DISCUSSION OF 'AN EFFICIENT LIQUIDITY SAVINGS MECHANISM' BY DARRELL DUFFIE, SRISHT FATEH SINGH AND CHAOJUN WANG

The following information was released by the Federal Reserve Bank of Dallas:

Sam Schulhofer-Wohl

July 09, 2026

I. Introduction

I would like to thank the workshop organizers for the opportunity to discuss this fascinating and innovative paper by Professor Darrell Duffie and co-authors (Duffie et al., 2026). These are my views and not necessarily those of the Federal Reserve Bank of Dallas or the Federal Reserve System.

The paper uses economic theory to find a way to process interbank payments efficiently in a real-time gross settlement (RTGS) system. An RTGS system settles each payment fully and finally when the bank sends it, simultaneously debiting the account of the sending bank and crediting the account of the receiving bank. Globally, RTGS systems replaced an earlier generation of deferred net settlement (DNS) systems that exposed banks to more risk. A DNS system accumulates interbank payment orders throughout the day and settles them all at once, on a net basis, at day's end. Much less cash than the gross amount of payments moves each day in a DNS system because a bank's incoming and outgoing payments tend to offset each other. However, if a bank fails during the day, other banks do not receive money they expected, and the shock can spread across the banking system.

RTGS systems require more liquidity than DNS systems and can be more sensitive to shortfalls of aggregate liquidity in the banking system. If a bank has abundant cash on hand, it will be willing to send outgoing payments early in the day, even before receiving incoming payments. But if a bank has relatively little cash, it may delay outgoing payments until more funds arrive. Payment delays can propagate as more banks find themselves low on funds, and the system can experience gridlock (Bech and Garratt, 2003; McAndrews and Kroeger, 2016).

In recent years, payment system operators have sought to combine the risk-reducing benefits of RTGS and the liquidity savings of DNS by matching offsetting payments while settling payments finally throughout the day. It is intuitively obvious how to create an algorithm that saves liquidity while settling payments in close to real time. The algorithm should queue up payment messages and, when the queue contains matching payments in opposite directions, settle both at once. A more sophisticated algorithm could also look for multilateral matching opportunities, such as instances when A sends to B, who sends to C, who sends to A. There are also choices to be made about how to prioritize payments in the queue, whether to search for matches continuously or at defined intervals, and so on. Liquidity savings mechanisms (LSMs) now exist within RTGS systems in many countries, including the United States, where the private-sector CHIPS system operated by a consortium of large banks employs a continuous netting approach (McAndrews and Vartin, 2022).

But while it is straightforward to construct algorithms that can save some liquidity, it is less obvious what algorithm would save the most liquidity or how payment system operators should choose among candidate algorithms. This is where Duffie and co-authors come in. Instead of inventing an algorithm, they propose a mechanism for processing payments.

II. Mechanism design

When Stefania D'Amico asked me to discuss an as-yet-uncirculated paper by Darrell about payments, I expected it would concentrate on the interaction of payments with areas where I have done research, such as monetary policy implementation or the Fed's balance sheet. I was surprised to open the paper and find a discussion of Vickrey-Clarke-Groves mechanisms (Vickrey, 1961; Clarke, 1971; Groves, 1973). But while I have never done research on mechanism design, I was fortunate to have as my graduate school advisor a pioneer of that literature, Robert M. Townsend (see, e.g., Harris and Townsend, 1981). So, I will try to build a bridge between the central banking I'm familiar with and the mechanism design theory I learned in my youth.

The word "mechanism" is a term of art. As formulated by Leonid Hurwicz in his Nobel Prize-winning work (Hurwicz, 1973; Prize Committee of the Royal Swedish Academy of Sciences, 2007), a mechanism is a system that allocates resources to participants as a function of messages they send each other. An art auction is a mechanism: The bidders shout out prices, raise paddles or scratch their ears, and the auctioneer awards the painting to whoever's shouting, paddling or scratching signals the highest bid. A command economy is a mechanism: The central planner tells everyone what to do. A payment system is likewise a mechanism: Banks send messages specifying how much money to transfer, and the system determines which payments to process and when.

