Charta is on Google Play: practise on the go. Get the app
Charta

For a trade, derive the spread risk factor and the cost expected on execution; for a position sold down across multiple days, compute the resulting liquidity adjustment applied to VaR.

FRM® Part II · Liquidity and Treasury Risk Measurement and Management · Liquidity and Leverage · FRM2.LTR.2.f

The idea

Liquidating a position costs about half the bid-ask spread, because you trade from the mid price to the bid or ask. With relative spread s, the expected cost is half the position's value times mean s: Expected cost = ½ × value × mean s. To allow for wide spreads at a confidence level, add z standard deviations of s to its mean: Spread risk factor = mean s + z × σs. In this simplified spread-based adjustment, ignoring market impact, the liquidity cost is half the position's value times that factor, and liquidity-adjusted VaR adds it to VaR: LVaR = VaR + liquidity cost. If the position is sold in T equal parts at the end of T days and daily returns are independent, the shrinking position is exposed for longer, and the loss variance scales with (T + 1)(2T + 1) ÷ 6T rather than T. Liquidation-adjusted VaR scales one-day VaR by the square root of that factor: VaR_T = one-day VaR × √[(T + 1)(2T + 1) ÷ 6T]. Typical mistakes are charging the full spread and scaling the whole position by the square root of T.

Formulas

Spread risk factor

Spread factor = μs + z × σs

μs
the mean relative bid-ask spread
z
the standard normal multiplier for the confidence level
σs
the standard deviation of the relative spread

Allows for spreads wider than average. The liquidity cost is then half the position value times this factor; omitting either term understates it.

Liquidity cost at a stressed spread

Cost = ½ × V × (μs + z × σs)

V
the position value at mid
μs
the mean relative bid-ask spread
z
the standard normal multiplier for the confidence level
σs
the standard deviation of the relative spread

Added to VaR to give LVaR, ignoring market impact. Dropping the ½ charges a round trip, not a one-way exit.

VaR of a position sold in T equal daily parts

VaR_T = VaR₁ × √[(T + 1) × (2T + 1) ÷ (6T)]

VaR₁
the one-day VaR of the full position
T
the number of days over which equal parts are sold

Assumes independent daily returns and a sale at the end of each day. Scaling the whole position by √T overstates the risk of a position that shrinks.

A practice question

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

Ignoring price impact, which expression gives the expected cost of liquidating a position at the quoted bid, relative to its mid-price value?

  1. The full proportional bid-ask spread multiplied by the position valueWhy not: Charges the full spread, which is the cost of a round trip, rather than the half-spread paid in one direction.
  2. One half of the proportional bid-ask spread multiplied by the position value - correct
  3. The daily volatility multiplied by the square root of the holding period and by the position valueWhy not: This is a market VaR style formula, not a transaction cost.
  4. The position value divided by the average daily trading volumeWhy not: This is a measure of days or scale to liquidate, not a cost in value terms.

A sale is made at the bid, which is half a spread below the mid price, so the expected cost is 1/2 x spread x value; the full spread applies only to a buy and sell round trip.

A harder one

A position is worth USD 25 million. The relative bid-ask spread has a mean of 0.30% and a standard deviation of 0.10%. Using a 99% confidence multiplier of 2.33, what is the liquidity cost of closing the position, closest to?

  1. USD 133,250Why not: Omits the factor one half and so charges the full spread.
  2. USD 37,500Why not: Uses only the mean spread and omits the spread risk term z x sigma.
  3. USD 29,125Why not: Uses only the volatility term z x sigma and omits the mean spread.
  4. USD 66,625 - correct

Spread risk factor = 0.30% + 2.33 x 0.10% = 0.533%. Cost = 0.5 x 25,000,000 x 0.00533 = 66,625.

The hardest tier

A fund holds a USD 80 million position with daily volatility of 2% and plans to sell it in 4 equal parts at the end of each of the next 4 days. Assume independent normal daily returns with zero mean and a 99% multiplier of 2.33. The loss variance equals the sum of daily variances of the position still held during each day. What is the 99% liquidation-period VaR (ignoring spread costs), closest to?

  1. USD 7.46 millionWhy not: Holds the whole position for the full four days and scales the one-day VaR by the square root of 4.
  2. USD 5.10 million - correct
  3. USD 9.32 millionWhy not: Adds the daily VaRs of the declining position, assuming perfect correlation between days.
  4. USD 3.73 millionWhy not: Uses the one-day VaR of the full position and ignores the remaining exposure on later days.

Exposure is 80, 60, 40, 20 over the four days, so variance is proportional to (1 + 0.5625 + 0.25 + 0.0625) = 1.875 = (T+1)(2T+1)/(6T). VaR = one-day VaR 3.728 x sqrt(1.875) = 5.10.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Liquidity and Leverage