Exposure profiles of common trades · RiskAtoms Academy

By the end you will be able to

1Explain the humped exposure profile of an interest rate swap as diffusion against amortisation, and find its peak.
2Explain why FX forwards and cross-currency swaps have exposure rising to maturity.
3Contrast bought and sold options, and payer and receiver swaps.
4Approximate EE and PFE from the standard deviation of a trade's value, and check them against the simulation.

Why it matters

Two £100m, five-year trades with Halvergate. One is an interest rate swap: its expected exposure peaks at £1.96m after about two years and then fades. The other is a cross-currency swap: its expected exposure keeps climbing to £7.30m in the last month. Same notional, same counterparty, five times the exposure. A credit officer who knows the shapes can tell at a glance which trades will use up a limit, and when.

Core explanation

Two forces shape every profile

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 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.

The cross-currency swap (X1) and the plain five-year swap (X2) are extras for this lesson, valued on the book's paths; they are not in the book and change none of its figures.

An exposure profile (L3.4.02: Exposure measures) is shaped by two forces. Diffusion: the further ahead you look, the further rates and exchange rates can have moved, so values spread out, roughly like $\sqrt t$ . Amortisation: as a trade's payments are made, less is left to exchange, so there is less value to build up. Which one wins, and when, sets the shape.

Interest rate swaps: the hump

A swap exchanges many payments over its life (L1.7.04: Interest rate swaps). Early on, rates have not had time to move. Late on, few payments remain. In between, exposure peaks. That is the hump. For Halvergate's seven-year swap it peaks at £2.96m after about 3 years. Each annual payment knocks a little off, giving a sawtooth. A simple formula, the dashed line, tracks the simulation closely; the formal version derives it.

Simulated EE (2,000 paths)Normal approximation φ(0)·N σ √t A(t)

FX forwards: rising to the end

A forward has a single exchange, at maturity (L1.7.02: Forwards). Nothing amortises, so diffusion wins all the way: exposure rises like $\sqrt t$ and drops to zero only when the forward settles. Halvergate's one-year forward on $65m peaks at £1.76m just before settlement. Because GBPUSD is lognormal (M-C07: The lognormal distribution), its EE has an exact, Black-Scholes-like formula, which matches the simulation within a few per cent.

Cross-currency swaps: a forward on the whole notional

A cross-currency swap (L1.7.05: Currency and cross-currency swaps) adds a notional exchange at the end at the original exchange rate. That final exchange is a long-dated FX forward on the full principal, and it never amortises. So the profile looks like an FX forward's, rising to maturity, with a small interest rate hump on top. The extra X1 (receive $130m at 4.60%, pay £100m at 4.00%, five years) peaks at £7.30m, against £1.96m for a plain £100m five-year swap. It is one reason why cross-currency swaps between banks are usually collateralised.

Options: bought or sold

Once the premium is paid, a bought option can only be an asset: its value is never negative. So its exposure is its whole value, and its EE is roughly the premium growing at the risk-free rate until expiry. The bank's put on Saltney is worth £18.98m today and has EE £19.89m in a year. A sold option is the mirror: the seller can only owe, so its exposure is zero. Saltney, which sold the put, has no exposure to the bank on it: its EE on the trade is £0m throughout.

Payer against receiver

On the book's flat curve, a payer swap and a receiver swap have almost mirror-image profiles. Real yield curves usually slope upwards. Then the fixed rate is above the early floating fixings and below the later ones, so a payer expects to pay out early and be paid later: its expected value drifts up over the swap's life, and so does its exposure. On an illustrative curve rising from 3% to 5% over ten years (an extra, par rate 3.93%), the payer's expected value is £2.20m in three years and its EE peaks at £5.80m, against £3.51m for the receiver.

Payer: bank's EEReceiver: bank's EEPayer: expected value

ExploreThe book's trades, one at a time

Simulated on the book's 2,000 paths. Illustrative.

Trade

Receive-fixed swap, £100m, 7 years at 4.00% (HAL)

Bank's EE to the counterpartyCounterparty's EE to the bank

Show the bank's 95% PFE too

Bank's peak EE£2.96m

… reached after2.96 years

Counterparty's peak EE to the bank£2.78m

… reached after2.96 years

A seven-year swap: the hump peaks around year 2 to 3 (T/3 ≈ 2.3), with a small drop at each annual payment.

