Exposure measures

The credit committee asks: how much could RiskAtoms Bank lose to Corriemoor Bank? The desk can give five honest answers. Nothing today (£0m). On average £6.25m in a year. Up to £43.9m in a bad case, about three years out. £4.45m averaged over the coming year. Or £6.24m for the capital calculation. They are all correct; each answers a different question. This lesson defines them and computes each one on the book.

The previous lesson (L3.4.01: What counterparty credit risk is) showed that a netting set's exposure at a future date is a random quantity, $E(t) = \max(V(t), 0)$. The bank simulates the market forward (here on 2,000 paths, the method of L3.2.11: Monte Carlo VaR), values the trades on every path and date, and so gets 2,000 possible exposures for each date. Every exposure measure is a way of summarising those numbers.

One date: the distribution of exposure

Here is Corriemoor's exposure in one year across the 2,000 paths. On 48.9% of them the netting set is worth less than nothing to the bank, so the exposure is zero. The rest spread out to the right.

Two summaries of one date's distribution matter most:

• Potential future exposurePotential future exposure (PFE): a high quantile (often 95% or 99%) of the exposure distribution at a future date. Basel calls it peak exposure. (PFE) is a high quantile (M-C09: Quantiles), the exposure exceeded on only 5% (or 1%) of paths. For Corriemoor in a year: £19.5m at 90%, £24.3m at 95%, £35.3m at 99%. It is the exposure cousin of VaR (L2.5.06: Value at risk: the idea), and it drives credit limits.
• Expected exposureExpected exposure (EE): the mean of the exposure distribution at a future date, counting the paths with zero exposure. (EE) is the mean (M-C02: Expectation), zeros included: £6.25m. It is what a default would cost on average, before recovery, and it drives pricing (CVA in L3.4.10: CVA basics) and capital.

Today the distribution is a single number, the current exposureCurrent exposure: max(V, 0) today, the replacement cost if the counterparty defaulted now.: zero for the three netting sets that are at market, and £18.98m for Saltney, whose put the bank owns.

All dates: the exposure profile

Repeat for every date and you get an exposure profileExposure profile: a measure such as EE or PFE plotted against future dates.. Corriemoor's EE climbs to £10.89m at about 3 years; its 95% PFE peaks at £43.9m. Two time averages compress a profile into one number:

• Expected positive exposureExpected positive exposure (EPE): the time-weighted average of EE over a period, usually the first year. (EPE): the average of EE over a period. Over the first year, Corriemoor's is £4.45m.
• Effective EPEEffective EPE: the time-weighted average of effective EE over the first year (or to maturity, if sooner).: the same average of effective EEEffective EE: EE made non-decreasing. At each date it is the larger of that date's EE and the effective EE before it., a version of EE that is never allowed to fall. Corriemoor's EE rises through its first year, so the two barely differ: £4.46m.

Why never allowed to fall? Halvergate's FX forward matures at one year, and its EE drops when it does. But a client that hedges its dollar sales will very likely do another forward. Effective EE holds the exposure at its earlier peak to cover that rollover riskRollover risk: short-dated trades that mature are usually replaced by new ones, so exposure that seems to fall away often comes back.. The worked example shows it.

The four netting sets

Saltney's numbers are dominated by the put the bank owns: worth £19.0m already, it is all exposure. Ebbsworth's single ten-year swap has a one-year EPE of £5.67m but a peak PFE of £51.1m, four years out: the one-year measures used for capital look only at the near future.

Why alpha is 1.4

Capital models for loans treat exposure at default as a fixed number. Effective EPE turns a random, moving exposure into such a number, a loan-equivalent. But it is an average: it ignores that exposures to many counterparties can rise together when markets move, that a few large counterparties make the portfolio lumpy, and that exposure may grow just as the counterparty weakens (wrong-way risk, L3.4.11: Wrong-way risk). AlphaAlpha (α): the multiplier, 1.4 by default, that turns effective EPE into exposure at default under the internal model method. scales effective EPE up to cover these. It is defined as the ratio of economic capital from a full simulation of exposures and defaults to the capital you get by treating EPE as a fixed loan amount. Industry studies in 2003 put it near 1.1 for large, diversified portfolios; the Basel Committee set 1.4, conservatively, and lets supervisors ask for more.

