Simple models for credit risk: formula sheet

Key formulas

Expected loss

EL = PD × LGD × EAD

Assumes PD, LGD and EAD are independent point estimates over the same period.

Loss given default

LGD = 1 − recovery rate

Recovery rate is the fraction of exposure recovered, as a proportion of EAD.

Credit spread

s = y(risky) − y(risk-free)

Use yields of the same term and similar features, such as coupon and currency.

Approximate spread from default

s ≈ λ × LGD

Simple approximation where λ is the annual default intensity (or PD for small values). It ignores risk premium and liquidity.

Transition matrix row condition

Σⱼ pᵢⱼ = 1 for each row i

Each row covers all possible end ratings, including default.

n-year transition probabilities

P(n) = Pⁿ

Valid when the rating process is a time-homogeneous Markov chain.

Equity payoff at T

E_T = max(V_T − D, 0)

Equity is a call on assets with strike D.

Debt payoff at T

B_T = min(V_T, D) = D − max(D − V_T, 0)

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

Merton equity value

E_0 = V_0 N(d1) − D e^(−rT) N(d2)

Black-Scholes call with S = V_0, K = D. No dividends on assets.

d1 and d2

d1 = [ln(V_0 ÷ D) + (r + σ²/2)T] ÷ (σ√T); d2 = d1 − σ√T

σ is the volatility of asset value, not of equity.

Debt value

B_0 = V_0 − E_0 = D e^(−rT) N(d2) + V_0 N(−d1)

Also B_0 = D e^(−rT) − put value.

Risk-neutral default probability

P(V_T < D) = N(−d2)

For the real-world probability, use μ in place of r in d2.

Yield and credit spread

B_0 = D e^(−yT), so y = −(1 ÷ T) ln(B_0 ÷ D); spread = y − r

Continuous compounding. The spread is always positive when default is possible.

Survival probability (constant hazard)

P(τ > t) = e^(−λt)

τ is the default time. Time to default is exponential with mean 1/λ.

Survival probability (time-varying hazard)

P(τ > t) = exp(−∫₀ᵗ λ(s) ds)

For piecewise constant λ, add λ × length over each period inside the exponent.

Default probability by time t

P(τ ≤ t) = 1 − e^(−λt)

Use the risk-neutral λ for pricing and the real-world λ for real-world risk measurement.

Hazard rate definition

λ(t) = f(t) ÷ S(t)

f is the density of τ and S is the survival function. This is the same idea as force of mortality.

Defaultable zero-coupon bond, recovery of face value at maturity

Price = e^(−rT) × [R + (1 − R) e^(−λT)]

Assumes constant r and λ, independent, with recovery R of face value paid at T if default occurs before T.

Defaultable zero-coupon bond, zero recovery

Price = e^(−(r + λ)T)

The spread equals λ when R = 0.

Approximate credit spread

s ≈ (1 − R) × λ

Approximation for small λ. Rearrange to get λ ≈ s ÷ (1 − R).

One-step transition probability

p(i,j) = P(X(t+1) = j | X(t) = i)

Entries of matrix P. Every row sums to 1 and every entry is ≥ 0.

Markov property

P(X(t+1) = j | X(t) = i, X(t−1), ..., X(0)) = P(X(t+1) = j | X(t) = i)

Future depends on the present rating only.

Chapman-Kolmogorov

p(i,j; n) = Σk p(i,k; m) × p(k,j; n−m)

In matrix form Pⁿ = Pᵐ × Pⁿ⁻ᵐ. Valid for a time-homogeneous chain.

Multi-period matrix

P(n) = Pⁿ

Default column of Pⁿ gives cumulative default probability by time n, if default is absorbing.

Distribution at time n

π(n) = π(0) × Pⁿ

π(0) is a row vector of starting probabilities. Multiply row vector on the left.

Absorbing default state

p(D,D) = 1

Default row is (0, ..., 0, 1). Defaults then accumulate and never reverse.

Credit spread (definition)

s = y − r

y is the yield on the risky bond and r is the yield on the risk-free bond of the same term and cash flows.

Approximate spread from default and recovery

s ≈ λ(1 − R)

λ is the annual default probability or hazard rate, R the recovery rate. It is an approximation that works best for small λ. It gives the expected-loss part only.

