On capital requirement v loan to deposit ratio

(This is a technical article justifying the assertion that if banks are lending with loan to deposit ratio above 100% then no capital requirements are going to prevent the financial system collapse.)

The banks are lending with loan to deposit ratio (L/D). Their balance sheets grow at exponential pace adding X*L/D^n at each deposit multiplication cycle, where X is an initial deposit. The value of the balance sheets at some point in time equals the sum of values at each cycle over a number of cycles from initial deposit until this point in time plus the initial deposit.

The regulatory approach is that banks should hold a certain fixed percentage of the value of their balance sheets as a reserve (R).

1. "Fractional-reserve banking" - case of loan to deposit ratio < 100%, i.e. in fraction representation L/D < 1

(Please note that R is also represented in fraction rather than percentage.)

If a L/D is below 100% (in a deposit multiplication) then the value of banks’ balance sheets at cycle n is represented by a formula: X*((1-L/D^n)/(1-L/D)). Accordingly the upper limit of the value of banks’ balance sheets generated from initial deposit X is X*(1/(1-L/D)), i.e. it is finite.

Let us assume that reserve R is required by a regulator. Then the difference between banks’ balance sheets and the reserves is X*((1-L/D^n)/(1-L/D)) – R* X*((1-L/D^n)/(1-L/D)), i.e. (1 – R)* X*((1-L/D^n)/(1-L/D)). Therefore the upper limit, i.e. when n tends to infinity, is (1 – R)*X*(1/(1-L/D)), i.e. it is finite.

It should also be noted that the difference between a loan given out and reserve generated, according to a prevailing requirement, at each deposit multiplication cycle tends to zero.


Blue line shows growth of balance sheets at L/D = 90%

Red line shows growth of reserves at L/D = 90% and R = 20%

Green line shows the growth of difference between balance sheets and reserves (i.e. balance sheets not covered by reserves) at L/D = 90% and R = 20%

Please note that L/D = 90% generates automatically a reserve 10% so another 10% would have to come from something else.

2. "Depleting-reserves banking" - case of loan to deposit ratio > 100%, i.e. in fraction representation L/D > 1

(Please note that R is also represented in fraction rather than percentage.)

If a L/D is above100% (in a deposit multiplication) then the value of banks’ balance sheets at cycle n is represented by a formula: X*((1-L/D^n)/(1-L/D)). Accordingly there is no upper limit of the value of banks’ balance sheets generated from initial deposit X since X*((1-L/D^n)/(1-L/D)) tends to infinity.

Let us assume that reserve R is required by a regulator. Then the difference between banks’ balance sheets and the reserves is X*((1-L/D^n)/(1-L/D)) – R* X*((1-L/D^n)/(1-L/D)), i.e. (1 – R)* X*((1-L/D^n)/(1-L/D)). Therefore there is no upper limit since (1 – R)*X*((1-L/D^n)/(1-L/D)) tends to infinity.

It should also be noted that the difference between a loan given out and reserve generated, according to a prevailing requirement, at each deposit multiplication cycle tends to infinity. It is practically very important, albeit irrelevant matter in this analysis, where this reserve is going to come from if L/D is above 1.


Blue line shows growth of balance sheets at L/D = 160%

Red line shows growth of reserves at L/D = 160% and R = 20%

Green line shows the growth of difference between balance sheets and reserves (i.e. balance sheets not covered by reserves) at L/D = 160% and R = 20%

The green line shows the growth of a pyramid scheme in the banking system despite the compliance with 20% reserve requirement.

Please note that when L/D is above 100% there is no reserve generated from a deposit multiplication cycle, so 20% reserve would have to come from something else. But this issue is irrelevant to this proof.

ANALYSIS:

It does not matter how large R is, even if it is ridiculously high like 99.999999%, this would only slightly slow the growth process, as 0.000001% coefficient of exponential function a^x (where x > 1) makes very little practical difference compared if R was 0%. It is still a pyramid scheme.

The difference between banks’ balance sheets and reserves grows to infinity at exponential pace. when L/D is above 1. Such process constitutes a pyramid scheme that is bound to collapse. Therefore even assuming that banks’ reserves are as good and as liquid as cash, since the difference between banks’ balance sheets and reserves grows without any upper limit at exponential pace, no fixed R (even 99.999999%) will prevent a pyramid building.

Consequently even assuming that banks reserves are as good as cash (or cash only) banks balance sheets are turned into toxic waste by lending with loan to deposit ratio above 100% since if it were not the case, the difference between banks' balance sheets and reserves would grow to without any bounds to infinity at an exponential pace (which is not possible).

If banks are lending with loan to deposit ratio (L/D) above 100% the financial system will still collapse being a pyramid scheme, regardless of how arbitrary high the reserve requirement (R) is and the banks’ compliance with it.

NOTES:

1. Please note that this proof does not even rely on the fact that banks’ reserves were turned into a toxic waste.

2. It is in my view criminally negligent that financial professionals responsible for running and oversight of the system especially regulators like Lord Turner do not appear to have a slightest clue about such basic and rudimentary arguments. A reader is invited to draw his/her own conclusions on the future of the financial system and indeed the economy.

3. The case of loan to deposit ratio equal 100% is left to an interested reader.

4. I would like to thank Stuart AJ, whose inquiry prompted finalisation and improvement of this proof.