Inferred Loss Rate as a Credit Risk Measure in the Bulgarian Banking System ^†

Abstract

The loss rate of a bank’s portfolio traditionally measures what portion of the exposure is lost in the case of a default. To overcome the difficulties involved in its computation due to, e.g., the lack of private data, one can utilize an inferred loss rate (ILR). In the existing literature, it has been demonstrated that this indicator has sufficiently close properties to the actual loss rate to facilitate capital adequacy analysis. The current study provides complete mathematical proof of an earlier-stated conjecture, that ILR can be instrumental in identifying a conservative upper bound of the capital adequacy requirement of a bank credit portfolio, using the law of large numbers and other techniques from measure-theory-based probability. The assumptions required in this proof are less restrictive, reflecting a more realistic view. In the current study, additional empirical evidence of the usefulness of the indicator is provided, using publicly available data from the Bulgarian National Bank. Despite the definite conservativeness of the capital buffer implied from the analysis of ILR, the empirical analysis suggests that it is still within the regulatory limits. Analyzing ILR together with the Inferred Rate of Default, we conclude that the indicator provides signals about a bank portfolio’s credit risk that are relevant, timely, and adequately inexpensive.

1. Introduction

The International Financial Reporting Standard IFRS 9 and the ensuing changes in Bulgaria’s accounting rules have mandated that all banks conduct regular forecasts of expected credit loss (ECL) for credit portfolios. A key parameter for this estimation is the loss-given-default ratio, LGD, which measures what proportion of the exposure’s value will be lost in the event of a default.

In addition, estimating the LGD ratio is essential in navigating the regulatory requirements because it is instrumental in verifying the sufficiency of capital buffers for meeting unexpected credit losses. For larger banks, particularly those applying the internal-rating-based approach to credit risk estimation, this challenge was already familiar and they were well-equipped to handle it. Following the introduction of BASEL II and, later, BASEL III, the ample regulatory literature supported them in developing and applying best practices in this matter (cf., e.g., [1,2]).

In addition, several fundamental studies, including [3,4], lay out the theoretical framework for loss rate management under the value-at-risk framework, which is hereinbelow briefly described.

Let ${\alpha }_q\left(X\right)$ denote the q-th quantile of a given random variable X:

$${\alpha }_q\left(X\right)=inf\{z:P(X\le z)\ge q\}.$$

 Definition 1. 

(i) The loss rate of a loan i over a given time interval (usually 12 months) is a random variable

$$U_i=\left\{\begin{array}{cc}0\hfill & \mathrm{if} \mathrm{the} \mathrm{loan} i \mathrm{survives} \mathrm{the} \mathrm{whole} \mathrm{period}\hfill \\ LG{D}_i\hfill & \mathrm{in} \mathrm{the} \mathrm{event} \mathrm{of} \mathrm{default} \mathrm{in} \mathrm{the} \mathrm{period}\hfill \end{array}\right.$$

 (ii) 

The loss rate of an n-loan portfolio, L, is the value-weighted average of the loss rates $U_i$ of all participating exposures $i=1,\cdots ,n$;

 (iii) 

The q-th value-at-risk, ${VaR}_q$, of a bank’s credit portfolio is the q-th quantile of the loss rate:

$${VaR}_q={\alpha }_q\left(L\right).$$

The currently applied regulatory regime is based on credit ratings. A bank, aiming to preserve its current credit rating, has a target probability of survival q, e.g., 95, 99, 99.9%, etc. Hence, the bank needs to maintain the capital adequacy level of (i.e., to demonstrate the capacity to finance with equity) at least the q-th value-at-risk, ${VaR}_q$.

A review of the shortcomings of this policy and a critical analysis of the properties of VaR as a risk indicator in comparison with the expected shortfall can be found in [5,6].

Following this guidance, nevertheless, studies often implement Merton-type asymptotic factor models (cf. [7,8]). In such models, the value of a specific obligor is assumed to follow a geometric random walk and is, hence, associated with a latent random variable $V_i$ with standard normal distribution. Furthermore, $V_i$ is decomposed as

$$V_i=w_{i}X+{\theta }_i{ϵ}_i.$$

on a single systemic risk factor X and an ${ϵ}_i$ representing the idiosyncratic risk. All ${ϵ}_i$ and X are independent, identically distributed (iid) with $N(0,1)$.

As demonstrated in [9,10] under certain conditions, including a fine granularity of the portfolio, ${VaR}_q$ can be asymptotically approximated with the q-th quantile of the expected loss rate, ${\alpha }_q\left(E\left[L\right|X]\right)$. Furthermore, under additional assumptions of the monotonicity of the relation between the loss rate and the macroeconomic factor X, ${VaR}_q$ is approximated by the loss rate’s expectation $E\left(L\right|X={\alpha }_q\left(X\right))$, conditional on X assuming its q-th quantile.

In attempting to follow such a process, smaller banks experience significant costs due to the complexity of the requirements and insufficient or lower-quality data. For example, measuring realized LGD is challenging for smaller institutions because of scarce historical data on losses. Banks utilize various estimation techniques to fill the data gaps in such cases, e.g., superimposing time intervals, as suggested by [11]. Further discussion on empirical difficulties in validating LGD can be found in [12,13].

Empirically, as an indicator, expected inferred loss rate, ILR, is computed on a portfolio-level compounding previously made estimates of LGD (potentially on individual exposure level). Similarly, the validity of the LGD estimate, as proposed in [11], is a statement of collective, rather than individual accuracy. The study of ILR, as a proxy of the actual loss rate of a portfolio, was partly motivated by the possibility of developing a methodology for the forecasting and cross-validation of portfolios’ LGD estimates that would circumvent the difficulty in measuring realized LGDs.

Other studies of risks, focusing on the Bulgarian banks, including [14,15,16], concentrate on the idiosyncratic characteristics of the local environment.

In contrast, the ILR is designed to study the systemic effect. It was introduced in [17], hypothesizing that it can provide a conservative measure of the actual loss rate. Because of its asymptotical nature, this study ignores the idiosyncratic characteristics usually analyzed in individual credit risk estimation. Regarding the individual behavior of the debtors, such risks are acknowledged in two ways:

• No assumption is made about individual losses being identically distributed. Because of the infinitely fine granularity of the portfolio, the effect of an individual debtor’s behavior is infinitesimal and shown to be inconsequential. If the collective behavior of the bank’s clients affects the loss rate, this is considered a part of the systemic risk factor;
• Assumptions 5 and 6 in Section 2 require only a partial continuity of the relation between the macro-factor X and the probability distribution of expected ILR. Generally, this functional dependence may have some jumps and flat sections due to individual defaults.

