FRM Exam Part II · Liquidity Risk Management

Optimal Liquidation and Endogenous Liquidity for FRM Part II

Updated 11 October 2026 · Fact-checked

Optimal liquidation is choosing how fast to sell a position so that total cost is lowest. Selling fast raises price impact (endogenous liquidity cost). Selling slowly leaves you exposed to price risk. You minimise the sum of expected impact cost and a risk penalty, so the best horizon balances the two.

Understand Optimal Liquidation and Endogenous Liquidity

Liquidity has two parts. Exogenous liquidity is the market-wide cost that no single trader controls, such as the normal bid-ask spread. Endogenous liquidity is the cost created by your own trade. A large sale pushes the price down against you. This is price impact. It matters most when your position is large relative to daily volume, or when many firms sell the same assets at once.

Price impact has two forms. Temporary impact is the extra concession you pay to trade quickly. It fades once you stop. Permanent impact is a lasting price shift, because your trade reveals information or changes supply and demand. Faster trading raises temporary impact per share. Total permanent impact depends on the size sold, not the speed.

The trade-off is simple. If you sell everything at once, you pay high impact but have no price risk afterwards. If you spread sales over many days, impact per day is small, but the unsold part stays exposed to price moves. The longer you wait, the more volatility can hurt you, and the loss grows with the square root of time for a constant volatility.

The Almgren-Chriss framework formalises this. You choose a trading schedule that minimises expected cost plus a risk-aversion multiple of the variance of cost. A risk-neutral trader sells evenly over a long horizon to cut impact. A more risk-averse trader sells faster, front-loading the trades. The set of best schedules forms an efficient frontier of expected cost against cost variance.

For FRM, you also need the link to liquidity-adjusted VaR. Endogenous liquidity means a normal VaR understates loss when you must exit a large or crowded position. The liquidation horizon you choose is the key input. A longer horizon means more price risk in VaR but a smaller impact cost.

Key formulas to remember

Linear temporary impact cost (equal slices): Cost per share = η × (shares per period ÷ period length); η is the temporary impact coefficient. Trading faster raises the cost per share in proportion to the trading rate.
Permanent impact cost (linear): Permanent cost ≈ ½ × γ × X²; X is total shares sold, γ is the permanent impact per share. It does not depend on the speed of selling. Applies in the linear Almgren-Chriss setup.
Temporary cost of an even sale over T: Temporary cost = η × X² ÷ T; Even sale of X shares over time T. A longer T cuts this cost.
Price risk of the unsold position: Risk over horizon = σ × √T × (average unsold shares); For an even schedule the average unsold is about X ÷ 2. The variance of cost is σ² × X² × T ÷ 3 in the continuous version.
Objective function: Minimise E[cost] + λ × Var[cost]; λ is risk aversion. Higher λ gives a faster liquidation.
Time-horizon scaling of volatility: σ(T days) = σ(1 day) × √T; Assumes independent returns and constant volatility.
Liquidity-adjusted VaR (exogenous spread): LVaR = VaR + ½ × spread × position value; Adds half the proportional spread as a liquidation cost. Endogenous impact is added on top if the position is large.

How to solve Optimal Liquidation and Endogenous Liquidity questions

Use this order for any optimal liquidation or endogenous liquidity question.

1Identify the position size, the daily volume, and the participation rate you are allowed.
2Separate the cost types: spread (exogenous), temporary impact and permanent impact.
3Compute the impact cost for each candidate horizon using the given impact formula.
4Compute the price risk for each horizon, using σ × √T on the unsold portion.
5Combine them as expected cost plus the risk penalty, or as impact cost plus a VaR-style loss at the stated confidence level.
6Pick the horizon with the lowest total, or state the direction of the trade-off if no numbers are given.
7Interpret: say whether faster selling suits a more risk-averse or more crowded-market setting.

Quickest way: Fast comparison of two horizons

When to use it: When the question gives two or three horizons and asks which has the lowest total cost.

Write impact cost and price-risk cost for each horizon in one line each.
Remember impact falls as the horizon lengthens and risk rises, so only the total decides.
Check the answer direction: higher risk aversion or higher volatility pushes the best horizon shorter.
Eliminate options that move the wrong way, then compute only the survivors.

