Portfolio Credit Risk: formula sheet

Key formulas

Single-name expected loss

EL = PD × LGD × EAD

Uses expected values. Assumes PD, LGD and EAD are treated as independent unless stated.

Portfolio expected loss

EL_P = Σ EL_i

Additive. Does not depend on default correlation.

Single-name unexpected loss (LGD and EAD fixed)

UL = EAD × LGD × √(PD × (1 − PD))

Bernoulli default standard deviation. With stochastic LGD, add the LGD variance term.

Two-name portfolio UL

UL_P = √(UL_1² + UL_2² + 2 × ρ × UL_1 × UL_2)

ρ is the correlation of default events (losses). UL_P is at most UL_1 + UL_2, with equality when ρ = 1.

Default correlation

ρ = [P(A ∩ B) − PD_A × PD_B] ÷ √(PD_A(1 − PD_A) × PD_B(1 − PD_B))

P(A ∩ B) is the joint default probability. Default correlation is typically small, often a few percent, even when asset correlation is higher.

Economic capital

Economic capital = Credit VaR (at confidence level) − EL

Capital covers unexpected loss only. EL is covered by provisions and pricing.

Independent-default limit

UL_P = √(Σ UL_i²) when ρ = 0

Diversification benefit is largest here.

Expected loss (single exposure)

EL = PD × LGD × EAD

PD, LGD and EAD must use the same horizon. LGD = 1 − recovery rate.

Expected loss (portfolio)

EL(portfolio) = Σ EL(i)

Expected losses always add, whatever the correlation.

Unexpected loss (single exposure, fixed LGD and EAD)

UL = EAD × LGD × √(PD × (1 − PD))

Standard deviation of the default loss. Assumes LGD is known and constant. If LGD is random, add its variance.

Unexpected loss with random LGD

UL = EAD × √(PD × σ(LGD)² + LGD² × PD × (1 − PD))

σ(LGD) is the standard deviation of LGD, assumed independent of default.

Portfolio UL (two exposures)

UL(p) = √(UL₁² + UL₂² + 2 × ρ × UL₁ × UL₂)

ρ is the default correlation. UL(p) ≤ UL₁ + UL₂, with equality only when ρ = 1.

Economic capital

EC = Credit VaR(α) − EL

Credit VaR here is the loss quantile at confidence α. Some texts call this the unexpected loss at that confidence.

Risk contribution

RC(i) = UL(i) × ∂UL(p)/∂UL(i), and Σ RC(i) = UL(p)

For the two-exposure case RC₁ = UL₁ × (UL₁ + ρ × UL₂) ÷ UL(p). Contributions add to the portfolio UL.

Default correlation

ρ_D = [P(A∩B) − PD_A × PD_B] ÷ √[PD_A(1 − PD_A) × PD_B(1 − PD_B)]

P(A∩B) is the joint default probability over the same horizon. Result lies between −1 and 1 but is capped by the PDs.

Joint default probability

P(A∩B) = PD_A × PD_B + ρ_D × √[PD_A(1 − PD_A) × PD_B(1 − PD_B)]

With ρ_D = 0 it reduces to independence: PD_A × PD_B.

Gaussian copula joint default

P(A∩B) = M₂(N⁻¹(PD_A), N⁻¹(PD_B); ρ)

M₂ is the bivariate standard normal CDF; ρ is the asset (latent) correlation, not default correlation.

Default threshold

Default if Z ≤ N⁻¹(PD)

Z is the standard normal latent variable for the name.

Conditional PD, one-factor Gaussian copula

PD(M) = N[(N⁻¹(PD) − √ρ × M) ÷ √(1 − ρ)]

M is the common factor, standard normal. A low M (bad economy) raises conditional PD.

Latent variable, one-factor model

Z_i = √ρ × M + √(1 − ρ) × ε_i

M and ε_i are independent standard normals. Pairwise correlation between Z_i and Z_j is ρ.