The formal version

Exposure of a normal value. Suppose a trade's value at $t$ is normal, $V \sim \mathcal N(\mu, \sigma^2)$ (M-C06: The normal distribution). Write $V = \mu + \sigma Z$ , with $Z$ standard normal and density $\varphi$ . Then

\begin{aligned} \text{EE} = \mathbb E[\max(V, 0)] &= \int_{-\mu/\sigma}^{\infty} (\mu + \sigma z)\,\varphi(z)\,dz \\ &= \mu\,\Pr\!\left(Z > -\tfrac{\mu}{\sigma}\right) + \sigma \int_{-\mu/\sigma}^{\infty} z\,\varphi(z)\,dz \\ &= \mu\,\Phi\!\left(\tfrac{\mu}{\sigma}\right) + \sigma\,\varphi\!\left(\tfrac{\mu}{\sigma}\right) \qquad \text{since } \textstyle\int_a^\infty z\varphi(z)\,dz = \varphi(a) \\ \mu = 0:\quad \text{EE} &= \sigma\,\varphi(0) = \frac{\sigma}{\sqrt{2\pi}} \approx 0.3989\,\sigma \\ \text{PFE}_q &= \max\bigl(\mu + \sigma\,\Phi^{-1}(q),\ 0\bigr), \qquad \text{PFE}_{95\%} = 1.645\,\sigma \text{ when } \mu = 0 \end{aligned}

So for an at-market trade everything follows from $\sigma_V(t)$ , the standard deviation of its value at each date.

FX forward, exactly. In the book, GBPUSD $X$ is lognormal, so $Y = 1/X(t)$ is lognormal too (M-C07: The lognormal distribution): $\ln Y \sim \mathcal N(m, s^2)$ with $s = \sigma_X\sqrt t$ . A forward to receive dollars $U$ for pounds $G$ at $T$ is worth $V = aY - b$ , with $a = U e^{-r_{\$}(T-t)}$ and $b = G e^{-r_{\text{£}}(T-t)}$ . The expected positive part of a lognormal less a constant is the Black-Scholes calculation:

\begin{aligned} \text{EE}(t) &= \mathbb E[\max(aY - b, 0)] = a\,\mathbb E[Y]\,\Phi(d_1) - b\,\Phi(d_2) \\ \mathbb E[Y] &= e^{m + s^2/2}, \qquad d_1 = \frac{m + s^2 - \ln(b/a)}{s}, \qquad d_2 = d_1 - s \end{aligned}

With $s = \sigma_X\sqrt t$ , this rises with $t$ until settlement. For Halvergate's forward it gives £1.238m at six months against £1.213m simulated, and for Corriemoor's three-year forward £5.477m at two and a half years against £5.473m.

Options. A bought option's value is never negative, so $\text{EE}(t) = \mathbb E[V(t)]$ . Under the book's risk-neutral index drift that is $V(0)\,e^{rt}$ : £19.75m in a year for the put, against £19.89m simulated.

Try it

Pick a trade type and a maturity and watch the shape. Lengthen the swap: the peak moves out to stay near $T/3$ . Switch to a cross-currency swap and see the end of the profile take off. Sell the option and the bank's exposure disappears.

Try itShape an exposure profile

Trade (£100m)

The bank's side

Maturity6 years

Rate volatility (normal, a year)90bp

Bank's EECounterparty's EE to the bank

Bank's peak EE£2.33m

… after2.96 years

Peak as a share of notional2.33%

Counterparty's peak EE£2.33m

Approximations, not the simulation: swaps φ(0)·σ_V(t) on the flat 4.00% curve; FX forwards exact under the book's lognormal GBPUSD; the cross-currency swap a normal approximation; the put's value grown at the 4% rate. Illustrative.

Worked example

Halvergate's swap at two and a quarter years

T1: the bank receives 4.00% on £100m for seven years. Rate volatility 90bp a year (normal), flat curve at 4.00%. Estimate EE and 95% PFE at $t = 2.25$ , near the peak, and check against the simulation.

From the annuity to EE

Step 1 of 5

1
Payments left and the annuity
At 2.25 years, 5 annual payments remain, due in 0.75, 1.75, …, 4.75 years:
$A(2.25) = \sum_{j=0}^{4} 1.04^{-(0.75 + j)} = 4.4957$

Same check for the cross-currency swap

The normal approximation for X1, with the FX and rate moves combined through their correlation of 0.3, gives £7.19m in the last month against £7.30m simulated, and £5.01m against £5.11m at about two and a half years.

