Measuring Credit Risk: formula sheet

Key formulas

Loss given default

LGD = 1 − Recovery rate

Recovery rate is the fraction of exposure recovered after default.

Expected loss

EL = PD × LGD × EAD

Use the same horizon for PD as for the loss you want. Expected losses add across loans.

Unexpected loss, single loan (fixed LGD)

UL = EAD × LGD × √(PD × (1 − PD))

Standard deviation of loss when LGD is known with certainty.

Unexpected loss, single loan (random LGD)

UL = EAD × √(PD × σLGD² + LGD² × PD × (1 − PD))

σLGD is the standard deviation of LGD, assumed independent of default and EAD fixed.

Portfolio unexpected loss, two loans

ULp = √(UL1² + UL2² + 2ρ × UL1 × UL2)

ρ is the correlation between the two loss outcomes. Perfect correlation gives UL1 + UL2.

Exposure on a credit line

EAD = Drawn + CCF × Undrawn

CCF is the credit conversion factor, the expected fraction of the undrawn amount drawn by default.

LGD and recovery rate

LGD = 1 − RR

Both are measured as a percentage of exposure at default.

Expected loss

EL = PD × LGD × EAD

PD is the probability of default over the horizon. EAD is the exposure at default.

Loss in currency

Loss = EAD × (1 − RR)

Applies once default has occurred.

Netted exposure

Net exposure = max(Σ contract values, 0)

Only valid with legally enforceable netting. Without it, exposure = Σ max(value, 0).

Collateral-adjusted exposure

Exposure after collateral = max(EAD − collateral × (1 − haircut), 0)

The haircut reduces the collateral value you can count on.

Guarantee (substitution) approach

P(loss) ≈ P(borrower default and guarantor default)

For independent parties this is PD(borrower) × PD(guarantor). Positive correlation raises it.

Seniority order

Secured > Senior unsecured > Subordinated > Equity

Higher priority usually means higher recovery and lower LGD.

Row sum property

Σj P(i → j) = 1 for every starting rating i

Each row of the matrix covers all outcomes, including default, so it must total 100%.

Absorbing default state

P(D → D) = 1, P(D → any other) = 0

Used in standard exam matrices. Default is never exited.

Two-year transition matrix

P(2) = P × P

Valid under the Markov and time-homogeneity assumptions. Use matrix multiplication, not squaring each entry.

n-year matrix

P(n) = Pⁿ

The default column of Pⁿ gives the cumulative n-year default probability for each starting rating.

Two-year default probability (one starting rating)

PD(2) = Σk P(i → k) × P(k → D)

Sum over every rating k reached after year 1, including D itself where P(D → D) = 1.

Marginal (unconditional) default probability

Marginal PD in year n = Cumulative PD(n) − Cumulative PD(n−1)

This is the unconditional probability of defaulting in year n. It is not conditional on surviving to year n. The conditional PD in year n = [Cumulative PD(n) − Cumulative PD(n−1)] ÷ [1 − Cumulative PD(n−1)].

Survival probability

S(n) = 1 − Cumulative PD(n)

Probability of not having defaulted by the end of year n.

Equity as a call option

E = V·N(d1) − D·e^(−rT)·N(d2)

V = asset value, D = face value of zero-coupon debt, r = risk-free rate (continuous), T = maturity.

d1 and d2

d1 = [ln(V ÷ D) + (r + σ²/2)·T] ÷ (σ·√T); d2 = d1 − σ·√T

σ is asset volatility, not equity volatility.

Risk-neutral default probability

P(default) = N(−d2)

Probability that V_T < D under risk-neutral measure.

Distance to default (real-world)

DD = [ln(V ÷ D) + (μ − σ²/2)·T] ÷ (σ·√T); P(default) = N(−DD)

Uses the expected asset return μ instead of r.

Value of risky debt

B = V − E = D·e^(−rT)·N(d2) + V·N(−d1) = D·e^(−rT) − Put

Debt is risk-free debt less a put on the assets.

Yield and credit spread

y = −(1 ÷ T)·ln(B ÷ D); spread = y − r

Continuous compounding. Spread rises with leverage and asset volatility.

Equity volatility link

σ_E·E = N(d1)·σ·V

Used to solve for unobservable V and σ from equity data.

KMV simple distance to default

DD = (V − default point) ÷ (σ·V); default point = short-term debt + 0.5 × long-term debt

EDF is read from an empirical mapping of DD to default rates.

Survival probability (constant hazard)

Q(t) = e^(−λt)

Probability of no default up to time t. λ is the hazard rate per year.

Cumulative default probability

PD(t) = 1 − e^(−λt)

Close to λt when λt is small.

Credit spread approximation

s ≈ λ × (1 − R) = λ × LGD

s is the spread over the risk-free rate. R is the recovery rate. Approximation, best for short maturities and small spreads.

Hazard rate from spread

λ ≈ s ÷ (1 − R)

Gives the risk-neutral hazard rate when the spread is used.

Average hazard rate from cumulative PD

λ̄ = −ln(1 − PD(t)) ÷ t

Use when you are given cumulative default probability.