The construction of ILR utilizes the stages of the impairment of exposures introduced by IFRS 9:

• Stage 1 designates all loans that perform within expectations for their originally assigned risk class. For such loans, ECL is computed over a 12-month horizon;
• Stage 2 classifies loans that have exhibited significant increases in their risk. Usually, this is a narrower class of loans; for them, ECL is computed over the loan’s lifetime;
• Stage 3 is the group of already defaulted loans. The loss for these loans is considered certain.

Assume that the bank has an infinite pool of credit exposures $\{E_i:i=1,2,\dots \}$ that is structured in a filtration of portfolios.

$${\mathcal{P}}_n=\{E_1,E_2,\dots ,E_n\}\subset {\mathcal{P}}_{n+1}\subset \cdots .$$

Let $a_i$ denote the exposure-at-default, EAD, of the i-th loan, i.e., the estimated amount on the loan when a default occurs. These are considered known deterministic values. For a portfolio of n exposures, we denote its EAD by

$$A_n=\sum _{i=1}^n{a}_i.$$

 Definition 2. 

(i) The inferred loss-rate of the i-th exposure is a random variable $W_i$, which depends on the stage of the exposure:

• Stage 1: $W_i=U_i$, the loss rate over 12-month horizon;
• Stage 2:

  $$W_i=\left\{\begin{array}{cc}0\hfill & \mathrm{if} \mathrm{the} \mathrm{loan} i \mathrm{survives} \mathrm{the} \mathrm{whole} \mathrm{term} \mathrm{of} \mathrm{the} \mathrm{loan}\hfill \\ LG{D}_i\hfill & \mathrm{in} \mathrm{the} \mathrm{event} \mathrm{of} \mathrm{default} \mathrm{the} \mathit{lifetime} \mathit{of} \mathit{the} \mathit{exposure}\hfill \end{array}\right.$$
• Stage 3: $W_i=LG{D}_i$;

 (ii) 

The random variable

$${\mathit{Y}}_n={\displaystyle \frac{{\sum }_{i=1}^n{a}_i{W}_i}{A_n}}$$

is the inferred loss rate of the portfolio ${\mathcal{P}}_n$.

Without any loss of generality, we assume that all $U_i$ and $W_i$ are random variables, defined on the same probability space. We do not assume, however, that they are identically distributed.

For convenience, denote, by $r_i\left(X\right)=E\left[W_i\right|X]$ and by

$$R_n\left(X\right)=E\left[{\mathit{Y}}_n\right|X]={\displaystyle \frac{{\sum }_{i=1}^n{a}_i{r}_i\left(X\right)}{A_n}}$$

(1)

the condition on X means of the inferred loss on exposure and portfolio level, respectively.

My main results are Theorems 1 and 2.

 Theorem 1. 

Under Assumptions 1–5, as stated in Section 2, for sufficiently large n and for an arbitrary $ϵ>0$, we have the following:

 (i) 

A capital reserve in the amount $c_i=r_i\left({\alpha }_q\left(X\right)\right)+ϵ$ per currency unit for each exposure $E_i$ is sufficient to cover the value at risk at the level of the probability of survival q for a portfolio ${\mathcal{P}}_n$ of sufficiently large numerosity n;

 (ii) 

The respective capital reserve proportion for the whole portfolio, ${\mathcal{P}}_n$, is

$${\mathcal{C}}_n=R_n\left({\alpha }_q\left(X\right)\right)+ϵ={\displaystyle \frac{{\sum }_{i=1}^n{a}_i{c}_i}{A_n}}+ϵ.$$

In [17], Theorem 1 was stated as a proposition and the proof was sketched under rather strict assumptions about continuity. In Section 2, a complete proof of this fact is given under a less restrictive, more realistic, set of assumptions.

To further demonstrate the similarity between ILR and the usual loss rate, my goal is to use $R_n\left({\alpha }_q\left(X\right)\right)$ as an approximation of ${VaR}_q$. By definition, the inferred loss is both a more frequent and a costlier event than the actual insolvency during the 12-month horizon. Hence, the capital adequacy requirements obtained in this way will be more conservative than the usual rating-based capital rules. Section 2 provides a proof that, under certain conditions of regularity, asymptotically, $R_n\left({\alpha }_q\left(X\right)\right)$ equals ${\alpha }_q\left({\mathit{Y}}_n\right)$ in the following sense.

 Theorem 2. 

Under Assumptions 1–6, we have the following:

$$\begin{array}{c}\underset{n\to \infty }{lim}P({\mathit{Y}}_n\le R_n\left({\alpha }_q\left(X\right)\right))=q\end{array}$$

(2)

$$\begin{array}{c}\underset{n\to \infty }{lim}{\alpha }_q\left({\mathit{Y}}_n\right)-R_n\left({\alpha }_q\left(X\right)\right)=0\end{array}$$

(3)

My results closely parallel the development of techniques for studying the loss rate in [9,10,18].

As demonstrated in [17], the expected ILR can be computed using publicly available data reported regularly to the regulator.

The Bulgarian regulator, Bulgarian National Bank, BNB, discloses quarterly information on the quality of loans by bank groups and by the type of debtor. This report reveals the share of performing and non-performing loans in the cumulative exposure in the banking system. The total size of ECL is also disclosed. Let $EC{L}_n\left(x\right)$ denote the conditional expected credit loss for a portfolio of size n, conditional on $X=x$. One can compute the expected inferred loss rate, as defined by (1), based on these data, by setting

$$R_n\left(x\right)={\displaystyle \frac{EC{L}_n\left(x\right)}{A_n}}.$$

(4)

This technique is used in [17] to provide empirical evidence of the usefulness of the ILR indicator. The main finding is that the capital adequacy threshold, computed using the expected ILR, is well within the regulatory requirement, alleviating some of the concerns about the excessive conservativeness of the indicator. This computation is performed for the cumulative portfolios of the whole banking system and Group 2, consisting of the smaller banks in the country.

Since then, 11 additional quarterly data points have become available for analysis. In Section 3, the older empirical results are confirmed with the new data and extended to Group 1, comprising the top-five banks, allowing a more explicit comparison of the risk profiles of the two bank groups.

Section 3.3 provides an analysis of the dependence of the ILR on macroeconomic factors, confirming empirically that, for corporate loans, the ILR is a decreasing function of gross domestic product. The ILR for retail loans exhibits an increasing behavior on the unemployment rate.

To summarize, the focus of the current study is twofold: first, to provide a complete mathematical justification of the capacity of ILR to estimate unexpected credit loss; second, to empirically provide evidence of the validity of the assumptions made for the proofs and to investigate the extent of the inherent conservative bias of the estimation of LGD, using ILR.

