Being “K” the random variable that represents the number of defaults, “k” the number of defaults, “n” the number of obligors, “λ” the default probability and “ω” the confidence level; and assuming that “K” follows a standard binomial distribution, the probability of having no more than “k” defaults inside a risk class with “n” obligors is given by:

P ( K ≤ k ) = ∑ i = 0 k ( n i ) λ i ( 1 − λ ) n − i ≥ 1 − ω (1)

If a standard Poisson distribution is used to represent “K”, the probability is:

P ( K ≤ k ) = ∑ i = 0 k e − n λ ( n λ ) i / i ! ≥ 1 − ω (2)

When default events are completely independent (ρ = 0%), “upper confidence bounds”⁴ for both binomial and Poisson distributions can be computed analytically through beta and gamma approximations or numerically. They provide the same outputs because the binominal distribution is proportional to the beta distribution and the Poisson distribution is proportional to the gamma distribution:

Binomial ( n , k ) ∝ Beta ( α , β ) ,     α = k + 1   and   β = n − k ( n k ) λ k ( 1 − λ ) n − k ∝ Γ ( α + β ) / [ Γ ( α ) Γ ( β ) ] λ α − 1 ( 1 − λ ) β − 1 (3)

and

Poisson ( n λ ) ∝ Gamma ( α , β ) ,     α = k + 1   and   β = 1 / n e − n λ ( n λ ) k / k ! ∝ β α / Γ ( α ) e − β λ λ α − 1 (4)

The theoretical environment must be drastically changed by positive asset correlation values, “ρ”. Instead of basic and unrealistic reduced form models that rely on default independence, complex and adequate structural models should be used.

J ( λ , y , ρ ) represents the probability function of the sample data resulting from the binomial function of “λ” or the Poisson function of “λ”, ϕ ( y ) represents the standard normal probability density function of “Y”, Φ ( . ) represents the standard normal cumulative distribution function, and Φ − 1 ( λ ) represents the inversed standard normal cumulative distribution function for “λ”. The meanings of “K”, “k”, “n”, “λ”, and “ω” are the same above mentioned.

Therefore, the probability of having no more than “k” defaults inside a risk class with “n” obligors is provided by:

P ( K ≤ k ) = ∫ − ∞ + ∞ [ ∑ i = 0 k J ( λ , y , ρ ) i ] ϕ ( y ) d y ≥ 1 − ω , (5)

with

J ( λ , y , ρ ) i = ( n i ) [ G ( λ , y , ρ ) ] i [ 1 − G ( λ , y , ρ ) ] n − i (6)

or

J ( λ , y , ρ ) i = e − n G ( λ , y , ρ ) [ n G ( λ , y , ρ ) ] i / i ! , (7)

if the probability function of the sample data follows a binomial distribution or a Poisson distribution, respectively, and

G ( λ , y , ρ ) = Φ { [ Φ − 1 ( λ ) − y ρ ] / 1 − ρ } (8)

The function G ( λ , y , ρ ) distinguishes the formulae of reduced form models from the structural models. A brief explanation of that crucial function and its origin can be found in Appendix A.

The default probability of the posterior distribution is derived from the likelihood and prior densities matching, as previously indicated. Multiple prior functions, ranging from less prudent to over conservative, are identified in this subsection. They reflect various elements or beliefs about the effective default probability.

When there is no understanding of the behavior of posterior default probability, it is common to use a non-informative prior. The most non-informative prior is a flat prior, concretely a uniform distribution between 0 and 1. However, prior functions are useful in most circumstances since they represent the default risk profile, in which case distributions need to be parametrized.

Because of its versatility in expressing the uncertainty of the default probability, the beta distribution is a widely used parametrized prior distribution. Let “Λ” be the random variable of the default probability “λ”. Assuming “Λ” follows a beta distribution, Λ ~ Beta ( α , β ) , with α , β > 0 and λ ∈ ] 0 , 1 [ , it is simple to adapt this distribution to subjective information about the mean (or average) and variance of default probability using the hyperparameters “α” and “β”. The mean and variance of a beta distribution are respectively:

E ( Λ ) = α / ( α + β ) (11)

V ( Λ ) = α β / [ ( α + β ) 2 ( α + β + 1 ) ] (12)

When no (objective or subjective) information about the default probability is available, a beta distribution with α = β = 1 can be taken because it represents a uniform distribution between 0 and 1, making it a non-informative prior.