Regulatory lens

Regulators build these shapes into the standardised approach. SA-CCR (M4.4: Counterparty, operational and large exposures) gives each trade an add-on built from its notional, a supervisory factor for its asset class and, for interest rate and credit trades, a supervisory duration that grows with maturity. FX trades get a much higher factor than interest rate trades (4% against 0.5%), reflecting the full-notional FX risk this lesson shows. Banks using internal models must simulate the profiles, as here, and show they capture each product's risk factors. Full exposure simulation, including sloping curves and stochastic volatility, is in M6.6: Counterparty risk and XVA.

Spark Atoms

Spark AtomA third of the way through

The T/3 rule holds for any swap maturity on a flat curve, and it is a handy check on a model: a ten-year swap whose simulated EE peaks in year 8, or in month 2, deserves a second look.

Spark AtomWhy exposure is a few per cent of notional

Halvergate's £100m seven-year swap peaks at £2.96m of EE, about 3.0% of notional. The cross-currency swap of the same size peaks near 7%. Notional tells you the size of the payments, not the size of the risk.

Key takeaways

Diffusion makes exposure grow like √t; amortisation shrinks it as payments run off.
Swaps: a hump, EE ≈ 0.4 × N σ √t A(t), peaking near T/3, with a sawtooth at payment dates.
FX forwards and cross-currency swaps: exposure rises to maturity, because the final exchange never amortises.
Cross-currency swaps carry far more exposure: £7.3m against £2.0m for a plain £100m five-year swap.
Bought options: exposure is the whole value. Sold options: none, once the premium is paid.
On an upward-sloping curve a payer swap carries more exposure than a receiver.

Reaction

Check what stuck

Question 1 · objective 4

An at-market trade's value in two years is roughly normal with mean zero and standard deviation £10m. What is its expected exposure then, in £m (two decimals)?

£m

Question 2 · objective 4

And its 95% PFE then, in £m (two decimals)?

£m

Question 3 · objective 1

Roughly when does a nine-year at-market interest rate swap's expected exposure peak, in years?

years

Question 4 · objective 2

Why is a cross-currency swap's exposure so much larger than an interest rate swap's on the same notional?

AThe principals are swapped back at the original exchange rate, so the whole notional carries FX risk until maturityBCross-currency swaps have higher interest ratesCThey are never collateralisedDTheir notional is paid in two currencies, so it counts twice

Question 5 · objective 3

The bank has sold an option and been paid the premium. What is its counterparty exposure on the option?

ANone: the bank can only owe on it; it is the buyer who is exposed to the bankBThe premiumCThe option's delta times notionalDThe same as if it had bought the option

Glossary

Diffusion: Diffusion: the spread of possible market levels widening with time, roughly in proportion to the square root of the horizon.
Amortisation (of exposure): Amortisation (of exposure): the shrinking of what is left to exchange as a trade's payments are made, so less value can build up.
Hump: The humped profile of a swap: exposure rises as rates diffuse, then falls as payments run off, peaking about a third of the way through.
Sawtooth: Sawtooth: the small drops in a swap's exposure on each payment date, as one payment leaves the trade.
Notional exchange: Notional exchange: in a cross-currency swap, the two principals are swapped at the start and swapped back at the end at the original exchange rate.
Payer swap: Payer swap: the bank pays fixed and receives floating. It gains when rates rise.
Receiver swap: Receiver swap: the bank receives fixed and pays floating. It gains when rates fall.

Deep Dives & Origins

Deep Dives

J. Gregory, The xVA Challenge, 4th ed., Wiley, 2020, chapter 11, “Future Value and Exposure”, with the typical exposure profiles of common products. Intermediate.
J. C. Hull, Options, Futures, and Other Derivatives, 11th ed., Pearson, 2021, chapter 9, “XVAs”, on credit value adjustment and its relatives. Intermediate.

Origins

M. Pykhtin and S. Zhu, “A guide to modelling counterparty credit risk”, GARP Risk Review, July/August 2007, pp. 16–22.
Basel Committee on Banking Supervision, Basel Framework, CRE52 (standardised approach to counterparty credit risk), BIS.

Watch this lesson

Video version

Prefer to watch? The same lesson as a 12-minute video will sit here, made from this page's script.