Central Counterparty Risk
Matthias Arnsdorf
Central Counterparty Risk
Introduction
Variation and Initial Margin in the ISDA Credit Support Annex
Variation and Initial Margin Required by Central Counterparty Clearing Houses
Margin Requirements for OvertheCounter Derivatives: A Supervisory Perspective
The Emergence and Concepts of the SIMM Methodology
The ISDA Standard Initial Margin Model Backtesting Framework
The Impact of Margin on Regulatory Capital
XVA for Margined Trading Positions
Modelling Forward Initial Margin Requirements for Bilateral Trading
Forward Valuation of Initial Margin in Exposure and Funding Calculations
Margin Value Adjustment for CCPs with QSimulated Initial Margin
Bilateral Exposure in the Presence of Margin
Central Counterparty Risk
Robust Computation of XVA Metrics for Central Counterparty Clearing Houses
Efficient Initial Margin Optimisation
Procyclicality in SensitivityBased Margin Requirements
Systemic Risks in Central Counterparty Clearing House Networks
12.1 INTRODUCTION
Central counterparties (CCPs) are a key part of the financial system. They have increased in significance since the 2007–9 financial crisis and are viewed as a key mitigant of credit risk and contagion while also providing increased transparency to the derivatives market.
As discussed in Chapter 2, CCPs are designed to reduce counterparty risk by holding high levels of collateral and by mutualising losses among clearing members. However, in extreme, stressed markets, the CCP funds may be insufficient to cover the portfolio losses of a defaulting clearing member. In these cases clearing members are exposed to concentrated tail risk and can incur losses on their default fund contributions and trade exposures. This does not necessarily require the default of the CCP itself.
This chapter is an extension of the framework introduced in Arnsdorf (2012, 2014), where we explore methodologies for quantifying the risk that a bank has when facing a CCP. Here we shall summarise the calculation of stress and expected exposures and look at how these measures can be applied in practice. In particular, we consider applications to stress tests such as the regulatory prescribed Comprehensive Capital Analysis and Review (CCAR) calculations.
The main risk transmission mechanism of the CCP is loss mutualisation. Each member insures the excess portfolio losses in the case of a member default. This is different in principle from a standard bilateral exposure. In particular, the loss amount that will be incurred by a clearing member is not related to the exposure on the member’s own portfolio. Instead, it is based on a mutualised loss allocation.
The difficulty in estimating the potential loss facing a CCP is that it depends on the portfolio characteristics of all other clearing members. These are not public. However, a typical CCP will report aggregate data on collateral amounts as well as details regarding the CCP’s risk methodology. We shall see that this is sufficient for obtaining reasonably accurate exposure measures. This implicitly leverages the fact that collateral levels are risk based and determined by internal CCP models.
Remarkably, we find that the expected exposure that a bank has to a CCP can be expressed simply as a function of the bank’s own posted margin. This is fundamentally due to the CCP’s loss allocation mechanism. In particular, if a bank has a large risk position with a CCP and is thus required to post a high level of initial margin, then the bank will also be allocated a larger fraction of any CCP losses.
We find that the expected losses facing a CCP are small in typical markets. However, if markets are stressed and, in particular, if we take into account wrongway risk and contagion, then losses can be become significant. The methodologies presented in this chapter can provide the tools to analyse and monitor CCP risk. As more transactions migrate to CCPs, this will become an increasingly significant area of credit risk.
12.2 CENTRAL COUNTERPARTY STRUCTURE AND RISK
We begin our discussion by summarising how a clearing house is designed and how this relates to the risks that a clearing member has when dealing with a CCP (Arnsdorf 2014). A more detailed discussion can be found in Chapter 2.
12.2.1 Risk waterfall
A typical CCP has a multilayer capital structure to protect itself and its members from losses due to member defaults. In general, we distinguish between the following types of collateral.

 Variation margin: variation margin is charged or credited daily to clearing member accounts to cover any portfolio marktomarket (MtM) changes.

 Initial margin: initial margin is posted by clearing members to the CCP. This is to cover any losses incurred in the unwinding of a defaulting member’s portfolio. Typically, the margin is set to cover all losses up to a predefined confidence level in normal market conditions.

 CCP equity: a typical CCP will have an equity buffer provided by shareholders. The position of the equity buffer in the capital structure can vary between CCPs.

 Default fund (funded): every member contributes to the clearing house default fund. This acts as a form of mutualised insurance for uncollateralised losses.

