# Journal of Risk

**ISSN:**

1465-1211 (print)

1755-2842 (online)

**Editor-in-chief:** Farid AitSahlia

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Need to know

- This paper explores and develops a new analytical framework of dealing with risky collateral.
- It calculates the required initial margin amount using risky collateral by solving a quadratic inequality.
- The paper also introduces a geometric structure to compare the required IM with risky and non-risky collateral.

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Abstract

The provision of initial margin (IM) for noncentrally cleared derivatives has gained prominence in financial markets as a way to mitigate counterparty credit risk. IM pro- tects transacting parties from the potential increase in future exposure that could arise from the portfolio value change during the time that it takes to close out and replace the portfolio following a counterparty default. The Basel Committee on Banking Supervision prescribes IM as the ten-day value-at-risk (VaR) of the portfolio at the 99th percentile confidence level. Current industry standard VaR approaches such as parametric or historical VaR methods necessitate an assumption that IM is posted in cash or cash-equivalent assets. Although many counterparty-credit-risk-related models exist in the academic literature, there has been little focus on achieving a theoretical basis for calculating margin with consideration of market risk of the collateral. In this paper, we explore the complication of calculating the IM amount required when collateral comprises risky assets in a parametric VaR framework. We show that the required IM amount can be calculated by solving a quadratic inequality. We also introduce a geometric structure to compare the IM amounts calculated using risky and nonrisky collateral.

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Introduction

## 1 Introduction

Counterparty credit risk modeling has attracted increasing attention in recent years, particularly since the 2008 economic crisis. An abundance of papers and books have been published on this topic, including Jarrow and Yu (2001), Kraft and Steffensen (2007), Brunnermeier (2009), Duffie and Zhu (2011), Duffie and Singleton (2012), Gregory (2012), Arora et al (2012), Bielecki and Rutkowski (2013) and Brigo et al (2013). The developments in approaching counterparty credit risk have highlighted the challenges of defining and quantifying all the associated risk factors, incorporating them in the pricing of derivatives and consequently managing and mitigating counterparty credit risk.

Collateralization has served as one of the most effective approaches to mitigating counterparty credit risk. Specific to over-the-counter (OTC) derivatives markets, collateral exists to fulfill two different obligations: variation margin and initial margin. Variation margin (VM) is intended to provide protection for the current in-the-money position of the unsecured portfolio. It is typically required and rebalanced on a daily basis, given a certain threshold. Initial margin (IM) is intended to provide protection from potential losses in the unsecured portfolio during the closeout period following a counterparty default. The closeout period refers to the time it takes for the surviving party to unwind or settle its affected trades. As the market moves during this closeout period, the VM by itself may not be sufficient to cover the trading portfolio exposure if its value increases, which could lead to losses for the surviving party.

While the calculation of VM is based on mark-to-market (MTM) that requires little or no additional modeling, the calculation of IM can depend on the choice of risk model and is justifiably subject to more attention from both regulators and market participants. For noncentrally cleared (ie, bilateral) OTC derivatives, the Basel Committee on Banking Supervision and the International Organization of Securities Commissions proposed that the IM reflects an extreme but plausible increase in potential future exposure that is consistent with a one-tailed 99% confidence interval over a ten-day period (BCBS–IOSCO 2013). For centrally cleared trades, the IM methodology adopted by a central counterparty clearing house (CCP) should, according to CPSS–IOSCO standards,^{1}^{1}CPSS is the Committee on Payment and Settlement Systems (now the Committee on Payments and Market Infrastructures, or CPMI). meet an established single-tailed confidence level of at least 99% of the estimated distribution of future exposures. Sidanius and Zikes (2012) showed that, under certain normal market conditions and assumptions and holding the current notional value of trades fixed, the total IM for centrally and bilaterally cleared trades ranges between USD200 billion and USD800 billion. The International Swaps and Derivatives Association (ISDA) also conducts the industry-wide margin survey annually among derivative dealers and end users to examine the state of collateral use and management for both cleared and noncleared derivative transactions.^{2}^{2}See, for example, the 2014 and 2015 ISDA Margin Surveys at http://bit.ly/2EajPAQ. In addition to the margin survey effort, the ISDA has conducted a project (International Swaps and Derivatives Association 2016) in collaboration with its members to develop a standardized IM methodology (SIMM) that can be used by market participants globally to meet the margin requirements across different jurisdictions. This took effect on September 1, 2016.

From a modeling perspective, the IM calculation essentially leads to the adoption of an appropriate method to calculate the 99% ten-day VaR for the portfolio in question. To be equivalent in value to the VaR in order to serve as protection against potential losses, the collateral needs to be risk-free in the sense that its value should be preserved regardless of market movements. Otherwise, even in cases where the amount of IM consisting of risky collateral is sufficiently large, the firm may still be exposed to losses, especially during a financial stress period, if the market value of the received collateral moves adversely. For this reason, cash-equivalent assets are deemed to be preferred collateral instruments. However, a broader set of eligible collateral is permitted by BCBS–IOSCO (2013), provided that the collateral amount is appropriately adjusted by taking into account the risk of the collateral security. Although many counterparty-credit-risk-related models exist in the academic literature, much less work has been done to achieve a viable theory for calculating margin amounts using risky collateral. Market participants and regulators typically use a so-called schedule-based haircut approach (shown in the online appendix) to determine the collateral amount based on the type of securities used. The haircut approach is, to some extent, a rule of thumb, without precise analysis. There is also limited research on the development of model-based haircut determination. Cossin and Hricko (2003) provide a framework to analyze haircut determination and the impact of risky collateral on credit risk. They look in-depth at the case of risky forwards, an analysis that could be generalized to swaps and other instruments. Lillo and Pirino (2015) present a model that allows the computation of a haircut incorporating the liquidity risk of the collateral and, most importantly, possible systemic effects. Lou (2016) adopts a credit approach to solve for haircuts such that the exposure to market risk obtains a near-cash credit quality. In Lou’s research, double exponential jump–diffusion-type processes are used to model the underlying asset price dynamics, which are shown to be able to produce haircut levels consistent with supervisory haircuts.

