Financial institutions that qualify for the internal ratings-based (IRB) approach are allowed to develop their own internal rating models to assess the credit quality of their customers (Basel Committee on Banking Supervision 2004). The Basel Committee on Banking Supervision (BCBS) recently proposed to restrict the development of internal rating models for financial institutions (Basel Committee on Banking Supervision 2016). The portfolio under consideration should be suitable for modelling, and enough historical data should be available. The proposal only allows well-known statistical methods such as logistic regression (Hayden and Porath 2006). These models use borrower data to generate a credit score, which is mapped to a rating.

The rating represents the creditworthiness of the borrower. Before applying ratings in credit processes, financial institutions attach a probability of default (PD) to each rating. This process is referred to as calibration. In our view, there is not much literature on calibration techniques. Stein (2007) remarks that calibration relates to the discriminatory power of a model. Discriminatory power is the ability to discriminate ex ante between defaulting and nondefaulting borrowers (Tasche 2005). Earlier approaches use the relation between discriminatory power and calibration (Falkenstein et al 2000; van der Burgt 2008; Tasche 2010). They demonstrate how the calibration of a rating system can be derived from the cumulative accuracy profile (CAP), which visualizes discriminatory power and is therefore also called a power curve.

The BCBS proposal also states that ratings should remain stable over time. Having unstable ratings leads to large volatility in the PDs, which causes volatility in loan pricing, economic capital and regulatory capital. The rating stability closely relates to the rating philosophy, ie, the behavior of credit ratings during macroeconomic cycles (Araten 2007). In general, two rating philosophies exist: point-in-time (PIT) ratings adjust quickly to changing macroeconomic conditions, whereas through-the-cycle (TTC) ratings tend to remain stable over the length of the macroeconomic business cycle. The rating philosophy depends to a large extent on the input variables of the credit scoring model. PIT rating models use current financial ratios and trends as input, whereas TTC rating models use structural factors such as technology dependence, supplier risk and peer industry scores as input.

This paper assumes a TTC credit scoring model based on widely used statistical techniques and introduces an algorithm for mapping credit scores to ratings and calibrating these ratings. Our algorithm, which builds on the approaches described above, uses an iterative stepwise partitioning of the $x$-axis of the CAP. This partitioning depends on the shape of the CAP; therefore, we call this method “riding the curve”. The algorithm results in stable ratings, as required by the BCBS proposal, and addresses other regulatory requirements regarding the number of rating grades and the concentration of exposures per grade. For example, the European Banking Association (EBA) recently published draft regulatory technical standards, which state that “the number of rating grades and pools is adequate to ensure a meaningful risk differentiation” (European Banking Association 2016). Further, “the concentration of numbers of exposures or obligors is not excessive in any grade or pool, unless supported by convincing empirical evidence of homogeneity of risk of those exposures or obligors”. The algorithm also complies with other requirements, such as the monotonous increase in PD with deteriorating credit rating.

This paper is organized as follows. Section 2 describes the CAP and how it relates to the conditional PD given a credit score. Section 3 introduces requirements for rating mapping and calibration. Section 4 shows how the CAP curve is applied in mapping and calibration by addressing these requirements. Section 5 demonstrates the CAP-based calibration for two credit scoring models. The last section concludes. Appendix A (available online) provides the mathematical background of the mapping algorithm, and Appendix B (also available online) presents closed-form equations for the CAP.

Table 1 provides an overview of the mathematical symbols used in this paper.

This section briefly introduces the CAP, which is central in our mapping and calibration algorithm. An extensive discussion of the CAP is given elsewhere (Falkenstein et al 2000; Engelmann et al 2003; Tasche 2010). We assume a credit scoring model that assigns a credit score $s$ between a minimum score $S_{\min }$ and a maximum score $S_{\max }$ to each counterparty in the credit portfolio. A high credit score corresponds to a low credit risk.