The current study does not discuss the construction of reasonable forecasting models for ILR. This task would require a more careful analysis of the underlying random processes. The time series of expected ILR are expected to have expressed autocorrelation since the portfolio inherits many of its constituents from one moment to the next. Some cyclical effects are to be expected, particularly in smaller banks. As more data become available with time, more advanced techniques from the VaR literature (cf. e.g., [19]) will be applicable in the study of the distribution of ILR.

Another aspect left to the future is the utilization of ILR for the validation of LGD models. Such analysis would require comparing the expected ILR for a given sub-portfolio to the available data of realized losses. This study, however, would require private bank data.

A further study of ILR, together with other credit risk indicators, would involve the application of the techniques of event study and principal component analysis (similarly to, e.g., refs. [20,21,22]) to analyze the effect of various system-wide events and regulatory changes. Monte Carlo simulations, like the ones used in [23,24], would be a suitable approach to this problem. The development of our methodology in this direction will be left to future study.

2. Asymptotic Estimation of the Inferred Loss Rate

We focus on the relation between the $qth$ quantiles ${\alpha }_q\left({\mathit{Y}}_n\right)$, ${\alpha }_q\left(R_n\left(X\right)\right)$, and ${\alpha }_q\left(X\right)$ of ILR, the conditional expectation of ILR, and the systemic risk factor. I make the following assumptions, reflecting the specific characteristics of the credit loss environment.

Generally, it is justified to assume that credit loss is contractually bounded. In studies where an actuarial approach is taken, the loss rate never exceeds 1; in reality, however, banks may incur expenses greater than the original value of the loan. The upper limit of the loss may be affected further if one accounts for the market value of the exposure. For this reason, we make the following assumption.

 Assumption 1. 

The inferred loss rates $W_i$ are non-negative and bounded.

In modeling credit risk, it is common to assume that exposure failures are independent events outside of the influence of a macro-factor X.

 Assumption 2. 

The random variables $W_i$ are, conditionally on $X,$ independent.

The following condition guarantees that the size of even the largest exposure vanishes to 0 as the number of exposures increases infinitely.

 Assumption 3. 

The EADs $a_i>0$ form a sequence of positive numbers, such that $A_n\uparrow \infty$ and ${\sum }_{i=1}^{\infty }{\left({\displaystyle \frac{a_i}{A_i}}\right)}^2$ is convergent.

The following is a technical assumption, common for asymptotic credit risk models (cf., e.g., [8]).

 Assumption 4. 

The systematic risk factor X is one-dimensional.

A monotonicity condition similar to Assumption 5 is commonly expected from the systemic risk factors. For example, Ref. [25] demonstrates that, in the Bulgarian banking system, the probability of default for retail and corporate loans is an increasing function of the unemployment rate and a decreasing function of the gross domestic product, respectively.

 Assumption 5. 

There is an open interval $B=(\underline{\alpha },\overline{\alpha })$ surrounding the q-th quantile of X, ${\alpha }_q\left(X\right)\in B$, and number $n_0$, such that, for any $n>n_0$, we have the following:

 (a) 

$R_n\left(x\right)$ is non-decreasing in $(\underline{\alpha },\overline{\alpha })$;

 (b) 

${sup}_{x\le \underline{\alpha }}{R}_n\left(x\right)\le {inf}_{x\in B}{R}_n\left(x\right)$;

 (c) 

${sup}_{x\in B}{R}_n\left(x\right)\le {inf}_{x\ge \overline{\alpha }}{R}_n\left(x\right)$.

 Remark 1. 

Some assumptions postulated in [17] have been weakened in the current version:

 (a) 

The systemic factor X is not required to be an absolutely continuous random variable in Assumption 4;

 (b) 

Assumption 5 does not require that expected loss rates for all exposures are monotone functions of X. This would simplify the proof; however, this does not reflect the possibility of the portfolio containing loans to entities with countercyclical business rhythm. Furthermore, a strict monotonicity of $R_n\left(x\right)$ is not assumed, which better reflects the actual conditions of credit portfolio management, compared to the assumptions made in [17].

 Proposition 1. 

Under the above assumptions, conditional on $X=x$,

$${\mathit{Y}}_n-R_n\left(x\right)\to 0,$$

almost surely as $n\to \infty$.

The proof makes use of the strong law of large numbers in the following form.

 Lemma 1 

(cf. [26], Theorem 6.7). Let $\left\{Z_n\right\}$ be a sequence of independent random variables. If $b_n\uparrow \infty$ and ${\sum }_{i=1}^{\infty }{\displaystyle \frac{Var{Z}_i}{b_i^2}}$ is convergent, then

$${\displaystyle \frac{{\sum }_{i=1}^n{Z}_i-E\left({\sum }_{i=1}^n{Z}_i\right)}{b_n}}\to 0,$$

almost surely as $n\to \infty$.

 Proof of Proposition 1. 

Let $b_n=A_n$ and $Z_n=a_n{W}_n$. Since ${\mathit{Y}}_n={\sum }_{i=1}^n{\displaystyle \frac{Z_i}{b_n}}$, checking the requirements of Lemma 1 would suffice.

For any value x of X, we have

$$\sum _{i=1}^{\infty }{\displaystyle \frac{Var\left(Z_i\right|x)}{b_i^2}}=\sum _{i=1}^{\infty }{\left({\displaystyle \frac{a_i}{A_i}}\right)}^{2}Var\left(W_i\right|x)$$

Assumption 2 implies that, for any i, $Var\left(W_i\right|x)\le K$ for some constant K. Hence,

$$\sum _{i=1}^{\infty }{\displaystyle \frac{Var\left(X_n\right|x)}{b_i^2}}\le K\sum _{i=1}^{\infty }{\left({\displaystyle \frac{a_i}{A_i}}\right)}^2$$

Assumption 3 implies that the requirements of Lemma 1 are satisfied. □

Proposition 1 is not precisely what we seek. With the following proposition, we aim to approximate the $qth$ quantile of ${\mathit{Y}}_n$ with the respective one of $R_n\left(X\right)$.

 Proposition 2. 

If $F_n$ is the cumulative distribution function (cdf) of ${\mathit{Y}}_n$, then, for any $ϵ>0$,

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{F}_n({\alpha }_q\left(R_n\left(X\right)\right)+ϵ)\ge q\end{array}$$

(5)

$$\begin{array}{c}\hfill \underset{n\to \infty }{lim}{F}_n({\alpha }_q\left(R_n\left(X\right)\right)-ϵ)\le q\end{array}$$

(6)

 Proof. 