Risky zero-coupon price, recovery of face value at maturity

P = e^(−rt) × [R + (1 − R) e^(−λt)]

Constant hazard rate λ, continuous compounding. It assumes recovery is a fraction R of face value paid at maturity. Equivalent to e^(−rt) × [e^(−λt) + R(1 − e^(−λt))].

Zero recovery case

P = e^(−(r + λ)t), so s = λ

Spread equals the hazard rate exactly when R = 0 and compounding is continuous.

Spread from prices (continuous compounding)

s = −(1/t) × ln(P_risky ÷ P_riskfree)

Use when you have risky and risk-free zero-coupon prices for the same maturity.

Spread decomposition

Observed spread = expected loss + risk premium + liquidity premium (+ other)

Used to explain why observed spreads exceed spreads implied by historical defaults.

CDS premium, rough approximation

CDS spread ≈ λ(1 − R)

Under risk-neutral λ. A fair CDS premium equals the present value of expected protection payments divided by the risky annuity of premiums.

Quick revision

• Expected loss = probability of default × loss given default × exposure.
• Loss given default = 1 − recovery rate, when recovery is a fraction of exposure.
• Structural models link default to firm asset value falling below the debt level.
• In the Merton model, equity behaves like a call option on the firm's assets.
• In the Merton model, risky debt equals risk-free debt minus a put option on the assets.
• Reduced form models treat default time as random, driven by a hazard rate.
• For a constant hazard rate λ, survival to time t is exp(−λt).
• In a transition matrix, each row sums to 1 and default is usually an absorbing state.
• Multi-period transition probabilities come from powers of the one-period matrix, under the time-homogeneous Markov assumption.
• Credit spread = risky yield − risk-free yield.
• Spread is not only expected loss; it also includes compensation for risk and other factors.
• State your assumptions before computing, and interpret the answer at the end.

Common mistakes

• Using recovery rate in place of LGD in the expected loss formula. Fix: Always write LGD = 1 − recovery first, then substitute.
• Treating the credit spread as equal to the expected loss rate only. Fix: State that observed spreads also include a risk premium and compensation for illiquidity, so they usually exceed the expected loss part.
• Using N(d1) or N(−d1) as the default probability. Fix: Default probability is N(−d2). N(d1) is the delta of the equity call, not a probability of default.
• Using equity volatility instead of asset volatility in d1 and d2. Fix: The model assumes assets follow GBM. σ is asset volatility. Equity volatility is higher than asset volatility because equity is a leveraged claim.
• Using the real-world default probability to price a bond. Fix: Price only with risk-neutral probabilities. Use real-world ones only when the question asks about actual expected losses.
• Writing default probability as λT for any T. Fix: Use 1 − e^(−λT). Use λT only if the question says the period is small and approximation is acceptable.
• Raising each entry of P to the power n instead of using matrix multiplication. Fix: Pⁿ means P × P × ... × P. Use row-by-column multiplication.
• Multiplying a starting vector on the wrong side, as P × π. Fix: With row-vector probabilities and row-sum-1 matrices, write π(0) × Pⁿ.
• Treating the whole spread as default compensation. Fix: Say that this is the expected-loss part only. Observed spread also contains risk and liquidity premiums.
• Using s = λ when recovery is not zero. Fix: Always divide by (1 − R) when finding λ from s: λ ≈ s ÷ (1 − R).

Exam tips

• Write the formula in notation before substituting. Marks go for method even if the arithmetic slips.
• In written answers, define default risk and spread risk separately, then link them through spread.
• For transition questions, list the paths in a short table of lines. This avoids missing the absorbing default path.
• State assumptions such as independence of PD, LGD and EAD, and the Markov property.
• Check units: PD per year, EAD in rupees, LGD as a proportion.
• Write 'equity = call on assets, strike D' at the start. Examiners give marks for setting up the option analogy, even if the arithmetic goes wrong.
• Show d1, d2, N(d1) and N(d2) clearly. Most marks are for the method, so state the formula in standard notation before substituting.
• If the question gives equity or a put price, use the shortcut B_0 = V_0 − E_0. Do not recompute Black-Scholes.