Credit Value at Risk: formula sheet

Key formulas

Expected loss (single exposure)

EL = PD × LGD × EAD

PD is probability of default over the horizon. LGD is loss given default as a fraction of EAD. EAD is exposure at default.

Expected loss (portfolio)

EL(portfolio) = Σ EL(i)

Expected losses add up exactly, with no diversification effect and no correlation needed.

Credit VaR

Credit VaR(α) = the α-percentile of the loss distribution

Measured as total loss over the horizon, for example 99.9% over one year.

Unexpected loss (VaR-based)

UL = Credit VaR(α) − EL

The tail loss beyond the mean. Economic capital is usually set equal to this.

Economic capital

Economic capital = Credit VaR(α) − EL

Check the question. If it says capital covers the whole VaR, use the VaR. Standard practice is VaR minus EL.

Unexpected loss (standard deviation, single exposure, fixed EAD and LGD)

UL = EAD × LGD × √(PD × (1 − PD))

Applies when LGD is a known constant. This is the standard deviation of loss, a different definition of UL.

Portfolio UL is not additive

UL(portfolio) ≤ Σ UL(i)

Holds when default correlations are below 1. Diversification reduces portfolio UL.

Expected loss

EL = PD × LGD × EAD

Use the same horizon for PD, usually one year. Assumes PD, LGD and EAD are independent for the point estimate.

LGD and recovery rate

LGD = 1 − Recovery rate

Recovery is a fraction of exposure, ideally net of costs and discounted to default.

Exposure at default with undrawn limit

EAD = Drawn + CCF × Undrawn

CCF is between 0% and 100%. Undrawn = Limit − Drawn.

Expected loss rate

EL rate = EL ÷ EAD = PD × LGD

Useful for comparing loans of different size.

Portfolio expected loss

EL(portfolio) = Σ PDᵢ × LGDᵢ × EADᵢ

Expected losses add up across exposures, with no diversification effect.

Multi-year PD

Cumulative PD over n years = 1 − (1 − PD)ⁿ

Valid only if the annual PD is constant and defaults are independent across years.

Portfolio expected loss

EL_P = Σ EADi × PDi × LGDi

Additive. Correlation has no effect on it.

Default correlation (two obligors)

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

p12 is the joint default probability. Independent defaults give p12 = p1 × p2 and ρD = 0.

Joint default probability

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

Rearranged from the correlation formula. Positive correlation raises joint default above p1 × p2.

Loss variance of a two-name portfolio (fixed LGD)

σ² = Σ Σ Li × Lj × Cov(Di, Dj), where Li = EADi × LGDi

Cov(Di, Di) = pi(1 − pi). Cov(Di, Dj) = ρD × σDi × σDj, with σDi = √[pi(1 − pi)].

Homogeneous portfolio loss volatility

σ_P = L × √[p(1 − p)] × √[N × (1 + (N − 1)ρD)]

Total loss volatility for N equal exposures L, same p, same pairwise ρD. Dividing by N gives the average per-loan volatility, L × √[p(1 − p)] × √[(1 + (N − 1)ρD) ÷ N]. As N rises, the per-loan figure approaches L × √[p(1 − p)] × √ρD.

Herfindahl index

H = Σ wi²

Equal weights give H = 1/N. Higher H means more concentration.

Risk contribution

RCi = wi × ∂σ_P/∂wi = wi × Cov(Ri, R_P) ÷ σ_P

Contributions sum to total portfolio risk σ_P (Euler allocation).

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, T = maturity.

d1 and d2

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

Uses asset volatility σV, not equity volatility.

Risk-neutral default probability

PD = N(−d2)

Probability that V_T < D under the risk-neutral measure.

Link between equity and asset volatility

σE·E = N(d1)·σV·V

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

Value of risky debt

B = V − E = D·e^(−rT) − Put(V, D)

Debt equals risk-free bond minus a put on assets.

Distance to default (real-world)

DD = [ln(V ÷ D) + (μ − σV²÷2)·T] ÷ (σV·√T)

μ = expected asset return. Real-world PD = N(−DD). KMV maps DD to an empirical EDF instead.

Credit spread in Merton

s = −(1 ÷ T)·ln(B ÷ D) − r

B is the market value of risky debt, so the spread is the yield on B minus r.

Expected bond value

E(V) = Σ pᵢ × Vᵢ

pᵢ is the transition probability to state i, including default. Vᵢ is the revalued bond in that state.

Value in default

V_default = Recovery rate × Face value

Recovery is often stated as a percentage of face value. Check whether the question includes accrued coupon.

Revalued bond if it survives

Vᵢ = Σ CFₜ ÷ (1 + fᵢ,ₜ)ᵗ

Use the forward zero rates for the end rating i, for the remaining cash flows, including the coupon paid at the horizon.

Credit VaR (relative to mean)

Credit VaR = E(V) − V at the (1 − c) percentile

Measured from the expected value. Some questions ask for loss from the current value, so read the wording.

Asset return thresholds

Z_k = N⁻¹(cumulative probability from default up to and including state k)

Order the states from worst (default) upward. The default threshold is Z_default = N⁻¹(PD). The threshold for the next state up is N⁻¹(PD + p of that state), and so on. Each threshold is the standard normal quantile of the running total of transition probabilities.

Standard deviation of value

σ = √[Σ pᵢ × (Vᵢ − E(V))²]

Used for single-bond volatility and sometimes for approximating VaR.

Poisson probability of n defaults

P(n) = e^(-μ) × μ^n ÷ n!

μ is the expected number of defaults. n = 0, 1, 2, ... Mean and variance are both μ.

Expected number of defaults

μ = Σ p_i

Sum of obligor default probabilities. Valid as an approximation when each p_i is small.

