# Default

See also [risk reduction](/vlzc8wOPTsu18oMWc64wpw?both), [late payment](/iIVkdpz_S76a_6FexHZWAw) and [glossary of terms and notation](/6kEVz305TOmyM2MUrY0oRQ).

In the [late payment](/iIVkdpz_S76a_6FexHZWAw?both) entry default is introduced as one of three possible generic outcomes (alongside timely payment and late payment) with respect to invoice settlement. As with late payment, two categories of transaction need to be considered; bilateral fiat or mutual credit payments between trading partners and fiat payments to or from the [trade credit club's](https://docs.google.com/document/d/1ofHJpalh83z0YAGBXrJHOOS-wOPYck24VMZwfqch7o4/edit#) 'system' account at the end of each settlement period to keep members' balances within their [limits](/AYLZ5n-WTG6eiRVSlh0qOw).

Unlike late payment, it is possible to unambiguously quantify the monetary impact of a default event by simply using the amount that was due (there may of course be knock-on effects, but it seems unlikely there is a simple and universally applicable way to estimate them and so considering only direct losses seems most appropriate). ['Expected loss'](https://en.wikipedia.org/wiki/Expected_loss) (consistent with the common defintion of 'risk' as 'probability of event mutliplied by impact of event') is calculated as the product of the [probability of default](https://en.wikipedia.org/wiki/Probability_of_default), [loss given default](https://en.wikipedia.org/wiki/Loss_given_default) ('the share of an asset that is lost if a borrower defaults') and the [exposure at default](https://en.wikipedia.org/wiki/Exposure_at_default) ('the gross exposure... upon default of an obligor'). Note that any calculation based on expected values assumes [ergodicity](/JL-hFK7BRjuxx6zbz3O6Qg), which may not hold. Each factor is discussed for each category of transaction below.

## Bilateral payments

### Probability of default

In the 'Bilateral payments' section of the [late payment](/iIVkdpz_S76a_6FexHZWAw) entry two example probability distributions of time intervals associated with settlement of an invoice are given, which are reproduced below.

In the first (which is meant to illustrate a possible set of outcomes when an invoice is settled in fiat outside of a trade credit club), default is considered to occur if the invoice has not been settled after a period three times the length of the stated terms (e.g. 90 days if 30 days were given to pay). In the second (illustrating outcomes within a trade credit club), invoices must be settled in mutual credit units (MCU) by the end of the club's settlement period (assumed for this example to be 30 days later, i.e. the invoice is issued on the first day of the current 30-day period), and the requirements for timely payment are more stringent (the section 'trade credit settled in mutual credit' in the late payment entry gives a list of suggestions for how the use of mutual credit might transform the distribution in Figure 1 into something resembling Figure 2).


*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.*


*Figure 2: A possible MCU payment interval probability distribution. The probablity of each generic outcome is the same as in Figure 1, but the club settlement period and use of mutual credit result in an altered timeline for late payments to be classified as defaults.*

In both figures the probability of default is conservatively assumed to be the same. However, the list of [benefits to members](/_P1DHaeIShuTZh6gpShPnA) suggests there may be reasons to believe that membership could help lessen the chances of a business failing and hence defaulting on all its obligations. In particular:
* Reduced [cost of finance](/PKmOIRXAQ6OEDE_mBYRB0Q).
* [Increased liquidity](/hZgux1eRScGqbyOi--DU1Q).
* [Intelligence sharing](/d_pInVkpRs2VWG3IaE_B-A).
* [Uncertainty management](/Bfp5PVh2Q7uifmgzevsQfQ).

Some of the factors listed in the 'trade credit settled in mutual credit' section of the [late payment](/iIVkdpz_S76a_6FexHZWAw) entry that could make settlement in MCU quicker than in fiat also suggest that default on a mutual credit obligation may be still less likely.

If the probability of a business $M_i$ defaulting on a fiat payment for transaction $l$ to another business $M_j$ is given by $P_{(i,j)_l}(DE)$, and the equivalent probability if the transaction were conducted in MCU is $P_{(i,j)_{l'}}'(DE)$, then if $P_{(i,j)_{l'}}'(DE)<P_{(i,j)_l}(DE)$ then this constitutes a risk reduction for $M_j$.

Finally, since the club settlement period is shorter than the threshold for considering a bilateral fiat payment to be in default, then this does at least mean that any parties that are affected by default will know about it sooner, reducing the uncertainty they face and [administrative costs](/brbI6dTeQ2ukhrUW7GoKWw) in the form of time spent chasing payment.

### Exposure given default

This is simply the amount of the expected payment, denoted $q_{(i,j)_l}$ if denominated in fiat or $q_{(i,j)_{l'}}'$ in MCU. Although MCU will presumably be pegged to fiat and $q_{(i,j)_l}$ and $q_{(i,j)_{l'}}'$ therefore represent equal purchasing power, if the lost payment is denominated in MCU it will have less of an impact on the ability to meet non-negotiable fiat obligations, e.g. taxes.

### Loss given default

Under bilateral fiat settlement terms, is it usual that the entire value of the invoice is lost, or is some often recovered one way or another? If so, typical fraction can be expressed as $\rho$.

In the context of a mutual credit club, the other members could [agree](/jTFZpns5QBixTAwG8Rl9-w) to cover a fraction $\rho'$ of the mutual credit loss, perhaps up to some limit. Are there any possible disadvantages to this?