 Default fund (unfunded): in addition to the default fund contributions that have been posted to the CCP, each clearing member is usually committed to providing further funds if necessary. The maximum amount of additional funds that can be called upon depends on the CCP. In some cases the liability is uncapped.
Losses arising from a member default will first be covered by the defaulting member’s initial margin and default fund contribution. Excess losses will then be charged against the CCP’s equity and, ultimately, the mutualised default fund. If all funds are used up and there are still outstanding losses, then the CCP could find itself in default.
There is an active debate on how to deal with losses that exceed the CCP’s resources, ie, to determine what happens “at the end of the waterfall”. One main proposal is that losses are allocated to surviving clearing members based on their outstanding trade exposure. This is referred to as “variation margin haircutting”. It is beyond the scope of this chapter; the interested reader is referred to Financial Stability Board (2017) for an overview. Below we look at the main sources of CCP risk in more detail.
12.2.2 Default fund risk
The main risk to a CCP is that a clearing member defaults. In this case the CCP needs to auction the portfolio of the defaulting member so that it can rematch all positions and continue operation. The initial margin and default fund contribution of the defaulted member will be used to cover any losses that are realised during the auction.
Typically, we expect that the collateral posted by a clearing member should be sufficient to cover any losses in default. However, in cases of extreme market stress or if the collateral has not been sized adequately there may be excess losses. Such excess losses will be covered by the CCP equity and the default fund contributions of the surviving members.
The main point is that default fund losses are mutualised, and the size of the loss that each individual surviving clearing member will bear is proportional to the size of its default fund contribution. In particular, the loss amount is not related to the size of the outstanding exposure. Indeed, a surviving clearing member could have a negative outstanding position with the CCP but still face a loss in the case of a clearing member default. This is fundamentally different to counterparty risk in a bilateral, overthecounter (OTC), transaction.
Following a default, all surviving clearing members are required to recapitalise the default fund under CCP rules. We refer to the time between the default and the recapitalisation of the default fund as the “recapitalisation” or “allocation” period. This can depend on the CCP, but is typically around 30 days. If there are multiple member defaults before the default fund is recapitalised, then there will be fewer surviving members to cover any excess losses. This introduces a nonlinearity or convexity in the risk a bank has to clearing member defaults, which means that the default fund risk will increase as a function of default correlation between clearing members.
12.2.3 CCP default risk
If losses due to member defaults exceed all funded and unfunded default fund contributions, then the CCP can be in default itself. Given the high levels of collateralisation that a CCP can draw upon, this is an extreme tail event.
A CCP can potentially also default due to events that are not related to clearing member defaults. In the literature these are referred to as nondefault losses. These could be due to operational risk issues, or market risk related to external CCP investments, for example. However, given the structure of a CCP, this is unlikely. Hence, in the following we assume that all losses are due to clearing member defaults.
The details of how a CCP will be resolved will depend on the CCP (for a discussion see, for example, Financial Stability Board 2017). The relevant point here is that once all CCP resources are exhausted losses may no longer be mutualised and the losses that surviving members will incur can be determined by their exposure to the CCP. This is the case, for example, if variation margin haircutting is used in the resolution of the CCP.
Nevertheless, here we shall assume that all losses are mutualised and are allocated to surviving members in relation to their default fund contributions. This is because default fund losses are by far the most likely cause of losses to a clearing member, and the default of the CCP itself is expected to be a low probability event.
Note, however, that we are not assuming that losses are capped or that a CCP default cannot occur. In particular, all losses that are realised due to the default of the clearing members will need to be covered by the surviving members. Our assumption is only with regard to how these losses are allocated. We can view the allocation on the basis of the default contributions as an approximation, even if in reality losses are allocated on the basis of exposure after the CCP default. This is because the default fund size of any given member will be driven by the size of their portfolio with the CCP.
12.3 MODELS OF EXPOSURE
In this section we recap the methodologies for quantifying CCP risk introduced in Arnsdorf (2014) and show how we can estimate the size of a loss that can occur due to a clearing member default. This is the credit exposure that a CCP has facing each of its members prior to any allocation to the surviving members. We shall look at several different approaches to calculating this exposure.
To make this problem more concrete, let us assume that we are dealing with a CCP with N + 1 clearing members denoted by CM_{k}, where the index k ranges from 0 to N. In the following risk calculations we shall be taking the perspective of the clearing member CM_{0}. We now assume that member CM_{k} has defaulted at time τ_{k} with portfolio value V_{k}(τ_{k}). Post default, the portfolio will be unwound over the liquidation period ∆_{l}.^{1}^{1}This is typically of the order of two to five days, given that cleared products tend to be very liquid. We assume that up to the default time the portfolio was perfectly collateralised with variation margin. The loss on the portfolio in the liquidation period is given by
∆V_{k}(τ_{k}) ≡ V_{k}(τ_{k} + ∆_{l}) − V_{k}(τ_{k})
Here and in the following we use the convention that a positive value of ∆V_{k} indicates a loss to the portfolio.
The losses are collateralised by the defaulting member’s initial margin M_{k}(τ_{k}) and default fund contribution^{2}^{2}This is only the funded part of the default fund contribution. D_{k}(τ_{k}) at the time of default. The excess, uncollateralised, loss is given by
U_{k}(τ_{k}) ≡ (∆V_{k}(τ_{k}) − M_{k}(τ_{k}) − D_{k}(τ_{k}))^{+}
and will be covered in the first instance by any available CCP equity cushion. All remaining excess losses will be allocated between the surviving clearing members. In the following we shall make the conservative assumption that the CCP equity cushion is negligible.
The loss U_{k} is a random variable and is scenario dependent. To calculate an exposure we need to specify a risk measure and apply this to the distribution of U_{k}. We shall explore two such typical measures in this chapter.

Expected loss, Ū_{k}(t): at any time t, this is defined as the expected uncollateralised loss conditional on default of CM_{k}
Ū_{k}(t) ≡ E[U_{k}  t = τ_{k}] (12.1) 
Stressed loss, Û_{k}: this is the uncollateralised loss on default of clearing member CM_{k} assuming default happens today in conjunction with an extreme market shock also applied to today’s market. We shall discuss specific examples in the following sections.
The difficulty in estimating U_{k} is that we as a clearing member do not have any detailed information about the portfolios of other clearing members. Typically, however, aggregate level data about the CCP resources is public. In particular, we note that CCPs are expected to provide a minimum set of quantitative data as part of the Committee on Payment and Settlement Systems and International Organization of Securities Commissions’ “Principles for Financial Market Infrastructures”. Details on the disclosure requirements and frequency are provided in Committee on Payment and Settlement Systems–Board of the International Organization of Securities Commissions (2015). For our purposes the following information will be useful.

 Number of clearing members: the number of clearing members facing the CCP.

 Default fund size: the total value of funded and unfunded commitments to the default fund.

 Cover level: a statement on whether the CCP is subject to a minimum “Cover 1” or “Cover 2” requirement.

 Liquidation period: the number of business days within which the CCP assumes it will close out a default.

 Largest single stress loss: the estimated largest aggregate stress loss (in excess of initial margin) that would be caused by the default of any single participant and its affiliates in extreme but plausible market conditions.

 Total initial margin: the total initial margin (IM) held by the CCP.

