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FRM Exam Part II · Liquidity Risk Management

Bid-Ask Spread and Liquidity-Adjusted VaR (LVaR)

Updated 11 October 2026 · Fact-checked

Liquidity-adjusted VaR adds a liquidation cost to ordinary VaR. With a constant spread, LVaR = VaR + 0.5 × spread × position value. With an exogenous stochastic spread, the cost uses mean spread plus a multiple of spread volatility. Compute VaR first, then the cost, then add them.

Understand Bid-Ask Spread and Liquidity-Adjusted VaR

Ordinary VaR values a position at the mid-price. But you cannot sell at the mid. You sell at the bid and buy at the ask. The gap between them is the bid-ask spread, and it is a real cost of trading out of a position.

The relative spread is (ask − bid) ÷ mid-price. The cost of closing a position is about half the spread, because the mid sits halfway between bid and ask. So the liquidation cost for a position of value P is 0.5 × relative spread × P.

Liquidity-adjusted VaR (LVaR) is VaR plus this cost. In the constant spread approach, the spread is treated as a fixed, known number. The cost is added directly to VaR. This is simple but ignores that spreads widen in stress.

In the exogenous spread approach, the spread is random, and it is outside the trader's control. It does not depend on the size of your trade. You model it as having a mean and a standard deviation. The cost is taken at a high-percentile spread, for example mean plus k standard deviations. This captures the widening of spreads in bad markets.

The endogenous approach is different. There the price impact depends on your own trade size, so large positions move the market against you. The exogenous and constant approaches ignore this. They suit small positions in liquid markets. Exams ask you to tell these apart and to compute the first two.

Key formulas to remember

Relative bid-ask spread
S = (Ask − Bid) ÷ Mid, where Mid = (Ask + Bid) ÷ 2
Expressed as a fraction or percentage of the mid-price.
Liquidation cost, constant spread
Cost = 0.5 × S × P
P is the position value at mid. Half the spread is paid on exit.
LVaR, constant spread
LVaR = VaR + 0.5 × S × P
Add the cost to VaR. Use the same currency and the same horizon for VaR.
LVaR, exogenous spread
LVaR = VaR + 0.5 × (μ + k × σ) × P
μ is the mean relative spread, σ its standard deviation, and k the multiplier for the confidence level (for example about 2.33 at 99% if the spread is assumed normal, or a value the question gives you).
Normal VaR for a position
VaR = z × σ_return × P
z is 1.645 at 95% and 2.326 at 99% (one-tailed). Scale σ by √t for a t-day horizon.

How to solve Bid-Ask Spread and Liquidity-Adjusted VaR questions

Use the same sequence for any LVaR question. Do the market-risk part and the liquidity part separately, then add.

  1. 1Identify the position value P at the mid-price, and the currency.
  2. 2Compute ordinary VaR: z × daily volatility × √(horizon days) × P, unless VaR is given.
  3. 3Find the spread inputs. If you get bid and ask, compute S = (ask − bid) ÷ mid. If you get a quoted relative spread, use it as is.
  4. 4Decide the approach. A single spread means constant. A mean and a standard deviation of the spread means exogenous.
  5. 5Compute the liquidation cost: 0.5 × S × P for constant, or 0.5 × (μ + kσ) × P for exogenous.
  6. 6Add the cost to VaR to get LVaR.
  7. 7Check the units. Percent versus decimal and horizon must match. Then read the answer: LVaR is always larger than VaR when the spread is positive.

Quickest way: Cost-first shortcut

When to use it: Use it when the question gives VaR directly and you only need to pick the right LVaR among four options.

  1. Write down the cost = 0.5 × spread × position. Do this in your head in rupees-style round numbers.
  2. Add it to the given VaR.
  3. If the spread is stochastic, replace the spread with mean + k × standard deviation first.
  4. Eliminate any option below VaR, since LVaR must be greater than VaR.
  5. Eliminate any option that uses the full spread instead of half, which usually doubles the cost.

Common mistakes in Bid-Ask Spread and Liquidity-Adjusted VaR

  • Using the full spread instead of half the spread as the cost.

    You see the spread as the cost per trade, but the mid-price is already halfway between bid and ask.

    Fix: Always multiply by 0.5 when the position is valued at mid and you liquidate once.

