FRM Exam Part II · Liquidity Risk

Liquidity-Adjusted VaR (LVaR) for FRM Part II

Updated 11 October 2026 · Fact-checked

Liquidity-adjusted VaR (LVaR) is ordinary VaR plus the cost of closing out a position at the bid-ask spread. In the constant spread approach, LVaR = VaR + ½ × spread × position value. In the exogenous spread approach, the spread is random, so the cost uses a high spread quantile: ½ × (mean spread + k × spread volatility) × position value.

Understand Liquidity-Adjusted VaR (LVaR)

Standard VaR values a position at the mid-price. But you cannot sell at the mid-price. A seller receives the bid, and a buyer pays the ask. The gap is a real cost of getting out. Liquidity-adjusted VaR (LVaR) adds this cost to VaR so the risk figure reflects what you could actually lose when you liquidate.

The literature splits liquidity risk into two kinds. Exogenous liquidity is the spread set by the market. It is the same for all traders, and your own trade does not move it. Endogenous liquidity depends on your own trade size. Selling a large block pushes the price down further, so the cost rises with position size. The two main LVaR methods in the readings, constant spread and exogenous spread, cover only exogenous liquidity.

In the constant spread approach, you assume the bid-ask spread stays fixed at its average. The liquidation cost is half the spread times the position value, because you give up half the spread relative to the mid-price. You add this cost to VaR.

In the exogenous spread approach, the spread itself is random. It has a mean and a standard deviation. You treat the worst-case spread at your confidence level as the cost driver, usually by adding a multiple of the spread volatility to the mean spread. This captures spreads that widen in stress. It gives a larger LVaR than the constant spread approach whenever the spread has volatility.

Both methods are simple. They assume the position can be closed in one trade at the quoted spread and that spread and price changes do not interact. Endogenous models relax the first point but are harder to apply.

Key formulas to remember

Liquidity cost (constant spread): LC = ½ × S̄ × P; S̄ is the mean relative spread (spread ÷ mid-price). P is position value. You pay half the spread when closing out.
LVaR, constant spread: LVaR = VaR + ½ × S̄ × P; Add the cost to VaR at the same confidence level and horizon. Use the same currency for both.
LVaR, exogenous spread: LVaR = VaR + ½ × (S̄ + k × σS) × P; σS is the standard deviation of the relative spread. k is the multiplier for the confidence level, e.g. 2.33 for 99% under normality.
Parametric VaR (input): VaR = z × σ × P; σ is return volatility over the horizon. Scale daily σ by √T for T days. z is 1.645 at 95% and 2.33 at 99% (one-tailed).
Liquidity-adjusted VaR as a ratio: LVaR ÷ VaR = 1 + liquidity cost ÷ VaR; Useful to state the percentage uplift from liquidity.

How to solve Liquidity-Adjusted VaR (LVaR) questions

Use this order for any LVaR question, whichever approach the question names.

1Identify the approach: constant spread, exogenous spread, or an endogenous idea. Check what the question gives you for the spread.
2Find the position value P in one currency. If there are several positions, calculate each cost and add them.
3Compute the ordinary VaR: z × σ × P, scaling σ to the horizon with √T. If VaR is already given, use it.
4Convert the spread to a relative spread if it is given in currency units: spread ÷ mid-price.
5Compute the liquidity cost. Constant: ½ × S̄ × P. Exogenous: ½ × (S̄ + k × σS) × P.
6Add the liquidity cost to VaR to get LVaR.
7Sense-check: LVaR must exceed VaR, and the exogenous result must exceed the constant result when σS > 0.

Quickest way: Half-spread shortcut

When to use it: Use when the question gives VaR and spread data directly and you need LVaR fast.

Write the cost as ½ × (spread term) × position value. Never skip the ½.
Spread term: just the mean for constant, mean plus k × volatility for exogenous.
Add to the VaR given. Do not rescale the cost by √T unless the question says so.
Match your answer to the option that is larger than VaR, then confirm by arithmetic.

