regulators. In this case the optimal location of liquidity centersmight be exogenously biased.

Main results. Our model is based on two main assumptions:

(1) Transferring liquidity from one branch to another incurs a certain time delay (Assumption 2).(2) Having set up a liquidity center reduces liquidity production cost (Assumption 5).

Clearly, these assumptions are very mild and reflect minimal requirements and facts about producing and trans-ferring liquidity. Our analysis proceeds in three steps: 1-branch model, 2-branch model and n-branch model.

We confirm the traditional trade-off between fixed cost and volume (in form of expected demand) for the1-branch model. This would still hold for the 2-branch setup, if remotely produced liquidity was immediatelyavailable. The assumption of a time delay introduces an information advantage for liquidity centers: liquidity cannot only be produced cheaper (on the interbank market), but also after knowing the realized liquidity demand.This avoids any penalty cost of under-/ overproduction. We show that even nodes with an absolute comparativedisadvantage in producing liquidity might be set up as liquidity centers in order to reduce emergency liquiditycost. As part of our analysis we derive a price for the information on the exact demand in a branch. Thus,the liquidity center can be interpreted as an option on cheap and immediate liquidity. In that spirit, the optionprice (= reduced cost) of a liquidity center is identical to the price of information. We see that the network coststructure is supermodular, the law of diminishing returns holds and so the price of information depends on theliquidity center locations already put in place (see Section 2.1). The price of information constitutes indeed avolume independent price of a risk that cannot be hedged or eliminated (except for setting up a liquidity centerin the considered branch); a basis risk in that setting. The price of information implies a threshold volatility: aliquidity center is economically advantageous if the demand volatility is larger than this threshold and vice versa.

The n-branch model is formulated as an optimization problem which implicitly values the (liquidity center)options and determines the optimal allocation of liquidity centers. The marginal prices of information, i.e., theoption prices can be recovered using this approach (see Section 3.4). The model presented in Section 3.2 isscenario based in order to allow for the inclusion of highly unlikely stress test scenarios and correlation effects.In Section 4 we present a deterministic model for the special case of normally distributed independent demands.Stochasticity in this case is directly incorporated via the price of information. We show computational feasibilityfor both models, even for large branch networks.

Throughout the exposition we assume that the bank aims for expected cost minimization in accordance withshareholder value maximization but we would like to stress that risk measures such as value-at-risk or expectedshortfall can be easily incorporated into the model using the approach presented in [7].

The outline of the article is as follows. Section 2 discusses the problem of managing liquidity in general andformulates necessary assumptions and constraints for locating liquidity centers. We will then derive the analyticalcharacterization for the 1-branch and 2-branch models. Section 3 extends the 2-branch to a n-branch model thatis formulated as a network flow problem with a certain facility location characteristic. In order to do so wefirst introduce the necessary notions and recall some basic facts. Then we transform the liquidity center locationproblem into a corresponding mixed integer linear program and provide an economic interpretation of the dualvariables and the reduced costs in the context of our location problem. Section 4 shows how the scenario-basedmodel of section 3 can be reformulated as deterministic optimization problem for normally distributed liquiditydemands. Section 5 applies the models to show the effects of volatility and computational feasibility for real-world banking groups. Section 6 concludes.

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2. MANAGING LIQUIDITY

We stated that a bank is liquid if no central bank account has a negative balance at the end of the day. Thisimplies that a bank has to manage the liquidity demand for each central bank account on a daily basis. Inparticular, each central bank account is exposed to a stochastic liquidity demand that the bank must cover bygenerating additional liquidity. Liquidity can be produced at the interbank market by specialized departments,called liquidity centers, or at the central bank by any branch.

The advantages of a liquidity center are many-fold:

Branches running a liquidity center can produce liquidity cheaper. A branch that runs a liquidity center can produceliquidity cheaper as compared to the case where no center has been set up. We assume that a liquidity centerregularly accesses local markets leading to lower production costs. Without a liquidity center, the infrastructure(knowledge, systems) does not allow to access the interbank market, but only the central bank at elevated rates(i.e., emergency rates). The access to local markets generates contacts and reduces the time and cost to produceliquidity.

Locally produced liquidity is instantaneously available. Liquidity produced in the branch itself can immediately beused. If liquidity was produced remotely in a different liquidity center, there is a substantial time delay betweenthe initiation of the liquidity production and the final settlement. This delay arises from the operational stepsthat have to be performed to produce liquidity remotely. In our setup, the network graph models the delay viathe edge structure. The further introduction of different time zones disperses branches on the (absolute) timescale. In particular, it shifts opening and closing of branches and restricts the time that one branch can requesta liquidity production in another branch. However, this is only relevant for the intraday production of expectedliquidity as the end-of-day liquidity shortage (unexpected liquidity demand) can only be locally produced. Aremote production requested end-of-day would be settled too late to cover the central bank account of therequesting branch. Time zones therefore reduce the effective time window for production of expected liquiditydemand, but do not affect the decisions that depend on unexpected liquidity demand. We account for that byassigning time zones to branches.

Production of liquidity in the same currency as the demand. A liquidity center can produce liquidity in the samecurrency as the liquidity demand of that branch. The production in another branch and the subsequent transferwould require the conversion into the currency of the destination branch. The conversion could be costly due toelevated FX-rates or convertibility limits. In our setup, conversion cost are part of the transportation cost.

Local production avoids operational risks and different jurisdictions. The local production avoids all operationalrisks related to the transfer of liquidity. The transfer is sensitive to disruptions of the payment system and errorsin account details and amounts, and other country risk related issues. Additionally, we might face restrictions onthe transfer of funds due to different jurisdictions. The operational risk could be reduced by limiting the demandthat can be remotely produced. Our model incorporates such limits.

A remote production is only favorable if the remote branch has a comparative advantage in liquidity produc-tion. One might think of economies of scale (volume) and scope (reduction in operational risk due to frequentmarket accesses) that constitute comparative advantages.

A large banking group faces the problem of how to organize its liquidity production. Which setup is optimal:one global liquidity center, a set of centers, or many local centers optimal? And which branches should beupgraded to a liquidity center? We assume that the optimal setup is the one that minimizes expected overallproduction cost which is in accordance with shareholder value maximization. After having provided the economicrationale, we formalize the optimization problem starting with the 1-branch model.

1-branch model. Let Q denote the set of scenarios that we consider. A bank is liquid if its incoming cash flows(C F+) cover its outgoing cash flows (C F−) end-of-day in each scenario q:

(2.1) C F+q ≥ C F−q

Note that these cash flows result from customer products, i.e., they are driven by customer behavior. Analternative formulation of (2.2) after netting the customer cash flows (cf. Figure 2.1) as liquidity demand (d):

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Funding requirement

FIGURE 2.1. Funding requirement

(2.2) dq = C F−q − C F+q

We consider stochastic liquidity demand and thus dq might vary with the scenarios q ∈ Q. Due to banks’liquidity strategies, the demand is usually positive in the short- and negative in the long-run. Therefore, liquiditymanagement has to produce liquidity x in the short-run to cover the demand, i.e.,

xq − dq ≥ 0.

Producing liquidity incurs a cost of cA for each unit of liquidity for a liquidity center and of cN for a non-liquidity center. A liquidity center produces liquidity using a variety of instruments on the interbank market. Anon-liquidity center can only raise emergency liquidity at elevated cost at the central bank. We assume that theliquidity production on the interbank market is cheaper than with the central bank (cA < cN ) . The downside ofa liquidity center are the required fixed cost in staff, systems and rents.