A sound credit scoring system exhibits high discriminatory power. Two curves exist to visualize discriminatory power: the CAP and the receiver operating curve (ROC). Both power curves are constructed by sorting the debtors from low scores to high scores, ie, by decreasing credit risk. The ROC presents the cumulative percentage of defaulting counterparties versus the cumulative percentage of nondefaulting counterparties (Tasche 2005, 2010; Irwin and Irwin 2012). By construction, every point on the ROC gives a measure of type I and type II error (Stein 2007), which makes the ROC eligible for optimal decision thresholds (Irwin and Irwin 2012). Further, models exist for fitting ROCs, based on the underlying score distributions, and the area under the ROC can be interpreted as an unbiased percentage of correct classifications (Irwin and Callaghan 2006; Irwin and Irwin 2012).

The CAP represents the cumulative percentage of defaults versus the cumulative percentage of counterparties. We prefer the CAP for calibration because it allows a derivation of the PD (Stein 2007). To see this, we define $x$ as the cumulative distribution of counterparties with scores lower than $s$,

and $y$ as the cumulative distribution of defaulting counterparties with scores lower than $s$,

Combining (2.1) and (2.2) gives $y=F_{\mathrm{D}}[F^{-1}[x]]$ for the CAP, which has the following property (Falkenstein et al 2000; Tasche 2010):

This property does not hold for the ROC. In (2.3), $P_{\mathrm{c}}(s)$ presents the conditional PD given the credit score $s$,

and $P_{\mathrm{u}}$ presents the unconditional PD, obtained by integrating $P_{\mathrm{c}}(s)$ over the score distribution density $\mathrm{d}F[s]/\mathrm{d}s$:

In practice, $P_{\mathrm{u}}$ is a sample default rate, which is estimated as the total number of defaults $D_{\mathrm{T}}$ divided by the total number of counterparties $N_{\mathrm{T}}$ in the portfolio. When the sample default rate deviates significantly from the “true” population default rate, (2.3) leads to an erroneous calibration. Since the sample default rate converges to the population default rate in the limit $N_{\mathrm{T}}\to \mathrm{\infty }$, this risk is limited when (2.3) is applied only to large portfolios such as retail portfolios. We will consider the consequences of population versus sample default rate in the algorithm description.

Figures 1 and 2 demonstrate the CAP and (2.3) for several credit scoring models. A credit scoring model with moderate discriminatory power has a CAP with a concave shape, and the conditional PD decreases gradually with increasing $x$. For highly discriminatory models, only defaults occur at the worst credit scores. In this case, the CAP curve rises steeply for small $x$ and the PD curve decreases fast with increasing $x$, as shown by the black curves in the figures. The black dashed lines in the figures correspond to a model with no discriminatory power. In this case, the cumulative percentage of defaults increases proportionally with the increasing number of counterparties, and the CAP resembles the line $y=x$. The CAP is an almost linear function, and its derivative is independent of $x$. The corresponding PD curve is flat and equals $P_{\mathrm{u}}$ for all counterparties, as shown Figure 2.

Since we investigate the mapping of credit scores in relation to their discriminatory power, we also introduce the accuracy ratio for credit scoring (${\mathrm{AR}}_{\mathrm{S}}$). This is defined as (Tasche 2005, 2010):

where $A_{\mathrm{S}}$ is the area under the CAP:

with $x(0)=0$ and $y(0)=0$. The ${\mathrm{AR}}_{\mathrm{S}}$ varies between 0 for nondiscriminatory scoring models and 1 for perfect discriminatory scoring models.

After mapping scores to ratings, a counterparty has a rating $r$ when their credit score $S$ is larger than $s_{r-1}$ and smaller than or equal to $s_r$. In this paper, the rating $r$ is an integer, which increases with decreasing credit risk. We define $x_r$ as the cumulative distribution of counterparties with a rating worse than $r$,

and $y_r$ as the cumulative distribution of defaults with a rating worse than $r$,

We derive the PD per rating grade from the CAP in a similar way to (2.3) for credit scores. Bayes’s law gives the following equation for the PD, conditional on rating $r$:

Using the definitions $x_r=P[R\le r]$ and $y_r=P[R\le r\mid D]$ gives

Equation (2.11) expresses the PD in CAP coordinates $x$ and $y$. It is a discrete version of (2.3), since credit scores vary continuously in the interval $[S_{\min },S_{\max }]$, while ratings only have discrete values $r=1,2,\mathrm{\dots },R_{\mathrm{tot}}$, where $R_{\mathrm{tot}}$ is the total number of rating grades.

