Revising SA-CCR

TL;DR

This paper enhances the SA-CCR framework by introducing RSA-CCR, which employs cashflow decomposition within a 3-Factor Gaussian Market Model to improve its self-consistency and risk sensitivity.

Contribution

It presents a novel revision of SA-CCR, making it more accurate and theoretically sound through advanced modeling techniques.

Findings

RSA-CCR improves risk sensitivity over traditional SA-CCR

The method achieves better self-consistency in risk measurement

Empirical tests show enhanced accuracy in risk estimation

Abstract

From SA-CCR to RSA-CCR: making SA-CCR self-consistent and appropriately risk-sensitive by cashflow decomposition in a 3-Factor Gaussian Market Model

Tables

Table 1. Table 2.1: Issues with SA-CCR. Major issues are those that potentially give rise to materially incorrect capital magnitude. Medium issues potentially give rise to material mis-assignment of risk netting, e.g. curve risk being assigned to basis risk.

Impact	Issue	Trades affected	Source
major	lack of self-consistency	linear	trades not decomposed to cashflows
major	lack of appropriate risk sensitivity	linear	trades not decomposed to cashflows
major	dependence on economically-equivalent confirmations	linear	trades not decomposed to cashflows
medium	ambiguity of risk assignment	linear	trades not decomposed to cashflows, and no explicit model
medium	lack of clear extensibility	non-vanilla	no explicit model
medium	lack of parameter transparency	linear	parameter not given in model terms

Table 2. Table 2.2: Notation. Horizontal lines separate notation used in different sections.

Symbol	Meaning
EAD	exposure at default
$\alpha$	regulatory multiplier, set at $1.4$
EEPE	effective expected positive exposure (this is the IMM-CCR EEPE)
RC	replacement cost
PFE	potential future exposure (over one year)
$C$	haircut value of net collateral held calculated using NICA methodology
TH	threshold
MTA	minimum transfer amount
NICA	net independent collateral amount
floor	regulatory floors, e.g. set at 5% in Equation 2.1
${\text{MF}}_i$	maturity factor for trade $i$
$d_i^{\left(a\right)}$	adjusted notional for trade $i$ w.r.t. asset class $a$
${\text{SF}}_i^{\left(a\right)}$	supervisory factor for trade $i$ w.r.t. asset class $a$
MPOR	margin period of risk
$r_t$	short rate
$a$	mean reversion speed
${\varphi }_t$	deterministic mean reversion target
$\sigma$	volatility
$W_t$	Brownian motion
$P(t,T)$	price at $t$ of zero coupon bond with maturity $T$
${\delta }_{\tau }$	day count fraction of underlying money market index

Table 3. Table 2.3: Equivalent derivatives positions with different add-ons. ATM swap is 10Y USD, 100M notional. Split means a 3Y swap and a separate forward starting swap (starting at 3Y for 7Y) with the same fixed rate as the ATM 10Y swap.

Instrument	SA-CCR add-on
ATM Swap	3,934,693
FRA replication	3,433,691
Split at 3Y	3,654,794

Table 4. Table 2.4: Material add-on for zero economic position. Position is ATM swap 10Y USD 100M notional hedged by strip of FRAs.

Instrument	SA-CCR add-on
ATM net of FRA replication	1,646,936

Table 5. Table 2.5: EEPE versus moneyness and direction calculated using one factor Hull-White model calibrated to SA-CCR mean reversion 0.05; flat volatility 0.0189. Under SA-CCR the top five trades have the same add-on, as to the bottom five. ATM swap is 10Y USD. All instruments have USD 100M notional

Instrument	Pay	Receive
ATM+500bps	5,778,184	3,898,079
ATM+100bps	4,271,792	3,707,917
ATM	3,903,699	3,668,881
ATM-100bps	3,539,968	3,634,569
ATM-500bps	2,166,837	3,577,668
Fwd Start ATM 1y-10y +500bps	4,712,043	4,891,493
Fwd Start ATM 1y-10y +100bps	3,932,090	3,931,231
Fwd Start ATM 1y-10y	3,737,350	3,691,418
Fwd Start ATM 1y-10y -100bps	3,542,763	3,451,756
Fwd Start ATM 1y-10y -500bps	2,766,729	2,495,425

Table 6. Table 3.1: Correlation structure of SA-CCR interest rates add-ons by instrument maturity T 𝑇 T , where ρ 1 = 0.7 subscript 𝜌 1 0.7 \rho_{1}=0.7 and ρ 2 = 0.3 subscript 𝜌 2 0.3 \rho_{2}=0.3 .

			$5\le T$
	1	${\rho }_1$	${\rho }_2$
	${\rho }_1$	1	${\rho }_1$
$5\le T$	${\rho }_2$	${\rho }_1$	1

Table 7. Table 3.2 : Add-ons components for the elementary cashflows listed in Definition 3.2 . Hedging sets for rates cashflows are Rates and Basis per currency. Hedging sets for inflation cashflows are inflation per currency. Derivations of these values follow the Floating example in Section 4 and details are given in the accompanying Technical appendix, including Floating and CMS examples in full.

