Last lesson you learned the one-sentence definition of counterparty risk: when a trade is worth money to you and your counterparty defaults, you lose that money — and exposure is , breathing as the market moves. That’s the what. This lesson is the how much. Because “they might not pay” is useless to a risk desk until it has a number, and exposure isn’t one number — it’s a whole distribution that stretches out into the future. By the end you’ll be able to read an exposure profile like a doctor reads an EKG: the swap’s tidy hump, the FX forward’s relentless climb, and the flat line that collateral draws through both. Let’s put numbers on the fear.
Before you read — take a guess
You have a single swap with one counterparty. Today it's worth +$2m to you. Before any modelling — what's the most honest statement about how much that counterparty could cost you if they default?
Exposure is replacement cost: max(value, 0)
Analogy. A counterparty trade is a half-finished home renovation with a contractor. If you’ve paid in advance and they’ve done little of the work — the deal is “in the money” to you — and they vanish, you’re out the money: you must hire someone else to finish, at today’s prices. That re-hiring cost is your loss. But if you still owe them for work already done and they vanish, you lose nothing on their disappearance — you simply… still owe the bill. The loss only flows one way: toward whoever is owed.
The precise definition. Exposure is replacement cost: if your counterparty defaults right now, what does it cost to replace the trade at market? If the trade is worth to you, you must go re-establish it with someone solvent and pay to do so — that’s your loss. If it’s worth to you (you’re underwater on it), replacing it actually frees you, so the default costs you nothing. Hence the one-sided formula you met last lesson:
The trade’s value is two-sided — it swings positive and negative as markets move. But your exposure is one-sided — it only ever counts the positive part. That asymmetry is the whole reason counterparty risk is hard: you eat the bad case (they default when they owe you) and get no credit for the good case (they default when you owe them — you still pay).
Worked example. One interest-rate swap, two snapshots:
| Snapshot | Mark-to-market to you | If they default now, you… | Exposure |
|---|---|---|---|
| Rates moved your way | +$2m | must replace at market, costing $2m | $2m |
| Rates moved against you | −$2m | replace and walk free — but you still owe $2m | $0 |
In the first row you’re owed $2m, so default costs you $2m. In the second you owe $2m, so their default is — bluntly — not your problem; you still have to settle up. Same swap, same $2m magnitude, two completely different exposures, because the clips off the half where you’re the debtor.
The asymmetry is the trap
The instinct is “the swap is worth $2m either way, so my risk is $2m either way.” Wrong. When you’re the one who owes, their default doesn’t refund your debt — you lose nothing from them defaulting and gain nothing either. Exposure throws away the favourable half of the value distribution. That’s why you can’t just look at net mark-to-market and call it risk: you have to model the positive part, and the positive part alone.
Fill in the definition of exposure.
Pick the right option for each blank, then check.
Exposure is the if your counterparty defaults, equal to . The trade's value V is , swinging both ways, but exposure is — it counts only the part where the counterparty owes you.
Current exposure vs. potential future exposure
Before you read — take a guess
Your risk system shows current exposure to a counterparty of $2m. Why is that single number a dangerously incomplete picture of the risk?
The definition. Split exposure into two questions: how much now and how much could it become.
Current exposure (CE) is the easy one — it’s just . You can read your marks off the screen, take the positive part, done. It’s known, deterministic, no modelling required.
The hard, interesting part is potential future exposure. Default is almost never today — it’s a future event, landing weeks, months, or years out. Between now and then the trade’s mark wanders, and by the time the counterparty actually folds, the exposure could be much larger than it is now. So we can’t get away with a snapshot. We have to model the whole distribution of future exposure at each future date: simulate thousands of market paths, revalue the trade along each one, take at every step, and look at the spread of outcomes.
Worked example. A swap is , i.e. $2m, today. We simulate it forward one year and get a distribution of possible exposures: most paths cluster near $2–3m, but a chunky tail reaches $6m, and some paths leave it worthless (the mark went negative, exposure clipped to $0). The current exposure is a flat $2m. The future exposure is that entire fan of outcomes — and the risk that bites you is whether the counterparty defaults on a high path or a low one. One snapshot can’t tell you that; a distribution can.