Mechanism designers seek to create mechanisms that will generate optimal resource allocations. Such an exercise has three components. First, designers must examine the constraints on feasible resource allocations. The auctioneer can't give the painting to more than one bidder. Second, designers must consider the participants' incentives, both those the mechanism creates and those in the broader economic environment. Will bidders offer as much for the painting as they are willing to pay, or do they have a reason to shade their bids? What induces them to attend the auction? Third, designers must specify their own objectives. By what yardstick do we judge whether a resource allocation is optimal? Should the auction aim to raise as much money as possible, or to deliver the painting to whoever most enjoys seeing it on the wall?

I applaud Darrell and co-authors for bringing the mechanism design approach to LSM design. Thinking about constraints, incentives and objectives beautifully clarifies what LSMs do, in a way that does not come out when one starts by discussing a specific algorithm for stepping through a queue, processing payments from smallest to largest, or what have you. The paper carefully identifies three mechanisms that are optimal in the sense of both minimizing banks' costs and maximizing payment flow, under different constraints on the use of overdrafts and reserve balances. This work simultaneously advances our theoretical understanding of LSMs and provides practical ideas for improving wholesale payment systems.

The baseline mechanism Darrell and co-authors propose is straightforward. At the start of the day, each bank has a list of payments it needs to make and some money in its account at the central bank. Each bank tells the payment system operator which payments it is willing to make through the LSM and a limit on the net debit the LSM can make to the bank's account. The LSM finds the combination of submitted payments that maximizes the gross total value of payments processed through the LSM. A bank that receives a net inflow from the LSM pays a fee, while a bank that has a net outflow receives a reward. These fees and rewards encourage banks to send money through the LSM. Finally, payments that the LSM did not select or that banks did not submit to it flow through an ordinary RTGS system.

Behind the elegant mechanism is a lot of careful work. The proof of efficiency has to keep track of all the details of how payments flow through a network of many banks, and how making or stopping one payment would influence all the others.

The paper also considers alternatives in which the LSM cannot debit banks' reserves at all or, to the contrary, is allowed to overdraft banks' accounts. The authors view their baseline mechanism as most relevant for practical purposes. However, I found all three mechanisms potentially relevant in practice, depending on the institutional context. A private-sector operator that implements an LSM in a shared account, along the lines of CHIPS, will have limited access to reserve balances and cannot allow overdrafts without reintroducing settlement risk. If the payments operator is distinct from the central bank but does not use a shared account, as in the case of Canada, the LSM might more easily access reserve balances, but overdrafts may remain problematic. If the central bank operates the payment system and integrates it into the credit architecture, it would be more feasible to allow use of both reserve balances and overdrafts.

I will organize my comments in those same three categories relevant to all mechanism designs: constraints, incentives and objectives.

III. Constraints

Relative to real-world payment systems, the paper relaxes one important constraint and tightens two others.

All-or-none payments

It is axiomatic that payments settle in their entirety or not at all. If you are buying a house, you or your mortgage lender must wire the full purchase price to the seller. If you wire half the money, you will create a mess. The seller is not going to give you the deed but has a lot of your money. It is better to delay sending any money until you can send all of it.

However, searching for the optimal combination of complete payments is an integer programming problem, which is computationally extremely difficult. Therefore, the paper allows the system to send partial payments.

There are three ways to think about a mechanism with partial payments. One possibility would be to change real-world systems to allow partial payments. The authors note this would require changes to payment message formats and accounting systems. It might also require changes to contracts between counterparties, so the implementation effort could extend well beyond the payment system. Second, the paper suggests that if each payment represents a collection of sub-payments, a partial payment could come close to satisfying the all-or-none constraint with respect to the sub-payments. But the United States has separate payment rails for wholesale and retail payments. In this environment, wholesale payments are often single large-value payments, such as one taxpayer's remittance to the Treasury or a corporation's settlement of a large invoice, that cannot be further disaggregated. Third, the paper's mechanism establishes an upper bound on the efficiency that any all-or-none mechanism can deliver. The authors show the efficiency loss from an all-or-none constraint can theoretically be very large. In future work, it would be valuable for the authors to quantitatively investigate how much the all-or-none constraint reduces efficiency when applied to actual payment patterns.

Time of day

The paper assumes all payments are known to banks and potentially available to process at the same time. In practice, payment needs arise throughout the operating day (for the United States, a 22-hour period). Real-world LSMs face tradeoffs in determining how frequently to match payments against each other. Collecting payment requests over a longer time generates more opportunities for matching but slows the flow of funds. Processing a payment at one point in time also precludes matching it against payment requests that may arrive later. Because the paper's environment constrains all payment needs to arise simultaneously, the paper cannot explore these tradeoffs.

