Rating-Based Credit Risk Models

Why Map Ratings to Default Probabilities?

Credit ratings summarize an assessment of an obligor’s credit quality. They are useful for organizing decisions, but a grade such as A or BB is not itself a probability. To use ratings in a credit model, an insurer or other creditor needs a defined mapping from each grade to the probability that an obligor in that grade will default over a specified period.

That mapping turns a qualitative or ordinal classification into a quantitative model input. It can support underwriting limits, monitoring priorities, and loss estimates. The result is only meaningful when the rating definition, default definition, observation date, and probability horizon are consistent. This topic is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: A rating-based obligor default model assigns an obligor to a credit rating grade $r$ at a specified assessment date and associates that grade with a probability $\operatorname{PD}_r(H)$ of default within horizon $H$. The probability is conditional on the obligor having grade $r$ at the assessment date. A rating is a category; $\operatorname{PD}_r(H)$ is the numerical model quantity assigned to that category.

The model is usually applied to one obligor at a time. It does not, by itself, describe how a rating may change during the horizon or how defaults of multiple obligors depend on one another. Its central question is narrower: given the grade now, what default probability is assigned for the stated horizon?

Define the Rating-to-Probability Mapping

A mapping is a table or rule that assigns a probability to each grade. Its entries must refer to the same horizon and the same definition of default. For example, an illustrative one-year mapping might assign different probabilities to grades A, BBB, and BB. Those values are hypothetical; an organization must use an appropriate source and calibration for its own purpose.

Model component	What must be specified
Rating grade	The grade assigned to the obligor on the assessment date
Horizon	The period over which default is being measured, such as one year
Default event	The operational definition of default used in the data and model
Probability source	External estimates, internal experience, or a documented combination
Application assumptions	Whether exposure and loss severity are treated as fixed or modeled as variable

Rating grades are generally ordinal: a stronger grade indicates better assessed credit quality than a weaker grade. They are not automatically equally spaced. A numerical code such as 1, 2, 3, 4 does not imply that the change from grade 1 to 2 represents the same change in credit risk as the change from grade 3 to 4. The probability mapping supplies numerical estimates; the grade numbers alone do not.

External ratings may be mapped using published or licensed default studies, subject to their definitions and scope. An organization may instead use internal ratings and its own observed default experience. In either case, the model should not treat a rating as a guarantee: an obligor in a strong grade can default, and an obligor in a weak grade can remain current.

Worked Example: Applying an External-Grade Mapping

A hypothetical insurer assesses a corporate obligor as BBB on January 1. Its documented one-year mapping assigns BBB a default probability of 0.60%. The insurer estimates exposure at default (EAD) at $2,400,000 and uses a fixed loss given default (LGD) of 45% for this facility. Calculate the one-year expected credit loss. Assume a one-year horizon, a Bernoulli default indicator with probability $p=0.006$, fixed EAD and LGD, and no discounting.

Method. First calculate the loss if default occurs. The conditional loss is $ \$2{,}400{,}000 \times 0.45 = \$1{,}080{,}000 $.

The probability of default is $0.60\%=0.006$, not 0.60. Multiplying the default probability by the conditional loss gives $0.006 \times \$1{,}080{,}000 = \$6{,}480$.

Result. The one-year expected credit loss is $6,480. As a check, the conditional loss is 45% of EAD, and 0.60% of that loss is $0.45 \times 0.006=0.0027$, or 0.27% of EAD. Indeed, $0.0027 \times \$2{,}400{,}000=\$6{,}480$.

Common mistake avoided. Treating 0.60% as 0.60 would produce an expected loss of $648,000, which is 100 times too large. Convert a percentage to a decimal probability before multiplying.

Calibrating a Grade-Level Probability

A direct way to estimate a grade-level default probability is to follow a cohort of obligors that all had the same grade at the start of a defined period. If $N_r$ obligors begin in grade $r$, and $D_r$ of them default within horizon $H$, the observed frequency is $\widehat{\operatorname{PD}}_r(H)=D_r/N_r$. This is an empirical estimate, not a guarantee that the future probability will equal the historical frequency.