  • Mixing percentage and decimal spreads, for example using 0.4 instead of 0.004.

    Spreads are quoted as percentages and the formula needs a decimal.

    Fix: Convert 0.4% to 0.004 before multiplying.

  • Forgetting to scale the VaR to the holding horizon.

    Daily volatility is given, but the question asks for a 10-day LVaR.

    Fix: Multiply daily VaR by √10 before adding the cost. Do not scale the liquidation cost by √10.

  • Using only the mean spread in the exogenous approach.

    The mean feels like the natural input.

    Fix: The exogenous approach adds a multiple of the spread volatility to reflect widening spreads in stress. Use μ + kσ.

  • Calling the exogenous approach endogenous.

    Both adjust for liquidity, so the terms blur.

    Fix: Exogenous means spread is independent of your trade size. Endogenous means your own trade moves the price. Constant and exogenous spread models ignore trade-size impact.

  • Computing the spread relative to the bid or ask rather than the mid.

    The quoted prices are the only numbers in sight.

    Fix: Divide (ask − bid) by the mid, (ask + bid) ÷ 2.

Worked examples

Example 1

A bank holds a USD 20 million equity position. Daily return volatility is 1.5%. The bid is 49.90 and the ask is 50.10. Compute the 1-day 99% LVaR using the constant spread approach with normal VaR (z = 2.33).

Show the solution
  1. Mid = (49.90 + 50.10) ÷ 2 = 50.00.
  2. Relative spread S = (50.10 − 49.90) ÷ 50.00 = 0.20 ÷ 50.00 = 0.004, or 0.4%.
  3. VaR = 2.33 × 0.015 × 20,000,000 = 0.03495 × 20,000,000 = USD 699,000.
  4. Liquidation cost = 0.5 × 0.004 × 20,000,000 = USD 40,000.
  5. LVaR = 699,000 + 40,000 = USD 739,000.

Answer: LVaR = USD 739,000 (VaR of USD 699,000 plus liquidation cost of USD 40,000).

Example 2

A EUR 10 million bond position has a 10-day 99% VaR of EUR 400,000. The relative bid-ask spread has a mean of 0.30% and a standard deviation of 0.10%. Using the exogenous spread approach with k = 3, compute LVaR.

Show the solution
  1. Spread at the chosen level = μ + kσ = 0.30% + 3 × 0.10% = 0.60%, or 0.006.
  2. Liquidation cost = 0.5 × 0.006 × 10,000,000 = EUR 30,000.
  3. VaR is already for 10 days, so no scaling is needed.
  4. LVaR = 400,000 + 30,000 = EUR 430,000.

Answer: LVaR = EUR 430,000. Using only the mean spread would give a cost of EUR 15,000 and understate LVaR.

Exam tips

  • Read whether the spread is a single number or has a mean and standard deviation. That tells you which approach to use.
  • Check the answer options. If two options differ by exactly the cost, you probably need to decide between half and full spread.
  • Know the conceptual split. Constant and exogenous approaches treat spread as independent of your trade size. Endogenous models allow your own liquidation to move prices.
  • Expect a conceptual question on the limits: LVaR with spread cost ignores large-trade price impact and the time needed to liquidate.
  • Do the arithmetic in the position currency and keep decimals until the last step.

Practice questions from Liquidity Risk Management

Bid-Ask Spread and Liquidity-Adjusted VaR in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Bid-Ask Spread and Liquidity-Adjusted VaR: frequently asked questions

What is the formula for liquidity-adjusted VaR in FRM Part II?

For a constant spread, LVaR = VaR + 0.5 × S × P, where S is the relative spread and P is the position value. For an exogenous spread, replace S with the mean spread plus k times the spread's standard deviation. The VaR part is computed first in the usual way.

Why is half the bid-ask spread used?

The position is valued at the mid-price, which lies halfway between bid and ask. Selling at the bid gives up half the spread relative to mid. So the cost of one exit is half the spread.

What is the difference between exogenous and endogenous liquidity?

Exogenous liquidity is the market-wide spread that you take as given. It does not depend on your trade. Endogenous liquidity depends on your own trade size, because a large sale moves the price against you.

Do I scale the liquidity cost with the square root of time?

No. Scale the market-risk VaR by √t to match the horizon. The spread cost is a one-time cost of exit, so it is added as it is.