Leverage and Balance Sheet Management: Reserve Ratios and VaR

Having previously considered how banks intermediate between investors and their investments, FRB and bank runs, and having shown that overlending is likely to be an equilibrium result of the financial market, I now turn my attention to how banks, among others manage their resources, specifically their balance sheet. This post topically follows the previous one, in trying to find a path towards, at least partial, financial enlightenment. Whereas the first post of this series focused on the insights of Diamond’s 1997 article and the second one led me to consider De Meza and Weiss 1981 contributions as well as those that followed it, this post will mostly consider the research that Adrian and Shin (2009 and 2010) have been conducting at the federal reserve of New York.

Their insight was derived from the fact that different types of agents manage their balance sheets according to different dynamics. Although, as expected, Households (HH) manage their balance sheet in such a way as to decrease leverage once the value of assets increases, this observation does not hold for commercial banks, investment banks or even for non-financial corporations. This has to do with the fact that commercial banks are subject to leverage limiting compulsory reserve ratios, while investment banks, whose balance sheets is “marked-to-market” with assets which are “held for trading”, are not. Moreover, non-financial corporations have a mix of “marked-to-market” and “held to maturity” assets. This implies a relatively concentrated and inelastic relationship of leverage growth vis a vis asset growth, for commercial banks, as well as for non-financial corporations ( although in a more dispersed manner), while a positive relationship is present for investment banks.

The implications are manyfold:

  • First, the authors use descriptive statistics and regression results to illustrate the consequences of thinking of liquidity as the amount of assets that can be leveraged.
  • Secondly, it is yet another observation that HHs consume intertemporally, as described by the permanent income hypothesis and the life cycle thirties.
  • Thirdly, it points to the necessity of dealing with “businesses” differently and in a segmented manner.
  • Finally, it shows yet again, that if left unregulated financial markets will overlend and create bubbles by behaving in a herdlike pro-cyclical manner.
  • Below I summarise the relatively convincing, descriptive, theoretical and empirical facts presented by the authors.


Before any complicated discussion of the statistical, mathematical and econometric dynamics at play, I think it’s appropriate to explain the concepts under consideration. I won’t ramble on for long about it and you can find a ton of much more thorough information around the web (here, here and here for example). However, I remember that grasping the basic concepts was my biggest difficulty at the time, so let’s start from the beginning.

 What is a Balance Sheet?

  • A balance sheet is a means of accounting for assets, liabilities and equity. In practical terms it is a table that is constantly updated whenever a company engages in commercial, financial or even barter activity. The size of a balance sheet is always the size of its assets, so thaBalance sheet = Assets= Liabilities + equity

What are Assets?

  • Assets are those things that you either own outright or to which you have a right to. They could be machines, buildings, stocks owned by the company, cash, inventories, etc. I understand that they are divided between Current Assets and Fixed Assets, among others.

What are Liabilities?

  • Liabilities are what you owe others, and so represent a responsibility to others, something that an individual, company or country are literally “liable” for. These could be commercial and financial debt, accounts payable (debt to suppliers), deferred taxation, etcWhat is Equity?
  • Shareholder’s equity corresponds to the shares of the company that have been issued. They represent a specific type of responsibility that the company has towards its shareholders of providing them with growth in share value and dividends for their investment.
  • For individuals the concept of equity is best understood as the assets worth net of liability, and so is a bit more complicated. It is best described with a numerical example. If I bought a house worth €100K which I paid by depositing €10K and by borrowing €90K, then the €10K that I paid would be my equity (the assets worth net of liability), whereas the €90K would be the liability and the €100K would be my assets

What is Leverage?

  • Leverage is the size of my balance sheet in relation to equity. So if, A= D+E, where “A” are assets, “D” are liabilities and “E” is equity, then leverage, “L”, can be best described as

What is Liquidity?

  • Liquidity “is an asset’s ability to be sold without causing a significant movement in its price and with minimum loss of its value. Money, or cash, is the most liquid asset,[because it serves no other purpose and has no other value than as a means of exchange.](…) [It] can be used immediately to perform economic actions like buying, selling, or paying debt, meeting immediate wants and needs.”
  • However the issue is slightly more complicated, in the sense that this is a relatively difficult and variably measured concept. Understanding it, and the measurement implications, is the point of a series of article by Adrian, Sin and others. Below I shall focus on the 2009 and 2010 studies mentioned before. There, the authors argue that “the authors argue that “liquidity should be understood in terms of the growth of balance sheets (i.e. as a flow), rather than as a stock.” Below I attempt to show why


The main descriptive facts that those authors uncovered can be easily summarized by the following figures, taken from Adrian and Shin 2010, describing the relationship between assets and leverage growth in HHs, nonfinancial corporations, commercial banks and dealer/brokers (aka investment banks).

