The margin period of risk · RiskAtoms Academy

Know these first

L3.2.13: Scaling and aggregating risk

Math Atoms

∑ The normal distribution

By the end you will be able to

1Explain why a netting set with daily, zero-threshold collateral still carries exposure, and trace the margin period of risk from the last call met to close-out.
2Derive EE ≈ 0.4 σ √MPoR and PFE = σ √MPoR Φ⁻¹(q) for a normal change in value, and check them against the simulation.
3Apply the regulatory floors: 5 and 10 business days, 20 days for large or hard-to-close sets, doubling after disputes, F + N − 1.
4Measure how a longer margin period of risk raises collateralised exposure on the book.

Why it matters

Corriemoor posts variation margin every day with a zero threshold. If its calls were always met, the bank would never be owed more than it held. Yet its expected exposure six months out, under the CSA, is £1.37m, not zero. The reason is timing. When a counterparty defaults, the last call it paid was set days earlier, and the trades keep moving until the bank can close them out. That stretch of time is the margin period of risk, and it is where most collateralised exposure lives.

Core explanation

The gap collateral cannot close

Illustrative dataRiskAtoms Bank plc's OTC derivatives book

Eight trades with four invented counterparties, in pounds, on one made-up valuation day, valued on 2,000 simulated paths (seed 34040) of a flat GBP rate (4.00% today), GBPUSD (1.30) and an equity index (8,000) every half month (about ten business days) out to ten years. The firms, trades and market model are made up for teaching; none is real data. The same book runs through every lesson in Module 3.4, so the figures agree.

This lesson uses the book's collateral model: the collateral at each date is set from the value one or more grid steps earlier. One step is 1/24 of a year, 10.4 business days, which the lesson calls ten days.

Under a CSA (L3.4.05: Collateral and the CSA) collateral follows the value with a lag. While the counterparty is healthy the lag is a day or so and harmless. The danger is the lag at default. Regulators define the margin period of risk (MPoR) as the time from the most recent exchange of collateral until the trades are closed out and the resulting market risk is re-hedged. Over that time the bank holds collateral sized on an old value, and every move in its favour is a move it may not recover.

TimelineAnatomy of a margin period of risk (illustrative timing)

1 / 5

Day 0

The last call met

Corriemoor pays the day's variation margin. The bank's collateral matches the value of the trades as of yesterday's close. From here on, nothing more arrives.

The rule of thumb

With a zero threshold and no minimum transfer amount, what the bank holds at default is the value at the start of the MPoR. Its exposure is the gain since then, if there is one. Over a short period the change in value $\Delta V$ is roughly normal with mean zero (M-C06: The normal distribution), and its standard deviation grows with the square root of time, as in L3.2.13: Scaling and aggregating risk. The expected positive part of a normal with mean zero is $\varphi(0) \approx 0.4$ times its standard deviation, which gives the rule of thumb

\text{EE} \approx 0.4\,\sigma\sqrt{\text{MPoR}}, \qquad \text{PFE}_{95\%} \approx 1.645\,\sigma\sqrt{\text{MPoR}}

Check it on Corriemoor six months out. Across the book's 2,000 paths, the change in the netting set's value over one step has a standard deviation of £3.47m. The rule of thumb gives $0.3989 \times \text{£}3.47\text{m} = \text{£}1.39\text{m}$ ; the simulated EE with ideal collateral is £1.37m. Averaged over the first year, the rule of thumb is £1.37m against £1.36m simulated. It runs about 1% high because the paths are not all alike: the spread of the change differs from path to path (the FX forward's sensitivity depends on the exchange rate), and a mix of normals has a slightly smaller average gain than one normal with the same overall spread. For Saltney's option book the gap is wider, £1.71m against £1.57m, because an option's value is not linear in the market.

Simulated EE, ideal collateral, ≈10 daysRule of thumb, ≈10 daysSimulated EE, ideal collateral, ≈21 daysRule of thumb, ≈21 days

Doubling the MPoR multiplies the spread by 1.414 in the simulation, against $\sqrt 2 = 1.414$ : the square-root-of-time rule holds well over a few weeks. The dips are the swaps' annual payments. At three years the FX forward matures and settles; the book's simple collateral model shows a one-off spike there, which the charts leave out.

The book under longer MPoRs

EPE, first year, under each CSA	≈10 days	≈21 days	≈31 days	≈42 days
Corriemoor (threshold 0)	£1.34m	£1.86m	£2.26m	£2.58m
Saltney (IA £3m)	£0.62m	£1.16m	£1.59m	£1.94m
Ebbsworth (threshold £10m)	£3.65m	£3.87m	£4.10m	£4.31m

Corriemoor's exposure grows almost exactly with the square root of the MPoR, because the MPoR is all that stands between it and zero. Saltney's £3m independent amount absorbs the first moves, so its exposure is small at ten days but grows faster in proportion. Ebbsworth's is dominated by its £10m threshold, so a longer MPoR adds less in proportion.

