*This article presents a rigorous analysis of many issues discussed on this blog already sometimes in a less formal manner. Especially a banking practice of lending with Loan to Deposit Ratio above 100% that has been shown to constitute a sufficient condition of causing liquidity shortage in the banking system (i.e. it was a sufficient condition that caused the current financial crisis).*

It is estimated that Cash (narrow money) constitutes around 2% of money circulation in the economy. The reminder 98%, sometimes referred to as broad money, is created by banks through Deposit – Loan Cycles. It is called Credit Creation. When money is paid into a Bank it is

either:

- a disbursement (cost) that a Bank has to pay out (but even, in this case, unless it is stored privately, it will end up in a Bank as someone else’s Deposit); a dividend to be paid by a Bank to its shareholders is also considered as a disbursement

- a Deposit; any other money than disbursement is considered as a Deposit paid into a Bank (e.g. if it is a Bank’s retained profit, Bank’s own money is a Bank’s Deposit on its own books; if a Bank uses its own money to buy an investment product, from Deposit – Loan Cycle perspective, in this model it is considered as if a Bank was lending money to itself).

A liquidity risk is a risk of a situation when a demand made by a Depositor to withdraw Cash (narrow money) cannot be met by a Bank. If Money Multiplier is 1 (or less) liquidity risk is 0%: $1 (or more) Cash Reserves covers every $1 Deposits on a Bank’s Loan/Deposit Balance Sheet. If Money Multiplier tends to infinity liquidity risk is 100% in a finite time: at the limit, $1 Cash would have to cover infinite amount of dollars of Deposits on a Bank’s Loan/Deposit Balance Sheet.

Liquidity risk is directly associated with a phenomenon called “bank run”, when depositors, in large numbers, would like to withdraw money from a Bank to either pay it to another bank or worse, from liquidity point of view, keep it privately. As some depositors cannot withdraw their money it results in destruction of a Bank’s credibility. Then even more depositors follow suit leading to a Bank’s collapse.

__1. Full Reserve Banking__

When a Bank retains all Deposits paid in, a Bank is not lending. It acts as a Product/Service Supplier of storing cash. This is also called 100% Reserve Banking.

For every $1 paid as Cash Deposit, Bank’s Loan/Deposit Balance Sheet shows Loan = $0, Deposit = $1, Cash Reserves = $1. Money Multiplier is 1 (i.e. in every day’s language money is multiplied by 1, i.e. it is not multiplied, no Credit is created).

**Conclusion: Full Reserve Banking is a case of complexity of no growth (**

*O*(1)). Ignoring theft and fraud, the liquidity risk of Full Reserve Banking is 0% (i.e. 100% Deposits paid in are always in a Bank and can be withdrawn on demand at any time.)__2. Fractional Reserve Banking__

When a Bank lends part of Deposits paid in, a Bank is Creating Credit. A proportion of a Loan to a Deposit from which a Loan is given is called Loan to Deposit Ratio (which may also be expressed in percentage terms: for example, LTD = 0.9 is equivalent to 90%.).

For example, for initial $1 paid in if the Loan to Deposit Ratio is LTD, a Loan given is $1 x LTD. Generally in the money-based economy it can be assumed that, on the whole, this Loan, $1 x LTD, will end up in a Bank as a Deposit and then it is re-lent again. Assuming the same Loan to Deposit Ratio, the next Loan is $1 x LTD

^{2}. Generally after n iterations from the initial $1, the Loan given is $1 x LTD

^{n}. Since LTD is less 1 (100%) then it is a regressive geometric series with limit 0 and the total Credit Created from the initial $1 Cash is $1 x 1/(1-LTD).

Money Multiplier is a ratio of Deposits on a Bank’s Loan/Deposit Balance Sheet to Cash Reserves accumulated. It tells us how many times Cash (narrow money) was multiplied by Credit Creation process (into broad money). Therefore based Loan to Deposit Ratio, LTD, we can calculate a Money Multiplier, MM. It is: MM = 1/(1-LTD). Money Multiplier tells us how many Deposited dollars on a Bank’s Loan/Deposit Balance Sheet are covered by $1 Cash.

For example, let LTD = 0.9 (i.e. 90%), i.e. MM = 10, then for every $1 of initial Cash Deposit (narrow money), the first loan is $0.9, the second $0.81, the

*n*

^{th}$1 x 0.9

^{n}, the total value of Deposits on a Bank’s Loan/Deposit Balance Sheet is $10 and there are $9 of Credit Created (circulated in money-based economy) and Cash Reserves are $1.

