A Quantity Theory of Reserves

Before the GFC there was a well-defined framework for liquidity management. That framework no longer exists. The ad-hoc evolution of policy since 2007 undermined the old arrangements without creating a coherent replacement. A formal framework for the current policy arrangements is long over-due. The following attempts to bring some of the key features of current monetary arrangements towards a formal understanding – though by no means a complete understanding. The implications of such a framework are mostly left unaddressed. That is not because implications are not important, but because identification of the framework is a prerequisite for assessing implications.

One of the notable features of the new monetary arrangements is that monetary quantity is explicit in policy, yet any quantity theory of money seems to remain anathema to policy makers. Both ‘quantitative easing’ and ‘quantitative tightening’ state that quantities are monetarily relevant – the clue is in the description. Yet central bankers, and many others of influence, avoid the obvious reference. Fed chairman Ben Bernanke highlighted this disconnection when he famously quipped in 2012, “Well, the problem with QE is it works in practice, but it doesn’t work in theory.” An obvious rejoinder was he obviously had the wrong theory and was unwilling to consider a theory that was relevant to the circumstances.

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The problem persists. Non-conventional quantitative policies by central banks have narrow monetary consequences, so a quantitative approach is surely worth investigating. Yet for the most part, studies focus on the effects of a) interest rates and b) central bank assets – the acquisition of (QE) or reduction of (QT) large scale portfolios. Much less focus has been given to the parallel expansion of liabilities required to fund those assets. And it may be that Ben Bernanke’s admitted inability to link QE theory to practice may be overcome or assisted by including the behaviour of quantities of liabilities as well as assets.

A new monetary theory must begin with the origin of money itself with the central monetary authority. Monetary theory once commonly assumed that central bank liabilities formed the origin of money stock, upon which wider definitions of money were built. This is why central bank liabilities are known as Base Money, from which the concept of bank reserve requirements emerged.

Now, it is true that in many instances, reserve requirements are no longer considered essential to the monetary regime – the Fed formally abandoned reserve requirements in 2020. However, abandonment of formal requirements need not mean central bank reserves are functionally less relevant.  Demand for reserves continues even though the formal regulatory requirement has been suspended. Indeed, the suspension of formal reserve requirements may have revealed an underlying demand for reserves which had been partially obscured previously.  

As Milton Friedman wrote “The quantity theory is in the first instance a theory of the demand for money. It is not a theory of output, or of money income, or of the price level[1].’ Any new theoretical framework that seeks to address the current regime of ‘ample reserves’ must be able to explain the apparent ‘natural’ demand for reserves by the banking system that continued despite the abandonment of requirements.

Certainly, any new formal monetary framework would need to address some striking features of monetary quantities, namely:

In the sixteen years since the GFC, the expansion of the monetary base has been accompanied by an expansion of its use as settlement of financial transactions. The monetary base is sometimes known as ‘inside money’. ‘Inside money’ accurately reflects the current arrangements where compositional changes of the monetary base within central bank balance sheet are linked to the turnover of financial transactions. This behaviour continues the manner of financial final settlement that existed before the GFC, yet it is quantitatively distinct from earlier arrangements. As should be clear, sufficient difference in quantity results in qualitatively different structure. This should be acknowledged in any formal understanding of monetary framework.

As noted above, the Fed formally abandoned required reserves in March 2020, when it set reserve requirement ratios for transaction accounts to 0%. The prior rationale for reserve requirements was that they “provided a means for influencing the expansion of bank credit.” The massive expansion of reserves held by the banking system may reflect an increase in reserve demand by banks, but it is not associated with an obvious influence on credit to the rest of the economy. Modern-day demand for reserves suggests the provision of financial settlement services has been added, and perhaps supplanted, the provision of credit for commercial banks. Financial transactions settlement appears to constitute a key determinant for demand of central bank reserves – though it may not be the only demand, as argued in this paper by Fed researchers David Lopez-Salido and Annette Vissing-Jorgensen, which links reserve demand to level of deposits.