In contrast to most of the existing risky collateral modeling approaches that focus on haircut calculation, we provide a new method to account for the common risk factors between the unsecured portfolio and the collateral.^{3}^{3}In this paper, we explore the market risk associated with collateral assets, and for this reason consideration of credit risk of the collateral issuer as well as wrong-way risk in relation to the portfolio composition is beyond the scope of our investigation. In this framework, risk factor sensitivities are derived for the collateral and added to the sensitivities of the unsecured portfolio, providing a netted sensitivity for the collateralized portfolio. The method we explore in this paper is based on a parametric VaR framework. We show that the required IM amount can be found by solving a quadratic inequality. In certain cases, there may exist an upper bound for the IM amount, above which the collateral will stop acting as a risk mitigant. This finding agrees with the intuition that, when the value of the collateral asset received as IM is volatile, it is not optimal to hold an excessive amount of such a collateral asset. Otherwise, the IM’s ability to provide loss coverage for the portfolio is reduced or even eliminated by losses in the value of the collateral itself. Moreover, we introduce an abstract inner product space to the problem. Our results from the geometric structure generalize the intuition that, in a single-risk-factor case, if the value of risky collateral moves in the same direction as the value of the unsecured portfolio, then the required IM amount with this risky collateral is lower than the IM amount with nonrisky collateral.

The remainder of this paper is organized as follows. In Section 2, we demonstrate how to calculate the IM with risky collateral by solving a quadratic inequality. In Section 3, we derive the condition under which the IM amount calculated using risky collateral is less than that using nonrisky collateral. Finally, in Section 4, we show several numerical examples to further illustrate the properties and results derived in the previous sections.

## 2 Calculation of IM with risky collateral

### 2.1 Problem setup

Assume the column vector $X\in {\mathbb{R}}^{n}$ includes all market risk variables considered in the problem. ${\pi}_{\mathrm{U}}(t,{X}_{t})$ denotes the value of the unsecured OTC derivative portfolio at time $t$. ${\pi}_{\mathrm{VM}}(t,{X}_{t})$ denotes the value of the VM portfolio at time $t$. ${\pi}_{\mathrm{IM}}(t,{X}_{t})$ denotes the value of the IM portfolio at time $t$.

The IM amount ${\pi}_{\mathrm{IM}}(0)$ is calculated such that, when the counterparty defaults, the probability that the loss is not covered by the sum of VM and IM is within a target threshold. Specifically,

$$\mathbb{P}({\pi}_{\mathrm{U}}(\mathrm{\Delta}t)-{\pi}_{\mathrm{VM}}(\mathrm{\Delta}t)-{\pi}_{\mathrm{IM}}(\mathrm{\Delta}t)\ge 0)\le 1-\alpha ,$$ | (2.1) |

where $\alpha =99$% and $\mathrm{\Delta}t$ is the closeout period, eg, ten days. Note that we exclude the $X$ term to simplify the notation.

Let ${D}_{\mathrm{U}},{D}_{\mathrm{VM}},{D}_{\mathrm{IM}}\in {\mathbb{R}}^{n}$ denote the first-order sensitivities (eg, delta)^{4}^{4}Other first-order sensitivities such as vega can also be included in this framework. to the market risk vector associated with the unsecured, VM and IM portfolios, respectively. Their deltas are defined as the change in value of the aforementioned components of the collateralized portfolio with respect to a one-unit change in the corresponding risk factor. We assume the change in the market risk vector, $\mathrm{\Delta}{X}_{t}={X}_{t+\mathrm{\Delta}t}-{X}_{t}$, follows a normal distribution under the parametric VaR framework with mean zero and covariance matrix $\mathrm{\Sigma}(\mathrm{\Delta}t)$, ie,

$$\mathrm{\Delta}{X}_{t}\sim N(0,\mathrm{\Sigma}(\mathrm{\Delta}t)).$$ | (2.2) |

The normal assumption is preferred due to its mathematical tractability. However, it is worthwhile investigating more general fat-tailed distributions, especially for assets whose returns exhibit high skewness and leptokurtosis. Since the key interest and aim of this paper is to explore and develop a new analytical framework of dealing with risky collateral, we leave the technical complications involving more general fat-tailed distributions to a future work.

Under the normal assumption of risk factor moves, we have the first-order approximation

$$\begin{array}{cc}\hfill {\pi}_{\mathrm{U}}(\mathrm{\Delta}t)& \approx {\pi}_{\mathrm{U}}(0)+{D}_{\mathrm{U}}^{\mathrm{T}}\mathrm{\Delta}{X}_{t},\hfill \\ \hfill {\pi}_{\mathrm{VM}}(\mathrm{\Delta}t)& \approx {\pi}_{\mathrm{VM}}(0)+{D}_{\mathrm{VM}}^{\mathrm{T}}\mathrm{\Delta}{X}_{t},\hfill \\ \hfill {\pi}_{\mathrm{IM}}(\mathrm{\Delta}t)& \approx {\pi}_{\mathrm{IM}}(0)+{D}_{\mathrm{IM}}^{\mathrm{T}}\mathrm{\Delta}{X}_{t}.\hfill \end{array}\}$$ | (2.3) |

Here we ignore the theta sensitivity accounting for the aging of the product. Let $\pi (\mathrm{\Delta}t)$ be the collateralized portfolio value, ie, $\pi (\mathrm{\Delta}t)={\pi}_{\mathrm{U}}(\mathrm{\Delta}t)-{\pi}_{\mathrm{VM}}(\mathrm{\Delta}t)-{\pi}_{\mathrm{IM}}(\mathrm{\Delta}t)$ (here we assume full netting across margin and unsecured portfolio values; in practice, we would expect there to be no netting or only partial netting of margin against the unsecured portfolio allowed). Then the distribution of $\pi (\mathrm{\Delta}t)$ is

$$\pi (\mathrm{\Delta}t)\sim N({\mu}_{\pi},{\sigma}_{\pi}^{2}),$$ |

where ${\mu}_{\pi}={\pi}_{\mathrm{U}}(0)-{\pi}_{\mathrm{VM}}(0)-{\pi}_{\mathrm{IM}}(0)$ and ${\sigma}_{\pi}^{2}={({D}_{\mathrm{U}}-{D}_{\mathrm{VM}}-{D}_{\mathrm{IM}})}^{\mathrm{T}}\mathrm{\Sigma}(\mathrm{\Delta}t)({D}_{\mathrm{U}}-{D}_{\mathrm{VM}}-{D}_{\mathrm{IM}})$. Therefore, (2.1) becomes