**Conclusion: Fractional Reserve Banking, with Loan to Deposit Ratio above 0 and below 1, is a complexity case of no growth (**

*O*(MM)) of a Bank’s Loan/Deposit Balance Sheets to underlying Cash Reserves as Money Multiplier is a constant number in relation to Loan to Deposit Ratio. We observe that it entails liquidity risk below 100%, since it is always possible that demand for withdrawals, at one time, exceeds Cash Reserves.It is not a purpose of this paper to argue what level of liquidity risk is acceptable or beneficial for money-based economy and what factors ultimately determine this risk in reality. This may depend on many phenomena that affect human decision making process.

__3. No Reserve Banking__

We also observe that as Loan to Deposit Ratio is less than 1 and approaching it, the Money Multiplier tends to infinity, and nearly no Cash Reserves are created, the liquidity risk keeps increasing up to 100% at the limit. Nearly all the money paid in as Deposits are turned into Credits. Loans and Deposits tend to infinity on a Bank’s Loan/Deposit Balance Sheets, whilst Cash Reserves tend to remain finite and constant.

If Loan to Deposit Ratio is 1, at the limit, it is a linear growth of Loans and Deposits on a Bank Loan/Deposit Balance Sheet, Money Multiplier is infinity, Cash Reserves’ growth is 0 (i.e. they stay on the level which was when Loan to Deposit Ratio of 1 was started) and liquidity risk (in a finite time) tends to 100%. It is a case of trivial geometric series with common ratio 1 (which is also an arithmetic series).

**Conclusion: No Reserve Banking is a complexity case of linear growth (**

*O*(n)) to infinity of both Loans and Deposits on a Bank’s Loan/Deposit Balance Sheet. A Money Multiplier is infinity resulting in liquidity risk of 100% in a finite time. No Reserve Banking is also called 0% Reserve Banking.__4. Depleting Reserve Banking__

Depleting Reserve Banking, lending with Loan to Deposit Ratio above 1, is not possible, unless there are already Cash Reserves. It is, effectively, a Credit Creation with a “top up” from already existing reserves (this top up may come as a Loan from another bank’s reserves and a lending bank may consider borrowing bank’s debt papers as good as Cash). A Bank is Creating Credit by lending Deposits paid in and topping up from existing Cash Reserves (or a Loan from another bank’s reserves). A proportion of a Loan to an underlying Deposit is called Loan to Deposit Ratio.

For example, for initial $1 paid in if the Loan to Deposit Ratio is LTD, a Loan given is $1 x LTD. Generally in the money-based economy it can be assumed that, on the whole, this Loan, $1 x LTD, will end up in a Bank as a Deposit and then it is re-lent again. Assuming the same Loan to Deposit Ratio, the next Loan is $1 x LTD

^{2}. Therefore after n iterations starting with the initial $1, the Loan given is $1 x LTD

^{n}. Since LTD is above 1 (100%) then it is a progressive geometric series with exponential growth to infinity. The total Credit Created from the initial $1 Cash tends exponentially to infinity: $1 x ((LTD

^{n}– 1)/(LTD – 1))

As Loan to Deposit is above 1, and the geometric series is diverging, it is impossible to calculate Money Multiplier based on a ratio of Loans to Deposits on a Bank’s Loan/Deposit Balance Sheet at any one time. We also must know how many times a Deposit – Loan Cycle was executed at what Loan to Deposit Ratio. The general formula to calculate Money Multiplier (when Loan to Deposit Ratio is above 1) is:

*i*= 1, …. ,

*n*

LTD

_{i}is a Loan to Deposit Ratio

*ki*is a number of Deposit - Loan Cycles with a Loan to Deposit Ratio LTD

_{i}(LTD

_{i+1}is a Loan to Deposit Ratio that follows LTD

_{i}).

It is a side note but we observe that it looks unlikely that Deposit – Loan Cycles (Credit Creation) starting with initial $1 Cash are uniform processes in money-based economy. Therefore it may be practically impossible to calculate accurate and reliable Money Multiplier when Loan to Deposit Ratio is above 1.

