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Set the way credit exposure is characterised against VaR methods and outline further considerations in determining credit exposure.

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

The idea

VaR and credit exposure both describe a loss distribution, but they differ in construction. VaR looks at the change in value of a portfolio over a short horizon, such as one to ten days, and counts gains and losses. Exposure must be projected forward to every date up to the maturity of the trades, because default can occur at any time. Let V be the portfolio value at a future date. On each date exposure keeps only the positive side of that value, which gives Exposure = max(V, 0). A negative value therefore counts as zero. That forces a pricing model at each future date and path, the need to model cash flows paid before default, and netting and collateral. Exposure used for limits is usually calibrated to real-world data, while exposure used to price CVA is risk-neutral. Scaling a short-horizon VaR by the square root of time ignores amortisation and so overstates long-dated exposure.

Formulas

Potential future exposure of a normal value

PFE = μ + z × σ

μ
the mean of the mark-to-market at the date
z
the normal quantile, 2.326 at 99%
σ
the standard deviation of the mark-to-market at the date

PFE is the upper, gain-side quantile. A VaR-style loss quantile looks at the other tail, so the two differ whenever the mean is not zero.

A practice question

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

Which feature most clearly separates the measurement of credit exposure from a standard market risk VaR calculation?

  1. Exposure counts gains and losses equally in its distribution, whereas VaR counts only the gainsWhy not: Reversed: exposure keeps only the positive side, while VaR looks at losses.
  2. Exposure is projected over a long horizon, to the life of the trades, whereas VaR typically uses one to ten days - correct
  3. Exposure is always quoted at a 95% confidence level, whereas VaR is always quoted at a 99% confidence levelWhy not: Confidence levels are choices for either measure and are not what distinguishes them.
  4. Exposure ignores portfolio netting, whereas VaR allows offsetsWhy not: Exposure explicitly reflects legally enforceable netting agreements.

Default can occur at any time up to maturity, so exposure is a profile over a long horizon and is floored at zero, whereas VaR is a single short-horizon loss quantile on the whole portfolio.

A harder one

The mark-to-market of a forward position changes by an independent, zero-mean amount each day with a standard deviation of USD 0.5 million, and the forward has no interim cash flows. Using 252 trading days a year and the 99% normal quantile of 2.326, the one-year 99% PFE is closest to:

  1. USD 293.1 millionWhy not: Scales the daily figure by 252 instead of by the square root of 252.
  2. USD 3.7 millionWhy not: Uses a 10-day VaR horizon, not the one-year exposure horizon.
  3. USD 18.5 million - correct
  4. USD 22.2 millionWhy not: Uses 365 calendar days in the square root instead of 252 trading days.

Diffusion scales volatility with the square root of time: PFE = 2.326 x 0.5 x sqrt(252) = 2.326 x 0.5 x 15.87 = 18.46.

The hardest tier

An analyst estimates the PFE of a five-year interest rate swap by taking a 10-day 99% VaR and scaling it by the square root of time out to five years. Which problem does this create?

  1. It understates exposure at short horizons because square-root scaling produces too little volatility early onWhy not: Square-root scaling is reasonable for short horizons; the main flaw is late in the life.
  2. It cannot be done because VaR is undefined for interest rate productsWhy not: VaR is well defined for rate products; the problem is the extrapolation.
  3. It understates exposure at all dates because a 99% confidence level is too low for credit exposure purposesWhy not: The confidence level is a parameter; the structural error is neglecting the swap's pull to par.
  4. It ignores amortisation, so it overstates exposure later in the life as the swap's sensitivity shrinks - correct

Pure square-root-of-time scaling assumes volatility of value keeps growing, as for a forward. A swap's value sensitivity falls as payments are made, so true exposure peaks and returns to zero, and the scaled figure overstates exposure late in life.

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