Cashflow	Hedging set	Maturity bucket	Effective Notional	Start	End	Delta
Floating	Rates, Basis	$M(T_f$)	$\left(N+CF\left(T\right)\right)P(0,T)$	$0$	$T_f$	$-1$
		$M(T_f+\tau$)	$\left(N+CF\left(T\right)\right)P(0,T)$	$0$	$T_f+\tau$	$+1$
Compound	Rates , Basis	$M\left(\max ({\tau }_c,T_s)\right)$	$CF\left(T\right)P(0,T)$	0	$\max ({\tau }_c,T_s)$	$-1$
		$M\left(T_e\right)$	$CF\left(T\right)P(0,T)$	$0$	$T_e$	$+1$
CMS	Rates, Basis	$M_1$	$\frac{{\delta }_{\tau }}{\delta (T_s,T_e)}\left(N+CF\left(T\right)\right)P(0,T)$	$\min (1,T_s)$	$\min (1,T_e)$	$-1$
		$M_2$	$\frac{{\delta }_{\tau }}{\delta (T_s,T_e)}\left(N+CF\left(T\right)\right)P(0,T)$	$\min (\max (1,T_s),5)$	$\min (\max (1,T_e),5)$	$+1$
		$M_3$	$\frac{{\delta }_{\tau }}{\delta (T_s,T_e)}\left(N+CF\left(T\right)\right)P(0,T)$	$\max (5,T_s)$	$\max (5,T_e)$	$+1$
Inflation Floating	Inflation	$M\left(\max ({\tau }_I,T_s)\right)$	$\left(N+CF\left(T\right)\right)P(0,T)$	$0$	$M\left(\max ({\tau }_I,T_s)\right)$	$-1$
		$T_e$	$\left(N+CF\left(T\right)\right)P(0,T)$	$0$	$T_e$	$+1$
Inflation Compound	Inflation	$M\left(\max ({\tau }_I,T_s)\right)$	$CF\left(T\right)P(0,T)$	$0$	$M\left(\max ({\tau }_I,T_s)\right)$	$-1$
		$T_e$	$CF\left(T\right)P(0,T)$	$0$	$T_e$	$+1$

Table 8. Table 5.1: Vanilla Swaps with varying moneyness, percentage differences from Gaussian Market Model in last column. The ATM swap is referred to below as AS.

Instrument	SA-CCR	RSA-CCR with no discounting	RSA-CCR with market discount	HW1F Average Pay/Receive Add-ons	Shifted LMM 3F Average Pay/Receive Add-on
ATM Swap	4%	3%	-5%	0%	3,783,285
ATM+100 bps	-1%	3%	-5%	0%	3,984,416
ATM+500bps	-19%	1%	-6%	0%	4,829,860
ATM-100bps	10%	3%	-4%	-0%	3,588,130
ATM-500bps	37%	1%	-6%	-0%	2,875,001

Table 9. Table 5.2: Vanilla Swaps with different replications, percentage differences from HW1F in last column.

Instrument	SA-CCR	RSA-CCR with no discounting	RSA-CCR with market discount	HW1F Average Pay/Receive Add-ons	Shifted LMM 3F Average Pay/Receive Add-on
ATM Swap	4%	3%	-5%	0%	3,783,285
FRA Replication	-9%	3%	-5%	0%	3,783,285
ATM – FRAs	inf%	0%	0%	0%	0
split at 3Y	-3%	3%	-5%	0%	3,783,285

Table 10. Table 5.3: Amortising Swaps, percentage differences from Gaussian Market Model in last column.

Instrument	SA-CCR	Full approach with discount equal 1	RSA-CCR with market discount	HW1F Average Pay/Receive Add-ons	Shifted LMM 3F Average Pay/Receive Add-on
Amortising 10Y:5Y/100M;5Y/200M	-2%	1%	-5%	-2%	5,996,321
FRA replication	-8%	1%	-5%	-2%	5,996,321
5Y/200M and forward start 5Y5Y/100M	-4%	1%	-5%	-2%	5,996,321
10Y/100M and 5Y/100M	-5%	1%	-5%	-2%	5,996,321
Amortising minus 10Y/150M	-100%	-29%	-17%	-25%	352,986

Table 11. Table 5.4: Zero Coupon Swaps, percentage differences from Gaussian Market Model in last column.

Instrument	SA-CCR	Full approach with discount equal 1	RSA-CCR with market discount	HW1F Average Pay/Receive Add-ons	Shifted LMM 3F Average Pay/Receive Add-on
Zero Coupon ATM 10y	-2%	15%	-3%	-1%	3,999,402
Zero Coupon ATM+100bps 10 yr	-9%	15%	-3%	-1%	4,341,650
Zero Coupon ATM+500bps 10 yr	-27%	15%	-2%	-3%	5,368,783
Zero Coupon ATM-100bps 10 yr	8%	15%	-3%	-1%	3,657,255
Zero Coupon ATM-500bps 10 yr	72%	14%	-3%	-4%	2,290,554

Table 12. Table 5.5: Forward Staring Swaps, percentage differences from Gaussian Market Model in last column.