The set of prior functions, f ( λ ) , addressed in this document is listed below.

1) Uniform distribution

f ( λ ) = 1 / u ,       λ ∈ ] 0 , u ] , (13)

being “u” the maximum limit of “λ”. Four possibilities of “u” are tested: 1, 0.25, 0.1, and 0.01—the same values evaluated by Dirk Tasche in his work⁵. These values are also utilized in two other types of priors: linear growth and linear decrease, as shown below. A tighter representation should be used because when u = 1 the condition λ < u must be met, not λ ≤ u . This note is also applicable to prior functions with linear growth and linear decrease.

2) Linear growth

f ( λ ) = λ ,       λ ∈ ] 0 , u ] (14)

3) Linear decrease

f ( λ ) = 1 − λ / u ,       λ ∈ ] 0 , u ] (15)

4) Conservative⁶

f ( λ ) = 1 / ( 1 − λ ) ,       λ ∈ ] 0 , 1 [ (16)

5) Immoderate

f ( λ ) = 1 / λ ,       λ ∈ ] 0 , 1 [ (17)

6) Expert judgement

6.1) Base scenario (linear growth and linear decrease)

f ( λ ) = { 2 ( λ − m ) / [ ( M − m ) ( M o − m ) ]     λ ∈ ] m , M o ] 2 ( λ − M ) / [ ( M − m ) ( M o − M ) ] λ ∈ ] M o , M ] 0                                                                                             others λ , (18)

where “m” stands for the minimum of “λ”, “Mo” stands for the mode of “λ”, and “M” stands for the maximum of “λ”, as assumed by the expert who defines the prior distribution. The values for “m”, “Mo”, and “M” were initially set to be conservative: m = 0.01, Mo = 0.025, and M = 0.075⁷.

6.2) Beta distribution as a proxy

For the beta distribution, α = 6.67 and β = 175.31 were used. These two parameters were set to ensure that the beta distribution’s mean and variance matched those of the base scenario, as decided by expert opinion—prior 6.1—, resulting in μ = 0.03667 and σ² = 0.00019, respectively.

6.3) Normal distribution as a proxy

To check that the normal distribution’s mean is equal to the mean of the prior 6.1 and the variance is equal to 0.03667/1.645, one used μ = 0.03667 and σ² = 0.02229 for normal distribution⁸.

7) Beta distribution based on empirical default rate

The prior’s mean is assumed to be the observed default rate for each combination of “n” and “k”, and the prior’s variance is expected to be equal to the ratio between that rate (and thus the mean) and the number 1.645. These mean and variance, on the one hand, and Equation (11) and Equation (12), on the other hand, are used to compute the beta distribution’s parameters⁹. This prior will only be used in Section 3.4 to compare the classical and Bayesian approaches.

1) Bayesian Approach with Reduced Form Models

Let’s use “K” to represent the random variable of the default number once more. For any potential default probability, “λ”, each value “y” of a standard and normally distributed random variable, “Y”, and a specified asset correlation, “ρ”, the posterior probability computation based on the Bayes’ theorem¹⁰ of the occurrence exactly “k” and no more than “k” as for Equation (1) and Equation (2) defaults is as follows:

P ( K = k ) = H ( λ , y ) = f ( λ ) ∫ − ∞ + ∞ ( n k ) λ k ( 1 − λ ) n − k ϕ ( y ) d y     ⋅ 1 / [ ∫ 0 u f ( λ ) ∫ − ∞ + ∞ ( n k ) λ k ( 1 − λ ) n − k ϕ ( y ) d y d λ ] , (19)

if the likelihood function follows a binomial distribution, or

P ( K = k ) = H ( λ , y ) = f ( λ ) ∫ − ∞ + ∞ e − n λ ( n λ ) k / k ! ϕ ( y ) d y   ⋅ 1 / [ ∫ 0 u f ( λ ) ∫ − ∞ + ∞ e − n λ ( n λ ) k / k ! ϕ ( y ) d y d λ ] , (20)

if the likelihood function follows a Poisson distribution.

It’s worth noting that “u” is the upper limit of “λ”¹¹, f ( λ ) is the prior probability density function of “λ”as described in 2.3.1, and ϕ ( y ) is the standard normal probability density function of “Y”.

The H ( λ , y ) denominator, also known as prior predictive distribution, normalizes the posterior distribution function.