Since almost sure convergence implies convergence in probability, by Proposition 1, for any x and $ϵ>0$, we have

$$P\left(\right|{\mathit{Y}}_n-R_n\left(x\right)|<ϵ|x)\to 1,\mathrm{as} n\to \infty .$$

(7)

Therefore,

$$F_n(R_n\left(x\right)+ϵ|x)-F_n(R_n\left(x\right)-ϵ|x)\to 1,\mathrm{as}\to \infty .$$

Hence,

$$\begin{array}{cc}\hfill F_n(R_n\left(x\right)+ϵ|x)& \to 1,\mathrm{and}\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill F_n(R_n\left(x\right)-ϵ|x)& \to 0,\mathrm{as}\to n\infty .\hfill \end{array}$$

(9)

Let

$$S=\left\{x:R_n\left(x\right)\le {\alpha }_q\left(R_n\left(X\right)\right)\right\}$$

be the set of values of X for which $R_n\left(x\right)$ does not exceed its $qth$ quantile. By construction, $P(X\in S)\ge q$. The law of total probability implies

$$\begin{array}{cc}{F}_n({\alpha }_q\left(R_n\left(X\right)\right)+ϵ)\hfill & =F_n({\alpha }_q\left(R_n\left(X\right)\right)+ϵ|X\in S)P(X\in S)\hfill \\ \hfill & +F_n({\alpha }_q\left(R_n\left(X\right)\right)+ϵ|X\notin S)P(X\notin S)\hfill \\ \hfill & \ge F_n({\alpha }_q\left(R_n\left(X\right)\right)+ϵ|X\in S)q.\hfill \end{array}$$

(10)

By design, for any $x\in S$, the monotonicity of $F_n$ implies

$$F_n({\alpha }_q\left(R_n\left(X\right)\right)+ϵ|x)\ge F_n(R_n\left(x\right)+ϵ|x).$$

Hence, by bounded convergence theorem, (8) implies that

$$F_n({\alpha }_q\left(R_n\left(X\right)\right)+ϵ|X\in S)\to 1,\mathrm{as} n\to \infty .$$

Combined with (10), this implies (5).

To prove Equation (6), let

$$S\left(ϵ\right)=\left\{x:R_n\left(x\right)\le {\alpha }_q\left(R_n\left(X\right)\right)-{\displaystyle \frac{ϵ}{2}}\right\}.$$

By construction, $P(X\in S(ϵ\left)\right)<q$; hence,

$$\begin{array}{cc}\hfill F_n({\alpha }_q\left(R_n\left(X\right)\right)-ϵ)& =F_n({\alpha }_q\left(R_n\left(X\right)\right)+ϵ|X\in S\left(ϵ\right))P(X\in S)\hfill \\ \hfill & +F_n({\alpha }_q\left(R_n\left(X\right)\right)+ϵ|X\notin S\left(ϵ\right))P(X\notin S\left(ϵ\right))\hfill \\ \hfill & \le q+F_n({\alpha }_q\left(R_n\left(X\right)\right)-ϵ|X\notin S\left(ϵ\right))P(X\notin S\left(ϵ\right)).\hfill \end{array}$$

Notice that, if $x\notin S\left(ϵ\right)$, $R_n\left(x\right)>{\alpha }_q\left(R_n\left(X\right)\right)-ϵ/2$ implies

$$F_n({\alpha }_q\left(R_n\left(X\right)\right)-ϵ|x)\le F_n\left(R_n\left(X\right)-ϵ/2|x\right)\to 0,$$

as $n\to \infty$, thanks to (9).

An argument similar to the proof of (5) justifies (6). □

Assumption 5 allows us to refine further the approximation of quantiles with the following proposition.

 Proposition 3. 

For sufficiently large n,

$${\alpha }_q\left(R_n\left(X\right)\right)=R_n\left({\alpha }_q\left(X\right)\right).$$

 Proof. 

Assumption 5 implies that, for $n>n_0$, if $X\le {\alpha }_q\left(X\right)$, then $R_n\left(X\right)\le R_n({\alpha }_q\left(X\right)$. Hence,

$$P(R_n\left(X\right)\le R_n\left({\alpha }_q\left(X\right)\right))\ge P(X\le {\alpha }_q\left(X\right))\ge q.$$

Conversely, if $R_n\left(X\right)<R_n({\alpha }_q\left(X\right)-ϵ)$, then $X<{\alpha }_q\left(X\right)-ϵ$ and

$$P(R_n\left(X\right)<R_n\left({\alpha }_q\left(X\right)\right)-ϵ)\le P(X<{\alpha }_q\left(X\right)-ϵ)<q,$$

for any $ϵ>0$. Therefore,

$$R_n\left({\alpha }_q\left(X\right)\right)=inf\{y:P(R_n\left(X\right)\le y)\ge q\}={\alpha }_q\left(R_n\left(X\right)\right).$$

□

Together, Propositions 2 and 3 imply the following theorem.

 Proof Theorem 1. 

The actual loss of the portfolio is given by ${\sum }_{i=1}^n{a}_i{U}_i$ and will exceed the suggested capital adequacy requirement if

$$\sum _{i=1}^n{a}_i{U}_i>A_n{\mathcal{C}}_n.$$

Furthermore, for the inferred loss, we have

$$A_n{\mathit{Y}}_n=\sum _{i=1}^n{a}_i{W}_i\ge \sum _{i=1}^n{a}_i{U}_i$$

Hence,

$$\begin{array}{cc}\hfill P(\sum _{i=1}^n{a}_i{U}_i>A_n{\mathcal{C}}_n)& \le P(A_n{\mathit{Y}}_n>A_n{\mathcal{C}}_n)\hfill \\ \hfill & =P({\mathit{Y}}_n>R_n\left({\alpha }_q\left(X\right)\right)+ϵ)\hfill \\ \hfill & =1-F_n(R_n\left({\alpha }_q\left(X\right)\right)+ϵ)\le 1-q.\hfill \end{array}$$

□

This result is fully sufficient to attempt empirical work.

 Remark 2. 

This finding implies that capital adequacy, considered not from a regulatory but from an economic perspective, can be validated externally, based on publicly disclosed data.

A more stringent condition of regularity, given by Assumption 6, is required to prove that, asymptotically, $R_n\left({\alpha }_q\left(X\right)\right)$ equals the value-at-risk threshold ${\alpha }_q\left({\mathit{Y}}_n\right)$ of inferred loss rate at a probability of survival q.

 Assumption 6. 