We assume that shareholders require the minimization of expected cost. One obtains (2.3) if the bank runs aliquidity center

minx(q)∈R+

q∈Q

cA ·E[xq] + c f i x

s.t. xq ≥ dq ∀q ∈Q,(2.3)

and (2.4) if it does not

minx(q)∈R+

q∈Q

cN ·E[xq]

s.t. xq ≥ dq ∀q ∈Q.(2.4)

In the optimization problems (2.3) and (2.4), expected cost are minimized at the production volume xq∗ = d(second stage decision). Minimizing across (2.3) and (2.4) leads to the following decision rule (first stagedecision): a liquidity center is favorable, if it holds:

E[d] >c f i x

cN − cA,(2.5)

where E[.] denotes the expected value.6

FIGURE 2.2. Basic sets/structure for the 2-branch model

Equation (2.5) confirms the classical trade-off between fixed cost and the savings on variable (production) costfor the 1-branch model. The driving factor whether to run a liquidity center is the expected demand. Volatilityaspects do not matter because a liquidity center does not provide any informational advantage. Please note thatthe second-stage decision is a deterministic one as liquidity can be produced after the demand realizes.

2-branch model. The situation gets significantly more complicated if we consider different branches of a largebanking group and the possibility to have several liquidity centers. In order to keep analytical tractability, weextend (2.1) to two branches in a first step. The introduction of a second branch requires the following modeladjustments:

(1) Definition of basis sets: branches, nodes, graph(2) Node-specific variables and parameters(3) Assumptions on liquidity transport

As the n-branch model will be structurally identical, we introduce the notation for the n-branch model and specifythe 2-branch model as particular case.

Definition of basis sets: branches, nodes, graph. B is the set of all location/ branches. For our 2-branch model,we set B := {T, F} corresponding to Tokyo and Frankfurt. In liquidity management, time matters as the liquiditycondition has to be fulfilled before central banks close. In order to capture the critical dimension “time”, wedefine a discrete (world) time grid T := {0,6, 12,18, 24} and extend the branch set to a time-expended set ofnodes V ⊆ B×T . Hence, a node is a (branch, time) pair that describes the availability of a branch on the (world)time scale. For our 2-branch model, we consider that Tokyo is available from 0 to 12 and Frankfurt from 6 to 18leading to the node set (2.6):

(2.6) V := {T0, T6, T12, F6, F12, F18} ⊆ B× T.

The availability of branches and the node definition are depicted in Figure 2.2 (branches and node definition).

Node-specific variables and parameters. In the case of multiple branches, we do not have a single aggregateddemand but a liquidity demand dq(v) for each node v ∈ V and each scenario q ∈ Q . Furthermore, every nodecan produce liquidity xq(v). As in the 1-branch model, the cost depend on the status (center/ non-center), butalso on the node itself. This models the differences across financial markets: it is likely that Tokyo might neverbe able to produce liquidity as cheap as Frankfurt even by opening a liquidity center. Hence, setting up Tokyo

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as liquidity center reduces production cost only locally (from having to pay the central bank rate to paying theinterbank one). The production cost are defined to be:

c(v) =

¨

cA(v), v is liquidity center

cN (v), v is not a liquidity center.

For our 2-branch model, we have xq(v), dq(v) and cA/N (v) with v ∈ V as defined in (2.6).

Assumptions on liquidity transport. Liquidity can be transported across branches. Hence, our node set extendsto a time-expended liquidity network graph G := (V, E) as displayed in Figure 2.2. Allowing for liquidity flowsacross branches, the liquidity constraint for every node v ∈ V extends to the following flow inequality

xq(v) + inflowq(v)− outflowq(v)≥ dq(v)

i.e., the locally produced liquidity xq(v) plus the in-flow of liquidity from other locations inflowq(v) minus theout-flow of liquidity into other locations outflowq(v) must satisfy the demand dq(v).

With respect to liquidity transport, we make two assumptions:

(1) For every edge (v, w) ∈ E we have transfer costs or storage costs t(v, w) ∈ R+ depending on v, w beingthe same branch or different branches. Transfer cost account for cost incurred to convert from currencyof branch v to the currency of branch w and for non-contractual reductions, e.g., by corruption. (seealso Assumption 2)

(2) Remote liquidity production exhibits a time delay. The time delay covers the gross time between identi-fying a liquidity demand, communicating it to the other branch, the production in the other branch andtransport to the requesting branch. It is incorporated via the network edges. For our 2-branch exam-ple (see Figure 2.2), this is reflected by the fact that liquidity can only flow forward in time (see alsoAssumption 5).

The formal optimization problem for the n-branch bank is sketched in Figure 3.2. Section 3 discusses the n-branch model including the additional constraints that we incorporated to account for other factors like countryrisk and diversification.

The assumption on the transfer delay significantly changes the properties of our model. Therefore, we discussthe 2-branch model briefly without (deterministic case) and in detail with the delay assumption (stochastic case).

In order to isolate the effects, we make further assumptions for the 2-branch analysis:

(1) The liquidity demand for all branches at all times is zero except for Tokyo end-of-day, i.e., dq(v) = 0 forall v ∈ V \ {T12}.

(2) The liquidity demand for T is normally distributed: d(T12)∼ N(µ,σ) and represented by the scenariosq ∈Q. We can think of dq(T12) as a discretization of the normally distributed random variable d(T12).Note that µ and σ are not scenario dependent. We set xq(T ) := xq(T12).

(3) The branch F is set up as a liquidity center.(4) For readability, we set t(F, T ) := t(F6, T12)

Immediate liquidity transport (deterministic case). Without a delay, Tokyo can observe the liquidity demand andorder all its interbank liquidity in Frankfurt. The decision on the optimal volume is therefore deterministic as inthe 1-branch model: xq(T )∗ = d. Over- and under-production are zero. The center decision is made by choosingthe cheapest setup in terms of expected cost. The optimization problem (2.7) describes the setup “T is LiquidityCenter” and the optimal costs amount to

minxq (T )∈R+

q∈Q

cA(T ) ·E[xq(T )] + c f i x

s.t. xq(T ) ≥ dq(T )(2.7)

The optimization problem (2.8) describes the setup “T is not a Liquidity Center” and the optimal costs amountto:

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minxq (F)∈R+

q∈Q

(cA(F) + t(T, F)) ·E[xq(F)](2.8)

s.t.

xq(F) ≥ dq(T )

Thus, we set up Tokyo as liquidity center, if it holds:

c f i x

cA(F) + t(F, T )− cA(T )< E(d(T ))(2.9)

Again we encounter the well-known trade-off between fixed cost and savings in production cost as seen in the1-branch case with the exception that remote production got more expensive by the transfer cost. However, thecritical factor is the expectation of the demand, not its uncertainty (volatility).

Delayed liquidity transport (stochastic case). With the assumption of a delayed transfer, a new liquidity centerchanges the uncertainty structure of the bank: a liquidity center not only provides cheap, but also immediateaccess to interbank liquidity. Now, remote interbank liquidity does not only differ in its price from local interbankliquidity, but also in its information content: it has to be decided before the local demand is known. Therefore,remote production exhibits a new disadvantage: penalty costs due to over-/ under-production via the centralbank channel becomes necessary (almost surely). As the liquidity center can reduce uncertainty, the degree ofuncertainty (volatility demand) becomes a critical factor for the center decision. We will see that the informationon the unexpected demand has a price (price of information) and that there exists a volatility threshold thatseparates the case of not having a liquidity center in place from the one where it is economically advantageousto setup such a center in a given branch. In particular, it might be favorable to upgrade a branch to a liquiditycenter although its production cost (after transportation) are substantially higher as this ensures that not onlythe expected but also the unexpected demand can be produced at interbank cost and not at penalty cost. Thesavings on penalty cost might outweigh the fixed cost of the center. Note that the expected demand will still beproduced in the cheaper node.

To isolate the interplay between liquidity volatility and location decision, we make additional simplificationsfor the case at hand. Let q ∈Q denote the set of scenarios, then:

(1) The interbank production cost are the same for all nodes: cq(v) = c for all q ∈Q.(2) The transportation cost are set to zero: tq(v, w) = 0 for all q ∈Q and v, w ∈ V .(3) If T is run as a liquidity center, the local production is at cost c. If T is not run as a liquidity center,

local production is at pup and pop respectively as only emergency funds at central bank can be accessed.The cost for over-production are pop and the cost for under-production are pup for all nodes v ∈ V andscenarios q ∈ Q. Hence, we split up cN into pop and pup to account for cost asymmetry between over-and under-production.

