FRM Part II · FRM Exam Part II

Liquidity and Leverage: formula sheet

Full chapter guide

Key formulas

Mid-price: M = (Ask + Bid) ÷ 2; Reference price for all spread measures.
Quoted spread (absolute): Quoted spread = Ask − Bid; Displayed cost of a round trip for a small trade.
Relative quoted spread: Relative spread = (Ask − Bid) ÷ M; Allows comparison across assets with different prices.
Effective spread: Effective spread = 2 × D × (P − M), where D = +1 for a buy and −1 for a sell; P is the trade price and M the mid-price at trade time. Divide by M for the relative form.
Realized spread: Realized spread = 2 × D × (P − M_later); M_later is the mid-price some time after the trade. Effective spread minus realized spread equals the price impact.
Price impact: Price impact = 2 × D × (M_later − M); Effective spread = realized spread + price impact.
Liquidity cost in VaR (constant spread): LVaR = VaR + ½ × S × V; S is the relative spread, V the position value. Assumes a known, constant spread and liquidation at the half-spread cost.
Leverage: Leverage = Total assets ÷ Equity; Higher leverage means a given price fall wipes out more equity and forces larger sales.
Haircut and funding: Funding obtained = Collateral value × (1 − haircut); The equity you must supply per unit of asset equals the haircut. A rise in haircut forces deleveraging.
Maximum position under margin: Max position = Capital ÷ Margin rate; If the margin rate rises, the maximum position falls, which drives the margin spiral.
Funding gap: Funding gap = Cash outflows − Cash inflows over a horizon; A positive gap must be covered by buffers or new funding.
Balance-sheet leverage (equity multiplier): Leverage = Total assets ÷ Equity; Equity = assets − liabilities. A leverage of 10 means ₹1 of equity supports ₹10 of assets.
Debt-to-equity: D/E = Debt ÷ Equity = Leverage − 1; Holds when all non-equity funding is counted as debt.
Return on equity with leverage: ROE = ROA × (Assets ÷ Equity) − (Assets − Equity) ÷ Equity × cost of debt; Equivalent to ROE = r_A + (D/E) × (r_A − r_D). Leverage helps only if the asset return exceeds the cost of debt.
Embedded leverage: Embedded leverage = Exposure (notional or delta-adjusted) ÷ Funds invested; For an option, delta-adjusted exposure = delta × underlying value × quantity. Funds invested = premium paid.
Basel III leverage ratio: Leverage ratio = Tier 1 capital ÷ Total exposure measure; Non-risk-weighted. The exposure measure includes on-balance-sheet assets and specified off-balance-sheet items. Note it is the inverse direction of assets ÷ equity: higher is safer.
Equity change from asset shock: % change in equity ≈ Leverage × % change in assets; Ignores funding cost and income. Wipe-out occurs when the asset fall ≥ 1 ÷ Leverage.
Leverage multiple: L = Assets ÷ Equity = 1 + Debt ÷ Equity; Equity multiplier. Debt-to-equity is L − 1.
Return on equity with leverage: ROE = r_A + (r_A − r_D) × (D ÷ E) = L × r_A − (L − 1) × r_D; r_A is the asset return, r_D the cost of debt. Both forms give the same value.
Equity volatility: σ_E = L × σ_A; Valid when the borrowing cost is fixed (no uncertainty in r_D) and the leverage multiple is constant.
Equity loss for an asset fall: Equity return = −L × (asset fall) − (L − 1) × r_D; This is the ROE formula with a negative r_A. Equity is exhausted when the asset fall ≥ 1 ÷ L (ignoring interest).
Break-even asset return: r_A = r_D; Above it, leverage raises ROE. Below it, leverage lowers ROE.
Basel III leverage ratio: Leverage ratio = Tier 1 capital ÷ Total exposure measure; Not risk-weighted. Basel III minimum is 3%.
Equity VaR from asset VaR: VaR on equity ≈ L × asset VaR (as % of equity); In currency terms the loss is the same on the assets; the percentage of equity is scaled by L.
Haircut: Haircut = (Collateral value − Cash lent) ÷ Collateral value; Cash lent = Collateral value × (1 − haircut). Check whether the question quotes the haircut on collateral value.
Equity funded by haircut: Equity required = Asset value × haircut; The rest of the position is financed by the lender.
Maximum leverage: Assets ÷ Equity = 1 ÷ haircut; A 4% haircut allows 25 times; 10% allows 10 times.
Margin call on haircut rise: Extra equity = Asset value × (new haircut − old haircut); Assumes asset value is unchanged. If prices also fall, recompute with the new value.
Forced sale to restore leverage: Sale = Current assets − Equity ÷ haircut; Equity is after any losses. Sale proceeds repay debt.
Variation margin: VM = Mark-to-market change of the position; Paid by the party whose position lost value, usually daily.
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.
Leverage ratio: Leverage = Total assets ÷ Equity; Equity multiplier. Leverage of 20 means equity is 5% of assets.
Equity change from asset return: Return on equity ≈ Leverage × Asset return (ignoring funding cost); A 2% asset fall at leverage 20 is roughly a 40% equity fall.
Haircut: Haircut = 1 − (Loan ÷ Collateral market value); Higher haircut means less borrowing against the same collateral.
Maximum assets funded with a haircut: Assets = Equity ÷ Haircut; A 5% haircut permits leverage of 20. A 10% haircut permits only 10.
Sales needed to restore a target leverage: Required asset reduction = Current assets − (Equity × Target leverage); Assumes sale proceeds repay debt and equity stays fixed after the loss.
Spiral sequence: Shock → losses → lower equity or higher haircuts → forced sales → lower prices → more losses; Know the loop and where policy can break it.