There is an open interval $B=(\underline{\alpha },\overline{\alpha })$ surrounding the q-th quantile of X, ${\alpha }_q\left(X\right)\in B$, and number $n_0$, such that, for any $n>n_0$, in addition to the conditions in Assumption 5, we have the following:

 (d) 

The cdf of the systemic factor X is continuous and strictly increasing in B;

 (e) 

For any i, the function $r_i\left(x\right)$ is differentiable in B;

 (f) 

For sufficiently large n, $R_n^{\prime }\left(x\right)$ is bounded in a positive finite interval, $0<m\le R_n^{\prime }\left(x\right)\le M<\infty$ for $x\in B$.

 Remark 3. 

Assumption 6 aims to guarantee the continuity of $r_i\left(x\right)$ at the $qth$ quantile. This form is, also, less restrictive than the requirement imposed in [17].

Let $F_n$, $G_n$, and H be the cdfs of ${\mathit{Y}}_n$, $R_n\left(X\right)$, and X respectively. Assumption 6 implies that $R_n\left(x\right)$ is strictly increasing in B, i.e., for any $x\in B$, $R_n\left(X\right)\le R_n\left(x\right)$ is equivalent to $X\le x$. Hence,

$$G_n\left(R_n\left(x\right)\right)=H\left(x\right)\mathrm{for} \mathrm{any} x\in B.$$

(11)

 Proposition 4. 

For any value x of X, if $x\in B$, then

$$\underset{n\to \infty }{lim}{F}_n\left(R_n\left(x\right)\right)-H\left(x\right)=0.$$

The proof relies on the following lemma.

 Lemma 2. 

Let Y and Z be two random variables with cdfs $F_Y$ and $F_Z$, respectively. For any u and $\delta >0$,

$$|F_Y\left(u\right)-F_Z\left(u\right)|\le P(|Y-Z|>\delta )+max\{F_Y(u+\delta )-F_Y\left(u\right),F_Y\left(u\right)-F_Y(u-\delta )\}$$

(12)

 Proof. 

Notice that, if $Z\le u$ and $|Y-Z|\le \delta$, then $Y\le u+\delta$. Hence,

$$F_Z\left(u\right)-P\left(\right|Y-Z|>\delta )\le F_Y(u+\delta ),$$

and

$$F_Z\left(u\right)-F_Y\left(u\right)\le P\left(\right|Y-Z|>\delta )+F_Y(u+\delta )-F_Y\left(u\right).$$

(13)

Similarly, if $Z>u$ and $|Y-Z|\le \delta$, then $Y>u-\delta$. Hence,

$$1-F_Z\left(u\right)-P\left(\right|Y-Z|>\delta )\le 1-F_Y(u-\delta ),$$

and

$$F_Y\left(u\right)-F_Z\left(u\right)\le P\left(\right|Y-Z|>\delta )+F_Y\left(u\right)-F_Y(u-\delta ).$$

(14)

Combining (13) and (14), we obtain (12). □

 Proof of Proposition 4. 

Take $n_0$, B, m, and M provided by Assumptions 5 and 6 and fix $n>n_0$ and $x\in B$. Using (11), and applying Lemma 2 for ${\mathit{Y}}_n$ and $R_n\left(X\right)$, we have

$$\begin{array}{cc}\hfill & |F_n\left(R_n\left(x\right)\right)-H\left(x\right)|=|F_n\left(R_n\left(x\right)\right)-G_n\left(R_n\left(x\right)\right)|\hfill \\ \hfill & \le P\left(\right|L_n-R_n\left(X\right)|>\delta )+max\{G_n(R_n\left(x\right)+\delta )-G_n\left(R_n\left(x\right)\right),G_n\left(R_n\left(x\right)\right)-G_n(R_n\left(x\right)-\delta )\}\hfill \end{array}$$

for any $\delta >0$.

Next, take $\widehat{\delta }>0$, such that $\underline{\alpha }<x-\widehat{\delta }$ and $x+\widehat{\delta }<\overline{\alpha }$. For any $\delta <m\widehat{\delta }$, we have

$$x+{\displaystyle \frac{\delta }{m}}<x+\widehat{\delta }<\overline{\alpha }.$$

Hence, $x+{\displaystyle \frac{\delta }{m}}\in B$ and

$$R_n\left(x+{\displaystyle \frac{\delta }{m}}\right)-R_n\left(x\right)>m{\displaystyle \frac{\delta }{m}}=\delta .$$

This implies

$$G_n(R_n\left(x\right)+\delta )\le G_n\left(R_n\left(x+{\displaystyle \frac{\delta }{m}}\right)\right)=H\left(x+{\displaystyle \frac{\delta }{m}}\right).$$

Similarly, we can see that

$$G_n(R_n\left(x\right)-\delta )\ge G_n\left(R_n\left(x-{\displaystyle \frac{\delta }{m}}\right)\right)=H\left(x-{\displaystyle \frac{\delta }{m}}\right).$$

Therefore,

$$\begin{array}{cc}\hfill max\{G_n(R_n\left(x\right)+\delta )& -G_n\left(R_n\left(x\right)\right),G_n\left(R_n\left(x\right)\right)-G_n(R_n\left(x\right)-\delta )\}\hfill \\ \hfill & \le max\left\{H\right(x+\delta /m)-H(x),H(x)-H(x-\delta /m\left)\right\}\hfill \end{array}$$

By Assumption 6, H is a continuous strictly increasing function. Therefore, for any $ϵ>0$, there exist values of $\delta >0$ such that

$$max\{H(X+\delta /m)-H\left(X\right),H\left(X\right)-H(X-\delta /m)\}<{\displaystyle \frac{ϵ}{2}}.$$

On the other hand, Proposition 1 implies convergence in probability and, hence, for any $\delta >0$ and $ϵ>0$, there exists $N=N(\delta ,\eta )$ such that

$$P\left(\right|{\mathit{Y}}_n-R_n\left(X\right)|>\delta )<{\displaystyle \frac{ϵ}{2}}.$$

Therefore, we have that, for any $ϵ>0$, if $n>max\{N,n_0\}$,

$$|F_n\left(R_n\left(x\right)\right)-H\left(x\right)|<ϵ,$$

which concludes the proof. □

 Proof of Theorem 2. 

Applying Proposition 4 with $x={\alpha }_q\left(X\right)$ and noticing that $H\left({\alpha }_q\left(X\right)\right)=q$ proves (2).