Budget balance

The paper requires the mechanism's fees and rewards to be budget balanced. This is restrictive for two reasons. First, a payment system needs to cover its capital and operating costs. Second, budget-balanced fees and rewards can address externalities between banks that participate in the payment system but cannot address externalities vis-à-vis the rest of the economy. I'll come back to this when I discuss objectives below.

IV. Incentives

We can think of banks' incentives in two stages. The second stage is the one modeled in most of the paper. A bank begins the day with cash on hand and a list of payments and decides which payments to send to the LSM. The first stage is whatever happened before today that led to the bank having this much cash, needing to make these payments and participating in the LSM.

The incentives in both stages matter to the mechanism's effects and efficiency. I'll take them in reverse order.

Second-stage incentives

In deciding which payments to send to the LSM, the bank balances the fees and rewards in the LSM against the costs of late payments and overdrafts.

If overdrawing the central bank account were costless, banks would have less need to economize on liquidity. In practice, in the United States the Fed does not charge banks for intraday collateralized overdrafts, and the large banks that the paper expects to participate in the LSM have substantial collateral. Still, some banks reportedly seek to avoid overdrafts (Nelson et al., 2026). Reported reasons for reluctance to use intraday credit include concern that regulators or others would take the borrowing as a negative signal (Nelson, 2024) and operational frictions (Touhey, 2024; Zorc, 2024). The Fed provides intraday credit to "healthy depository institutions" (Board of Governors of the Federal Reserve System, 2023a), a policy that should mitigate stigma. The Federal Reserve Banks and Federal Reserve Board are also working to reduce operational frictions (Board of Governors of the Federal Reserve System, 2024). Increasing banks' willingness to use intraday credit could be a way to smooth payment flows without building a new LSM.

In the paper's model, the cost of late payments must be smaller than the LSM's rewards and fees to induce banks to use the LSM optimally. The paper assumes all payments are equally costly to delay. In practice, some payments are much more time-critical than others. Derivatives market participants must make margin payments at precise moments in the day (Marshall and Steigerwald, 2013), while a manufacturer's invoice can typically be paid anytime within 30 days.

Banks know the time-criticality of each payment; the payment system operator may not. If the LSM sets identical rewards and fees for all payments, banks have an incentive to submit only relatively less time-critical payments to the LSM. Selective submissions may prevent the LSM from generating an efficient allocation. The standard tool for solving this private-information problem in mechanism design is the revelation principle (Myerson, 1981): Ask banks to report the time-criticality of each payment, use this information to decide which payments to process and create incentives for banks to make truthful reports. It would be valuable for the authors to extend their analysis to private information about time-criticality.

Table 1: U.S. wholesale payment systems

System Operator Participants Average daily volume Average daily transactions Average transaction amount

CHIPS The Clearing House 42 $2.014 trillion 632,419 $3.18 million

Fedwire Federal Reserve >5,000 $4.593 trillion 869,187 $5.28 million

In the United States, another incentive consideration is that there are two wholesale payment systems, listed in Table 1. Fedwire Funds (hereafter, Fedwire) is operated by the Fed and used by banks of all sizes. It has no LSM. CHIPS is a private-sector system primarily used by large banks. It has an LSM that processes an average of 26 dollars of transactions per dollar of bank funding (The Clearing House, 2026b).

Table 2 shows that CHIPS' efficiency is more than double the average efficiency of the most efficient international systems with LSMs studied by Kabadjova et al. (2023). The combined efficiency of the U.S. payment system, a weighted average of CHIPS and Fedwire efficiencies, also compares favorably to international benchmarks.

In effect, then, CHIPS functions as an LSM for the U.S. payment system taken as a whole. Its presence changes the incentives banks would face if an LSM using the authors' mechanism were added to Fedwire. Banks would choose not just between LSM and regular RTGS, but between two different LSMs. It would be useful for the paper to analyze whether its efficiency result continues to hold if there is a competing LSM that has different fees and rules.

