Operationalization of the theory of forecasting default: a conceptual model

Makarov Il'ya Mihailovich Master's Degree; Department of Economics and Finance; Moscow University of Finance and Law Lawyer; KA 'ExlEdge' 115172, Russia, Moscow, Bolshye Kamenshchiki str., 2 ilya_mac@mail.ru

Introduction

When can a company be considered financially sound? How can the risk be measured and assessed (in terms of probability) that this financial health may deteriorate and that the business will default? What is the contractual strength of the borrower in relation to his creditors? What is the maximum debt that a company can withstand, taking into account its prospects? How can debt be restructured to restore the company's health? These issues form the cornerstones of finance.

To date, models assessing the risk of a company's default and, conversely, its financial health have not been able to produce reliable results, and their use and technical aspects vary in the world of finance (corporate finance, credit risk management, financial intermediation, structured finance, project finance, corporate restructuring, etc.). After several decades of research On this topic, the question of how to measure the probability that a company will not fulfill its contractual obligations on time remains unresolved ^[17]. Unfortunately, domestic research over the past ten years has not been used in practice in Russia, analysts of companies and financial institutions use the achievements of foreign scientists in this aspect of finance. analysis.

"The failure of default prediction models" ^[11] is mainly due to two factors. On the one hand, (1) modern valuation models underestimate financial constraints ^[15] and the dynamic interdependent behavior of operators; on the other hand, (2) soft proprietary information – usually of a predictive nature – is ignored in credit assessment systems ^[9], where the specific characteristics of the firm (asset volatility, growth opportunities, key partners, the management team, fixed costs, production capacity, transparency of information, etc.) are fundamental to the assessment ^[1]. For these purposes, it is necessary to integrate human judgment into the credit assessment process. In general, credit ratings have difficulty assessing the borrower's idiosyncratic risk ^[11]. This form of risk demonstrates its main importance in variations of expected credit losses at the individual level ^[16] and is fundamental for pricing purposes ^[13].

In fact, these errors arise from models that are either tailored to data that does not follow from any specific theory of the financial health of the company and the risk of its default, or based on an incorrect theory, which leads to an incorrect determination of the probability of default. Models built on the wrong foundation or without any foundation at all lead to errors. Translating the estimate obtained from these models into the probability of default is another methodological error, since it is based on a frequency analysis of past default behaviors (a frequency approach to probability), instead of taking into account the future default factors of the company in question. In other words, modern models study probability prediction instead of prediction probability. The use of these erroneous models causes serious market inefficiencies (procyclicity, credit crises, adverse selection, moral hazard, arbitrage of regulatory capital). Moreover, when a bank's regulatory capital is closely linked to these models, the effects of procyclicity may even increase ^[2], especially due to the low consistency of internal estimates of the probability of default by banks ^[18]. That is why there is a growing need for models that better take into account the complexity of the financial market and the potential of the borrower to respond to changes in real time and lead to an evolution in the field of credit scoring.

The real challenge is to develop a theory of the financial health of the company. We need a clear conceptual understanding of the phenomenon that causes financial difficulties for the company.: how they begin, develop and escalate to analyze and predict the borrower's future poor performance and assess the possibility (risk) of his inability to meet his obligations on time. Moreover, the task is to develop a model for quantifying the probability of a given event within a consistent probabilistic framework, where the factors of idiosyncratic risk — assessed using soft information and human skills — are fundamental to understanding.

In principle, the financial health of a company depends on maintaining a balance between its demand for credit and supply in the credit market. This is a function not only of the borrower's demand conditions — the subject of models developed to date — but also of the conditions of intermediaries that make up the loan supply segment (competition, yield curves, availability of information, analytical capabilities, etc.). In fact, all these factors directly affect the ability of a company to apply for a loan to remain in the market, since they determine the availability of resources that are or will be available to her , and also determine at what price these resources will be available in the future.

Based on these data, the article forms a theory and develops a model that tries to solve the above problems and answer the questions raised at the beginning of the article. The reasoning begins with the assumption that a company is considered financially sound as long as it is able to maintain balance in the financial system.

Default and the probability of its occurrence

Since we are interested in studying default in order to predict it, it is necessary to understand the genesis and dynamics of this phenomenon. Therefore, the central question is: how and when does the company end up in default?