A new formal framework needs a theory. If the data support the theory, then this may form the basis for a framework. Perhaps an appropriate theory would be the Quantity Theory of Reserves, an adaptation which seems to describe how monetary relationships currently behave.

One famous equation for the Quantity Theory of Money, formulated by Irving Fisher, states:

M * Vt = Pt *T

A reworking of this formula into the language of reserves & settlement may approximate the following:

Res * Vt = Pt *T

where: Res is the total stock of reserves available, Vt is the velocity of reserves during the observation period. Pt is price level of securities transactions during the observation period, T is an index of the aggregate securities transactions.

The equation suggests that a higher stock of reserves leads to higher level of prices of financial instruments settled by those reserves, assuming stable velocity of reserves. Alternatively, and perhaps more commonly, a higher level of demand for reserves from settlement requires a higher velocity of reserves, absent additional provision by the Fed.

The equation certainly suggests the level of reserves may be associated with the turnover of those reserves. This conjecture is in line with the data. The chart below shows the strong relationship post-GFC between the stock of reserves and the use of reserves in FedWire Funds Service. This supports the conjecture that ‘ample reserves’ regime meets the demand for greater movement of reserves between banks, including in final settlement of financial transactions. Reserves are not ‘dead money’.

Now, reserves have always been used as final settlement media. Daily turnover was typically multiples of the outstanding stock of reserves issued by the Federal Reserve. Given the massive increase in stock of reserves since 2007, their velocity must have fallen. Or, to put it another way, an increase in demand for reserves may have placed downward pressure on velocity.

The second chart confirms the trend velocity of reserves has fallen since the GFC. The downward trend is intermittently punctuated by periods of reserve ‘drought’, requiring increases in velocity – as occurred in the latter stages of the GFC (2009-2010), during balance sheet contraction (2018-2019), and during the ‘Dash-to-Cash’ in March 2020. On all occasions, reserve demand by banking system outstripped the availability of reserves. Invariably, such an increase in demand was followed by additions to the stock of reserves by the Federal Reserve – including the most recent rise above the long-term trend in March 2023. It may be imagined, therefore, that an increase in velocity is associated with an increase in demand for reserves above the amount required for settlement purposes.  The ‘liquidity preference’ of banks for a higher level of reserves requires reserve velocity to increase.

Both the left and the right sides of the Quantity equation can both be simplified into single identities; average daily value (ADV) in dollars of FedWire Funds Service transactions for the left side and average daily value in dollars of financial transactions for the right side. Both data are reported, so it is possible to estimate a relationship between securities turnover and reserves turnover. That takes us some of the way to testing the Quantity identity on reserve demand, if not actually confirming a theory.

It is important to note that a wide range of financial transactions may be settled by FedWire, and switches between different assets is an important part of overall investment function. It follows that an estimate of the influence of asset turnover on reserve turnover should reflect the broad categories of asset type (equities, corporate bonds, Treasuries, repo transactions). With those details, a model of FedWire Funds Service ADV as reflected in securities/repo turnover is possible.

The charts below illustrate the relationship. The fit is good, with normal distribution of residuals, and statistical significance shown by many of the independent variables especially equities and repo. This seems to underscore the relationship between securities transactions (independent variable) and demand for reserves via FedWire (dependent variable). The panel results are given at the end of the article.

We haven’t directly considered how to measure the effects of this relationship on asset prices themselves. Nonetheless, the provision of settlement media is functionally equivalent to the provision of liquidity and thus a key determinant of asset price levels. To date, discussion of the distributional effects on social welfare of this arrangement have been largely ignored.

There are some obvious points to highlight regarding asset prices and reserve demand. Sharp declines in asset prices (as in early 2020) led to a large rise in the dollar value of securities transacted, and to demand for reserves for settlement. This is a normal characteristic of negative price shocks in markets. This behaviour suggests acute demand for reserves is associated with the negative price behaviour while accommodation of additional demand by increased supply impacts assets positively. Stable asset prices may require a steady supply of reserves. This seems an alternative interpretation of the well-known ‘Fed put’ for asset prices.

