Financial Risk Manager (FRM) Credit Risk Measurement and Management Syllabus

Give the standard discretized formula for (unilateral) CVA.

$$CVA = (1-R)\sum_{i=1}^{n} EE(t_i)\, \cdot PD(t_{i-1}, t_i)\, \cdot DF(t_i)$$ where $R$ is recovery, $EE(t_i)$ is discounted expected exposure, $PD(t_{i-1},t_i)$ is the marginal default probability in each interval, and $DF$ is the discount factor.

What is Debt Valuation Adjustment (DVA) and how does it relate to CVA?

DVA is the mirror image of CVA from the entity's own default perspective — it is the expected benefit (to the entity) from its OWN potential default, based on the counterparty's exposure to it. Bilateral CVA $= CVA - DVA$. A worsening of one's own credit raises DVA, producing a counterintuitive accounting gain.

What is wrong-way risk (and right-way risk) in CVA?

Wrong-way risk occurs when exposure to a counterparty is positively correlated with that counterparty's default probability (exposure rises as creditworthiness falls), increasing CVA. Right-way risk is the opposite — exposure falls as default probability rises, reducing CVA.

What roles do netting and collateral play as credit risk mitigants?

Netting reduces exposure by offsetting positive and negative positions within a netting set. Collateral (margin) further reduces exposure by requiring the in-the-money party to post assets, so net exposure is limited to the uncollateralized portion plus exposure that builds during the margin period of risk.

What is a Credit Support Annex (CSA), and what key parameters does it specify?

A CSA is the ISDA Master Agreement annex governing collateral. Key parameters include the threshold (unsecured exposure allowed before collateral is posted), the minimum transfer amount (MTA), the independent amount, eligible collateral and haircuts, and the margin call frequency.

Define 'threshold' and 'minimum transfer amount' in a CSA.

Threshold is the level of uncollateralized exposure permitted before any collateral must be posted (a zero threshold means full collateralization). Minimum transfer amount (MTA) is the smallest collateral movement allowed, avoiding operationally costly tiny transfers.

What is the 'margin period of risk' (MPoR) and why does it matter?

MPoR is the time between the last successful collateral exchange and the close-out/replacement of a defaulted counterparty's trades (typically assumed ~10 business days for cleared/collateralized OTC). Exposure can build over this period, so even fully collateralized trades retain residual CVA.

Define Funding Valuation Adjustment (FVA).

FVA is the adjustment for the cost (FCA) and benefit (FBA) of funding the uncollateralized portion of a derivative over its life. It reflects that funding an in-the-money uncollateralized position must be raised at a rate above the risk-free rate, creating a funding cost.

Define Capital Valuation Adjustment (KVA).

KVA is the adjustment for the lifetime cost of holding regulatory (and economic) capital against a derivative position. It represents the return that shareholders require on capital that must be reserved for the trade over its life.

List the components of the 'XVA' family of valuation adjustments.

CVA (counterparty default), DVA (own default), FVA (funding cost/benefit), ColVA (collateral cost), KVA (capital cost), and MVA (initial margin funding). The fully adjusted price is risk-free value minus the sum of relevant XVAs.

Describe the cash flows of a single-name credit default swap (CDS).

The protection buyer pays a periodic premium (the CDS spread) on the notional until default or maturity. If a credit event occurs, the protection seller pays the loss (notional $\times (1-R)$), settled physically or in cash, and premium accrual stops (with accrued payment to default date).

State the par-spread condition that prices a CDS.

The par CDS spread equates the present value of the premium leg to the present value of the protection leg: $$s = \dfrac{(1-R)\sum_i DF(t_i)\,[Q(t_{i-1}) - Q(t_i)]}{\sum_i DF(t_i)\, \Delta t_i\, Q(t_i)}$$ where $Q(t)$ is the survival probability and $R$ is recovery.

What is the mark-to-market value of an existing CDS position to the protection buyer?

$MtM \approx (s_{\text{market}} - s_{\text{contract}}) \times RPV01 \times \text{Notional}$, where $RPV01$ (risky annuity / risky PV01) is the present value of 1 bp of premium accounting for survival. The buyer gains when market spreads widen above the contract spread.

What is the 'CDS-bond basis' and what does a negative basis mean?

The CDS-bond basis is the CDS spread minus the bond's credit spread (e.g., asset-swap or Z-spread). A negative basis means the CDS spread is lower than the bond spread, allowing a basis trade: buy the bond and buy CDS protection for an apparent arbitrage.

What is a CDS index (e.g., CDX, iTraxx) and how does it differ from a single-name CDS?