As is evident, a clear set of different patters exists depending on the sector under consideration:

  • Households (HH) manage their balance sheet in such a way as to decrease leverage once the value of assets increases, this observation does not hold for commercial banks, investment banks or even for non-financial corporations.
  • Commercial banks are subject to leverage limiting compulsory reserve ratios
  • Non-financial corporations have a mix of “marked-to-market” and “held to maturity” assets.
  • Finally, investment banks are not subject to such reserve requirement. Moreover, because their balance sheets are “marked-to-market” with assets which are “held for trading”, it is much more actively managed, leading to a positive relationship between asset and leverage growth. For the sake of clarity the authors provide the following figures for specific investment banks.

Commercial banks & reserve ratios

  • The situation with commercial banks is extremely easy to understand. In fact commercial banks are imposed a reserve ratio, meaning a maximum amount of leverage at any given moment.
  • Although this is all in relative terms, there are still insights from an absolute point of view. By this I mean that although although leveraging does not change, as a ratio, its components do. Returning to the Leverage equation from before

Investment banks & Value at Risk (VaR)

    • Clearly investment banks, or “broker/dealers” as Adrian and Shin 2010 prefer to call them do are not restricted by reserve ratios as is the case for commercial banks. This is a purely legal matter and there is nothing, other than the fear of financial capital flight, to stop legislators from waking up tomorrow and imposing similar limits on investment banking. However for now that is not the case and so it is necessary to understand how these organisations go about managing their balance sheet in a way that explains the relationships described above. To this effect I consider the Value at Risk (VaR) model of asset management. The ECB provides an excellent and in depth discussion of this issue in a 2001 working paper series article by Manganelli and Engle. There is also this very good and less technical discussion provided by New York University. Finally, Wikipedia also provides a very good summary of VaR. The discussion below is based on lecture notes freely available by Dr Peter Tinsley of the Department of Economics, Mathematics and Statistics at Birkbeck College, University of London.
    • This balance sheet management focuses on estimating left tail exposure to losses. In this sense the VaR corresponds to the maximum losses that the firm can endure to its equity/net value (E) before going bankrupt given a certain level of confidence.

    • Where “ – r*” represent the maximum negative returns, “A” is the value of assets and the subscript “C” stands for the confidence level with which the VaR is estimated, so that assuming that presuming that returns are normally distributed, say in a standard way with mean equal to 0 and variance equal to 1, r~N(0,1), then this can be seen as follows

               So that         

  • Where r* is determined by the choice of the confidence interval. In light of this assumed normal distribution for returns, we can then derive an expression for the unknown cut-off level of returns, r*, below which the company goes bankrupt.
  • We can calculate r*, first by normalizing the probability distribution with the assistance of the Central Limit Theorem(where r’ is the average return), and then by going from probabilities to critical values.

Where  is the critical value associated with a confidence level C and  is the VaR per unit of assets.

. For a typical 99% level of confidence, “C” = 0.99, this implies, following the table below, that the critical value is given by

This number is, of course, meaningless in that investment returns are never standard normally distributed. However, it does help to show that knowing the distribution of returns makes it easy to estimate the value at risk with a certain level of confidence. Given this, it is possible to calculate the level of leverage that is confidently consistent with non-bankruptcy:

If VaR is indeed adopted with a fixed level of confidence “C”, then the relationship between assets “A” and leverage “L” is such as to imply the positive stable relationship identified in the previous figures.

Notice that using a normal distribution is not strictly correct. Instead it would be more accurate to use fat tailed or stable distributions (which can be used with the help of Generalised Central Limit Theorem) for stock returns and left skewed tailed distribution for credit returns. If you are interested in this topic, Credit Suisse has an interesting credit portfolio analysis handbook with a good discussion of credit returns, while mathestate has a great guide to stock market returns and investopedia and Wikipedia both have good discussions of probability distributions. issue while


The authors then go on to study empirical effects. In the following table and equation from Adrian and Shin 2008 they show that monetary policy affects the size of asset growth and that Asset growth affects housing investment through lending.

The authors note that their insights differ from those of Bernanke and Gertler 1999 and  Kiyotaki and Moore 1997, in that those authors focus on the borrowing effects of monetary policy, whereas this is about the lending decision. They also distinghuish their work from that of Curdia and Woodford (2008) (comments here), who study the effect of credit spreads.


  • As argued in the introduction, Adrian & Shin’s research offers simple insights into the financial sector and in my opinion further arguments in support of banking regulation. They also introduce some predictability into the financial sector. During financial crises when assets are, at least partially wiped out, commercial banks will remain leveraged at the same level as before, but with less assets. This relative stability must be reflected into a fall of lending/borrowing in absolute terms. Similarly, VaR implies a much more pronounced reaction in investment banks. Because relative levels of leveraging increase during booms and fall during contractions, the movements in absolute terms will also be higher.
  • The authors also mention in a subtle wink of support that Bordo and Jeanne (2002) argue that banking balance sheets should be included in monetary policy rules in order to decrease financial crises.
  • This contradicts my previous ( and unsubstantiated) claim that during crisis commercial lending falls more than investment banking lending, which I argued could be due to scale effects and to lower levels of asymmetric information in the first vis a vis the second. I will look deeper into this issue.
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