FRM Exam Part II · Liquidity and Leverage

Transactions Cost and Liquidity-Adjusted VaR Explained

Updated 11 October 2026 · Fact-checked

Liquidity-adjusted VaR (LVaR) is ordinary VaR plus the cost of exiting a position. With exogenous liquidity, you add half the bid-ask spread times position value (and optionally a stressed spread). With endogenous liquidity, your own trade moves the price, so you also model market impact and the best liquidation horizon.

Understand Transactions Cost and Liquidity-Adjusted VaR

Standard VaR assumes you can sell any position at the mid-price, instantly. Real markets do not work that way. You sell at the bid, and a large sale can push the price down. LVaR adds these costs to VaR so the loss estimate is closer to what you would really lose.

Liquidity risk comes in two forms. Exogenous liquidity is the part set by the market as a whole. The bid-ask spread is the same no matter how much you trade, at least for normal sizes. You are a price taker. Endogenous liquidity is the part caused by your own trade. A big sale widens the spread or moves the price against you. It matters most for large positions and in stressed markets.

The simplest exogenous method adds a transactions cost. Selling at the bid instead of the mid costs half the spread per unit. So the cost of liquidation is 0.5 × spread × position value, where the spread is a proportion of the price. This cost is added to the VaR. A more careful version treats the spread as random. You use the mean spread plus a multiple of its standard deviation, so the cost reflects a bad-spread scenario.

Endogenous methods deal with the liquidation horizon. If you sell fast, you pay more impact cost but hold market risk for less time. If you sell slowly, impact is lower but price risk builds up. The optimal liquidation horizon balances the two. In practice, the VaR is scaled to the number of days needed to liquidate, and a liquidity cost is added for the impact.

In the exam, you are usually asked to compute LVaR with a spread cost, to say which approach (exogenous or endogenous) fits a case, or to explain how horizon changes the result. The hard part is not the algebra. It is keeping the units straight and remembering that the spread cost is a half-spread.

Key formulas to remember

Exogenous spread cost (constant spread): Liquidity cost = 0.5 × S × P; S is the relative spread = (ask − bid) ÷ mid. P is the position value. The 0.5 is because you give up half the spread to sell at the bid.
LVaR with constant spread: LVaR = VaR + 0.5 × S × P; VaR is measured at the mid-price. Both amounts are in the same currency and for the same position.
LVaR with stochastic spread: LVaR = VaR + 0.5 × (μS + k × σS) × P; μS is the mean relative spread, σS its standard deviation, and k the confidence multiplier (for example 2.33 at 99% if the spread is assumed normal).
Relative spread: S = (Ask − Bid) ÷ Mid, where Mid = (Ask + Bid) ÷ 2; Use the relative form. An absolute spread must be divided by the mid-price first.
Square-root-of-time scaling for horizon: VaR(T days) = VaR(1 day) × √T; Valid under independent, identically distributed returns with zero mean. Use it when the liquidation takes T days.
Liquidation horizon trade-off: Total cost = market risk over horizon + market impact cost; A longer horizon raises price risk and lowers impact. The optimal horizon minimises the total. This is a concept to recall, not a formula to compute.

How to solve Transactions Cost and Liquidity-Adjusted VaR questions

Use this order for any LVaR question. It stops you mixing up the approaches and the units.

1Identify the approach. If the spread is given as a market-wide figure and your trade does not move it, use the exogenous method. If the question mentions your own trade size, price impact or liquidation speed, it is endogenous.
2Compute the base VaR at mid-price: position value × z × volatility, scaled for the holding period.
3If the liquidation takes more than one day, scale the volatility by √T where the question says to.
4Convert the spread to a relative spread if it is quoted as bid and ask prices.
5Compute the liquidity cost as 0.5 × relative spread × position value. For a stochastic spread, use the mean plus k standard deviations in place of the spread.
6Add the liquidity cost to VaR to get LVaR. Do not add it inside the square root.
7State the result in currency and, if asked, as a percentage of position value. Check that LVaR is larger than VaR.

Quickest way: Half-spread add-on shortcut

When to use it: Use this when the question gives a spread and a position and asks for LVaR. Most exogenous questions fit.

