FRM Part II · FRM Exam Part II

Structured Credit Risk: formula sheet

Full chapter guide

Key formulas

Tranche thickness: Thickness = Detachment point − Attachment point; Both points are percentages of the total pool. A tranche from 5% to 15% is 10% thick.
Tranche loss: Tranche loss = min(max(Pool loss − Attachment, 0), Detachment − Attachment); Expressed in the same units as the pool. Divide by thickness for the percentage loss on the tranche.
Subordination: Subordination of a tranche = Attachment point of that tranche; It is the share of the pool that absorbs losses before the tranche takes any.
Overcollateralization: OC = (Pool balance − Liabilities issued) ÷ Pool balance; Equivalent to extra collateral beyond what funds the notes. It is a form of credit enhancement.
Excess spread: Excess spread = Pool interest income − (Coupons + Fees + Losses); Positive excess spread is the first line of defence against losses before tranches are written down.
Waterfall order: Fees → Senior interest → Mezzanine interest → Principal per structure → Equity residual; Losses are allocated in the reverse order: equity first, senior last. Always check the deal's stated order.
SMM from CPR: SMM = 1 − (1 − CPR)^(1/12); CPR is annual. SMM is the monthly prepaid fraction of the balance after scheduled principal.
CPR from SMM: CPR = 1 − (1 − SMM)^12; Do not simply multiply SMM by 12. That is only a rough approximation.
Prepayment amount: Prepayment = SMM × (Beginning balance − Scheduled principal); Apply SMM after removing scheduled principal.
PSA CPR for month t: CPR = 6% × (t ÷ 30) × (PSA ÷ 100) for t ≤ 30; CPR = 6% × (PSA ÷ 100) for t > 30; Standard 100% PSA ramps 0.2% per month to 6%.
Pass-through rate: Pass-through rate = WAC − servicing and other fees; Investors receive the net coupon.
Effective duration: D = (P₋ − P₊) ÷ (2 × P₀ × Δy); Use option-adjusted cash flows. MBS have negative convexity.
Tranche loss: Tranche loss = min(max(Portfolio loss − A, 0), D − A); A is the attachment point and D the detachment point, both in currency units or as % of the pool. The tranche is wiped out when portfolio loss reaches D.
Tranche loss as % of tranche: Tranche loss % = tranche loss ÷ (D − A); Thin tranches show large percentage losses for small pool losses. This is leverage.
Tranche thickness: Thickness = D − A; Equity tranche has A = 0. Thickness sets how much pool loss it can absorb.
Portfolio loss: Portfolio loss = Σ (exposure × LGD) over defaulted names; LGD = 1 − recovery rate. Use it to get the pool loss before tranching.
Correlation effect: Higher default correlation → equity tranche value up; senior tranche value down; Mezzanine effect is ambiguous and depends on its location in the structure.
Synthetic CDO tranche cash flows: Premium on outstanding tranche notional; protection payment = tranche loss; Tranche notional falls as losses are written down.
Tranche thickness: Thickness = Detachment point − Attachment point; Both points are percentages of the total pool notional.
Tranche loss rate: Tranche loss % = min(max(L − A, 0), D − A) ÷ (D − A); L is pool loss %, A is attachment, D is detachment. Result is the share of the tranche principal lost.
Tranche loss amount: Loss = min(max(L − A, 0), D − A) × Pool notional; Use this when the question asks for a currency loss.
Subordination (credit enhancement) for a tranche: Subordination = Attachment point; It equals the combined size of all tranches below it, as a percentage of the pool.
Overcollateralization: OC % = (Pool balance − Notes outstanding) ÷ Pool balance; Some texts divide by notes instead. Check the question's definition.
Excess spread: Excess spread = Pool interest income − Note interest − Fees − Losses (if stated after losses); Usually quoted annually as a percentage of the pool. Read whether losses are deducted.
Effective loss absorbed before tranche: Net pool loss = Gross loss − Excess spread used − OC used; Enhancement items reduce the loss that reaches tranches.
One-factor Gaussian copula: Xi = √ρ × M + √(1 − ρ) × Zi; M and Zi are independent standard normals. Pairwise asset correlation is ρ.
Default threshold: K = N⁻¹(PD); Firm defaults if Xi < K. Marginal PD is unchanged by ρ.
Conditional default probability: PD(M) = N[(N⁻¹(PD) − √ρ × M) ÷ √(1 − ρ)]; Defaults are independent given M. Low M raises PD(M).
Joint default probability of two firms: P(both default) = PD² + ρD × PD × (1 − PD), if both have the same PD; ρD is the default correlation (default indicators). It is usually smaller than the asset correlation ρ.
Tranche loss rule: Tranche loss = min(max(L − A, 0), D − A); L is pool loss, A attachment, D detachment, all in the same units.
Correlation effects: Equity value ↑ and senior value ↓ as ρ ↑; Pool expected loss is unchanged. Mezzanine is ambiguous.
Base correlation tranche: Tranche A–D = equity(0–D) − equity(0–A); Each equity piece uses its own base correlation.
Tranche loss: Tranche loss = min(max(L − A, 0), D − A); L = pool loss, A = attachment point, D = detachment point, all in the same units (for example % of pool). Tranche loss is in the same units as L; divide by (D − A) for the % of tranche notional lost.
Tranche loss as % of tranche: % tranche loss = min(max(L − A, 0), D − A) ÷ (D − A); Shows leverage: a thin tranche loses a large percentage for a small pool loss.
Tranche thickness: Thickness = D − A; Thinner tranches are more sensitive to pool losses.
Expected loss of the pool: EL = Σ (PD × LGD × EAD); Sum over assets. Tranche expected losses add up to the pool expected loss (before fees and excess spread).
Value of tranche: Value = Σ expected cash flow(t) × discount factor(t); Expected cash flows come after applying the waterfall to simulated loss scenarios. Discounting is at risk-neutral or market-consistent rates.
Correlation effect (rule of thumb): Higher default correlation: equity tranche value rises, senior tranche value falls; Holds in a standard copula model; mezzanine can move either way depending on position.
Originate-to-distribute logic: Originator income = fees on volume; credit loss borne by investors; This is the core incentive failure. Weak retained risk means weak screening and monitoring.
Risk retention (Dodd-Frank): Retained credit risk ≥ 5% of the securitized exposure (general rule); Sponsors may not hedge or transfer the retained risk. Qualified residential mortgages are exempt. The rule can be met by retaining a vertical or horizontal interest.
Tranche loss rule: Tranche loss = min(tranche size, max(0, pool loss − attachment point)); Use it to show why thin mezzanine tranches are wiped out by moderate pool losses.
Leverage effect: Return on equity ≈ asset return × (assets ÷ equity) − funding cost × (debt ÷ equity); A small fall in asset value causes a large fall in equity when assets ÷ equity is high.