The regulatory floors

A bank modelling exposure with its own model (the internal model method, M4.4: Counterparty, operational and large exposures) may not assume a short MPoR to flatter itself. The Basel Framework sets floors (CRE53.24 to 53.26):

Netting set, remargined daily	MPoR floor
Only repo-style transactions (securities financing)	5 business days
All other netting sets, including OTC derivatives	10 business days
More than 5,000 trades at any point in the previous quarter	20 business days
Illiquid collateral, or an OTC derivative that cannot easily be replaced	20 business days
More than two margin call disputes in the previous two quarters, each lasting longer than the MPoR	At least double the floor, for the next two quarters
Remargined every N days instead of daily	Floor F + N − 1 days

Sort itWhich floor applies?

A daily-margined netting set of gilt repos with a money market fund.
Corriemoor's three trades, margined daily in cash.
A dealer netting set that held 7,200 trades at one point last quarter.
A netting set containing a long-dated exotic option no other dealer will quote.
A daily-margined swap netting set with three long disputes in the past six months.

The formal version

Let $\Delta V$ be the change in the netting set's value over the MPoR $T$ (in years), with collateral equal to the value at the start (threshold, MTA and initial margin zero). The exposure at close-out is $\max(\Delta V, 0)$ .

In full, with $\Delta V \sim N(0, \sigma^2 T)$ and $z = x / (\sigma\sqrt T)$ :

\begin{aligned} \text{EE} = \mathbb{E}\bigl[\max(\Delta V, 0)\bigr] &= \int_0^{\infty} x\,\frac{1}{\sigma\sqrt T}\,\varphi\!\left(\frac{x}{\sigma\sqrt T}\right)dx \\ &= \sigma\sqrt T \int_0^{\infty} z\,\varphi(z)\,dz \\ &= \sigma\sqrt T\,\bigl[-\varphi(z)\bigr]_0^{\infty} \qquad \text{since } \varphi'(z) = -z\,\varphi(z) \\ &= \sigma\sqrt T\,\varphi(0) = \frac{\sigma\sqrt T}{\sqrt{2\pi}} \approx 0.3989\,\sigma\sqrt T \\ \text{PFE}_q &= \sigma\sqrt T\,\Phi^{-1}(q) \end{aligned}

Where σ comes from. For a netting set driven by the GBP rate and GBPUSD, with sensitivities $s_r$ (£ per bp) and $s_{fx}$ (£ per 1%), annual moves $\sigma_r$ bp and $\sigma_{fx}$ % and correlation $\rho$ , the variance of the value change is the two-factor version of L3.2.09: Parametric VaR for a portfolio:

\sigma^2 = (s_r\sigma_r)^2 + (s_{fx}\sigma_{fx})^2 + 2\rho\,(s_r\sigma_r)(s_{fx}\sigma_{fx})

Floors and frequency. If collateral is exchanged every $N$ days and the floor for daily margining is $F$ , the MPoR is at least $F + N - 1$ days, and EE scales with $\sqrt{F + N - 1}$ .

Try it

Pick a counterparty and stretch the MPoR from about ten to about forty business days. With ideal collateral, the dashed rule of thumb sits on the simulated EE. Switch to the counterparty's own CSA to see what a threshold or an independent amount changes.

Try itStretch the margin period of risk

Counterparty

Margin period of risk (business days)

Collateral terms

Simulated EE under these termsRule of thumb 0.4 σ√MPoR

EE at six months£1.37m

Against the shortest MPoR× 1.00

EPE over the first year£1.33m

Rule of thumb at six months£1.39m

√(MPoR ÷ shortest)1.00

Simulated on the book's 2,000 paths; one grid step is 1/24 of a year, about 10.4 business days, so the choices are one to four steps. The rule of thumb uses the spread of the simulated change over the MPoR at each date and is left out where a trade matures, since the value jumps there.

Worked example

Corriemoor, six months out

The netting set: pay-fixed £250m 10 years, receive-fixed £150m 5 years, sell $130m in 3 years. Today's sensitivities: £135,995 per bp rise in the GBP rate (PV01) and £873,803 per 1% rise in GBPUSD. The book's model: rate vol 90bp a year, GBPUSD vol 9% a year, correlation 0.3. MPoR one step, 10.4 business days ( $T = 1/24$ year).

From sensitivities to exposure

Step 1 of 6

1
Spread of the change, from the sensitivities
$\begin{aligned} s_r\sigma_r\sqrt T &= \text{£}135{,}995 \times 90 \times \sqrt{1/24} = \text{£}2.50\text{m} \\ s_{fx}\sigma_{fx}\sqrt T &= \text{£}873{,}803 \times 9 \times \sqrt{1/24} = \text{£}1.61\text{m} \\ \sigma\sqrt T &= \sqrt{2.50^2 + 1.61^2 + 2 \times 0.3 \times 2.50 \times 1.61} = \text{£}3.35\text{m} \end{aligned}$

Regulatory lens

The MPoR definition comes from the Basel Framework's counterparty credit risk definitions (CRE50.19) and the floors from its internal models method (CRE53.24 to 53.26); the EU's Capital Requirements Regulation defines the MPoR in the same words (Article 272). SA-CCR, the standardised method, also builds the MPoR into its treatment of margined netting sets. Both methods, and how they turn exposure into capital, are in M4.4: Counterparty, operational and large exposures. Central counterparties work to their own shorter horizons, at least five business days for OTC derivatives under EMIR, because they can auction a defaulter's book quickly (L3.4.08: Central counterparties). Margin period of risk modelling with stochastic collateral and cash flows during the MPoR is in M6.6: Counterparty risk and XVA.

