# Late payment

This is classified as a potential source of [risk reduction](/vlzc8wOPTsu18oMWc64wpw?both) for Trade Credit Club members.

As discussed in the risk reduction entry, late payment risk is analysed in terms of the probability distributions associated with the time intervals between invoices being issued and settled; to avoid potentially misleading assumptions, there is no attempt to quantify risk in fiat units.

There are two categories of payment that have to be considered; bilateral payments between trading partners, and (after the establishment of a trade credit club) payments to or from a dedicated club account that keeps members within their agreed mutual credit balances.

## Bilateral payments

In this section the replacement of fiat with mutual credit to settle bilateral [trade credit](/CncoUGlDQU-1ZRS5KpfLsg) invoices is considered. It is assumed that the numbers of transactions and their amounts are unchanged, and so the argument for late payment risk reduction is based on the plausibility of influencing behaviour such that the distributions of time intervals associated with invoice settlement for mutual credit become more favourable than the distributions associated with the erstwhile fiat payments.

### Trade credit settled in fiat

In a bilateral trade credit arrangement between two businesses $M_i$ and $M_j$, there will be some time interval $\Delta t$ between an invoice being issued and resolved, with the party issuing the invoice specifying some time period $\tau$ in which the counterparty should pay in fiat. Assuming that interval duration is a random variable, the likelihood of $M_i$ paying $M_j$ after an interval of $\Delta t$ is given by a set of probabilities $\{P_{i,j}(\Delta T=\Delta t)\}$, e.g. a 5% chance of an interval of one day being realised, a 3% chance for a two-day interval etc. This set defines a [probability mass function](https://en.wikipedia.org/wiki/Probability_mass_function) $p_{i,j}(\Delta t)$ which can in principle be estimated from historical payments data.

The distribution can be divided into three possible generic outcomes: 'timely payment' (the realised value of $\Delta t$ is less than or equal to $\tau$), '[default](/nO2IbqcMTbiNSY0AG0PCnA)' (defined here as $\Delta t>3\tau$) and 'late payment' ($\tau<\Delta t\leq3\tau$):

$$ P(TP)+P(LP)+P(DE)=1.
$$

The subset of the probability mass function associated with timely payment will be different to that associated with late payment; not only is it expected that most payments will be made on time, but the probabilities of specific outcomes (corresponding to different values of $\Delta t$ being realised) that constitute the function are distributed over a longer stretch of time (see [mixture distributions](https://en.wikipedia.org/wiki/Mixture_distribution)). Figure 1 shows a very simple probability mass function, with uniform distributions for each generic outcome and $\tau=$ 30 days.


*Figure 1: A possible fiat payment interval probability distribution. The probability of each generic outcome is given by the sum of all associated specific outcomes; $P(TP)=$ 0.93, $P(LP)=$ 0.06 and $P(DE)=$ 0.01.*

If there are $n_{i,j}$ payments due from $M_i$ to $M_j$, the probability distribution describing the possible sum values of payment intervals is the $n_{i,j}$-fold [convolution](https://en.wikipedia.org/wiki/Convolution_of_probability_distributions) of $p_{i,j}(\Delta t)$ with itself, which is written $p_{i,j}^{(n_{i,j})}(\Delta t)$. In some cases this may be analytically tractable; if $p_{i,j}(\Delta t)$ is uniformly distributed then $p_{i,j}^{(n_{i,j})}(\Delta t)$ [closely approximates a Gaussian distribution](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.489.7257&rep=rep1&type=pdf) for $n_{i,j}\geq$ 4, but for arbitrary distributions simple expressions may not be obtainable. To avoid simplifying assumptions, [Monte Carlo methods](https://en.wikipedia.org/wiki/Monte_Carlo_method) can be used to sample from $p_{i,j}(\Delta t)$ many times and observe the distribution of cumulative outcomes, yielding an estimate of $p_{i,j}^{(n_{i,j})}(\Delta t)$.

