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

Define the risk-neutral measure as used in credit pricing.

The risk-neutral measure ($Q$) is the probability measure under which all assets earn the risk-free rate and prices equal discounted expected payoffs. Risk-neutral default probabilities are used to price credit instruments and embed risk premia, not to forecast actual defaults.

Define the real-world (physical) measure for default probabilities.

The real-world (physical) measure ($P$) describes actual, statistically estimated default frequencies — e.g., from historical default data or rating-agency transition matrices. It is used for risk management, capital, and expected-loss estimation rather than no-arbitrage pricing.

Why are risk-neutral default probabilities typically larger than real-world default probabilities?

Risk-neutral PDs embed a risk premium: investors demand extra compensation for bearing systematic default risk, the uncertainty of recovery, and illiquidity. This premium inflates market-implied PDs above the actuarially expected (real-world) default rates.

For which purposes should real-world vs. risk-neutral PDs each be used?

Use risk-neutral PDs for pricing and valuation of credit instruments and CVA (no-arbitrage). Use real-world PDs for risk management, economic capital, expected loss/loss provisioning, and stress testing — anything requiring a forecast of actual default frequency.

Define a credit scoring model and name a classic statistical technique used.

A credit scoring model maps borrower characteristics into a numerical score predicting default likelihood. Classic techniques include logistic regression (logit), linear discriminant analysis (e.g., Altman's Z-score), and modern machine-learning classifiers.

What is Altman's Z-score, and what does a low value indicate?

Altman's Z-score is a discriminant-analysis credit model combining weighted financial ratios (working capital, retained earnings, EBIT, market equity, sales — all scaled by assets/liabilities) into a single score. A low Z-score signals high bankruptcy/default risk; high indicates financial health.

Distinguish through-the-cycle (TTC) from point-in-time (PIT) rating philosophies.

Through-the-cycle ratings aim to be stable over the business cycle, stressing borrowers to a downturn so ratings change little with conditions. Point-in-time ratings reflect the borrower's current condition and migrate with the cycle, producing more volatile but timelier default estimates.

What is an internal ratings-based (IRB) model, and what regulatory framework permits it?

An IRB model is a bank's internal system for assigning credit grades and estimating risk parameters (PD, and under Advanced IRB also LGD and EAD) for regulatory capital. Basel II/III's IRB approach permits qualifying banks to use these internal estimates.

Distinguish Foundation IRB from Advanced IRB.

Under Foundation IRB, banks estimate PD internally while LGD, EAD, and maturity are set by the regulator (supervisory values). Under Advanced IRB, banks estimate PD, LGD, EAD, and effective maturity using their own internal models, subject to supervisory approval.

What is a rating transition (migration) matrix?

A rating transition matrix gives the probabilities that an obligor in a given rating grade migrates to each other grade (or to default) over a fixed horizon, usually one year. Rows are starting ratings, columns are ending ratings/default, and each row sums to 100%.

In a one-year transition matrix, what does the last column typically represent and what property do rows satisfy?

The last column typically represents the probability of default (migration to the default state) over the year. Each row must sum to 1 (100%), since the obligor must end in exactly one of the possible states including default.

How are multi-year (cumulative) transition probabilities obtained from a one-year matrix, and under what assumption?

Under a time-homogeneous Markov assumption, the $n$-year transition matrix is the one-year matrix raised to the $n$th power: $M_n = M_1^{\,n}$. Cumulative default probabilities accumulate down the default column across horizons.

Name three standard techniques used to validate a credit rating / scoring model's discriminatory power.

The Cumulative Accuracy Profile (CAP) and its Accuracy Ratio (Gini), the Receiver Operating Characteristic (ROC) curve and its area (AUC), and the Kolmogorov-Smirnov (KS) statistic. Calibration is checked separately, e.g., with the Hosmer-Lemeshow or binomial/Brier tests.

What is the relationship between the Accuracy Ratio (AR/Gini) from a CAP curve and the AUC from an ROC curve?

They are linearly related: $AR = 2 \times AUC - 1$. An AR of 0 (AUC = 0.5) indicates no discriminatory power; an AR of 1 (AUC = 1) indicates perfect discrimination between defaulters and non-defaulters.

Distinguish discrimination from calibration in rating model validation.

Discrimination measures how well the model ranks/separates defaulters from non-defaulters (CAP/Gini, ROC/AUC, KS). Calibration measures whether predicted PDs match realized default rates in each grade (Hosmer-Lemeshow, binomial test). A model can rank well yet be poorly calibrated.

Define exposure at default (EAD) in the context of counterparty credit risk.

EAD is the expected amount owed to the surviving party at the time of the counterparty's default. For derivatives it is uncertain and market-driven, generally taken as the positive part (replacement cost) of the netted portfolio value at default.