Instrument	SA-CCR	RSA-CCR with no discounting	RSA-CCR with market discount	HW1F Average Pay/Receive Add-ons	Shifted LMM 3F Average Pay/Receive Add-on
Fwd Start ATM 1y-10y	2%	6%	-3%	1%	3,668,070
Fwd Start ATM 1y-10y +100bps	-4%	6%	-3%	1%	3,888,496
Fwd Start ATM 1y-10y +500bps	-22%	7%	-3%	1%	4,772,075
Fwd Start ATM 1y-10y -100bps	9%	6%	-3%	1%	3,447,875
Fwd Start ATM 1y-10y -500bps	46%	6%	-2%	2%	2,570,157

Equations

$$\text{EAD}=\alpha \text{EEPE}=\alpha (\text{RC}+\text{PFE})$$

$$\text{RC}=\max (V-C,\text{TH}+\text{MTA}-\text{NICA},0)$$

$${\text{PFE}}_{\text{SA-CCR}}:=$$

$$f(x):=$$

$${\text{AddOn}}_i^{\text{trade}}={\delta }_i{\text{SF}}_i^{(a)}{d}_i^{(a)}{\text{MF}}_i$$

$$d_i^{(\text{IR})}$$

$${\text{SD}}_i$$

$${\text{MF}}_i^{\text{unmargined}}:=$$

$${\text{MF}}^{\text{margined}}:=$$

$$\sum _{i=3,4}{\delta }_i{d}_i^{(\text{IR})}=$$

$$=$$

$${\text{AddOn}}_V(T)=\frac{1}{T}{\int }_0^T{\mathbb{E}}^{}\left[{\left(V(t)-V(0)\right)}^{+}\right]dt$$

$${\text{AddOn}}_V(T)=\frac{2}{3}\sigma \sqrt{\frac{T}{2\pi }}$$

$${\text{AddOn}}^{\text{trade}}=\delta N\cdot S{F}^{IR}\frac{e^{-0.05S}-e^{-0.05E}}{0.05}\text{MF}$$

$${\sigma }_{\text{HW}}:=\frac{3}{2}\sqrt{2\pi }S{F}^{IR}$$

$$d{r}_t=a({\varphi }_t-r_t)dt+{\sigma }_{HW}d{W}_t$$

$${\sigma }_{\text{ATM}}(0)=N\frac{{\sigma }_{HW}}{a}(e^{-a{T}_s}-e^{-a{T}_e})$$

$${\text{AddOn}}^{\text{trade}}$$

$$=N\cdot S{F}^{IR}\frac{e^{-a{T}_s}-e^{-a{T}_e}}{a}$$

$$dP(t,T)=r_{t}P(t,T)dt-\frac{\sigma }{a}(1-e^{-a(T-t)})P(t,T)d{W}_t$$

$$V(t)$$

$$=N\sum _{i=s+1}^{e}P(t,T_i){\delta }_{i}L(t,T_{i-1},T_i)-N\sum _{i=s+1}^{e}P(t,T_i){\delta }_{i}R$$

$${\sigma }_V(t)={\sigma }_{\text{ATM}}(t)+{\sigma }_{\text{Float}}(t)+{\sigma }_{\text{Fixed}}(t)$$

$${\sigma }_{\text{ATM}}(t)=$$

$${\sigma }_{\text{Float}}(t)=$$

$${\sigma }_{\text{Fixed}}(t)=$$

$${\sigma }_V(t)=N\sum _{i=s+1}^{e}P(t,T_i)\frac{\sigma }{a}(e^{-a(T_{i-1}-t)}-e^{-a(T_i-t)})={\sigma }_{\text{ATM}}(t)$$

$${\sigma }_{\text{ATM}}(t)=N\frac{\sigma }{a}(e^{-a(T_s-t)}-e^{-a(T_e-t)})$$

$${\sigma }_V(t)$$

$$+N\sum _{i=s+1}^e{\delta }_{i}P(0,T_i)\frac{\sigma }{a}(1-e^{-a{T}_i})\left(R-L(0,T_{i-1},T_i)\right)$$

$$dP(t,T)=r_{t}P(t,T)\cdot dt-\frac{\sigma }{a}(1-e^{-a\cdot (T-t)})\cdot P(t,T)\cdot d{Z}_t^{M(T)}$$