 IM confidence level: the singletailed confidence level targeted in the IM calculation.
In the following we shall show how we can use the information on the default fund and IM sizes to estimate the risk facing a CCP. This is possible because the collateral levels are determined by the CCP on the basis of the risk inherent in each member’s portfolio.
12.3.1 Defaultfundbased loss estimates
The most direct approach to estimate U_{k} is to make use of the CCP requirements for calculating default fund levels. In particular, for a CCP to be classed as “qualifying”, the calculation of the default fund level needs to meet (or exceed) a socalled “Cover 1” or “Cover 2” requirement.
In general, a “Cover n” requirement means that the default fund is large enough to cover the excess default losses of the n largest clearing members in a period of significant stress. Here “largest” means in terms of risk, ie, the members with the largest stress losses. Typically, CCPs will estimate the “Cover n” requirement based on the results of historical and theoretical stress scenarios applied to the current member’s portfolios, taking into account current levels of collateral. We can use it to provide a conservative estimate for the stress loss conditional on a clearing member default.
To illustrate this let us assume the CCP default fund is based on a “Cover 2” calculation, which is the most typical case. We denote the two largest clearing members by CM_{a} and CM_{b} and we set the stress losses for these members to be determined by the “Cover 2” requirement. This means that
Û_{a} + Û_{b} = D_{tot} − D_{a} − D_{b}
where D_{tot} is the total default fund size. Note that we need to subtract the (current) default fund contributions of D_{a} and D_{b}, as they are already used as collateral in the calculation of Û_{a} and Û_{b}.
The average of the largest losses is given by
$${\widehat{U}}_{ab}=\frac{1}{2}({D}_{tot}{D}_{a}{D}_{b})<\frac{1}{2}{D}_{tot}\overline{D}<\frac{1}{2}{D}_{tot}$$ 
where $\overline{D}$ is the average default fund level, which will be less than the default fund contributions, D_{a} or D_{b}, of the largest members.
In a typical CCP, the portfolios of a few members will dominate, and hence ${\widehat{U}}_{ab}$ will be a conservative measure of stress loss for all other clearing members. Hence, to generalise, we propose the following stress exposure measure for a CCP with a “Cover n” default fund
$${\widehat{U}}_{k}=\frac{1}{n}{D}_{tot}\overline{D}$$ 
This will be an upper bound for all members except the largest. This loss measure is easy to compute but it suffers from the fact that it does not distinguish between clearing members and is thus likely to overestimate losses for smaller clearing members. Alternatively, if published by the CCP, we can use the largest single stress loss directly to define Û_{k}. This, however, would be even more conservative than the approach above.
Our measure of Û_{k} is dependent on the quality and severity of the CCP default fund calculation. This is independently monitored by regulators as well as the financial institutions that are members of the CCP.
In the next section, we look at a more granular alternative.
12.3.2 Marginbased loss estimates
To obtain a more granular loss estimate we can make use of the fact that the initial margin charged by the CCP is risk based. In particular, we can show that under fairly general assumptions the default losses for a clearing member will be proportional to the posted IM. This means we can postulate
$${\widehat{U}}_{k}={M}_{k}\widehat{\beta}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\widehat{U}}_{k}={M}_{k}\overline{\beta}$$  (12.2) 
where $\widehat{\beta},\text{\hspace{0.17em}}\overline{\beta}$ are clearinghousespecific scaling factors that need to be determined.
This loss estimate relies on knowledge of individual clearing members’ IM contributions. In general, these are not available but can potentially be estimated based on the relative size of the members. We shall show later that for important applications, such as determining the total expected loss to a CCP, we do not need the memberspecific IM information.
A direct approach for determining $\widehat{\beta}$ or $\overline{\beta}$ is to use the bank’s own internal exposure models. For example, let us assume we are interested in the expected loss. We can first calculate the expected losses ${\overline{U}}_{0}$ that our portfolio with the CCP would incur using our own internal models. Given that we know our own IM posting, we can then calculate
$$\overline{\beta}=\frac{{\overline{U}}_{0}}{{M}_{0}}$$ 
A similar approach can be taken to determine $\widehat{\beta}$. This is explored further in Chapter 13.
Performing a full internal model calculation may be difficult and resource intensive. In the next section we shall show how $\overline{\beta}$ can be estimated with sufficient accuracy using a very simple model. This will justify in particular that the expected loss is proportional to the margin level.
12.3.2.1 Parameterised loss model
Here we present a simple model, first introduced in Arnsdorf (2012), that can be used to estimate potential losses U_{k} based on the member CM_{k}’s posted IM. This provides a straightforward calculation of loss distributions that can be used to understand the qualitative and quantitative behaviour of clearing house risk. Given the uncertainty in inputs that any calculation of CCP risk involves, the simple method illustrated here can provide a sufficient level of accuracy and can avoid the need to employ more complex inhouse models.
We start by defining a parametric form for the distribution of the portfolio losses ∆V_{k}. In particular, we propose to model the tail of the loss distribution as a Pareto distribution, which is both analytically tractable and able to capture the heavytailed nature of financial asset distributions.
To be more precise, we assume that for any future time t and for x ≥ M_{k}(t) the probability of losses exceeding x is given by
$$P[\Delta {V}_{k}(t)>x]={p}_{M}{\left(\frac{{M}_{k}(t)}{x}\right)}^{\alpha}$$ 
where α is the Pareto index that determines how fast the tail of the distribution decays. We assume here that this parameter is universal for all portfolios of the clearing members at a given CCP. This is because the nature of the distribution will largely be determined by the product set traded at the CCP and the associated underlying risk factors.
The distribution above is not conditional on the default of any particular clearing member. The unconditional probability of losses exceeding the initial margin is given by
$${p}_{M}=P[\Delta {V}_{k}(t)>{M}_{k}(t)]$$ 
This probability is typically set as a target confidence level by the CCP in calculating margin requirements. More specifically, the IM for a given member will be determined as a percentile in the historical loss distribution of the member’s portfolio. Hence, p_{M} is known for most CCPs. It is part of the minimum disclosure requirements listed in Section 12.3. A typical value of p_{M} is 1%. In other words, at any given point in time, there is a 1% chance that the portfolio value of a given clearing member will exceed the IM over the liquidation period.
We now use this distribution to calculate an expected loss measure. Further examples of how this can be applied to calculate a stress loss are given in Arnsdorf (2014).
12.3.2.2 Expected loss