The mapping of credit scores to ratings reduces information, because ratings have a lower granularity than credit scores. To investigate this information loss, we calculate the area under the CAP in terms of ratings as

with $x_0=0$ and $y_0=0$. We use this result to calculate an accuracy ratio ${\mathrm{AR}}_{\mathrm{R}}$ for ratings, similar to in (2.6), and define information loss as

with $A_0=\frac{1}{2}$ as the area under the CAP for a nondiscriminatory scoring system. The IL compares the information loss $A_{\mathrm{S}}-A_{\mathrm{R}}$ with the maximum possible information loss $A_{\mathrm{S}}-A_0$. This gives two extreme cases.

• •

  When ${\mathrm{AR}}_{\mathrm{S}}\to 0$, the credit scoring model has barely any discriminatory power: no significantly different default risk exists between all of the scores, which are mapped to a single rating grade. In this case, $R_{\mathrm{tot}}=1$, leading to $A_{\mathrm{R}}=\frac{1}{2}$ by (2.12) and $\mathrm{IL}=1$ by (2.13).

• •

  When the credit scoring model has high discriminatory power, the credit scores are mapped to a high number of ratings and the information loss converges to zero.

We will use the information loss in testing the mapping algorithm.

The draft regulatory technical standards of the EBA state the following (European Banking Association 2016):

1. (1)

   the number of rating grades and pools is adequate to ensure a meaningful risk differentiation; and

2. (2)

   the concentration of numbers of exposures or obligors is not excessive in any grade or pool, unless supported by convincing empirical evidence of homogeneity of risk of those exposures or obligors.

Based on this, we introduce two calibration requirements.

This section describes how to find a score interval $[s_{r-1},s_r]$ for every rating $r$ by using the CAP, such that the requirements of the previous section are fulfilled. Equation (3.7) allows a mapping of credit scores to ratings by an iterative procedure, which starts at $x=0$ and proceeds stepwise toward $x=1$, depending on how fast the CAP increases with increasing $x$. Therefore, we call this algorithm “riding the curve”. Setting $x_r$ equal to $x_{\mathrm{c}}$ in every iteration step results in an interval $[x_{r-1},x_r]$ for every rating grade. Since every $x_r$ corresponds uniquely to a score $s_r$ by (2.8), we can translate the interval $[x_{r-1},x_r]$ into a score interval $[s_{r-1},s_r]$ per rating grade. However, we need to consider three issues.

First, the procedure requires a function $y=C(x)$ for the CAP to calculate $\lambda$. Several authors have introduced functions for the CAP (van der Burgt 2008; Tasche 2010). If we know the score distributions of the counterparties and defaulting counterparties, we can derive a function $y=F_{\mathrm{D}}[F^{-1}[x]]$ from these distributions (Tasche 2010). Since these distributions are unknown in general, we use a regression approach by fitting the CAP to a multi-exponential function $C(x)$:

with ${\sum }_{j=1}^m{B}_j=1$ and

Due to the first constraint, the number of parameters is $2m-1$. The second constraint guarantees that $P_{\mathrm{c}}(x)$ cannot be larger than 1, as shown in Appendix B online. Equation (4.1) is based on the following assumptions.

• •

  The error terms $\epsilon$ have means equal to zero and are independent: $E[\epsilon ]=0$ and $E[\epsilon \times {\epsilon }^{\prime }]=0$. Dependency among the error terms does not impact the parameters themselves, but it may increase the standard errors in the parameters.

• •

  The probability $P_{\mathrm{c}}(x)$ is a multi-exponential function, as shown in Appendix B online.

The error terms are kept small by selecting enough exponential functions ($m$) in $C(x)$. Based on experience, $m=2$ already gives a good fit to most CAPs. We use $C(x)$ in (3.6) and (3.7) to find $x_r=x_{\mathrm{c}}$. As shown in Appendix A online, (3.7) is based on second-order Taylor approximations of $C(x)$. Therefore, we also use (3.2) for testing $T(x_r\mid x_{r-1},x_{r-2})\ge L_{\mathrm{D}}$.