Analytical outputs can be generated if no correlation of defaults among borrowers is assumed—reduced form models—, as mentioned in Subsection 2.2.1. Only a few circumstances in Bayesian estimation have an explicit analytical solution: when there is also independence among borrowers, and simultaneously when joining the prior function with the likelihood function yields a standard distribution.

Concretely, analytical forms occur in the following special cases: when the likelihood is a binomial distribution and the prior is a beta distribution, on the one hand, and when the likelihood is a Poisson distribution and the prior is a gamma distribution, on the other hand. The beta-binomial distribution and the gamma-Poisson distribution derive from those joining.

2) Bayesian Approach with Structural Models

It should be noted that the existence of correlation requires numerical outputs from stochastic treatments or simulation procedures. Being ρ > 0, the formulae stated in the preceding subsection must be changed:

P ( K = k ) = H ( λ , y , ρ ) = f ( λ ) ∫ − ∞ + ∞ L ( λ , y , ρ ) ϕ ( y ) d y     ⋅ 1 / [ ∫ 0 u f ( λ ) ∫ − ∞ + ∞ L ( λ , y , ρ ) ϕ ( y ) d y d λ ] , (21)

with

L ( λ , y , ρ ) = ( n k ) [ G ( λ , y , ρ ) ] k [ 1 − G ( λ , y , ρ ) ] n − k (22)

or

L ( λ , y , ρ ) = e − n G ( λ , y , ρ ) [ n G ( λ , y , ρ ) ] k / k ! , (23)

if the likelihood function is represented respectively by a binomial or a Poisson distribution, and G ( λ , y , ρ ) has the meaning expressed in Equation (8).

L ( λ , y , ρ ) denotes the probability of the sample data generated from binomial or Poisson likelihood functions of “λ”, for a risk class with “n” counterparties and “k” defaults. Φ ( . ) stands for the standard normal cumulative distribution function, and Φ − 1 ( λ ) is the inversed standard normal cumulative distribution function for “λ” (see 2.2.2).

The H ( λ , y , ρ ) denominator ensures that the posterior distribution function is normalized once again.

Table 6 and Table 7, on the one hand, and Table 8 and Table 9, on the other hand, show that there are no significant differences between the binomial distribution and the Poisson distribution used as the likelihood function of the Bayesian default probability, with the outcome being roughly the same for both the basic and correlated techniques (similar to the classical approach). Nonetheless, the Poisson distribution’s means are marginally higher than the binomial distribution’s because the Poisson distribution is a little skewer than the binomial distribution²³. Hence the matching percentiles associated with the mean in the binomial distribution are immaterially higher than the equivalent percentiles in the Poisson distribution.

When comparing Table 6 and Table 8 (or Table 7 and Table 9), it is clear that (regardless of “n” and “k”) the larger “ρ”, the greater the likelihood function’s mean and standard deviation. When the asset correlation is introduced, the likelihood function becomes considerably skewer than when no correlation is employed. Furthermore, one verifies the rule that the larger the risk group, the higher the coefficient of skewness and the smaller the mean and the standard deviation²⁴.

The lowering effect on the standard deviation when “n ” grows is explained by the fact that the percentage increase in “n ” is smaller than the modulus of the percentage decrease in “λ”.

There are capital savings with the integration of risk categories, as seen in the classical approach. Savings in the Bayesian context, assuming ρ = 12%, equate to 9% or 18% whether two risk classes (B + C) or three risk classes (A + B + C) are aggregated. For ρ = 0%, corresponding savings increase to 20% or 33%—34% (rather than 33%) regarding that the likelihood function follows a Poisson distribution—if two or three risk classes are aggregated²⁵.

A binomial distribution is suitably better than a Poisson distribution for depicting the number of defaults within a risk class containing a fixed number of obligors, as stated in Section 2.1. As a result, for the sake of simplicity, only outcomes based on the binomial distribution are now reported.

The prior functions discussed in 2.3.1 are used to find a group of statistical values for the posterior distributions: mean, median, mode, standard deviation and coefficient of skewness as well as four highest density intervals. Table 10 displays the outcomes of a 12% asset correlation, while Table 11 in Appendix B shows the results of no correlation²⁶. These tables indicate that different prior functions and asset correlations have a big impact on posterior probabilities²⁷.

Highest density intervals expressed in percentage.

