Knowledge Limited for Dummies….Didactics.

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Bertrand Russell with Alfred North Whitehead, in the Principia Mathematica aimed to demonstrate that “all pure mathematics follows from purely logical premises and uses only concepts defined in logical terms.” Its goal was to provide a formalized logic for all mathematics, to develop the full structure of mathematics where every premise could be proved from a clear set of initial axioms.

Russell observed of the dense and demanding work, “I used to know of only six people who had read the later parts of the book. Three of those were Poles, subsequently (I believe) liquidated by Hitler. The other three were Texans, subsequently successfully assimilated.” The complex mathematical symbols of the manuscript required it to be written by hand, and its sheer size – when it was finally ready for the publisher, Russell had to hire a panel truck to send it off – made it impossible to copy. Russell recounted that “every time that I went out for a walk I used to be afraid that the house would catch fire and the manuscript get burnt up.”

Momentous though it was, the greatest achievement of Principia Mathematica was realized two decades after its completion when it provided the fodder for the metamathematical enterprises of an Austrian, Kurt Gödel. Although Gödel did face the risk of being liquidated by Hitler (therefore fleeing to the Institute of Advanced Studies at Princeton), he was neither a Pole nor a Texan. In 1931, he wrote a treatise entitled On Formally Undecidable Propositions of Principia Mathematica and Related Systems, which demonstrated that the goal Russell and Whitehead had so single-mindedly pursued was unattainable.

The flavor of Gödel’s basic argument can be captured in the contradictions contained in a schoolboy’s brainteaser. A sheet of paper has the words “The statement on the other side of this paper is true” written on one side and “The statement on the other side of this paper is false” on the reverse. The conflict isn’t resolvable. Or, even more trivially, a statement like; “This statement is unprovable.” You cannot prove the statement is true, because doing so would contradict it. If you prove the statement is false, then that means its converse is true – it is provable – which again is a contradiction.

The key point of contradiction for these two examples is that they are self-referential. This same sort of self-referentiality is the keystone of Gödel’s proof, where he uses statements that imbed other statements within them. This problem did not totally escape Russell and Whitehead. By the end of 1901, Russell had completed the first round of writing Principia Mathematica and thought he was in the homestretch, but was increasingly beset by these sorts of apparently simple-minded contradictions falling in the path of his goal. He wrote that “it seemed unworthy of a grown man to spend his time on such trivialities, but . . . trivial or not, the matter was a challenge.” Attempts to address the challenge extended the development of Principia Mathematica by nearly a decade.

Yet Russell and Whitehead had, after all that effort, missed the central point. Like granite outcroppings piercing through a bed of moss, these apparently trivial contradictions were rooted in the core of mathematics and logic, and were only the most readily manifest examples of a limit to our ability to structure formal mathematical systems. Just four years before Gödel had defined the limits of our ability to conquer the intellectual world of mathematics and logic with the publication of his Undecidability Theorem, the German physicist Werner Heisenberg’s celebrated Uncertainty Principle had delineated the limits of inquiry into the physical world, thereby undoing the efforts of another celebrated intellect, the great mathematician Pierre-Simon Laplace. In the early 1800s Laplace had worked extensively to demonstrate the purely mechanical and predictable nature of planetary motion. He later extended this theory to the interaction of molecules. In the Laplacean view, molecules are just as subject to the laws of physical mechanics as the planets are. In theory, if we knew the position and velocity of each molecule, we could trace its path as it interacted with other molecules, and trace the course of the physical universe at the most fundamental level. Laplace envisioned a world of ever more precise prediction, where the laws of physical mechanics would be able to forecast nature in increasing detail and ever further into the future, a world where “the phenomena of nature can be reduced in the last analysis to actions at a distance between molecule and molecule.”

What Gödel did to the work of Russell and Whitehead, Heisenberg did to Laplace’s concept of causality. The Uncertainty Principle, though broadly applied and draped in metaphysical context, is a well-defined and elegantly simple statement of physical reality – namely, the combined accuracy of a measurement of an electron’s location and its momentum cannot vary far from a fixed value. The reason for this, viewed from the standpoint of classical physics, is that accurately measuring the position of an electron requires illuminating the electron with light of a very short wavelength. But the shorter the wavelength the greater the amount of energy that hits the electron, and the greater the energy hitting the electron the greater the impact on its velocity.