Figure 2 shows a hypothetical set of businesses which have some probability mass function associated with the time interval between $M_i$ settling an invoice issued by $M_j$. The arrows show the direction in which payment is expected to go, which is opposite to the direction in which credit has been extended, e.g. $M_2$ has supplied goods/services of value £250 to $M_1$ and the relevant payment interval probability distribution that $M_2$ is exposed to is therefore $p_{1,2}(\Delta t)$.


*Figure 2: A set of businesses with ongoing bilateral trade credit arrangements, with the amount due for each payment. The probability distributions of payment intervals are included for some transactions.*

If $M_j$ is due to receive payments from $N_{bi}^+$ bilateral partners, the sum of all the corresponding probability distributions $p_{N_{bi},j}^{(tot)}$ is again their convolution:

$$ p_{N_{bi}^+,j}^{(tot)}=p_{i,j}^{(n_{i,j})}*p_{i+1,j}^{(n_{i+1,j})}*...*p_{N_{bi}^+,j}^{(n_{N_{bi}^+,j})}.
$$

### Trade credit settled in mutual credit

If a [Trade Credit Club](https://docs.google.com/document/d/1jVtzd6_uXMJztHKcs3QBiDdtKBzrE8-0pVSLflQrHq8/edit#heading=h.nvotqw8e8hde) is set up, bilateral settlement can now occur in mutual credit units (MCU). For all payments for which fiat has been replaced by MCU, $p_{i,j}$ has been replaced with a new probability distribution $p_{i,j}'$ (the ' symbol denotes variables associated with MCU payments).

Recognising that the period $\tau$ for fiat payment is a matter of convention, it may be possible to establish a norm that MCU payments should be made within a shorter timeframe $\tau'$, resulting in an MCU payment interval probability distribution that is more skewed towards small values of $\Delta t$ than the fiat distribution it has replaced. This would translate directly into reduced risk for payment recipients, and could plausibly be achieved by several factors:
* Increased ease of making MCU payments compared to fiat ones.
* Lack of interest payments/fees on negative balances.
* Absence of artificial scarcity associated with fiat.
* [Reputation scoring](/5FokSVvRSjChYIPN2PuhcA).
* Regulations/penalties in accordance with the [club agreement](/jTFZpns5QBixTAwG8Rl9-w): although as a general principle it does not seem desirable for the club to oversee or intervene in bilateral trades, it should be specified that all MCU invoices issued must be paid before the day of club fiat settlement, otherwise members with balances just above $L_c^-$ may delay making MCU transfers in order to avoid becoming liable for a fiat payment to the club's ['system' account](/spMD7anJS0uiuVoF2OXE2w) (see next section). The funds thereby withheld would not be available to pay members with balances above $L_c^+$, which may or may not include the member the MCU payment is due to.
* Informal reputation.
* Understanding of mutual benefits of cooperation.
* Altruism.

The first five of these are specific to club membership, whilst the last three exist in any context but may be more effective within a club; an increase in shared information will make informal reputation more potent, mutual benefits grow with the club's success and altruism may also become more salient.

Although these mechanisms give good reason to believe bilateral payments in MCU will generally occur more rapidly than equivalent fiat payments, sufficient data will have to be gathered to establish whether or not this is true in practice. Figure 3 shows an example of what $p'(\Delta t)$ might look like; the probability of each generic outcome is the same as in Figure 1, but 'timely payment' now consists of settlement within five days and if it has not ocurred before the end of the club settlement period (assumed for this example to be 30 days later, i.e. the invoice is issued on the first day of the current period) then it is classified as a default.


*Figure 3: A possible MCU payment interval probability distribution. The probablity of each generic outcome is the same as in Figure 1, but the reduction of the timely payment window and the requirement to make MCU settlement before the end of the current club setttlement period result in a distribution skewed towards faster payment.*

If the convolutions $p_{N_{bi}^+,j}^{(tot)}$ and $p_{N_{bi}^{'+},j}^{'(tot)}$ have a common form which can be parameterised by their [moments](https://en.wikipedia.org/wiki/Moment_(mathematics)), then it becomes possible to make detailed comparisons of how payment intervals have changed. If this is not the case then it may still be possible to make quantitative statements by considering the relative frequencies of generic outcomes, e.g. the factor by which late payment probability has been changed.