Common mistakes in Optimal Liquidation and Endogenous Liquidity

Treating liquidity cost as only the bid-ask spread.
Basic LVaR uses only the spread, so students stop there.
Fix: For large positions, add price impact from your own trading. That is the endogenous part.
Saying a longer horizon always lowers total cost.
It lowers impact cost, and students forget the rising price risk.
Fix: Always compare both costs. Total cost has a minimum at an intermediate horizon.
Letting permanent impact depend on trading speed.
Students merge temporary and permanent impact.
Fix: In the standard linear model permanent impact depends on total size only. Speed changes temporary impact.
Scaling volatility linearly with time.
Mixing up the T-day scaling with the sum of positions.
Fix: Use σ × √T for the standard deviation, and σ² × T for the variance.
Assuming a risk-averse trader sells slower.
Students link caution with patience.
Fix: A more risk-averse trader wants less exposure to price moves, so sells faster and pays more impact.

Worked examples

Example 1

A fund must sell 2,00,000 shares. Temporary impact is η = ₹0.000002 per share per (share per day). It sells evenly over T days, so the temporary cost per share is η × (X ÷ T), where X = 2,00,000. Compute the total temporary impact cost if T = 4 days and if T = 10 days.

Show the solution

Use total temporary cost = η × X² ÷ T.
X² = 2,00,000² = 4 × 10¹⁰.
η × X² = 0.000002 × 4 × 10¹⁰ = 80,000.
For T = 4: 80,000 ÷ 4 = ₹20,000.
For T = 10: 80,000 ÷ 10 = ₹8,000.

Answer: Temporary impact cost is ₹20,000 over 4 days and ₹8,000 over 10 days. The longer horizon cuts impact cost, but the unsold shares carry price risk for longer.

Example 2

A bank must liquidate a ₹50,00,00,000 position evenly. Daily volatility is 1%. Horizon T = 4 days gives impact cost of ₹10,00,000; T = 9 days gives impact cost of ₹4,00,000. Take the price-risk measure as 1.65 × σ × √T × (position ÷ 2), with σ = 1%. Which horizon has the lower total cost?

Show the solution

Position ÷ 2 = ₹25,00,00,000.
For T = 4: √4 = 2. Risk = 1.65 × 0.01 × 2 × 25,00,00,000 = 1.65 × 0.02 × 25,00,00,000 = 1.65 × 5,00,000 = ₹8,25,000.
Total for T = 4: 10,00,000 + 8,25,000 = ₹18,25,000.
For T = 9: √9 = 3. Risk = 1.65 × 0.03 × 25,00,00,000 = 1.65 × 7,50,000 = ₹12,37,500.
Total for T = 9: 4,00,000 + 12,37,500 = ₹16,37,500.
Compare: 16,37,500 is lower than 18,25,000.

Answer: The 9-day horizon has the lower total cost, ₹16,37,500 against ₹18,25,000 for 4 days. Here the saving in impact outweighs the extra price risk.

Exam tips

Read whether the question asks for impact cost only or total cost including price risk. Many wrong options give only one part.
Track the direction first: higher volatility or risk aversion shortens the best horizon; higher impact coefficient lengthens it.
Know the words endogenous, exogenous, temporary and permanent. Options often swap them.
When a crowded trade or a market-wide sale is described, endogenous liquidity risk is the concept being tested.
Check units: shares per day against total shares, and whether volatility is daily or annual.

Practice questions from Liquidity Risk Management

Optimal Liquidation and Endogenous Liquidity: frequently asked questions

What is endogenous liquidity?

It is the liquidity cost caused by your own trading or by the combined actions of many market participants. Large sales move prices against the seller. Standard LVaR using only the spread ignores this.

How do I compute the liquidation horizon in FRM?

Usually it is position size divided by the volume you can trade per day, given a participation limit. If the position is 5,00,000 shares and you can trade 1,00,000 shares per day, the horizon is 5 days. Then scale volatility by √T for the price risk.

What is the Almgren-Chriss model in simple terms?

It chooses a selling schedule that minimises expected execution cost plus a penalty for cost variance. Risk-neutral traders sell evenly and slowly. Risk-averse traders sell faster and accept higher impact.

Does permanent impact depend on how fast I sell?

In the standard linear model, no. It depends on the total quantity sold. Temporary impact is the part that depends on the trading rate.