$${\delta }_{\tau }NL(T_f,T_f+{\delta }_{\tau })$$

$${\delta }_{\tau }N{S}_{\tau }(T_s,T_e)$$

$$N\left(\frac{I(T_e)}{I(T_s)}-1\right)$$

$$N\frac{I(T_e)}{I(T_s)}$$

Full text

1. 1 Introduction

2. 2 Issues with SA-CCR

3. 2.1 Reminder on SA-CCR structure

4. 2.2 Lack of self-consistency for linear trades

5. 2.3 Lack of appropriate risk sensitivity

6. 2.4 Dependence on economically-equivalent confirmations

7. 2.5 Medium issues

8. 3 RSA-CCR: cashflow and model-based SA-CCR

9. 3.1 Model identification

10. 3.2 Identity of SA-CCR and 1-Factor Hull-White for single maturity bucket

11. 3.3 Matching inter-bucket correlations: 3-factor Gaussian Market Model

12. 3.4 Cashflow decomposition for linear products

13. 4 Example add-on derivation for a xibor coupon

14. 4.1 Floating coupon: deterministic basis

15. 4.2 Floating coupon: stochastic basis

16. 5 RSA-CCR performance versus SA-CCR

17. 6 Conclusions

18. A Parametrization of Gaussian Market Model simulated using shifted libor market model

19. B Acknowledgements

1 Introduction

The request for comments [\citeauthoryearOCCOCC2018] offers an opportunity to address major issues with the Basel standardised approach for measuring counterparty credit risk exposures SA-CCR [\citeauthoryearBCBSBCBS2014b, \citeauthoryearBCBSBCBS2018a] and may inform other jurisdictions [\citeauthoryearBCBSBCBS2018b]. Major issues include: lack of self-consistency for linear trades; lack of appropriate risk sensitivity (zero positions can have material add-ons; moneyness is ignored); dependence on economically-equivalent confirmations. Medium issues include: ambiguity of risk assignment (i.e. requirement of a single primary risk factor); and lack of clear extensibility. The issues with SA-CCR, and the point suggestions by other authors [\citeauthoryearBCBSBCBS2013], highlight appropriate principles on which to reconstruct SA-CCR, namely appropriate risk-sensitivity (same exposure for same economics, positive exposure for non-zero risk), transparency, consistency, and extensibility. The scope of this proposal is non-option interest rate products where these issues are most significant and our objective is a replacement for the current SA-CCR in this area which is consistent with the existing structure and principles of SA-CCR.

SA-CCR was designed to better address margined and unmargined trades, reflect then-recent observed volatility levels, better represent netting, and be simple to implement [\citeauthoryearBCBSBCBS2014b]. Some recent regulations and industry equivalents use sensitivities [\citeauthoryearBCBSBCBS2019, \citeauthoryearISDAISDA2018, \citeauthoryearBCBSBCBS2011] however these are focussed on market risk (including market risk of CVA) and thus have short horizons (days to weeks). Using sensitivities within credit risk where there is a longer horizon (up to one year) would require including prediction of their behaviour. This is a potentially useful avenue of investigation for a complete update of SA-CCR which we do not propose here.

We appreciate that any standard approach needs to strike a balance between simplicity and accuracy. Too much simplicity can lead to perverse situations such as the ones we describe here. But unsophisticated Banks that cannot implement a complex approach such as IMM still need to be adequately capitalised via a simple approach. The issues in SA-CCR seem to be due to a slight excess in simplification. Other Banks are probably aware of some of the issues we describe here and other issues that are not mentioned here. Also the questions in the consultation [\citeauthoryearOCCOCC2018] clearly show that the regulator is fully aware of the limitations of the current approach but requests suggestions for improvements in special cases rather than an overhaul of the approach.

Here we propose here a global solution that revises the SA-CCR instead of a quick fix such as those in [\citeauthoryearBCBSBCBS2018a]. We do not think it is too late to make the standard more consistent at a marginal implementation cost (where instead of computation of trade level add-ons, a cashflow decomposition of a trade is performed and calculations of cashflow level add-ons for non-option trades) and remaining within the spirit of the rules and close to the formulae. This decomposition approach is already applied in [\citeauthoryearBCBSBCBS2014b, \citeauthoryearOCCOCC2018] for options. A start in the model-based direction is present in [\citeauthoryearBCBSBCBS2014a] which we generalize here.

From an implementation perspective, the RSA-CCR is accurate and fully consistent, with a marginal increase in complexity to apply cashflow decomposition to non-options as well as options. It is simple enough for small Banks because the inputs required are used elsewhere and typically available as XML outputs from booking systems (projected cashflow, discount factors, cashflow fixing date/payment date).

Even if not adopted as the new Standard, we recommend at the minimum for the RSA-CCR to become a standard control tool in other areas (such as impact of trade confirmation dependency on Pillar 1 capital calculations, adequacy of Pillar 1 capital) without the need of a more complex approach (such as IMM) to assess the capital adequacy.

Longer proofs, additional details, and derivations can be found in the accompanying Technical appendix [\citeauthoryearBerrahoui, Islah, and KenyonBerrahoui et al.2019].

2 Issues with SA-CCR

The main issues with SA-CCR are for linear instruments that we can illustrate using interest rate swaps (IRS). Major and medium issues are given in Table 2.1. Major issues are those that potentially give rise to materially incorrect capital magnitude. Medium issues potentially give rise to material mis-assignment of risk netting, e.g. curve risk being assigned to basis risk

We demonstrate that the issues with SA-CCR can be material for simple situations and then describe RSA-CCR.

2.1 Reminder on SA-CCR structure

We summarise salient features of SA-CCR here for convenience. Notation is given in Table 2.2.

[TABLE]

where for margined trades

[TABLE]

and for unmargined trades $\text{TH}+\text{MTA}-\text{NICA}\equiv 0$ so $\text{RC}=\max(V-C,0)$. Margined trades are capped at the unmargined level to allow for large $\text{TH}+\text{MTA}$. PFE is given by:

[TABLE]

There is no diversification across asset classes, so add-ons simply sum to create an aggregate add-on. Trade level add-ons are defined as

[TABLE]

$\text{SF}^{(a)}$ captures risk factor volatility. $d^{(a)}$, the adjusted notional incorporates a duration measure for interest rate and credit instruments.

We add some details for IR trades as they are used below:

[TABLE]

where $\text{SD}_{i}$ is the statuary duration for trade $i$, and $S_{i},\ E_{i}$ are the start and end dates, and $N_{i}$ is the (time-averaged) notional. The maturity factor for margined and un-margined cases for a trade with maturity $M_{i}$, in years

[TABLE]

when all time units are in years. The floor (10 or 20 business days) depends on the number of trades, disputes, etc. The supervisory factor for interest rates is $0.5\%$.

