CFA Level II · CFA Level II Exam

Credit Analysis Models: formula sheet

Full chapter guide

Key formulas

Loss given default: LGD = Exposure × (1 − Recovery rate); Recovery rate is the share of exposure recovered. LGD can also be quoted as a percentage: LGD % = 1 − recovery rate.
Expected loss: EL = PD × LGD × Exposure at default; If LGD is already a currency amount, EL = PD × LGD amount. Use the same time period for PD as for EL.
Credit spread: Credit spread = Yield on risky bond − Yield on risk-free bond of same maturity; Compensates for expected loss, a risk premium for uncertainty and liquidity effects.
Probability of survival over two years (given annual PDs): P(survive 2 yrs) = (1 − PD₁) × (1 − PD₂); PD₁ and PD₂ are conditional annual default probabilities, treated as independent here. Cumulative PD = 1 − survival probability.
Equity as a call on assets: E(T) = max(V(T) − K, 0); V = firm asset value, K = face value of zero-coupon debt. Equity holders own the call.
Risky debt at maturity: D(T) = min(V(T), K) = K − max(K − V(T), 0); Debt holders hold risk-free debt and are short a put on assets.
Value of risky debt today: D(0) = PV(K) − Put = V(0) − Call; PV(K) is K discounted at the risk-free rate. Put and call both have strike K and maturity T.
Put-call parity on assets: V + Put = Call + PV(K); Use it to get the put from a given call, or the reverse.
Credit spread: Spread = y − r, where D(0) = K ÷ (1 + y)^T; y is the yield on the risky debt. Use the same compounding as r. With continuous compounding, y = −ln(D(0) ÷ K) ÷ T.
Black-Scholes-Merton inputs: d1 = [ln(V ÷ K) + (r + σ² ÷ 2)T] ÷ (σ√T); d2 = d1 − σ√T; Call = V·N(d1) − K·e^(−rT)·N(d2); σ is asset volatility, r is continuously compounded. The put uses N(−d1) and N(−d2).
Risk-neutral default probability: P(V(T) < K) = N(−d2) = 1 − N(d2); This is under risk-neutral probabilities, not real-world ones.
Approximate default probability over a short period: PD ≈ λ × Δt; λ is the annual default intensity (hazard rate) and Δt is a short time in years. Valid only for small λΔt.
Survival probability with constant intensity: P(survive to t) = e^(−λt); Assumes λ is constant. Default probability to time t is 1 − e^(−λt).
Loss given default: LGD = 1 − Recovery rate; Recovery is an exogenous assumption in reduced-form models.
Approximate credit spread link: Spread ≈ λ × LGD = λ × (1 − R); A rough relationship under risk-neutral measures. Intensity here is the risk-neutral one.
Expected loss: EL = PD × LGD; Use the same time horizon for PD as for the loss you want.
Loss given default: LGD = 1 − Recovery rate; Recovery rate is a percentage of exposure. Use LGD, not recovery, in the CVA formula.
Credit valuation adjustment: CVA = Σ [ Expected exposure(t) × LGD(t) × POD(t) × DF(t) ]; Sum over every date t on which default can occur. POD(t) is the unconditional probability of default at t. DF(t) is the risk-free discount factor.
Risky value: Value of risky bond = Value of risk-free bond − CVA; Risk-free value uses the same promised cash flows discounted at risk-free rates.
Expected exposure for a bond: Exposure at t = Cash flow at t + PV at t of all later cash flows; This assumes default occurs just before the payment at t, so the payment at t is also at risk.
Unconditional default probability: POD(t) = Survival probability to t−1 × Conditional POD(t); Survival probability to t = Π (1 − conditional POD) over earlier periods. Check which type of POD the question gives.
One-period risk-neutral POD from prices: Risky price × (1 + r) = Face × (1 − p) + Face × Recovery × p, so p = [Face − Risky price × (1 + r)] ÷ [Face × LGD]; r is the risk-free rate for the period. Here risky price = Face ÷ (1 + risky yield). Exact for a one-period zero-coupon bond.
Spread approximation: Credit spread ≈ POD × LGD, so POD ≈ Spread ÷ LGD; A short-horizon approximation. It is not exact.
Expected loss: EL = PD × LGD × exposure; Use the same horizon for PD and the spread. Exposure is the amount at risk at default.
Loss given default: LGD = 1 − recovery rate; If the recovery rate is 40%, LGD is 60%.
Approximate credit spread: Spread ≈ PD × LGD + risk premium; A one-year, annualized approximation. The premium is what remains after expected loss.
Price change from spread change: %ΔPrice ≈ −Duration × ΔSpread + ½ × Convexity × (ΔSpread)²; Use spread duration. Express ΔSpread in decimals, so 25 bps is 0.0025.
Excess return over the benchmark: EXR ≈ (Spread × t) − (ΔSpread × EffSpreadDur) − (t × PD × LGD); Spread is the spread at the start of the period, t is the holding period in years, and PD is the annual default probability. The three terms are spread income, the price effect of the spread change, and expected credit loss. This is an excess return over the benchmark, not a total return.
Survival probability: Survival = 1 − PD; For multiple periods, multiply the conditional survival probabilities.