In particular, the decision on the production level in Tokyo at cost c if T is a liquidity center is taken end-of-dayafter the demand is known, whereas if T is not a liquidity center, it is taken in Frankfurt at T6/F6 (cf. Figure2.2) as Frankfurt has to produce in advance to compensate for the delay. We will now compare the wait-and-seedecision in Tokyo which can be made after the demand is known with the here-and-now decision to be made inFrankfurt prior to the knowledge of the demand in Tokyo, i.e., without complete information. In this sense theliquidity production in Tokyo can be understood as a just-in-time production whereas the production in Frankfurtfor Tokyo has to be planned (without complete information).

Let d denote the end-of-day liquidity demand in Tokyo, i.e., d := d(T12). The objective functions reads asfollows:

In case T is not a liquidity center, then the demand has to be satisfied by Frankfurt:

cost(F) = c · x(F)(2.10)

+ pup ·max(0, d − x(F))

+ pop ·max(0, x(F)− d).

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As the decision on the production level x(F) in Frankfurt for Tokyo is to be decided in advance, d is a randomvariable with respect to our decision x(F).

In case T is a liquidity center, then demand can be satisfied by local production in Tokyo just-in-time:

cost(T ) = c · x(T )(2.11)

+ pup ·max(0, d − x(T ))

+ pop ·max(0, x(T )− d).

Therefore, d is known whenever we have to decide on x(T ) and thus by producing x(T ) = d the aboveformula simplifies to

(2.12) cost(T ) = c · x(T ).

We will now compare the two costs for this specific setup and derive a price on the demand information.

2.1. Volatility as a driver of decentralization. In the following section we will show that volatility indeeddrives the degree of decentralization. We will see that having a liquidity center set up can be understood as anoption on cheap and just-in-time liquidity. Thus, it provides insurance against unexpected volatility demand andcan be considered as a hedge against volatility in liquidity demand. We will illustrate the effects of volatility ondecentralization below. More specifically, we compare the here-and-now performance with lack of information tothe wait-and-see performance and thus we effectively establish a price/valuation for the missing information.

We compare the marginal cost between both scenarios. The costs for the two scenarios are given in (2.10) and(2.12) respectively. Let P denote the probability function. We are interested in the expected cost for productionlevel x(F) in case Frankfurt produces the liquidity for Tokyo, i.e.,

E[cost(F)] = c · x(F)+ pup ·E[max(0, d − x(F))]

+ pop ·E[max(0, x(F)− d)].

For a given production level x(F), the expected value of having a liquidity center in place in T is then given by

value[x(F)] = E[cost(F)− cost(T )]

= c ·E[(x(F)− d)]

+ pup ·E[max(0, d − x(F))]

+ pop ·E[max(0, x(F)− d)].

We can further split up c ·E[(x(F)− d)] into

c ·E[(x(F)− d)] =−c ·E[max(0, d − x(F))] + c ·E[max(0, x(F)− d)]

and thus the value of the position at hand is given by

(2.13) value[x(F)] = (pup − c) ·E[max(0, d − x(F))] + (pop + c) ·E[max(0, x(F)− d)].

Note that cost(F) represents only the variable, volume dependent cost and the fixed cost c f i x are treated sepa-rately as this will be our option premium. We assume that pup− c ≥ 0; otherwise there is no incentive to producelocal liquidity at all. The combination of these two positions in (2.13) is a classical straddle option strategy withstrike x(F) as depicted in Figure 2.3. The lines in the figure represent the cost under information certainty, thecurve (straddle) the cost under information uncertainty. Cost can only be completely eliminated under certainty.Under uncertainty, cost persists even for the optimal production volume. Thus the cost can be decomposed intoa component that the decision maker can minimize by choosing the optimal production volume. The base costhowever cannot be eliminated by the optimal production. It can only be eliminated by opening a liquidity center(moving from uncertainty to certainty). We will show this now in detail.

The minimal value of the position is the one obtained for an optimal production level x(F)∗ in F as it isreasonable to assume that F is a rational utility maximizer and hence a cost minimizer, i.e.,

value[x(F)∗] = minx(F)∈R+

value[x(F)].

10

FIGURE 2.3. Straddle position comprised of over-production and production deficit

This value can therefore be considered the true value of the position. In the following we will provide an analyticalderivation of value[x(F)∗]. In order to determine the minimum of (2.13) we will use the following lemma. Let Φdenote the cumulative distribution function and φ the probability density function of the standard distribution.

Lemma 2.1. Let η be a normally distributed random variable with mean µ and standard deviation σ. Then

∂

∂ xE[max(0,η− x)] = Φ

�

x −µσ

�

− 1=−P[η≥ x].

Similarly we have∂

∂ xE[max(0, x −η)] = Φ

�

x −µσ

�

= P[η≤ x].

Proof. Note that

E[max(0,η− x)] =ˆ ∞

x(η− x)φ(η)dη=

ˆ ∞xηφ(η)dη− x

ˆ ∞xφ(η)dη.

We therefore obtain∂

∂ xE[max(0,η− x)] =−xφ(x)−

ˆ ∞xφ(η)dη+ xφ(x) =−

ˆ ∞xφ(η)dη.

By the definition of φ we have −´∞

x φ(η)dη = −(1 − Φ� x−µ

σ

�

) = Φ� x−µ

σ

�

− 1 = −P[η ≥ x]. The secondequality follows similarly. �

Using Lemma 2.1 we obtain the value of the straddle position and the optimal production level:

Proposition 2.2. Let η be normally distributed variable with mean µ and standard deviation σ. Further let p1, p2 >0 and consider the straddle position

value[x] = p1 ·E[max(0,η− x)] + p2 ·E[max(0, x −η)].

Then value[x∗] = minx∈R+ value[x] is attained for x∗ := Φ−1( p1

p1+p2)σ + µ. Further, the optimal percentile z∗ =

Φ( x∗−µσ) is obtained as a convex multiplier of −p1 and p2.

11

Proof. As we want to determine the optimal solution x∗ we have to solve ∂ value[x]∂ x

= 0. Using Lemma 2.1 weobtain

0=∂

∂ xvalue[x] = p1 ·

�

Φ�

x −µσ

�

− 1�

+ p2Φ�

x −µσ

�

= −p1 ·�

1−Φ�

x −µσ

��

+ p2Φ�

x −µσ

�

.

This establishes that the optimal percentile z∗ = Φ( x∗−µσ) is indeed obtained as a convex multiplier of −p1 and

p2. Further we deduce,

Φ�

x −µσ

�

=p1

p1 + p2

and thus the optimal solution is obtained for x∗ = Φ−1( p1

p1+p2)σ+µ and clearly, as we consider a straddle position

and p1, p2 > 0, at x∗ a minimum is attained. �

Note that if either p1 = 0 or p2 = 0 the optimal solution is x∗ = ±∞ respectively. With p1 = pup − c andp2 = pop + c we therefore obtain that for our specific example the optimal production level x(F)∗ is given byx(F)∗ = Φ−1( pup−c

pup+pop )σ+ µ. We will now relate the straddle position to the expected shortfall to finally valuateour position:

Lemma 2.3. Let η be a normally distributed random variable with mean µ and standard deviation σ. Then

E[max(0,η− x)] =−(E[η | η≥ x]− x) ·�

∂

∂ xE[max(0,η− x)]

�

and similarly, E[max(0, x −η)] = (x −E[η | η≤ x]) ·�

∂

∂ xE[max(0, x −η)]

�

.

Proof. Observe that

E[max(0,η− x)] =ˆ ∞

x(η− x)φ(η)dη

=ˆ ∞

xηφ(η)dη− x

ˆ ∞xφ(η)dη

= E[η | η≥ x] · P[η≥ x]− xP[η≥ x]= (E[η | η≥ x]− x) · P[η≥ x].