Spark Atoms

Spark AtomEvery dispute counts

The Basel Committee's FAQ on the dispute rule (CRE53.25) answers a natural question: is there a minimum size for a dispute to count? No. Every margin call dispute counts towards the “more than two” that doubles the floor, whatever its amount, as long as it lasted longer than the MPoR.

Spark AtomThe square root of time, measured

On the book's 2,000 simulated paths, the spread of Corriemoor's value change over twenty days is 1.4139 times its spread over ten. The textbook answer is √2 = 1.4142. The rule is only a rule, but over a few weeks of a swap book it is hard to beat.

Key takeaways

Collateral lags the value. At default, the bank holds collateral from the last call met while the trades move until close-out: the margin period of risk.
With ideal collateral, EE ≈ 0.4 σ √MPoR and PFE_q ≈ σ √MPoR Φ⁻¹(q); both grow with the square root of the MPoR.
On the book, Corriemoor's six-month EE is £1.39m by the rule of thumb and £1.37m simulated.
Basel floors: 5 days for repo-style sets, 10 for other daily-margined sets, 20 for sets above 5,000 trades or with illiquid collateral or hard-to-replace trades; doubled after repeated long disputes; F + N − 1 for margining every N days.
A 20-day MPoR raises Corriemoor's α × EEPE from £1.93m to £2.70m.
Thresholds and initial margin change the picture: a threshold adds exposure that the MPoR does not touch; initial margin absorbs the MPoR move.

Reaction

Check what stuck

Question 1 · objective 2

A fully collateralised netting set (threshold and MTA zero) has a value change over the margin period of risk with standard deviation £4m and mean zero. What is the expected exposure, in £m?

£m

Question 2 · objective 2

The same netting set moves from a 10-day to a 20-day margin period of risk. Using the rule of thumb, what is the new EE, in £m?

£m

Question 3 · objective 3

A netting set of OTC derivatives is remargined every 5 business days rather than daily. Under the Basel floor of F + N − 1, what is the minimum margin period of risk, in business days?

days

Question 4 · objective 3

A bank's netting set with a dealer has had three margin call disputes in the last two quarters, each lasting longer than the MPoR. The set has 800 OTC swaps and liquid collateral. What floor applies for the next two quarters?

A20 business days: double the 10-day floorB10 business daysC5 business daysD40 business days

Question 5 · objective 1

Under its CSA, Saltney's EE at six months is £0.66m, against £1.59m with no initial margin. Why lower?

AIts £3m independent amount absorbs most moves over the margin period of riskBSaltney has a lower credit spreadCIts trades do not move over ten daysDThe MPoR is shorter for hedge funds

Glossary

Margin period of risk (MPoR): Margin period of risk: the time from the most recent exchange of collateral with a defaulting counterparty until its trades are closed out and the resulting market risk is re-hedged.
Close-out: Terminating all the trades in a netting set after a default, valuing them, and netting them into one amount owed one way or the other.
Grace period: The days a party has, after notice of a failure to pay or deliver, to put it right before the failure becomes an event of default.
Early termination date: Early termination date: the date the surviving party designates for closing out all the trades after an event of default.
Re-hedge: Replacing the market risk of the closed-out trades with new trades, so the bank is no longer exposed to further moves.
Supervisory MPoR floor: The shortest margin period of risk a bank may assume in its exposure model, set by supervisors.
MPoR rule of thumb: With collateral up to date as of the last call met, EE ≈ 0.4 σ √MPoR, where σ is the annual standard deviation of the netting set's value.

Deep Dives & Origins

Deep Dives

J. Gregory, The xVA Challenge: Counterparty Risk, Funding, Collateral, Capital and Initial Margin, 4th ed., Wiley, 2020, chapter 7, “Margin (Collateral) and Settlement”, and the exposure chapters (11, “Future Value and Exposure”, and 15, “Quantifying Exposure”). Intermediate.
Basel Committee on Banking Supervision, Basel Framework, CRE53 Internal models method for counterparty credit risk, BIS. Advanced.

Origins

Basel Committee on Banking Supervision, Basel Framework, CRE50 Counterparty credit risk definitions and terminology, paragraph 50.19, and CRE53 Internal models method for counterparty credit risk, paragraphs 53.14 and 53.24 to 53.26 and the FAQ to 53.25, BIS.
Regulation (EU) No 575/2013 (Capital Requirements Regulation), Article 272 Definitions, as published on legislation.gov.uk.
Jolly Contrarian, commentary on the 1994 ISDA Credit Support Annex (New York law), Paragraph 7, and the 2002 ISDA Master Agreement, Section 5(a)(i) Failure to Pay or Deliver.
Commission Delegated Regulation (EU) No 153/2013, Chapter VI, Article 26 (time horizons for the liquidation period).

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Collateral and the CSA

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Initial margin and the uncleared margin rules