Learning From Financial Disasters: formula sheet

Key formulas

Net interest margin

Net interest margin = Interest income − Interest expense

For a thrift, this turns negative when funding costs reset above fixed asset yields.

Price change from duration

ΔP ≈ −D × Δy × P

D is modified duration. Use for small rate moves. A higher duration or higher leverage means larger losses.

Duration gap (approximate)

Duration gap = D(assets) − (L ÷ A) × D(liabilities)

A positive gap means equity loses value when rates rise. L ÷ A is liabilities over assets.

Leverage ratio

Leverage = Total assets ÷ Equity

Leverage multiplies the loss on equity: loss on equity ≈ leverage × loss on assets.

Reverse repo economics (as used by Orange County)

Net carry = Yield on securities bought − Repo rate paid

The strategy earns carry while short rates are low. It loses when the repo rate rises above the yield.

Leverage ratio

Leverage = Total assets ÷ Equity

A 25-to-1 leverage means a fall of about 4% in asset value wipes out equity (1 ÷ 25 = 4%).

Equity loss from asset fall

Equity loss % = Leverage × Asset fall %

Valid when liabilities are fixed in value and the fall applies to all assets, with leverage defined as assets ÷ equity.

Repo haircut

Cash borrowed = Collateral value × (1 − haircut)

A higher haircut means less cash for the same collateral, so more funding is needed from other sources.

Variation margin on a futures hedge

Cash call = Number of contracts × contract size × adverse price move

An adverse move is a price fall for a long futures position and a price rise for a short position. MG was long futures, so falling prices triggered cash calls, while a short position would gain on a fall.

Value at Risk (definition)

VaR(α) = loss level L such that P(Loss > L) = 1 − α

Gives a threshold, not the size of losses beyond it. Expected shortfall measures the average loss beyond VaR.

Model risk sources

Model risk = wrong assumptions + implementation error + misuse outside valid range + weak validation

A checklist, not a numeric formula. Classify each case into one or more sources.

Default correlation effect on a senior tranche

Higher default correlation → higher probability of many joint defaults → senior tranche loses more, equity tranche loses less (relative)

Direction rule. Senior tranches are short correlation; equity tranches are long correlation in value terms.

Normal VaR scaling

VaR(h days) = VaR(1 day) × √h

Holds only for independent, identically distributed returns with zero mean. It understates risk if returns are serially correlated or positions are illiquid.

Tranche loss allocation

Losses hit tranches in order: equity (first loss) → mezzanine → senior

A tranche is only hurt once all junior tranches are exhausted. Attachment point is where a tranche starts to lose; detachment point is where it is wiped out.

Tranche thickness

Thickness = Detachment point − Attachment point

Thin mezzanine tranches are very sensitive to pool losses. Express as a % of the pool.

Loss to a tranche

Tranche loss = min(max(Pool loss − Attachment, 0), Detachment − Attachment)

Use pool loss in the same units as the attachment and detachment points (% of pool or currency).

Leverage ratio

Leverage = Total assets ÷ Equity

A fall in asset value of 1 ÷ leverage wipes out equity. At 30x, a 3.33% fall eliminates equity.

Quick revision

• Savings and loan crisis: long-term fixed-rate assets funded by short-term deposits, so rising rates hurt.
• Orange County: leveraged bets on falling interest rates, funded through repo, with losses when rates rose.
• Metallgesellschaft: long-term fixed-price commitments hedged with short-dated futures, causing rollover and margin-call strain.
• LTCM: high leverage in convergence trades, with a liquidity squeeze when spreads widened.
• Lehman: heavy leverage, illiquid assets and reliance on short-term funding led to loss of confidence.
• Rogue trading cases point to weak segregation of duties and poor oversight of the back office.
• Model risk arises from wrong assumptions, wrong inputs, misuse or unchecked models.
• Securitization failures involved weak underwriting, complexity, ratings reliance and poor incentives.
• Funding liquidity risk is the risk of being unable to meet cash obligations as they fall due.
• Good governance needs strong culture, independent risk functions and clear accountability.
• Always identify the primary risk type first, then the secondary ones.

Common mistakes

• Naming the loss event as the cause, such as saying the cause was a market fall. Fix: Ask why the firm could not survive the move. The answer is usually leverage, weak limits or poor governance.
• Forcing each disaster into a single cause. Fix: Remember that most cases mix causes. Pick the one the question stresses and note the contributing ones.
• Saying the S&L crisis was caused by falling interest rates. Fix: S&Ls lost because rates rose. Funding costs went up while mortgage yields were fixed.
• Treating the S&L crisis as only a market risk problem. Fix: Add deposit insurance, moral hazard and regulatory forbearance. They raised the final cost.
• Treating funding liquidity and market liquidity as the same thing. Fix: Ask whether the problem is paying obligations (funding) or selling assets at a fair price (market).
• Saying MG failed because its hedge was unhedged speculation. Fix: MG's hedge was intended to offset customer contracts. The problem was the cash flow mismatch and the rolling of short-dated futures against long-dated commitments.
• Treating rogue trading as market risk because the loss came from positions. Fix: Classify the root cause. Failed controls, process and people mean operational risk, even when the loss comes from market moves.
• Mixing up the cases, for example attributing custody forgery to Barings. Fix: Memorize one signature fact per case: Barings error account, Daiwa custody records, Sumitomo copper, Société Générale fictitious hedges, London Whale model.
• Saying the Gaussian copula was 'wrong mathematics'. Fix: Say the mathematics was fine; the assumptions (single stable correlation, thin joint tails, calm data) and over-reliance on it were the problem.
• Claiming a higher correlation hurts all tranches. Fix: Higher correlation raises senior tranche risk and lowers equity tranche risk, because losses become all-or-nothing.

Exam tips

• Expect scenario questions that ask for a cause, a control failure or a lesson. Read which one is being asked before looking at the options.
• For each case you study, keep a one-line summary: risk type, root cause, remedy. That is faster to revise than a full narrative.
• Watch for options that are true in general but not in the scenario. The correct option fits the facts given.
• Link cases to frameworks you meet elsewhere: risk appetite, three lines of defense, independent valuation and stress testing. Remedies in answers usually come from these.
• Do not spend time on exact dates or loss amounts. Questions focus on what failed and why.
• Expect conceptual questions on cause and lesson. Pick the interest rate risk and leverage answer, not credit or fraud.
• Remember the direction: rates rose in both cases.
• For S&L questions, look for moral hazard and regulatory forbearance as aggravating factors.