For example, let LTD = 1.17 (i.e. 117%) then for every $1 of initial Cash Deposit (narrow money), the first loan is $1.17, the second $1.3689, the

*n*

^{th}$1 x 1.17

^{n}, i.e. for

*n*equal 220 the 220

^{th}Loan value is over $1 x 10

^{15}. The total value of Deposits on a Bank’s Loan/Deposit Balance Sheet is over $5.8 x 10

^{15}and there is also over $5.8 x 10

^{15}Credit Created and circulated in money-based economy. Cash Reserves are initial Cash Reserves minus $5.8 x 10

^{15}. Money Multiplier is over 5.8 x 10

^{15}. In other words, whatever the initial Cash Reserves had been at the start of Credit Creation with Loan to Deposit Ratio of 1.17 (117%), after 220 Deposit – Loan Cycles executions, starting with initial $1 Deposit, the Cash Reserves were depleted by over $5.8 x 10

^{15}.

It is a side note but in practice banks’ Cash Reserves, on a Bank’s books, may be replaced with credit collaterals or other non-Cash financial instruments, being considered as good as Cash. This may result in, nominally, retaining any required reserve ratio, but this reserve would not be in Cash. These financial instruments are a part of Products/Services Supply and their market value/price (in Cash) depends on Products/Services Demand (Money Supply), see the graph on page 2.

**Conclusion: Depleting Reserve Banking, with a Loan to Deposit Ratio above 1, LTD > 1, is a complexity case of exponential growth (**

*O*(LTD^{n})) of a Bank’s Loan/Deposit Balance Sheets to underlying Cash Reserves. As Money Multiplier tends to infinity the liquidity risk is 100% in a finite time.__5. Brief computational complexity analysis summary__

Let us consider Credit Creation as an algorithm that we would like to implement on a computer. (It is actually a very basic algorithm.)

The cases of Full Reserve Banking and Fractional Reserve Banking, of

*O*(k) where k is a constant equal 1/(1 - LTD), are tractable algorithms. They can be considered as contained algorithms: the requirement on resources is a constant multiple of an underlying parameter of Credit Creation (i.e. Cash).

No Reserve Banking (of

*O*(n)) is also a tractable algorithm but it takes more resources than Full Reserve Banking or Fractional Reserve Banking. In fact it would be a case of concern that at some point, in linear time depending on a number of executions of Deposit – Loan Cycles, the demand on resources will eventually exceed availability threshold. This is non-containable case as there is no upper limit of demand on resources in relation to underlying parameter of Credit Creation (i.e. Cash).

Depleting Reserve Banking, of

*O*(LTD

^{n}), is an intractable algorithm. The growth of demand on resources is exponential. This type of algorithms is considered impractical for implementation on computers. (In general, in computer science algorithms with complexity above polynomial are not accepted as general solutions to underlying problems.)

In the cases considered above, containable algorithms of Credit Creation present liquidity risk below 100%. They are of manageable risk as a part of a risk portfolio. Non-containable algorithms present liquidity risk of 100% in a finite time, i.e. it is unmanageable risk as liquidity shortage is a matter of a finite time. In the case of Depleting Reserve Banking, due to exponential growth of a Bank of Loan/Deposit Balance Sheet and Money Multiplier (and also exponential depletion of Cash Reserves) such finite time is assumed to be very short: short enough, in practice, for the liquidity shortage to occur. By computational complexity standards, Credit Creation using Depleting Reserve Banking is intractable, i.e. non-practical for implementation.

__6. Depleting Reserve Banking and loss of control of Money Multiplier__

Essential to managing liquidity risk is the knowledge of Money Multiplier (MM), i.e. how many dollars of Deposits on banks’ Loan/Deposit Balance Sheets are covered by $1 Cash. As already showed in the preceding sections of this article, if a Loan to Deposit Ratio is always below 1 then the total value of Deposits and total value of Loans on banks’ Loan/Deposit Balance Sheets are the basis for calculation of Money Multiplier: MM = Total Deposits/(Total Deposits – Total Loans).

If Loan to Deposit Ratio is above 1 then the above equation does not hold. In order to calculate Money Multiplier we need more information about each Deposit – Loan Cycle (which may be impractical or even impossible to gather). This is even more complicated: it is possible to execute any number of Loan – Deposit Cycles with Loan to Deposit Ratio above 1 (achieving any arbitrary high Money Multiplier in this process) and then with one cycle only reduce a ratio of total value of Loans to total value of Deposits below 1 on a Bank’s Deposit – Loan Balance Sheets. This may look like an “average” Loan to Deposit Ratio below 1. However this value used in the formula presented above, MM = Total Deposits/(Total Deposits – Total Loans), will not produce a true value of Money Multiplier (most likely, in practice, a real Money Multiplier will be much higher).