Define current exposure (CE) for a derivatives portfolio.

Current exposure is the larger of zero and the current market (replacement) value of the contracts with a counterparty: $CE = \max(V_t, 0)$ on a netted basis. It is the loss today if the counterparty defaulted immediately.

Define potential future exposure (PFE) and how it differs from current exposure.

PFE is a high-percentile (e.g., 95% or 97.5%) estimate of exposure at a future date, capturing how large exposure could plausibly become due to market moves. Unlike current exposure (a present value), PFE is a forward-looking, confidence-level worst-case measure.

Define expected exposure (EE) at a future time $t$.

Expected exposure is the probability-weighted average of positive exposure at future time $t$: $EE(t) = \mathbb{E}[\max(V_t, 0)]$. It is the mean of the exposure distribution (ignoring negative values) at that horizon.

Define expected positive exposure (EPE) and how it differs from EE.

EPE is the time-average of expected exposure over the life of the portfolio: $$EPE = \frac{1}{T}\int_0^{T} EE(t)\, dt.$$ Whereas $EE(t)$ is the expected exposure at a single horizon, EPE averages those expected exposures across the whole horizon.

What is effective expected exposure (Effective EE) and effective EPE, and why are they used?

Effective EE is a non-decreasing version of EE: $EEE(t) = \max_{s \le t} EE(s)$, preventing exposure from dropping as positions roll off. Effective EPE is the time-average of Effective EE. Basel uses Effective EPE to avoid understating rollover/short-maturity exposure when computing regulatory EAD.

List the typical drivers that increase potential future exposure.

Higher market volatility of underlying risk factors, longer time to maturity (more time for drift/diffusion), absence of netting and collateral, and contract structures whose value can diffuse widely. Mitigants like netting agreements, collateral/margin, and break clauses reduce PFE.

How do netting agreements reduce counterparty exposure?

Under a legally enforceable netting agreement, positive and negative mark-to-market values across trades with the same counterparty offset, so exposure is $\max(\sum_i V_i, 0)$ rather than $\sum_i \max(V_i, 0)$. This reduces both current and potential future exposure.

Define Credit Valuation Adjustment (CVA).

CVA is the market value of expected loss due to counterparty default on a derivatives portfolio — the difference between the risk-free (default-free) value and the true value accounting for counterparty default risk. It is effectively the price of the counterparty's default risk.

Write the standard (unilateral) discretized CVA formula.

$$CVA \approx LGD \sum_{i=1}^{n} EE(t_i)\, \cdot DF(t_i)\, \cdot \big[ PD(t_{i-1}, t_i) \big]$$ i.e. $CVA = (1-R)\sum_i$ discounted expected exposure $\times$ marginal risk-neutral default probability in each interval, summed over the time grid.

What three ingredients does unilateral CVA combine?

(1) Loss given default $LGD = 1-R$ of the counterparty; (2) the discounted expected exposure profile $EE(t_i)$ of the portfolio; and (3) the counterparty's marginal (risk-neutral) default probabilities over each time interval.

Distinguish unilateral CVA from bilateral CVA.

Unilateral CVA accounts only for the counterparty's default (the dealer treats itself as default-free). Bilateral CVA also accounts for the possibility that the reporting institution itself defaults, netting the counterparty's CVA against the institution's own default benefit (DVA).

Define Debit Valuation Adjustment (DVA).

DVA is the adjustment reflecting the institution's own default risk — the expected gain to the institution from the possibility that it defaults and fails to pay its negative-value (out-of-the-money) positions. It is a positive contribution to the institution's reported derivative value (an own-credit benefit).

State the relationship between bilateral CVA (BCVA), unilateral CVA, and DVA.

$$BCVA = CVA - DVA$$ Bilateral CVA equals the counterparty-default CVA minus the institution's own DVA. From the institution's accounting value, $V_{\text{adjusted}} = V_{\text{risk-free}} - CVA + DVA$.

Why is DVA controversial / counterintuitive in risk management?

DVA implies an institution books a gain when its own credit quality deteriorates (its credit spread widens), improving reported earnings precisely when it is more likely to fail. The gain is also hard to monetize/hedge, since a firm cannot easily profit from its own default, prompting regulators (Basel III) to exclude DVA from regulatory capital.

What is wrong-way risk in counterparty credit risk, and how does it affect CVA?

Wrong-way risk occurs when exposure to a counterparty is positively correlated with the counterparty's probability of default — exposure rises just as the counterparty becomes more likely to default. It increases CVA (and EE), because the largest exposures coincide with the highest default likelihood; right-way risk (negative correlation) reduces CVA.