One approach to bridging the gap between transactions and price may be found in Friedman’s definition of Income (Y) outlined in his introductory chapter of ‘Studies in the Quantity Theory of Money’. There he includes an equation for income as a function of bond interest rates, equity yields, expected inflation, wealth, real income, and other factors.  

Where: v = income velocity, rb = return from bonds, re = return from equities, P = initial price of assets, d = change (in price or time), w = stock of wealth, Y = income, u = utility, M = money (or in our case reserves). Rearranging the equation slightly gives the definition of velocity based on turnover of returns from financial assets. That seems at least an approximate description of financial transaction turnover and reserves even though Friedman’s terms apply more generally than we are focussed upon.

And Friedman’s approach overlaps with our interest in the type of actors involved. Friedman comments that his equation should be applied ‘solely for money held directly by ultimate wealth-owning units.’ Friedman views this as a drawback, because not all money demand stems from ‘ultimate wealth-owning units. Notably, business enterprises also demand money for productive investment. Yet, his parsimonious description of money demand may inadvertently have created a functional description of the demand for reserves as the settlement medium of modern ‘ultimate wealth-owning units’ – namely those who settle financial claims. Friedman is forced to consider money velocity in terms of final settlement and even defined the main asset classes settled.

Indeed, Friedman echoes earlier descriptions of velocity which refer only to final settlement as the relevant velocity. “Most of the pre-Fisher velocity theorists… defined money as consisting solely of gold and silver coin. They excluded bank notes and deposits on the ground that such instruments lack the unconditional power of specie to settle final transactions and thus are not money per se but rather devices to accelerate money’s velocity.[2]” Thus the only velocity of interest to classical monetary theorists was the velocity of final settlement. Today there is no question that the modern ultimate power of final settlement lies with reserve balances held at the Federal Reserve.  

It is worth noting that Reserves, ONRRP and Treasury General Account together comprise the largest contributors to the monetary base. These balance sheet items are also by far the most volatile in dollars terms. Normally, this volatility is due to substitution effects between the components. However, the recent steep rundown in ONRRP holdings is a reminder that overall levels change quickly. This itself would encourage banks to hold precautionary stocks of reserves above the level associated with theoretical equilibrium. Indeed, it is notable that the level of reserves deemed by regulators to promote financial stability is considerably lower than the level seen at the last overt demand for reserves by the banking system, around the time of the SVB bankruptcy in March 2023.

A factor supporting demand for reserves is the high interest rate paid by the Fed. However, the question of how Interest on Reserves translates into velocity of reserves is not immediately obvious, though it would certainly have a bearing. It may, of course, be overcome by expected returns from other assets, but this question, though important, remains unanswered. It may also be that return on reserves (interest rate) is less relevant than the demand for reserves.

I am grateful for comments by Professor Geoffrey Wood on early drafts of this article. Of course, any errors or misinterpretations are entirely of my own making.

Panel regression model for FedWire Funds ADV using key securities & repo ADV


Dependent variable:


`FedWire Funds ADV`


`ADV: Equities (CBOE, SIFMA)`                          0.683***


`ADV: Treasury (FRBNY, SIFMA)`                      -0.031


`ADV: Agency MBS (FINRA, SIFMA)`              -0.445***


`ADV: Agency (FINRA, SIFMA)`                        0.339***


`ADV: Corp nonconv (FINRA, SIFMA)`          0.170**


`ADV Tri-party: FedWire eligible`                     0.465***


Constant                                                                   0.075***



Observations                                                                 136           

R2                                                                                    0.776          

Adjusted R2                                                                   0.766          

Residual Std. Error                                                      0.456 (df = 129)     

F Statistic                                                                       74.490*** (df = 6; 129) 


Note:                                   *p<0.1; **p<0.05; ***p<0.01

[1] M. Friedman, The Quantity Theory of Money – A Restatement, University of Chicago Press, 1956

[2] “The Origins of Velocity Functions”, Thomas M. Humphrey, Federal Reserve of Richmond, Economic Quarterly Review Fall 1993

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