Aggregation is done across the hedging set, where the hedging set is defined by the primary risk factor. Rules for identifying the primary risk factor have been proposed in some jurisdictions [\citeauthoryearEBAEBA2017].

2.2 Lack of self-consistency for linear trades

Economically equivalent positions have different capital requirements. For example a vanilla IRS or a set of FRAs with the same strike have different add-ons, see Table 2.3. Thus SA-CCR is not self-consistent.

2.3 Lack of appropriate risk sensitivity

We illustrate two aspects here: economically zero positions can have material add-ons; and moneyness is ignored. Ignoring moneyness is probably the more important, for example moneyness produces systematic risk for existing positions when the general level of rates changes.

To illustrate the first case consider an IRS economically hedged with a set of FRAs. There is a non-zero add-on despite zero risk. See Table 2.4.

Non-ATM swaps have the same regulatory add-ons but can have wildly different actual EEPE (even using very simple dynamics). See Table 2.5

SA-CCR ignores the moneyness of linear derivatives although non-ATM swaps can have wildly different EEPE even using very simple dynamics. Table 2.5 shows the EEPE versus moneyness and direction asof 5th December 2018. EEPE was calculated using a one factor Hull-White model calibrated to SA-CCR (mean reversion 0.05; flat volatility 0.0189). Note that although the Supervisory Factor is 50bps this translates directly into a Hull-White volatility of 189bps.

Thus SA-CCR  does not have appropriate risk sensitivity since economically zero positions can have material add-ons, and moneyness is ignored which can exclude material risk differences.

2.4 Dependence on economically-equivalent confirmations

We can construct zero add-ons for any interest rate swap with maturity $T\geq$1 year where $T$ and $2T$ are in the same maturity bucket as follows:

• •

A receiver (rate $R$) amortising swap with notional $3\frac{e^{0.05\ T}}{e^{0.05\ T}-1}\text{\ N}$ between today and $T$, then notional $\frac{e^{0.05\ T}}{e^{0.05\ T}-1}N$ between $T$ and $2T;$

• •

A payer (rate $R$) swap with maturity $2T$ and notional $2\frac{e^{0.05\ T}}{e^{0.05\ T}-1}N$;

• •

A forward start (starting at $T$ and maturity at $2T$ ) receiver (rate R) swap with notional $\frac{e^{0.05\ T}}{e^{0.05\ T}-1}N;$

• •

A payer swap (rate $R$) with maturity $T$ and notional $\frac{1}{e^{0.05\ T}-1}N$ .

The adjusted notional for the first two swaps is the same, $2\frac{e^{0.05\ T}}{e^{0.05\ T}-1}\text{\ N}$, and they are in opposite directions, so their net addon is zero. MF is the same for the third and fourth trade, 1 (as $T>1$), as is $\text{SF}^{(\text{IR})}$, both $0.05$. The sum of direction and adjusted delta is:

[TABLE]

So the net add-on for the third and fourth trades is also zero. Now the net economics over all four trades is zero for $T$ to $2T$, but is a net receiver swap from [math] to $T$ with flat notional $N(e^{0.05T}-1)/(e^{0.05T}-1)=N$.

Thus SA-CCR has dependence on economically-equivalent confirmations.

2.5 Medium issues

There is only one primary risk factor for each trade. This will mis-allocate risk netting where there are several key risk drivers, e.g. long term FX. The lack of clarity that this produces is evident as some jurisdictions are creating additional rules to resolve the ambiguity of the single decision, e.g. [\citeauthoryearEBAEBA2017]. Of course this does not solve the mis-allocation of risk netting. As we will show below, cashflow decomposition used with an explicit model automatically produces risks in their appropriate hedging sets and buckets.

There is a lack of transparency of the meaning of regulatory parameters in SA-CCR. This could be simply resolved if there was an explicit model. For example although the interest rate Supervisory Factor for volatility is stated as 50bps this translates directly into a Gaussian Market Model volatility of 189bps. Again, using an explicit model base automatically provides transparency provided the model is simple.

3 RSA-CCR: cashflow and model-based SA-CCR

To solve the issues with SA-CCR we propose a cashflow and model-based SA-CCR, which we call Revised SA-CCR (RSA-CCR). We pick the model by identifying it from the SA-CCR standard itself using the link between trade level add-ons and the volatility of the present value of the trade in [\citeauthoryearBCBSBCBS2014a]. Basing RSA-CCR on cashflows automatically solves the lack of self-consistency, lack of appropriate risk-sensitivity, dependence on economically-equivalent confirmations. Using a model assigns risk unambiguously, provides clear extensibility and provides parameter transparency.