Bounds on default correlation

Maximum occurs when P(A∩B) = min(PD_A, PD_B)

Equal PDs allow a maximum of 1; unequal PDs cap it below 1.

Equity as a call option

E = V × N(d1) − D × e^(−rT) × N(d2)

Equity is a call on assets V with strike D (face value of zero-coupon debt).

d1 and d2

d1 = [ln(V ÷ D) + (r + σ_V²÷2)T] ÷ (σ_V√T); d2 = d1 − σ_V√T

Use r for risk-neutral values. Use μ in place of r for real-world probability.

Risk-neutral default probability

PD = N(−d2)

Probability that V < D at time T under the risk-neutral measure.

Distance to default

DD = [ln(V ÷ D) + (μ − σ_V²÷2)T] ÷ (σ_V√T)

Number of standard deviations from expected log asset value to the default point.

Real-world default probability

PD = N(−DD)

Valid only if asset returns are normal in logs. KMV replaces this with an empirical EDF.

Equity and asset volatility link

σ_E × E = N(d1) × σ_V × V

Used with the equity formula to solve for unobserved V and σ_V.

Value of risky debt

B = V − E = D × e^(−rT) − Put(V, D)

Lenders are long a risk-free bond and short a put on the assets.

KMV default point

Default point = short-term debt + 0.5 × long-term debt

Used instead of total debt in KMV's DD.

Simplified KMV distance to default

DD = (V − default point) ÷ (V × σ_V)

Approximation for a one-year horizon. It ignores drift and replaces ln(V ÷ default point) with (V − default point) ÷ V, so it is the gap as a share of assets, over asset volatility.

Asset return in the one-factor model

Aᵢ = √ρ · M + √(1 − ρ) · Zᵢ

M and Zᵢ are independent standard normal. Default if Aᵢ < N⁻¹(PD).

Default threshold

K = N⁻¹(PD)

N is the standard normal CDF. N⁻¹ is its inverse.

Conditional default probability

PD(M) = N[(N⁻¹(PD) − √ρ · M) ÷ √(1 − ρ)]

Given the market factor M. Lower M means higher default rate.

Worst-case default rate (WCDR)

WCDR(X) = N[(N⁻¹(PD) + √ρ · N⁻¹(X)) ÷ √(1 − ρ)]

Default rate not exceeded with confidence X. Basel uses X = 99.9%, so N⁻¹(X) ≈ 3.09.

Credit VaR and unexpected loss

Credit VaR = WCDR × LGD × EAD; UL = (WCDR − PD) × LGD × EAD

Expected loss is PD × LGD × EAD. Basel IRB capital is based on UL.

Useful normal values

N⁻¹(95%) ≈ 1.645; N⁻¹(99%) ≈ 2.326; N⁻¹(99.9%) ≈ 3.090

Quote these when the question gives no tables.

Expected loss

EL = PD × LGD × EAD

Common to all models. The portfolio EL is the sum of individual ELs.

Unexpected loss and credit VaR

Credit VaR = loss at chosen percentile − expected loss

Check whether the question defines VaR relative to expected loss or as the total loss quantile.

Poisson probability of n defaults

P(n) = e^(−μ) × μ^n ÷ n!

CreditRisk+ building block. μ is the expected number of defaults. Mean = variance = μ.

Merton distance to default (simplified)

DD = (asset value − default point) ÷ (asset value × asset volatility)

KMV maps DD to EDF empirically, not through the normal distribution.

CreditPortfolioView default rate

p = 1 ÷ (1 + e^(y)), where y is the macro health index

Here y is a health index built from macro variables. A lower y (a weak economy) gives a higher default rate p, and a higher y (a strong economy) gives a lower p. The logit keeps p between 0 and 1.

CDS payoff on a credit event

Payoff = Notional × (1 − Recovery rate)

Paid by the protection seller to the buyer. Assumes cash or physical settlement at the recovery value.