It’s worth noting that, according to the concept of expected value, the mean of the posterior probability of “λ”, μ λ , is computed by:

μ λ = ∫ 0 u λ f ( λ ) ∫ − ∞ + ∞ L ( λ , y , ρ ) ϕ ( y ) d y d λ               ⋅ 1 / ∫ 0 u f ( λ ) ∫ − ∞ + ∞ L ( λ , y , ρ ) ϕ ( y ) d y d λ , (25)

having f ( λ ) , L ( λ , y , ρ ) , and ϕ ( y ) the same meaning as before.

The effect described in the last phrase of the penultimate paragraph of 3.3.1 about the growth of “n” is validated for all priors used: the larger the risk group, the smaller the mean and the standard deviation, and the greater the coefficient of skewness.

When the prior function is a uniform distribution spanning from 0 to 1 (i.e., the same figures as the default probability assumes), the values for posterior probabilities are obviously the same as those provided by the likelihood function. This can be seen by comparing Table 8 and Table 10 likewise Table 6 and Table 11. By definition, default probabilities are tiny; empirical rates for the five scenarios range from 0% to 0.57%. As a result, a uniform distribution with values between 0 and 0.25 is expected to yield identical results (about the likelihood function and the posterior with uniform distribution as a prior). Even if the top limit of the uniform distribution is set to 0.1, there are no discernible differences.

Table 10 shows that the posterior with an immoderate prior—the polar opposite of the conservative prior—produces the lowest default probability, with the mean of 4.3 to 11.9 times lower than the posterior resulting from linear growth as a prior (being “u” constrained between 0 and 1) and between 2.4 and 5 times lower than the posterior resulting from the uniform distribution as a prior (being “u” between 0 and 1 too). Therefore, the most cautious or conservative prior is the linear growth function—the theoretical polar opposite of the linear decrease prior—, not the conservative prior itself.

The uniform distribution, linear decrease (both with “u” between 0 and 1), and conservative priors generate comparable posterior default probabilities for all the pairs of “n” and “k” studied. Those three sorts of functions will appear to be the most appropriate and beneficial priors, in contrast to immoderate and linear growth priors. Comments on expert knowledge priors—base scenario as well as beta and normal distributions as proxie—will be addressed later.

One returns to capital savings through risk group integration. Using the linear growth function with “u” between 0 and 1 as a prior and ρ = 12%, savings are substantially equivalent to savings at a 50% confidence level under the classical approach: 13% for n = 850 and 23% for n = 1000. Furthermore, a 95% confidence level comparison is required when ρ = 0%. This is because, at a 95% confidence level, savings obtained using the linear growth function as a prior are similar to those obtained using the classical approach: 28% for n = 850 and 44% for n = 1000, which are close to the 30% and 45% mentioned in 3.2, respectively.

Because the uniform distribution is a non-informative prior, posterior distribution outputs match to likelihood function outputs as aforementioned; corresponding capital savings have already been presented (in the last paragraph of 3.3.1). Being the means of posterior probabilities identical for uniform distribution, linear decrease, and conservative priors (ranging from 0 to 1), the savings are also equivalent. Immoderate prior yields the smallest savings.

Table 10 and Table 11 show a simple rule: the stronger the asset correlation, the wider the tail of the right side of the distribution (or the higher the coefficient of skewness)²⁸. However, this rule is not established in two instances: when u = 0.01 (and in some cases, when u = 0.1 for linear growth prior), and when expert information is the base of the prior function. The first exception is self-evident, as “λ” has a smaller upper limit—0.01 (or 0.1 is some cases)—than the broad range of values that default probabilities can tolerate (based on the binomial distribution).

The second exemption is explained by the nature of the prior subjected to expert judgment. The previous point concerning the 0.01 threshold also applies to priors based on that judgment. These judgments typically specify a much narrower range of values than the likelihood function’s available values²⁹.

It is important to remember that priors that provide information have a stronger influence on the posterior than priors that do not. Furthermore, as previously stated, a prior based on expert opinion is excessively rigid because the posterior probability density is constrained by the range of values for expert knowledge as a prior.

Table 12 compares the cumulative densities of prior and posterior distributions linked to expert information thresholds to the densities of different kinds of priors and their corresponding posterior distributions. The mean is utilized as a reference point for prior and posterior distributions. In the case of employing the normal distribution as a proxy for the base scenario of expert knowledge, μ = 0.03912 rather than 0.03667 because only positive values less than 1 were used³⁰.

(#) minimum of λ = 0.01, mode of λ = 0.025, and maximum of λ = 0.075.

