15.2 Measuring Credit Risk

Measuring Credit Risk

VRM–6 turns single-name credit ideas into portfolio loss measurement. The exam core is the distinction between expected and unexpected loss, the role of default dependence, workhorse models (binomial, Vasicek/Gaussian copula, CreditMetrics), capital attribution, and why derivatives defy simple loan-style capital formulas.

Economic Capital Versus Regulatory Capital

Economic capital (EC) is the firm’s own estimate of capital required to support credit (and other) risk at a chosen confidence level over a horizon—often one year at 99.9% for banking books, though firms vary. EC is used for risk-adjusted performance, limit setting, and internal capital adequacy assessment.

Regulatory capital is the amount supervisors require under Basel (or local) rules: standardized risk weights or internal-ratings-based (IRB) formulas with floors, buffers, and output floors. Regulatory capital can be higher or lower than EC for a given book; firms must hold at least the regulatory amount and typically manage to the more binding of EC, regulatory, and rating-agency constraints.

Default Dependence

If defaults were independent, diversification would drive portfolio credit risk toward a law-of-large-numbers mean. In practice, defaults cluster because of shared macro factors, industry shocks, and contagion. Default correlation (or factor loading in a structural/copula model) is therefore the dominant driver of unexpected loss at the portfolio level.

Conditional independence given a systematic factor is the usual modeling compromise: given the economy state, obligor defaults are independent draws; unconditionally they are positively dependent. That structure appears in Vasicek/ASRF-style models used in Basel IRB.

Expected Loss and Unexpected Loss

For a single exposure (or homogeneous pool approximation):

EL = PD × LGD × EAD

where PD is probability of default over the horizon, LGD is loss given default (fraction of exposure lost), and EAD is exposure at default.

Unexpected loss (UL) is a measure of loss dispersion—often the standard deviation of the loss distribution, or more generally the gap between a high quantile (VaR) and EL for capital:

Capital ≈ VaR_α − EL (or sometimes ES_α − EL)

at confidence α. Pricing and provisions target EL; capital and limits target UL / tail loss.

Worked EL example

A corporate facility has EAD = USD 20 million, PD = 1.5%, LGD = 40%.

EL = 0.015 × 0.40 × 20,000,000 = USD 120,000.

If one-year 99.9% credit VaR on this name (stand-alone, for illustration) is USD 2.0 million, a crude stand-alone capital proxy is 2,000,000 − 120,000 = USD 1.88 million. Portfolio diversification and dependence will change the incremental capital sharply.

Binomial Credit-Loss Mean and Standard Deviation

For n identical independent exposures each of size EAD, with default indicator Dᵢ ~ Bernoulli(PD) and fixed LGD, portfolio loss is

L = LGD × EAD × (D₁ + … + Dₙ)

If N = Σ Dᵢ ~ Binomial(n, PD) under independence,

E[L] = LGD × EAD × n × PD

SD(L) = LGD × EAD × sqrt(n × PD × (1 − PD))

Worked binomial example

Let n = 100, PD = 0.02, LGD = 0.45, EAD = USD 1 million each.

EL = 0.45 × 1,000,000 × 100 × 0.02 = USD 900,000

SD(L) = 0.45 × 1,000,000 × sqrt(100 × 0.02 × 0.98) = 450,000 × sqrt(1.96) = 450,000 × 1.4 = USD 630,000

Independence keeps UL (here SD) relatively moderate. Positive default correlation inflates the loss variance and the high quantiles—exam writers often ask you to reason directionally without a full correlated simulation.

Gaussian Copula and One-Factor Dependence

A Gaussian copula links uniform marginal default times (or default indicators) through a multivariate normal correlation structure. In credit, a common implementation is the one-factor Gaussian model:

Xᵢ = sqrt(ρ) × F + sqrt(1 − ρ) × εᵢ

where F and εᵢ are standard normals, ρ is asset (or latent-variable) correlation, and default occurs if Xᵢ < threshold calibrated to PD (threshold = Φ⁻¹(PD)).