Together with Lemma 2.1 we obtain

(E[η | η≥ x]− x) · P[η≥ x] =−(E[η | η≥ x]− x) ·∂

∂ xE[max(0,η− x)]

and the result follows. The second equality follows mutatis mutandis. �

Observe that the term E[η | η ≥ x] corresponds to the expected value of η given that it exceeds x which isequivalent to the expected shortfall for the level x . For a normally distributed random variable η the expectedshortfall has a particularly nice form:

Lemma 2.4. [see e.g., [7]] Let η be a normally distributed random variable with mean µ and standard deviation

σ. Then E[η | η≥ x] = µ−∂ 2

∂ x2 E[max(0,x−η)]∂

∂ xE[max(0,η−x)]

σ and similarly, E[η | η≤ x] = µ−∂ 2

∂ x2 E[max(0,x−η)]∂

∂ xE[max(0,x−η)]

σ.

Together with Lemma 2.1 we obtain E[η | η≥ x] = µ+φ� x−µ

σ

�

1−Φ� x−µ

σ

�σ and similarly, E[η | η≤ x] = µ−φ� x−µ

σ

�

Φ� x−µ

σ

�σ.

We will now combine Lemma 2.4 and Lemma 2.3:

Corollary 2.5. Let η be a normally distributed random variable with mean µ and standard deviation σ. Then

E[max(0,η− x)] = (µ− x)�

−∂

∂ xE[max(0,η− x)]

�

+∂ 2

∂ x2E[max(0, x −η)]σ

and similarly, E[max(0, x −η)] = (x −µ)�

∂

∂ xE[max(0, x −η)]

�

+ ∂ 2

∂ x2E[max(0, x −η)]σ.12

Using Corollary 2.5 we can valuate our straddle position

value[x(F)∗] = (pup − c) ·E[max(0, d − x(F))] + (pop + c) ·E[max(0, x(F)− d)]

= (pup − c)

�

(d − x(F)∗)�

−∂

∂ xE[max(0, d − x(F)∗)]

�

+∂ 2

∂ x2E[max(0, x(F)∗ − d)]σ

�

+ (pop + c)

�

(x(F)∗ − d)�

∂

∂ xE[max(0, x(F)∗ − d)]

�

+∂ 2

∂ x2E[max(0, x(F)∗ − d)]σ

�

.

and after rewriting and ordering terms we obtain

value[x(F)∗] = (x(F)∗ − d)�

(pop + c)�

∂

∂ xE[max(0, x(F)∗ − d)]

�

+ (pup − c)�

∂

∂ xE[max(0, d − x(F)∗)]

��

+�

pup + pop� ∂2

∂ x2E[max(0, x(F)∗ − d)]σ,(2.14)

with two summands that have a particularly nice economic interpretation. The first summand is the part thatcan be hedged by the rational cost minimizer and is proportional to the first derivative of value[x(F)]. By thechoice of x(F)∗ we thus have

(x(F)∗ − d)�

(pop + c)�

∂

∂ xE[max(0, x(F)∗ − d)]

�

+ (pup − c)�

∂

∂ xE[max(0, d − x(F)∗)]

��

= (x(F)∗ − d) ·∂

∂ xvalue[x(F)∗] = 0.

The second summand�

pup + pop� ∂ 2

∂ x2E[max(0, x(F)∗ − d)]σ constitutes the basis risk that we cannot hedge oreliminate. It is the price of information on the exact demand: having the information on exact demand availableis worth no more than

�

pup + pop� ∂ 2

∂ x2E[max(0, x(F)∗ − d)]σ. Therefore we have

value[x(F)∗] =�

pup + pop� ∂2

∂ x2E[max(0, x(F)∗ − d)]σ.

And so it pays off to set up T as a liquidity center if (2.15) holds:

(2.15) c f i x <�

pup + pop� ∂2

∂ x2E[max(0, x(F)∗ − d)]σ.

In the setting at hand we assumed that d is a normally distributed random variable and therefore we obtainin this particular case:

(2.16) value[x(F)∗] =�

pup + pop�φ

�

x(F)∗ −µσ

�

σ =�

pup + pop�φ

�

Φ−1�

pup + c

pup + pop

��

σ.

It is important to note that the price of information is linear in σ in this case and thus the unit price of information

is given by�

pup − pop�φ�

Φ−1�

pup+cpup+pop

��

, establishing relation between volatility and the degree decentraliza-tion. Further it is independent of the actual volume to be transferred.

In Figure 2.4 value[x(F)] is depicted for a fixed σ and 2pup = pop. The price of information is obtained forz∗ ≈ 80%. As expected, the optimal production level is located to the right of the 50%-percentile of d as beingliquidity short is more expensive then being liquidity long. Clearly, c f i x and value[x(F)∗] have to be consideredfor the same time horizon, i.e., if we calculate the value of the option on a time horizon of a year, we also haveto break down the fixed costs c f i x to the same horizon.

3. THE n-BRANCH MODEL

We will now formulate the liquidity center location problem as an optimization problem which can be solvedusing mixed-integer programming techniques (see e.g., [29, 34]). We first introduce the necessary preliminaries.

13

ExpectedCostasafunc0onofLiquidityProduc0onQuan0le

0

0,5

1

1,5

2

2,5

3

3,5

4

4,5

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

LiquidityProduc0onQuan0le

Expe

cted

Cost[€]

FIGURE 2.4. value[x(F)] vs. production quantiles

FIGURE 3.1. A network flow graph

3.1. Preliminaries. The formulation of the liquidity center location problem will be based on two commonproblem types in operations research and optimization. We will use network flow and facility location problemsin order to model our problem at hand. An extended discussion of these problems can be found in [34, 29].

Network flow problems. The first problem type is the network flow problem. Given a directed graph G = (V, E),we can transfer units of flow from v ∈ V to w ∈ V if (v, w) ∈ E. Such a network is depicted in Figure 3.1. Weare in particular interested in the case where we can generate flow in every node v ∈ V with cost c(v) per unit offlow and we have a flow demand d(v) for every node v ∈ V . Moreover, we have transportation cost t(v, w) forevery edge (v, w) ∈ E that are incurred whenever we transfer a unit of flow from v to w.

14

The objective is to produce flow with minimal cost so that all demands are met, i.e.:

min∑

v∈V

c(v)x(v) +∑

w∈V,(v,w)∈E

t(v, w) f (v, w)

s.t. x(v) +∑

w∈V,(w,v)∈E

f (w, v)−∑

w∈V,(v,w)∈E

f (v, w)≥ d(v) ∀v ∈ V.

The variables x(v)with v ∈ V are the (non-negative) production variables and the variables f (v, w)with (v, w) ∈ Eare the (non-negative) flow variables. In the absence of indivisibilities this problem can be solved efficiently usinglinear programming.

Facility location problems. The second type of problem is the (uncapacitated) facility location problem. Supposewe have m customers C := {C1, . . . , Cm} that we have to supply with a certain good and k potential locationsfor factories F := {F1, . . . , Fk} that could be the potential supplier. For each pair (Fi , C j) (with 1 ≤ i ≤ k and1 ≤ j ≤ m) we have a certain cost to ship the good from factory Fi to customer C j which we denote by t(i, j).This problem can be understood as given on a bipartite graph G = (C ∪ F , E) with (C , F) ∈ E if and only ifC ∈ C and F ∈ F . If the only constraint is that every customer demand has to satisfied, we assign each customerC j to factory Fi with t(i, j) minimal and thus obtain an overall optimal solution. Unfortunately, building andoperating a factory incurs additional costs (i.e., staff, infrastructure, etc). Thus we are interested in a trade-offbetween operational costs for the factories and shipping costs. Let c f i x be the fixed cost for operating a factory,let d j denote the demand for customer C j with 1 ≤ j ≤ m, and let M be the maximal demand in our instance.Then we can formulate the problem as follows:

min∑

1≤i≤k

c f i x b(i) +

∑

1≤ j≤m

t(i, j) f (i, j)

s.t. f (i, j)≤ M · b(i) ∀1≤ j ≤ m, 1≤ i ≤ k∑

1≤ j≤k

f (i, j)≥ d j ∀1≤ j ≤ m.