Write $V_j(t_k)$ for the netting set's value on path $j$ (of $N$) at grid date $t_k$. The exact measures are an expectation and a quantile of $E(t) = \max(V(t), 0)$; the simulation estimates them.

The quantile is read off the sorted exposures. With $N$ paths, the book puts the $q$-quantile at position $q(N-1)$ in the sorted list (counting from 0) and interpolates between neighbours, as in historical VaR. The EE estimate has a standard error of $s/\sqrt N$, where $s$ is the sample standard deviation of the exposures (with $N - 1$). The book integrates EPE with the trapezium rule on its half-month grid; Basel's sum over grid dates gives almost the same number on a grid this fine.

Because effective EE is a running maximum, $\text{Eff EE}_k \ge \text{EE}_k$ on every date, so effective EPE is never below EPE. The two are equal whenever EE rises throughout the first year.

Pick a netting set and a quantile. The chart shows its profile out to ten years; the slider shows the distribution of exposure on one date. Then re-run the simulation with fresh paths and see which numbers hold still.

Halvergate has no CSA: a seven-year receive-fixed swap on £100m and a one-year forward buying $65m. Both are at market today. The bank simulates 2,000 paths.

The Basel Framework's definitions (CRE50) match this lesson: current exposure, peak exposure (a high percentile, typically 95% or 99%), expected exposure, EPE, effective EE and effective EPE. Under the internal model method (CRE53), EAD is alpha times effective EPE, with alpha set at 1.4. Supervisors can require more, for example where the bank has few counterparties, high general wrong-way risk or highly correlated exposures, and a bank may estimate its own alpha with approval, subject to a floor of 1.2. Under Basel III the bank must also compute effective EPE with model parameters calibrated to a stressed period and hold whichever capital charge is larger, for the portfolio as a whole. Most banks use the standardised approach, SA-CCR, which has its own replacement cost and add-on and keeps the same alpha. Both are in M4.4: Counterparty, operational and large exposures; exposure simulation in depth is in M6.6: Counterparty risk and XVA.

• Current exposure is today's max(V, 0); everything else summarises the simulated distribution of future exposure.
• PFE is a high quantile (limits); EE is the mean, zeros included (pricing and capital).
• EPE averages EE over time; effective EE never falls, which covers rollover of maturing trades.
• EAD under the internal model method = 1.4 × effective EPE: Halvergate £2.15m → £3.00m.
• Corriemoor: EE in a year £6.25m, peak 95% PFE £43.9m, effective EPE £4.46m.
• Means settle quickly as paths are added; high quantiles need many more paths.

Current exposure (CE)

Current exposure: max(V, 0) today, the replacement cost if the counterparty defaulted now.

Potential future exposure (PFE)

Potential future exposure (PFE): a high quantile (often 95% or 99%) of the exposure distribution at a future date. Basel calls it peak exposure.

Expected exposure (EE)

Expected exposure (EE): the mean of the exposure distribution at a future date, counting the paths with zero exposure.

Expected positive exposure (EPE)

Expected positive exposure (EPE): the time-weighted average of EE over a period, usually the first year.

Effective EE

Effective EE: EE made non-decreasing. At each date it is the larger of that date's EE and the effective EE before it.

Effective EPE

Effective EPE: the time-weighted average of effective EE over the first year (or to maturity, if sooner).

Alpha (α)

Alpha (α): the multiplier, 1.4 by default, that turns effective EPE into exposure at default under the internal model method.

Exposure profile

Exposure profile: a measure such as EE or PFE plotted against future dates.

Rollover risk

Rollover risk: short-dated trades that mature are usually replaced by new ones, so exposure that seems to fall away often comes back.