$$\mathbb{P}(\pi (\mathrm{\Delta}t)\ge 0)=\mathrm{\Phi}\left(\frac{{\mu}_{\pi}}{{\sigma}_{\pi}}\right)\le 1-\alpha ,$$ |

and hence

$$\frac{{\mu}_{\pi}}{{\sigma}_{\pi}}\le {q}_{1-\alpha}=-{q}_{\alpha},$$ |

where ${q}_{\alpha}={\mathrm{\Phi}}^{-1}(\alpha )$ is the $\alpha $th percentile of the standard normal distribution. Note that the sensitivity vector for the IM portfolio ${D}_{\mathrm{IM}}$ implicitly depends on the IM ${\pi}_{\mathrm{IM}}(0)$ and its composition of risky assets. When the IM portfolio is not risky, ie, ${D}_{\mathrm{IM}}=0$, we can simply solve the above inequality for ${\pi}_{\mathrm{IM}}(0)$ as

$${\pi}_{\mathrm{IM}}(0)\ge {q}_{\alpha}{\sigma}_{\pi}+{\pi}_{\mathrm{U}}(0)-{\pi}_{\mathrm{VM}}(0).$$ | (2.4) |

As VM is required to be rebalanced daily with a zero threshold, ie, ${\pi}_{\mathrm{VM}}(0)={\pi}_{\mathrm{U}}(0)$, we can get the well-known VaR formula

$${\pi}_{\mathrm{IM}}(0)\ge {q}_{\alpha}{\sigma}_{\pi}.$$ |

However, when the collateral is risky, the right-hand side of (2.4) is also related to ${\pi}_{\mathrm{IM}}(0)$ through ${D}_{\mathrm{IM}}$ in ${\sigma}_{\pi}$, which makes the calculation of ${\pi}_{\mathrm{IM}}(0)$ more complex. To link ${D}_{\mathrm{IM}}$ with ${\pi}_{\mathrm{IM}}(0)$, we write

$${D}_{\mathrm{IM}}={\pi}_{\mathrm{IM}}(0){D}_{\mathrm{IM}}^{*},$$ | (2.5) |

where ${D}_{\mathrm{IM}}^{*}$ denotes the delta associated with the IM portfolio normalized to one unit of domestic currency. Specifically, we consider the composition of the IM portfolio as follows. First, we define $C(t)={({C}_{1}(t),\mathrm{\dots},{C}_{m}(t))}^{\mathrm{T}}$, where ${C}_{i}(t)$ is the value of one unit share of collateral asset $i$ at time $t$. Define $\beta ={({\beta}_{1},\mathrm{\dots},{\beta}_{m})}^{\mathrm{T}}$, where ${\beta}_{i}$ is the number of shares of collateral asset $i$ in the IM portfolio. We then have

${\pi}_{\mathrm{IM}}(0)$ | $={\beta}^{\mathrm{T}}C(0),$ | ||

${D}_{\mathrm{IM}}^{\mathrm{T}}$ | $={\beta}^{\mathrm{T}}{\displaystyle \frac{\partial C(0)}{\partial X}},$ |

where $\partial C(0)/\partial X\in {\mathbb{R}}^{m\times n}$ is the Jacobian matrix for the deltas. Hence, the delta for the normalized IM portfolio, ${D}_{\mathrm{IM}}^{*}$, is

$${D}_{\mathrm{IM}}^{*\mathrm{T}}=\frac{{D}_{\mathrm{IM}}^{\mathrm{T}}}{{\pi}_{\mathrm{IM}}(0)}=\frac{{\beta}^{\mathrm{T}}(\partial C(0)/\partial X)}{{\beta}^{\mathrm{T}}C(0)}=\frac{({\beta}^{\mathrm{T}}/\parallel \beta \parallel )(\partial C(0)/\partial X)}{({\beta}^{\mathrm{T}}/\parallel \beta \parallel )C(0)}.$$ |

Here we assume that the relative composition $\beta /\parallel \beta \parallel $ of the IM portfolio is an invariant. Therefore, ${D}_{\mathrm{IM}}^{*}$ depends only on the relative composition of IM portfolio and is invariant with respect to the magnitude of the IM portfolio. To illustrate this idea, let us consider a case in which a certain equity price $X$ is the only risk factor, and the IM portfolio can consist of cash, the equity and the at-the-money (ATM) European call option on this equity.^{5}^{5}Most liquid options are short term and not typically used as collateral in practice. Options are included here to illustrate the calculation of ${D}_{\mathrm{IM}}^{*}$. We assume a stock price ${X}_{0}=\$100$ per share, and the value of one share of the ATM European call option is ${C}_{0}=\$8$. The examples below demonstrate ${D}_{\mathrm{IM}}^{*}$ for four different compositions of IM portfolios.

- (1)
If all cash, then ${D}_{\mathrm{IM}}^{*}=0$.

- (2)
If all equity, then ${D}_{\mathrm{IM}}^{*}=1/100=0.01$.

- (3)
If all ATM call, then ${D}_{\mathrm{IM}}^{*}=0.5/8=0.0625$, assuming delta of the ATM call option is $0.5$.

- (4)
If 0.5 million dollars cash, 0.25 million dollars equity and 0.25 million dollars ATM call, then ${D}_{\mathrm{IM}}^{*}=\frac{1}{2}\times 0+\frac{1}{4}\times 0.01+\frac{1}{4}\times 0.0625=0.018125$.

After introducing ${D}_{\mathrm{IM}}^{*}$, the delta vector ${D}_{\mathrm{IM}}$ is uniquely determined by ${\pi}_{\mathrm{IM}}(0)$, which is the target we need to meet next. Assuming ${\pi}_{\mathrm{VM}}(0)={\pi}_{\mathrm{U}}(0)$ (due to the regulatory requirement that VM is rebalanced on a daily basis with zero threshold), (2.4) reduces to an inequality with only one unknown variable, ${\pi}_{\mathrm{IM}}(0)$, ie,

$${\pi}_{\mathrm{IM}}(0)\ge {q}_{\alpha}\sqrt{{({D}_{\mathrm{U}}-{D}_{\mathrm{VM}}-{\pi}_{\mathrm{IM}}(0){D}_{\mathrm{IM}}^{*})}^{\mathrm{T}}\mathrm{\Sigma}({D}_{\mathrm{U}}-{D}_{\mathrm{VM}}-{\pi}_{\mathrm{IM}}(0){D}_{\mathrm{IM}}^{*})}.$$ | (2.6) |