The cohort must be constructed carefully. Count each obligor once, assign the grade as of the same kind of assessment date, and observe each obligor over the same horizon. The default definition must match the definition used by the data source. Incomplete follow-up needs appropriate treatment: an obligor whose observation ends early cannot simply be counted as a non-default for the full horizon.

Worked Example: Estimating One-Year Probabilities From a Cohort

A hypothetical lender reviews a completed one-year study of obligors classified at the start of the year. Among 400 obligors initially rated A, 3 defaulted during the year. Among 250 initially rated BB, 10 defaulted. All 650 obligors had complete one-year observation, each was counted once, and the same default definition was used. Estimate the observed one-year default probability for each grade. Treat each obligor’s one-year default outcome as a Bernoulli observation; no interest or mortality assumptions apply.

Method. Divide defaults by the number of obligors in the same starting-grade cohort. For A: $3/400=0.0075$, or 0.75%.

For BB: $10/250=0.04$, or 4.00%. The BB cohort’s observed default frequency is $4.00\%/0.75\%=5.333$ times the A cohort’s frequency.

Result. The empirical one-year estimates are 0.75% for A and 4.00% for BB. The ordering is consistent with the intended interpretation that BB represents weaker credit quality than A. This comparison is a useful reasonableness check, not proof that the estimates are stable or that every future sample must have the same ordering.

Common mistake avoided. Using the total number of obligors as the denominator for each grade would give $3/650$ and $10/650$, which are not grade-specific default frequencies. The denominator for each estimate is the number that began in that grade.

Why Expected Loss Uses the Rating Probability

A rating-based default probability can be combined with exposure and loss severity to obtain expected loss when the necessary assumptions are made explicit. Let $I$ equal 1 if default occurs within horizon $H$, and 0 otherwise. Suppose EAD and LGD are fixed for the obligor over the calculation, and let $p=\operatorname{PD}_r(H)$. The loss is $I \times \mathrm{EAD} \times \mathrm{LGD}$.

E[I \times \mathrm{EAD} \times \mathrm{LGD}] = \mathrm{EAD} \times \mathrm{LGD} \times E[I] = \mathrm{EAD} \times \mathrm{LGD} \times p

The justification is that a Bernoulli indicator has expectation $E[I]=1p+0(1-p)=p$. When default does not occur, $I=0$ and the modeled default loss is zero; when it occurs, the loss is EAD times LGD. Therefore, averaging the two possible losses gives the formula. If EAD or LGD varies with default circumstances, multiplying separate averages need not give the correct result. In that case the model must account for their relationship with default and with each other.

Theorem: For a fixed EAD and fixed LGD, with default probability $p=\operatorname{PD}_r(H)$ over horizon $H$, expected loss over that horizon is $p \times \mathrm{EAD} \times \mathrm{LGD}$. This result concerns the expectation of loss; it does not describe the full distribution of losses or the chance of a loss exceeding a threshold.

Rating-Based Decisions at the Obligor Level

Ratings support decisions only when the numerical mapping is connected to a defined use. A policy might set review priorities by grade, set exposure limits, or require additional analysis above a specified mapped probability. Such a rule is a management choice, not a mathematical consequence of the rating itself. The mapping and decision rule should be documented separately so that a change in policy is not mistaken for a change in default risk.

The following example illustrates why rating is assigned to the obligor, while exposure and loss severity can differ by facility. Assume both facilities share the same obligor and therefore the same default event.

Worked Example: One Rating, Two Facilities

A hypothetical borrower is assigned internal grade BB. The model’s one-year probability for BB is 3.0%. The borrower has two facilities: Facility 1 has EAD of $400,000 and LGD of 50%; Facility 2 has EAD of $600,000 and LGD of 40%. Assume one common borrower-level default event, fixed facility EADs and LGDs, and no discounting. Calculate the borrower’s combined one-year expected loss. The default indicator is Bernoulli with parameter $p=0.03$.

