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Show why trading liquidity risk matters and work out its size using the cost of liquidating a position and liquidity-adjusted VaR (LVaR).

FRM® Part II · Liquidity and Treasury Risk Measurement and Management · Liquidity Risk · FRM2.LTR.1.a

The idea

Trading liquidity risk is the risk that a position cannot be closed quickly without a price concession. It matters because conventional VaR values positions at the mid-price as if they could be sold at no cost. The relative spread is the ask minus the bid, divided by the mid-price. The cost of liquidating a position is half the relative spread times the position value, written Cost = 0.5 × Spread × Position. It is half because you give up only the distance from mid to bid. In this simplified spread-based adjustment, liquidity-adjusted VaR is VaR plus this cost, written LVaR = VaR + Cost. With a variable spread, the cost uses the mean spread plus k standard deviations of the spread in place of the average. If liquidation takes T days, scale VaR by the square root of T. Typical mistakes are using the full spread and ignoring price impact from a large position. The exam wants VaR plus half the spread times the position, but market impact, position size and a longer sale than assumed need further modelling.

Formulas

Relative bid-ask spread

Spread = (Ask − Bid) ÷ Mid

Ask
the ask price
Bid
the bid price
Mid
the mid-price, (ask + bid) ÷ 2

Expressed as a proportion of the mid-price, so a 0.40 point spread on a price of 100 is 0.40%, not 0.04%.

Cost of liquidating a position

Cost = 0.5 × Spread × Position

Spread
the relative bid-ask spread; with a variable spread, the mean plus k standard deviations
Position
the market value of the position at mid

A seller gives up only the distance from mid to bid, so half the spread. Charging the full spread is the classic slip, and the cost ignores price impact from a large position.

Liquidity-adjusted VaR

LVaR = VaR + Cost

VaR
the conventional VaR at mid-price, scaled by √T if liquidation takes T days
Cost
the half-spread cost of liquidating the position

The simplified spread-based adjustment, so LVaR is always above VaR. Market impact and a longer sale than assumed need further modelling.

A practice question

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

A trader holds USD 40 million of a bond quoted at 99.80 bid and 100.20 ask. Relative to valuing the position at mid-price, the closest estimate of the cost of liquidating it is:

  1. USD 80,000 - correct
  2. USD 160,000Why not: Uses the full spread (0.4% x 40 million) instead of half the spread.
  3. USD 8,000Why not: Unit slip: reads the 0.40 point spread as 0.04% of value rather than 0.40%.
  4. USD 40,000Why not: Halves twice (a quarter of the spread); the cost is only one half-spread.

Mid = 100.00, relative spread = 0.40/100 = 0.40%. Liquidation cost = 1/2 x 0.40% x 40 million = USD 80,000, the distance from mid to bid.

A harder one

A fund must sell a position several times the normal daily volume, and dealers are known to move quotes against large sellers. Which statement about a spread-based LVaR (VaR plus half the quoted spread times position) is correct?

  1. It overstates the cost because the quoted spread already includes the price impact of very large block trades.Why not: Quoted spreads apply to normal trade sizes; they do not include the impact of a block many times the usual volume.
  2. It is exact, because liquidation cost grows in direct proportion to the size of the position being sold.Why not: Assumes linear cost; price impact rises more than proportionally with size, which the spread approach ignores.
  3. It is irrelevant because only price volatility, and not spreads or impact, determines losses when a position is liquidated.Why not: Ignores the liquidation cost that LVaR is designed to add; spread and impact matter as well as volatility.
  4. It captures exogenous liquidity at normal sizes but understates cost, since the fund's own selling creates endogenous price impact. - correct

Spread-based LVaR treats liquidity as exogenous (given by the market). Endogenous liquidity risk is the extra price move caused by the trader's own size, which the simple half-spread cost does not capture.

The hardest tier

A dealer holds USD 80 million of a bond. Daily volatility is 1.5% and a normal 99% quantile of 2.326 applies over a 10-day horizon (square-root-of-time scaling). The relative spread has a mean of 0.30% and a standard deviation of 0.12%, and the liquidity cost is set at half of (mean + 3 x standard deviation) times the position. LVaR is closest to:

  1. USD 8.95 millionWhy not: Uses only the mean spread and omits the 3 standard deviation allowance for spread variability.
  2. USD 9.36 millionWhy not: Uses the full spread (mean + 3 sd) instead of half of it.
  3. USD 9.09 million - correct
  4. USD 3.06 millionWhy not: Uses 1-day VaR although the question specifies a 10-day horizon.

10-day VaR = 2.326 x 1.5% x sqrt(10) x 80 = 8.828 million. Stressed spread = 0.30% + 3 x 0.12% = 0.66%; cost = 1/2 x 0.66% x 80 = 0.264 million. LVaR = 9.09 million.

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