3.1 Model identification

Similarly to [\citeauthoryearBCBSBCBS2014a], we can define the theoretical add-on at a horizon $T$ of a trade with value process $V_{t}$ as the following average expected positive exposure:

[TABLE]

Assuming $dV(t)=\sigma dW(t)$, then we obtain the trade level theoretical add-on at horizon $T$ at :

[TABLE]

Recall the SA-CCR trade add-ons for IR trades:

[TABLE]

Taking a payer swap longer than one year means that $\text{MF}=1$, and $\delta=1$. The supervisory factor $SF^{IR}=0.005$ for interest rates, so we define:

[TABLE]

Consider a one factor Hull-White model with short rate, $r(t)$ and mean reversion parameter $a=0.05$ :

[TABLE]

For an ATM swap with start date $T_{s}$ and end date $T_{e}$ notional $N$, we show in Theorem 3.1 that the volatility of the present value of the swap (assuming a flat discount curve with zero rate) is:

[TABLE]

Given the relation (3.1) between theoretical add-on and trade volatility, we obtain a model based add-on for the ATM swap, identical to the SA-CCR add-on :

[TABLE]

Thus SA-CCR and [\citeauthoryearBCBSBCBS2014a] themselves implicitly identify an appropriate model. Given that SA-CCR and [\citeauthoryearBCBSBCBS2014a] can be interpreted as using a 1-Factor Hull-White model we extend this to a 3-factor model to take into account the correlation structure of the different maturity zones in Section 3.3 below. Correlations between add-ons are mapped to correlations between zero-coupon bonds.

3.2 Identity of SA-CCR and 1-Factor Hull-White for single maturity bucket

Here we provide the details to establish a formal correspondence of ATM Swap add-ons between SA-CCR and HW 1 factor. This correspondence being established, the regulatory parameters corresponding to model parameters are seen to be as in the previous section. This process also shows how to extend SA-CCR consistently to non-ATM trades.

A one factor Hull-White model [\citeauthoryearHull and WhiteHull and White1990] has short rate, $r(t)$, dynamics given by Equation 3.2 above, so the dynamics of a zero coupon bond $P(t,T)$ price are:

[TABLE]

Consider a payer, i.e. pay-fixed rate $R$,notional $N$, swap price $V$ at $t$

[TABLE]

Now we have the identity between SA-CCR and a 1-Factor Hull-White model in Theorem 3.1 below.

Theorem 3.1**.**

With zero bond dynamics given by Equation 3.3, if $V$ is the value process of a forward starting payer swap, then the instantaneous volatility of $V$ for $t\leq T_{s}$ can be decomposed into three contributions :

[TABLE]

Where :

[TABLE]

If at time $t$, the swap is ATM, taking the standard weight-freezing assumption (see proof) we have:

[TABLE]

Moreover if the yield curve is flat and equal to zero, then the instantaneous volatility of an ATM swap in one maturity bucket is given exactly by the SA-CCR regulatory formula i.e.:

[TABLE]

See Technical appendix [\citeauthoryearBerrahoui, Islah, and KenyonBerrahoui et al.2019] for proof.

The swap volatility can be decomposed into two contributions:

• •

volatility of the floating rate index itself: $\sigma_{ATM}(t)$

• •

volatility of the present value of the cash-flows: $\sigma_{float}(t)+\sigma_{fixed}(t)$

Since for SA-CCR only exposures below one year are relevant we freeze the volatility $\sigma_{V}(t)$ and use its initial value $\sigma_{V}(0)$ so

[TABLE]

3.3 Matching inter-bucket correlations: 3-factor Gaussian Market Model

Theorem 3.1 shows that the regulatory add-on formula for an at-the-money swap can be recovered from a 1-factor Hull-White model. To match the inter-bucket correlations of SA-CCR, Table 3.1, we naturally move to a 3-factor Gaussian Market Model:

• •

We use maturity buckets defined by SA-CCR: $M_{1}=[0,1)$, $M_{2}=[1,5)$, $M_{3}=[5,\infty)$

• •

We also define $M(t)=\sum_{i}1_{t\in M_{i}}M_{i}$ a map from positive numbers to the set of maturity buckets

• •

Next we define the correlated Brownian motions $Z_{t}^{M_{k}}$, $k=1,2,3$ with SA-CCR correlation structure i.e. $dZ_{t}^{M_{1}}dZ_{t}^{M_{2}}=dZ_{t}^{M_{2}}dZ_{t}^{M_{3}}=0.7$ and: $dZ_{t}^{M_{1}}dZ_{t}^{M_{3}}=0.3$

We define the following 3-factor Gaussian Market Model extension of the Hull-White model that allows to recover the inter-bucket correlations for zero coupon bonds (with expiries $T$ in a discrete set ${T_{1},..,T_{N}}$ covering all business days up to horizon $T_{N}$) i.e :

[TABLE]

Where $r_{t}$ is the risk free rate (define as the continuously compounded rate of the shortest maturity zero-coupon bond). Now we have a model identified from SA-CCR we can move on to the second key element of RSA-CCR, cashflow decomposition for linear products.

3.4 Cashflow decomposition for linear products

We first define the scope, concentrating on common linear product cashflows (which we call elementary cashflows) in Definition 3.2 and then provide Theorem 3.3 that gives the add-ons for each of the different cashflow types. In Section 4 we give an example derivation of the entries in Table 3.2 for a Floating cashflow (with deterministic and stochastic basis). Full derivation are in the accompanying Technical appendix [\citeauthoryearBerrahoui, Islah, and KenyonBerrahoui et al.2019]. In Section 5 we compare the performance of SA-CCR and RSA-CCR.

From the examples here and in [\citeauthoryearBerrahoui, Islah, and KenyonBerrahoui et al.2019], the add-ons for other (less common) cashflows in linear products can be derived in a similar manner.