Unconditional default probability in year t

Q(t−1) − Q(t) = e^(−λ(t−1)) − e^(−λt)

Default occurs in year t only if the firm survived to its start.

Risky bond price (zero-coupon)

P = e^(−(r + s)T) where s = λ × LGD

Uses continuous compounding. r is the risk-free rate.

Expected loss (portfolio)

EL = Σ (PDᵢ × LGDᵢ × EADᵢ)

EL is additive across exposures. It does not depend on correlation.

Default correlation (two names)

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

This is the correlation of the two 0/1 default indicators. It is usually much smaller than asset correlation.

Joint default probability

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

The same formula rearranged. Independence means ρ_D = 0.

Vasicek asset return

Aᵢ = √ρ × M + √(1−ρ) × Zᵢ

M is the common factor and Zᵢ is the idiosyncratic factor. Both are standard normal and independent. Default occurs if Aᵢ < N⁻¹(PD).

Conditional PD given M

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

A low (bad) M raises the conditional default rate.

Worst-case default rate

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

X is the confidence level, e.g. 0.999. N⁻¹(0.999) ≈ 3.09 and N⁻¹(0.99) ≈ 2.33. Applies to a large, granular portfolio with one common factor.

Credit VaR and economic capital

Credit VaR = WCDR × LGD × EAD; Economic capital = Credit VaR − EL

Assumes constant LGD and a large homogeneous portfolio.

Basel IRB capital charge (no maturity adjustment)

K = LGD × [WCDR(99.9%) − PD]

Capital per unit of EAD. Multiply by EAD for the amount.

Variance of default count (n equal names)

Var = n × PD × (1−PD) × [1 + (n−1) × ρ_D]

Assumes equal PD and equal pairwise default correlation ρ_D. It shows why correlation raises risk.

Quick revision

• Expected loss = PD × LGD × EAD.
• LGD = 1 − recovery rate, with both stated on the same exposure basis.
• Unexpected loss is the variability of loss around its expected value, usually measured as a standard deviation.
• Credit VaR is a loss quantile minus expected loss when you measure unexpected loss, so read the question's definition.
• Netting, collateral and guarantees reduce exposure or loss given default.
• In a transition matrix, each row sums to 1 and default is absorbing.
• Multi-year migration probabilities come from multiplying the matrix, not from adding annual numbers.
• Merton model: equity is a call option on firm assets, and debt equals a risk-free bond minus a put.
• In the Merton model, N(−d2) is the risk-neutral probability that assets end below the debt face value.
• Higher asset volatility or higher leverage raises Merton default probability.
• Reduced-form models treat default as a random event driven by a hazard rate.
• Spread ≈ hazard rate × (1 − recovery rate) as a rule of thumb.
• Higher default correlation leaves expected loss unchanged but fattens the tail of the loss distribution.

Common mistakes

• Using LGD as the recovery rate. Fix: Always convert: LGD = 1 − recovery. Underline which one the question gives.
• Calling unexpected loss the loss above the expected loss in a bad scenario. Fix: In this topic UL is the standard deviation of loss. Credit VaR minus EL is a separate quantity that is sometimes used as economic capital.
• Treating recovery rate and LGD as the same number. Fix: They are complements: LGD = 1 − RR. Check which one the question asks for.
• Ignoring haircuts on collateral. Fix: Multiply collateral by (1 − haircut) before subtracting it from exposure.
• Squaring each matrix entry to get the two-year matrix. Fix: Use matrix multiplication: row times column, summing over intermediate ratings.
• Forgetting the default-to-default term when computing two-year PD. Fix: Add P(i → D) × 1. Cumulative PD includes year-1 defaulters.
• Using equity volatility as σ in d1 and d2 Fix: The formulas need asset volatility. Equity volatility is larger because of leverage. Convert using σ_E·E = N(d1)·σ·V.
• Using N(d2) as the default probability Fix: N(d2) is the probability of repayment. Default probability is N(−d2) = 1 − N(d2).
• Treating spread-implied PD as the real-world probability of default Fix: Remember spreads include risk premia and liquidity effects, so risk-neutral PD is generally higher than historical PD.
• Using λ = spread without dividing by LGD Fix: Always use λ = s ÷ (1 − R). With a 40% recovery, λ is spread ÷ 0.6.

Exam tips

• Read whether the question gives recovery rate or LGD before doing anything else.
• Expect at least one question that tests whether you know EL is covered by provisions and pricing, while UL is covered by capital.
• For two-loan portfolios, check whether the correlation is given. If ρ = 1, simply add the ULs; if ρ = 0, use the square root of the sum of squares.
• Eliminate options by size: EL is PD × loss given default, so it must be far smaller than EAD for low PD.
• Read whether the recovery rate is given on the whole claim or only on the unsecured portion.
• Always floor exposure at zero after netting and collateral.
• Know the direction of effects: seniority and collateral reduce LGD, correlation between guarantor and borrower raises joint default risk.
• Expect conceptual items on the link between default rates and recovery rates in downturns.