A CDS index is a standardized, equally weighted portfolio of single-name CDS (e.g., CDX.NA.IG has 125 names; iTraxx Europe 125 names). It trades with a fixed coupon and upfront, offers diversified credit exposure, and on a member's default that name is removed and settled, with notional reduced.

What is a tranched CDS index product, and what defines a tranche?

Tranched index products slice an index's portfolio loss distribution into tranches defined by attachment and detachment points (e.g., 0–3% equity, 3–7%, 7–10%, ...). A tranche absorbs portfolio losses between its attachment and detachment points; once cumulative losses exceed detachment, the tranche is wiped out.

Order CDO/index tranches by risk: equity, mezzanine, senior.

Equity (first-loss) tranche is riskiest — absorbs initial losses, highest spread. Mezzanine is intermediate. Senior/super-senior is least risky — only hit after lower tranches are exhausted, lowest spread. Risk and required yield decrease as you move up the capital structure.

Distinguish a cash CDO from a synthetic CDO.

A cash CDO holds actual debt assets (bonds/loans) whose cash flows fund the tranches. A synthetic CDO gains credit exposure by selling protection via CDS rather than owning the assets; tranche investors effectively sell protection on portions of a reference portfolio's loss distribution.

What is the one-factor Gaussian copula model used for in credit?

It models default correlation among many obligors. Each obligor's latent asset variable is $X_i = \sqrt{\rho}\,M + \sqrt{1-\rho}\,Z_i$, where $M$ is a common factor and $Z_i$ idiosyncratic, both standard normal. Default occurs when $X_i$ falls below a threshold mapped from its marginal PD; $\rho$ controls correlation.

In the one-factor Gaussian copula, what is the conditional default probability given the common factor $M$?

$$PD_i(M) = N\!\left(\dfrac{N^{-1}(PD_i) - \sqrt{\rho}\,M}{\sqrt{1-\rho}}\right)$$ Conditional on $M$, defaults are independent (the conditional-independence framework), which makes large-portfolio loss distributions tractable.

What is 'implied correlation' (compound vs base correlation) for tranches?

Implied correlation is the copula correlation that reprices a tranche to its market quote. Compound correlation is solved per individual tranche (can be non-unique/non-monotonic for mezzanine). Base correlation uses cumulative equity-style tranches (0–X%), giving a unique, monotonic, interpolatable curve — the 'correlation smile'.

What is a securitization, and what is the purpose of a special purpose vehicle (SPV)?

Securitization pools assets (e.g., loans, receivables) and issues tranched securities backed by their cash flows. The SPV is a bankruptcy-remote entity that holds the assets, isolating them from the originator's credit so investors rely on the pool, not the originator.

List common forms of credit enhancement in securitizations.

Internal: subordination/tranching (overcollateralization), excess spread, reserve/cash collateral accounts. External: third-party guarantees, surety bonds/monoline wraps, and letters of credit. Enhancement raises the credit quality of senior tranches above the pool average.

What is the 'waterfall' in a securitization structure?

The waterfall is the contractual priority of payments: collected cash flows pay senior tranches first (interest then principal), then mezzanine, then equity/residual, with triggers that can divert cash. Losses are absorbed in reverse order — equity first, senior last.

Define default correlation and explain its effect on a credit portfolio's loss distribution.

Default correlation measures the tendency of obligors to default together. Higher correlation does not change expected loss but fattens the tail of the portfolio loss distribution — raising the probability of both very low and very high losses — which increases unexpected loss and credit VaR.

What is concentration risk, and what are its two main types?

Concentration risk is the elevated risk from imperfect diversification. Name (single-obligor) concentration arises from large exposures to individual borrowers; sector/industry/geographic concentration arises from correlated exposures to a common risk factor. Both increase tail risk beyond what granular models assume.

Contrast the CreditMetrics and CreditRisk+ portfolio models.

CreditMetrics is a ratings-migration (mark-to-market) model using a transition matrix and asset-value correlations to simulate value changes including downgrades. CreditRisk+ is an actuarial default-mode model treating defaults as Poisson events with gamma-distributed intensities, yielding an analytic loss distribution (default/no-default only).

Distinguish 'default mode' from 'mark-to-market mode' credit portfolio models.

Default-mode (two-state) models recognize loss only on actual default over the horizon. Mark-to-market (multi-state) models also capture value changes from rating migrations and spread moves, so they record gains/losses from up/downgrades even without default.

Define economic capital for credit risk and its relation to expected and unexpected loss.