By squaring both sides of (2.6) and restricting ${\pi}_{\mathrm{IM}}(0)>0$, (2.6) can equivalently be written as the following quadratic inequality for ${\pi}_{\mathrm{IM}}(0)$:

$$(1-{q}_{\alpha}^{2}{D}_{\mathrm{IM}}^{*\mathrm{T}}\mathrm{\Sigma}{D}_{\mathrm{IM}}^{*}){\pi}_{\mathrm{IM}}^{2}(0)+2{q}_{\alpha}^{2}{D}_{\mathrm{IM}}^{*\mathrm{T}}\mathrm{\Sigma}{D}_{\mathrm{C}}{\pi}_{\mathrm{IM}}(0)-{q}_{\alpha}^{2}{D}_{\mathrm{C}}^{\mathrm{T}}\mathrm{\Sigma}{D}_{\mathrm{C}}\ge 0,$$ | (2.7) |

where we set ${D}_{\mathrm{C}}={D}_{\mathrm{U}}-{D}_{\mathrm{VM}}$ to simplify the notation. ${D}_{\mathrm{C}}\in {\mathbb{R}}^{n}$ represents the deltas of the unsecured portfolio netted with the deltas of the VM portfolio. In the following subsections, we solve inequality (2.7) for ${\pi}_{\mathrm{IM}}(0)$.

### 2.2 Single-risk-factor solution

Inequality (2.7) has a simple solution if the unsecured, VM and IM portfolios are sensitive to only a single risk factor. This section describes the solution for the simple one-dimensional case, which is expanded to the general multidimensional case in the next section. Set $\mathrm{\Sigma}={\sigma}^{2}\in {\mathbb{R}}^{+}$.

- (1)
If $1-{q}_{\alpha}^{2}{\sigma}^{2}{({D}_{\mathrm{IM}}^{*})}^{2}>0$, ie, $$, the IM ${\pi}_{\mathrm{IM}}(0)$ needs to satisfy the following inequality:

$${\pi}_{\mathrm{IM}}(0)\ge \frac{{q}_{\alpha}\sigma |{D}_{\mathrm{C}}|-{q}_{\alpha}^{2}{\sigma}^{2}{D}_{\mathrm{C}}{D}_{\mathrm{IM}}^{*}}{1-{q}_{\alpha}^{2}{\sigma}^{2}{({D}_{\mathrm{IM}}^{*})}^{2}}.$$ (2.8) When ${D}_{\mathrm{IM}}^{*}{D}_{\mathrm{C}}\ge 0$, the above inequality (2.8) can be further simplified as

${\pi}_{\mathrm{IM}}(0)$ $\ge {\displaystyle \frac{{q}_{\alpha}\sigma |{D}_{\mathrm{C}}|}{1+{q}_{\alpha}\sigma |{D}_{\mathrm{IM}}^{*}|}}.$ When $$, (2.8) is reduced to ${\pi}_{\mathrm{IM}}(0)$ $\ge {\displaystyle \frac{{q}_{\alpha}\sigma |{D}_{\mathrm{C}}|}{1-{q}_{\alpha}\sigma |{D}_{\mathrm{IM}}^{*}|}}.$ - (2)
If $$, ie, $|{D}_{\mathrm{IM}}^{*}|>1/({q}_{\alpha}\sigma )$, to ensure inequality (2.7) has positive solutions, we need ${D}_{\mathrm{IM}}^{*}{D}_{\mathrm{C}}>0$. Under this condition, the IM ${\pi}_{\mathrm{IM}}(0)$ should satisfy the following inequality:

$$\frac{{q}_{\alpha}\sigma |{D}_{\mathrm{C}}|-{q}_{\alpha}^{2}{\sigma}^{2}{D}_{\mathrm{C}}{D}_{\mathrm{IM}}^{*}}{1-{q}_{\alpha}^{2}{\sigma}^{2}{({D}_{\mathrm{IM}}^{*})}^{2}}\le {\pi}_{\mathrm{IM}}(0)\le \frac{-{q}_{\alpha}\sigma |{D}_{\mathrm{C}}|-{q}_{\alpha}^{2}{\sigma}^{2}{D}_{\mathrm{C}}{D}_{\mathrm{IM}}^{*}}{1-{q}_{\alpha}^{2}{\sigma}^{2}{({D}_{\mathrm{IM}}^{*})}^{2}},$$ which can be further simplified as

$$\frac{{q}_{\alpha}\sigma |{D}_{\mathrm{C}}|}{1+{q}_{\alpha}\sigma |{D}_{\mathrm{IM}}^{*}|}\le {\pi}_{\mathrm{IM}}(0)\le \frac{{q}_{\alpha}\sigma |{D}_{\mathrm{C}}|}{-1+{q}_{\alpha}\sigma |{D}_{\mathrm{IM}}^{*}|}.$$ The existence of the upper bound is due to the fact that when the collateral in the IM portfolio is volatile (in the sense $|{D}_{\mathrm{IM}}^{*}|>1/({q}_{\alpha}\sigma )$) its value at the end of the closeout period, ${\pi}_{\mathrm{IM}}(\mathrm{\Delta}t)={\pi}_{\mathrm{IM}}(0)(1+{D}_{\mathrm{IM}}^{*}\mathrm{\Delta}{X}_{t})$, has a high probability of being negative due to the linear assumption (2.3). In this case, holding the IM portfolio with its value larger than the upper bound will result in a higher probability of the IM amount not being able to protect the change in the value of the unsecured portfolio, ie, a large value of $\mathbb{P}({D}_{\mathrm{C}}\mathrm{\Delta}{X}_{t}-{\pi}_{\mathrm{IM}}(\mathrm{\Delta}t)\ge 0)$ compared with the required $1-\alpha $ in (2.1).

However, if we constrain the collateral to be positive valued, eg, equities or bonds, the counterintuitive upper bound can be released. It can be shown that, as long as ${\pi}_{\mathrm{IM}}(0)$ is greater than the lower bound in both of the above cases, we have

$$\mathbb{P}({\pi}_{\mathrm{U}}(\mathrm{\Delta}t)-{\pi}_{\mathrm{VM}}(\mathrm{\Delta}t)-{\pi}_{\mathrm{IM}}(\mathrm{\Delta}t)\ge 0\text{and}{\pi}_{\mathrm{IM}}(\mathrm{\Delta}t)0)\le 1-\alpha .$$

We refer the reader to the online appendix for the details of this property.