Table 2: Efficiency of selected payment systems

Jurisdiction System Has LSM? Efficiency

United States CHIPS Yes 26

United Kingdom CHAPS Yes 12.55

United States CHIPS + Fedwire Yes 11.1

Mexico SPEI Yes 9.22

Canada LVTS (pre-2021 system) (2006-2018 data) Yes 8.46

Canada Lynx (post-2021 system) Yes 8.4

Canada LVTS (pre-2021 system) (2020-2021 data) Yes 7.0

Colombia CUD Yes 5.36

United States Fedwire No 4.55

Switzerland SIC Yes 4.31

Eurozone TARGET2 Yes 4.13

Brazil STR Yes 3.28

Denmark Kronos Yes 2.52

First-stage incentives

Banks' first-stage choices include whether to participate in the LSM, whether to try to influence the mix of payments that will need to be sent and received on future days, and how much cash to hold overnight and begin the next day with.

The authors analyze the first of those first-stage choices. They show that, for the version of the LSM that can overdraw banks' accounts, each bank finds it rational to join, assuming all others do so as well.

Banks' business models and customers are not static. Banks can choose to enter, exit, scale up or scale down lines of business depending on costs and profits from those activities. One relevant cost is for the intraday liquidity a business requires. When the only available payment system uses RTGS, all intraday liquidity needs are equally costly. Once an LSM is available, intraday liquidity needs that can offset inside the LSM are less expensive than intraday liquidity needs that will not offset. On the margin, then, an LSM could give banks an incentive to pursue activities in which payment flows are more nettable and avoid those in which payment flows are less nettable. For example, businesses that generate large, unpredictable and time-sensitive payment flows, such as derivatives and securities clearing, might become relatively less attractive in the presence of an LSM. Whether this incentive is efficient for the economy depends on the social costs of intraday liquidity. If intraday liquidity is socially costly, encouraging banks to use it in nettable ways is desirable. However, if society can provide intraday liquidity at low cost, deterring the use of intraday liquidity for non-nettable activities could inefficiently distort banks' business models.

The presence of the LSM also reduces the marginal value of cash on hand at the start of the day. Banks could respond by reducing their start-of-day cash balances. Reducing banks' desired cash balances might be one objective of an LSM, a point I'll return to in a moment, but this incentive also affects how well the LSM might achieve other objectives.

V. Objectives

The authors take as their objective to minimize the cost of overdrafts and late payments in the banking system. It turns out the way to do this, in the baseline mechanism, is to maximize the value of payments the LSM can process given banks' start-of-day central bank balances. Equivalently, since every optimization problem has a dual, the mechanism minimizes the aggregate bank reserves needed to process a given volume of payments.

The quest to reduce the payments demand for bank reserves originates in recent discussions of options and tradeoffs associated with reducing the size of the Fed's balance sheet, of which reserves are one component (Anderson et al., 2026; Barr, 2026; Duffie, 2026; Logan, 2026; Logan and Schulhofer-Wohl, 2026). Observers argue that a more efficient payment system could allow for a smaller Fed balance sheet in two ways. Banks might demand fewer reserves (Levy, 2025). In addition, a more efficient payments system could be more resilient to temporary drops in reserve levels or increases in reserve needs (Duffie, 2026).

I would make three points here. One, given the other drivers of reserve demand and the presence of CHIPS, it is uncertain how much adding an LSM to Fedwire can reduce reserve demand. Second, reducing the steady-state demand for reserves is not the same as making the system more resilient. Third, as discussed by Logan (2026), the Fed's balance sheet exists to serve the public. Accordingly, it is worthwhile to ask what payment system design best serves the public.

Potential effect of a Fedwire LSM on reserve demand

Banks hold reserves not only to make payments under ordinary circumstances but also to manage the risk of unexpected outflows, to meet regulatory and supervisory requirements (which relate partly to the risk of unexpected outflows), and potentially as investments (Logan and Schulhofer-Wohl, 2026). An LSM can reduce business-as-usual (BAU) reserve needs. It is less likely to reduce reserve needs in stress scenarios, because stress scenarios may involve one-way outflows, like a run on the bank, that a netting mechanism cannot offset.

