FRM Part II · FRM Exam Part II

Estimating Default Probabilities: formula sheet

Full chapter guide

Key formulas

Marginal (unconditional) default probability: Marginal(t) = Q(t) − Q(t−1); Q(t) is the cumulative default probability by the end of year t. Q(0) = 0.
Survival probability: S(t) = 1 − Q(t); Probability of no default through year t.
Hazard rate (annual conditional default probability): h(t) = [Q(t) − Q(t−1)] ÷ [1 − Q(t−1)]; Default in year t given survival to the start of year t.
Cumulative from annual hazards: 1 − Q(t) = Π (1 − h(i)) for i = 1 to t; Survival is the product of yearly conditional survival probabilities.
Constant continuous hazard rate λ: Q(t) = 1 − e^(−λt); λ = −ln(1 − Q(t)) ÷ t; Use when the question says a constant hazard (default intensity). Approximately λ ≈ spread ÷ (1 − recovery rate).
Row sum condition: Σj P(i, j) = 1 for every row i; Use it to check a matrix or to find a missing entry. The default row is (0, ..., 0, 1) when default is absorbing.
Two-year transition matrix: P(2) = P × P; Matrix multiplication under the Markov assumption. Row of the first matrix times column of the second.
Entry of a two-year matrix: P2(i, k) = Σj P(i, j) × P(j, k); Sum over every intermediate rating j after year 1, including default (j = D) when k is default.
n-year matrix: P(n) = P^n; The default column of P^n gives the cumulative default probability over n years for each starting rating.
Marginal default probability in year 2: PD(year 2, unconditional) = PD(2 yr cumulative) − PD(1 yr); Probability of defaulting in year 2 and not before.
Conditional default probability in year 2: PD(year 2 | survived year 1) = [PD(2 yr) − PD(1 yr)] ÷ [1 − PD(1 yr)]; Divide by the survival probability. This is the figure comparable to a hazard rate.
Constant-hazard shortcut (no migration): PD(n yr) = 1 − (1 − q)^n; Only if the annual default probability q is the same each year. A matrix with migration gives a different answer, so use P^n when asked.
Credit spread: s = y − y_f; Corporate yield minus risk-free yield, same maturity and same compounding. Convert bp to decimals: 150 bp = 0.015.
Spread, hazard rate and recovery (approximation): s ≈ λ × (1 − R); λ is the average annual risk-neutral hazard rate. It is an approximation, and works best for modest spreads.
Hazard rate from spread: λ ≈ s ÷ (1 − R); Higher assumed recovery means a higher implied hazard rate for the same spread.
Cumulative default probability (constant hazard): Q(T) = 1 − e^(−λT); Survival probability is e^(−λT). For small λT, Q(T) ≈ λT.
Default probability from zero-coupon prices: Q(T) = [1 − B_corp ÷ B_rf] ÷ (1 − R); Assumes recovery is a fraction R of the face value paid at maturity. With continuous compounding, B_corp ÷ B_rf = e^(−sT).
Average hazard rate from cumulative probability: λ = −ln(1 − Q(T)) ÷ T; Use this to turn a cumulative default probability back into an annual hazard rate.
Credit triangle: s ≈ λ × (1 − R); s is the annual CDS spread in decimal (120 bp = 0.012). It is an approximation that assumes a constant hazard rate.
Hazard rate from spread: λ = s ÷ (1 − R); Gives the average annual default intensity. Lower assumed recovery gives a lower implied hazard rate. A 40% recovery is a common standard assumption.
Survival probability: Q(t) = e^(−λt); Constant hazard rate. Probability of no default up to time t.
Cumulative default probability: PD(t) = 1 − e^(−λt); For small λt this is close to λt, but use the exponential when asked for precision.
Forward (period) hazard rate: λ(T1,T2) = (λ̄2 × T2 − λ̄1 × T1) ÷ (T2 − T1); λ̄ is the average hazard rate to each maturity. This is the quick bootstrap between two maturities.
Bootstrap condition: PV of premium payments = PV of expected protection payout; A par CDS has zero value at start. Premium leg includes the accrual paid on default.
CDS-bond basis: Basis = CDS spread − bond spread; Bond spread = bond yield minus the risk-free (or swap) rate, or the asset swap spread. Positive means CDS is the more expensive measure of credit risk.
Par asset swap spread: Asset swap spread ≈ bond yield − swap rate (par bond); Approximation for a bond at par. Non-par bonds need an adjustment for the premium or discount.
Average hazard rate from spread: λ ≈ s ÷ (1 − R); s = credit spread (annual, decimal), R = recovery rate. A risk-neutral estimate, approximate.
Cumulative default probability: Q(t) = 1 − e^(−λt); Assumes a constant hazard rate λ over t years.
Risk-neutral vs real-world relationship: λ(risk-neutral) = λ(real-world) × hazard rate ratio, usually ratio > 1; The ratio is the risk-premium effect; exceeds 1 in most markets. Not a fixed constant.
Spread decomposition: Spread ≈ expected loss component + risk premium + liquidity and other components; Only the expected loss part relates to real-world PD × (1 − R).
Expected loss rate (real-world): EL rate ≈ PD(real-world) × (1 − R); Spread minus this is the excess spread, a measure of compensation for risk.
Equity as a call option: E = V·N(d1) − D·e^(−rT)·N(d2); V = asset value, D = face value of debt, r = risk-free rate, T = maturity, N = standard normal CDF.
d1 and d2: d1 = [ln(V ÷ D) + (r + σV²÷2)·T] ÷ (σV·√T); d2 = d1 − σV·√T; σV is asset volatility. Use the same inputs in both.
Risk-neutral default probability: PD = N(−d2); This is the probability that V is below D at T under the risk-neutral measure.
Real-world default probability: PD = N(−d2*), with d2* = [ln(V ÷ D) + (μ − σV²÷2)·T] ÷ (σV·√T); μ is the expected asset return. Use it for real-world PD.
Equity volatility link: σE·E = N(d1)·σV·V; Used with the equity formula to solve for the unobservable V and σV.
Distance to default (simple): DD = (V − default point) ÷ (V·σV); Number of asset standard deviations above the default point, over one year. Default point is often short-term debt + 50% of long-term debt in KMV.
Risky debt value: B = V − E = D·e^(−rT) − Put(V, D); Debt holders are long a risk-free bond and short a put on assets.
Original Altman Z-score (public manufacturers): Z = 1.2·X1 + 1.4·X2 + 3.3·X3 + 0.6·X4 + 1.0·X5; X1 = working capital ÷ total assets; X2 = retained earnings ÷ total assets; X3 = EBIT ÷ total assets; X4 = market value of equity ÷ book value of total liabilities; X5 = sales ÷ total assets. Ratios are entered as decimals.
Original zone cut-offs: Z > 2.99 safe; 1.81 ≤ Z ≤ 2.99 grey; Z < 1.81 distress; Lower Z means higher default risk. These apply to the original model only.
Z'-score (private firms): Z' = 0.717·X1 + 0.847·X2 + 3.107·X3 + 0.420·X4 + 0.998·X5; X4 uses book value of equity instead of market value. Cut-offs are about 1.23 and 2.90. Know that the version changes weights and cut-offs.
Logit model: PD = 1 ÷ (1 + e^−(b0 + b1·x1 + … + bn·xn)); Output lies between 0 and 1. A positive coefficient on a variable raises PD as that variable rises.
Probit model: PD = N(b0 + b1·x1 + … + bn·xn); N is the standard normal CDF. Same idea as logit with a different link function.
Recovery rate (bonds): Recovery rate = Post-default market price ÷ Face value; Price is typically observed about 30 days after default. Expressed as a % of face value.
Loss given default: LGD = 1 − Recovery rate; Applies per unit of exposure. A 35% recovery gives a 65% LGD.
Expected loss: EL = PD × LGD × EAD; EAD is exposure at default. Use LGD, not recovery, in this formula.
Seniority ordering: Senior secured > Senior unsecured > Subordinated > Junior subordinated; Ordering of average recoveries and priority of claim. Individual cases can differ.
Default–recovery relationship: Corr(default rate, recovery rate) < 0; Negative on average. It is an empirical tendency, not an exact rule.