## Central fiat settlement

In order to facilitate [circularity of trade](/CMlnVwUjR6-EPA4jHyqsqw) and keep members within their desired mutual credit [account balance limits](/AYLZ5n-WTG6eiRVSlh0qOw), periodic settlement in fiat is necessary. The settlement process involves two sets of transactions; each member that has an [MCU balance](/R1iAIsZoRRmJgIV4NlHQZw) lower than their negative 'clearing' limit has to make a fiat payment into the system account, and each member with a balance greater than the positive clearing limit receives fiat payment from it.

In both cases the member's balance is brought back within the relevant limit. Because MCU balances are the net result of [gross income](/tkEQAtNMTtWFnRDexj83QQ) and [gross expenditure](/4JynudQ7Srq5-3oHmGx8dg), as determined by transactions that should have been processed before the day of club-wide fiat settlement, no more than one additional transaction will be necessary for each member. The combination of the netting off process (embodied in close-of-period MCU balances) with credit limits mean that the amount of fiat that must change hands is reduced compared with the bilateral scenario, providing an immediate reduction in risk and [cashflow benefits](/gup34BGbQUSLhpaillAXEw).

These payments can be mediated by a designated system account operated by a service provider, but this status as a hub in a multilateral process means the system account is much more likely to experience late payment at the end of a settlement period than any individual member (unless that member expects to receive payments from a very large fraction - half is a reasonable approximation - of the other members of the club). Because members expecting to receive fiat are exposed to the resulting risk, there is a need to mitigate any potential cashflow issues with the system account itself. Without mitigation:

* Club membership will not be attractive to businesses who do not already do a significant amount of trade with existing members; for example, if they are only exposed to the risk of late payment by one member when they are outside the club, joining could substantially increase their exposure.
* If it is not clear how late payments are handled, confidence in equitable outcomes could suffer.

Several potential mechanisms to ensure payments to and from the system account are reliably made in a timely manner are considered below, such that the risk to fiat recipients associated with the settlement transaction is minimised.

Finally, the regularity of the process provides a degree of [uncertainty management](/Bfp5PVh2Q7uifmgzevsQfQ) for all members.

### System payment interval probabilities

When joining the network, each member is assigned two types of account balance limit; 'trading' limits $L_t^\pm$ and 'clearing' limits $L_c^\pm$. The trading limits specify how far into credit or debit a member can go over the course of the club's fiat 'settlement period' (a trade cannot be carried out in MCU if it would result in either party's trading balance limits being exceeded). The clearing limits determine the number to which each member's balance must be brought by the settlement process. The settlement period is the time between fiat payments to or from the club's system account, which acts as an intermediary to redistribute fiat as necessary.

At the end of each settlement period, members which have a balance below their negative clearing limit $L_c^-$ are expected to make a single fiat transaction into the system account to bring them back within the limit; they are not considered creditworthy for the 'excess' over any time greater than the settlement period and so ultimately have to pay for what they have received for these credits in fiat. Any member which has exceeded their positive clearing limit $L_c^+$ will expect to receive enough fiat to bring them back in line with their desired exposure; they are effectively cashing out their excess credits to bring their exposure to [systemic risk](/H3VYE7o3QwWqkw3dw4l3GQ) back into line with their desired level.

As shown in Figure 4, the system account acts as an intermediary for this settlement process; credit flows from accounts in surplus to accounts in deficit, and fiat flows the other way. This can be thought of as the account in deficit buying credits from the account in surplus, with the result that the excess credits are taken out of circulation (in the final stage the magnitude of balances has been reduced from 2100 MCU to 2000 MCU, equal to the sum of clearing limits).


*Figure 4: Before settlement, both accounts have exceeded their clearing limits; there is for the left- and right-hand accounts respectively an exposure to and liability for credit in excess of what is desired by each party. To settle, the system account mediates an exchange of fiat for MCU that brings both accounts back within their limits and takes excess credit out of circulation.*

Figure 5 shows the same set of bilateral transactions as Figure 2 with the addition of transactions to and from the system account. It is assumed that every member that has spent more credit than they have received is liable to make a fiat payment and each member that has received more than they spent is due fiat payment, although in practice trading activity and choice of balance limits may mean that this is not the case. Self-evidently, net spenders of credit ($M_1$, $M_4$ and $M_7$) have achieved complete de-risking with respect to receipt of fiat payments.