What is true in the subatomic sphere ends up being true – though with rapidly diminishing significance – for the macroscopic. Nothing can be measured with complete precision as to both location and velocity because the act of measuring alters the physical properties. The idea that if we know the present we can calculate the future was proven invalid – not because of a shortcoming in our knowledge of mechanics, but because the premise that we can perfectly know the present was proven wrong. These limits to measurement imply limits to prediction. After all, if we cannot know even the present with complete certainty, we cannot unfailingly predict the future. It was with this in mind that Heisenberg, ecstatic about his yet-to-be-published paper, exclaimed, “I think I have refuted the law of causality.”

The epistemological extrapolation of Heisenberg’s work was that the root of the problem was man – or, more precisely, man’s examination of nature, which inevitably impacts the natural phenomena under examination so that the phenomena cannot be objectively understood. Heisenberg’s principle was not something that was inherent in nature; it came from man’s examination of nature, from man becoming part of the experiment. (So in a way the Uncertainty Principle, like Gödel’s Undecidability Proposition, rested on self-referentiality.) While it did not directly refute Einstein’s assertion against the statistical nature of the predictions of quantum mechanics that “God does not play dice with the universe,” it did show that if there were a law of causality in nature, no one but God would ever be able to apply it. The implications of Heisenberg’s Uncertainty Principle were recognized immediately, and it became a simple metaphor reaching beyond quantum mechanics to the broader world.

This metaphor extends neatly into the world of financial markets. In the purely mechanistic universe of classical physics, we could apply Newtonian laws to project the future course of nature, if only we knew the location and velocity of every particle. In the world of finance, the elementary particles are the financial assets. In a purely mechanistic financial world, if we knew the position each investor has in each asset and the ability and willingness of liquidity providers to take on those assets in the event of a forced liquidation, we would be able to understand the market’s vulnerability. We would have an early-warning system for crises. We would know which firms are subject to a liquidity cycle, and which events might trigger that cycle. We would know which markets are being overrun by speculative traders, and thereby anticipate tactical correlations and shifts in the financial habitat. The randomness of nature and economic cycles might remain beyond our grasp, but the primary cause of market crisis, and the part of market crisis that is of our own making, would be firmly in hand.

The first step toward the Laplacean goal of complete knowledge is the advocacy by certain financial market regulators to increase the transparency of positions. Politically, that would be a difficult sell – as would any kind of increase in regulatory control. Practically, it wouldn’t work. Just as the atomic world turned out to be more complex than Laplace conceived, the financial world may be similarly complex and not reducible to a simple causality. The problems with position disclosure are many. Some financial instruments are complex and difficult to price, so it is impossible to measure precisely the risk exposure. Similarly, in hedge positions a slight error in the transmission of one part, or asynchronous pricing of the various legs of the strategy, will grossly misstate the total exposure. Indeed, the problems and inaccuracies in using position information to assess risk are exemplified by the fact that major investment banking firms choose to use summary statistics rather than position-by-position analysis for their firmwide risk management despite having enormous resources and computational power at their disposal.

Perhaps more importantly, position transparency also has implications for the efficient functioning of the financial markets beyond the practical problems involved in its implementation. The problems in the examination of elementary particles in the financial world are the same as in the physical world: Beyond the inherent randomness and complexity of the systems, there are simply limits to what we can know. To say that we do not know something is as much a challenge as it is a statement of the state of our knowledge. If we do not know something, that presumes that either it is not worth knowing or it is something that will be studied and eventually revealed. It is the hubris of man that all things are discoverable. But for all the progress that has been made, perhaps even more exciting than the rolling back of the boundaries of our knowledge is the identification of realms that can never be explored. A sign in Einstein’s Princeton office read, “Not everything that counts can be counted, and not everything that can be counted counts.”