Quick revision

Securitization moves assets into an SPV, which funds itself by issuing tranches.
Losses hit the equity tranche first, then mezzanine, then senior.
Tranche loss = min(max(pool loss − attachment point, 0), detachment point − attachment point).
Subordination is the main form of internal credit enhancement; excess spread and overcollateralization are others.
Prepayment risk matters for MBS: faster or slower prepayments change average life and returns.
A synthetic CDO takes credit exposure through credit default swaps instead of owning the loans.
Higher default correlation tends to increase risk for senior tranches and tends to reduce it for equity tranches.
The Gaussian copula links defaults with one correlation parameter and understates tail dependence.
Ratings of structured products depend heavily on model and correlation assumptions, so they can shift sharply.
CDO squared and re-securitizations amplify model risk and concentration.
The subprime crisis combined weak underwriting, originate-to-distribute incentives, rating errors and funding runs.
Senior tranches are not risk free; they carry systemic risk when many assets default together.

Common mistakes

Treating a synthetic securitization as a true sale. Fix: In a true sale the assets legally move to the SPV. In a synthetic deal the assets stay put and only credit risk moves, typically through credit derivatives.
Dividing a tranche loss by the pool size instead of the tranche size. Fix: Subtract the attachment point, cap at thickness, then divide by thickness to get the percentage loss on the tranche.
Dividing CPR by 12 to get SMM. Fix: Use SMM = 1 − (1 − CPR)^(1/12). Dividing by 12 is only an approximation.
Applying SMM to the beginning balance instead of the balance after scheduled principal. Fix: Subtract scheduled principal first, then multiply by SMM.
Ignoring recovery and using exposure lost as the pool loss. Fix: Multiply each default by LGD = 1 − recovery before allocating to tranches.
Dividing tranche loss by pool size instead of tranche size. Fix: Tranche loss % = tranche loss ÷ (D − A). State the denominator before computing.
Dividing tranche loss by the pool size instead of the tranche size. Fix: Compute the loss in pool percentage first, then divide by thickness (D − A) to get the tranche loss rate.
Treating a pool loss above the attachment point as a full tranche loss. Fix: Cap the loss at D − A. Only L ≥ D wipes out the tranche.
Saying higher correlation raises the expected loss of the pool. Fix: Expected pool loss depends only on PD, LGD and exposure. Correlation changes the shape of the loss distribution, not its mean.
Claiming senior tranches gain from higher correlation. Fix: Higher correlation fattens the tail. Senior tranches only lose in the tail, so they lose value.

Exam tips

Draw the tranche stack in your head from the bottom: equity, mezzanine, senior. Most loss questions are solved this way.
Read whether the deal is true sale or synthetic before answering any question on balance sheet or capital relief.
When an option says the senior tranche is immune to losses, treat it as wrong unless the loss is below its attachment point.
Link the originate-to-distribute model to weak underwriting incentives. This is a frequent conceptual question.
Check units: loss on a tranche as a percentage of the pool is not the percentage loss on the tranche.
Check whether a speed is monthly or annual before any calculation.
For PSA questions, find the month first and apply the ramp or the cap accordingly.
Expect scenario questions: say whether contraction or extension risk dominates, and what happens to duration and convexity.