We want to calculate an expected excess loss, Ū_{k}, conditional on default. For this we need to take into account the fact that the default is likely to happen in a stressed environment, ie, we need to take into account wrongway risk as well as contagion. We denote the collateral levels at the time of default by M_{k}^{∗} ≡ M_{k}(τ_{k}) and D^{∗}_{k} ≡ D_{k}(τ_{k}), respectively, where τ_{k} denotes the default time of member CM_{k}. Once a clearing member has defaulted, market volatilities may increase. This means that the probability of exceeding the IM over the liquidation period may also be larger than pM. Let us denote the margin breach probability conditional on default by
$${p}^{+}\equiv P[\Delta {V}_{k}(t)>{M}_{k}^{*})t={\tau}_{k}]$$ 
Here we have assumed for simplicity that the breach probability does not depend on the identity of the defaulting member. Thus, we calculate the expected excess loss as
$$\begin{array}{lll}{\overline{U}}_{k}(t)\hfill & \equiv \hfill & \text{E[(}\Delta {V}_{k}(t){M}_{k}^{*}{D}_{k}^{*}{)}^{+}t={\tau}_{k}]\hfill \\ \hfill & =\hfill & {p}^{+}{\displaystyle {\int}_{{M}_{k}^{*}+{D}_{k}^{*}}^{\infty}(x{M}_{k}^{*}{D}_{k}^{*})\alpha {M}_{k}^{*\alpha}{x}^{(\alpha +1)}\text{\hspace{0.17em}}\text{dx}}\hfill \\ \hfill & =\hfill & \frac{{p}^{+}}{\alpha 1}{\left(\frac{{M}_{k}^{*}}{{M}_{k}^{*}+{D}_{k}^{*}}\right)}^{\alpha}({M}_{k}^{*}+{D}_{k}^{*})\hfill \end{array}$$ 
which is well defined for α > 1. Assuming a fixed ratio r of the initial margin to the default fund contribution that is independent of k, ie
$$r\equiv \frac{{D}_{tot}}{{M}_{tot}}\approx \frac{{D}_{k}}{{M}_{k}}$$ 
we have
$${\overline{U}}_{k}(t)={M}_{k}^{*}\frac{{p}^{+}}{\alpha 1}{(1+r)}^{1\alpha}$$ 
and we find that the expected loss is proportional to the initial margin level. We also note that the time dependency in the expected loss is only due to the changes in the margin over time. We shall discuss this further in subsequent sections.
To use this model for risk calculations we need to determine the parameters ${M}_{K}^{*},\text{\hspace{0.17em}}{p}^{+}$ and α. We look at how we can do this in the following sections.
12.3.2.3 The Pareto index
The Pareto index determines the tail behaviour of the loss distribution and is asset class dependent. We assume that the index does not depend strongly on the detailed portfolio composition. We can calibrate the index by fitting to the tails of the historical return distributions for the relevant markets as shown in Figure 12.1. The Pareto parameter typically ranges between 3 and 4, which implies significant heavytailed behaviour for the financial time series we have analysed. For comparison, a Gaussian tail corresponds roughly to a Pareto index of 7 in the examples below.
12.3.2.4 Wrongway risk and collateral levels at default
Margin and default fund levels can vary over time due to changing market conditions and changes in the CCP portfolios.
For the purposes of this chapter we shall ignore the effect of a changing portfolio. Effectively, we are making a socalled “constant risk” assumption. This means that we assume that the portfolio is continuously rebalanced so that the risk profile remains stable over time. This assumption is appropriate for many applications and also allows us to focus on the main results regarding clearing house risk. If required, more specific assumptions can be incorporated, given a model of the future projected margin levels. This is explored further in Chapter 13.
Here we assume that the main reason that margin level may be different in the future is that the default of a clearing member is likely to occur in a stressed market environment. This is a wrongway risk effect. In particular, we assume that at the time of the member CM_{k}, the initial margin is given by
$${M}_{k}^{*}\equiv w{M}_{k}$$ 
where w is the “wrongway factor”, which gives the ratio of the stressed margin over today’s margin level.
To estimate w we again make use of the fact that for a typical CCP the initial margin is given by a quantile in the distribution of the portfolio returns. This means that margin levels are approximately proportional to the volatility of the returns over the liquidation period. These are exactly proportional in the idealised case that the portfolio distribution is Gaussian.
More precisely, if we denote the market volatility today by σ and the estimated volatility at the time of default by σ ^{∗}, then we have
$$w=\frac{{M}_{k}^{*}}{{M}_{k}}\approx \frac{{\sigma}^{*}}{\sigma}$$ 
The wrongway factor can thus be estimated conservatively by taking the ratio of a historical volatility from a period of extreme stress to today’s volatility. A typical level for w is between 2 and 3, but this will clearly depend on the current market regime.
12.3.2.5 Contagion and the breach probability p^{+}
The default of a large clearing member can have a contagion effect on the market. This means that the default event can cause the volatility to jump from a level σ^{∗} just before default to a new level σ ^{+} after default, where σ ^{+} > σ ^{∗} (see Pykhtin and Sokol 2013).
The probability of losses exceeding the margin, ${M}_{k}^{*}$, in the liquidation period following default is given by p^{+}. This will be driven by σ ^{+}, and in general we have p^{+} > pM.
To estimate p^{+}, we write the unconditional distribution of portfolio loss used by the CCP to determine margin levels as a function, f_{k}, of a Gaussian variable, W_{t}, ie, ∆V_{k} = f_{k}(W_{t}). We need to relate the volatility of W_{t} post default, σ ^{+}, to the volatility at default, σ ^{∗}. We do this by introducing the contagion parameter γ > 1 given by
$${\sigma}^{+}=r{\sigma}^{*}$$ 
The probability p^{+} can now be calculated as
$$\begin{array}{lll}{p}^{+}(r)\hfill & \equiv \hfill & P[\Delta {V}_{k}>{M}_{k}^{*}\text{default]}\hfill \\ \hfill & =\hfill & \Phi \left(\frac{{f}_{k}^{1}({M}_{k}^{*})}{{\sigma}^{+}}\right)\hfill \\ \hfill & =\hfill & \Phi \left(\frac{{\Phi}^{1}({p}_{M})}{r}\right)\hfill \end{array}$$ 
Note that p^{+} does not depend on σ ^{+} but only on the stress factor γ, as well as pM, which is set by the CCP.
The breach probability p^{+} is very sensitive to γ. The contagion stress factor can be estimated by comparing historical volatilities before and after stress events. Typical values for γ range between 2 and 3. For γ = 2 we have p^{+} = 12% and for γ = 3 we get p^{+} = 22%.
12.4 LOSS ALLOCATION
In the previous section we calculated the potential losses to a CCP due to the default of a clearing member. To the extent that the losses are not covered by the CCP’s own equity, the losses will be mutualised and allocated to the surviving clearing members. In this section we are interested in calculating the allocation of losses to the clearing member CM_{0}. As before, we assume that there is no CCP own equity to absorb any of the losses, and also that all losses will be mutualised.
Let us denote a default scenario at time t by s_{t}. This is the set of all defaulting members over an allocation period [t, t + ∆_{r}] in the scenario. We can then write the total loss during an allocation period as follows
$${U}_{tot}({s}_{t})={\displaystyle \sum _{k\in {s}_{t}}{U}_{k}({s}_{t})}$$ 
In the following we shall drop the subscript t for ease of notation.
To calculate the portion of this total loss that is allocated to clearing member CM_{0} we follow the typical convention that the loss allocation is proportional to CM_{0}’s contribution to the default fund as a fraction of the total remaining fund. In particular, the allocated fraction of the loss A_{0}(s) is given by
