What an Exposure Profile Measures
A counterparty exposure profile summarizes how much a portfolio could be owed by one counterparty at future dates. It is built from simulated future portfolio values, not from the portfolio’s current value alone. At each date, a simulated value below zero represents an amount owed by the institution; for unsecured positive exposure, that path contributes zero rather than a negative amount.
The simulation method and probability measure should match the purpose of the calculation. For example, a valuation application and a risk-management application may use different scenario measures. The resulting expected exposures are therefore conditional on the chosen model, measure, valuation assumptions, and set of trades. The calculations below do not include collateral.
An exposure profile is the set of date-specific results \(\mathrm{EE}(t)\), and often \(\mathrm{PFE}_q(t)\), across the dates being studied. EE answers, “What is the average positive exposure at this date?” PFE answers, “What exposure level is not exceeded in approximately \(q\) of the modeled scenarios at this date?” Neither quantity is itself a default probability or a loss given default.
The term EPE is not always implemented with the same date weights. For a continuous profile over a horizon of length \(T\), a time-average definition is \(\frac{1}{T}\int_0^T \mathrm{EE}(t)\,dt\). With discrete dates, a model may use specified time weights, or a stated numerical approximation to that integral. If dates are equally weighted instead, the result is a date-average. State the convention rather than treating those summaries as interchangeable.
From Simulated Values to Exposure
For each future date, apply the positive-part operation to each simulated portfolio value first, then average the resulting exposures using the simulation probabilities. If there are \(N\) equally weighted simulations, each path has probability \(1/N\). With unequal scenario probabilities \(w_s\), the weights must be nonnegative and sum to one.
This order matters. Averaging the signed values first and then taking the positive part would calculate \(\max(E[V_t],0)\), not the expected positive exposure. Negative values in some scenarios do not cancel positive exposures in other scenarios when calculating EE.
Worked Example: Netting Two Trades Across Simulated Paths
A hypothetical agreement covers two trades with the same counterparty. At one future date, four equally likely simulations give the following values, in thousands of dollars. Assume the trades may be netted under the agreement for this calculation. Find the gross trade-by-trade EE, the netted EE, and the netted 75% PFE. For this discrete example, define a quantile as the smallest exposure whose cumulative probability is at least the stated level.
| Simulation | Trade A | Trade B |
|---|---|---|
| 1 | 80 | -20 |
| 2 | 50 | 30 |
| 3 | -30 | 90 |
| 4 | -70 | -10 |
Method. For gross exposure, take the positive part of each trade separately and add the results on each path. For exposure after netting, first add the trade values on each path and then take the positive part.
Gross exposure by path. The four amounts are \(\max(80,0)+\max(-20,0)=80\), \(\max(50,0)+\max(30,0)=80\), \(\max(-30,0)+\max(90,0)=90\), and \(\max(-70,0)+\max(-10,0)=0\). Their mean is \((80+80+90+0)/4=62.5\) thousand dollars.
Netted exposure by path. The combined values are \(80-20=60\), \(50+30=80\), \(-30+90=60\), and \(-70-10=-80\). Taking positive parts gives \(60,80,60,0\). Thus netted EE is \((60+80+60+0)/4=50\) thousand dollars.
Netted PFE. Sort the netted exposures: \(0,60,60,80\). Each has probability \(0.25\), so cumulative probabilities are \(0.25,0.50,0.75,1.00\). The smallest exposure with cumulative probability at least \(0.75\) is \(60\) thousand dollars. Therefore, the 75% PFE is \(60\) thousand dollars.
Check. Netted EE is \(12.5\) thousand dollars below gross EE, and the 75% quantile is one of the simulated netted exposures. The common mistake is to net trade values only after taking each trade’s positive part; that computes the gross amount, \(62.5\), rather than the exposure under the assumed netting set, \(50\).
Why Netting Changes Exposure
Netting is applied to values within a covered set of transactions before the positive-part operation. It is not justified merely because two positions have opposite signs in a spreadsheet: the modeled netting set must correspond to transactions that can be treated as one enforceable set for the intended analysis. In the example, the ability to net was an explicit assumption.
The mathematical reason netting cannot increase pathwise exposure, when applied to the same trades, follows from a simple inequality. For trade values \(V_1,\ldots,V_m\), the positive part of their sum is no greater than the sum of their positive parts.