The behavioral analogue to the Uncertainty Principle is obvious. There are many psychological inhibitions that lead people to behave differently when they are observed than when they are not. For traders it is a simple matter of dollars and cents that will lead them to behave differently when their trades are open to scrutiny. Beneficial though it may be for the liquidity demander and the investor, for the liquidity supplier trans- parency is bad. The liquidity supplier does not intend to hold the position for a long time, like the typical liquidity demander might. Like a market maker, the liquidity supplier will come back to the market to sell off the position – ideally when there is another investor who needs liquidity on the other side of the market. If other traders know the liquidity supplier’s positions, they will logically infer that there is a good likelihood these positions shortly will be put into the market. The other traders will be loath to be the first ones on the other side of these trades, or will demand more of a price concession if they do trade, knowing the overhang that remains in the market.

This means that increased transparency will reduce the amount of liquidity provided for any given change in prices. This is by no means a hypothetical argument. Frequently, even in the most liquid markets, broker-dealer market makers (liquidity providers) use brokers to enter their market bids rather than entering the market directly in order to preserve their anonymity.

The more information we extract to divine the behavior of traders and the resulting implications for the markets, the more the traders will alter their behavior. The paradox is that to understand and anticipate market crises, we must know positions, but knowing and acting on positions will itself generate a feedback into the market. This feedback often will reduce liquidity, making our observations less valuable and possibly contributing to a market crisis. Or, in rare instances, the observer/feedback loop could be manipulated to amass fortunes.

One might argue that the physical limits of knowledge asserted by Heisenberg’s Uncertainty Principle are critical for subatomic physics, but perhaps they are really just a curiosity for those dwelling in the macroscopic realm of the financial markets. We cannot measure an electron precisely, but certainly we still can “kind of know” the present, and if so, then we should be able to “pretty much” predict the future. Causality might be approximate, but if we can get it right to within a few wavelengths of light, that still ought to do the trick. The mathematical system may be demonstrably incomplete, and the world might not be pinned down on the fringes, but for all practical purposes the world can be known.

Unfortunately, while “almost” might work for horseshoes and hand grenades, 30 years after Gödel and Heisenberg yet a third limitation of our knowledge was in the wings, a limitation that would close the door on any attempt to block out the implications of microscopic uncertainty on predictability in our macroscopic world. Based on observations made by Edward Lorenz in the early 1960s and popularized by the so-called butterfly effect – the fanciful notion that the beating wings of a butterfly could change the predictions of an otherwise perfect weather forecasting system – this limitation arises because in some important cases immeasurably small errors can compound over time to limit prediction in the larger scale. Half a century after the limits of measurement and thus of physical knowledge were demonstrated by Heisenberg in the world of quantum mechanics, Lorenz piled on a result that showed how microscopic errors could propagate to have a stultifying impact in nonlinear dynamic systems. This limitation could come into the forefront only with the dawning of the computer age, because it is manifested in the subtle errors of computational accuracy.

The essence of the butterfly effect is that small perturbations can have large repercussions in massive, random forces such as weather. Edward Lorenz was testing and tweaking a model of weather dynamics on a rudimentary vacuum-tube computer. The program was based on a small system of simultaneous equations, but seemed to provide an inkling into the variability of weather patterns. At one point in his work, Lorenz decided to examine in more detail one of the solutions he had generated. To save time, rather than starting the run over from the beginning, he picked some intermediate conditions that had been printed out by the computer and used those as the new starting point. The values he typed in were the same as the values held in the original simulation at that point, so the results the simulation generated from that point forward should have been the same as in the original; after all, the computer was doing exactly the same operations. What he found was that as the simulated weather pattern progressed, the results of the new run diverged, first very slightly and then more and more markedly, from those of the first run. After a point, the new path followed a course that appeared totally unrelated to the original one, even though they had started at the same place.

Lorenz at first thought there was a computer glitch, but as he investigated further, he discovered the basis of a limit to knowledge that rivaled that of Heisenberg and Gödel. The problem was that the numbers he had used to restart the simulation had been reentered based on his printout from the earlier run, and the printout rounded the values to three decimal places while the computer carried the values to six decimal places. This rounding, clearly insignificant at first, promulgated a slight error in the next-round results, and this error grew with each new iteration of the program as it moved the simulation of the weather forward in time. The error doubled every four simulated days, so that after a few months the solutions were going their own separate ways. The slightest of changes in the initial conditions had traced out a wholly different pattern of weather.

