Liquidity Risk Management: formula sheet

Key formulas

Bid-ask spread (relative)

Relative spread = (Ask − Bid) ÷ Mid price, where Mid = (Ask + Bid) ÷ 2

A wider relative spread means lower market liquidity. The cost of selling one unit is about half the spread.

Leverage

Leverage = Assets ÷ Equity

A loss spiral raises leverage when equity falls faster than assets are cut.

Margin or haircut and funding capacity

Assets financeable = Equity ÷ Margin rate (haircut as a fraction of asset value)

If the margin rises, the same equity supports fewer assets. This drives the margin spiral.

Core linkage rule

Funding stress → forced sales → lower prices and market liquidity → higher margins and weaker collateral → more funding stress

Know the direction of each link. Questions often ask which step comes next.

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.

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.

Net funding gap (per bucket)

Gap = Cash inflows − Cash outflows

Negative means a shortfall in that bucket.

Cumulative gap

Cumulative gap(t) = Σ Gap(i) for buckets i = 1 to t

Compare with the buffer at each horizon, not just the last one.

Stressed buffer

Buffer = Σ [Market value of asset × (1 − haircut)]

Include only unencumbered, readily saleable assets.

Survival horizon

Longest horizon t where Buffer + Cumulative gap(t) ≥ 0

Cumulative gap is negative when outflows exceed inflows. Use stressed flows.

Liquidity Coverage Ratio

LCR = Stock of HQLA ÷ Total net cash outflows over 30 days ≥ 100%

Net outflows = outflows − min(inflows, 75% of outflows).

Loan-to-deposit ratio

Loans ÷ Deposits

A higher ratio means more reliance on wholesale funding.

Liquidity Coverage Ratio

LCR = Stock of HQLA ÷ Total net cash outflows over next 30 days ≥ 100%

HQLA must be unencumbered and is measured after haircuts and caps.

Net cash outflows

Net outflows = Total outflows − min(Total inflows, 75% × Total outflows)

Inflows are capped at 75% of outflows, so net outflows are at least 25% of gross outflows.

Net Stable Funding Ratio

NSFR = Available stable funding ÷ Required stable funding ≥ 100%

Horizon is one year. ASF and RSF are each the sum of balances × factors.

HQLA composition

HQLA = Level 1 + Level 2A × (1 − 15%) + Level 2B × (1 − haircut)

Level 2B haircuts are 25% to 50% depending on the asset. Level 2 ≤ 40% of HQLA; Level 2B ≤ 15%.

Level 2 cap

Max Level 2 = (2 ÷ 3) × Level 1

Equivalent to Level 2 being 40% of total HQLA. Ignores adjustments for unwinding secured transactions.

Funding gap

Funding gap = Cash outflows − Cash inflows over a horizon

A positive gap must be covered by the liquidity buffer or contingent sources.

Survival horizon

Survival horizon = time until cumulative net outflows exceed the liquidity buffer

Longer is safer. Compare it to the stress period required by policy.

Loan-to-deposit ratio

Loan-to-deposit ratio = Loans ÷ Deposits

A high ratio suggests reliance on wholesale funding.

Repo cash raised

Cash = Collateral market value × (1 − haircut)

A higher haircut means less cash from the same collateral.

Leverage

Leverage = Total assets ÷ Equity

Higher leverage means smaller price falls can force asset sales and margin calls.

Quick revision

• Funding liquidity is the ability to meet obligations; market liquidity is the ability to trade without moving price.
• The two types reinforce each other: funding stress forces asset sales, and falling prices tighten funding.
• Liquidation cost for a position is roughly its value × half the relative spread.
• LVaR = VaR + liquidity cost, so it is always at least as large as VaR.
• Endogenous liquidity comes from your own trades; exogenous liquidity is market-wide.
• Selling fast raises price impact; selling slowly adds price risk during the delay.
• Stress tests should combine market shocks with funding outflows and haircut increases.
• LCR = stock of HQLA ÷ total net cash outflows over 30 days, with a minimum of 100%.
• NSFR = available stable funding ÷ required stable funding, with a minimum of 100%, over a one-year horizon.
• LCR is a short-term stress buffer; NSFR limits structural maturity mismatch.
• Short-term wholesale funding and rollover reliance are classic crisis weak points.
• A contingency funding plan sets triggers, funding sources and responsibilities before stress occurs.

Common mistakes

• Treating market liquidity and funding liquidity as the same thing. Fix: Market liquidity concerns trading an asset. Funding liquidity concerns meeting cash and collateral needs. Ask which side the stem describes.
• Saying a wide bid-ask spread is funding liquidity risk. Fix: Spreads measure market liquidity. A funding problem appears as failed rollover, margin calls or a cash shortfall.
• Using the full spread instead of half the spread as the cost. 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. Fix: Convert 0.4% to 0.004 before multiplying.
• Treating liquidity cost as only the bid-ask spread. Fix: For large positions, add price impact from your own trading. That is the endogenous part.
• Saying a longer horizon always lowers total cost. Fix: Always compare both costs. Total cost has a minimum at an intermediate horizon.
• Using the unhaircut market value of assets in the buffer. Fix: Multiply each asset by (1 − haircut) before adding. Stress raises haircuts.
• Treating contractual flows as the stressed case. Fix: Always apply behavioural run-off and draw-down assumptions given in the question.
• Mixing up the horizons, using 1 year for LCR or 30 days for NSFR. Fix: Remember LCR = 30 days, liquidity under stress; NSFR = 1 year, funding structure.
• Counting Level 2 assets at full value or ignoring the 40% cap. Fix: Haircut first, then test Level 2 ≤ 40% of HQLA and Level 2B ≤ 15%.

Exam tips

• Start every question by labelling the risk as market or funding. Many wrong options swap the two.
• Know both Brunnermeier-Pedersen spirals and the trigger of each: losses versus higher margins.
• Expect link questions: which condition makes spirals stronger? Think crowded positions, high leverage and short-term funding.
• For margin arithmetic, compute assets before and after, then take the difference. Do not rush the division.
• 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.