The left side is zero. Every term on the right is nonnegative, so the inequality holds.
For each trade, \(V_j\leq\max(V_j,0)\). Adding these inequalities gives \(\sum_j V_j\leq\sum_j\max(V_j,0)\). The left side of the displayed inequality is then \(\sum_j V_j\), so the inequality again holds.
Because the pathwise inequality holds in every simulation, averaging with nonnegative scenario weights preserves it. Therefore, under the same scenarios and weights, netted EE cannot exceed gross EE.
This result is specific to the comparison of the same positions under the stated netting assumption. It does not establish that a legal netting right exists, and it does not mean every risk measure or every model output must decline by the same amount. For a PFE comparison, the pathwise inequality also implies that netted exposure is no greater than gross exposure in each matched scenario; the corresponding ordered quantiles therefore cannot be greater under the same scenario probabilities.
Profiles Across Dates and EPE
Calculate exposure separately at each future date. The scenarios at different dates may be linked along simulated paths, but EE at a date is calculated from the distribution of values at that date. To calculate a date-average EPE, average the date-specific EE values using the chosen date weights. If the dates represent unequal time intervals, an unweighted average generally does not represent a time-average.
Worked Example: Calculating a Discrete Exposure Profile
A hypothetical portfolio has five equally likely simulated paths. The table gives the portfolio’s net value, in thousands of dollars, at three equally spaced future dates. At each date, calculate EE and the 80% PFE. Then calculate the EPE defined here as the equally weighted average of the three date-specific EE values. Use the smallest simulated exposure with cumulative probability at least \(0.80\) as the PFE.
| Path | \(V_{0.5}\) | \(V_{1.0}\) | \(V_{1.5}\) |
|---|---|---|---|
| 1 | 40 | -20 | 100 |
| 2 | 100 | 60 | -50 |
| 3 | -30 | 120 | 80 |
| 4 | 0 | 40 | 150 |
| 5 | 60 | -10 | -20 |
At \(t=0.5\). Positive exposures are \(40,100,0,0,60\). Their sum is \(200\), so \(\mathrm{EE}(0.5)=200/5=40\) thousand dollars. Sorted exposures are \(0,0,40,60,100\). The fourth value reaches cumulative probability \(4/5=0.80\), so \(\mathrm{PFE}_{0.80}(0.5)=60\) thousand dollars.
At \(t=1.0\). Positive exposures are \(0,60,120,40,0\). Their sum is \(220\), giving \(\mathrm{EE}(1.0)=220/5=44\) thousand dollars. The sorted exposures are \(0,0,40,60,120\); the fourth value is \(60\), so \(\mathrm{PFE}_{0.80}(1.0)=60\) thousand dollars.
At \(t=1.5\). Positive exposures are \(100,0,80,150,0\). Their sum is \(330\), giving \(\mathrm{EE}(1.5)=330/5=66\) thousand dollars. The sorted exposures are \(0,0,80,100,150\); the fourth value is \(100\), so \(\mathrm{PFE}_{0.80}(1.5)=100\) thousand dollars.
EPE under the stated convention. The date-average is \((40+44+66)/3=50\) thousand dollars. Thus the profile reports EE values of \(40,44,66\), PFE values of \(60,60,100\), and a discrete-date EPE of \(50\), all in thousands of dollars.
Check. Each EE lies between the smallest and largest exposure at its date. The common mistake is to take the positive part of the average signed portfolio value at a date. That would let negative paths offset positive paths and would not produce the expected positive exposure.
Potential Future Exposure and Scenario Weights
PFE is a percentile of the exposure distribution at a particular date, not an average and not the maximum except when the chosen quantile or discrete simulation happens to select the maximum. For a finite set of simulations, the quantile convention matters, especially when scenarios have unequal probabilities or the target percentile falls between cumulative probabilities. State whether the convention is the inverse empirical distribution, interpolation, or another specified rule.
