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# System prepended metadata

title: Account balance circularity

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# Account balance circularity

This is categorised as an [account metric](/xnUSdbbaQS-wn0KmIYRh-Q).

In a mutual credit system, participants' [account balances](/R1iAIsZoRRmJgIV4NlHQZw) must be kept with [limits](/AYLZ5n-WTG6eiRVSlh0qOw). Maintaining a balance close to either limit results in reduced scope for buying or selling, whilst holding a large positive balance corresponds to exposure to [systemic risk](/H3VYE7o3QwWqkw3dw4l3GQ?both). It will therefore generally be in the interests of each member to maintain a balance close to zero much of the time.

This is also desirable from the perspective of the other network participants; in the short term a large negative balance indicates that the member has received more value than it has supplied to them and a large positive balance means less liquidity available to facilitate trade, whilst in the longer term it prevents the emergence of harmful [balance inequalities](/LVypX41ZTJCDQrcnbZ3aKw) and hoarding of credit.

It is therefore useful to have a metric that reflects the tendency of an account balance to return to or hover around zero. Since all credits issued by a participant must ultimately be redeemed with them, there is also a sense in which such a metric reflects the 'circularity' of that account's activity. If the notion of 'circular money for a circular economy' has merit, then this could also provide a point of departure for analysing the effect of mutual credit networks on the [material circular economy](/cDUb3gPOTJmm7RH1HNCDsA). 

This note introduces the 'zero crossing frequency' as an appropriate measure of account health in the three respects described above; individual utility, desirability from the perspective of other participants and propensity for circularity.

## Zero crossing frequency

The tendency of a network participant $M_i$'s account balance to remain close to zero can be easily quantified by the number of times $n_{i_0}$ it passes through or attains zero relative to number of payments $n_i$ it has made to other club members over some [analysis period](/conspoQ5TNmH5pfZqp-LfQ). This defines the account's 'zero crossing frequency', used interchangably with its 'circularity':

$$f_i = \frac{n_{i_0}}{n_i}.
$$

Some example transaction sequences are given in the tables below.

In the first case network members $M_1$ and $M_2$ with initial balances of $b_1=b_2=0$ make an alternating series of payments of equal value, starting with $M_1$ spending $q$ credits with $M_2$ (note that a new member's starting balance of zero should not be counted as an instance of $n_{i_0}$):

| $n_1$ | $n_2$ | $b_1$ | $b_2$ | $n_{1_0}$ | $n_{2_0}$ | $f_1$ | $f_2$ |
| - | - | - | - | - | - | - | - |
| 1 | 0 | $-q$ | $q$ | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 1 | 1 | 1 | 1 |
| 2 | 1 | $-q$ | $q$ | 1 | 1 | 0.50 | 1 |
| 2 | 2 | 0 | 0 | 2 | 2 | 1 | 1 |
| 3 | 2 | $-q$ | $q$ | 2 | 2 | 0.67 | 1 |
| 3 | 3 | 0 | 0 | 3 | 3 | 1 | 1 |
| 4 | 3 | $-q$ | $q$ | 3 | 3 | 0.75 | 1 |
| 4 | 4 | 0 | 0 | 4 | 4 | 1 | 1 |


When both parties have made an equal number of payments then $f_1=f_2=$ 1, reflecting the perfect symmetry of exchange in both number and amount of payments. Where $n_1 \neq n_2$, as the number of payments increases the value of $f$ for the party that first spent credit tends towards one. This can be interpreted as an instantaneous asymmetry in balances being given progressively less 'weight' in the calculation of $f$ as a history of equal exchange is built up over the analysis period. 

In the second example, $M_1$ spends $n_1 \times q$ credits over the course of $n_1$ successive payments, followed by a single payment from $M_2$ that restores both balances to zero:

| $n_1$ | $n_2$ | $b_1$ | $b_2$ | $n_{1_0}$ | $n_{2_0}$ | $f_1$ | $f_2$ |
| - | - | - | - | - | - | - | - |
| 1 | 0 | $-q$ | $q$ | 0 | 0 | 0 | 0 |
| 2 | 0 | $-2q$ | $2q$ | 0 | 0 | 0 | 0 |
| 3 | 0 | $-3q$ | $3q$ | 0 | 0 | 0 | 0 |
| 4 | 0 | $-4q$ | $4q$ | 0 | 0 | 0 | 0 |
| 5 | 0 | $-5q$ | $5q$ | 0 | 0 | 0 | 0 |
| 5 | 1 | 0 | 0 | 1 | 1 | 0.2 | 1 |

This sequence results in a low value of $f_1$, reflecting an uninterrupted series of payments that take $M_1$'s balance ever further from zero. Again this can be interpreted as weighting by history; it implies that a situation where one party is reliant on a single large repayment to balance continuous expenditure is less desirable than one with a greater degree of exchange symmetry, even if the final balances are the same.

These results illustrate that a high zero crossing frequency reflects desirable activity for each account; the first example is the 'ideal' scenario and invariably results in a high and increasing values of $f$ as the pattern of symmetrical exchange is sustained, whilst in the second an essentially opposite situation is constructed and shown to result in low and decreasing values of $f$ as imbalances persist. It seems plausible to assume that cases corresponding to an intuitive sense of intermediate 'desirability' will result in values of $f$ between these two extremes, but unless this can be proved rigorously counter-examples should be sought.

These findings generalise in a straightforward manner to more complex sets of trading relationships among multiple parties; any member that does not see its balance return to zero on a regular basis (relative to the number of transactions it engages in) will likely either soon reach one of its balance limits or is overly reliant on payments from a single partner (or possibly a set of partners that it transacts with in sync) to counteract its other relationships.