Two clocks ticking
Counterparty risk runs on two clocks at once: when does the mark move (market risk — the value drifts around) and when do they default (credit risk — the counterparty fails at some unknown future date). Exposure measurement is where those two clocks collide. You’re asking, “across all the futures where they default, how deep in the money is the trade at the moment they fail?” That’s why a single current number isn’t enough — you need the exposure’s shape across every future date a default could land on.
Match each exposure idea to what it actually is.
Pick a term, then click its definition.
Expected Exposure (EE) and EPE
Before you read — take a guess
At a future date t, your simulation gives many possible trade values — some positive, some negative. To compute expected exposure EE(t), what do you average?
The definition. Expected Exposure is the average exposure at a future time — specifically, the mean of the positive part of value across the simulated distribution:
The order of operations is everything. You clip first (take on each path, turning every negative into a zero) and average second. Do it the other way — average first, then clip — and the negative paths cancel out positive exposure on other paths, leaving you with a number that badly understates the risk. Clip, then average.
The worked example. Forget thousands of paths; do three by hand. At time , three simulated values for one trade:
| Path | Value | Exposure |
|---|---|---|
| A | +$6m | $6m |
| B | −$2m | $0 |
| C | +$3m | $3m |
Clip first: the exposures are $6m, $0, $3m. Average second (in $m):
So the expected exposure at one year is $3m.
Now watch what the wrong order does. Average the raw values: , i.e. $2.33m — path B’s negative leaked in and reduced the answer, even though a counterparty defaulting on path B costs you nothing (you owe them). The $2.33m understates the true $3m exposure. Clip, then average. Always.
EPE — squashing the curve to one number. is a curve — one value per future date. Expected Positive Exposure (EPE) flattens that curve into a single summary by averaging over the life of the trade:
If reads $2m, $3m, $3m, $2m, $0 across five annual points, EPE is roughly their average, about $2m. EPE is the “typical exposure over the whole life” — and, as you’ll see, it’s the headline number that feeds pricing (the CVA you’ll meet two lessons from now).
Clip then average — the #1 exposure mistake
The single most common error in hand-built exposure calculations is averaging raw P&L and then taking the positive part. That lets in-the-money and out-of-the-money paths net against each other, which is exactly the netting exposure does not give you for a single trade’s positive part. A defaulting counterparty doesn’t refund you on the paths where you owe them. Take per path, first; average after. If you remember one thing from this section, make it the order.
Fill in the EE / EPE machinery.
Pick the right option for each blank, then check.
Expected exposure EE(t) is the average of across the simulated paths at time t — you clip , then average. EPE is the of the EE curve across the life, collapsing the whole profile into a single number that feeds .
Potential Future Exposure (PFE)
Before you read — take a guess
A risk officer wants to set a credit limit on a counterparty — a line they should never plausibly breach. Which exposure measure should they use, and why?
The definition. Potential Future Exposure (PFE) answers a different question from EE. EE asks “on average, how exposed am I?” PFE asks “in a plausibly bad case, how exposed could I get?” It’s a high quantile of the exposure distribution at a horizon — typically the 95th or 97.5th percentile. Not the worst case (that’s unbounded and useless), and not the average (that’s EE) — the bad-but-believable case, the kind of number you’d be embarrassed to be caught above.
Why a quantile, not a max. The single worst simulated path is noise — one freak draw tells you nothing actionable. The 95th percentile says: “only 1 path in 20 is worse than this.” That’s a line you can manage to. PFE is the headroom number: it’s what credit limits are built on. A desk gets a PFE limit per counterparty, and if a new trade would push the PFE profile above that line, the trade is blocked or needs more collateral.
Worked example. At a one-year horizon, sort the simulated exposures from low to high. Say 20 representative paths come out (in $m):
The average of these is the EE — work it out and it’s about $3.1m. The 95th percentile sits near the top: with 20 points, the 95th percentile is around the 19th value, so , i.e. $7m. Notice the gap: EE is $3.1m but PFE is $7m — more than double. That gap is the whole point. You price the trade off the $3.1m (the average pain), but you limit the counterparty off the $7m (the plausible bad day). Confuse the two and you’ll either misprice or under-cap.
Two numbers, two jobs — don't mix them up
EE feeds pricing. PFE feeds limits. Say it twice. The average exposure (EE/EPE) is what you charge for — CVA is essentially “EE integrated against the probability of default.” The quantile exposure (PFE) is what you bound — limits keep any single counterparty from getting big enough to hurt you on a bad day. Same simulated distribution, two different statistics, two different jobs. A risk system that prices off PFE will overcharge; one that limits off EE will under-protect.