Economic capital for credit risk equals unexpected loss = Credit VaR at a chosen confidence minus expected loss: $EC = \text{VaR}_\alpha - EL$. Expected loss is covered by reserves/pricing; economic capital is the buffer held against unexpected losses up to the target solvency level.

What is Credit VaR?

Credit VaR is the maximum portfolio credit loss not exceeded at a given confidence level over a horizon (often 1 year). Formally, $\text{VaR}_\alpha$ is the $\alpha$-quantile of the credit loss distribution; economic capital is typically this quantile in excess of expected loss.

State the Basel IRB capital requirement's reliance on the Asymptotic Single-Risk-Factor (ASRF) model and its key property.

The IRB formula is derived from the ASRF model, which assumes a single systematic factor and an infinitely granular (idiosyncratic-risk-free) portfolio. Its key property is portfolio invariance: each loan's capital depends only on its own characteristics (PD, LGD, M), not on the rest of the portfolio.

Give the Basel IRB conditional (stressed) PD formula used to compute capital.

$$PD_{cond} = N\!\left(\dfrac{N^{-1}(PD) + \sqrt{R}\,N^{-1}(0.999)}{\sqrt{1-R}}\right)$$ where $R$ is the asset correlation and $0.999$ is the supervisory confidence level. Capital is based on $LGD \times (PD_{cond} - PD)$, i.e., the unexpected-loss portion, times a maturity adjustment.

In the Basel IRB approach, how does asset correlation $R$ depend on PD for corporate exposures?

Correlation decreases as PD rises, interpolating between 0.24 and 0.12: $$R = 0.12\,\dfrac{1-e^{-50\,PD}}{1-e^{-50}} + 0.24\left(1 - \dfrac{1-e^{-50\,PD}}{1-e^{-50}}\right)$$ The rationale is that high-PD borrowers' defaults are driven more by idiosyncratic factors.

Distinguish the Foundation IRB (F-IRB) from the Advanced IRB (A-IRB) approach.

In F-IRB, the bank estimates only PD; supervisory values are used for LGD, EAD, and maturity. In A-IRB, the bank estimates PD, LGD, EAD, and effective maturity (M) using its own internal models, subject to supervisory approval and minimum standards.

How can securitization serve as a loan portfolio management tool?

By securitizing loans, a bank can transfer credit risk off balance sheet, free up regulatory capital, manage concentration limits, improve liquidity and funding, and originate-to-distribute. It lets a bank rebalance its portfolio toward target risk/return without selling individual relationships.

Why are recovery rates (LGD) negatively correlated with default rates over the cycle, and what is the implication?

In downturns defaults rise while collateral and asset values fall, so recovery rates drop just when default rates peak — LGD and PD are positively correlated (recoveries are cyclical/procyclical). The implication is that using average (cycle-neutral) LGD understates downturn losses, so Basel requires downturn LGD estimates.

What is the difference between market-value (price-based) and ultimate (workout) recovery measures?

Market-value recovery uses the defaulted instrument's trading price shortly after default (typically ~30 days), reflecting market expectations and easy to observe. Ultimate recovery is the actual discounted value of cash/securities received through the workout/bankruptcy process, realized over a longer, uncertain horizon.

How does seniority and security affect recovery rates in the debt structure?

Recovery rates rise with seniority and collateral: secured > senior unsecured > senior subordinated > subordinated > junior subordinated. Higher-priority and collateralized claims recover more in default, so LGD assignment depends on facility rank and collateral, not just the obligor.

What is the formula linking unexpected loss (UL) of a single facility to PD and LGD volatility?

For a single exposure, $UL = EAD \times \sqrt{PD \cdot \sigma_{LGD}^{2} + LGD^{2} \cdot \sigma_{PD}^{2}}$, where $\sigma_{PD}^{2} = PD(1-PD)$ for a Bernoulli default. UL is the standard deviation of loss, distinct from the EL mean.

What is a synthetic securitization's 'single-tranche CDO' (bespoke tranche), and what hedging concept does it require?

A bespoke single-tranche CDO sells protection on one custom tranche of a chosen reference portfolio rather than the whole capital structure. The dealer delta-hedges using single-name CDS or index tranches; the tranche's sensitivity to the underlying spread is its 'delta', and correlation risk must also be managed.

What does a credit transition (migration) matrix represent, and what is a key empirical regularity?

A transition matrix gives the probabilities of an obligor moving between rating grades (including default) over a horizon (usually 1 year). Empirically, diagonal elements are large (ratings are sticky), default probabilities rise steeply for lower grades, and matrices show momentum (downgrades tend to be followed by further downgrades).