Expected loss

EL = Σ p_i × v_i

v_i is exposure net of recovery, i.e. loss given default in currency terms.

Count variance with random default rate

Var(N) = μ + Var(Λ)

Λ is the random mean default count with mean μ. Variance exceeds the mean, so the count is overdispersed. Gamma mixing gives a negative binomial.

Band expected loss in units

ε_j = v_j × μ_j

v_j is the loss per default in exposure units for band j. μ_j is the expected defaults in band j.

Panjer recursion (single factor, fixed rate)

A(0) = e^(-μ), A(n) = (1 ÷ n) × Σ ε_j × A(n − v_j), summed over bands with v_j ≤ n

A(n) is the probability that portfolio loss equals n exposure units. Here μ is the total across bands.

Latent variable

Xᵢ = √ρ × M + √(1 − ρ) × Zᵢ

M, Zᵢ independent standard normals. Default if Xᵢ < N⁻¹(PD).

Conditional default probability

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

Valid for a given factor value M. Falls as M rises.

Worst case default rate (Vasicek)

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

Large homogeneous portfolio. X is the confidence level, e.g. 99.9%.

Credit VaR

Credit VaR = WCDR × LGD × EAD

Total loss at the confidence level. Unexpected-loss capital subtracts PD × LGD × EAD.

Unexpected loss capital (Basel IRB core)

K = LGD × [WCDR(99.9%) − PD]

Before the maturity adjustment and other scaling in the Basel formula.

Credit VaR (unexpected loss form)

Credit VaR = Loss at confidence level α − Expected loss

Some texts quote the percentile loss itself. Read the question to see which is asked. Economic capital is usually the unexpected-loss form.

Expected loss

EL = PD × LGD × EAD

Expected loss is covered by provisions and pricing. Capital covers losses above it.

Vasicek conditional default probability

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

α is the confidence level, ρ the asset correlation, N the standard normal CDF. Higher ρ raises tail loss.

Expected number of exceedances in backtest

Expected exceedances = T × (1 − confidence level)

At 99.9% with 10 annual observations, expected exceedances = 0.01. This is why backtesting has little power.

Quick revision

• Expected loss = PD × LGD × EAD; it is the mean of the loss distribution.
• Unexpected loss (UL) is the standard deviation of losses. Economic capital (sometimes called unexpected loss at the tail) is credit VaR minus expected loss. They are different measures.
• Credit VaR is a high quantile of a skewed loss distribution with a fat right tail.
• LGD = 1 − recovery rate, and recovery is often uncertain and tied to the credit cycle.
• Higher default correlation leaves expected loss unchanged but raises tail loss.
• Merton: equity is a call option on firm assets; default occurs if assets fall below debt at maturity.
• KMV uses distance to default and maps it to an empirical expected default frequency.
• CreditMetrics uses a rating migration matrix and revalues the portfolio at the horizon, usually one year.
• CreditRisk+ models default counts with a Poisson-type distribution and ignores migration.
• Vasicek tail default rate = N[(N⁻¹(PD) + √ρ × N⁻¹(q)) ÷ √(1 − ρ)], where q is the confidence level.
• In the Vasicek formula, for a high confidence level (e.g., 99.9%) and a typical low PD, a higher ρ raises the tail default rate.
• Credit models are hard to backtest because defaults are rare and horizons are long.

Common mistakes

• Treating the credit loss distribution as normal and using a normal multiple of the standard deviation. Fix: Remember the distribution is skewed with a fat right tail. Use the stated percentile, not a z-score, unless the question says to assume normality.
• Setting economic capital equal to credit VaR without subtracting EL. Fix: Use economic capital = credit VaR − EL unless the question defines it otherwise.
• Using the full limit as EAD on a revolving facility Fix: Use EAD = drawn + CCF × undrawn, and read the limit and drawn amounts carefully.
• Using the recovery rate as LGD Fix: Always write LGD = 1 − recovery before multiplying.
• Saying higher default correlation raises expected loss. Fix: Expected loss is a sum of individual expected losses. Correlation changes only the volatility and tail, such as unexpected loss and credit VaR.
• Confusing asset correlation with default correlation. Fix: Asset correlation drives the latent variable in Merton or Vasicek models. Default correlation is the correlation of default indicators and is usually much smaller. Check which one the question gives.
• Using equity volatility in d1 and d2. Fix: The formulas use asset volatility σV. Asset volatility is lower than equity volatility because of leverage.
• Treating N(d2) as the default probability. Fix: N(d2) is the probability of no default. Default probability is N(−d2) = 1 − N(d2).
• Measuring credit VaR from the current bond value when the question defines it from the mean. Fix: Read the definition in the question. CreditMetrics classically uses the expected value as the reference.
• Discounting with one curve for all states. Fix: Revalue with the forward curve of the end rating. The change in spread is what drives the value change.

Exam tips

• Questions often hide EL inside a loss table. Compute the probability-weighted mean before anything else.
• Read the confidence level carefully. In a discrete table, VaR is the smallest loss whose cumulative probability reaches that level.
• Expect concept items asking why credit VaR differs from market VaR: skew, fat tail, longer horizon, higher confidence and illiquid positions.
• Know who covers what: pricing and provisions cover EL, economic capital covers UL, and losses beyond VaR fall on shareholders and creditors.
• When a stem says economic capital, check whether it is internal (model-based) or regulatory (Basel rules).
• Questions often hide EAD in a limit and drawn amount. Compute EAD before anything else.
• If a recovery rate is given, convert it to LGD before multiplying.
• Expect conceptual items on downturn LGD, CCF behaviour and why EL differs from unexpected loss.