$${A}_{0}(s)=\frac{{D}_{0}}{{D}_{tot}{\displaystyle {\sum}_{j\in s}{D}_{j}}}$$ 
and the loss L_{0} that is incurred by CM_{0} is just
L_{0}(s) = A_{0}(s)U_{tot}(s)
This expression can be used directly if we are interested in a scenariobased loss estimate, eg, if we wish to calculate the loss assuming only a single member defaults.
12.4.1 Expected allocation
We can also calculate an expected allocation factor by considering a probabilityweighted sum over allocations in all scenarios. This is particularly useful if we are calculating an expected loss to the entire CCP.
We shall show how the CCP risk can be decomposed into a linear sum of exposures to each clearing member together with a suitable convexity adjustment. The convexity adjustment corrects for the fact that we can have multiple clearing member defaults in an allocation period. In particular, it captures the default correlation risk that is introduced via the allocation mechanism.
We begin by writing the expected allocated loss, L_{0}(t), over the period [t, t + ∆_{r}] as an average over all possible scenarios s ∈ S
$${\overline{L}}_{0}(t)\equiv \text{E[}{L}_{0}(s)]={\displaystyle \sum _{s\in S}P(s){A}_{0}(s){\overline{U}}_{tot}(s)}$$  (12.3) 
where P(s) is the probability of the scenario s and S is the set of all possible default scenarios. The expected total loss is denoted Ū_{tot}(s) ≡ E[U_{tot}(s)].
We can reexpress Equation 12.3 as a sum over the exposures to the individual clearing members as follows
$${\overline{L}}_{0}(t)={\displaystyle \sum _{s\in S}P(s){A}_{0}(s){\displaystyle \sum _{k\in s}{\overline{U}}_{k}}}$$  (12.4)  
$\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}={\displaystyle \sum _{s\in S}P(s){A}_{0}(s){\displaystyle \sum _{k=1}^{N}{\text{I}}_{k\in s}{\overline{U}}_{k}}}$  (12.5)  
$\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}={\displaystyle \sum _{k=1}^{N}{\overline{U}}_{k}{\displaystyle \sum _{s\in S}{\text{I}}_{k\in s}P(s){A}_{0}(s)}}$  (12.6) 
where we have introduced the indicator function I_{x}, which is 1 if x is true and 0 otherwise.
We now split A_{0}(s) into a linear part and a scenariodependent correction. For s ∈ S and for any k ∈ s we write
$$\begin{array}{l}{A}_{0}(s)=\frac{{D}_{0}}{{D}_{tot}{\displaystyle {\sum}_{j\in s}{D}_{j}}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{{D}_{0}}{{D}_{tot}{D}_{k}}(1+{B}_{k}(s))\end{array}$$ 
where we have introduced the correction term
$${B}_{k}(s)\equiv \frac{{\displaystyle {\sum}_{j\in s\backslash \left\{k\right\}}{D}_{j}}}{{D}_{tot}{\displaystyle {\sum}_{j\in s}{D}_{j}}}$$ 
where s\ {k} denotes all defaulted members in the scenario s excluding member CM_{k}. The correction term B_{k}(s) adjusts for the fact that more than one clearing member can default during an allocation period. It is 0 unless there are at least two member defaults in the scenario s. We can use this to rewrite the righthand side of Equation 12.6 as
$${D}_{0}{\displaystyle \sum _{k=1}^{N}\frac{{\overline{U}}_{k}}{{D}_{tot}{D}_{k}}\left({\displaystyle \sum _{s\in S}{\text{I}}_{k\in s}P(s)+{\displaystyle \sum _{s\in S}{\text{I}}_{k\in s}P(s){B}_{k}(s)}}\right)}$$ 
where ${\sum}_{s\in S}{\text{I}}_{k\in s}P(s)$ is the sum of the probabilities of all scenarios in which the member CM_{k} defaults. This is just the marginal default probability of the member CM_{k}, which is given by
$$\sum _{s\in S}{\text{I}}_{k\in s}P(s)}={\lambda}_{k}{\Delta}_{r$$  (12.7) 
where λ_{k} is the default intensity of the member CM_{k} (which we assume is constant here).
Putting this together, we can now express the expected loss as
$${\overline{L}}_{0}(t)={D}_{0}{\displaystyle \sum _{k=1}^{N}\frac{{\overline{U}}_{k}}{{D}_{tot}{D}_{k}}(1+{\epsilon}_{k}){\lambda}_{k}{\Delta}_{r}}$$  (12.8) 
where the correction term
$${\epsilon}_{k}=\frac{1}{{\lambda}_{k}{\Delta}_{r}}{\displaystyle \sum _{s\in S}^{}P(s){\text{I}}_{k\in s}{B}_{k}(s)}$$ 
is a probabilityweighted sum over scenarios with more than one default. It corrects for the fact that the loss allocation is higher in multiple default scenarios and depends on the default correlation as well as the size of the allocation period, ${\Delta}_{r}$. It can be calculated given a model for joint member defaults.
Default intensity (bp)  
Correlation (%)  50  100  200  400  800 
0  0  1  1  2  5 
20  3  4  7  10  16 
40  12  16  21  27  35 
60  35  42  50  60  72 
80  108  122  139  160  185 
99.99  13.3  13.4  13.4  13.3  13.5 
Note: this is based on a clearing house with 15 members. All default fund contributions are assumed equal. All values are given in percent, except for the 99.99% correlation, for which absolute values are given.We use a Gaussian copula to generate joint default probabilities. 
Number of members  
Correlation (%)  10  15  20 
0  1  1  2 
20  5  7  9 
40  16  21  25 
60  41  50  56 
80  118  139  153 
99.99  8.7  13.4  18 
Note: this default intensity is 200 basis points. All values are given in percent, except for the 99.99% correlation, for which absolute values are given. 
In the following we assume that the correction factor does not depend strongly on the clearing member, and is given by a homogeneous factor ε. This also allows us to drop the t dependency of ${\overline{L}}_{0}$ .
In Tables 12.1 and 12.2 we show typical levels of ε as a function of the default correlation, the member default intensity and the number of clearing members. The allocation period is set to ∆_{r} = 30 days. We see that ε increases rapidly with correlation. The sensitivity to the intensity and the number of clearing members is comparatively low.
An instructive limit case to Equation 12.8 is obtained if we assume that default fund contributions, default intensities and the excess loss ${\overline{U}}_{k}=\overline{U}$ are the same for all members. In this case the loss is simply given by
$${\overline{L}}_{0}=\overline{U}(1+\epsilon )\lambda {\Delta}_{r}$$ 
If the correlation is 0%, then ε ≈ 0 and the allocated expected loss is simply the expected loss of one member. When the correlation approaches 100%, (1 + ε) will tend to N. This means that we expect all members (apart from CM_{0}) to default at the same time, and the entire loss is allocated to CM_{0}.
We can write Equation 12.8 in the following more intuitive form
$${\overline{L}}_{0}={\displaystyle \sum _{k=1}^{N}{\overline{U}}_{k}{A}_{0,k}{\lambda}_{k}{\Delta}_{r}}$$ 
where
$${A}_{0,k}\equiv \frac{{D}_{0}}{{D}_{tot}{D}_{k}}(1+\epsilon )$$ 
is an effective allocation factor.
Hence, we see that the expected loss over a period of length ∆_{r} is given by the sum of expected exposures ${\overline{E}}_{0,k}$ to all clearing members
$${\overline{L}}_{0}={\displaystyle \sum _{k=1}^{N}{\overline{E}}_{0,k}{\lambda}_{k}{\Delta}_{r}}$$  (12.9) 
and we can identify the expected (allocated) exposure, ${\overline{E}}_{0,k}$, of member CM_{0} versus member CM_{k} as
$${\overline{E}}_{0,k}\equiv {\overline{U}}_{k}{A}_{0,k}$$  (12.10) 
Similarly, we can define a stressed allocated exposure as
$${\overline{E}}_{0,k}\equiv {\overline{U}}_{k}{\widehat{A}}_{0,k}$$  (12.11) 
where ${\widehat{A}}_{0,k}$ is an allocation factor calculated with a correction term based on stressed default probabilities.
12.5 APPLICATIONS
In the previous sections we calculated the potential losses given a clearing member default as well as the allocation of the losses to the surviving members. We have provided different loss estimation methodologies based on either the default fund size or the size of individual margin contributions. In this section we shall show how these results can be applied to provide expressions for different measures of CCP risk for the clearing member CM_{0}. Further applications, to CCP capital calculations in particular, are described in Arnsdorf (2014).