### Expected loss

The expected loss on an invoice issued by $M_j$ to $M_i$ for trade $l'$ (with payment expected in MCU) is therefore given by:

$$ \epsilon_{(i,j)_{l'}}' = P_{(i,j)_{l'}}'(DE)\times (1-\rho') \times q_{(i,j)_{l'}}'.
$$

If there are $n_{i,j}'$ expected MCU payments from $M_i$ to $M_j$ and $M_j$ is due to receive payments from $N_{bi}^{'+}$ bilateral partners over one settlement period, then the total expected MCU loss for $M_j$ will be given by:

$$ E' = \sum_{i=1}^{N_{bi}^{'+}} \sum_{l'=1}^{n_{i,j}'} \epsilon_{(i,j)_{l'}}' \quad \quad (1)
$$

with an equivalent expression for expected loss $E$ on fiat payments. If $P_{(i,j)_{l'}}'(DE)<P_{(i,j)_l}(DE)$ and $\rho'>\rho$ (as discussed above), then $E'<E$. However, if the club's [systemic risk](/H3VYE7o3QwWqkw3dw4l3GQ?both) is not well-managed, then MCU default probabilities may become significantly correlated such that Equation 1 is no longer valid. On the other hand, it is to be hoped that the systemic risk associated with the club is preferable to that imposed by full dependence on the fiat economy.

## Central fiat settlement

The settlement process involves two sets of transactions; each member that has an MCU balance lower than their [negative clearing limit](/AYLZ5n-WTG6eiRVSlh0qOw) 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 (see the section 'system payment interval probabilities' in the [late payment](/iIVkdpz_S76a_6FexHZWAw) entry for a more detailed description of this process).

As per the example given in Table 1 of the late payment entry, many members may not be exposed to any fiat default risk (either because they themselves are due to make a payment or because they have finished the period with an MCU balance within their clearing limits). However, because the fiat settlement is a multilateral process the intermediating system account is much more likely to experience a default at the end of a settlement period than any individual member.

The nature of club membership means that participants will unavoidably be exposed to the consequences of default by other members they did not have direct dealings with. This should be made clear in the [club agreement](/jTFZpns5QBixTAwG8Rl9-w?both), which should also establish mechanisms for reducing the impacts of defaults and make it clear how any losses are to be apportioned.

### Probability of default

The list of benefits that are suggested to reduce the probability of default on all obligations also apply to fiat payments to the system account. If the probability of member $M_i$ defaulting on a payment to the system account is given by $P_{i,sys}(DE)$, the probability of the system account experiencing default from _ members is given by:

$$
$$

### Exposure given default

The system account's exposure is simply the expected payment amount.

### Loss given default

Unless some of the expected amount can be recovered (presumably through the same means that might be pursued in the bilateral case), or the club is federated with others that have agree to cover part of its losses, the system account will experience a 100% loss of the payment amount.

The potential for mitigation of the impacts on club members arises from mechanisms that allow the loss given default experienced by them to be shared. The [system account fiat buffer](/d2m7xva0QzOZ80y2uBvBVg) proposed as a way of decoupling the timing between the system account receiving and being able to make payments in the [late payment](/iIVkdpz_S76a_6FexHZWAw) entry (see the 'system payment interval probabilities' section) could be drawn down in the event that a late payment turns into a default. Because all parties would have already received payment on settlement day, when the default is finally 'declared' it would no longer have an immediate disruptive effect on any individual business.

There are several possibilities for apportioning losses:

* If the buffer is provisioned equally by all members (perhaps as a kind of membership deposit) and is sufficient to cover the loss, then if the club has $N$ members the loss given default is $(1-\rho)/N$ per member, where as before $\rho$ is the fraction of the expected payment that is recovered by the system account.
* If the buffer is provisioned equally by all members but is not sufficient to cover the loss, then if losses beyond the buffer are also shared equally, then loss given default per member is still $(1-\rho)/N$.
* If the buffer is sufficient to cover the loss but has to be replenished by members in proportion to how much credit they received from the defaulting party over the settlement period before the payment became overdue, then $M_j$'s loss given default by $M_i$ is:
$$\frac{\sum_{l'=1}^{n_{i,j}'} q_{(i,j)_{l'}}'}{\sum_{j=1}^{N_{bi}^{'-}} \sum_{l'=1}^{n_{i,j}'} q_{(i,j)_{l'}}'}
$$
where $l'$ is an index over $n_{i,j}'$ (the number of MCU payments from $M_i$ to $M_j$) and $N_{bi}^{'-}$ is the number of bilateral parties to whom $M_i$ has made MCU payments. Such an arrangement would provide an incentive for each member to consider whether to accept MCU from other members (and hence a self-regulating defence against incentives to [game balance limits](/fNhqlAK8TNqmDiXtEuLUWg)), but has disadvantages; it is complicated to calculate liability in the case of multiple defaults (opening up the potential for disputes), and since all credit is extended to each member by the entire club may be perceived as running counter to the spirit of mutuality. Furthermore, creditworthiness is meant to be assessed when a member joins the club as part of a multilateral process, and a mechanism should also be in place to periodically adjust limits in accordance with changes to business performance.
* If liability is proportional but the buffer is not sufficient to cover the loss,

### Expected loss

The loss to the buffer could either be replenished by all remaining members equally, or perhaps according to some other arrangement. In this way the loss given default to each individual business is reduced, at the cost of an increased frequency with which default on fiat settlement payments is (indirectly) encountered. In theory it is possible to maintain a buffer large enough to cover any default that has a non-negligible chance of occuring, but in reality a club may decide to allow for some probability of a 'non-insured' default (or combination of defaults) if the required fiat reserves are too large. If this were the case, then these losses would also have to be apportioned; see the [fiat buffer](/d2m7xva0QzOZ80y2uBvBVg) entry for discussion.