Banks do not publicly report how much reserve demand comes from BAU needs versus liquidity risk management. However, in the September 2023 Senior Financial Officer Survey, the Fed asked banks about the importance of various factors in determining their lowest comfortable level of reserves (Board of Governors of the Federal Reserve System, 2023b). Banks gave the highest average rating, 4.4 on a 5-point scale, to "satisfying liquidity stress-testing metrics (meeting projected outflows under stressed market conditions)." At a rating of 3.7, "meeting routine intraday payment or settlement needs" tied with the need to meet projected outflows over windows of more than one day.

As Professor Wenxin Du observed in her discussion (Du, 2026) of Duffie (2026) at the Brookings Papers on Economic Activity conference, CHIPS likely already captures a substantial share of the most nettable payment flows in the U.S. payment system. CHIPS participants are large banks in the core of the payments network. CHIPS participants also already choose which payments to send there and which to Fedwire. Nettability is one factor that would motivate sending a payment to CHIPS. Payments sent to Fedwire may be more likely to be ones where delay is unacceptable or there is no potential offsetting flow. The reduction in the U.S. banking system's liquidity needs from adding an LSM to Fedwire, when CHIPS already has such a mechanism, is therefore likely to be smaller than the comparison of a hypothetical efficient LSM to RTGS alone would suggest.

Another conceivable use of the authors' mechanism in the U.S. would be to replace the current LSM in CHIPS. If the authors can get transaction-level payments data from CHIPS, it would be valuable to simulate their mechanism on these data and learn how close CHIPS comes to the theoretical benchmark the mechanism provides. This analysis could also illuminate how much of CHIPS' high efficiency relative to international benchmarks comes from its algorithm and how much comes from banks sending relatively more nettable payments to CHIPS.

Resilience and elasticity

There are occasional days when payment flows surge or bank reserve balances dip, such as in connection with tax payment deadlines and Treasury auction settlements. Starting from an exogenously specified level of reserves, the financial system can better manage a drop in reserves or surge in payments if an LSM is available, because the LSM reduces the reserves needed to process payments.

However, the starting level of reserves is not exogenous. With an LSM in place, the marginal value of start-of-day reserves is lower, incentivizing banks to hold fewer reserves. This is exactly the means by which an LSM can help reduce the central bank's balance sheet. But it implies the system would begin each day closer to the level where stress would emerge. In equilibrium, then, an LSM may or may not make the system more elastic. It would be useful to extend the paper's model to analyze this effect.

Public benefits and costs

The paper defines efficiency as minimizing banks' total costs for overdrafts and late payments. On its face, this definition seeks an LSM that is efficient for banks.

When is the LSM also efficient for the economy as a whole? I'll highlight three considerations: whether banks' costs for overdrafts and late payments match the costs of banks' counterparties; whether there are other externalities between the banking system and the rest of the economy; and whether it is costly to implement the LSM.

At minimum, for an LSM that minimizes banks' late payment and overdraft costs to be efficient economywide, the cost of late payments to banks must equal the cost of late payments to banks' customers, and the cost of overdrafts to banks must equal the central bank's cost of providing intraday credit. The condition on late payment costs appears reasonable. The condition on overdraft costs is more challenging. As discussed above, some banks reportedly treat collateralized intraday credit as costly, even though the Fed provides it for free. If the Fed's pricing reflects the social cost, then the paper's LSM would underuse intraday credit from a societal perspective. In this case, a more efficient LSM's fees and rewards would encourage banks to use more intraday credit.

The paper's LSM is efficient for banks because its rewards and fees offset the externalities that banks create for each other when they send or delay payments. To be efficient for the economy, the LSM would also need to offset any externalities the banking system creates for the rest of the economy in processing payments. These externalities could be positive or negative: Faster payments might foster innovation and benefit consumers and businesses but could also facilitate fraud (World Bank Group, 2023). Governments worldwide have been seeking to foster faster payments (World Bank Group, 2021), suggesting that governments may see the externalities as positive on net. If so, an efficient LSM might need to subsidize the sending of payments. Such a subsidy could require relaxing the balanced budget constraint or balancing the budget through a two-part tariff.

Lastly, adding an LSM to Fedwire would have costs. Developers would need to produce, test and maintain the computer code, both at the Fed and at the participating banks. This code would need to be very robust, because U.S. dollar wholesale payments form the foundation of the financial system. There are also the costs for banksespecially the thousands of smaller banksof learning about the new system and deciding which payments to submit. An efficient LSM would weigh these costs against the liquidity gains.

