Credit Risk: formula sheet

Key formulas

Expected loss

EL = PD × LGD × EAD

Use the same horizon for PD and EAD. EL is in currency units. EL rate = PD × LGD.

Loss given default

LGD = 1 − Recovery rate

Recovery is expressed as a share of exposure. Check whether the question gives recovery or LGD.

EAD for a credit line

EAD = Drawn + CCF × Undrawn

CCF is the credit conversion factor, the expected share of the undrawn limit that is drawn by default.

Unexpected loss, single exposure (fixed LGD and EAD)

UL = EAD × LGD × √(PD × (1 − PD))

This is the standard deviation of loss. It assumes LGD is a known constant.

Unexpected loss with random LGD

σ² of loss = EAD² × [PD × σ_LGD² + LGD² × PD × (1 − PD)]

Assumes LGD is independent of default and EAD is fixed. UL is the square root.

Portfolio expected loss

EL_portfolio = Σ EL_i

EL is additive across exposures, whatever the correlation.

Portfolio UL, two exposures

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

ρ is the correlation of losses. UL is not additive unless ρ = 1.

Economic capital (credit VaR basis)

Economic capital = Credit VaR at confidence level − EL

Credit VaR is the quantile loss. Capital covers only the unexpected part.

Row sum of a transition matrix

Σj P(i → j) = 100% for each starting grade i

Includes the default column. Use it to find a missing entry.

Two-year transition probability

P(i → k, 2 years) = Σj P(i → j) × P(j → k)

This is the matrix product of the one-year matrix with itself. Assumes a constant Markov matrix.

Cumulative PD from the default column

PD(i, 2 years) = Σj P(i → j) × P(j → D), with P(D → D) = 100%

Default is absorbing. Include the direct one-year default and the paths through other grades.

Marginal (unconditional) default probability

Marginal PD in year 2 = Cumulative PD(2) − Cumulative PD(1)

This is the unconditional probability of defaulting in year 2 specifically. A conditional PD would divide it by the probability of surviving year 1, which is 1 − Cumulative PD(1).

Logistic scorecard PD

PD = 1 ÷ (1 + e^−(b0 + b1x1 + … + bnxn))

Maps a linear score to a probability between 0 and 1.

Equity as a call option

E = V·N(d1) − D·e^(−rT)·N(d2)

V is asset value, D is face value of debt, r is the risk-free rate, T is maturity.

d1 and d2

d1 = [ln(V ÷ D) + (r + σV²÷2)·T] ÷ (σV·√T); d2 = d1 − σV·√T

Use σV, the asset volatility, not equity volatility.

Risk-neutral default probability

P(V_T < D) = N(−d2)

For the real-world probability, replace r with μ in d2 (the numerator becomes ln(V ÷ D) + (μ − σV²÷2)·T).

Risky debt value

B = D·e^(−rT) − Put(V, D)

Equals V − E. Put value is the credit risk cost.

Equity and asset volatility link

σE·E = N(d1)·σV·V

Used with the equity formula to solve for V and σV.

Distance to default (simple)

DD = (V − default point) ÷ (V·σV), for a one-year horizon

Number of standard deviations of assets above the default point, using current asset value V over a one-year horizon.

Constant hazard rate survival

Q(t) = e^(−λt); PD(0,t) = 1 − e^(−λt)

λ is the intensity of default.

Credit spread approximation

λ ≈ s ÷ (1 − R)

s is the credit spread, R the recovery rate. An approximation.

Credit exposure of one trade

Exposure = max(V, 0)

V is the mark-to-market value to you. Negative value means zero exposure.

Expected exposure and EPE

EE(t) = E[max(V(t), 0)]; EPE = average of EE(t) over the period

EPE is a time average of EE. It is not a percentile.

Potential future exposure

PFE(t) = the q-th percentile of max(V(t), 0), e.g. q = 95% or 99%

Used for limits. It is always at or above EE at the same date.