The variables b(i) are binary decision variables, i.e., either 0 or 1 that indicate if factory Fi is has been built. Thevariables f (i, j) with 1 ≤ i ≤ k and 1 ≤ j ≤ m are the (non-negative) flow variables and indicate the amount thathas been shipped from factory i to customer j. This problem is harder to solve in general due to the decisionvariables b(i), but can be tackled for reasonable sizes with state-of-the-art mixed integer programming solvers.

3.2. The liquidity center location problem. We are now ready to cast the sketched liquidity center locationproblem in terms of an optimization problem. Note that the simplifying assumptions made in the 2-branchsection are relaxed here. In particular, demands are not forced to zero. Furthermore, the demand is distributedaccording to the chosen scenarios. In particular, it is not necessarily normally distributed as assumed before. Nobranch is predefined as liquidity center. The interbank production cost are node-specific and transportation costare not necessarily zero.

We use a network flow model to distribute liquidity across the different locations. The decision to open aliquidity center or not adds a facility location aspect to the problem at hand (see Section 3.1). Our formulationuses a very limited number of integer variables making even large instances accessible to solvers that are availabletoday. In fact we have one binary variable for each potential liquidity center.

Let B = {b1, . . . , bn} denote the set of branches and let T be a discretization of the 24 hour horizon, e.g.,T = {0,6, 12,18, 24}. We consider the time-expansion of B by T and consider V ⊆ B×T where we add only nodesaccording to the active times of the branches. Thus each node that we consider is a (location,time) pair. In viewof our example the node F18 would correspond to Frankfurt at 6pm with respect to an arbitrary but fixed chosenlocal time. We further define the set of edges E to be given by

((v, g), (w, h)) ∈ E :⇔¨

v = w available and g is the immediate predecessor of hv 6= w both available and g +τ≤ h (+ extra requirements)

,

where τ is the delay due to remote liquidity production. The extra requirements could be, e.g., some additionalrequirements regarding the stability of the payment system. Effectively, the definition of the edge set E allowsfor determining which branch in the branch network is allowed to transfer liquidity to which other branch and

15

min∑

v∈Vq∈Q

P(q)

cq(v)xq(v) +∑

w∈V(v,w)∈E

tq(v, w) f (v, w) + pop(v)penop,q(v) + pup(v)penup,q(v)

+∑

b∈B

c f i x(b)l(b)

s.t.

xq(v) +∑

w∈V,(w,v)∈E

f (w, v)−∑

w∈V,(v,w)∈E

f (v, w) +penup,q(v)− penop,q(v) = dq(v) ∀v ∈ V, q ∈Q(3.1)

f (v, w) ≤ cr(v)dq(w) ∀(v, w) ∈ E, q ∈Q(3.2)

M · l(b) ≥∑

v∈V,q∈Q∃t∈T :v=(b,t)

xq(v) ∀b ∈ B(3.3)

xq(v), penup,q(v), penop,q(v) ∈ R+ ∀v ∈ V, q ∈Q

f (v, w) ∈ R+ ∀(v, w) ∈ E

l(b) ∈ {0,1} ∀b ∈ B

FIGURE 3.2. Liquidity center location problem

we can freely add or delete edges to account for special structural properties. A typical edge set without anyadditional requirements is depicted in Figure 2.2. Note that liquidity can only flow in the direction of time.

Using the notation from above we can formulate the liquidity center location problem as a mixed integerlinear program as shown in Figure 3.2. The variables xq(v) denote the produced liquidity (with v ∈ V ) for eachscenario q ∈Q — these are indeed scenario dependent as each branch can adjust its production just in time. Theflow variables indicating the transfer of liquidity from node v to node w are denoted by f (v, w) (with (v, w) ∈ E).These variables do not depend on the scenario as the transport takes, and flows in the direction of, time andthus has to be decided in advance. This condition incorporates the non-anticipativity known from stochasticprogramming approaches. Whether we open a liquidity center or not is expressed by the indicator variables l(b)(with b ∈ B). The parameter M , as part of a classical big-M formulation, has to be chosen sufficiently large, sothat M · l(b) ≥

∑

v∈V,∃t∈T :v=(b,t) x(v) does not limit the production in case of l(b) = 1. The variables penop,q(v)and penup,q(v) with v ∈ V measure the over-production and under-production respectively and are thus scenario-dependent with q ∈Q. We have three types of constraints in our model. Constraint (3.1) is the flow equality thatenforces that a demand dq(v) has to be covered by local production and inflow minus outflow. The differenceis the over-/under-production. In Constraint (3.2) we account for the country risk which ultimately might affectthe success probability of a transfer. We thus enforce certain transport limits, i.e., liquidity inflow from a remotelocation v with a high country risk and hence concentration risk must not cover more than cr(v) percent of thedemand. This constraint also ensures a certain diversification. The third condition, Constraint (3.3), ensures thatwe can only generate cheap liquidity in a branch b if it is indeed a liquidity center.

3.3. Stochasticity. To this end we decided to focus on stochastic demand. It is perfectly possible though tofurther extend the model to include more stochastic parameters or declare certain decision variables scenario-dependent. One candidate for such a change could be the country risk that could be argued to depend signifi-cantly on the scenario considered. Another potential candidate could be the costs for over- and under-productionwith a similar argumentation.

3.4. Recovering the option prices and economic interpretation. Given a liquidity center setup C ⊆ B we cancompute the marginal price of information for each branch b0 ∈ B \ C in the following way. We first computethe optimal solution to the liquidity center location problem (see Figure 3.2) and we obtain the optimal liquiditycenter locations (l(b))b∈B. Let cost∗ denote the optimal objective function value. For every b ∈ B with l(b) = 1,we fix this variable to 1, i.e., we add a constraint of the form l(b) = 1 to our optimization problem. Further weadd a constraint l(b0) = 1 , i.e., we force the liquidity center to be set up. Note, that we confine ourselves to thecase b0 ∈ B \ C; the case that b0 ∈ C follows analogously by adding the constraint l(b0) = 0 and performing thesame computation.

16

Now we solve the augmented model again and obtain an optimal objective function value cost for the aug-mented model. Let the difference of the two values be denoted by 4 :=cost− cost∗. It holds,

c f i x(b0)≥ cost− cost∗ ≥ 0.

As cost∗ was the objective function value of the overall optimal solution without fixing certain variables we havecost− cost∗ ≥ 0. On the other hand observe that just adding b0 to the network without any production assignedwe obtain a solution whose objective function value is no worth than cost∗ + c f i x(b0). Now, if the optimalobjective function value for the augmented optimization problem satisfies cost < cost∗ + c f i x(b0) the additionalvalue generated from setting up b0, i.e., the marginal price of information for the configuration C is given by

value(b0, C) := c f i x(b0)−4.

When dealing with linear optimization problems (without integer variables) 4 is well approximated by thereduced cost of the variable l(b0). The decision of setting up a liquidity center or not is a binary one thoughand so we have to solve the two optimization problems separately and compare the objective function value.Nonetheless, the solver uses the reduced cost information to decide which location can appear in the optimallocation assignment. In particular if we have

costLP + costred(b0)> cost+

with costLP being the optimal objective function value where the integrality constraint on l(b) has been relaxedto l(b) ∈ [0, 1] for all b ∈ B, costred(b0) being the reduced cost of b0 associated with the relaxed solution, andcost+ being the objective function value of any feasible solution to the original problem, then b0 cannot be aliquidity center in any integral optimal solution.

Once an optimal location arrangement is chosen, the optimization problem simplifies to a linear program. Inthis case we have the full duality theory available and obtain marginal prices for the demand in any node v ∈ V .The dual prices for the flow equalities (cf. (3.1)) provide the cost for an additional unit of demand on average(for a given node v ∈ V ). Further we can extract the cost of concentration risk from inequalities (3.2) and theassociated dual prices.