Method. Calculate the loss on each facility conditional on the borrower defaulting. Facility 1 loss is $\$400{,}000 \times 0.50=\$200{,}000$. Facility 2 loss is $\$600{,}000 \times 0.40=\$240{,}000$.

The combined conditional loss is $\$200{,}000+\$240{,}000=\$440{,}000$. Apply the single borrower-level default probability: $0.03 \times \$440{,}000=\$13{,}200$.

Result. The combined one-year expected loss is $13,200. As a check, the facility-level expected losses are $0.03 \times \$200{,}000=\$6{,}000$ and $0.03 \times \$240{,}000=\$7{,}200$; their sum is $13,200.

Common mistake avoided. Assigning an independent 3.0% default event to each facility changes the model: it treats two facilities of one borrower as if they belonged to separate obligors. The calculation here applies the borrower’s one default event to both facility losses.

Calibration Choices and Common Traps

The same grade can be associated with different probability estimates depending on the population, time period, and purpose of calibration. A through-the-cycle estimate aims to represent credit quality across a range of economic conditions, while a point-in-time estimate is intended to reflect conditions at a particular time. These approaches answer different questions; neither label alone establishes that an estimate is suitable for every decision. A model user should understand which approach the mapping represents.

Keep the following distinctions clear when building or applying the model:

Grade versus probability. A grade is a category. Use the documented mapping to obtain a probability; do not treat the grade label or numerical grade code as a probability.
Horizon versus annual rate. A one-year probability is defined over one year. It should not be silently reused as a probability for a different horizon.
Rating date versus later information. Use the grade as of the stated assessment date. A grade assigned after observing subsequent events does not provide an unbiased estimate for the original cohort.
Default frequency versus certainty. Historical frequency is an estimate based on observed data. Small cohorts can produce unstable estimates, and a zero observed default count does not prove that the true probability is zero.
Obligor versus facility. A borrower can have multiple facilities. A borrower-level rating normally describes the obligor’s credit quality, while facility-level exposure and loss severity may differ.
Expected loss versus total risk. Expected loss is an average. It is not the maximum possible loss and does not by itself measure how variable or concentrated losses may be.

A sound rating-based model therefore documents its grade definitions, assessment date, default event, probability horizon, data source, and intended use. It also tests whether the assigned probabilities remain sensible for the population being modeled. These controls help prevent false precision: a percentage with several decimal places is not necessarily a well-supported estimate.

Key takeaway: A rating-based obligor model consists of a grade assigned at a stated date and a calibrated default probability for a stated horizon. Use that probability only with compatible exposure, severity, and default assumptions.

Check Your Understanding

For each problem, state the calculation and check that the grade, horizon, and denominator match the information given.

A hypothetical one-year mapping assigns grade A a default probability of 0.40%. An obligor has fixed EAD of $750,000 and fixed LGD of 35%. Under a Bernoulli default model with no discounting, calculate one-year expected loss. (Answer: $1,050)
In a completed one-year cohort, 6 of 300 obligors initially rated BB default. All had complete follow-up and were counted once. Estimate the observed one-year BB default probability. (Answer: 2.00%)
A borrower has two facilities and one common one-year default event with probability 2.5%. Facility A has EAD $200,000 and LGD 40%; Facility B has EAD $300,000 and LGD 60%. Assume fixed amounts and no discounting. Calculate combined expected loss. (Answer: $6,500)
A hypothetical internal mapping assigns BBB a one-year probability of 1.2%. For a facility with EAD $1,500,000 and fixed LGD 40%, calculate the conditional loss if default occurs and the one-year expected loss. Assume a Bernoulli default indicator and no discounting. (Answer: Conditional loss $600,000; expected loss $7,200)
A completed one-year study records 12 defaults among 600 obligors that began in grade B. A separate group of 200 obligors in another grade is also in the data. Estimate the grade B probability, then calculate expected loss for a grade B exposure with EAD $4,000,000 and LGD 45%, assuming fixed EAD and LGD. (Answer: Grade B probability 2.00%; expected loss $36,000)

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