The exposure profile of a fund's currency hedge, month by month, at 95 and 99 per cent, with expected exposure beside it and what daily margining does to both.
Potential future exposure (PFE) is a high quantile of what a derivative could be worth to you at a future date, counted only when it is positive. For an illustrative fund that sells USD 200 million forward for one year against the euro, with 8 per cent EUR/USD volatility, the 95 per cent PFE rises from USD 7.69 million after one month to USD 27.40 million at maturity, 13.70 per cent of notional. Daily margining cuts that peak to USD 5.31 million.
Worked in full in Financial Risk Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
A forward is worth zero on the day it is struck. After that it is an asset to one side and a liability to the other. If the dollar weakens, the bank owes the fund the difference, and if the bank defaults the fund loses it. PFE answers the credit officer's question: how large can that claim plausibly become, and when? It is a profile over time, not one number, and its peak is what is set against the counterparty limit.
A euro-denominated fund holds USD assets and hedges them by selling USD 200,000,000 one year forward at 1.1000 dollars per euro, the market forward on the trade date. EUR/USD follows a lognormal path with 8 per cent annual volatility; the forward rate is treated as a martingale and discounting is ignored, which over one year moves the figures by a few per cent at most. Quantiles are 95 per cent (z of 1.645) and 99 per cent (2.326). Exposure is measured in dollars, the currency of the notional; a limit set in euros would convert each scenario at its own future rate, which trims the upper tail somewhat.
| Input | Value |
|---|---|
| USD sold forward | 200,000,000 |
| Contract rate, USD per EUR | 1.1000 |
| EUR/USD volatility | 8% |
| Tenor, months | 12 |
| Margin period of risk if collateralised, business days | 10 |
Value to the fund at time t = USD 200m × (Ft / 1.1000 − 1), positive when the euro strengthens
Ft / K = exp(σ√t × z − σ²t / 2) at the chosen quantile z
PFE95(12 months) = 200m × (exp(0.08 × 1.645 − 0.0032) − 1) = 200m × 13.70% = USD 27.40m
Expected exposure EE(t) = 200m × (2N(σ√t / 2) − 1), the at-the-money option formula
Excel for PFE: =Notional*(EXP(Vol*SQRT(t)*NORM.S.INV(0.95)-Vol^2*t/2)-1); for EE: =Notional*(2*NORM.S.DIST(Vol*SQRT(t)/2,TRUE)-1)
The 95 per cent case at twelve months corresponds to EUR/USD at 1.2507. The linear shortcut, notional times z times volatility times the square root of time, gives 13.16 per cent, USD 26.32 million, which understates the lognormal upside slightly.
| Horizon | PFE 95% | % of notional | PFE 99% | Expected exposure | Fund owes bank, 95% |
|---|---|---|---|---|---|
| 1 month | 7.69 | 3.84% | 10.98 | 1.84 | 7.51 |
| 3 months | 13.43 | 6.72% | 19.33 | 3.19 | 12.89 |
| 6 months | 19.15 | 9.58% | 27.76 | 4.51 | 18.06 |
| 9 months | 23.61 | 11.80% | 34.41 | 5.53 | 21.97 |
| 12 months | 27.40 | 13.70% | 40.13 | 6.38 | 25.22 |
Exposure grows with the square root of time: the three-month PFE is 49.0 per cent of the twelve-month figure. Peak PFE is at maturity because nothing settles before then. Averaged over the year, the expected positive exposure is USD 4.49 million, the figure that drives a CVA charge (the swap version is in how to calculate CVA on an interest rate swap). The last column is the mirror image: the 5 per cent case in which the dollar strengthens and the fund owes the bank USD 25.22 million at maturity, which is a liquidity problem rather than a credit one, and the one more likely to break a fund.
Under a credit support annex with daily variation margin, the bank posts collateral as the value moves, so the fund is exposed only to the move between the last margin call and the close-out of a defaulted bank, the margin period of risk. With 10 business days out of 250, t is 0.040 of a year:
| EUR/USD volatility | Peak PFE, uncollateralised | EPE, uncollateralised | PFE, daily margin |
|---|---|---|---|
| 6% | 20.35 | 3.37 | 3.97 |
| 8% | 27.40 | 4.49 | 5.31 |
| 10% | 34.58 | 5.61 | 6.65 |
| 12% | 41.90 | 6.73 | 7.99 |
Margining removes 80.6 per cent of the peak at 8 per cent volatility, and roughly the same share at every other level in the table. The practical consequence shows up against limits. If the fund's risk policy allows USD 20.0 million of PFE per bank, the uncollateralised hedge uses 137.0 per cent of it: the fund can hedge at most USD 146 million with that bank. Collateralised, usage is 26.5 per cent and the same limit supports about USD 754 million. Thresholds and minimum transfer amounts in the CSA add uncollateralised exposure back on top of the margin-period figure, so the 5.31 million is a floor, not the answer. A lender that secures the hedge on a subscription facility often uses a single add-on of this kind rather than a full profile, as worked in how much subscription line capacity an FX hedge uses.
Using the current mark-to-market as the exposure. On the trade date it is zero, and a limit check against zero passes any size. The other mistake is quoting the regulatory exposure figure as the risk measure: SA-CCR, explained in how to calculate SA-CCR exposure at default, is a standardised capital formula with supervisory factors, not a quantile of this trade's distribution, and it answers a different question.
Model the forward's value at each date, take the 95 or 99 per cent positive tail, and read the peak against the limit. A one-year USD 200 million hedge at 8 per cent volatility peaks at USD 27.40 million uncollateralised and 5.31 million under daily margin. The book's own value at risk, stress and liquidity numbers for one fund, side by side, are in the free workbook for this case.
PFE is a quantile, the exposure that will not be exceeded with a stated confidence, and is used for credit limits. Expected exposure is the average of the positive values and feeds CVA and capital. On the worked USD 200 million forward at twelve months, 95 per cent PFE is USD 27.40 million while expected exposure is 6.38 million.
Because the rate can drift further the longer the horizon, so the spread of possible values grows roughly with the square root of time and nothing amortises before settlement. At three months the 95 per cent PFE is 49.0 per cent of its twelve-month level, close to the square root of a quarter, then it drops to zero once the contract settles.
With daily variation margin, exposure is limited to the move over the margin period of risk, the time to close out a defaulted counterparty. With 10 business days, the 95 per cent PFE on the worked hedge falls from USD 27.40 million to 5.31 million, 80.6 per cent less, before thresholds and minimum transfer amounts are added back.
This article is one calculation from Financial Risk Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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