3.5. Supermodular cost structure. In the n-branch model, possible decisions are not just those of setting upan additional liquidity center but also decisions of the either-or-type, e.g., setting up liquidity centers in eitherlocations A, B, and C or in locations X and Y . These decisions could still be valuated in an option framework,but determining the value is more complicated. Further the overall structure of the problem changes as now thevalue of the option is contingent on the number or liquidity centers already in place. More specifically we havea supermodular cost structure: Say we have C1 ⊆ C2 ⊆ B where B is the set of branches and the branches in C1and C2 have been set up as liquidity center. Further let b ∈ B \ C2, i.e., a branch that has not yet been set up as aliquidity center. Then it holds

cost(C1 ∪ {b})− cost(C1)≤ cost(C2 ∪ {b})− cost(C2)

as the liquidity center b added to the large network C2 might generate less savings than when added to a smallernetwork C1 although the fixed cost is identical in both cases. Therefore when considering a branch network thelaw of diminishing returns (also referred to as the diseconomies of scale) holds and thus we have to consider themarginal price of information for a specific configuration at hand. Due to that specific cost structure there is anatural saturation on the number of liquidity setups.

3.6. Scenario generation. An important aspect when dealing with stochastic models where the stochastic be-havior is introduced via scenarios is the actual generation of scenarios. Similar to the calculation of potentialfuture exposure for trading positions via a Monte-Carlo approach, the actual choice of the scenarios can (andusually does) affect the solution of the optimization problem. The generated scenarios should reflect the risk ap-petite of the bank and should be consistent with liquidity stress testing scenarios. The Federal Deposit InsuranceCorporation (FDIC) wrote [16]:

Bank managers must focus on adequate liquidity during both normal times and times of stress.Liquidity managers are rightly concerned with profitable, efficient operations in normal eco-nomic environments. The best managers use scenario analysis to balance the inverse relation-ship between liquidity and earnings during good times, but will also spend time evaluating the

17

min c·∑

v∈V

x(v) +∑

w∈V(v,w)∈E

t(v, w) f (v, w)

+∑

b∈B

c f i x(b)l(b) + (1− l(b)) ·

∑

v∈V∃t∈T :v=(b,t)

PI(v)

s.t.

x(v) +

∑

w∈V,(w,v)∈E

f (w, v)−∑

w∈V,(v,w)∈E

f (v, w)

≥ l(b) ·E[d(v)] + (1− l(b)) · P L(v)(4.1)

∀v ∈ V, b ∈ B, t ∈ V : (b, t) = v

f (v, w) ≤ cr(v)P L(w) ∀(v, w) ∈ E

M · l(b) ≥∑

v∈V∃t∈T :v=(b,t)

x(v) ∀b ∈ B

x(v) ∈ R+ ∀v ∈ V

f (v, w) ∈ R+ ∀(v, w) ∈ E

l(b) ∈ {0, 1} ∀b ∈ B

FIGURE 4.1. Deterministic formulation for normally distributed demand

impact of stressful, low-probability liquidity events. When evaluating liquidity risk and isolatingthe liquidity component rating, examiners are primarily concerned with the risk managementinformation derived from management’s evaluation of more extreme liquidity scenarios.

We do not want to further address the scenario generation. A scenario reduction approach for stochastic pro-gramming that approximates the potentially large amount of (stress testing) scenarios can be found in [20, 14].

4. A DETERMINISTIC FORMULATION OF THE N -BRANCH MODEL FOR NORMALLY DISTRIBUTED DEMAND

In this section we want to consider the special case that all demands are normally distributed and all the otherparameters are deterministic. We further assume that the demands are independent and the actual productioncosts c(v) = c for some c is identical for all v ∈ V . Under these assumptions we can use the price of informationdirectly to obtain a deterministic model that is equivalent to the stochastic model presented in Section 3.2. LetPI(v) be the price of information (cf. (2.16)) and further let P L(v) denote optimal amount of remote liquidityrequested (cf. Proposition 2.2) for each node v ∈ V . With the notation from Section 3.2 we can formulatea deterministic model as specified in Figure 4.1. Note that we do not have any scenario dependency in thisdeterministic model.

It is easy to verify that the formulation does indeed describe the optimal configuration. For any given nodev ∈ V with demand d(v) the optimal production level of remote liquidity P L(v) is obtained by Proposition 2.2.In Constraint (4.1) we require that at least this amount is the net inflow into v given that v (or more specificallythe associated b) is not set up as a liquidity center. If v ∈ V is setup as a liquidity center, then we require thatwe produce at least the expected demand E[d(v)] but potentially more as we might produce liquidity for othernodes as well. The objective function slightly changes as well, as we now (exclusively) either pay the fixed costfor a branch b ∈ B to set it up as a liquidity center or we pay the price of information PI(v) for each associatedv ∈ V which is the expected extra cost for not having a liquidity center in place.

This approach is a convenient way to remove scenario dependency from the model given that the underlyingvariables are independent and normally distributed. It would be interesting to consider this approach for genericn-stage stochastic programming with simple recourse provided that the variables are normally distributed or thedistributions admit a closed form solution for the price of information.

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100

200

300

400

Center Opening depends on Demand Volatility

Tokyo:

-200

-100

0

100

0 50000 100000 150000 200000 250000 300000 350000

Analytical Mixed Integer Programming Formulation (100 Scen)

demand volatility

Tokyo:

No Ly Center

Ly Center

FIGURE 5.1. Comparison analytical formula vs. numerical model

5. COMPUTATIONAL RESULTS

This section discusses computational results for the n-branch models presented in Section 3 and 4. In order toverify the correctness of the n-branch formulation we performed several tests, one of which was the comparisonto the analytical 2-branch model of Section 2. Implementing the simplifications for the 2-branch model for asuitable number of approximation scenarios of the normal distribution, the n-branch and the analytical 2-branchmodel generate identical results with negligible errors (see Figure 5.1).

For the demonstration of the effect of volatility we devised an exemplary 10-branch instance. Details on theparameters and their description are summarized in Table 1. We distinguish external and internal parametersdepending whether their calibration is based on market (external) or bank (internal) estimates. Note that ourparameter structure is rich as it covers the following dimensions:

(1) FX- and operational risk (via transportation cost)(2) Country risk (via transportation cost)(3) Concentration risk(4) Over-/ and under-production(5) Different interbank market conditions

We strongly believe that these components have to constitute the minimal requirements for any reasonable coststructure when dealing with liquidity center location decisions. A more differentiated cost structure accountingfor more risk drivers is recommended for all real world applications. We are currently cooperating with a bankto obtain the necessary cost information to create a more sophisticated cost structure for a specific bank and itsbranch network. Its analysis will be subject to an upcoming article.

For a 10-branch instance, the number of parameters is already large: For each node, we have 10 x 7 parametersfor c f i x(v), c(v), etc. according to Table 1 plus 100 transportation parameters (10 x 10 transportation cost

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Type Cost Description

External Fixed Cost measure salaries, rentand technical infrastructure

External Production Cost measure cost to raise liquidityon interbank market

External Underproduction measure cost to produce liquidityCost at central bank

External Overproduction measures yield reduction ofCost investing in liquid assets

External Transportation measure (a) currency conversion costCost and (b) losses due to corruption

Internal Concentration % of demand to which extend remote branchRisk Factor is allowed to cover faced demand

Internal E[Demand] Measures the expectedper branch liquidity demand per branch

Internal Volatility[Demand] Measures the uncertaintyper branch of the E[Demand]-forecast

TABLE 1. Model Parameters and Proxies

Branch Time Zone c c f i x pup pop Demand Currency Corruption / TransferLiquidity Country risk Limits

Tokyo 1 2 2 1 1 2 2 1 1Singapore 1 2 1 1 1 2 2 1 1

Dubai 1 3 1 1 1 2 3 2 2Frankfurt 2 1 2 1 1 2 1 1 1Brussels 2 2 2 1 1 2 1 1 1London 2 1 3 1 1 2 1 1 1Zurich 2 1 3 1 1 2 1 1 1

Sao Paulo 3 3 1 1 1 2 3 3 3New York 3 1 3 1 1 2 1 1 1Toronto 3 2 2 1 1 2 1 1 1

TABLE 2. 10-branch model categorized parameters

matrix). In order to keep things tractable, we assign categories to parameters in a first step. The categories aremapped to numerical values in a second step.