Sort each fact under the exposure measure it belongs to.
Place each item in the right group.
- Its time-average over the life is a single pricing summary
- A new trade is blocked if it pushes this profile too high
- It is the input that CVA is built on
- It is the average of max(V, 0) across paths
- It is a high quantile, e.g. the 95th percentile
- It caps how big a single counterparty can get
The shape of exposure: diffusion vs. amortisation
Before you read — take a guess
An interest-rate swap's exposure profile rises, peaks somewhere in the middle of its life, then falls back to zero at maturity. What pair of opposing forces produces that hump?
The two forces. Why does an exposure profile have a shape at all? Two forces pull in opposite directions across a trade’s life:
- Diffusion (pushes exposure UP). The further into the future you look, the more the underlying could have moved, and that uncertainty grows roughly like — the same square-root-of-time spreading you’ve seen in volatility scaling. More dispersion in the mark means a fatter positive tail, means more exposure. Diffusion alone would make exposure rise forever.
- Amortisation (pulls exposure DOWN). But trades don’t last forever, and many pay themselves down as they go. Every coupon or settlement on a swap is one more cash flow exchanged and gone — leaving less remaining value left to replace if the counterparty later defaults. As maturity approaches, there’s almost nothing left, so exposure must collapse to zero at maturity.
The interest-rate swap: a hump. A swap feels both forces. Early in its life, diffusion dominates — uncertainty is piling up faster than the cash flows are amortising — so exposure rises. Later, the swap has paid down most of its flows, amortisation takes over, and exposure falls back to zero at maturity. The result is the swap’s signature hump, peaking roughly one-third of the way through the life. (That isn’t folklore: for the simple proxy , the maximum is exactly at .)
The FX forward: a monotonic climb. Now strip out amortisation. An FX forward has a single exchange of currencies at maturity — no intermediate cash flows, nothing paid down along the way. So amortisation never gets to act. Only diffusion is left, and diffusion only ever pushes up. The exposure therefore rises monotonically, biggest on the very last day, when the accumulated uncertainty about the exchange rate is at its maximum and the whole notional is about to change hands. Same family of math, opposite shape — and the difference is entirely “does this trade pay itself down, or does it all land at the end?”
The island below lets you flip between the two and watch the shapes diverge. The defaults open on the swap’s hump; toggle to the FX forward and watch the curve straighten into a climb.
- Peak PFE
- 2.50×
- EPE (avg EE)
- 0.69×
A swap feels two opposing forces: diffusion (uncertainty about the mark grows like √t, pushing exposure up) and amortisation (each settled payment leaves less left to replace, pulling it down). The result is a hump — peaking about a third of the way in, then sliding back to zero at maturity when nothing is left to exchange.
Toggle between Interest-rate swap and FX forward and watch where each one peaks: the swap humps and slides back to zero at maturity (diffusion up, amortisation down), while the FX forward climbs all the way to a maturity peak (diffusion only — nothing amortises). The solid line is EE (the pricing average); the dashed band is PFE (the bad-case quantile for limits) — notice PFE sits well above EE, the same gap you computed by hand. Then tick 'Add collateral (CSA)' to see the entire profile collapse to a thin band — a preview of the netting-and-collateral machinery coming up in the next two lessons.
The shape is a fingerprint
Once you’ve internalised these two forces, you can guess a trade’s exposure shape before you simulate it. Lots of intermediate cash flows that pay down the notional? Expect a hump — diffusion early, amortisation late. One big bullet exchange at the end with nothing in between? Expect a monotonic climb to a maturity peak. A forward-starting trade that doesn’t begin for two years? Expect exposure pinned near zero until it starts, then rising. The profile is a fingerprint of the trade’s cash-flow timing, read through the lens of diffusion-versus-amortisation.
Fill in the two forces that shape an exposure profile.
Pick the right option for each blank, then check.
Exposure is shaped by two opposing forces: , which pushes exposure up as uncertainty grows like the square root of time, and , which pulls it down as cash flows settle and there's less left to replace. An interest-rate swap feels both, giving a that peaks about a third of the way in. An FX forward has no amortisation — one exchange at maturity — so it .
