Friday, November 20, 2009

Exponential Monetary Growth under Basel 2

I have been reading this working paper by Trond Andresen.

ABSTRACT: A Basel-type bank regulation regime has the side effect of endogenous money growth. The growth rate turns out to be inversely proportional to the required minimum capital/asset ratio. This money growth contributes to avoiding debt crises, as opposed to non-bank lending which increases debt but not money stock, and is therefore dangerous in the long run. Banks often prefer to sell loans onwards. It is shown that this doesn't only decrease the bank's risk, it may also imply faster asset growth for the selling bank by allowing an increase in the flow of new extended loans.

I will try and present the equations of the first part, offer a generalization of the framework, and briefly discuss the results.

The papers look at the balance sheet of an idealized bank, it has assets, and liabilities. Double entry accounting states:

Assets + Cash = Debt + Equity. (0)

Which can be read as all assets and cash need to be coming from debt or paid-in equity. This equations holds in time as the bank starts earning money. Equity is then defined as liquidation value, where you liquidate all assets, add cash, pay debt and what is left is distributed to equity.

The quantities studied are:

A, the amount of assets the bank holds. This is the amount of loans that are extended. A loan is something that pays ia*A to the banks (think of your mortgage, a 7%). It cost you ia*A ($/YR). This model captures bad debt in the form of lambda*A. Lambda is the rate of default on debt. Think about sub-prime, lambda was assumed to be 15%, it turned out to be 30%. It also models repayments as r*A, r being the portion of loans repaid per year. The assets also grow at the rate of new loans which we call l.

The equation of growth of A (noted A') is straightforward

A' = l - r*A -lamda*A (1)

And reads "The rate of growth of the assets is the amount of new loans minus the loans that are repaid minus the loans that default". If you can read the above equation as the sentence, nothing more complicated will be coming your way.

The equation for the growth (rate of change) of Liabilities is the following

L' = l -r*A - beta*(ia*A-il*L) - ec (2)

The first element is l is the same the new loans. The reason they show up here as well is that all cash that is lent by the banks is deposited back in the banks.

r*A is the amount of loans that are repaid. To repay a loan you pay with money from your deposit. So it is substracted here.

The bank makes money on the difference between the rates in and out. It gets paid a flow ia*A (think your mortgage at 7%). In turn, it pays, il*L to you on your deposits (think 3%). That is captured by the flow (ia*A-il*L). The flow from operation is the profit margin of this flow, which we call beta. Hence the beta*(ia*A-il*L).

Here, compared to the Andresen original, we have introduced a new variable ec as the amount of paid-in capital raised during the period. This is a flow of new equity issued.

We define C as the amount of cash the bank has on hand. Cash grows as the company makes it from operations per the rate above or it is raised on the markets. Cash in turn is drained as it is loaned out, we call this flow lc. This is captured by the following equation on the first derivative, or rate of growth of C which we note C':

C' = beta*(ia*A-il*L) + ec -lc (3)

new loans are created all the time, this is where debt money is lent into existence we call this flow ln. The total amount of loans at each period is:

l = lc+ln (4)

These five equations define a banking sector that emits debt money, raises capital, generates earnings.

As a further requirement we impose a Basel 2 constraint on the system. It is defined with the capital ratio, where capital/assets is not to go below a threshold we call k.
C/A = k (5)

Equations (1-4,5) define a simple model of a bank under Basel rules.

To study it, we can at this stage run computer simulations without loosing generality. However we can introduce further simplifications to find the Andresen equations.

Specifically if we impose C'= 0 and C=0

(5) <=> (A-L+C)/A = k <=> A(1-k)=L

Or a hard linear relation between assets and liabilities, expressed by k.


(3)<=> lc = beta*(ia*A-il*L) + ec

Which simply states that all cash-flow from profits and paid-in equity are recycled as loans in this assumption.

Then (1) resolves to

A' = (beta*i-lambda)/k*A + ec/k (6)

With i = beta*(ia-il*(1-k))

(6) is the slightly generalized form of the Andresen differential equation. ec is in practice a function of time, the amount of cash raised by the banks over a period of time but if we simplify by assuming ec=0 (no cash raised) then (6) resolves to the Andresen case which has the solution:

A = A0*exp(gt) (7)


g = (beta*i-lambda)/k) (8)

Assuming all cash flow from operations is relent out, assets will grow exponentially with time under Basel 2. QED.

A comment on debt money growth rates

Look at equation (8)

If beta*i > lambda, then g > 0 and debt money grows exponentially.

If however bad debts spikes beyond the capacity of the business to generate free cash, then

lambda > beta*i <=> g<0

In this case, the debt-money supply needs to shrink exponentially to satisfy Basel. In other words, Basel compliance means a deflationary dynamic after a bad debt shock. A shrinking money supply can trigger a deflationary depression.

Equation (6) offers some relief but means new equity needs to be raised to satisfy A'>0.

Next we will look at central banks and how deficit spending can stabilize the money supply.


pcleddy said...
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Marcf said...
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Juha Lindfors said...

Liked it.

So who's running the simulations on these that go beyond mere derivative plotting into independent agents maximizing their incentives and recording all the fantastic ways the banking system could blow up?

Now that would be fun to watch.

Marcf said...

I have seen work by Steve Keen taking similar models and letting them rip in Mathematica. He doesn't attempt at solving algebraically, he just inputs stuff and let's it run its course.

Anonymous said...