Intrigued by his chance observation, Lorenz wrote an article entitled “Deterministic Nonperiodic Flow,” which stated that “nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states.” Translation: Long-range weather forecasting is worthless. For his application in the narrow scientific discipline of weather prediction, this meant that no matter how precise the starting measurements of weather conditions, there was a limit after which the residual imprecision would lead to unpredictable results, so that “long-range forecasting of specific weather conditions would be impossible.” And since this occurred in a very simple laboratory model of weather dynamics, it could only be worse in the more complex equations that would be needed to properly reflect the weather. Lorenz discovered the principle that would emerge over time into the field of chaos theory, where a deterministic system generated with simple nonlinear dynamics unravels into an unrepeated and apparently random path.

The simplicity of the dynamic system Lorenz had used suggests a far-reaching result: Because we cannot measure without some error (harking back to Heisenberg), for many dynamic systems our forecast errors will grow to the point that even an approximation will be out of our hands. We can run a purely mechanistic system that is designed with well-defined and apparently well-behaved equations, and it will move over time in ways that cannot be predicted and, indeed, that appear to be random. This gets us to Santa Fe.

The principal conceptual thread running through the Santa Fe research asks how apparently simple systems, like that discovered by Lorenz, can produce rich and complex results. Its method of analysis in some respects runs in the opposite direction of the usual path of scientific inquiry. Rather than taking the complexity of the world and distilling simplifying truths from it, the Santa Fe Institute builds a virtual world governed by simple equations that when unleashed explode into results that generate unexpected levels of complexity.

In economics and finance, institute’s agenda was to create artificial markets with traders and investors who followed simple and reasonable rules of behavior and to see what would happen. Some of the traders built into the model were trend followers, others bought or sold based on the difference between the market price and perceived value, and yet others traded at random times in response to liquidity needs. The simulations then printed out the paths of prices for the various market instruments. Qualitatively, these paths displayed all the richness and variation we observe in actual markets, replete with occasional bubbles and crashes. The exercises did not produce positive results for predicting or explaining market behavior, but they did illustrate that it is not hard to create a market that looks on the surface an awful lot like a real one, and to do so with actors who are following very simple rules. The mantra is that simple systems can give rise to complex, even unpredictable dynamics, an interesting converse to the point that much of the complexity of our world can – with suitable assumptions – be made to appear simple, summarized with concise physical laws and equations.

The systems explored by Lorenz were deterministic. They were governed definitively and exclusively by a set of equations where the value in every period could be unambiguously and precisely determined based on the values of the previous period. And the systems were not very complex. By contrast, whatever the set of equations are that might be divined to govern the financial world, they are not simple and, furthermore, they are not deterministic. There are random shocks from political and economic events and from the shifting preferences and attitudes of the actors. If we cannot hope to know the course of the deterministic systems like fluid mechanics, then no level of detail will allow us to forecast the long-term course of the financial world, buffeted as it is by the vagaries of the economy and the whims of psychology.

Asset Backed Securities. Drunken Risibility.

Asset Backed Securities (ABS) are freely traded financial instruments that represent packages of loans issued by the commercial banks. The majority are created from mortgages, but credit card debt, commercial real estate loans, student loans, and hedge fund loans are also known to have been securitized. The earliest form of ABS within the American banking system appears to stem from the creation of the Federal National Mortgage Association (Fannie Mae) in 1938 as part of amendments to the US National Housing Act, a Great Depression measure aimed at creating loan liquidity. Fannie Mae, and the other Government Sponsored Enterprises buy loans from approved mortgage sellers, typically banks, and create guaranteed financial debt instruments from them, to be sold on the credit markets. The resulting bonds, backed as they are by loan insurance, are widely used in pension funds and insurance companies, as a secure, financial instrument providing a predictable, low risk return.

The creation of a more general form of Mortgage Backed Security is credited to Bob Dall and the trading desk of Salmon brothers in 1977 by Michael Lewis (Liar’s Poker Rising Through the Wreckage on Wall Street). Lewis also describes a rapid expansion in their sale beginning in 1981 as a side effect of the United States savings and loans crisis. The concept was extended in 1987 by bankers at Drexel Burnham Lambert Inc. to corporate bonds and loans in the form of Collateralized Debt Obligations (CDOs), which eventually came to include mortgage backed securities, and in the form of CDO-Squared instruments, pools of CDO.

