External and Internal Credit Ratings: formula sheet

Key formulas

Investment grade boundary

Investment grade: S&P/Fitch ≥ BBB-; Moody's ≥ Baa3

One notch below (BB+ / Ba1) is speculative grade.

Scale order (S&P/Fitch)

AAA > AA > A > BBB > BB > B > CCC > CC > C > D

+ and - modifiers sit within AA to CCC categories. Example: BBB+ > BBB > BBB-.

Scale order (Moody's)

Aaa > Aa > A > Baa > Ba > B > Caa > Ca > C

Modifiers 1, 2, 3 apply within Aa to Caa. Example: Baa1 > Baa2 > Baa3.

Equivalent mapping

AAA=Aaa, AA=Aa, A=A, BBB=Baa, BB=Ba, B=B, CCC=Caa; CC ≈ Ca and C ≈ C (approximate); D (S&P/Fitch default) has no Moody's equivalent

AA+ = Aa1, AA = Aa2, AA- = Aa3, and so on. The CC to Ca and C to C links are approximate equivalences, not exact ones. C is the lowest Moody's rating. Moody's has no D category.

Issuer vs issue rating

Issue rating = issuer rating adjusted for seniority, security and expected recovery

Subordination usually lowers it; strong security can raise it.

Row sum check

Σj P(i → j) = 100% for every starting rating i

Includes the default column and any not-rated column. Use it to find a missing entry.

Two-year transition probability

P2(i → k) = Σj P(i → j) × P(j → k)

Matrix multiplication of row i with column k. This assumes a time-homogeneous Markov chain.

n-year matrix

Mn = M × M × ... × M (n times)

Default column of Mn gives cumulative PD for each starting rating.

Two-year cumulative PD

CPD2(i) = Σj P(i → j) × CPD1(j), with CPD1(Default) = 100%

Fast way: you only need the one-year default column, not the whole matrix.

Marginal PD

MPD(t) = CPD(t) − CPD(t−1)

Probability of defaulting in year t, measured from today.

Survival probability

S(t) = 1 − CPD(t)

Probability of no default up to year t.

Conditional (forward) PD

CondPD(t) = MPD(t) ÷ S(t−1) = [CPD(t) − CPD(t−1)] ÷ [1 − CPD(t−1)]

Default in year t given survival to the start of year t.

Constant annual hazard

CPD(t) = 1 − (1 − h)^t, so h = 1 − (1 − CPD(t))^(1/t)

Use only when told to assume a constant annual default probability. Continuous form: CPD(t) = 1 − e^(−λt).

TTC vs PIT: what moves

TTC: grade stable, realized default rate per grade varies with cycle | PIT: borrower grades and borrower-level PDs move with the cycle; the PD and realized default rate per grade stay roughly stable

This is the core contrast. Memorise which quantity is stable in each approach.

Horizon and conditions

TTC = stressed or full-cycle view, long horizon | PIT = current conditions, short horizon (often 1 year)

Use this to classify a rating system from a description.

Procyclicality link

Downturn → PIT downgrades → higher PD and capital → less lending → weaker economy

Greater with PIT ratings. TTC reduces the amplification but is slower to react.

Rating momentum

P(downgrade | previous downgrade) > P(downgrade | previous upgrade or no change)

Observed empirically in agency transition data; it reflects gradual adjustment and is a feature of stability-oriented ratings.

Expected loss

EL = PD × LGD × EAD

Use PD as a decimal over a one-year horizon. EL is the average loss, not the capital buffer.

LGD and recovery rate

LGD = 1 − Recovery rate

Recovery is measured as a share of exposure at default, usually after costs.

EAD with undrawn commitment

EAD = Drawn + CCF × Undrawn

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

Foundation vs advanced IRB

F-IRB: bank estimates PD. A-IRB: bank estimates PD, LGD, EAD and, usually, maturity

In F-IRB, LGD and the CCFs used for EAD are supervisory values, and maturity is also set by the supervisor.

Rating philosophy

PIT = responds to current conditions. TTC = through the cycle, more stable

PIT grades migrate more with the cycle; TTC grades migrate less.