Write down VaR. If it is not given, compute it as P × z × σ.
Compute 0.5 × spread × P. Check whether the spread is already relative or in price units.
Add the two figures.
Sanity check: the liquidity cost should be small next to VaR for liquid assets, and the final LVaR must exceed VaR.
If the options differ by the factor of two, you probably forgot the 0.5 or used the full spread.

Common mistakes in Transactions Cost and Liquidity-Adjusted VaR

Using the full spread instead of half the spread
The spread is the headline number, so students apply it directly.
Fix: You sell at the bid, and the mid is half a spread away. Always multiply by 0.5 for a one-way exit.
Mixing absolute and relative spreads
A spread of 0.40 on a price of 50 is easy to treat as 40%.
Fix: Divide by the mid-price first: 0.40 ÷ 50 = 0.8%. Then apply it to position value.
Confusing exogenous and endogenous liquidity
Both involve costs of trading, so they sound alike.
Fix: Ask whether the trader's own size changes the price. If no, exogenous. If yes, endogenous.
Scaling the liquidity cost by √T
Students scale the whole LVaR by the horizon.
Fix: Only the VaR part is scaled by √T. The spread cost is added after scaling.
Saying a longer liquidation horizon always lowers LVaR
Slower selling reduces impact, so it seems safer.
Fix: A longer horizon lowers impact cost but raises market risk. The optimal horizon is where the total is smallest.
Adding the spread cost to the VaR of a different confidence level or position
Numbers from different parts of a case get combined carelessly.
Fix: Match the position value and horizon in both parts before adding.

Worked examples

Example 1

A trader holds USD 20 million of a stock. The 1-day 99% VaR at mid-price is USD 600,000. The stock has bid 49.80 and ask 50.20. Calculate the LVaR using the constant-spread exogenous method.

Show the solution

Mid = (49.80 + 50.20) ÷ 2 = 50.00.
Relative spread S = (50.20 − 49.80) ÷ 50.00 = 0.40 ÷ 50.00 = 0.8%.
Liquidity cost = 0.5 × 0.008 × 20,000,000 = 80,000.
LVaR = 600,000 + 80,000 = 680,000.

Answer: LVaR = USD 680,000

Example 2

A EUR 10 million bond position has a 1-day 99% VaR of EUR 150,000. The relative bid-ask spread is assumed normal with a mean of 0.30% and a standard deviation of 0.10%. Use a multiplier of 2.33 for the spread. Calculate the LVaR.

Show the solution

Worst-case relative spread = 0.30% + 2.33 × 0.10% = 0.30% + 0.233% = 0.533%.
Liquidity cost = 0.5 × 0.00533 × 10,000,000 = 26,650.
LVaR = 150,000 + 26,650 = 176,650.

Answer: LVaR = EUR 176,650

Exam tips

Expect a short numeric question: VaR plus 0.5 × spread × position. Check the units before you add.
When a case mentions a large position, thin market or fire sale, the answer is almost always about endogenous liquidity.
The optimal liquidation horizon question is conceptual. Link the trade-off to impact cost versus price risk, and do not claim that slower is always better.
Wrong options are often built from the full spread or an unscaled VaR, so recompute before you trust a close option.
In stress, spreads widen and become more volatile. A stochastic-spread LVaR is higher than a constant-spread LVaR for the same mean.

Practice questions from Liquidity and Leverage

Transactions Cost and Liquidity-Adjusted VaR: frequently asked questions

What is the difference between exogenous and endogenous liquidity in LVaR?

Exogenous liquidity is set by the market and does not change with your trade, so the spread is taken as given. Endogenous liquidity depends on your own trade size, which moves the price against you. Endogenous effects matter most for large positions and in stress.

How do I calculate LVaR with a spread cost?

Compute VaR at mid-price, then add 0.5 × relative spread × position value. If the spread is random, use the mean spread plus a multiple of its standard deviation. The sum is the LVaR.

Why do we use half the bid-ask spread?

Mid-price is halfway between bid and ask. Selling at the bid means you give up half the spread relative to mid. A full round trip would cost the whole spread, but LVaR looks only at liquidation.

What is the optimal liquidation horizon?

It is the selling period that minimises total cost. Selling quickly raises market impact, while selling slowly leaves you exposed to price moves for longer. The optimum balances these two costs.