Figure 1 graphically depicts the pattern of eleven types of default rates as a function of the number of obligors, when k = 2 and ρ = 12%: posterior default probabilities for nine prior functions (UD, LG, LD, C, I, EJ-BS, EJ-BD, EJ-ND, and EJ-ABS), probabilities for the likelihood of the default distribution (BL), and historical default occurrences (EDR). The most important topics of that figure are described below.

When expert information is used as a prior, the general rule that the larger the risk group, the less relevant the prior distribution in defining the posterior distribution, and the stronger the convergence of the posterior distribution to the likelihood function is not obvious. That rule and, in the case of expert judgment as a prior, the exception as well as the trend associated with an alternative base scenario for expert judgment (EJ-ABS) are depicted in the figure.

In other words, when the size of the risk class, “n”, grows, the ratio between posterior probability with prior based on expert opinion and likelihood probability climbs significantly. The likelihood probabilities decrease greatly when “n” rises, which is not the case with the posterior based on expert information as a prior. By contrast, when posterior probability is calculated from any other prior, the ratio remains roughly constant because both posterior and likelihood probabilities are sensitive to “n” in almost equal proportions. Similarly, the ratio of empirical default rate to likelihood probability is quite stable across “n”.

When “k” and “ρ” are fixed, the figure shows that posterior default probabilities tend to empirical rates as the number of obligors grows, as expected by the law of large numbers. There is no such coherence when posterior probabilities are generated from any prior that relies on expert judgment.

From n = 50 to n = 500, with k = 2 and ρ = 12%, the empirical default rate falls by 360 basis points (bp), from 4% to 0.4%. The likelihood decreases by 770 bp, from 9.77% to 2.07%, corresponding to same drop as the posterior with uniform distribution between 0 and 1 as a prior. For posteriors with linear growth, conservative, linear decrease, and immoderate functions, the respective declines are 1,072, 825, 721, and 441 bp. For posteriors with hypothetical scenarios related to 1%/2.5%/7.5% (EJ-BS) and 0.3%/1%/2% (EJ-ABS)—for the minimum/mode/maximum values of the prior default probability—, the decreases are only 101 and 16 bp, respectively.

Finally, Figure 1 shows that the concavity verified in general for posterior functions of default probabilities does not exist when expert prior knowledge is provided, regardless of the values taken by the base scenario as a prior. To demonstrate the (in)flexibility of expert distributions, one notes that coefficients of variation—the mean and variance of these coefficients are measured via the 10 means (one for each n-value³¹)—in two aforesaid expert scenarios are vastly smaller: 0.1 or 0.04 regarding respectively EJ-BS or EJ-ABS. The coefficients of variation for posteriors with other priors are substantially greater, ranging from 0.48 (linear growth) to 0.7 (immoderate).

The prior density of the default probability for EJ-BS has a mean of 3.67% and a standard deviation of 1.39%, whereas EJ-ABS as a prior has a mean of 1.1% and a standard deviation of 0.35%. Although these two scenarios are quite different, they both display the same inflexibility: large risk class sizes are insufficient to ensure a substantial convergence of the posterior distribution to the likelihood function, and thus the prior is stronger than the likelihood. In light of the foregoing concerns, selecting a prior based on expert information should be approached with caution.

The 0.01 and 0.1 thresholds are retaken. The two formers of the four thresholds of “u” used for uniform, linear growth, and linear decrease priors—1, 0.25, 0.1, and 0.01—will be sufficient because they reflect the default probability profile. The others, particularly the latter (0.01), have some difficulties due to differences in probability profiles. The following is a quick rundown of these difficulties.

There are a number of inconsistencies when the upper threshold assumed for “λ” is believed to be low. As previously mentioned (after the analysis of Table 10 and Table 11), the stronger the asset correlation, the higher the coefficient of skewness. However, this rule does not work with u = 0.01 and (for linear growth function as a prior) with u = 0.1.

The default distribution is right-skewed, regardless of the asset correlation percentages that are connected. Nevertheless, the coefficient of skewness might be negative when u = 0.01 (either for ρ = 0% or ρ = 12%).

The higher the asset correlation, the broader the range of the highest density interval; this is, as correlation increases, the distance between the upper and lower bounds of the shortest credible interval growths. At times, such a rule is not verified when u = 0.01. Furthermore, in some circumstances with u = 0.01, that range is wider with ρ = 0% than with ρ = 12%.