Accuracy ratio

AR = (Area between model CAP and diagonal) ÷ (Area between perfect CAP and diagonal)

Ranges from 0 (no better than random) to 1 (perfect). Also called the Gini coefficient or Somers' D in this context.

AR and AUC link

AR = 2 × AUC − 1

AUC is the area under the ROC curve. AUC = 0.5 is random, so AR = 0.

Perfect model CAP area

Area above diagonal = 0.5 × (1 − D), where D = overall default rate

Used to compute AR from a model area. A lower default rate means a larger perfect-model area.

Binomial mean and standard deviation of defaults

E(defaults) = n × p; σ = √(n × p × (1 − p))

Assumes independent defaults and one PD for a grade of n borrowers.

Normal approximation z-score

z = (Observed defaults − n × p) ÷ √(n × p × (1 − p))

Reasonable when n × p is not small. Compare with the critical value, for example 1.645 for a one-sided 95% test.

Observed default rate

Default rate = Number of defaults ÷ Number of borrowers at start of period

Compare this with the predicted PD for each grade.

Quick revision

• Investment grade means BBB-/Baa3 or higher; below that is speculative grade.
• Each row of a transition matrix sums to 100%.
• Default is an absorbing state: once there, a borrower stays there.
• Multi-year transitions come from multiplying the one-year matrix by itself, assuming it is stable over time.
• Expected loss = PD × LGD × EAD.
• TTC ratings look past the cycle, so they are stable and change slowly.
• PIT ratings reflect current conditions, so they move with the cycle and give more procyclical capital.
• Agency ratings are criticised for lag, cliff effects, conflicts of interest and reliance on issuer-paid models.
• Internal ratings can use bank-specific data and allow a rating for every borrower, including unrated ones.
• Discriminatory power asks whether a system separates defaulters from non-defaulters.
• Calibration asks whether predicted PDs match observed default rates.
• Backtest by comparing predicted PDs with realised default frequencies over time.

Common mistakes

• Treating BB+ or Ba1 as investment grade. Fix: The lowest investment grade is BBB- / Baa3. BB+ / Ba1 is already speculative.
• Confusing Moody's Baa with Ba. Fix: Baa is the BBB category (investment grade); Ba is the BB category (speculative).
• Reading a column as the starting rating instead of a row. Fix: Rows are 'from', columns are 'to'. The row must sum to 100%. If a column sums to 100% instead, the table is probably transposed, so check the labels.
• Squaring each entry to get the two-year matrix. Fix: Use row-by-column multiplication: add up P(i→j) × P(j→k) over all intermediate ratings j, including the path through default.
• Saying TTC ratings have stable default rates over the cycle. Fix: Under TTC the grade is stable and is associated with a long-run (cycle-neutral or stress-oriented) PD, while the realized default rate for that grade rises in recessions and falls in booms.
• Calling TTC ratings more procyclical than PIT ratings. Fix: Procyclicality in capital is greater for PIT systems. Agency cliff effects are a separate issue and TTC remains the more stable approach.
• Saying ratings are always slower than markets because agencies are careless. Fix: Explain that stability is intentional. The cost is lag relative to spreads and CDS prices.
• Claiming the issuer-pays model proves agencies knowingly gave false ratings. Fix: Say it creates a potential conflict and rating-shopping pressure. Do not state it as proof of intent.
• Using the full undrawn limit as EAD, or ignoring it Fix: Use EAD = drawn + CCF × undrawn.
• Entering LGD as the recovery rate Fix: Always compute LGD = 1 − recovery before using it.

Exam tips

• Memorise the S&P to Moody's mapping table and the BBB- / Baa3 line; many questions reduce to this.
• Read whether the question says issuer or issue rating before answering.
• Count notches by writing out the ladder, especially across categories.
• Watch for qualitative questions on issuer-pays conflicts and cliff effects when a bond drops to speculative grade.
• Do not assume a rating equals a specific default probability unless a table is given.
• Read the question for the exact measure: cumulative, marginal or conditional. Examiners often put the wrong ones among the answer options.
• Use the default-column shortcut for two-year PD. It cuts the work to one short sum.
• Always check that cumulative PD rises with the horizon and that a lower rating has a higher PD. This catches most row or column slips.