Annual CDS premium

Premium = CDS spread × Notional

Paid by the protection buyer until a credit event or maturity. Spread is quoted in basis points per year; 100 bp = 1%.

Expected loss

EL = PD × LGD × EAD

Use it to measure the benefit of a hedge or sale on the hedged amount.

Net exposure after a hedge

Net exposure = Exposure − Hedged notional

Valid when the hedge matches the exposure. Mismatches leave basis and maturity risk.

Concentration share

Concentration % = Exposure to a name or sector ÷ Total portfolio (or capital)

Compare to the limit. A breach needs reduction, sale or hedging.

Net carry on a hedged loan

Net spread = Loan spread − CDS spread

Ignores funding cost and capital. A positive value means the hedge costs less than the loan earns.

Quick revision

• Expected loss = PD × LGD × EAD; it is a cost of business, not a risk to be capitalised.
• Unexpected loss is the volatility of credit loss; economic capital covers tail loss beyond expected loss at a chosen confidence level.
• Higher default correlation fattens the tail of the loss distribution, even if expected loss stays the same.
• Diversification lowers risk only to the extent that correlations are below one; credit loss distributions stay skewed with a fat right tail.
• In Merton, equity is a call option on firm assets and debt is risk-free debt minus a put option.
• Distance to default measures how many standard deviations assets sit above the default point; a larger value means lower default risk.
• KMV maps distance to default to an expected default frequency using empirical default data.
• Vasicek: conditional default probability rises when the common factor is bad and when asset correlation is higher.
• The large homogeneous portfolio assumption removes idiosyncratic risk, leaving only systematic risk.
• A Gaussian copula links marginal default probabilities through correlation but has weak tail dependence.
• Credit VaR is a loss quantile; subtract expected loss from it to get the unexpected loss capital.
• Mitigants such as collateral, netting, credit default swaps and limits reduce loss but can add basis, counterparty or wrong-way risk.

Common mistakes

• Adding unexpected losses across names Fix: Only EL adds. UL needs the correlation formula, and the sum is an upper bound reached only when ρ = 1.
• Saying correlation changes portfolio expected loss Fix: Correlation changes the spread and tail of the loss distribution, not its mean.
• Treating economic capital as the full credit VaR. Fix: Use EC = Credit VaR − EL unless the question states a different definition.
• Adding individual ULs to get portfolio UL. Fix: Use the correlation formula. Simple addition holds only when ρ = 1.
• Using asset correlation in the default correlation formula. Fix: Check the wording. Asset or latent correlation goes into the copula. Default correlation goes into the covariance-style formula with PDs.
• Assuming default correlation equals asset correlation. Fix: Remember that default correlation is much smaller for low PDs. Asset correlation of 30% might give a default correlation of only a few percent.
• Using r when the question asks for real-world default probability. Fix: Real-world PD and DD use the expected asset return μ. N(−d2) with r is the risk-neutral PD.
• Forgetting the −σ²÷2 term in the drift. Fix: Write the full numerator every time: ln(V ÷ D) + (μ − σ_V²÷2)T.
• Subtracting √ρ · N⁻¹(X) in the WCDR formula. Fix: The worst market factor is M = −N⁻¹(X), so the sign becomes plus. WCDR must be above PD.
• Using ρ instead of √ρ. Fix: Always take √ρ and √(1 − ρ) before substituting.

Exam tips

• Expect a question testing that EL is additive and UL is not. Read the options for this trap first.
• When a vignette mentions one sector or region dominating the book, think concentration and systematic risk, not name count.
• Check whether the question gives default correlation or asset correlation, and use it accordingly.
• For capital questions, subtract EL from the stated percentile loss.
• Sanity check numerical UL: it must sit between the independent case and the fully correlated sum.
• Read whether the question gives LGD or recovery rate. This is the most common trap.
• Check whether the question wants credit VaR or economic capital. They differ by EL.
• When correlation is below 1, expect the portfolio UL to be lower than the sum of stand-alone ULs. Use this to eliminate options.