Common mistakes in Liquidity-Adjusted VaR (LVaR)

Using the full spread instead of half the spread
Students think the cost of a trade equals the spread.
Fix: A mid-price valuation is half a spread away from either the bid or the ask. Use ½ × spread.
Using an absolute spread with a relative formula
Spread is quoted as ₹ or $ in the question, but the formula uses a percentage.
Fix: Either use half the absolute spread × number of units, or convert to a relative spread first. Do not mix them.
Leaving out the spread volatility term in the exogenous approach
Students stop after computing the mean spread cost.
Fix: If a spread standard deviation is given, add k × σS to the mean spread before halving.
Treating the spread cost as an endogenous effect
Both ideas involve liquidity, so they get blended.
Fix: Constant and exogenous spread methods ignore trade size impact. Endogenous liquidity is the price effect of your own large trade.
Scaling the liquidity cost by √T along with VaR
Students apply the square-root rule to the whole LVaR.
Fix: Only the volatility inside VaR is scaled. The spread cost is a one-off cost of closing the position.
Using the wrong z for the confidence level
Mixing 95% and 99% values or one- and two-tailed values.
Fix: Use 1.645 for 95% and 2.33 for 99% one-tailed. Use the same level for the spread multiplier k.

Worked examples

Example 1

A bank holds a position worth $20 million. Daily return volatility is 1.5%. The 99% one-day VaR uses z = 2.33. The mean relative bid-ask spread is 0.40%. Using the constant spread approach, find the one-day 99% LVaR.

Show the solution

VaR = 2.33 × 1.5% × $20,000,000 = 2.33 × $300,000 = $699,000.
Liquidity cost = ½ × 0.40% × $20,000,000 = ½ × $80,000 = $40,000.
LVaR = $699,000 + $40,000 = $739,000.

Answer: LVaR = $739,000. The liquidity cost adds about 5.7% to VaR ($40,000 ÷ $699,000).

Example 2

A portfolio is worth €10 million. Its 99% one-day VaR is €400,000. The relative spread has mean 0.50% and standard deviation 0.20%. Use the exogenous spread approach with k = 2.33. Find LVaR and compare with the constant spread LVaR.

Show the solution

Worst-case spread = 0.50% + 2.33 × 0.20% = 0.50% + 0.466% = 0.966%.
Liquidity cost = ½ × 0.966% × €10,000,000 = ½ × €96,600 = €48,300.
Exogenous LVaR = €400,000 + €48,300 = €448,300.
Constant spread cost = ½ × 0.50% × €10,000,000 = €25,000, so constant LVaR = €425,000.
Difference = €448,300 − €425,000 = €23,300.

Answer: Exogenous LVaR = €448,300, which is €23,300 above the constant spread LVaR of €425,000, because spread volatility is included.

Exam tips

Questions often give VaR already. Do not recompute it; add the half-spread cost.
Look for the word 'exogenous' or 'constant' to pick the formula. Endogenous wording signals a conceptual question about position size.
Check units: percentage spreads must be turned into a currency cost using position value.
Eliminate options smaller than VaR first. LVaR can never be below VaR in these methods.
Expect conceptual items too: why LVaR rises in stress, and which approach ignores your own trade size.

Practice questions from Liquidity Risk

Liquidity-Adjusted VaR (LVaR): frequently asked questions

What is the LVaR formula in FRM Part II?

For the constant spread approach, LVaR = VaR + ½ × mean relative spread × position value. For the exogenous spread approach, replace the mean spread with mean plus k times the spread standard deviation. Both add the liquidation cost to normal VaR.

Why do we use half the bid-ask spread?

VaR values the position at the mid-price. Selling happens at the bid and buying at the ask, each half a spread from mid. So the expected cost of closing is half the spread.

What is the difference between exogenous and endogenous liquidity risk?

Exogenous liquidity is market-wide and set by conditions, such as the quoted spread. It does not depend on your trade. Endogenous liquidity depends on your own position size, since a large sale moves the price against you.

Is LVaR always higher than VaR?

In the constant and exogenous spread methods, yes, because you add a non-negative cost. The exogenous result is higher than the constant one when the spread has positive volatility and k is positive.