On the assumption that there have been no changes in behaviour associated with fiat settlement practices, payment interval probability distributions for payments to the system account $p_{i,sys}$ will be similar to the distributions associated with each member's erstwhile bilateral fiat settlement behaviour, e.g. $p_{7,sys}$ will be approximately the same as $p_{7,1}$ or $p_{7,4}$ (assuming $M_7$ treats their various counterparties similarly).


*Figure 5: In a trade credit club bilateral payments are now made in MCU with payment interval probability distribution of $p_{i,j}'$ and fiat settlement is mediated by a system account (some fiat payments are labelled). If settlement behaviour has not changed, members expecting fiat may experience long payment intervals. Club mechanisms can be designed to mitigate this.*

Because one payment at most is due to the system account from each member ($n_{i,sys}\leq$ 1), the first convolution when analysing the bilateral case is unnecessary, and so the total probability distribution of payment intervals the system account is exposed to is given by:

$$ p_{N,sys}^{(tot)}=p_{i,sys}*p_{i+1,sys}*...*p_{N,sys}
$$

where $N$ is the number of members in the club, including only distributions associated with members which are due to make a payment on this occasion.

If $m_{\Delta T=\Delta t}$ is the number of times a particular value of $\Delta t$ is realised from the set of payments to the system account, the probability of the account receiving at least one payment that has an interval of $\Delta t$ associated with it is given by:

$$ P_{N,sys}(m_{\Delta T=\Delta t}\geq1)=1-\prod_{i=1}^N (1-P_{i,sys}(\Delta T=\Delta t)).
$$

Assuming that the system account cannot make any payments until it has received all payments due to it, then:

$$ P_{sys,j}(\Delta T=\Delta t)=P_{N,sys}(m_{\Delta T=\Delta t}\geq1). \quad \quad (1).
$$

This means that every member expecting fiat may now experience long payment intervals due to the behaviour of members they would otherwise not have had dealings with (e.g. $M_6$ is dependent on $M_1$ for timely payment).

There are several potential mechanisms (in addition to those considered in the comparison between bilateral payments in fiat and MCU) for improving the probability distributions for payments to the system; automated transfers could be made a [condition of membership](/jTFZpns5QBixTAwG8Rl9-w?both), for example (in which case anything other than $\Delta t=$ 0 would classify as late payment). Reputational motives to make timely payments will be even more intense than they are for bilateral payments, since all members will be aware that long intervals could disrupt fiat payments to multiple parties.

As a safeguard, maintenance of a [system account fiat buffer](/d2m7xva0QzOZ80y2uBvBVg) can loosen the coupling expressed in Equation 1 between the intervals at which the system account receives and makes payments; if there are any late payments then immediate settlement in full or in part to members expecting fiat may still be possible, dependent on the amount in the buffer and the sum of payments that are overdue.

### System payment amounts

Because of the netting off process and the willingness of the club to indefinitely extend credit up to $L_c^-$ to them, members that are liable for fiat have to pay less to the system account than they do to their partners in Figure 2; fiat payment amount is their total MCU expenditure reduced by their total MCU income and their negative clearing limit:

$$ q_{i,sys}=\sum_{j=1}^{N_{bi}^{'-}} \sum_{l'=1}^{n_{i,j}'} q_{(i,j)_{l'}}'-\sum_{j=1}^{N_{bi}^{'+}} \sum_{k'=1}^{n_{j,i}'} q_{(j,i)_{k'}}'+L_{i_c}^- \\= Q_{i,N_{bi}^{'-}}'-Q_{N_{bi}^{'+},i}'+L_{i_c}^- \quad \quad (2)
$$

where $N_{bi}^{'-}$ is the number of bilateral parties to whom $M_i$ must make MCU payments (recall $N_{bi}^{'+}$ is the number of members that $M_i$ receives payments from), $l'$ is an index over $n_{i,j}'$ (the number of MCU payments from $M_i$ to $M_j$) and $k'$ is an index over $n_{j,i}'$ (the number of MCU payments from $M_j$ to $M_i$). Note that $L_{i_c}^-$ takes a negative value.