Quick revision

Expected loss = exposure × PD × LGD, with LGD = 1 − recovery rate.
Structural models view equity as a call option on firm assets.
In structural models, risky debt equals a risk-free bond minus a put option on assets.
Equity holders are long the call and gain from higher asset volatility. Lenders are short the put, so higher volatility raises the put value, lowers risky debt value and raises its spread.
Reduced-form models treat default as a surprise that occurs at a random time, governed by a (possibly stochastic) default intensity, also called the hazard rate.
Structural models need firm asset value and volatility; reduced-form models are typically calibrated to observable market data such as bond or CDS spreads.
Risky bond value = PV of promised cash flows at risk-free rates minus CVA, where CVA is the PV of expected losses on those cash flows. Alternatively, discount the promised cash flows at the risky yield (risk-free rate plus credit spread).
CVA is the sum of discounted expected losses across periods.
Compute each period's expected loss as exposure at that date × marginal (unconditional) PD for that period × LGD. Do not use the cumulative PD. Exposure may differ by date. Marginal PD = probability of surviving to the prior period × conditional probability of default in that period.
Wider credit spreads compensate for higher expected loss and risk premium.
Know the PD types. Cumulative PD is the probability of default by a date. Marginal (unconditional) PD is the probability of default in one period, seen from today. Conditional PD is the probability of default in a period given survival to its start. Match the type to the period you are using.
Answer every question; there is no penalty for wrong answers.

Common mistakes

Using the recovery rate in place of LGD in the expected loss formula. Fix: Always compute LGD = 1 − recovery rate before multiplying.
Treating PD and LGD as the same thing. Fix: PD is how likely default is. LGD is how much is lost if default happens. Ask 'likelihood or size?'
Calling equity a put option and debt a call Fix: Equity gets V − K only when assets exceed debt, so it is a call. Debt holders are short the put.
Using face value K instead of PV(K) in the debt formula Fix: Risky debt today = PV(K) − put. Discount K at the risk-free rate over T first.
Saying reduced-form models link default to the firm's asset value. Fix: Remember: structural = asset value vs debt. Reduced-form = exogenous intensity, so default can be a surprise.
Treating recovery as something the model derives from asset value. Fix: In reduced-form models, recovery is an exogenous assumption you are given.
Using the recovery rate instead of LGD in the CVA formula. Fix: Write LGD = 1 − recovery as the first line of your working. With 40% recovery, LGD is 60%.
Using conditional default probabilities as if they were unconditional. Fix: If the table says 'conditional' or 'given survival', multiply by the survival probability to the prior date. The unconditional year-2 POD is (1 − PD1) × conditional PD2.
Using the recovery rate in place of LGD when computing expected loss. Fix: Always convert first: LGD = 1 − recovery. Then EL = PD × LGD.
Treating the whole spread as compensation for expected loss. Fix: Expected loss is only part of the spread. The remainder is the risk premium, and sometimes liquidity.

Exam tips

Underline whether the vignette gives recovery rate or LGD. Many wrong options come from mixing them.
Check the horizon before multiplying. Annual PD answers an annual question only.
When asked what the spread compensates for, remember expected loss plus risk premium and liquidity, not expected loss alone.
If a question asks which credit has the higher expected loss, compare PD × LGD × exposure for each; do not rank on PD alone.
Write V = D + E and the payoffs at the top of your rough work before reading the questions. Most answers follow from them.
When a vignette gives the call value, use debt = V − call. Do not recompute the model.
For direction questions, go through the put: volatility, leverage and maturity up means put up, debt down, spread up. Equity moves the other way on volatility.
Read whether d2 or N(d2) is given. N(d2) is the no-default probability. N(−d2) is the default probability.