Netted exposure

Netted = max(V1 + V2 + ... + Vn, 0) ≤ Σ max(Vi, 0)

Holds only if netting is legally enforceable in the relevant jurisdiction.

Net-to-gross ratio

NGR = Netted exposure ÷ Gross exposure

A lower ratio means a bigger netting benefit.

Unilateral CVA (discrete)

CVA ≈ LGD × Σ [discounted EE(ti) × marginal PD(ti-1, ti)]

Assumes exposure and default are independent. LGD = 1 − recovery rate.

Bilateral adjustment

Bilateral CVA = CVA − DVA

DVA gains when your own credit worsens, which is controversial.

Collateralised exposure

Exposure ≈ max(V(t) − C, 0), where C is collateral held at the time of default

The key driver is the margin period of risk, which sets how far V can move before close-out.

Expected loss

EL = PD × LGD × EAD

Per exposure. Portfolio EL is the sum of individual ELs, with no correlation effect.

Credit VaR (unexpected loss at confidence level)

Credit VaR = Loss quantile at α − Expected loss

Some texts quote the quantile itself as the VaR. Read the question for which definition applies.

Two-asset default correlation from joint default

ρD = (p12 − p1·p2) ÷ √[p1(1 − p1) × p2(1 − p2)]

p12 is the joint default probability. Default correlations are usually small, even when asset correlations are high.

Vasicek worst-case default rate

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

N is the standard normal CDF, ρ the asset correlation, α the confidence level. Conditional on the systematic factor at its (1 − α) tail. Large homogeneous portfolio assumed.

Vasicek loss at confidence level

Loss = EAD × LGD × WCDR

Subtract EL = EAD × LGD × PD to get unexpected loss. Basel IRB capital is of this form, with further adjustments.

Poisson default probability (CreditRisk+)

P(n defaults) = e^(−μ) × μⁿ ÷ n!

μ is the expected number of defaults. Mean and variance both equal μ.

Portfolio loss variance (two exposures)

σ² = σ1² + σ2² + 2ρ·σ1·σ2

Loss standard deviation rises with correlation. Used for unexpected loss.

CDS payout on credit event

Payout = Notional × (1 − Recovery rate)

Also equals notional × (1 − market price as a fraction of par) for cash settlement.

Credit triangle

Spread ≈ λ × (1 − R), so λ ≈ Spread ÷ (1 − R)

λ is the annual hazard rate. It is an approximation that assumes a flat hazard rate and spread. Spread is in decimals (120 bp = 0.012).

CDS valuation (par spread)

s × Risky annuity = (1 − R) × PV of default-weighted discount factors

Premium leg PV equals protection leg PV at the par spread.

Value of an existing CDS (approximate)

Value to buyer ≈ (Current spread − Contract spread) × Notional × Risky annuity

Positive for the protection buyer if spreads have widened.

Cumulative default probability (constant hazard)

P(default by T) = 1 − e^(−λT)

Survival probability is e^(−λT).

Tranche loss

Tranche loss (as % of tranche) = min[max(L − A, 0), D − A] ÷ (D − A)

L = pool loss %, A = attachment point, D = detachment point.

Tranche size

Tranche size = (D − A) × Pool notional

Dollar tranche loss = tranche loss % × tranche size.

Conservation of expected loss

Pool expected loss = Σ (tranche expected loss in pool terms)

Correlation changes the split between tranches, not the total.

Correlation effect on tranches

Higher correlation → equity tranche risk falls; senior tranche risk rises

Mezzanine behaviour is mixed and depends on the structure.

Risk-weighted assets

RWA = Exposure × Risk weight

Exposure is EAD after any permitted credit risk mitigation. Sum across all exposures.

Minimum capital

Capital = RWA × capital ratio

Basel minimum total capital is 8% of RWA, before buffers. Common Equity Tier 1 minimum is 4.5%.

Expected loss

EL = PD × LGD × EAD

Covered by provisions, not by IRB capital.

