Market liquidity risk · RiskAtoms Academy

By the end you will be able to

1Compute the cost of exiting a book at the bid-ask spread, position by position.
2Estimate liquidation horizons from position size and daily volume, and spot large and concentrated positions.
3Weigh price impact against market risk when deciding how fast to sell, and derive the risk of an even exit.
4Compute a liquidity-adjusted VaR for the desk and say which positions drive it.

Why it matters

The desk's VaR assumes every position could be sold tomorrow at today's mid price. The futures could. The £80m of corporate bonds could not: the five issues trade about £8m a day between them, and dealers quote them 50bp apart. Selling them takes weeks and costs money, and in a crisis the spread widens just when the desk needs to sell. That is market liquidity risk, and the VaR does not see it.

Core explanation

What it costs to get out

Illustrative dataRiskAtoms Bank plc's Markets desk

The module's trading book: eight positions in four books (Equities, Rates, FX and commodities, Credit) driven by eight risk factors, with 750 days of seeded factor moves that include a stressed spell (days 480 to 539) and four jump days. Every position is revalued in full. The numbers are made up for teaching, in pounds, on one valuation day; they are not market data.

The spreads, spread volatilities and daily volumes in this lesson are an illustrative extra: the shared book has no liquidity data. The positions, their sensitivities and the £1.94m VaR are the module's.

L1.1.07: Market liquidity defined market liquidity: how easily an asset can be sold quickly near its fair price. (Funding liquidity, a firm's ability to pay its bills, is M3.5: Liquidity risk and funding.) For a trading desk it has two parts. Exogenous liquidity risk is the bid-ask spread everyone faces, and how much it can widen. Endogenous liquidity risk is the extra cost of your own size: selling a large position moves the price.

The desk's exit costs

Position	Spread (quote)	Half-spread cost	Size ÷ ADV	Days to exit
UK index futures (long £60m)	1 bp of notional	£3,000	0.02	1
US equity basket ($39m)	6 bp of value	£9,000	0.06	1
UK index puts (1,000 contracts)	0.6 vol points	£37,227	0.50	2
10-year gilt (£100m face)	3 bp of price	£14,820	0.10	1
2-year swap, pay fixed (£300m)	0.3 bp of rate	£8,487	0.01	1
Sell $26m forward	1 bp of notional	£1,000	0.00	1
Brent futures (200,000 barrels)	2 bp of notional	£1,231	0.01	1
Corporate bonds (£80m)	50 bp of price	£200,000	10.00	40

Each spread is quoted the way its market quotes it, and turned into pounds with the right unit: basis points of notional or price for most, vol points times vega for the puts, basis points of rate times DV01 for the swap. Selling everything at the bid would cost £0.27m against the mid prices the desk marks at; the corporate bonds alone are £200k. The last column assumes the desk trades at most 25% of the average daily volume, its participation rate. Everything is out in a day or two except the corporate bonds, whose liquidation horizon is 40 days.

Sort itLiquid or not?

How long would each position take to exit without moving the market? Sort them, then check against the table's logic.

£60m of UK index futures
1,000 exchange-traded index puts, half a day's volume
£80m of five corporate bonds trading £8m a day
£300m notional 2-year sterling swap
A 30% stake in one small company's only bond issue

Liquidity-adjusted VaR

Anil Bangia, Francis Diebold, Til Schuermann and John Stroughair (1999) proposed a simple fix for the exogenous part: add to VaR the cost of exiting at a bad spread, the average spread plus a multiple $a$ of its volatility. They chose $a$ from the data, between 2.0 and 4.5 for the markets they studied; this lesson uses $a = 3$ . For the desk the cost is £0.69m, so the liquidity-adjusted VaR is £2.63m, 35% above the plain VaR. The corporate bonds supply 73% of the adjustment.

Adding the two assumes the worst spread and the worst market move come on the same day. That is cautious, and in a crisis it is also what tends to happen: spreads widen as prices fall.