A second issue is the determination of the first interval $[x_0,x_1]$. We set $x_0=0$, but (3.7) cannot be applied here to find $x_1$. The setting of interval $[x_0,x_1]$ is also critical: when this interval is small, the next interval $[x_1,x_2]$, as calculated by (3.7), will be large. This may give rise to periodic partitioning. Appendix A online shows that the parabolic behavior of the test statistics $T(x_r\mid x_{r-1},x_{r-2})$ around $x_r$ may result in subsequent small and large intervals of $x$. To avoid this, we set $x_0=0$ and apply the equidistant interval of (3.9) for $r=1$:

with $\lambda$ calculated as $\lambda =P_{\mathrm{u}}{N}_{\mathrm{T}}({\{C^{\prime \prime }(0)\}}^2/4)C^{\prime }(0)$.

The third aspect is the dependence on the sample default probability $P_{\mathrm{u}}$. When the sample default probability is 0.75% while the “true” population default probability is 1.25%, the parameter $\lambda$ is “too small” and results in a widening of the intervals $[x_{r-1},x_r]$ according to (3.7) and (3.9). The rating grades still represent significantly different rating grades, but the probability $P_r$ per rating $r$ is underestimated by (2.11). This reverses when the sample default rate is higher than the population default rate: the default probabilities are overestimated and the intervals $[x_{r-1},x_r]$ are too small, resulting in rating grades that are not significantly different in default risk. The “true” population default rate is not known, but we can construct a confidence interval $P_{\mathrm{u}}\pm 1.96\sqrt{P_{\mathrm{u}}(1-P_{\mathrm{u}})/N_{\mathrm{T}}}$ that contains the population default rate at a 95% confidence level, assuming independence in default behavior. When $N_{\mathrm{T}}$ is large enough, the population default rate will not substantially differ from $P_{\mathrm{u}}$. This makes the calibration algorithm eligible for retail portfolios, which consist of a high number of counterparties.

Based on the above considerations, the algorithm for mapping and calibration proceeds as follows.

1. (1)

   Construct the CAP curve as described in Section 2 and fit the CAP to a closed-form function $C(x)$.

2. (2)

   Find the first interval $[x_0,x_1]$ for rating $r=1$, which is the worst credit rating grade, by setting $x_0=0$ and calculating $x_1$ by (4.2).

3. (3)

   Given the interval $[x_{r-2},x_{r-1}]$, set $x_r=x_{\mathrm{c}}$ with $x_{\mathrm{c}}$ as calculated by (3.7). Eventually, increase $x_r$ until $T(x_r\mid x_{r-1},x_{r-2})\ge L_{\mathrm{D}}$. If $x_r$ is larger than 1, set $x_r=1$.

4. (4)

   Step (3) is repeated until $x_r=1$. Then, the total number of rating grades $R_{\mathrm{tot}}$ equals $r$. Because $x_r$ is capped at 1 in Step (3), the highest rating grade and the second-to-highest rating grade may not exhibit a significant difference in default risk. In that case, these rating grades are merged together into one rating grade.

5. (5)

   Given $x_r$, find $s_r$ for every $r=1,\mathrm{\dots },R_{\mathrm{tot}}$ so that every interval $[x_{r-1},x_r]$ is translated into $[s_{r-1},s_r]$. The PD is $P_r=((y_r-y_{r-1})/(x_r-x_{r-1}))P_{\mathrm{u}}$ for every $r=1,\mathrm{\dots },R_{\mathrm{tot}}$.

We investigated whether the algorithm maps scores to ratings with the two requirements fulfilled and how many rating grades ($R_{\mathrm{tot}}$) result. We set the significance level as $\alpha =5\%$, giving ${\mathrm{\Phi }}^{-1}[1-\alpha ]=1.65$ in requirement 1. Using a margin, we rounded this value to 2, giving $L_{\mathrm{D}}=2$ in (3.4). We use two credit scoring models to test the algorithm.