We note that in both cases the required IM ${\pi}_{\mathrm{IM}}(0)$ is within a range. When considering the exact amount of margin to be posted, we refer to the minimum required IM amount, defined as

$${\pi}_{\mathrm{IM}}=\text{inf}(\{{\pi}_{\mathrm{IM}}(0)\}).$$ |

For each of the above two cases, the minimum requirement of the IM ${\pi}_{\mathrm{IM}}$ is

$$ |

### 2.3 General solution

To analyze the quadratic inequality (2.7) in a multifactor case, we need to introduce a geometric structure to the problem. This is motivated by the fact that the covariance matrix $\mathrm{\Sigma}$ is assumed to be positive definite. Thus, we can consider the problem in an inner product space $({\mathbb{R}}^{n},\u27e8\cdot ,\cdot \u27e9)$, with the associated inner product defined as $\u27e8x,y\u27e9={x}^{\mathrm{T}}\mathrm{\Sigma}y$.

For ease of notation, we define the following:

$$\begin{array}{cc}\hfill {D}_{\mathrm{C}}^{\mathrm{T}}\mathrm{\Sigma}{D}_{\mathrm{C}}& ={\parallel a\parallel}^{2},\hfill \\ \hfill {D}_{\mathrm{IM}}^{*\mathrm{T}}\mathrm{\Sigma}{D}_{\mathrm{IM}}^{*}& ={\parallel b\parallel}^{2},\hfill \\ \hfill {D}_{\mathrm{IM}}^{*\mathrm{T}}\mathrm{\Sigma}{D}_{\mathrm{C}}& =\u27e8a,b\u27e9=\parallel a\parallel \parallel b\parallel \mathrm{cos}\theta ,\theta \in [0,\pi ],\hfill \\ \hfill {q}_{\alpha}& =q.\hfill \end{array}\}$$ | (2.9) |

Then inequality (2.7) can be written as

$$(1-{q}^{2}{\parallel b\parallel}^{2}){\pi}_{\mathrm{IM}}^{2}(0)+2{q}^{2}\u27e8a,b\u27e9{\pi}_{\mathrm{IM}}(0)-{q}^{2}{\parallel a\parallel}^{2}\ge 0.$$ | (2.10) |

- (1)
We first consider the case when the leading coefficient of the quadratic inequality $1-{q}^{2}{\parallel b\parallel}^{2}>0$. Note that, given $q$, according to a confidence level $\alpha $, the magnitude $q\parallel b\parallel $ describes how risky the unit value IM portfolio is. As the condition $1-{q}^{2}{\parallel b\parallel}^{2}>0$ implies that the IM portfolio’s VaR is less than the IM portfolio value itself, ie, $$, we may infer that the unit value IM portfolio is not volatile. The solution to (2.7) is

$${\pi}_{\mathrm{IM}}(0)\ge \frac{-{q}^{2}\u27e8a,b\u27e9+\sqrt{{q}^{4}{\u27e8a,b\u27e9}^{2}+(1-{q}^{2}{\parallel b\parallel}^{2}){q}^{2}{\parallel a\parallel}^{2}}}{1-{q}^{2}{\parallel b\parallel}^{2}}.$$ (2.11) This can easily be derived by noting that the determinant is positive and that the corresponding quadratic equation must have one positive root and one negative root. We discard the degenerate solution of ${\pi}_{\mathrm{IM}}(0)$ less than or equal to the negative root.

- (2)
Next, we consider the case when the leading coefficient $$. This implies the unit value IM portfolio is volatile. Note that ${\pi}_{\mathrm{IM}}(0)$ has to be positive, as indicated by (2.6). To ensure (2.7) has a positive solution, we further require

$${q}^{4}{\u27e8a,b\u27e9}^{2}+(1-{q}^{2}{\parallel b\parallel}^{2}){q}^{2}{\parallel a\parallel}^{2}\ge 0$$ (2.12) and

$$\u27e8a,b\u27e9>0.$$ (2.13) Condition (2.12) guarantees that inequality (2.7) has real roots of the same sign, which implies that the IM portfolio cannot be too volatile in the sense that

$$ (2.14) The second inequality in (2.14) can be further understood as

${q}^{2}{\parallel b\parallel}^{2}\le 1+{\displaystyle \frac{{q}^{2}{\u27e8a,b\u27e9}^{2}}{{\parallel a\parallel}^{2}}}$ $\iff {q}^{2}{\parallel a\parallel}^{2}{\parallel b\parallel}^{2}(1-{\mathrm{cos}}^{2}\theta )\le {\parallel a\parallel}^{2}$ $\iff {\mathrm{sin}}^{2}\theta \le {\displaystyle \frac{1}{{q}^{2}{\parallel b\parallel}^{2}}}$ $$ So, this condition is a restriction on the angle between $a$ and $b$ rather than a restriction on the magnitude of $b$. It describes how much the value of the IM portfolio moves in the same direction as the unsecured portfolio. If (2.12) does not hold, the IM portfolio itself is too risky and we cannot offset the potential loss of the entire portfolio.

Condition (2.13) results from the fact that the two roots of inequality (2.7) must be positive. It indicates that the IM portfolio has to move in the same direction as the remainder of the unsecured portfolio that is not offset by VM.

With conditions (2.12) and (2.13), the solution to (2.6) is

$${x}_{1}\le {\pi}_{\mathrm{IM}}(0)\le {x}_{2},$$ (2.15) where

${x}_{1}$ $={\displaystyle \frac{-{q}^{2}\u27e8a,b\u27e9+\sqrt{{q}^{4}{\u27e8a,b\u27e9}^{2}+(1-{q}^{2}{\parallel b\parallel}^{2}){q}^{2}{\parallel a\parallel}^{2}}}{1-{q}^{2}{\parallel b\parallel}^{2}}},$ ${x}_{2}$ $={\displaystyle \frac{-{q}^{2}\u27e8a,b\u27e9-\sqrt{{q}^{4}{\u27e8a,b\u27e9}^{2}+(1-{q}^{2}{\parallel b\parallel}^{2}){q}^{2}{\parallel a\parallel}^{2}}}{1-{q}^{2}{\parallel b\parallel}^{2}}}.$ We note that the distance between the lower and upper bounds, $|{x}_{1}-{x}_{2}|$, is

$$|{x}_{1}-{x}_{2}|=\frac{2q\parallel a\parallel \sqrt{1-{q}^{2}{\parallel b\parallel}^{2}{\mathrm{sin}}^{2}\theta}}{{q}^{2}{\parallel b\parallel}^{2}-1}.$$ It can be seen that this distance shrinks when $\theta $ approaches $\mathrm{arcsin}(1/q\parallel b\parallel )$ from below. When $q\parallel b\parallel \to \mathrm{\infty}$, this distance approaches zero. This is an undesired feature that implies the effective IM range actually vanishes as the IM assets become increasingly risky.