Depending on the magnitude of implementation costs and the perceived benefits of reducing liquidity needs, it might not be efficient, in this broader sense, to wring out every dollar of payments from the available supply of reserves. A simpler mechanism might be easier to deploy robustly and deliver "good enough" payments efficiency. In that case, the mechanism in this paper would represent a benchmark for how many payments can be processed if processing payments is the only object, but other mechanisms might remain attractive in practice.

In my view, the implications for small banks are particularly relevant in analyzing these trade-offs for the United States. As we have seen, the U.S. wholesale payment system includes both a public operator, the Fed, and a private operator, The Clearing House. A core rationale for the Fed's provision of payments services is to make payment services "available to all depository institutions, including smaller institutions in remote locations that other providers might not choose to serve" (Board of Governors of the Federal Reserve System, 2001). Congress also requires the Fed in its payments activities to "give due regard to competitive factors and the provision of an adequate level of such services nationwide" (Monetary Control Act, 1980).

The paper suggests that "an LSM would likely only be used by a subset of large U.S . banks." However, while Fedwire and CHIPS both process trillions of dollars of payments per day, as shown in Table 1, Fedwire has two orders of magnitude more participants. In assessing how to efficiently process Fedwire payments, therefore, the question is not how to process large banks' payments efficiently in an economy with no other wholesale payment system, but rather how to efficiently process payments for all banks in an economy where a large-bank consortium already has an LSM.

Table 3 shows that the United States has many more banks than most major economies, both in absolute number and relative to population and GDP. There are more than 8,000 depository institutions (DIs) in the United States. Three-quarters of U.S. commercial banks have assets under $1 billion, according to call report data compiled by Correia, Luck and Verner (2026).

Table 3: Number of depository institutions in selected economies

Jurisdiction Depository institutions Depository institutions per million population Depository institutions per $1 billion GDP

Norway (all DIs) 316 56.2 0.56

United States 8,623 25.2 0.34

Norway (domestic DIs) 90 16.0 0.16

Eurozone 4,649 13.0 0.20

Sweden 130 12.3 0.20

United Kingdom 672 9.7 0.18

New Zealand 41 7.7 0.27

Canada 265 6.4 0.10

Japan 586 4.8 0.10

Australia 125 4.5 0.06

The benefits and costs of an LSM may vary with bank size. Small banks may have fewer resources to devote to analyzing payment choices and may be more sensitive to the risk of paying large fees to an LSM. On the other hand, small banks do not face the same liquidity regulations as large banks and might have more flexibility to reduce reserve holdings if the payment system were more efficient. It would be valuable to investigate how adding an LSM to Fedwire would affect payment system access and costs as a function of bank size. Would an LSM in Fedwire support a level playing field for banks of all sizes (Logan, 2024), or would it tilt the playing field one way or another? If there is a tilt, would allowing banks to opt out of the LSM solve the problem, or is a different remedy needed?

What's so valuable about the authors' research strategy is that it opens up a clear avenue for addressing questions of this type. Tell Darrell and his colleagues the constraints, incentives and objectives, and they can work out the optimal mechanism. Without mechanism design, it would be much more difficult to know what each LSM algorithm achieves and what challenges it may create.

VI. Conclusion

I applaud the authors for bringing an innovative and clarifying theoretical approach to the design of LSMs. The paper provides an elegant benchmark for efficient LSM algorithms and demonstrates the power of mechanism design in this space. If I have raised some questions about the mechanism, it is mainly because the clarity of the authors' approach makes it possible to ask better questions in the first place. The authors' strategy opens up possibilities for using mechanism design to create payment systems that address different constraints, incentives and objectives. I look forward to reading the literature that I hope this paper launches.

Notes

I thank Amy Chapel, Stefania D'Amico and Rosie Levy for helpful comments.

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Zorc, Beth. 2024. Comment letter of Institute of International Bankers on Request for Information on Operational Aspects of Federal Reserve Extensions of Discount Window and Intraday Credit, Board of Governors of the Federal Reserve System Docket No. OP-1838, ID FR-0000-0137-01-C113. Dec. 9.

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About the speaker

Sam Schulhofer-Wohl is senior vice president and senior advisor to the president of the Federal Reserve Bank of Dallas.

The views expressed are those of the speaker and should not be attributed to the Federal Reserve Bank of Dallas or the Federal Reserve System.