Likewise, because of the netting off process and their willingness to hold MCU up to $L_c^+$ for longer than the club settlement period, members that receive fiat are due less from the system account (their total MCU income minus total MCU expenditure and $L_c^+$) than they are from their bilateral partners in Figure 2:

$$ q_{sys,i}=\sum_{j=1}^{N_{bi}^{'+}} \sum_{k'=1}^{n_{j,i}'} q_{(j,i)_{k'}}'-\sum_{j=1}^{N_{bi}^{'-}} \sum_{l'=1}^{n_{i,j}'} q_{(i,j)_{l'}}'-L_{j_c}^+\\= Q_{N_{bi}^{'+},i}'-Q_{i,N_{bi}^{'-}}'-L_{j_c}^+. \quad \quad (3)
$$

Each members's trading activity is entirely under their control (assuming there is sufficient [club variety](/ocvl5vrOTluL59u82b3PnA) and [trade volume](/banCtXUuQ9C76sGhroS0dA) for them to find suitable buyers and sellers) and they will also have some say over the size of the fiat buffer, which (in the event of another party making late payment on an occasion they are due to receive fiat) will also infuence the amount they can expect to receive from the system account on settlement day.

However, the scope for large late payment amounts also depends upon the clearing and trading limits of all members of the club, and so each member can be indirectly affected by the balance limits of all other members; unless limits are collectively negotiated a clear [principal-agent problem](https://en.wikipedia.org/wiki/Principal%E2%80%93agent_problem) arises. By the same token, each member's limits are only under their control to some extent. Nevertheless, trading and clearing limits constitute quantitative risk control parameters that do not exist outside the context of the club. See [account balance limits](/AYLZ5n-WTG6eiRVSlh0qOw) and [parasitic strategies](/fNhqlAK8TNqmDiXtEuLUWg) for an in-depth discussion.

Using the bilateral transactions in Figure 5, Table 1 shows the balance of each account and the amount by which it exceeds arbitrarily chosen (but uniform across the set of members) symmetric positive and negative clearing limits. Balances below $L_c^-$ result in liability for a payment to the system account of an excess given by Equation 2, balances above $L_c^+$ in an expected receipt given by Equation 3.

| Member | $L_c^-$ (MCU) | $L_c^+$ (MCU) | Balance (MCU) | Excess (£) |
| -------- | -------- | -------- | -------- | -------- |
| $M_1$ | -240 | 240 | -970 | -730 |
| $M_2$ | -240 | 240 | 250 | 10 |
| $M_3$ | -240 | 240 | 220 | 0 |
| $M_4$ | -240 | 240 | -250 | -10 |
| $M_5$ | -240 | 240 | 60 | 0 |
| $M_6$ | -240 | 240 | 230 | 0 |
| $M_7$ | -240 | 240 | -480 | -240 |
| $M_8$ | -240 | 240 | 940 | 700 |

*Table 1: Balances and excesses beyond clearing limits for each member using the transactions in Figure 5. Because of the netting off process and non-zero clearing limits not all members need interact with the system account (their excess is zero).*

For this choice of clearing limits several members have an excess of zero and therefore will not need to transact with the system account. Fiat due to such members with a positive balance (e.g. $M_5$) is now zero; total fiat de-risking can be achieved by *any* member that balances buying and selling in MCU within their clearing limits (not just net spenders of credit, as was the case before netting off and clearing limits were taken into account). Those who have to make fiat payments are liable for less than they were under bilateral fiat relations in Figure 2, and those who are due fiat expect less. This result is completely independent of all fiat and MCU payment interval probability distributions.

Figure 6 depicts the residual system transactions and their amounts (with labels removed from bilateral payments for clarity). Note also that the amount received by the system account is £270 greater than it has to pay out; this adds to the buffer suggested as a means of decoupling the timing of payments to and from the system account from each other, although the existence of such a surplus is clearly contingent on activity.