Similar to the single-risk-factor case, the existence of upper bound ${x}_{2}$ appears counterintuitive. It can be understood that since the collateral received as IM is volatile, the positions in these volatile assets cannot be too large. Otherwise, IM’s ability to provide loss coverage will be mitigated or even eliminated by its own loss. However, if we assume the value of collateral asset $i$ in the IM portfolio is always positive, then we can release the upper bound ${x}_{{2}_{i}}$ implied by ${x}_{2}$ through the invariant composition ${\beta}^{\mathrm{T}}/\parallel \beta \parallel $. We refer the reader to the online appendix for the details of this property.

- (3)
Finally, if the leading coefficient $1-{q}^{2}{\parallel b\parallel}^{2}=0$, the quadratic inequality becomes a linear inequality:

$$2{q}^{2}\u27e8a,b\u27e9{\pi}_{\mathrm{IM}}(0)-{q}^{2}{\parallel a\parallel}^{2}\ge 0.$$ The solution is

$$\{\begin{array}{cc}\hfill {\pi}_{\mathrm{IM}}(0)\ge \frac{1}{2}\frac{{\parallel a\parallel}^{2}}{\u27e8a,b\u27e9}& \text{if}\u27e8a,b\u27e90,\hfill \\ \hfill \text{no positive solution}& \text{if}\u27e8a,b\u27e9\le 0.\hfill \end{array}$$ It can be shown that the solution of the above linear inequality is consistent with the limiting case of both scenarios (1) and (2).

## 3 Comparison of initial margin with risky and nonrisky collateral

In this section, we investigate the conditions under which the IM amount with risky collateral is less than the IM amount with nonrisky assets (eg, cash and cash equivalents). Specifically, let ${\pi}_{\mathrm{IM}}^{\mathrm{R}}$ denote the value of the IM portfolio using risky collateral, and let ${\pi}_{\mathrm{IM}}^{\mathrm{NR}}$ denote the value of the IM portfolio using nonrisky collateral. We identify scenarios under which $$.

### 3.1 Single risk factor

We start from a simple case with only one risk factor. If the collateral is nonrisky, the minimum requirement of the IM is

$${\pi}_{\mathrm{IM}}^{\mathrm{NR}}={q}_{\alpha}\sigma |{D}_{\mathrm{C}}|.$$ |

If the collateral is risky, then, as discussed in Section 2.2, the minimum requirement of the IM is

$$ |

It is clear that, when the unsecured portfolio and the risky collateral are comoving (ie, if ${D}_{\mathrm{IM}}^{*}{D}_{\mathrm{C}}\ge 0$), ${\pi}_{\mathrm{IM}}^{\mathrm{R}}\le {\pi}_{\mathrm{IM}}^{\mathrm{NR}}$. Thus, we have the following theorem.

###### Theorem 3.1.

As long as the solution of ${\pi}_{\mathrm{IM}}^{\mathrm{R}}$ exists, the required IM using risky collateral ${\pi}_{\mathrm{IM}}^{\mathrm{R}}$ will be less than that using nonrisky assets ${\pi}_{\mathrm{IM}}^{\mathrm{NR}}$ if the unsecured portfolio and the risky collateral are comoving.

The economic intuition of Theorem 3.1 is not surprising, since IM is intended to cover the potential MTM increase of the OTC transaction during the closeout period following a counterparty default. When holding a certain security as IM, it will be favorable to the collateral holder that when the MTM value of the unsecured transaction gets larger, the collateral value increases synchronously. The increase in the IM provides extra compensation for the enlarged credit loss exposure. Therefore, the comoving risky collateral is able to provide greater protection than an equal value of cash collateral. As a consequence, less comoving risky collateral is required to provide the same level of protective power, which is prescribed by the regulator as ten-day 99% VaR. We can consider an extreme case as an example. Suppose the unsecured portfolio has a market value of 10 dollars but the portfolio is volatile with a ten-day 99% VaR of 100 dollars. Therefore, the cost of posting cash collateral as IM is 100 dollars. However, if there exists a security that replicates the unsecured portfolio, and we use this security as IM, it can completely cover the loss when the counterparty defaults. The cost of posting such IM is then only 10 dollars, which is much lower than the cost of cash IM. The same idea can be applied to other risky collateral that has a small market value but is highly correlated with the unsecured portfolio.

### 3.2 Multiple risk factors

Note that, in either of the cases described in Section 2.3, the minimum amount of IM with risky collateral is the positive root of the quadratic equation (2.7), that is,

$${\pi}_{\mathrm{IM}}^{\mathrm{R}}=\frac{-{q}^{2}\u27e8a,b\u27e9+\sqrt{{q}^{4}{\u27e8a,b\u27e9}^{2}+(1-{q}^{2}{\parallel b\parallel}^{2}){q}^{2}{\parallel a\parallel}^{2}}}{1-{q}^{2}{\parallel b\parallel}^{2}}.$$ | (3.1) |

Having said that, the minimal amount of IM with nonrisky collateral is

$${\pi}_{\mathrm{IM}}^{\mathrm{NR}}=q\parallel a\parallel .$$ | (3.2) |

Theorem 3.2 describes the scenarios under which the IM amount using risky collateral is less than that using nonrisky assets. The proof is given in the online appendix.

###### Theorem 3.2.

Let ${\pi}_{\mathrm{IM}}^{\mathrm{R}}$ be the minimal amount of IM with risky collateral, and ${\pi}_{\mathrm{IM}}^{\mathrm{NR}}$ be the minimal amount of IM with nonrisky collateral. Depending on the volatility level $q\mathit{}\mathrm{\parallel}b\mathrm{\parallel}$ of the IM portfolio, the comparison between ${\pi}_{\mathrm{IM}}^{\mathrm{R}}$ and ${\pi}_{\mathrm{IM}}^{\mathrm{NR}}$ can be summarized as follows.