From a profile to one number
Before you read — take a guess
Regulators need a single exposure number per netting set to plug into a capital formula. Which family of measures does that regulatory 'Exposure at Default' (EAD) come from?
From a curve to a scalar. You’ve now got a whole profile — EE at every future date, a PFE band, a shape driven by diffusion and amortisation. Beautiful, but a capital formula can’t eat a curve; it wants one number. So the last move is to collapse the profile into a single regulatory summary.
The bridge is the expected-exposure family you already met. Start with EPE (the time-average of EE). Regulators tweak it into effective EPE — essentially EPE computed on a non-decreasing version of the EE profile, so a trade’s exposure measure can’t drop just because the next trade hasn’t been added yet (it stops banks from gaming roll-off). Then scale by a regulatory multiplier (the famous alpha, ) to get Exposure at Default (EAD) — the single figure that feeds the capital charge.
We’re not deriving that here — the full SA-CCR machinery that computes EAD gets its own lesson later in this course. For now, just hold the chain in your head: simulate → EE profile → EPE → effective EPE → × alpha → EAD → capital. Every box in that chain is a way of squeezing the same underlying object — the distribution of over time — down to whatever shape the next consumer needs: a curve for intuition, an average for pricing, a quantile for limits, a single scalar for capital.
One distribution, many faces
Everything in this lesson is a different view of one object: the distribution of across future dates. EE is its average at each date. EPE is that average over time. PFE is its high quantile. Effective EAD is the regulatory scalar. You don’t compute four unrelated things — you compute one exposure distribution and then read off whichever statistic the question demands. Master the distribution and every exposure metric is just a lens you hold up to it.
Putting it together
Exposure is replacement cost — — and it’s brutally one-sided: value swings both ways, but you only ever lose when the counterparty owes you. Current exposure is today’s known snapshot; the real risk is potential future exposure, because default lands later, when the mark may have wandered far higher, so we model the whole future distribution. From that distribution we read two headline statistics: Expected Exposure , the mean of — clip each path first, average second — whose time-average EPE feeds pricing (CVA); and Potential Future Exposure, a high quantile (95th/97.5th percentile), which feeds credit limits. The profile’s shape is a tug-of-war: diffusion (uncertainty ~, pushing up) versus amortisation (cash flows settling, pulling down). A swap feels both and humps to a peak about a third of the way in, then back to zero at maturity; an FX forward has no amortisation and climbs monotonically to a maturity peak. And to feed capital, the same distribution collapses one last time — EPE → effective EPE → × alpha → EAD — a single number waiting on the SA-CCR lesson ahead.
Big picture
Measuring exposure
- Measuring Exposure = views of max(V, 0)
- Exposure = replacement cost
- Exposure = max(V, 0)
- Value is two-sided; exposure is one-sided
- +$2m owed to you → $2m loss; −$2m → $0
- Current vs. future
- Current exposure = max(MtM now, 0), known
- Default lands later → model the future
- Simulate paths, revalue, clip at max(V,0)
- EE & EPE → pricing
- EE(t) = mean of max(V,0): clip first, average second
- 3-path example: (6 + 0 + 3)/3 = $3m
- EPE = time-average of the EE curve
- EPE feeds CVA (pricing)
- PFE → limits
- A high quantile (95th / 97.5th percentile)
- The plausible bad case, not the average
- Sets credit limits; blocks oversized trades
- Shape: diffusion vs. amortisation
- Diffusion (~√t) pushes exposure UP
- Amortisation pulls it DOWN to zero at maturity
- Swap = hump, peaks ~1/3 of the way in
- FX forward = monotonic climb to maturity peak
- Profile → one number
- EPE → effective EPE (non-decreasing)
- × alpha (≈ 1.4) → EAD
- EAD feeds capital — SA-CCR lesson ahead
- Exposure = replacement cost
Recap: measuring exposure
A swap is worth −$3m to you (you owe the counterparty). They default. What is your exposure on this trade?
Check your answer to continue.
Next — Netting & Collateral — that thin band you saw when you ticked the collateral box wasn’t a rounding error. It’s the single most powerful lever in counterparty risk: netting agreements that let offsetting trades cancel, and CSAs that demand margin as the mark moves. We’ll show how they shrink those humps and climbs down to almost nothing — and exactly how much “almost” is.