Quick revision

Ratings are through-the-cycle opinions; they change slowly and lag market information.
Historical default rates are real-world estimates and usually rise as ratings fall.
Cumulative default probability is not the sum of annual rates; use survival probabilities.
Transition matrices: multiply for multi-period results; default is an absorbing state.
Spread ≈ hazard rate × (1 − recovery), so λ ≈ spread ÷ (1 − R) as an approximation.
Survival probability to time t is e^(−λt) for a constant hazard rate.
Risk-neutral default probabilities are usually higher than real-world ones, since spreads include a risk premium and other effects.
Use risk-neutral for pricing and valuation; use real-world for scenario analysis and loss estimation.
Merton: equity is a call option on firm assets, and debt is risk-free debt minus a put.
Higher asset volatility or leverage means a smaller distance to default and a higher default probability.
Altman Z-score is a discriminant model that combines financial ratios; lower scores signal higher distress risk.
Higher assumed recovery implies a higher default probability for the same spread.

Common mistakes

Treating the cumulative rate for year 3 as the probability of defaulting in year 3. Fix: Cumulative is default by year 3. For year 3 alone, take Q(3) − Q(2).
Computing the hazard rate as marginal ÷ 1, ignoring survival. Fix: Divide by 1 − Q(t−1). Hazard is conditional on survival.
Squaring each entry instead of multiplying matrices Fix: Use row-times-column multiplication. Each two-year cell is a sum over all intermediate ratings.
Leaving out the default path in year 1 when computing two-year PD Fix: Include the term P(i, D) × 1. Because default is absorbing, the two-year PD is the one-year PD plus the paths that default in year 2.
Treating the credit spread as the default probability. Fix: Spread is expected loss, not probability. Always divide by (1 − R) to get λ.
Multiplying by (1 − R) instead of dividing. Fix: Solve for the unknown first. For λ, divide the spread by (1 − R). A λ smaller than the spread is a red flag.
Dividing the spread by R instead of (1 − R). Fix: Loss given default is (1 − R). With R = 40%, divide by 0.6, not 0.4.
Forgetting to convert basis points to decimals. Fix: Divide by 10,000 first. 120 bp = 0.012. A result above 1 for a hazard rate is a warning sign.
Using risk-neutral PDs to forecast actual defaults or expected losses Fix: They overstate actual default likelihood. Use real-world PDs for loss forecasts, stress tests and capital.
Using real-world PDs to value a CDS or bond relative to market prices Fix: Valuation consistent with traded prices needs risk-neutral inputs. Real-world PDs would produce prices that offer arbitrage against the market.

Exam tips

Read whether the question says unconditional, marginal or conditional. These words decide the formula.
Annual versus continuous hazard: if the question gives λ, use e^(−λt). If it gives yearly percentages, multiply survival terms.
Expect interpretation items: speculative grade hazards often decline with time, investment grade hazards often rise.
Remember that historical rates are real-world, and spread-implied rates are risk-neutral and higher.
Do the subtraction first, then the division. Writing both steps stops slips.
Questions are usually small matrices (two to four states). Write the row and column as lists and compute only the cell asked for.
Always include the default-in-year-1 path with weight 1. Wrong answer options are often built by leaving it out.
Read carefully whether the question wants cumulative, marginal (unconditional) or conditional default probability.