12.5.1 Stress exposure
A basic risk measure is the stressed or peak exposure. By this we mean an extreme potential loss, Ê_{0},k, which CM_{0} can incur due to the default of clearing member CM_{k}. The loss calculation is assumed to be as of today, ie, based on today’s margin level and today’s market conditions. As we have shown in the previous section, this is given by
$${\widehat{E}}_{0,k}={\widehat{U}}_{k}{\widehat{A}}_{0,k}={D}_{0}{\widehat{U}}_{k}\frac{1+\widehat{\epsilon}}{{D}_{tot}{D}_{k}}$$ 
where $\widehat{\epsilon}$ is a correction factor calculated with stressed default intensities.
To estimate the exposures we need to calculate ${\widehat{U}}_{k}$. If we use the simple method based on the CCPs default fund presented in Section 12.3.1 we have
$${\widehat{U}}_{k}^{(n)}=\frac{1}{n}{D}_{tot}\overline{D}$$ 
where n is the cover requirement of the total default fund. In the typical case that n = 2 we can write
$${\widehat{E}}_{0,k}^{(2)}=\frac{{D}_{0}}{2}\frac{{D}_{tot}2\overline{D}}{{D}_{tot}{D}_{k}}(1+\widehat{\epsilon})$$  (12.12) 
This can be conservatively approximated by
$${\widehat{E}}_{0,k}^{(2)}=\frac{1}{2}{D}_{0}(1+\widehat{\epsilon})$$  (12.13) 
This leads to the very simple rule of thumb that the peak exposure a bank has to any clearing member is roughly onehalf of its own default fund contribution.
The key point here is that, owing to the allocation mechanism, the loss due to a clearing member default can be estimated using our own default fund contribution. Note that the loss expression above does not in fact depend on the identity of CM_{k} at all. This is because we are basing the loss estimate on the loss of the largest members. For any average clearing member this is clearly conservative.
The stress losses derived above can be aggregated to other exposures a bank has with the same counterparties. This provides a consistent risk measure across OTC and cleared transactions.
12.5.2 Expected loss due CCP membership
In addition to the stress exposure, we can also calculate an expected loss reserve or expected cost due to membership of a CCP. This is essentially given by integrating the allocated loss in Equation 12.9 over time. To calculate a loss reserve we additionally need to consider the fact that the CCP might recover part of the claim it has towards a defaulting clearing member. Here we assume that the claim is pari passu with other bilateral derivatives claims on the counterparty. This means we expect a standard recover rate, R, on any default claim by the CCP.
The expected loss reserve, EL_{0}(t), which CM_{0} needs to hold against the expected losses a CCP faces over the period [t, ∆_{r}], is hence given by
$${\text{EL}}_{0}(t)=(1R){\overline{L}}_{0}(t)=(1R){\displaystyle \sum _{k=1}^{N}{\overline{E}}_{0,k}{\lambda}_{k}{\Delta}_{r}}$$  (12.14) 
If we assume that defaulting clearing members are replaced such that the overall risk profile of the CCP stays roughly constant over time, then we can calculate the expected loss from today, t_{0}, up to time T as
$${\text{EL}}_{0}({t}_{0},T)=(1R){\displaystyle \sum _{k=1}^{N}{\overline{E}}_{0,k}{\lambda}_{k}T}$$  (12.15) 
The default intensity, λ_{k}, can be implied from market credit default swap (CDS) quotes or estimated based on realised historical default frequencies. If we are using market implied default intensities, then EL_{0}(t_{0}, T) can be interpreted as the cost of buying protection against the expected losses faced by the CCP. This is analogous to a counterparty credit valuation adjustment (CVA) charge. Whether CVA is held against CCP exposures depends on fair value considerations and the applicable accounting framework. If we use historical default probabilities, on the other hand, then EL_{0}(t_{0}, T) is just an expected loss similar to a loan loss reserve.
Using the parametric loss model, the allocated (expected) exposure can be written as
$${\overline{E}}_{0,k}={M}_{k}\frac{w{p}^{+}}{\alpha 1}{(1+r)}^{1\alpha}\frac{{D}_{0}}{{D}_{tot}{D}_{k}}(1+\epsilon )$$  (12.16) 
To obtain a simplified expression, we first assume that default intensities are homogeneous across members and take the average value λ. This means we can write
$${\text{EL}}_{0}(T)=(1R)\lambda T\frac{w{p}^{+}}{\alpha 1}{(1+r)}^{1\alpha}(1+\epsilon ){\displaystyle \sum _{k=1}^{N}{M}_{k}}\frac{{D}_{0}}{{D}_{tot}{D}_{k}}$$ 
If we make use of the assumption that the ratio of initial margin to the default fund contribution is given by r, then we can write the above equation entirely in terms of the initial margin
$${\text{EL}}_{0}({t}_{0},T)=(1R)\lambda T\frac{w{p}^{+}}{\alpha 1}{(1+r)}^{1\alpha}(1+\epsilon ){M}_{0}{\displaystyle \sum _{k=1}^{N}\frac{{M}_{k}}{{M}_{tot}{M}_{k}}}$$ 
Finally, if CM_{0} is a representative member, then it is reasonable to replace M_{k} with M_{0} in the denominator. This is conservative if CM_{0} has a large margin posting. We then obtain
$$\sum _{k=1}^{N}\frac{{M}_{k}}{{M}_{tot}{M}_{k}}\approx {\displaystyle \sum _{k=1}^{N}\frac{{M}_{k}}{{M}_{tot}{M}_{0}}=\frac{{M}_{tot}{M}_{0}}{{M}_{tot}{M}_{0}}=1}$$ 
and hence
$${\text{EL}}_{0}({t}_{0},T)\approx (1R)\lambda {M}_{0}\frac{w{p}^{+}}{\alpha 1}\frac{(1+\epsilon )}{{(1+r)}^{\alpha 1}}$$ 
For reasonable values of α ≈ 3, r ≈ 10% and ε ≈ 20% we also find that (1 + ε)/(1 + r)^{α−1} ≈ 1.
Putting this all together, we finally get a simple expression for the expected loss
$${\text{EL}}_{0}({t}_{0},T)\approx (1R){M}_{0}\frac{w{p}^{+}}{\alpha 1}\lambda T$$  (12.17) 
This can be interpreted as a CVA charge or an expected loss reserve, depending on the choice of default intensity.
Expression 12.17 is remarkable in that expected loss to a CCP can be estimated based on our own posted margin, even though the risk is due to the default of other clearing members. This is due to the allocation mechanism. In particular, Expression 12.17 has the same form as an expected loss on a (nonsegregated) posted margin with a risk weight
$$W=\frac{w{p}^{+}}{\alpha 1}$$ 
which only depends on the CCP specific margin confidence level as well as the parameterisation of the market factor distributions underlying the products traded at the CCP. These ideas are developed further in Chapter 13.
As described in Section 12.3.2, the risk weight W could also be estimated using a bank’s internal exposure model, in which case we would avoid needing to estimate p^{+} and α. The simple parametric model above is, however, likely to be sufficient for most applications.
In the next section we shall see how these results apply in stress testing, and provide quantitative loss estimates.
12.5.3 Stress testing
One important application of the methodology described above is stress testing. For this purpose we are interested in a forecasted loss over a fixed time horizon T in a severe stress scenario. A particular example of this is the CCAR stress test mandated by the US regulators. We now look at how this can be estimated in more detail.
We assume a stress scenario consisting of a large instantaneous market shock at the beginning of the forecast period. The stress persists for the entire duration of the period. This is the typical setup in the CCAR stress test, where the forecast period is nine quarters.
Losses due to CCP membership can manifest themselves in two ways.