The parameter categories for each branch are shown by table 2.In general, category 1 corresponds to low cost, category 2 to medium and category 3 to high cost. Similarly

for the demand we have category 1 for low demand, 2 for medium, and 3 for high demand. With respect to timezones, 1 refers to the 0/12-, 2 to the 6/18- and 3 to the 12/24-slot on the world time grid (see bottom of Figure5.2). The time zones are chosen according to the geography of the location. The categories have been assignedbased on their proxies or by expert judgement.

The second step, i.e. the mapping of these categories to numerical values can be found in the appendix.

5.1. The effects of volatility. The base case with low volatility leads to one liquidity center, Frankfurt, (depictedin yellow) and the remaining branches get their liquidity from Frankfurt (see Figure 5.2).

The actual optimization model can be separated into two parts: a deterministic and a stochastic one. Thedeterministic component optimizes the production of expected demand based on the well-known trade-offs offixed cost on the one hand and production and transportation cost on the other hand. A branch with low pro-duction cost, low fixed cost and low transportation cost will serve as global producer and distributor. If a branch

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FIGURE 5.2. Optimal center allocation (base case)

has an advantage in production but not in fixed cost/ transportation cost, it depends on the volume whether thisbranch or another with an advantage in fixed cost and/or variable cost (production + transportation) will be setup as liquidity center (Mechanism 1). High transportation cost favor the autonomy of branches, i.e., the networkdecomposes into a network of stand-alone branches (Mechanism 2). Apart from these basic trade-offs we alsoincorporate a time zone aspect: as the branches of time zone 1 and 3 do not overlap in terms of availability, incase of an overall setup with only one global center, this one must be in time zone 2 (Mechanism 3). The sto-chastic part of our model favors the set up of liquidity centers if liquidity uncertainty increases due to increasedvolatility or if the liquidity uncertainty costs (price of information) increase due to the increasing penalty cost forover- and underproduction (Mechanism 4).

Due to the high expected demand (500.000) in the base case, the production cost is a key parameter. Frankfurt,London, Zurich, and New York can produce at low cost (category 1) and are thus candidates for liquidity centers(Mechanism 1). As all four candidates have the same transportation cost but Frankfurt has an advantage infixed cost, Frankfurt operates as a liquidity center and produces the major demand in liquidity and the branchescover the remaining just-in-time demand via the central bank. If the branch with the production and fixed costadvantage would be in time zone 3 (e.g., the New York branch), New York would operate as a liquidity center, butalso the most profitable branch in time zone 1 would have to be opened as New York cannot produce (withoutovernight-roll over) for time zone 1. The branches Dubai and Sao Paulo suffer from high transportation cost(Currency/ Corruption/ Transfer Limits: Dubai (3/2/2) and Sao Paulo (3/3/3)). Nonetheless, they are not setup as liquidity centers and do not produce their own liquidity as their interbank markets are small implying highlocal production cost (category 3).

The base case is almost deterministic as the demand volatilities are far below the break even volatility (seecondition (2.15)). If the volatility increases, remote production becomes more expensive due to the increasedexpected penalty cost; the price of information increases. We assume that the volatility in Zurich, London,Brussels, and New York increases to 200.000 (Mechanism 4). Now it is economically advantageous to runFrankfurt, but also Zurich, London, and New York as liquidity centers (cf. Figure 5.3) Although, Brussels forexamples faces the same uncertainty, it is not set up as a liquidity center as it has a higher break even volatilitydue to its increased interbank market production cost (category 2). Hence, the relative advantage to produce

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FIGURE 5.3. Volatility effect promotes setting up centers

FIGURE 5.4. Opening of all centers due to high volatility and/ or penalty cost

locally is lower in Brussels. Increased penalty cost of Brussels (for example category 2) would reverse thesituation. Looking at the transfer volumes, London surpasses Frankfurt as a production and distribution hub.

Further increasing volatility and/or penalty cost might successively lead to setting up more and more locationsas liquidity centers as indicated in Figure 5.4.

Thus, our model captures the well-known mechanism of the deterministic component, but also successfullyincorporates the component arising from stochastic liquidity demand. External parameters can be estimated frommarket data via proxies. The data on the demand though has to be provided by the bank. It is imperative thatbanks estimate the distribution function of the demands and in particular the demand volatility in each branch;estimating only the expected demand neglects volatility as an important key factor for the center allocation.

5.2. Computational feasibility. One of the challenges is the numerical instability that might arise in some casesas the order of magnitudes of the volumes to be transferred (billions) and the production cost (basis points) mightdiffer significantly. Thus the big-M formulations might behave numerically unstable. For our computationalfeasibility tests we mainly used the commercial solver CPLEX 11.1.0 [1] but we also compared the results to

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Number of Number of Solution Standardbranches scenarios time [secs] Deviation

(10 instances for each size) (geometric mean)

10 4 0.01 0.0020 4 0.01 0.0030 4 0.04 0.0140 4 0.08 0.0150 4 0.12 0.0360 4 0.20 0.0210 100 1.30 0.0920 100 4.30 0.4030 100 10.29 0.9340 100 17.94 1.9350 100 26.47 19.2060 100 23.81 13.50

TABLE 3. Computational results (Model Section 3)

Instance Solution Standardset group time [secs] Deviation

(60 branches) (geometric mean)1 0.65 0.122 0.89 0.213 0.69 0.214 0.94 0.445 0.69 0.11

TABLE 4. Computational results (Model Section 4)

those obtained by using Gurobi 1.1.1 [19], and the non-commercial scip 1.1.0 [2], all of which are state-of-the-art mixed integer linear programming solvers. With all three solvers optimal or near-optimal solutions can beobtained within a short amount of time and the solution process was numerically stable. We report the timingsfor the runs performed with CPLEX 11.1.0.

In Table 3 we provide timings for solving the model from Section 3 for various sizes. We vary the number ofbranches in the network as well as the number of scenarios. For the scenarios we decided to provide timings fora rather coarse approximation with 4 scenarios only which can be solved very fast and for 100 scenarios whichallow for a sufficiently fine model of the stochastic demand structure. The instances for each (network size,branch size) combination were generated according to a cost structure similar to the one in Section 5 and wegenerated different cost mappings and scenarios randomly. We would like to point out that due to the imposedcost structure the actual instances are not random instances: we randomly generate the costs and spreads froma distribution and vary the category keys; the overall cost structure is preserved. For each instance set group wegenerated 10 instances. In all cases an optimal solution can be computed in less than 1 minute and the standarddeviation is small. One particularity that we observed is that indeed the geometric mean of the solution times forthe 50 nodes was higher than the for 60 nodes model. This was due to one instance in the 50 nodes instance setwhich had a higher overall solution time of 59.58 seconds.

In Table 4 we provide timings for instances with 60 nodes for the model from Section 4. Each instance groupconsisted of 10 instances. As expected, the model can be solved extremely fast to optimality even for largeinstances. In all cases the solution times were below 2 seconds and an optimal assignment could be computed.We report geometric means and the standard deviations in the table.

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6. CONCLUSION

Our paper studies how to optimize the liquidity production in a global banking group. More precisely weask whether liquidity should be produced in one (central) branch, in several regional ones or (locally) in eachbranch. This question naturally translates into a two-stage optimization problem: the first-stage decision is anorganizational one: which branches should be extended to liquidity centers with direct access to cheap interbankliquidity (and requiring fixed cost) and which branches should be run as simple branches with expensive centralbank access only? The second-stage decision is an operational one: once the organizational setup is fixed theoptimal production and transport volumes have to be decided. We are primarily interested in the organizationaldecision that trades off the marginal cost (fixed cost) against its marginal benefits. The stochastic driver of themodel is the liquidity demand in each branch. The problem is formulated as a Mixed Integer Program and solvedaccordingly.

In Section 2 we confirm that in the 1-branch bank the determining factor to run a liquidity center is theexpected liquidity demand. This reflects the well-known trade-off between fixed cost and the saving on (variable)production cost.