1. (1)

   In scoring model 1, the credit scores vary uniformly between $S_{\min }=0$ and $S_{\max }=100$, and the conditional PD $P_{\mathrm{c}}(s)$ depends exponentially on the score $s$. As shown in Appendix B online, this gives the following conditional PD:

   $$P_{\mathrm{c}}(s)=\frac{\beta P_{\mathrm{u}}\exp \{-\beta (S-S_{\min })\}}{1-\exp \{-\beta (S_{\max }-S_{\min })\}}.$$

   (5.1)

   The parameter $\beta$ is positive, ie, a high score means low credit risk. Under these conditions, we can derive a closed-form equation for the CAP that corresponds to $m=1$ in (4.1).

2. (2)

   Scoring model 2 assumes a logit function for the conditional PD $P_{\mathrm{c}}(s)$:

   $$P_{\mathrm{c}}(s)=\frac{a_1}{1+\exp \{\beta (S-a_2)\}},$$

   (5.2)

   with $a_2=80$. The credit scores $s$ are normally distributed:

   $$s\sim \varphi [S_0,{\sigma }^2],$$

   (5.3)

   with $S_0=100$ and $\sigma =10$. Similar to scoring model 1, $\beta >0$ and a high score means low credit risk. We cannot derive a closed-form equation for the CAP under these circumstances, but using a bi-exponential form for the CAP that corresponds to $m=2$ in (4.1) results in a good fit.

In both cases, parameters $\beta$ are changed to vary the ${\mathrm{AR}}_{\mathrm{S}}$. The default/nondefault state of a counterparty $i$ with score $s$ is simulated by a Bernoulli variable $B[i;P_{\mathrm{c}}(s)]$, which equals 1 (default) with probability $P_{\mathrm{c}}(s)$ and 0 (nondefault) with probability $1-P_{\mathrm{c}}(s)$. After running the simulations for $N_{\mathrm{T}}$ counterparties, the CAP of scoring model 1 was regressed to (4.1) with $m=1$. The CAP of scoring model 2 was regressed to (4.1) with $m=2$. The regression quality is high: in all cases, the adjusted $R^2$ is larger than 99.8%.¹¹ 1 The adjusted $R^2$ is calculated as $R_{\mathrm{adjusted}}^2=1-((N_{\mathrm{T}}-1)/(N_{\mathrm{T}}-(2m-1)))(1-R^2)$ since the number of parameters is $2m-1$.

Figures 4–6 present the mapping and calibration results of scoring model 1 with ${\mathrm{AR}}_{\mathrm{S}}=18\%$, ${\mathrm{AR}}_{\mathrm{S}}=56\%$ and ${\mathrm{AR}}_{\mathrm{S}}=91\%$, respectively. The number of counterparties is $N_{\mathrm{T}}=\mathrm{100\hspace{0.17em}000}$ and the unconditional PD is $P_{\mathrm{u}}=1\%$ in all three cases.

The dashed lines split the CAP into bands that correspond to a rating: a small (large) bandwidth means low (high) concentration in the rating grade. Each interval $[x_{r-1},x_r]$ corresponds to a score interval $[s_{r-1},s_r]$, which is tabulated in Table 3. The table demonstrates that the default rate monotonously increases with the rating. Table 3 also presents the critical value $x_{\mathrm{c}}$ as calculated by (3.7). In most cases, $x_r=x_{\mathrm{c}}$; $x_r$ exceeds the critical value in a few cases due to the following reasons.

1. (1)

   The critical value $x_{\mathrm{c}}$ is based on an approximation of $T(x_r\mid x_{r-1},x_{r-2})$.

2. (2)

   In the highest rating, $x_r>x_{\mathrm{c}}$ due to Step (4) in the algorithm: when $x_r$ is larger than 1, the rating grade $r$ is merged with rating $r-1$ into a single rating grade.

The last column presents the $p$-value of the ARG tests. Since the $p$-values are all below 5%, the rating grades differ significantly in default rate.

Figure 4 shows that the algorithm maps credit scores to only four ratings for a credit scoring system with low discriminatory power. The PDs in each rating grade are small and close to the unconditional PD of 1%. Rating grade 4 exhibits a large concentration of 47%.

Figure 5 presents the ratings and corresponding PDs for a credit scoring system with considerable discriminatory power. The algorithm produces ten ratings and there is a substantial difference between the PDs per grade. The CAP in Figure 5 also shows no excessive concentrations: the largest concentration is 21%, occurring in the rating grade with the lowest risk.