To prove (3), suppose that $\widehat{\delta }$ is chosen as in the proof of Proposition 4. Notice that for any $\delta >0$, and $n>n_0$, if $\delta <M\widehat{\delta }$, then

$$R_n\left({\alpha }_q\left(X\right)\right)-R_n\left({\alpha }_q\left(X\right)-{\displaystyle \frac{\delta }{M}}\right)\le M{\displaystyle \frac{\delta }{M}}=\delta .$$

Hence,

$$F_n(R_n\left({\alpha }_q\left(X\right)\right)-\delta )\le F_n(R_n({\alpha }_q\left(X\right)-\delta /M).$$

(15)

Since ${\alpha }_q\left(X\right)-\delta /M>{\alpha }_q\left(X\right)-\widehat{\delta }>\underline{\alpha }$, ${\alpha }_q\left(X\right)-\delta /M\in B$, we have, by Proposition 4,

$$|F_n\left(R_n({\alpha }_q\left(X\right)-\delta /M)\right)-H({\alpha }_q\left(X\right)-\delta /M)|\to 0,\mathrm{as} n\to \infty .$$

Assumption 6 implies that $H({\alpha }_q\left(X\right)-\delta /M)<H\left({\alpha }_q\left(X\right)\right)=q$, so there exists $N_1$ such that, if $n>N_1$,

$$F_n\left(R_n({\alpha }_q\left(X\right)-\delta /M)\right)<q.$$

Therefore, (15) implies $F_n(R_n\left({\alpha }_q\left(X\right)\right)-\delta )<q$, so $R_n\left({\alpha }_q\left(X\right)\right)-\delta <{\alpha }_q\left({\mathit{Y}}_n\right)$.

Similarly,

$$R_n\left({\alpha }_q\left(X\right)+{\displaystyle \frac{\delta }{M}}\right)-R_n\left({\alpha }_q\left(X\right)\right)\le \delta ,$$

and, likewise, there exists $N_2$ such that, if $n>N_2$, $R_n\left({\alpha }_q\left(X\right)\right)+\delta >{\alpha }_q\left({\mathit{Y}}_n\right)$.

Thus, for $n>max\{n_0,N_1,N_2\}$, we have

$$|{\alpha }_q\left({\mathit{Y}}_n\right)-R_n\left({\alpha }_q\left(X\right)\right)|<\delta ,$$

for any $\delta <M\widehat{\delta }$. Since $\delta$ can be chosen arbitrarily close to 0, this concludes the proof of (3). □

3. Empirical Analysis

3.1. Capital Buffers

As documented in [17], although conservative, the capital adequacy limit deduced from the inferred loss rate analysis is still within the regulatory limits imposed on Bulgarian banks. In this subsection, we report the expected inferred loss rate’s summary statistics and confirm the findings of [17] using all the newly available data.

Table 1 represents the system’s structure and displays the bank groups’ current ratings.

Table 1. Average insolvency rates (in %) in the respective rating classes for the Bulgarian banking system. Of the 23 banks, 9 have been rated by Fitch Ratings, Inc., 3 by Moody’s Ratings, and the rest by the Bulgarian Agency for Credit Rating. One bank has the last available rating from 2021; the rest were rated in 2024. Comparative rating scales have been applied and the average default rates have been taken from [27], based on annual reports from 1981–2023. Column 2 reports the ratings range, and column 3 shows the number of banks in each group. Summary statistics of actual default rates are reported in columns 4–8. The year 2008, the most recent major financial crisis, and the last available year, 2023, are also reported. Based on this, the average survival rate in all groups is greater than 99%.

	Ratings	Count	Min	Max	Mean	Median	STD	2008	Latest (2023)
Bank System	B+–A+	23	0.01	2.44	0.38	0.31	0.57	0.72	0.16
Group 1	BB+–A-	5	0.00	1.52	0.21	0.13	0.36	0.53	0.10
Group 2	BB-–BBB+	12	0.00	2.62	0.36	0.27	0.61	0.65	0.14
Group 3	B+–A+	6	0.04	2.83	0.57	0.55	0.67	1.04	0.24

The Bulgarian banking system consists of 23 banks structured in 3 groups. The top-five banks constitute Group 1, which currently accounts for about 80% of the banking business in the country. Group 3 consists of six branches of foreign banks. These are excluded from our analysis because they are supervised differently; however, they are still included in the data for the total banking system. The remaining 12 banks form Group 2. Based on the historic defaults in the respective rating classes, using [27], one can compute that the expected survival rate in all the groups is greater than 99%.

Since 2008, the Bulgarian National Bank periodically collects reports from individual banks on the quality of their credit portfolios. Banks report the gross carrying amounts in a two-way classification: by status, as performing/non-performing, and by the type of borrower. The total expected credit loss is also reported by type of borrower. The individual banks’ reports are not publicly available. The BNB website publishes the quarter-end results, compiled by bank groups in the report Information on non-performing exposures and accumulated impairment. (A link to the BNB reports is given in an endnote.) Through the years, the BNB report has changed its format; however, it has always provided data sufficient to compute the expected inferred loss rate using Formula (4) for the portfolios of loans to non-financial corporations and households in the two bank groups and the whole bank system.

Table 2 presents the summary statistics of the computed expected ILR. The available information permits the construction of a time series of 66 quarterly data points. The required capital adequacy ratio in the EU is 8%, increased to 18.5% by the Bulgarian regulator. In addition, the Bulgarian National Bank requires six institutions (all banks in Group 1 and the largest one in Group 2) to maintain an individual buffer from 0.5 to 1.75%.

Table 2. Summary statistics of the expected inferred loss rate based on data from the Bulgarian National Bank’s Information on non-performing exposures and accumulated impairment. Quarterly data are available from Q2/2008 till Q3/2024. Expected ILR is reported by debtor segments: Corporate (non-financial corporations) and Retail (loans to households). The reported maximums and 99th and 95th percentiles are compared with the survival rates of the banks based on their ratings. Group 2 banks have smaller ILR, which is consistent with the theory that, despite being perceived as riskier, their competitors in Group 1 have a bigger risk appetite, being the leaders in the market.

	Obs	Mean	St. Dev.	Min	p95	p99	Max
Corporate
Bank System	66	6.8%	3.1%	1.6%	12.0%	12.2%	12.2%
Group 1	66	7.8%	3.7%	1.8%	14.3%	14.3%	14.8%
Group 2	66	5.8%	2.5%	1.6%	9.6%	9.8%	9.8%
Retail
Bank System	66	6.5%	2.0%	3.0%	9.4%	9.5%	9.6%
Group 1	66	6.8%	2.3%	2.7%	10.0%	10.2%	10.2%
Group 2	66	5.4%	2.2%	2.2%	8.0%	8.2%	8.3%

As indicated in Table 1, in both groups, the banks’ average probability of survival, based on rating, is greater than 99%. Since only 66 data points have been available, based on Theorem 1, the maximal expected ILR is used to infer the capital adequacy threshold. Table 2 demonstrates that the banking system’s maximal expected inferred loss rate is 12.2%, decomposing to 14.8% and 9.8% in the two groups, not overly conservative compared to the current regulation. The 99th and 95th percentiles are reported for comparison.