*Figure 6: The residual system transactions necessary to restore the balances of all parties to within their clearing limits. Only two members ($M_2$ and $M_8$) are now exposed to any fiat payment risk at all; every other member has achieved total de-risking. Furthermore, the amount due to each of the remaining fiat recipients is less than the sum of their erstwhile bilateral fiat payments received in Figure 2.*

Overall, there seem to be plausible grounds to believe that replacing bilateral fiat payments with MCU and settling the fiat payments necessary to maintain MCU balances within desired limits via a dedicated account (with collective action mechanisms to influence payment behaviour) will bring significant late payment risk benefits for each member.

Realising this potential is partly down to the service provider - coordination will be needed, and maladministration could in fact be a source of increased fiat payment risk. However, if managed well then because the system's record will be available (in contrast to bilateral fiat payment histories before the club was set up), reduced [uncertainty](/Bfp5PVh2Q7uifmgzevsQfQ) may make [new trading activity](/Xcuzv6TaSRG5tpvC1NCwMg) within the club more likely, leading to additional benefits.

## Risk over multiple settlement periods

The example above shows that bilateral payment intervals for MCU should be preferable to fiat intervals for all members, and any member can achieve total fiat de-risking with respect to trade in the network by ensuring their MCU balance is within their clearing limits at the end of the settlement period. Furthermore, even members who do not do this can still expect some risk reduction from reduced fiat transaction amounts and participating in a framework that facilitates collective action.

Because all members must approximately balance buying and selling over time, members who receive fiat from the system account in one period may not in the next, and those who don't at some point will. This means that even though some members in the example (those expecting fiat payments from the system account) were exposed to greater risk than the other members they will not be in all other periods, and so over an extended time horizon their risk exposure should tend towards the club average (i.e. lower than that experienced by comparable businesses outside the club).

## Necessary data

To determine whether a reduction in late payment risk is a benefit for each potential club member in reality, the following data needs to be acquired (ideally on multiple ocassions spanning many settlement periods):

* How frequently each member received late fiat payments in bilateral trade relations, the lengths of the associated delays, payment amounts and what the agreed settlement periods were (broken down by trading partner) - survey.
* How frequently each member receives late MCU payments in bilateral trade relations, the lengths of the associated delays, payment amounts and (if applicable) what the agreed settlement periods were (broken down by trading partner) - survey.
* How frequently each member receives late fiat payments from the system account, the lengths of the associated delays and payment amounts - club ledger.

To determine if perceptions reflect any changes and what mechanisms are considered responsible:
* Whether each member perceives each aspect (frequency, delay length and amount) of late payments risk to have decreased or increased - survey.
* What factors each member thinks have been responsible for any changes - survey.
* Whether each member perceives overall late payment risk to have decreased or increased - survey.
* The factors each member thinks have been responsible for any changes - survey.

Supplementary data:
* How frequently each member made late fiat payments in bilateral trade relations, the lengths of the associated delays, payment amounts and what the agreed settlement periods were (broken down by trading partner) - survey.
* How frequently each member makes late MCU payments in bilateral trade relations, the lengths of the associated delays, payment amounts and what the agreed settlement periods were (broken down by trading partner) - survey.
* How frequently each member makes late fiat payments to the system account, the lengths of the associated delays and payment amounts - club ledger.
* Details of account limits, the club fiat settlement period and the size of the system fiat buffer and deployment policy (if applicable) - club ledger & agreement.
* Whether each member perceives each aspect of late payments made to have decreased or increased - survey.
* What factors each member thinks have been responsible for any changes - survey.
* Whether each member thinks there has been a reduction or increase in each aspect of late payments made by other parties - survey.
* What factors each member thinks have been responsible for any changes - survey.

Although in the example risk was compared under bilateral and club conditions for identical trading activity, an increase in trade is one of the aims of setting up a club. This means it may not be possible to control for activity when gathering data. However, normalising late payment rates, delays and amounts to value exchanged (or number of trading relations) should provide an appropriate correction so that details of late payments received/made per unit value exchanged (or per trading relation) before and after the club was established can be meaningfully compared.