A bank can incur realised losses due to CCP member defaults; we can calculate the expectation of such losses in the stress scenario.

If an accounting CVA reserve is held against the CCP exposure, then this will increase due to the instantaneous market shock. This will lead to marktomarket losses on the balance sheet. If a CVA reserve is held, then we do not need to separately estimate realised losses, as they would be offset (in expectation) by a reduction in the reserve over time.
In both cases we can apply the methodologies discussed in this chapter. The two loss metrics differ only in the default probability estimation, which can be based on market CDS data (in the CVA case) or on historical estimates (in the realised expected loss case).
12.5.3.1 The first margin period
We first need to divide the forecast period [t_{0}, T] into two parts (ie, [t_{0}, t_{∆r}] and [t_{∆r} , T], with t_{∆r} = t_{0} + ∆_{r}), as they will need separate treatment. We assume that the market shock occurs at the beginning of the first period, and initial margin and default fund contributions are only rebalanced after the end of the period to adjust for the increased market risk. This means that the CCP will be undercollateralised in the initial period.
We want to calculate the stressed expected loss in the first period from the perspective of CM_{0}. We denote this by EL _{0}(t_{0}, t_{∆r} ). As before, this loss will be due to the defaults of other clearing members and can be written as a sum over individual stressed expected exposures
$${\widehat{EL}}_{0}({t}_{0},\text{\hspace{0.17em}}{t}_{{\Delta}_{r}})=(1\widehat{R}){\displaystyle \sum _{k=1}^{N}{\widehat{E}}_{0,k}({t}_{0},{t}_{{\Delta}_{r}}){\widehat{\lambda}}_{k}{\Delta}_{r}}$$ 
This is equivalent to Equation 12.15, but we need to take into account that the various quantities entering the expression are stressed after the market shock.
The stressed default intensities are denoted by ${\widehat{\lambda}}_{k}$ and should be determined based on the severity of the stress scenario. Typically, a stress scenario will include a specification for the credit stress. We recall that the choice of base default intensity will depend on whether we are interested in a CVA loss or a realised expected loss.
For simplicity we shall assume in the following that in the stress period the recovery rate R = 0.
We now focus on the expected exposure in the first period, which will also be stressed due to the market shock. To estimate the stressed expected exposure we can use the two measures we discussed in previous sections.
The simplest measure to use is the defaultfundbased expression for the stress loss given in Equation 12.12 with ${\stackrel{\stackrel{\wedge}{=}}{EL}}_{0,k}({t}_{0},\text{\hspace{0.17em}}{t}_{{\Delta}_{r}})={\widehat{E}}_{0,k}$. This assumes that any clearing member default will result in a loss given by a fraction of the default fund. For the typical “Cover 2” case and using the approximation in Equation 12.13, this results in the following expected loss over the first period
$${\widehat{EL}}_{0}({t}_{0},\text{\hspace{0.17em}}{t}_{{\Delta}_{r}})=\frac{1}{2}{D}_{0}(1+\widehat{\epsilon}){\displaystyle \sum _{k=1}^{N}{\widehat{\lambda}}_{k}{\Delta}_{r}=\frac{1}{2}{D}_{0}(1+\widehat{\epsilon}})N\widehat{\lambda}{\Delta}_{r}$$  (12.18) 
where the last equality assumes that the individual default intensities can be approximated by a single representative stressed intensity, $\widehat{\lambda}$.
The main assumption we make in determining the stressed exposure above is that the stress scenarios used by the CCP in the determination of the default fund are similar in severity to the instantaneous shock that is part of the loss forecast. Equation 12.18 is likely to be a very conservative measure, as we are assuming that every clearing member default will result in the same loss as the largest members. Given more detailed information about the CCP composition, the approach can be suitably modified to give more realistic estimates.
Alternatively, we can use the initialmarginbased approach to estimate the stressed expected loss. We do this by making a suitable choice of wrongway risk and contagion parameters. Let us assume that the most representative volatility for the products traded at the CCP is given by σ before stress. After the instantaneous market shock we assume that this volatility increases to $\widehat{\sigma}$. Let us denote the ratio of the volatilities by ${R}_{\sigma}\equiv \widehat{\sigma}/\sigma $. In the first period we know that the initial margin is given by the current margin levels. This means that the wrongway factor is given by w = 1. After the shock, however, the probability of losses exceeding the posted initial margin levels will increase, as volatilities will have increased. This is equivalent to the contagion effect described earlier, and we can set γ = R_{σ} .
If we make the same assumptions as in Section 12.5.2, this results in the following equation for the first period loss
$${\widehat{EL}}_{0}({t}_{0},\text{\hspace{0.17em}}{t}_{{\Delta}_{r}})={M}_{0}\frac{{p}^{+}({R}_{\sigma})}{\alpha 1}\widehat{\lambda}{\Delta}_{r}$$  (12.19) 
which can be estimated simply. We note in particular that the ratio of stressed to unstressed volatilities, R_{σ} , will be specified as part of the stress scenario.
It can be instructive to compare Equations 12.18 and 12.19. In both cases the loss is estimated as a fraction of collateral posted by us to the CCP. In the former case the effective collateral is given by ND_{0}, and in the latter case this is M_{0}. A typical ratio of margin to default fund contribution is M_{0}/D_{0} ≈ 10. Hence, the effective collateral levels in both approaches will be similar for CCPs with around 10 major clearing members, which is a reasonable number. The effective collateral level the in defaultfundbased approach increases with the number of clearing members, while that given by the marginbased formula does not. This is because the former approach is conservative in assuming that each default will cause the same maximum loss. Hence, the defaultfundbased approach becomes unsuitable in the presence of many clearing members.
The collateral scaling factor is given by the cover level in Equation 12.18 and is $\frac{1}{2}$ for a “Cover 2” CCP. In the marginbased approach the scaling factor is given by p^{+}/(α − 1). For a volatility stress of 300% and pM = 1%, the breach probability will be around 22%. If we assume a conservative alpha of 3, then the scale factor is given by 11%. This is less than the 50% we have in the default fund case, but of the same order of magnitude.
Hence, we conclude that two approaches are comparable. However, the defaultfundbased estimation is conservative, as expected, particularly if N is large.
12.5.4 Future margin periods
At the end of the first margin period, the collateral levels in the CCP will have been updated. This means in particular that the initial margin posted by all members will have increased to cover the increased risk due to the instantaneous market shock. It also means that the expected loss in future periods is much smaller than in the initial period.
To demonstrate this in more detail we now show how the future period loss can be estimated in the marginbased approach. As discussed, there are two key points we need to take into consideration.