However, we want to study banking groups. To study the key mechanics of our model, we started with thesmallest possible banking group, the 2-branch group. We assume that (interbank) liquidity can be exchangedbetween branches. This implies additional transportation cost and a time delay. The time delay introduces aninformational dimension to our model: a liquidity center does not only provide cheaper, but also immediateaccess to liquidity (via interbank). A non-center branch must order its interbank liquidity from a center branchbefore knowing the exact liquidity demand. Any over-/ under-production must be locally invested/ producedat unattractive central bank rates. We interpret these cost as penalty cost. As a liquidity center is effectivelya vehicle to resolve information uncertainty we define the cost differential between center/ non-center as priceof information. We show that the price of information increases with the demand uncertainty (volatility). Incontrast to the 1-node branch, the volatility is the driving factor behind the decision of opening a liquiditycenter. We derive a threshold volatility beyond which the branch should be extended to a liquidity center asthe avoidance of penalty cost outweighs the fixed cost. Hence, fixed cost are traded-off against penalty cost(volatility). The center opening only saves cost on the production quantity that is decided after the demandrealizes (e.g., peak productions). This implies that production quantity that has been decided before (e.g. theexpected part of the demand) is not traded-off against fixed cost. Where this quantity is produced depends on thetrade-off remote production cost (i.e., production of other node + transportation cost) versus local productioncost. We might therefore see the opening of liquidity centers although they have absolute disadvantages inlocal interbank production (even after transportation cost) compared to other centers. The rationale for thesecenters is not the production of expected portions of liquidity demand, but of (unexpected) peaks. The expectedportions are remotely produced at centers with more attractive interbank rates. We also show that it is the delayassumption that makes volatility an important factor in the center decision. Without delay, a center does notreduce uncertainty and therefore the production decision is made under certainty implying that the expectationof the demand, but not its volatility is the key driver for the center decision.

After having studied the volatility effect in the 2-branch model, we extended the model to a n-branch modelin two formulations: scenario-based (Section 3) and as a deterministic formulation (Section 4).

Section 5 demonstrated the applicability of the model to real world problems. In contrast to our (pathological)2-branch model, all factors came into play: different fixed cost, transportation cost, production cost, and demandvolatilities. We used a 10-branch model to explain the interplay of these factors. In particular, we identified lowproduction cost as a centralization factor, high transportation and penalty cost, time zones and demand volatilityas decentralization factors. In order to study the computational feasibility, we investigated the performance forseveral network sizes (10 - 60 branches) and granularities (4 to 100 scenarios). We can state that even modelsof large banking groups (60 branches) and of high granularity (100 scenarios) can be solved in less than 30seconds using the commercial solver CPLEX 11.1.0. As expected, the deterministic formulation can be solvedmuch quicker: for large banking groups (60 branches) within a second.

Hence, Section 5 demonstrates that the implementation challenge is not computation, but calibration. Moreprecisely, a bank must estimate the interbank production, transportation, penalty and fixed cost for each branch/market. Apart from these exogenous cost, it has to estimate the demand distribution for each branch.

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Our model can be extended in several ways. First, the current model horizon of one business day should beextended to several days with daily (second-stage) volume decisions. Such a multi-period model is able to capturethe different holiday and weekend calenders of financial centers as well as seasonal patterns of liquidity demand.Second, it is interesting to study the impact of a stochastic cost structure on the center decision. Market turmoilsusually translate in both large liquidity demands and higher (production, transportation, penalty) cost. As fixedcost are scenario-invariant, we expect that a positive correlation between cost and demand favors decentralizedsetups. Third, the current objective function describes a risk-neutral banking group whereas existing bankinggroups are risk-averse. Hence, the incorporation of a risk measure in the objective function would make theanalysis more realistic.

Finally, our model does not explicitly capture liquidity constraints resulting from monetary policy (e.g. mini-mum reserves) or from regulatory liquidity models (e.g. minimum liquidity buffers). The consideration of thesereal-world constraints might provide further insight how a large banking group should organize their liquidityproduction.

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APPENDIX

Within Section 5 we used categories for the model parameters. Table 5 shows the values that we assigned tothe categories and references that we consulted for indicative figures.

CategoryCost Symbol Proxies/ References 1 2 3

Interbank Production c Market Size of Interbank markets 0.50% 1.50% 2.50%Fixed Cost c f i x Global Office Occupancy Cost, 50 300 1000

Source: DTZ (2007),Wage Comparison Index, UBS(2008)

Under-production Cost pup Rates of Standing Lending 2.00% 3.00% 6.00%Facilities of Central Banks

Over-production Cost pop Rates of Standing Depositing 0.50% 1.00% 3.50%Facilities of Central Banks

Demand, E[] µ Internal bank estimate 500000 500000 500000Demand, volatility σ Internal bank estimate 2000 50000 200000Currency Liquidity t(w, v) Average bid-ask spreads of FX-swaps 0.03% 0.13% 3.03%

Corruption/ Country Risk t(v, w) Corruption Perception Index, 0% 2.00% 5.00%Transparency International (2007)

Transfer Limits cr(v) Internal bank policy/ 0.00% 5.00% 30.00%external regulatory requirement

TABLE 5. Mapping of parameter categories to parameter values

The interbank production rates refer to the average funding cost that a bank has to pay overnight. Category1 refers to a large interbank market with production cost of 50 bps. The smaller the interbank market (lowerliquidity), the higher are the funding rates. Therefore, we assume additional spreads of 100 bps and 200 bps fora medium and a small interbank market respectively.

The fixed cost are the volume-independent cost to setup a liquidity center (salaries, rent, systems). We assumethat the annual cost in a branch with low salaries, low rents and low system cost is around 18.000 monetary units(e.g. Sao Paulo), 108.000 for a medium (e.g. Brussels) and 360.000 for an expensive location (e.g. New York).

The underproduction cost are the rates to raise liquidity at the central bank overnight. Again, the centralbank rates vary according to (central bank) sizes. We assume that 2.00 % are to be paid in a large, 3.00 %in a medium and 6.00% in a small central bank (market). Note that the underproduction cost must be higherthan the interbank rates as interbank rates are considered to be one of the benefits of a liquidity center. If theunderproduction cost were lower than the interbank rates, there is no rationale to open a liquidity center ascheap and immediate funds can be produced at the central bank.

The over-production cost measure the opportunity cost for liquid assets that are not used for liquidity produc-tion. We assume that they are of 0.50% for large, 1.00% for medium and 3.50% for small interbank markets.

In order to neutralize the effects of expected demand, we set the expected demand constant across all cat-egories and only varied the demand uncertainty (volatility). We assume that a stable liquidity demand has astandard deviation of 2000, a medium volatile one of 50000 and an unstable of 200000 units overnight.

Transportation cost have two components: FX-conversion cost and cost that reflect the country risk. Countryrisk summarizes corruption and credit risk of the country. To each branch we assigned a currency liquidityand corruption category. As proxy for the currency liquidity we used bid-ask spreads. We assumed for liquidcurrencies 3 bps, for medium 13 bps and for illiquid currencies around 300 bps. For low risk-countries, weassume 0%, for medium risk-countries 2% and for high risk-countries 5%. We used the corruption index as anindication. However, the figures are ad hoc assumptions.

The transportation cost for a transport between branch w and branch v are the sum of liquidity currency of wplus v and the country risk spreads for w plus v.

The transfer limits are introduced to capture the risk of technical failure of a branch. Such failures would havea high impact if liquidity production was based in one single node even if it is optimal from a cost perspective.

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As our cost perspective does not account for that technical dimension, we use transfer limits to promote adiversification of liquidity production to reduce the risk of technical failures.

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TECHNISCHE UNIVERSITÄT DARMSTADT, DEPARTMENT OF MATHEMATICS / KDB KRALL DEMMEL BAUMGARTEN / GERMANY

E-mail address: pokutta@mathematik.tu-darmstadt.de

FRANKFURT SCHOOL OF FINANCE AND MANAGEMENT / KDB KRALL DEMMEL BAUMGARTEN / GERMANY

E-mail address: c.schmaltz@kdb.eu

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