Worked Example: Weighted Scenarios and a Tail Quantile
A hypothetical counterparty portfolio is simulated at a future date under five scenarios. The scenario weights reflect the model probabilities used for this calculation. The table gives the net portfolio value in thousands of dollars before taking its positive part. Calculate EE and the 80% and 95% PFEs. Define each PFE as the smallest exposure whose cumulative scenario weight is at least the requested confidence level.
| Scenario | Net value | Weight | Exposure | Cumulative weight (sorted) |
|---|---|---|---|---|
| A | -30 | 0.10 | 0 | 0.10 |
| B | 20 | 0.20 | 20 | 0.30 |
| C | 50 | 0.25 | 50 | 0.55 |
| D | 90 | 0.25 | 90 | 0.80 |
| E | 160 | 0.20 | 160 | 1.00 |
Check the weights. They sum to \(0.10+0.20+0.25+0.25+0.20=1.00\), so they define a probability distribution for this calculation. The exposure in each row is \(\max(V,0)\).
Expected exposure. Weight each positive exposure by its scenario probability: \(0(0.10)+20(0.20)+50(0.25)+90(0.25)+160(0.20)=0+4+12.5+22.5+32=71\) thousand dollars.
80% PFE. The cumulative weight reaches \(0.80\) at exposure \(90\). Therefore, \(\mathrm{PFE}_{0.80}=90\) thousand dollars.
95% PFE. Cumulative weight at exposure \(90\) is only \(0.80\), which is below \(0.95\). The next exposure is \(160\), at which cumulative weight reaches \(1.00\). Therefore, \(\mathrm{PFE}_{0.95}=160\) thousand dollars under the stated discrete quantile convention.
Check. The 95% PFE exceeds the 80% PFE, as it should. The common mistake is to select the exposure at the 95th percentile by scenario row number, ignoring unequal weights. Here scenario weights—not row counts—determine the cumulative distribution.
Interpretation and Common Traps
Exposure profiles are useful because they retain both the size and timing of positive counterparty values. EE offers an average view; PFE gives a selected upper quantile; EPE compresses a profile into a horizon summary under a specified time-weighting convention. These are different summaries and generally cannot be substituted for one another.
- Taking positive parts in the wrong order. For a netting set, add covered transaction values on each path, then take the positive part. Taking positive parts trade by trade measures gross exposure instead.
- Allowing negative paths to cancel positive paths in EE. Calculate \(\max(V_t^{(s)},0)\) for every path before averaging.
- Ignoring the quantile convention. Discrete simulations may not have an observation exactly at the requested percentile. State how the quantile is selected, and use scenario weights when they are unequal.
- Mixing up confidence level and time horizon. A 95% PFE at one date is a quantile of exposure at that date under the modeled distribution; it is not, by itself, a 95% probability statement about exposure over the entire horizon.
- Reporting EPE without its time weights. A simple average across dates and an approximation to a time-integral can differ, particularly when dates are unevenly spaced.
- Treating modeled netting as a legal conclusion. The calculation relies on the stated netting-set assumption. Whether the positions can legally be netted is a separate matter.
These definitions and calculations are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus. When presenting a result, identify the future date, currency or scale, scenario weighting, netting assumption, and—in the case of EPE or PFE—the aggregation or quantile convention.
Check Your Understanding
For each problem, use the stated discrete quantile rule: select the smallest exposure whose cumulative probability is at least the requested level.
- Four equally likely simulations give net portfolio values of \(-40,20,60,100\) dollars at one date. Calculate EE and the 75% PFE. (Answer: EE = $45; 75% PFE = $60.)
- Two trades in an assumed netting set have simulated values \((70,-50),(-20,80),(30,10)\) dollars in three equally likely scenarios. Calculate gross trade-by-trade EE and netted EE. (Answer: Gross EE = $190/3 = $63.33; netted EE = $40.)
- Three equally weighted-by-probability exposure scenarios have exposures of $0, $40, and $100, with probabilities \(0.20,0.50,0.30\), respectively. Calculate EE and the 80% PFE. (Answer: EE = $50; 80% PFE = $100.)
- A model reports EE of $20, $50, and $80 thousand at successive dates. The specified EPE uses normalized date weights \(0.25,0.50,0.25\). Calculate EPE. (Answer: EPE = $50 thousand.)
- Four scenarios have net values of \(-20,10,50,120\) dollars and weights \(0.10,0.20,0.40,0.30\). Calculate EE and the 70% and 90% PFEs. (Answer: EE = $58; 70% PFE = $50; 90% PFE = $120.)