Quick revision

Market liquidity is the ability to trade quickly at low cost without moving the price.
Funding liquidity is the ability to meet cash and collateral demands when due.
Quoted spread = ask − bid; relative spread = (ask − bid) ÷ mid-price.
Cost of selling at the bid is half the spread per unit relative to mid.
Leverage ratio = total assets ÷ equity; debt-to-equity is a different ratio.
Leverage magnifies gains and losses; volatility of equity returns rises roughly in proportion to leverage.
Levered return = asset return × leverage − borrowing rate × (leverage − 1).
A haircut is the part of collateral value that cannot be borrowed against; the borrower funds it with own capital.
Higher haircuts and margin demands in stress force deleveraging and asset sales.
LVaR adds a liquidity cost, often half the spread times position value, to ordinary VaR.
Funding and market liquidity reinforce each other in a liquidity spiral.
Embedded leverage in derivatives can be high even when balance sheet leverage looks low.

Common mistakes

Treating the effective spread as the same as the quoted spread. Fix: Quoted comes from displayed bid and ask. Effective comes from the actual trade price against mid, so it can be smaller (price improvement) or larger (walking the book).
Forgetting to multiply by 2 in effective and realized spreads. Fix: Effective and realized spreads are defined as twice the signed deviation, so they are comparable to a full quoted spread.
Treating funding liquidity and market liquidity as the same thing. Fix: Funding is about meeting obligations with cash or collateral. Market is about selling an asset near fair value.
Confusing the loss spiral with the margin spiral. Fix: Loss spiral starts with losses reducing capital. Margin spiral starts with higher margins or haircuts due to higher volatility or uncertainty.
Confusing debt-to-equity with assets-to-equity. Fix: Remember assets ÷ equity = 1 + D/E. Read which one the question asks for.
Using notional instead of delta-adjusted exposure for options. Fix: Embedded leverage of an option uses delta × underlying ÷ premium unless the question defines it otherwise. Notional overstates the real exposure.
Using the debt-to-equity ratio instead of the leverage multiple to scale volatility. Fix: Scale by Assets ÷ Equity. Debt-to-equity is one less than that.
Charging the cost of debt on the whole asset base. Fix: Interest is paid only on the borrowed amount, which is (L − 1) times equity.
Treating the haircut as the interest rate or the lender's fee. Fix: The haircut is the collateral cushion. The repo rate is the price of the cash.
Swapping initial and variation margin. Fix: Initial margin is a buffer for future moves and default close-out. Variation margin settles current mark-to-market gains and losses.

Exam tips

Always compute the mid-price first. Most spread questions are one step after that.
Watch the word half. Many answer options differ only by a factor of 2.
Know the link: effective spread = realized spread + price impact. Questions often give two and ask for the third.
For concept questions, match the term to the dimension: spread = tightness, volume at quote = depth, speed of recovery = resilience.
Remember the LVaR assumption of a constant spread. If spread varies, the basic add-on understates the risk.
Read the first trigger in the vignette: losses mean loss spiral, margin or haircut increases mean margin spiral.
Expect questions linking short-term wholesale funding and maturity mismatch to funding liquidity risk.
Watch for answers claiming the two liquidity risks are independent. They are wrong in stress.