Therefore, concluding this computational complexity analysis, Credit Creation with Loan to Deposit Ratio above 1 must not be practiced since:

- exponential growth of banks’ Loan/Deposit Balance Sheets is intractable

- liquidity risk, in a finite time, is 100%

- practical macro control of growth of Money Multiplier is lost

It is a side note but due to exponential growth of a Bank’s Loan/Deposit Balance Sheet and exponential depletion of reserves, Depleting Reserve Banking is a classic example of a pyramid scheme.

It is also a side note but the present liquidity crisis and resulting economic crisis was preceded by a prolonged period of Credit Creation with Loan to Deposit Ratio above 1.

__7. Consequences of Depleting Reserve Banking on Mark-to-market and Value-at-risk (VaR)__

Depleting Reserve Banking results in Money Multiplier growing at exponential pace without any upper bound (i.e. infinity is the limit) such that each unit of cash has to satisfy ever growing demand on banks balance sheet. In a finite time (in practice, due to exponential growth, very short time), banks run out of cash to service their payment obligations like deposit withdrawals: all cash becomes tied to servicing such obligations and the shortage keeps growing presenting liquidity shortage. Consequently there is a decreasing volume of money (cash) to pay of any non-cash financial (and other) products. This volume decreases at exponential pace (i.e. practically very fast) to zero [as Money Multiplier grows at exponential pace to infinity, cash available to service non-cash obligations decreases to zero]. Hence a Mark-to-market price of any non-cash financial products also decreases to zero, as there is a decreasing volume of cash available to complete a transaction. In such scenario a probability of a non-cash asset losing 100% of its value in a finite time tends to 100%. In other words, as a result of Depleting Reserve Banking, for any arbitrary high loss less than 100% of a non-cash asset, there is always a finite time horizon (in practice, due to exponential characteristics, very short) such that probability of such loss is 100% (it is a certainty).

**Conclusion: The effect of Depleting Reserve Banking is such that, if continued, it turns all non-cash financial products into worthless assets (so-called “toxic waste”): Value-at-risk (VaR) as a loss of 100% of a value of non-cash financial products is practically 100% in a finite time.**

__8. Financial__

*perpetuum mobile*Depleting Reserve Banking (i.e. Credit Creation with Loan to Deposit Ratio above 1) can lead to unusual, unintuitive effects. As noted above a Bank, as a Credit Creator, makes profit, on the whole, by paying out less for taking Deposits than charging Creditors for Loans. It is not typical for a Bank’s customer to expect to be paid more in interest on a Deposit put in a Bank than to pay for a Credit taken out. It is even less typical to expect a Bank to make a profit in such a scenario. It would be a perfect business model, “win – win “, for a customer and a Bank: financial

*perpetuum mobile*.

Let L be a Loan taken by a Bank customer which she immediately deposits in a Bank. Let I

_{1}be an Interest Rate paid on a Loan by a customer to a Bank and I

_{2}be an Interest Rate paid on a Deposit by a Bank to a customer. I

_{1}< I

_{2}. Let D be a total Deposits accepted by a Bank from which it Creates Credits, C = D x LTD. Let LTD > I

_{2}/I

_{1}. (A customer’s Deposit and Loan, both equal L, are, in practice, much smaller than D.)

Customer’s perspective:

L x I

_{2}- L x I

_{1}= L x (I

_{2}– I

_{1}) > 0: since I

_{2}> I

_{1}a customer makes profit.

Bank’s perspective:

C x I

_{1}- D x I

_{2}= D x LTD x I

_{1}- D x I

_{2}= D x (LTD x I

_{1}– I

_{2}) > 0: since LTD > I

_{2}/I

_{1}a Bank makes profit too.

A Bank is making profit since it is lending out more than is taking in Deposits with LTD > I

_{2}/ I

_{1}. But such lending is unsustainable: it can only last as long as Cash Reserves are sufficient to top up Loans. However Cash Reserves are depleting at exponential pace. Hence this financial

*perpetuum mobile*will have to come to a halt.

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