Large positions: fast and costly, or slow and risky

Bangia's adjustment assumes the desk can sell at the quoted spread. A position ten times daily volume cannot. Selling it all at once would push the price down: price impact. Selling slowly keeps impact low but leaves the rest exposed to the market for longer. With a simple impact model (selling one day's volume in a day costs 0.5% of the amount sold), selling the bonds in one day costs £4.20m; spread over 40 days, only £0.30m, but the 99% market risk over the exit rises from £0.49m to £1.83m. The best plan balances the two: 10 days, with cost plus risk of £1.57m. This is the trade-off Robert Almgren and Neil Chriss formalised for optimal execution.

The formal version

Bangia's cost of liquidity. For a position of size $V$ with relative spread mean $\bar s$ and volatility $\sigma_s$ :

\begin{aligned} \mathrm{COL} &= \tfrac12\,V\,(\bar s + a\,\sigma_s) \\ \mathrm{LVaR} &= \mathrm{VaR} + \sum_i \mathrm{COL}_i \end{aligned}

Cost against risk. Selling $X$ evenly over $h$ days in a market trading $\mathrm{ADV}$ a day, with temporary impact proportional to the daily trade ( $\eta \times$ the share of a day's volume sold):

\begin{aligned} \text{impact cost} &= \sum_{k=1}^{h}\frac{X}{h}\cdot\eta\,\frac{X/h}{\mathrm{ADV}} = \frac{\eta\,X^2}{h\,\mathrm{ADV}} \\ \text{risk}_{99\%} &= z\,\sigma X\sqrt{\frac{(h+1)(2h+1)}{6h}} \\ \min_h \quad & \tfrac12\,\bar s\,X + \frac{\eta\,X^2}{h\,\mathrm{ADV}} + z\,\sigma X\sqrt{\frac{(h+1)(2h+1)}{6h}} \end{aligned}

The impact term falls like $1/h$ ; the risk term grows like $\sqrt{h/3}$ for large $h$ . With the desk's numbers ( $X = \text{£}80\text{m}$ , $\mathrm{ADV} = \text{£}8\text{m}$ , $\eta = 0.5\%$ , one-day P&L standard deviation $\sigma X = \text{£}211{,}807$ ) the minimum is at $h = 10$ . Almgren and Chriss solve the richer problem in which the selling speed can change from day to day.

Try it

Slide the horizon from 1 to 40 days and watch the blue cost curve fall and the purple risk curve rise. Then make the market less liquid by raising the impact: the best horizon moves out, because rushing now costs more.

Try itPlan the exit from the corporate bonds

Days to sell (h)1

Price impact of selling one ADV in a day0.5%

Spread + impact cost£4.20m

99% market risk over the exit£0.49m

Cost + risk£4.69m

Best horizon10 days, £1.57m

Blue: half the bid-ask spread plus price impact, η X² / (h × ADV). Purple: 2.326 × σ × √((h + 1)(2h + 1)/(6h)). Dashed: the sum.

Worked example

The desk's liquidity-adjusted VaR

From VaR to LVaR

Step 1 of 5

1
Start from the market VaR
One-day 99% delta-normal VaR: £1.94m (L3.2.09: Parametric VaR for a portfolio). It assumes exit at mid prices.

What the desk can do

Cap corporate bond holdings at a multiple of daily volume (a concentration limit), charge the desk for the liquidity it uses, hold a reserve for exit costs in the valuation (L3.2.18: Valuation control and the daily risk report), and run stress tests with wider spreads and longer horizons.