- (1)
When $q\parallel b\parallel \le 1$ ,

$$ - (2)
When $$ ,

$$ - (3)
When $q\parallel b\parallel \ge \sqrt{2}$ ,

$$\theta \in \{\begin{array}{cc}[0,\mathrm{arcsin}\left(\frac{1}{q\parallel b\parallel}\right)]\hfill & \iff {\pi}_{\mathrm{IM}}^{\mathrm{R}}\le {\pi}_{\mathrm{IM}}^{\mathrm{NR}},\hfill \\ (\mathrm{arcsin}\left(\frac{1}{q\parallel b\parallel}\right),\pi ]\hfill & \iff \mathit{\text{no solution to}}{\pi}_{\mathrm{IM}}^{\mathrm{R}}.\hfill \end{array}$$

###### Remark 3.3.

Theorem 3.2 can be illustrated by Figure 1, in which the two shaded areas depict the regions where $$ and ${\pi}_{\mathrm{IM}}^{\mathrm{R}}>{\pi}_{\mathrm{IM}}^{\mathrm{NR}}$. A further observation is that the curves in Figure 1 have first-order smoothness when connecting critical points $A$ and $B$. This can be shown by straightforward calculus.

As discussed in Theorem 3.1, when there is only one risk factor, the condition for $$ is that the collateral and the unsecured portfolio are comoving, ie, ${D}_{\mathrm{C}}{D}_{\mathrm{IM}}^{*}>0$. This condition cannot be easily extended to the multifactor case without introducing the angle $\theta $ between collateral and unsecured portfolio in a geometric space.

In the single-factor case, $\theta $ is degenerated to only two cases, $\theta =0,\pi $, and the comoving condition ${D}_{\mathrm{C}}{D}_{\mathrm{IM}}^{*}>0$ is equivalent to $\theta =0$. It is natural to extend the comoving definition to the multifactor case as $\theta \in [0,\pi /2)$ and expect that a similar result to Theorem 3.1 holds, ie, if $\theta \in [0,\pi /2)$, then ${\pi}_{\mathrm{IM}}^{\mathrm{R}}\le {\pi}_{\mathrm{IM}}^{\mathrm{NR}}$. However, as shown in Theorem 3.2 and Figure 1, the comoving condition needs to be defined more stringently as $\theta \in [0,\mathrm{arccos}(q\parallel b\parallel /2))$. This range is narrower than $[0,\pi /2)$ since the nontrivial boundary $\mathrm{arccos}(q\parallel b\parallel /2)$ is less than $\pi /2$. It should also be noted that the boundary depends on the volatility of the IM portfolio and finally decreases to $\pi /4$ when $q\parallel b\parallel =\sqrt{2}$.

The economic intuition behind this mathematically derived result is not as straightforward as the single-factor case. The angle $\theta $ between the risky collateral and the unsecured portfolio indicates the level of comoving. As this angle moves from 0 to the boundary $\mathrm{arccos}(q\parallel b\parallel /2)$, the comoving condition weakens. Eventually, the comoving condition does not hold when the angle is larger than $\mathrm{arccos}(q\parallel b\parallel /2)$. As a result, the risky collateral leads to a smaller IM amount when it is highly comoving with the unsecured portfolio; then, the IM amount gets increasingly larger as the comoving condition gets weaker, and finally the risky assets are incapable of serving as collateral for IM purposes when the angle moves toward $\pi $.

## 4 Numerical examples and applications

In this section, we give several numerical examples to further illustrate the properties and results derived in the previous sections.

### 4.1 Setup of risky assets

Risk factor (RF) | RF attributes |
---|---|

Stock price ($) | 100 |

Stock lognormal volatility (%) | 30 |

Interest rate (%) | 2 |

Interest rate volatility (%) | 20 |

Equity/IR correlation (%) | 10 |

Asset | Asset attributes |

Stock | |

European equity call option | ATM, $\text{strike}=\text{\$100}$, $\text{maturity}=\text{0.25Y}$, $\text{LNV}=\text{30}$% |

European equity put option | ATM, $\text{strike}=\text{\$100}$, $\text{maturity}=\text{0.25Y}$, $\text{LNV}=\text{30}\%$ |

Payer swap (pay) | $\text{MTM}=\text{\$10\hspace{0.17em}000}$, $\text{DV01}=\text{\$5000}$ |

Receiver swap (rec) | $\text{MTM}=\text{\$10\hspace{0.17em}000}$, $\text{DV01}=-\text{\$5000}$ |

Bond | $\text{MTM}=\text{\$1000}$, $\text{duration}=\text{15Y}$ |

Cash |

For illustrative purposes, we assume that there are two asset classes to be considered in the problem: equity and interest rate (IR). The hypothetical assets that can be included in the unsecured portfolio or IM portfolio are an equity, a European equity call option, a European equity put option, an IR payer swap, an IR receiver swap, a bond and cash. The assumed values for the attributes of these risk factors and assets are listed in Table 1.

For simplicity, we assume that the stock price follows a lognormal distribution and that the options are ATM options. The swap has two defined attributes, MTM and DV01, and the bond has two similar attributes, MTM and duration. The key quantity we investigated was the ratio of IM to the unsecured portfolio value, ie,

$$\text{initial margin ratio (IMR)}=\frac{{\pi}_{\mathrm{IM}}}{{\pi}_{\mathrm{U}}(0)}.$$ |

### 4.2 Scenario analyses

#### Scenario 1

We first examine a scenario where there is only one asset in the unsecured portfolio (UP) and one asset in the IM portfolio (IM). We paired all possible combinations of assets, picking one as the unsecured portfolio asset and the other as the collateral asset in the IM portfolio. We note that some combinations are not practical, such as those using call, put and swaps as collateral. However, the aim of these numerical examples is not to show the immediate practical application but to illustrate the mathematical result and demonstrate it is consistent with business intuitions. Therefore, we provide a full combination of test cases using different assets as unsecured transaction and collateral with the intention of covering all scenarios discussed in Theorem 3.2.