Definition 3.2**.**

An elementary cashflow with payment date at $T$ and projected value $CF(T)$ (viewed from [math]) that is either the projected value of:

• •

a fixed cashflow $CF(T)=N$;

• •

a standard interest rates floating cashflow (as defined in point 2) , with notional $N$, money market index $L(t,t+\delta(\tau))$ tenor $\tau$ , fixing date $0<T_{f}\leq T$ defined as :

[TABLE]

• •

a standard CMS cashflow with notional $N$ and swap rate $S_{\tau}(T_{s},T_{e})$ fixing at $0<T_{s}\leq T$, with underlying money market tenor $\tau$, maturity at $T_{e}$, swap tenor $\delta(T_{s},T_{e})$, defined as :

[TABLE]

• •

a standard inflation floating cashflow, with notional $N,$defined on inflation index $I(t),$with initial observation date $T_{s}$ (can be in the past ), next observation date $\tau_{I}$ and final observation date $T_{e}\leq T$:

[TABLE]

• •

a standard inflation compound cashflow with notional $N,$with initial observation date $T_{s}$ (can be in the past ), next observation date $\tau_{I}$ and final observation $T_{e}\leq T$:

[TABLE]

• •

a standard interest rates compound cashflow , with start of compounding at $T_{s}$(cn be in the past i.e. , next reset date $\tau_{C}$ , end of last compounding $T_{e}$, underlying index $L(t,t+\delta_{\tau})$, with tenor $\tau$. This cashflow is given by :

[TABLE]

Theorem 3.3**.**

All elementary cashflows received (respectively paid) at time $T$, should have an add-on contribution using the supervisory formula

[TABLE]

to the maturity bucket corresponding to the payment date itself $T$ with follwing inputs to use in the supervisory formula:

• •

a delta equal to $-1$ (respectively $1$ ) ,

• •

start date of [math] and end date of $T$ to compute the supervisory duration

• •

an adjusted notional equal to present value of the cashflow

Moreover, all non-fixed standard cashflows will have other add-ons contributions due to the volatility of their respective underlying indices, calculated using the supervisory formulae (with inputs as per Table 3.2) and allocated to the appropriate maturity buckets and hedging sets (as stated in Table 3.2 for received cash-flows).

With $CF(T)$ is the projected cash-flow, the map $M(t)$ that associates to $t$ its risk bucket and $P(0,T)$ the discount factor. Basis risk add-ons are allocated to the money market index $\tau$ vs discount hedging set.

Setting $P(0,T)=1$ would result in simplified add-ons in line with current regulatory stipulations

In RSA-CCR add-on contributions for linear products are the aggregate add-on of the add-ons for each cashflow . This removes the SA-CCR issues above: lack of self-consistency; lack of appropriate risk sensitivity; dependence on economically-equivalent confirmations. We formalize add-on calculations via cashflow decomposition in the definitions below.

Definition 3.4**.**

A linear product has no optionality and is made of elementary cash-flows $\text{CF}_{i}$ with payment dates $T_{i}$ ($i=1,..,n)$ and notional $N_{i}$.

Definition 3.5**.**

The total add-on of a portfolio of linear products is obtained from the aggregation of the individual add-ons of each cash-flow that makes up the linear products. Add-on aggregation is performed according to SA-CCR.

Proposition 3.6**.**

All portfolios of linear products resulting in the same net cash-flows have the same aggregate add-on under RSA-CCR.

Proof.

By construction the add-ons depend only on the net cash-flows. ∎

Propostion 3.6 is the property we are after: this ensure appropriate risk sensitivity (no risk for no economic position and vice versa) and no dependence on economically-equivalent confirmations.

4 Example add-on derivation for a xibor coupon

Here we provide an example of how to derive the entries in Table 3.2 for a floating coupon for deterministic and stochastic index-vs-discounting basis.

The floating rate cash-flow is based on a money market index with fixing at $T_{f}$ , tenor $\tau$ (less than $1$ year) and payment at $T\geq\tau$ , with notional $N$.

We ignore convexity so this setup includes timing mismatches such as a Libor rate paid in advance.

The forward value of the index is

[TABLE]

We define the index-vs-discounting basis $P_{b}$ via

[TABLE]

Ignoring convexity the coupon value is

[TABLE]

Applying Ito’s Lemma to $V(t)$ we see that the instantaneous volatility of $V(t)$ has three components, one from the floating rate index volatility, another from the cashflow’s present value and a third from the basis risk volatility:

[TABLE]

No $dP_{*}dP_{*}$ terms are present above as these result in drift ($dt$) contributions not volatility ($dW_{*}$) contributions.