Regulatory lens

The Fundamental Review of the Trading Book builds market liquidity into the capital measure: each risk factor is given a liquidity horizon of 10, 20, 40, 60 or 120 days, and expected shortfall is scaled up for the longer ones (L2.5.08: Expected shortfall). The detail is in M4.3: Market risk capital (FRTB) and CVA. Prudent valuation, the subject of the next lesson, requires adjustments for close-out costs and for concentrated positions; that is in M4.7: Accounting for risk. Funding liquidity and the liquidity coverage ratio are in M3.5: Liquidity risk and funding and M4.5: Liquidity and IRRBB regulation.

Spark Atoms

Spark AtomA quarter of the risk

Bangia and his co-authors found that ignoring the bid-ask spread could understate market risk in emerging markets by as much as 25 to 30%. In the most liquid currency markets they looked at, the adjustment was only a percent or so.

Spark AtomWhen the gilt market seized up

On 28 September 2022 the Bank of England announced temporary purchases of long-dated gilts to restore orderly market conditions, warning that continued dysfunction in that market would pose a material risk to UK financial stability. Even a government bond market can run short of buyers.

Key takeaways

Exiting at the bid costs half the spread; in pounds, use the market's own quote times the right unit (notional, vega, DV01).
Exogenous liquidity risk is the spread and its volatility; endogenous risk is your own price impact.
LVaR = VaR + Σ ½V(s̄ + aσₛ). For the desk: £1.94m + £0.69m = £2.63m, mostly from the corporate bonds.
Liquidation horizon = size ÷ (participation × daily volume). An even exit over h days has variance σ²X²(h+1)(2h+1)/(6h).
Selling fast costs impact, selling slowly costs risk; the best horizon minimises their sum.

Reaction

Check what stuck

Question 1 · objective 1

A £25m bond position has a bid-ask spread of 40bp of price. What does it cost to sell it at the bid, against the mid price, in £?

£

Question 2 · objective 2

A desk holds £30m of a bond that trades £10m a day, and it never trades more than 20% of daily volume. How many days does it take to exit?

Question 3 · objective 3

Selling evenly over 10 days, by what factor is the variance of the P&L over the exit larger than one day's variance on the full position (two decimals)?

Question 4 · objective 3

Why is there a best number of days to sell a large position?

ABecause spreads are lower on some days of the weekBSelling fast costs price impact; selling slowly leaves the position exposed to market moves for longer. Their sum has a minimum.CBecause VaR is always lowest after ten daysDThere is none: faster is always better

Question 5 · objective 4

Which position on the desk adds most to its liquidity-adjusted VaR?

AThe UK index futuresBThe 2-year swapCThe corporate bondsDThe FX forward

Glossary

Exogenous liquidity risk: Liquidity risk that comes from the market itself, the same for every trader: the bid-ask spread and how much it varies.
Endogenous liquidity risk: Liquidity risk that comes from the size of your own position: selling a lot moves the price against you.
Bid-ask spread: The gap between the price at which dealers buy (bid) and sell (ask). Exiting at the bid costs half the spread against the mid price.
Average daily volume (ADV): Average daily volume: how much of an instrument trades in a normal day.
Participation rate: The share of a day's market volume a desk allows itself to trade, to limit its price impact.
Liquidation horizon: The number of days needed to exit a position without moving the market too much.
Price impact: The extra price move caused by your own selling, larger the faster you sell relative to normal volume.
Liquidity-adjusted VaR: VaR plus the cost of exiting the positions at a stressed bid-ask spread.

Deep Dives & Origins

Deep Dives

R. Almgren and N. Chriss, “Optimal execution of portfolio transactions”, Journal of Risk 3(2), 2001, pp. 5–39. Advanced.
K. Dowd, Measuring Market Risk, 2nd ed., Wiley, 2005, the chapter on liquidity risk. Intermediate.

Origins

A. Bangia, F. X. Diebold, T. Schuermann and J. D. Stroughair, “Modeling liquidity risk, with implications for traditional market risk measurement and management”, Wharton Financial Institutions Center working paper 99-06, 1999.
Bank of England, “Bank of England announces gilt market operation”, news release, 28 September 2022.

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