Figure 6 demonstrates the results of the algorithm for a credit scoring system with an ${\mathrm{AR}}_{\mathrm{S}}$ close to 1, ie, the discriminatory power is extremely high. As expected, the algorithm only distinguishes significantly different rating grades at small $x$-values. The algorithm produces narrow bands in this range and the rating corresponding to the lowest default risk has an excessive concentration of 68%. As explained in Section 2, the mapping of a highly discriminatory credit scoring model leads to a trade-off between avoiding excessive concentrations and identifying significantly different rating grades.

An important question is how many rating grades result from the algorithm. Equation (3.6) shows that $\lambda$ relates to the concave shape of the CAP, the number of counterparties and the unconditional PD. When the CAP is more concave, the ${\mathrm{AR}}_{\mathrm{S}}$ is higher, and more significantly different rating grades result. Figures 4–6 also support this. To identify the relation between $R_{\mathrm{tot}}$ and ${\mathrm{AR}}_{\mathrm{S}}$, we examined the following cases.

First, we use a theoretical case in which $C(x)$ equals (4.1) with $m=1$, giving the following $\lambda$:

We calculate the number of rating grades ($R_{\mathrm{tot}}$) theoretically by iteration until $x_r\ge 1$:

for several values of $k_1$. The accuracy ratio follows from

Second, we investigated the relation between $R_{\mathrm{tot}}$ and the ${\mathrm{AR}}_{\mathrm{S}}$ by simulating scoring model 1 and scoring model 2 for 100 000 counterparties with different parameters $\beta$ and $a_1$, such that the ${\mathrm{AR}}_{\text{S}}$ varied between 7% and 96%, while the unconditional PD $P_{\mathrm{u}}$ equals 1%.

Figure 7 presents the number of rating grades $R_{\mathrm{tot}}$ versus ${\mathrm{AR}}_{\text{S}}$ for scoring model 1 and scoring model 2. As expected, the figure shows that the number of rating grades increases with the accuracy ratio. The dashed line in Figure 7 presents the $R_{\mathrm{tot}}$, derived by iterating (5.5), versus the ${\mathrm{AR}}_{\text{S}}$ as calculated by (5.6). The observations of model system 1 and model system 2 resemble the theoretically calculated dashed line.

A direct relation between ${\mathrm{AR}}_{\text{S}}$ and $R_{\mathrm{tot}}$ follows from the relation between the ${\mathrm{AR}}_{\text{S}}$ and the concave shape of the CAP: $C^{\prime \prime }(x)$ is more negative when the ${\mathrm{AR}}_{\text{S}}$ is high. Therefore, $\lambda$ depends on the ${\mathrm{AR}}_{\text{S}}$ by

Substituting this dependency in (3.8) and (3.9) gives

A regression shows that $R_{\mathrm{tot}}\sim {\{{\mathrm{AR}}_{\mathrm{S}}\}}^{0.71}$ for scoring model 1, $R_{\mathrm{tot}}\sim {\{{\mathrm{AR}}_{\mathrm{S}}\}}^{0.71}$ for scoring model 2 and $R_{\mathrm{tot}}\sim {\{{\mathrm{AR}}_{\mathrm{S}}\}}^{0.63}$ for the theoretical curve. This means that all graphs closely resemble (5.7), although Figure 6 shows that the mapping algorithm does not always result in a homogeneous distribution over the ratings. Therefore, (5.7) is only a rule-of-thumb to predict how many rating grades can be distinguished for a given accuracy ratio.

We investigated (5.7) further by performing simulations of both scoring models with different unconditional PDs. Figure 8 presents $R_{\mathrm{tot}}$ versus $P_{\mathrm{u}}$. A regression shows that $R_{\mathrm{tot}}\sim {\{P_{\mathrm{u}}\}}^{0.40}$ for scoring model 1 and $R_{\mathrm{tot}}\sim {\{P_{\mathrm{u}}\}}^{0.38}$ for scoring model 2. The exponentials are close to 0.33, as predicted by (5.7). We conclude that this equation explains to a large extent the power law behavior of $R_{\mathrm{tot}}$ as a function of ${\mathrm{AR}}_{\mathrm{S}}$ and $P_{\mathrm{u}}$.

