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Describe and calculate the credit exposure measures of maximum exposure, effective expected positive exposure, expected positive and negative exposure, potential future exposure, expected exposure and expected mark-to-market.

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

The idea

Credit exposure is the positive part of a contract's future mark-to-market value, because a firm loses only when the counterparty owes it money. At a future date, expected exposure (EE) is the average of max(V, 0) across scenarios. Expected negative exposure (ENE) is the average of min(V, 0). Expected mark-to-market is their sum, EE + ENE, since the negatives are not floored. Potential future exposure (PFE) is a high percentile, such as 97.5% or 99%, of the exposure distribution. Its peak over time is the maximum exposure. Expected positive exposure (EPE) averages EE through time. Effective EPE first forces the EE profile never to fall (a running maximum) and then averages over the first year. The usual mistake is averaging negative values into EE. Remember: EE is a mean, PFE is a quantile.

Formulas

Expected exposure at a date

EE = E[max(V, 0)]

V
the simulated mark-to-market value of the netting set at that date

Average over all scenarios, with negative values counted as zero but kept in the count. Averaging the negatives in gives expected mark-to-market instead.

Expected mark-to-market

Expected MtM = EE + ENE

EE
expected exposure, the average of max(V, 0)
ENE
expected negative exposure, the average of min(V, 0), a negative number

Holds because every value splits into its positive and negative parts. ENE is negative, so subtracting it is the usual sign error.

Expected exposure of a normally distributed value

EE = μ × N(μ ÷ σ) + σ × n(μ ÷ σ)

μ
the mean of the value at that date
σ
its standard deviation
N
the standard normal cumulative distribution
n
the standard normal density

Value normally distributed and uncollateralised. With μ = 0 it reduces to about 0.399 × σ; ignoring a non-zero mean is the trap.

A practice question

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

Which description matches potential future exposure (PFE) at a given future date?

  1. A high percentile, such as the 97.5th or 99th, of the positive mark-to-market distribution at that date - correct
  2. The probability-weighted average of the positive mark-to-market values across all scenarios at that date, shown as one figure per dateWhy not: That is expected exposure (EE), a mean, not a high quantile.
  3. The average of the expected exposure profile over the first year, after the profile has been made non-decreasingWhy not: That is effective expected positive exposure, a time-average rather than a quantile at one date.
  4. The cost of replacing the contract at today's market prices if the counterparty were to default immediatelyWhy not: That is current exposure at time zero, not a forecast of future exposure.

PFE is a percentile of the future exposure distribution at each date (for example 99%), so it is a worst-case style measure. EE is the mean of max(V,0), and current exposure is the value today.

A harder one

A bank simulates six equally likely values of a netting set's mark-to-market (USD million) at the one-year date: 14, 8, 3, -2, -6 and -9. What is the expected exposure (EE) at that date, closest to?

  1. 1.33Why not: Averages all six values, negatives included: that is the expected mark-to-market, not EE.
  2. 4.17 - correct
  3. 8.33Why not: Divides the positive values by 3, the number of positive scenarios, instead of by all 6 scenarios.
  4. 7.00Why not: Averages the absolute values of all scenarios, treating the negative values as exposure.

EE = average of max(V,0) over all six scenarios = (14 + 8 + 3 + 0 + 0 + 0)/6 = 25/6 = 4.17. The negative scenarios count as zero exposure but still count in the denominator.

The hardest tier

The mark-to-market of an uncollateralised trade at a future date is normally distributed with mean USD 2 million and standard deviation USD 10 million. Use a normal pdf at 0.2 of 0.391 and a cdf at 0.2 of 0.579. Its expected exposure at that date is closest to which value?

  1. USD 5.07 million - correct
  2. USD 3.99 millionWhy not: Uses sigma/sqrt(2 pi), which is only right when the mean is zero; ignores the mean of 2.
  3. USD 2.00 millionWhy not: Reports the expected mark-to-market, ignoring that negative values are floored at zero.
  4. USD 25.26 millionWhy not: Reports the 99th percentile (mean + 2.326 sigma), which is PFE rather than EE.

For V ~ N(m, s), EE = m x N(m/s) + s x n(m/s) = 2 x 0.579 + 10 x 0.391 = 1.158 + 3.91 = 5.07, using the normal cdf N and pdf n at 0.2.

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