Conditional on F = f, defaults are independent with conditional PD

PD(f) = Φ( (Φ⁻¹(PD) − sqrt(ρ)·f) / sqrt(1 − ρ) )

Large negative F (bad systematic state) raises conditional PDs together—producing the heavy right tail of portfolio losses.

Vasicek Portfolio Loss Distribution

The Vasicek (ASRF) limiting loss rate for an infinitely granular portfolio with uniform PD and correlation ρ has closed form. The α-quantile of the loss rate (before LGD) is

VaR_α(loss rate) = Φ( (Φ⁻¹(PD) + sqrt(ρ)·Φ⁻¹(α)) / sqrt(1 − ρ) )

and credit loss quantile ≈ LGD × that rate for unit exposure. Basel IRB risk-weight formulas are built on this idea with regulatory ρ(PD) functions and a 99.9% confidence level.

Worked Vasicek-style reading

Suppose PD = 1%, ρ = 0.20, α = 99.9%, LGD = 45%, EAD = USD 100 million. You would evaluate Φ⁻¹(0.01) ≈ −2.33, Φ⁻¹(0.999) ≈ 3.09, then

conditional quantile factor = (−2.33 + sqrt(0.20)·3.09) / sqrt(0.80)

sqrt(0.20) ≈ 0.447; sqrt(0.80) ≈ 0.894

numerator ≈ −2.33 + 0.447 × 3.09 ≈ −2.33 + 1.38 ≈ −0.95

ratio ≈ −0.95 / 0.894 ≈ −1.06 → Φ(−1.06) ≈ 0.1445

Loss quantile ≈ 0.45 × 0.1445 × 100m ≈ USD 6.5 million, versus EL = 0.01 × 0.45 × 100m = USD 0.45 million. Capital-style UL ≈ 6.5m − 0.45m ≈ USD 6.05 million in this stylized infinite-granularity case. Real portfolios need granularity adjustments and concentration add-ons.

CreditMetrics

CreditMetrics (J.P. Morgan’s classic framework) builds a portfolio loss distribution from rating migrations, not only default. Each obligor’s value at horizon depends on the rating transition (including default with recovery). Correlated rating moves are driven by correlated asset returns (again often Gaussian). Mark-to-market credit VaR captures spread/migration risk for bonds and loans that are marked or economically sensitive to ratings—broader than default-mode models that only care about default versus survival.

Euler Contributions and Capital Allocation

Portfolio capital K(L) (e.g., VaR or ES of portfolio loss) must be allocated to desks, names, or facilities for limits and pricing. Euler allocation uses the homogeneity of risk measures: for a positive-homogeneous measure ρ,

ρ(L) = Σᵢ ∂ρ/∂wᵢ · wᵢ

so risk contributions RCᵢ = wᵢ · ∂ρ/∂wᵢ sum to total capital. Intuition: the contribution is exposure times marginal impact of a small increase in that exposure. Concentrated names and highly systematic names receive large Euler shares even if stand-alone EL is modest.

Derivatives Versus Loans: Why Capital Is Harder

Loan EAD is often contractual (drawn + credit-conversion factors for undrawn). Derivatives EAD is uncertain: mark-to-market paths, netting sets, collateral, and early termination drive exposure profiles (EE, PFE). Wrong-way risk links rising exposure to rising counterparty PD (e.g., selling put protection to a hedger who defaults when markets crash). CVA capital, SA-CCR, and internal model methods exist precisely because loan-style PD×LGD×EAD with fixed EAD is insufficient.

Quantification Challenges

Credit-risk quantification struggles with: sparse default data (especially low-PD portfolios); cyclical bias in PD/LGD; uncertain dependence (correlation is not directly observed); recovery variation; name concentration; and model risk in copulas and migration matrices. Stress testing and expert overlays remain necessary complements to any point-estimate capital number.

Master the EL/UL split, the binomial independent benchmark, the one-factor Gaussian/Vasicek tail, CreditMetrics’ migration view, and Euler thinking—and you have the VRM–6 toolkit.