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Identify what drives the credit exposure profile calculation and summarise how posting collateral alters exposure.

FRM® Part II · Credit Risk Measurement and Management · Future Value and Exposure · FRM2.CR.19.c

The idea

The exposure profile of a trade is driven by two opposing forces. Diffusion is the growth in uncertainty of the underlying risk factors with time. For a trade with no cash flows the standard deviation of value grows with the square root of time, so exposure rises. Amortisation is the fall in remaining sensitivity as payments are made and maturity nears, so exposure falls. The balance gives each product its shape. Other inputs are volatility, correlation, the shape of the forward curve and netting. Let V be the trade value and C the collateral held. Collateral reduces exposure by the amount held, so the exposure that remains is Exposure = max(V − C, 0). It does not remove exposure, because the value can still move between the last margin call and close-out. Collateralised exposure therefore depends on the margin period of risk rather than trade maturity.

Formulas

Exposure of a trade with no interim cash flows

PFE(t) = PFE(1) × √t

PFE(1)
the exposure at the one-year date
t
the future date, in years

Diffusion only, as for a forward. A swap amortises, so this overstates its exposure late in life; scaling linearly with t is the other trap.

Exposure with collateral held

Exposure = max(V − C, 0)

V
the portfolio value at close-out
C
the collateral actually held, set at the last successful margin call

Collateral reduces exposure but lags the value by the margin period of risk, so a residual remains. Flooring V before deducting C is wrong.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which pair of effects mainly shapes the exposure profile of a derivative through its life?

  1. Counterparty default probability and loss given default, which scale the size of the exposure profileWhy not: PD and LGD turn exposure into expected loss; they do not shape the exposure profile.
  2. The bid-offer spread and the day-count convention, which fix the level of exposure at inceptionWhy not: Trading conventions do not drive the time shape of future exposure.
  3. Diffusion, which raises uncertainty with time, and amortisation, which cuts sensitivity as payments are made - correct
  4. Rating migration and recovery assumptions, which determine the future mark-to-market of the contractWhy not: These are credit quality inputs, not drivers of the contract's future value.

Exposure is a market-risk quantity: the underlying's volatility grows with time (diffusion), while scheduled payments and approaching maturity reduce how much value can still move (amortisation).

A harder one

An FX forward with no interim cash flows has a 99% PFE of USD 8 million at the one-year date. Assuming the distribution of the mark-to-market grows with the square root of time, the 99% PFE at the four-year date is closest to:

  1. USD 32 millionWhy not: Scales PFE linearly with time (8 x 4) instead of with the square root of time.
  2. USD 8 millionWhy not: Assumes PFE stays flat, ignoring that diffusion raises uncertainty over time.
  3. USD 4 millionWhy not: Divides by the square root of 4, treating the longer date as lower exposure as if amortisation applied.
  4. USD 16 million - correct

With diffusion only, PFE is proportional to sqrt(t): PFE(4) = 8 x sqrt(4/1) = 16. There is no amortisation because no cash flows are paid before maturity.

The hardest tier

A bank holds two five-year par swaps with the same notional, one paying fixed (payer) and one receiving fixed (receiver), each with a different counterparty. The forward curve is upward-sloping and the bank values off today's forwards. Which swap has the larger expected exposure profile?

  1. The payer swap, because forward rates later in its life exceed the par rate, so its expected value is positive - correct
  2. The receiver swap, because it receives the par rate that is higher than the floating rate in later periodsWhy not: Reverses the sign: with an upward curve the later floating payments exceed the fixed rate, benefiting the payer.
  3. They are identical, because the two swaps have equal notional and maturityWhy not: Symmetry holds only when the curve is flat; curve slope tilts expected values.
  4. Neither, because the expected exposure of a par swap is zero at every dateWhy not: The value is zero only at inception; forward-implied expected values are non-zero later.

Along an upward-sloping curve the par fixed rate sits below the later forward rates. At future dates the payer's remaining floating leg is expected to be worth more than the fixed leg, so its EE is higher and the receiver's lower.

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