Unsecured portfolio | ||||||||
---|---|---|---|---|---|---|---|---|

IM | Stock | Call | Put | Pay | Rec | Bond | Cash | |

Stock | 12.21 | 106.64 | 129.03 | 92.29 | 94.92 | 2.85 | 0.00 | |

Call | 6.28 | 54.85 | NA | NA | NA | NA | 0.00 | |

Put | NA | NA | 52.63 | NA | NA | NA | 0.00 | |

Pay | 29.00 | 253.37 | 377.80 | 48.10 | 1266.85 | 38.01 | 0.00 | |

Rec | 47.28 | 413.10 | 231.72 | 1266.85 | 48.10 | 1.44 | 0.00 | |

Bond | 13.95 | 121.86 | 110.82 | 95.33 | 90.18 | 2.71 | 0.00 | |

Cash | 13.90 | 121.47 | 111.09 | 92.68 | 92.68 | 2.78 | 0.00 |

The results are presented in Table 2. The columns and rows represent the assets in the unsecured and IM portfolios, respectively. The percentage is expressed as the required IM amount relative to the unsecured portfolio value. For instance, in row 1, column 2, the unsecured portfolio consists of a call option and the IM portfolio consists of its underlying equity. Here the required IM amount is 106.64% of the unsecured portfolio value.

Cash can be treated as the standard benchmark collateral, and we can identify what is “good” and “bad” collateral for each unsecured portfolio in Table 2. For example, assume the unsecured portfolio is a call option. Then, the required IM amount is 121.47% of the unsecured portfolio value if cash is selected as collateral, while the required IM amount is only 106.64% of the unsecured portfolio value if an equity is selected as collateral. However, if the unsecured portfolio is a put option, then additional collateral is required if IM is equity as opposed to cash. These results indicate that using equity as IM is more efficient if the unsecured portfolio is a call option and less efficient if the unsecured portfolio is a put option. This observation is consistent with our theorem that, if the two assets move in the same direction, then the IM amount with risky collateral is less than the IM amount for nonrisky collateral.

The following additional observations, consistent with our expectations, are apparent from the results in Table 2.

- •
For entries with NA in the table, the IM portfolio is too risky to offset the unsecured portfolio. For example, the ATM call and put options are both too risky to offset each other.

- •
Due to the low correlation between the equity and IR risk factors, the IM amounts for unsecured portfolios with IR assets (payer swap, receiver swap or bond) are similar no matter whether an equity or cash is used as the collateral in the IM portfolio. Similarly, the IM amounts for unsecured portfolios with equity assets (equity, equity call option and equity put option) are similar no matter whether a bond or cash is used as the collateral in the IM portfolio.

#### Scenario 2

The unsecured portfolio in this scenario consists of a long equity position. The collateral to be considered for IM are an equity, an equity call option and cash. In this scenario, we examine the impact of volatility on the IM amount requirement.

Figure 2 presents the required amount of IM for an unsecured portfolio consisting of a long equity position across a wide range of equity volatilities. The amount of nonrisky collateral required as IM increases linearly with respect to increases in volatility. The amount of risky collateral required as IM is less than the required amount of nonrisky collateral because the collateral value moves in the same direction as the unsecured portfolio value. One interesting observation is that the required IM amount is initially less if the call option is selected as collateral than the IM amount if the equity is selected as collateral, but, as volatility increases, the two IM amounts converge. This observation can be attributed to the fact that the value of a call option converges to the value of the underlying stock as volatility increases.

#### Scenario 3

Scenario 2 was a one-risk-factor scenario, where both the unsecured portfolio and the IM portfolio shared the same risk factor. In this scenario, we consider a case where the unsecured portfolio is sensitive to one risk factor, while the IM portfolio is sensitive to a different risk factor.

To be specific, we select the payer swap as our unsecured portfolio and a combination of an equity call option and an equity as the collateral in the IM portfolio. We address the case where the value of the collateral assets moves in the same direction as the unsecured portfolio ($\u27e8a,b\u27e9>0$).

The call option comprises 85% of the value of collateral in the IM portfolio, and the equity comprises the remaining 15%, with its volatility set at 30%. This corresponds to ${q}^{2}{\parallel b\parallel}^{2}=1.1096$ and falls into scenario (2) of Theorem 3.2. Figure 3 shows the required amount of collateral with respect to the correlation between the equity stock price and the interest rate risk factor.

The results demonstrate that when the correlation is less than 30% the collateral cannot serve the purpose of IM requirement. This corresponds to the case where $\theta \in (\mathrm{arcsin}(1/q\parallel b\parallel ),\pi /2)$, and therefore there is no solution to ${\pi}_{\mathrm{IM}}^{\mathrm{R}}$.

The risky collateral can be used for IM purposes when the correlation is greater than 30%. However, the risky collateral is inefficient (a greater amount is required than of nonrisky collateral) when the correlation is less than 54%. As the correlation increases, the gap between these two decreases gradually and the risky collateral becomes more efficient than the nonrisky collateral for correlations greater than 54%. This three-phase result indicates that, consistent with our Theorem 3.2, correlation plays a critical role in the selection of risky collateral for IM requirements.

## 5 Summary

In this paper, we explored the complication of calculating IM requirements in a parametric VaR framework when the collateral consists of risky assets. We showed that the required IM amounts with risky collateral can be found by solving a quadratic inequality (see (2.7)). Two IM ranges, (2.11) and (2.15), exist according to the risk level of the IM portfolio. We also compared the IM requirements calculated using risky and nonrisky collateral. The main theorems in Section 3 provide the conditions under which the amount of risky collateral required for IM would be less than the amount of nonrisky collateral. These conditions were described using the geometric structure of an inner product space generated by the covariance matrix of the market risk factors of the unsecured portfolio and the IM portfolio.

We note that the linear assumption of the value of the IM portfolio in (2.3) allows the collateral value to be negative. This is not a realistic assumption for assets such as equities or long equity options. A future enhancement of the work in this paper could be to extend the problem to a nonnegative distribution assumption for positive-valued collateral assets.

We also note that the normal assumption in (2.2) is not ideal, especially for assets whose returns exhibit high skewness and leptokurtosis. More general fat-tailed distributions can be investigated.

## Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper. The information and views set out in this paper are those of the authors and do not necessarily reflect the official opinion of Ernst & Young LLP.

## Acknowledgements

The authors thank Qingji Yang, Young Wang, Scott Underberg, Seha Islam and Sotirios Malliaros for valuable comments and suggestions.

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