Analysis of the systemic effects of Asset Backed Security has concentrated chiefly on their ability to improve the quantity of loans, or loan liquidity, which has been treated as a positive feature by Greenspan. It has also been noted that securitization allowed banks to increase their return on capital by transforming their operations into a credit generating pipeline process. Hyun Song Shin has also analysed their effect on bank leverage and the stability of the larger financial system within an accounting framework. He highlights the significance of loan supply factors in causing the sub-prime crisis. Although his model appears not to completely incorporate the full implications of the process operating within the capital reserve regulated banking system, it presents an alternate, matrix based analysis of the uncontrolled debt expansion that these instruments permit.

The systemic problem introduced by asset backed securities, or any form of sale that transfers loans made by commercial banking institutions outside the regulatory framework is that they allow banks to escape the explicit reserve and regulatory capital based regulation on the total amount of loans being issued against customer deposits. This has the effect of steadily increasing the ratio of bank originated loans to money on deposit within the banking system.

The following example demonstrates the problem using two banks, A and B. For simplicity fees related to loans and ABS sales are excluded. It is assumed that the deposit accounts are Net Transaction accounts carry a 10% reserve requirement, and that both banks are ”well capitalised” and that the risk weighted multiplier for the capital reserve for these loans is also 10.

The example proceeds as a series of interactions as money flows between the two banks. The liabilities (deposits) and assets (loans) are shown, with loans being separated into bank loans, and Mortgage Backed Securities (MBS), depending on their state.

Initial Conditions: To simplify Bank B is shown as having made no loans, and has excess reserves at the central bank to maintain the balance sheet. The normal inter-bank and central bank lending mechanisms would enable the bank to compensate for temporary imbalances during the process under normal conditions. All deposit money used within the example remains on deposit at either Bank A or Bank B. On the right hand side of the table the total amount of deposits and loans for both banks is shown.

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Step 1: Bank A creates a $1000 Mortgage Backed Security from the loan on its balance sheet.

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Step 2: The securitized loan is sold to the depositor at Bank B. $1000 is paid to Bank A from that depositor in payment for the loan. Bank A now has no loans outstanding against its deposits, and the securitized loan has been moved outside of banking system regulation. Note that total deposits at the two banks have temporarily shrunk due to the repayment of the loan capital at A. The actual transfer of the deposits between the banks is facilitated through the reserve holdings which also function as clearing funds.

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Step 3: As Bank A now has no loans against its deposits, and is within its regulatory capital ratios, it can make a new $1000 loan. The funds from this loan are deposited at Bank B. The sum of the deposits rises as a result as does the quantity of loans. Note that the transfer of the loan money from Bank A to Bank B again goes through the reserve holdings in the clearing system and restores the original balance at Bank B.

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Step 4: Bank A securitizes the loan made in Step 3 repeating Step 1.

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Step 5: Bank A sells the MBS to the depositor at Bank B, repeating Step 2.

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Step 6: Bank A makes a new loan which is deposited at Bank B, repeating Step 3.

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Step 7: Bank A securitizes the loan made in Step 6, repeating Step 4.

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Since the Deposit and Loan positions of the two banks are identical in all respects in Steps (1,4), (2,5), (3,6) and (4,7) the process can continue indefinitely, resulting in expansion of the total commercial bank originated loan supply independent of central bank control.

This is a simplified version of the flows between loans, deposits, and asset backed securities that occur daily in the banking system. At no point has either bank needed recourse to central bank funds, or broken any of their statutory requirements with respect to capitalisation or reserve requirements where they apply.

The problem is the implicit assumption with reserve based banking systems that bank originated loans remain within the banking system. Allowing the sale of loans to holders outside of the regulated banking system (i.e. to entities other than regulated banks) removes these loans from that control and thus creates a systemic loophole in the regulation of the commercial bank loan supply.