Margin levels have adjusted to match the increased volatilities. We recall that the ratio of stressed to unstressed volatilities is given by R_{σ} . We can incorporate this in our expected loss model using the wrongway factor w by setting w = R_{σ} .

We do not expect volatilities to increase further, conditional on default, given the fact that we are already in an extremely stressed market environment. This means that we can set the contagion factor to γ = 1.
Using the same approximations as before this means that the loss over a future period [t, t + ∆_{r}] with t > t_{∆r} is given by
$${\widehat{\text{EL}}}_{0}(t,t+{\Delta}_{r})={M}_{0}\frac{{R}_{\sigma}{p}_{M}}{\alpha 1}\widehat{\lambda}{\Delta}_{r}$$ 
where again we assume that the default intensities of the clearing members remain constant in time. We also assume that the initial margin level M_{0} remains constant throughout the forecast period, which as before is a consequence of our “constant risk” assumption. The main alternative is to assume that the portfolio is fixed as of today and runs off over time, which would result in a lower expected loss. This can be incorporated given a model of the amortisation of the initial margin over time.
The ratio of the first period to any future period is given by
$$\frac{1}{{R}_{\sigma}}\frac{{p}^{+}({R}_{\sigma})}{{p}_{M}}$$  (12.20) 
If, as before, we assume R_{σ} = 300% = w and pM = 1% and thus p^{+} ≈ 22%, then we find that the initial period expected loss will be around seven times higher than the expected loss in any future period.
12.5.5 Total loss
Let us combine these results to provide an approximate estimate of a total expected loss ${\widehat{EL}}_{0}({t}_{0},\text{\hspace{0.17em}}T)$ for clearing member CM_{0} facing a CCP over a forecast period [t_{0}, T]. We assume an instantaneous market shock at t_{0} = 0, where volatilities are increased by a factor R_{σ} .
Using the marginbased approach described above we have
$${\widehat{EL}}_{0}({t}_{0},\text{\hspace{0.17em}}T)=\frac{\widehat{\lambda}}{\alpha 1}[{R}_{\sigma}{p}_{M}(T{\Delta}_{r})+{p}^{+}({R}_{\sigma}){\Delta}_{r}]{M}_{0}$$ 
We now examine the behaviour of this expression for a forecast horizon of two years. To do this we first need to choose a reasonable level of the base default intensity λ. This will depend on whether we are interested in a riskneutral CVAtype calculation or a historical expected loss estimation. A typical riskneutral default intensity for a clearing member is around 200 basis points (bp), which is what we shall use in the following. Default intensities based on realised default frequencies would be an order of magnitude lower.
Given a base intensity, we need to determine $\widehat{\lambda}$. In the example below we shall assume that the credit stress is the same as the volatility stress. This means $\widehat{\lambda}={R}_{\sigma}\lambda $.
(a) α = 3  
∆_{r}  
R_{σ}  p^{+} (%)  1W  1M  2M  3M  2Y 
1  1  2  2  2  2  2 
2  12  8  10  11  13  49 
3  22  19  23  27  32  131 
4  28  34  40  48  56  224 
5  32  53  61  73  84  321 
(b) α = 4  
∆_{r}  
R_{σ}  p^{+} (%)  1W  1M  2M  3M  2Y 
1  1  1  1  1  1  1 
2  12  6  6  8  9  33 
3  22  13  15  18  21  88 
4  28  23  27  32  37  150 
5  32  35  41  48  56  214 
Note: we consider α = 3 and α = 4. Results are based on an unstressed riskneutral default intensity of 200 bp. 
In Table 12.3 we show how the expected loss as a proportion of the initial margin posted varies as a function of the volatility stress R_{σ} and the recollateralisation period ∆_{r} for two different levels of α.
We can see that the expected losses are small as a proportion of the initial margin in normal market conditions. However, the loss increases rapidly as volatilities are stressed. For banks with large positions at several CCPs, the risk will accumulate and can be become significant.
12.6 CONCLUSION
Membership of a CCP entails credit risk. This is primarily because of loss mutualisation, which is a key feature of a CCP’s risk management. In particular, each clearing member is required to maintain contributions to a default fund, which are at risk in the case of a clearing member default. An important point is that default fund contributions are at risk even if the CCP itself does not default. Furthermore, the losses incurred by a bank are not related to the open positions the bank has with the CCP. This is fundamentally different from a bilateral OTC exposure.
In this chapter we have explored different ways of analysing and quantifying the risk a financial institution faces due to clearing house membership. The difficulty in doing this stems from a lack of detailed information on the composition of clearing member portfolios.
We have shown how we can leverage CCP risk management practice to estimate exposures contingent on clearing member defaults. In particular, we presented two main approaches to calculating default fund losses. The first provides a stressloss measure and uses directly the fact that CCP default contributions are based on socalled “cover ” requirements, which state that the fund must withstand the default losses of a predefined number of member defaults under stress conditions. The second method uses a parameterised model of the clearing member portfolio loss distributions. This can be used to estimate expected losses as well as stress losses. Key drivers of the risk are default contagion, wrongway risk and correlation between member defaults.
We find that, due to the CCP loss allocation mechanism, the expected loss that a bank facing a CCP has can be expressed simply and is proportional to the bank’s own posted initial margin. This means that the expected exposure to a CCP has a very familiar form, even though the loss mechanism in a CCP is different from a standard counterparty portfolio exposure.
The framework for understanding CCP risk introduced here can be used by financial institutions to manage clearing house risk and to estimate costs of clearing house membership. We have looked at a variety of applications including stress testing as required by the CCAR. As the volume of transactions cleared at CCPs grows, this will become an increasingly important area of credit risk management.
The statements, views and opinions expressed in this chapter are the author’s own and do not necessarily reflect those of their employer, JPMorgan Chase & Co, its affiliates, other employees or clients.
References
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