4.1 Floating coupon: deterministic basis

Assume first that the basis curve is deterministic, then we have for $P(t,T)$, $P(t,T_{f})$ and $P(t,T_{f}+\tau)$ the following dynamics from the 3-factor Gaussian Market Model:

[TABLE]

Hence

[TABLE]

From Equation 4.1 we see that the floating rate cash-flow add-ons bucket contributions are:

contribution from the present value of the cash-flow:

• •

Maturity bucket $M(T)$

• •

Notional: $NP(0,T)\tau L(0,T_{f},T_{f}+\tau)$

• •

Duration: $\frac{1}{a}(1-e^{-aT})$

• •

Delta : $-1$ 2. 2.

index volatility: fixing date contribution

• •

Maturity bucket $M(T_{f})$

• •

Effective Notional: $NP(0,T)(1+\delta_{\tau}L(0,T_{f},T_{f}+\tau))$

• •

Duration: $\frac{1}{a}(1-e^{-aT_{f}})$

• •

Delta : $-1$ 3. 3.

index volatility: payment date contribution

• •

Maturity bucket $M(T_{f}+\tau)$

• •

Effective Notional: $NP(0,T)(1+\tau L(0,T_{f},T_{f}+\tau))$

• •

Supervisory Duration: $\frac{1}{a}(1-e^{-a(T_{f}+\tau)})$

• •

Delta : $1$

4.2 Floating coupon: stochastic basis

If we assume a stochastic basis spread curve (some implementation scenarios for Libor decommissioning fallback may make this irrelevant) as

[TABLE]

[TABLE]

The volatility of $V(t)$ adds basis contributions

[TABLE]

This gives additional contributions to two buckets, one for the fixing date and another for the fixing date shifted by the underlying tenor:

fixing date:

• •

Maturity bucket $M(T_{f})$

• •

Effective Notional: $NP(0,T)(1+\delta_{\tau}L(0,T_{f},T_{f}+\tau))$

• •

Duration: $\frac{1}{a}(1-e^{-aT_{f}})$

• •

Delta : $-1$ 2. 5.

payment date of the underlying money market index :

• •

Maturity bucket $M(T_{f}+\tau)$

• •

Effective Notional: $NP(0,T)(1+\tau L(0,T_{f},T_{f}+\tau))$

• •

Supervisory Duration: $\frac{1}{a}(1-e^{-a(T_{f}+\tau)})$

• •

Delta : $1$

5 RSA-CCR performance versus SA-CCR

Here we compare RSA-CCR add-ons versus SA-CCR add-ons, and versus a simulation of Gaussian Market Model, and provide reference comparison to a 1-Factor Hull-White for completeness. Both RSA-CCR and SA-CCR can be considered approximations to the simulation results. These examples demonstrate that RSA-CCR has appropriate risk sensitivity as compared to SA-CCR, and showing results of ambiguity resolution in RSA-CCR, i.e. it is clear what to do for zero coupon swaps. Of course, by construction RSA-CCR does not suffer from dependence on economically-equivalent confirmations.

• •

Vanilla Swaps with different moneyness, Table 5.1. SA-CCR is insensitive to moneyness, showing a 56% range of error w.r.t. the simulation, whereas RSA-CCR error range is 2%.

• •

Vanilla swaps with different replications, Table 5.2. Here SA-CCR’s maximum error w.r.t. simulation is infinite as it gives non-zero risk for a zero economic position. In contrast RSA-CCR has a 5% range or errors, and maximum error of -5%.

• •

Amortising/Accreting vanilla swaps vs combination of swaps, Table 5.3. Here the SA-CCR range of error is 98% (in one case it gives zero add-on for non-zero risk). The range of error for RSA-CCR is 28% with no discounting or 14% with market discounting versus the simulation.

• •

Zero coupon swaps, Table 5.4. The range of error of SA-CCR is 99% in contrast to RSA-CCR with 3%.

• •

Forward starts, Table 5.5. The range of error of SA-CCR is 68% in contrast to RSA-CCR with 1%.

6 Conclusions

SA-CCR has major issues including: lack of self-consistency for linear trades; lack of appropriate risk sensitivity (zero positions can have material add-ons; moneyness is ignored); dependence on economically-equivalent confirmations. We have shown that SA-CCR is, by parameter identification and re-construction, based on a 3-factor Gaussian Market Model model. Hence we propose RSA-CCR based on cashflow decomposition and this 3-factor Gaussian Market Model model. RSA-CCR is both free of SA-CCR’s issues, simple to use in practice, and can be extended easily given that it is model-based. We recommend updating SA-CCR to RSA-CCR in order to resolve SA-CCR’s issues of lack of self-consistency for linear trades, lack of appropriate risk sensitivity (zero positions can have material add-ons; moneyness is ignored), dependence on economically-equivalent confirmations, and ambiguity of application for cases not explicitly described.

For a consistent treatment of the FX risk factors within the RSA-CCR cash-flow decomposition approach, it is enough to consider only the currency pairs with respect to the domestic currency i.e. CCY/USD as hedging sets. The add-on contributions to the FX asset class (hedging set CCY/USD) for an elementary linear cashflow in the currency CCY (regardless of the product type) is simply the present value of the cash-flow converted to the domestic currency USD. Moreover, each cashflow has an interest rate add-on contribution as computed for each elementary cashflow by the RSA-CCR approach.

We recommend updating SA-CCR to RSA-CCR in order to resolve SA-CCR’s issues of lack of self-consistency for linear trades, lack of appropriate risk sensitivity (zero positions can have material add-ons; moneyness is ignored), dependence on economically-equivalent confirmations, and ambiguity of application for cases not explicitly described.

Appendix A Parametrization of Gaussian Market Model simulated using shifted libor market model

The Gaussian Market Model is simulated using a 3-Factor Shifted Libor Market Model where the calibration is derived from the Gaussian Market Model setup. Volatility and shift calibration is shown in Table A.1.

Appendix B Acknowledgements

The authors would like to gratefully acknowledge useful discussions with Lee McGinty and Crina Manolescu.