The introduction of loans sales has consequently created a novel situation in those modern economies that allow them, not only in causing a significant expansion in total lending from the banking sector, but also in changing the systemic relationship between the quantity of money in the system to the quantity of bank originated debt, and thereby considerably diluting the influence the central bank can exert over the loan supply. The requirement that no individual bank should lend more than their deposits has been enforced by required reserves at the central bank since the 19th century in Europe, and the early 20th century in the USA. Serendipitously, this also created a systemic limit on the ratio of money to bank originated lending within the monetary system. While the sale of Asset Backed Securities does not allow any individual bank to exceed this ratio at any given point in time, as the process evolves the banking system itself exceeds it as loans are moved outside the constraints provided by regulatory capital or reserve regulation, thereby creating a mechanism for unconstrained growth in commercial bank originated lending.

While the asset backed security problem explains the dramatic growth in banking sector debt that has occurred over the last three decades, it does not explain the accompanying growth in the money supply. Somewhat uniquely of the many regulatory challenges that the banking system has created down the centuries, the asset backed security problem, in and of itself does not cause the banks, or the banking system to ”print money”.

The question of what exactly constitutes money in modern banking systems is a non-trivial one. As the examples above show, bank loans create money in the form of bank deposits, and bank deposits can be used directly for monetary purposes either through cheques or more usually now direct electronic transfer. For economic purposes then, bank deposits can be regarded as directly equivalent to physical money. The reality within the banking system however is somewhat more complex, in that transfers between bank deposits must be performed using deposits in the clearing mechanisms, either through the reserves at the central bank, or the bank’s own asset deposits at other banks. Nominally limits on the total quantity of central bank reserves should in turn limit the growth in bank deposits from bank lending, but it is clear from the monetary statistics that this is not the case.

Individually commercial banks are limited in the amount of money they can lend. They are limited by any reserve requirements for their deposits, by the accounting framework that surrounds the precise classification of assets and liabilities within their locale, and by the ratio of their equity or regulatory capital to their outstanding, risk weighted loans as recommended by the Basel Accords. However none of these limits is sufficient to prevent uncontrolled expansion.

Reserve requirements at the central bank can only effectively limit bank deposits if they apply to all accounts in the system, and the central bank maintains control over any mechanisms that allow individual banks to increase their reserve holdings. This appears not to be the case. Basel capital restrictions can also limit bank lending. Assets (loans) held by banks are classified by type, and have accordingly different percentage capital requirements. Regulatory capital requirements are divided into two tiers of capital with different provisions and risk categorisation applying to instruments held in them. To be adequately capitalised under the Basel accords, a bank must maintain a ratio of at least 8% between its Tier 1 and Tier 2 capital reserves, and its loans. Equity capital reserves are provided by a bank’s owners and shareholders when the bank is created, and exist to provide a buffer protecting the bank’s depositors against loan defaults.

Under Basel regulation, regulatory capital can be held in a variety of instruments, depending on Tier 1 or Tier 2 status. It appears that some of those instruments, in particular subordinated debt and hybrid debt capital instruments, can represent debt issued from within the commercial banking system.

Annex A – Basel Capital Accords, Capital Elements Tier 1

(a) Paid-up share capital/common stock

(b) Disclosed reserves

Tier 2

(a) Undisclosed reserves

(b) Asset revaluation reserves

(c) General provisions/general loan-loss reserves

(d) Hybrid (debt/equity) capital instruments

(e) Subordinated debt

Subordinated debt is defined in Annex A of the Basel treaty as:

(e) Subordinated term debt: includes conventional unsecured subordinated debt capital instruments with a minimum original fixed term to maturity of over five years and limited life redeemable preference shares. During the last five years to maturity, a cumulative discount (or amortisation) factor of 20% per year will be applied to reflect the diminishing value of these instruments as a continuing source of strength. Unlike instruments included in item (d), these instruments are not normally available to participate in the losses of a bank which continues trading. For this reason these instruments will be limited to a maximum of 50% of tier 1.

This is debt issued by the bank, in various forms, but of guaranteed long duration, and controlled repayment. In effect, it allows Banks to hold borrowed money in regulatory capital. It is subordinate to the claims of depositors in the event of Bank failure. The inclusion of subordinated debt in Tier 2 allows financial instruments created from lending to become part of the regulatory control on further lending, creating a classic feedback loop. This can also occur as a second order effect if any form of regulatory capital can be purchased with money borrowed from within the banking system