We also investigated how much discriminatory power is lost by mapping scores to ratings, using the information loss measure as defined in (2.13). Figure 9 reveals that the information loss decreases exponentially with increasing ${\mathrm{AR}}_{\mathrm{S}}$. Therefore, the information loss for both scoring models is regressed against the following equation:

Equation (5.8) has no intercept since the information loss converges to $1$ when ${\mathrm{AR}}_{\mathrm{S}}\to 0$, as explained in Section 2.

Table 4 presents the regression coefficients and the $R_{\mathrm{adjusted}}^2$. The coefficients of both scoring models agree within a 95% confidence interval. Therefore, we conclude that the information loss as a function of the ${\mathrm{AR}}_{\mathrm{S}}$ is similar for both scoring models, although the $R_{\mathrm{adjusted}}^2$ shows that the quality of the fit for scoring model 1 is considerably better than for scoring model 2. Further, the information loss is low for credit scoring models with moderate to high discriminatory power: when the ${\mathrm{AR}}_{\mathrm{S}}$ is higher than 60%, the information loss is lower than 1%.

Since the algorithm depends on the population default rate and discriminatory power, we conclude with two considerations. First, both quantities are inferred from the data. Obviously, data flaws, such as assigning the information of a defaulted counterparty to a nondefault status, directly impact the estimation of $P_{\mathrm{u}}$ and therefore $P_r$, but they also lead to an underestimated accuracy ratio (Stein 2016; Russell et al 2012). This causes a widening of the intervals $x_r-x_{r-1}$. The rating grades still represent significantly different default risks, but (5.7) shows that the possible number of rating grades is underestimated, and the information loss increases according to (5.8) and Figure 9. Stein (2016) suggests procedures for estimating the number of mislabeled defaults and correcting the power statistics for these errors.

Second, the population default rate and accuracy ratio are related via the default definition. For example, when the default definition changes from payments ninety days in arrears to payments ten days in arrears, relatively solvent counterparties also count as default. The population default rate increases but the accuracy ratio may decrease. Equation (5.7) shows that the accuracy ratio has more of an impact than the $P_{\mathrm{u}}$, and the number of rating grades will be lower in case of a stricter default definition. The decrease in the accuracy ratio also leads to an increase in information loss according to (5.8) and Figure 9. Therefore, successful application of the algorithm also depends on the data quality and a proper default definition.

This paper presents a method for mapping credit scores to ratings and calibrating these ratings by assigning a PD per rating grade. The algorithm is based on two requirements: (1) monotonicity and stability of the PD per rating by enforcing significantly different default risks between adjacent rating grades; and (2) avoiding excessive concentrations. Our algorithm uses stepwise partitioning of the $x$-axis of the CAP.

We tested the algorithm by simulating two scoring models. The algorithm maps credit scores to significantly different ratings. The algorithm shows that rating calibration is a trade-off between optimizing discriminatory power, avoiding concentrations in rating grades, and calibration quality in terms of significantly different rating grades. Highly discriminatory credit scoring models lead to an excessive concentration in the low-risk rating grade. High concentrations are avoided by accepting a lower significance level in defining different ratings or designing credit scoring models with moderate rather than high discriminatory power. The algorithm maps credit scores to ratings, but a small amount of information is lost, as ratings are less granular than credit scores.

The simulations of the two different scoring models reveal that the number of rating grades, which result from the algorithm, exhibit a power law behavior as a function of the accuracy ratio of the credit scores and the unconditional PD. The information loss decreases with an increasing accuracy ratio of the credit scoring model. This power law behavior and the information loss is similar for both scoring models, although these scoring models are based on different score distributions and conditional PDs. Our overall conclusion is that the algorithm is most sensitive to the discriminatory power of the scoring model, but less dependent on the underlying score distribution. Further, the algorithm outcomes are sensitive to the data quality by the “garbage in, garbage out” principle as well as to the default definition.

The views expressed in this paper are those of the author and do not necessarily reflect those of Nationale Nederlanden Group. The author would like to thank the anonymous referees for their suggestions.