Table 2 confirms the finding that ILR is generally higher in Group 1 than in Group 2. This indicates a higher risk appetite in the leading banks in the system.

3.2. Loss-Given-Default Ratio and Net Inferred Loss Rate

Recall that, for any exposure, the expected credit loss over a certain horizon is computed as

$$EC{L}_i\left(x\right)=P{D}_i\left(x\right)·LG{D}_i\left(x\right)·EA{D}_i,$$

where $P{D}_i\left(x\right)$ is the point-in-time probability of default and and $LG{D}_i\left(x\right)$ is the expectation of the exposure’s $LGD$ conditional on the state of the macroecomomic factor $X=x$.

Since system-wide rates of default are unavailable, I approximate them using the inferred rate of default, $IR{D}_i\left(x\right)$, as introduced by [25].

This approach was introduced in [17] to define inferred loss-given-default, ILGD, to gauge the conservative effect of defaulted exposures on the ILR.

The exposures in Stage 3 are loans that have already defaulted. These are part of the bank’s portfolio; however, by regulation, a great part of their expected credit loss (the specific provisions have already been registered as realized loss on the bank’s balance sheet. For this reason, they should be excluded from the calculation of the expected loss rate.

As above, I approximate the exposure at default $EA{D}_i$ with the current balance of the loans. Hence, on a portfolio level, we have

$$ECL\left(x\right)=IRD\left(x\right)·ILGD\left(x\right)·A_n^{PE}+ILGD\left(x\right)·A_n^{NPE},$$

where $A_n^{PE}$ and $A_n^{NPE}$ denote the EAD of the performing and the non-performing loans, respectively.

From here, the inferred loss-given-default rate is expressed as

$$ILGD\left(x\right)={\displaystyle \frac{ECL\left(x\right)}{IRD\left(x\right)·A_n^{PE}+A_n^{PE}}}.$$

(16)

Then, the expected net inferred loss rate, defined as

$$R_n^{PE}\left(x\right)=IRD\left(x\right)·ILGD\left(x\right)$$

represents the ILR net of the effect of defaulted exposures. For the defaulted exposures, the loss has been realized and ILR is

$$R_n^{NPE}=LGD\left(x\right).$$

Notice that, by definition, $R_n\left(x\right)$ is the value-weighted average of $R_n^{PE}\left(x\right)$ and $R_n^{NPE}\left(x\right)$.

Table 3 exhibits the summary statistics of the expected net inferred loss rate.

Table 3. Summary statistics of the expected net inferred loss rate. The results are classified by type of debtor and by bank group (cf. Table 1 for details). The 95th and 99th percentiles are reported for comparison. The losses of already-defaulted loans have been disregarded in the computation of this ratio. The net inferred loss rate provides a lower bound for the loss rate.

	Obs	Mean	St. Dev.	Min	p95	p99	Max
Corporate
Bank System	66	1.2%	0.8%	0.3%	2.8%	2.8%	2.8%
Group 1	66	1.3%	1.0%	0.3%	3.7%	3.9%	4.0%
Group 2	66	1.1%	0.7%	0.2%	2.8%	2.9%	3.2%
Retail
Bank System	66	1.3%	0.7%	0.6%	2.9%	3.0%	3.0%
Group 1	66	1.3%	0.8%	0.5%	3.1%	3.2%	3.2%
Group 2	66	1.3%	0.8%	0.5%	2.9%	3.0%	3.0%

The net inferred loss rate is visibly lower than the total ILR and is likely less than the actual loss rate. Here are some reasons for its possible inaccuracy:

1.

Not all expected losses of defaulted loans are accounted as fully realized. By considering them fully forgone, we lower the numerator of the ratio;

2.

The current size of the loan is an approximation of the EAD for two reasons: (a) For revolving credit lines only, a part of the off-balance may be converted to on-balance due, which will increase the value; (b) As the loan matures before the default, the debtor will likely repay a portion. This adjustment applies to the vast majority of the loans and will decrease the size of the EAD, i.e., the denominator of the ratio.

For these reasons, it is fair to believe that the expected net inferred loss ratio is only a lower bound of the actual loss rate. For all portfolios currently, it is sufficient to maintain a capital buffer of 4% to cover all unexpected credit risk, which, incidentally, falls within the regulatory Capital Adequacy Ratio for Common Equity Tier 1 capital of 4.5% according to Article 92 of the Capital Requirements Regulation [28] and the Capital Requirements Directive of EU [29], reflecting the BASEL III rules (cf. [30]).

3.3. Monotonicity of the Macroeconomic Factors

Recall that Assumptions 5 and 6 in Section 2 introduced some requirements of the local strict monotonicity of the macroeconomic factor. In this section, I study the question of the monotonicity of $R_n\left(x\right)$ as a function of x using OLS regression.

For this purpose, I construct the same macroeconomic variables used in [25] to forecast the inferred probability of default:

1.

Corporate loans. The annual gross domestic product, measured quarterly, is denoted by $GD{P}_t$. The macroeconomic variable

$$X_t={\displaystyle \frac{GD{P}_t}{GD{P}_{t-4}}}-1$$

is the annual growth of $GD{P}_t$;

2.

Retail loans. The unemployment rate, measured quarterly, is denoted by $U{R}_t$. The macroeconomic factor

$$X_t={\displaystyle \frac{U{R}_t}{U{R}_{t-4}}}-1$$

is the annual growth of $U{R}_t$.

As in [25], it is expected that the macroeconomic effect will be visible with some lag. We test several specifications:

$$Z_t=\alpha +\beta X_{t-L}+{ϵ}_t,$$

(17)

(a)

In Table 4 and Table 5, taking the expected inferred loss rate as dependent variable $Z_t$ with lag $L=4$ and $L=6$ quarters;

Table 4. Results of the regressions of expected inferred loss rate on a macroeconomic factor for the corporate portfolios. Regression (17) is performed with two specifications: (1) with lag $L=4$ and (2) with $L=6$; with $L=0$, the result had no statistical significance. Student’s t-statistic is reported for each estimate. The estimates for $\beta$ are marked for significance: with (*) and (**)—the significance at the 5% and at 1% level, respectively. Since we are interested in monotonicity, one-sided hypotheses were tested. The negative significant coefficients point out that the expected ILR is a decreasing function of the macroeconomic factor. Specification (2) with 18 months’ lag fits the data better, as evidenced by the higher significance of the $\beta$ and a better $R^2$.

