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Contrast the liquidity horizons the FRTB assigns across asset classes, and show how a bank derives its expected shortfall using those different horizons.

FRM® Part II · Market Risk Measurement and Management · Fundamental Review of the Trading Book · FRM2.MR.18.b

The idea

FRTB assigns each risk factor a liquidity horizon, the time needed to exit or hedge a position in stressed markets. It has five buckets: 10, 20, 40, 60 and 120 days. Illustratively, large-cap equity prices and major currency rates sit at 10 days, while credit spread volatility and some commodity factors sit at 120, but the prescribed classification governs. The bank computes a base ES over a 10-day period, written T, then reruns it shocking only factors whose horizon is at least each longer bucket. Write ESP for the base ES, ESj for the rerun for bucket j, and LHj for that bucket's horizon. Each bucket from the 20-day one upward adds a term of this form: Term = ESj × √((LHj − LHj−1) ÷ T). The liquidity-adjusted ES then combines the base ES and the squared terms: ES = √(ESP² + Σ Term²). The ES uses a 97.5 percent one-tailed level. Stress calibration scales a reduced-set stress ES by the full-set to reduced-set current ES ratio, under prescribed rules. Common mistakes are using LHj ÷ T instead of the difference between horizons, and adding the terms without squaring.

Formulas

FRTB liquidity-adjusted expected shortfall

ES = √(ESP² + Σ (ESj × √((LHj − LHj−1) ÷ T))²)

ESP
the 10-day ES with all risk factors shocked
ESj
the 10-day ES shocking only factors with horizon at least LHj
LHj
the liquidity horizon of bucket j, from 20 days upward
LHj−1
the next shorter liquidity horizon
T
the base horizon, 10 days

Uses the difference between successive horizons, not LHj alone, and squares each term before adding. ES is at 97.5 percent.

FRTB stress-calibrated expected shortfall

Stressed ES = ESRS × (ESFC ÷ ESRC)

ESRS
the ES on the reduced factor set in the stress period
ESFC
the ES on the full factor set in the current period
ESRC
the ES on the reduced factor set in the current period

Used when the full set lacks stress-period data; the ratio scales the reduced-set stress ES up to the full portfolio.

A practice question

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

Which set of liquidity horizons, in days, does the FRTB use in its expected shortfall calculation?

  1. 1, 5, 10, 30 and 60Why not: Takes pre-FRTB short horizons; the shortest FRTB horizon is 10 days and the longest is 120.
  2. 10, 20, 40, 60 and 120 - correct
  3. 5, 10, 20, 60 and 250Why not: Invents a 5-day bucket and uses a one-year (250-day) horizon, which belongs to the default risk charge, not ES.
  4. 10, 30, 60, 90 and 180Why not: Uses evenly spaced quarterly-style horizons that are not FRTB's buckets.

FRTB defines five horizon buckets of 10, 20, 40, 60 and 120 days. Each risk factor is mapped to one bucket according to how long it takes to exit or hedge in stressed conditions.

A harder one

In the FRTB formula ES = sqrt( ES_T(P)² + the sum over j of [ES_T(P,j) x sqrt((LH_j - LH_j-1)/T)]² ), with T = 10 days, what does ES_T(P,j) represent?

  1. The 10-day ES from shocking only factors with a liquidity horizon of at least LH_j, others held fixed - correct
  2. The ES measured directly over a horizon of LH_j days from historical LH_j-day returnsWhy not: The terms are 10-day ES figures scaled by the square-root rule across horizon increments, not ES measured over LH_j days.
  3. The 10-day ES of only those factors whose liquidity horizon is exactly LH_jWhy not: The set is cumulative: factors with horizons at least LH_j, not exactly LH_j.
  4. The stress-period ES of the reduced set of risk factors with adequate historical dataWhy not: Confuses ES_T(P,j) with ES_R,S, the reduced-set stress ES used for stress calibration.

ES_T(P,j) shocks only the factors that remain illiquid beyond LH_(j-1), i.e. horizons of at least LH_j, over the 10-day base horizon. Each is scaled by sqrt((LH_j - LH_j-1)/T) and the pieces are combined as a root sum of squares.

The hardest tier

A portfolio's 10-day ES values by liquidity horizon bucket, in USD million, are: ES_T(P) = 10 for all factors, and ES_T(P,j) = 8, 6, 4 and 2 for factors with horizons of at least 20, 40, 60 and 120 days. Which is closest to the FRTB liquidity-adjusted ES, with T = 10 days?

  1. USD 14.8 millionWhy not: Omits the square-root scaling by horizon increments, so every term has weight 1.
  2. USD 22.7 millionWhy not: Scales each term by sqrt(LH_j/T) instead of sqrt((LH_j - LH_j-1)/T), counting earlier horizon days again.
  3. USD 37.0 millionWhy not: Adds the scaled terms linearly instead of combining them as a root sum of squares.
  4. USD 17.1 million - correct

Increments (LH_j - LH_j-1)/T are 1, 2, 2 and 6. ES = sqrt(10² + 8² x 1 + 6² x 2 + 4² x 2 + 2² x 6) = sqrt(100 + 64 + 72 + 32 + 24) = sqrt(292) = 17.09.

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