IRB conditional PD (Vasicek)

WCDR = N[ (N⁻¹(PD) + √ρ × N⁻¹(0.999)) ÷ √(1 − ρ) ]

N is the standard normal CDF, ρ the asset correlation. 0.999 is the confidence level.

IRB capital requirement (before maturity adjustment)

K = LGD × (WCDR − PD)

Unexpected loss per unit of EAD. Wholesale exposures also get a maturity adjustment.

IRB RWA

RWA = K × 12.5 × EAD

12.5 is 1 ÷ 8%. Older Basel text also applied a scaling factor of 1.06.

Capital to RWA conversion

RWA = Capital requirement × 12.5

Use when a problem gives a capital charge and asks for RWA.

Quick revision

• Expected loss = PD × LGD × EAD.
• LGD = 1 − recovery rate, when recovery is expressed as a fraction of exposure.
• Expected loss is a cost of doing business; unexpected loss is the volatility of losses around it and is what capital covers.
• Structural models treat equity as a call option on firm assets and default as assets falling below debt.
• Reduced-form models treat default as a random event with a hazard rate, often linked to credit spreads.
• Counterparty exposure is positive-only: you lose only if the counterparty owes you net value at default.
• Netting and collateral reduce exposure; netting works within legally enforceable agreements.
• CVA is the market value of counterparty credit risk on derivatives.
• Higher default correlation fattens the loss tail in a portfolio without changing expected loss.
• Credit VaR is a loss quantile at a confidence level over a horizon, measured relative to expected loss or to zero depending on the definition used.
• Securitization tranches absorb losses in order; equity takes first loss and senior tranches last.
• Basel capital is based on risk-weighted assets, with standardised and internal ratings-based approaches.

Common mistakes

• Using LGD as the recovery rate, or the reverse. Fix: Always write LGD = 1 − recovery before you multiply.
• Forgetting the undrawn commitment when finding EAD. Fix: For lines of credit use Drawn + CCF × Undrawn. If no CCF is given, check whether the question states a draw assumption.
• Squaring each entry of the matrix instead of doing a matrix product. Fix: Multiply row by column and sum over intermediate grades.
• Forgetting that default is absorbing and omitting P(D → D) = 100%. Fix: Always add the direct one-year default path with weight 1 for a two-year PD.
• Using equity volatility in d1 and d2. Fix: Use asset volatility σV. Equity volatility is higher for a levered firm: σE = N(d1)·σV·V ÷ E.
• Treating N(d2) as the default probability. Fix: N(d2) is the probability the firm repays (under the relevant measure). Default is N(−d2) = 1 − N(d2).
• Averaging negative values when computing EE. Fix: Apply max(V, 0) in each scenario first, then average.
• Calling EPE a percentile or PFE an average. Fix: EPE is a time-average of EE. PFE is a high percentile at a date. Use EPE for pricing and capital, PFE for limits.
• Treating Credit VaR as the full loss quantile when the question asks for unexpected loss. Fix: Check the wording. If the question says relative to expected loss, subtract EL.
• Assuming credit losses are normally distributed. Fix: Remember the loss distribution is skewed with a fat right tail. Normal approximations understate the tail.

Exam tips

• Read whether the question gives recovery or LGD, and whether PD is annual or cumulative, before touching a calculator.
• Expect conceptual items too: who covers EL (pricing, provisions) and who covers UL (capital).
• When a portfolio question mentions correlation, the answer is a UL below the sum of individual ULs. Eliminate options that simply add them.
• Know that Basel IRB uses PD, LGD, EAD and maturity as inputs to risk weights, and that maturity adjustments raise capital for longer exposures.
• If an option shows EL larger than EAD × LGD, discard it at once.
• Write out the paths explicitly. Most errors come from skipping the default-to-default path.
• Read whether the question wants cumulative or marginal PD before calculating.
• For TTC versus PIT, expect conceptual MCQs on procyclicality, rating stability and Basel capital.