Specification	Bank System	Group 1	Group 2	Bank System	Group 1	Group 2
	(1)	(1)	(1)	(2)	(2)	(2)
$\alpha$	0.078	0.090	0.065	0.080	0.093	0.066
t-value	16.55	16.01	17.02	17.91	17.38	18.04
$L4.X$	−0.288	−0.369	−0.198
t-value	−2.06 *	−2.21 *	−1.75 *
$L6.X$				−0.345	−0.451	−0.214
t-value				−2.60 **	−2.84 **	−1.97 *
$R^2$	6.6%	7.5%	4.9%	10.5%	12.2%	6.3%

Table 5. Results of the regressions of expected inferred loss rate on a macroeconomic factor for the retail portfolios. Regression (17) is performed with two specifications: (1) with lag $L=4$ and (2) with $L=6$; with $L=0$, the result had no statistical significance. Student’s t-statistic is reported for each estimate. The estimates for $\beta$ are marked for significance: with (*) and (**)—the significance at the 5% and 1% level, respectively; the result marked with (^†) is insignificant. A one-sided significance test is applied. The positive significant coefficients point out that the expected ILR is an increasing function of the macroeconomic factor. Specification (2) with 18 months’ lag fits the data better, as evidenced by the higher significance of the $\beta$ and a better $R^2$.

Specification	Bank System	Group 1	Group 2	Bank System	Group 1	Group 2
	(1)	(1)	(1)	(2)	(2)	(2)
$\alpha$	0.067	0.070	0.056	0.067	0.071	0.057
t-value	27.24	24.81	32.09	28.13	25.54	33.68
$L4.X$	0.022	0.027	0.007
t-value	1.88 *	2.04 *	0.84 †
$L6.X$				0.032	0.039	0.014
t-value				2.87 **	3.06 **	1.82 *
$R^2$	5.6%	6.6%	1.2%	12.7%	14.1%	5.5%

(b)

In Table 6, taking the expected net inferred loss rate as dependent variable $Z_t$ with lag $L=0$.

Table 6. Results of the regressions of expected net inferred loss rate on macroeconomic factors for the corporate and retail portfolios. Regression (17) is performed with expected net inferred loss rate taken as the dependent variable $Z_n$ and no lag, $L=0$. Student’s t-statistic is reported for each estimate. The estimates for $\beta$ are marked for significance: with (*) and (**)—the significance at the 5% and 1% level, respectively. The $\beta$s exhibit pronounced significance in all portfolios, suggesting the monotonicity of the expected ILR for performing loans. The specification for the retail portfolios fits the data better, as evidenced by the higher significance of the $\beta$ and a better $R^2$.

	Corporate			Retail
	Bank System	Group 1	Group 2	Bank System	Group 1	Group 2
$\alpha$	0.014	0.015	0.013	0.012	0.013	0.013
t-value	11.57 **	9.83 **	11.73 **	18.65 **	14.87 **	14.46 **
X	−0.094	−0.098	−0.100	0.019	0.018	0.014
t-value	−2.63 **	−2.07 *	−2.92 **	5.90 **	4.22 **	3.21 **
$R^2$	9.8%	6.3%	11.8%	35.6%	22.0%	14.1%

Table 4 exhibits the result of the regressions for the corporate portfolios, and Table 5 presents the regressions for the retail portfolios. Both Table 4 and Table 5 demonstrate higher significance and a better fit of the specification with $L=6$ lags, suggesting that this version better reveals the dependence between the variables.

In both Table 4 and Table 5, the results for Group 2 are less significant and with smaller explanatory power than those for Group 1. This is likely due to smaller banks managing their defaulting clients with more individual attention. The fact that the results for the whole system resemble Group 1’s is undoubtedly due to Group 1 constituting the largest share of the banking system.

Comparing Table 4 with Table 5, we can see that, in the whole system and Group 1, the models fit the data for the retail portfolios better. This can be seen as evidence supporting the theory that banks, in general, evaluate their corporate clients better and are more reluctant to leave them due to the random pressures of the macroeconomic elements. For Group 2 banks, this cannot be observed due, perhaps, to the fact that, in that group, there is only one bank, TBI, that specializes in retail lending; the rest have select portfolios consisting of mostly residential mortgages.

Table 6 presents the results of the regressions with the expected net inferred loss rate taken as the dependent variable $Z_n$ in Equation (17).

As in Table 4 and Table 5, we notice that the retail model fits better, likely for the same reason as above. Comparing the two groups, we find that, in Table 6, the performing loans in retail follow the same general pattern as in Table 4 and Table 5, with Group 2 less significant than Group 1. For corporate loans, things are reversed, with Group 2 showing higher significance and $R^2$. Note that, in Group 1, by far the largest corporate lender is Unicredit, which is the only bank in the country preparing their financial reports with an internal-rating-based forecasting model certified by the ECB.

The results of this analysis have been repeated with various lags (not reported) with high statistical significance. The best fitting specification is with no lag. The difference, both in lag and significance, between performing and non-performing loans is highly visible, and this is likely due to the difference in the banks’ treatment of these two categories of loans.

The overall low values of $R^2$ in all the regression exhibited in Table 4, Table 5 and Table 6 indicate that a significant portion of the variance of the expected ILR remains unexplained by the macroeconomic factor, which is most likely due to the character of the random process. This analysis is left to the future.

4. Conclusions

We have revisited the notion of inferred loss rate defined in [17], providing full theoretical proofs and, further, more extensive empirical evidence.

A framework for the asymptotic estimation of the ILR was set up by providing detailed proofs of all the outstanding hypotheses. As a result, the expected ILR can be used as a tool to construct a conservative upper bound of the capital adequacy buffer using measurable and regularly reported quantities.

The empirical analysis performed with data for the systemic portfolios of corporate and retail loans reported by the regulator reveals the following:

1.

The capital adequacy threshold, obtained using ILR, despite being conservative, is well within the regulatory limits. This suggests that the conservative bias of the ILR limit does not render it useless when making approximate judgments;

2.

ILR as an indicator of credit risk reveals characteristic differences in credit risk management between banks of different sizes and credit standing;

3.

As a credit risk indicator, ILR is capable of discerning differences in the treatment of corporate and retail loans, as well as performing and non-performing loans;

4.

The expected ILR computed from publicly available data is monotone as a function of macroeconomic factors, as postulated in Assumption 5.

This analysis suggests that ILR, as a credit risk indicator, can be used by practitioners in approximating LR when monitoring and managing the bank’s credit portfolios. Regulators and auditors can utilize it for benchmarking and cross-sectional comparison between institutions.