Each company implements its strategic plan based on the business continuity hypothesis, which assumes the support of all its creditors. When a company is faced with a more complex scenario compared to the initial plan, it causes a shortage of funds, which affects budget management ^[8]. If these requirements become so burdensome (imagine a serious, extraordinary event) that the company's budget is in a difficult position, default can occur immediately. It is very difficult to predict such events. Most likely, default represents a slower trajectory of financial stress — and therefore less difficult to predict — which results in a gradual decline in the strategic and competitive strength of the company ^[4]. By focusing on this aspect of default, we estimate the so-called "expected" probability of default (PD), or more precisely, to what extent the company (and its strategic plan) is expected to fail and be unable to repay its debts.

In practice, the gradual decline in competitiveness leads to greater cash shortages than planned, which, as already mentioned, fall on budget management. In particular, it is "seasonal and revolving loans" that provide businesses with short-term financing for the purchase of inventories, receivables, purchase of materials and all other monetary needs, including debt service (principal and interest), to act as a liquidity buffer and insurance ^{[14; 19]}. This is in line with the principles of finance, which have recently highlighted the advantages of flexibility of short-term credit instruments and their interest rate structure compared to long-term loans.

If the shortage of funds increases and is not eliminated, corporate difficulties deepen, and credit lines are used more intensively. The company revises its basic projects depending on the degree of difficulties, sometimes even radically (for example, changing sales and procurement policies in certain business areas, anticipating recapitalization, changing some managers, etc.) ^[6]. If again the plan encounters worse conditions than expected, large cash shortages are also incorrectly allocated to short-term debt (unless the plan is changed, for example, a bond issue is provided). The rate on these credit lines — usually variable — increases due to the greater risk perceived by the lender (contractually), which contributes to a deterioration in financial conditions and an increase in cash outflows associated with debt servicing itself ^[3]. As a result, companies experiencing financial difficulties with liquidity cannot but continue to use short-term credit instruments. Banks, as "providers of liquidity of last resort", should assess the need to provide additional or reserve credit lines.

If financial difficulties continue, credit lines are almost completely exhausted and this signals a higher level of difficulties for lenders, which in turn leads to a further increase in interest rates and limited access to credit lines ^[12]. Even medium- and long-term loans behave in a similar way (credit restrictions and deterioration of contractual terms) due to violation of contractual obligations (which aggravates the company's situation at this critical moment).

In such difficult phases, when a company and its creditors decide whether to "turn off the oxygen", we observe the phenomenon of "zombie lending": banks tend to extend short-term credit lines to support monetary needs that cannot be postponed — even when conditions are violated — as long as they believe in the minimum profitability of their investments. Then there comes a moment when no bank in the market is ready to provide/extend short—term credit lines anymore, because no one considers it profitable at any interest rate; thus: "when the payment deadline comes and the company does not have enough funds to fulfill it, it finds itself in default." The Company itself or a third party decides that the difficulties are irreversible and decides to terminate the activity or demand its termination.

This path of difficulties for any company can develop more or less intensively and quickly; thus, each company has its own (high or low) probability of default. But that's not all. Each strategic plan can lead to default sooner or later and along different trajectories; therefore, each plan that a company could present on the market has its own probability of falling into default. Therefore, an important conclusion is that the expected probability of default concerns the likelihood that a company may fail to implement its current plan.

Trying to theorize the path of difficulties in the way described above, we can argue that the "expected" default event is the top of the trajectory of an unforeseen chain of events that lead to stress on short-term instruments of a company unable to find a strategy to get out of difficulties. Analyzing this definition, we can say the following: default occurs when the company is in the following conditions:

(a) faces unforeseen cash flow needs (payment to suppliers, employees, interest, etc.);

(b) no lender is willing to support the company by providing short-term instruments;

(c) it is unable to develop a reliable alternative restructuring plan.

The probability of this combined event occurring is very difficult to estimate. With the company's business plan and forecasts of its future economic and competitive activities at hand, we can predict points (a) and (b) what we are going to do. The same cannot be predicted regarding potential restructuring changes by managers, shareholders, or even third parties (including the state) in difficult circumstances — point (c). If this is true, we need to make a strong assumption to continue: that the company does not have the opportunity to develop alternative plans for the submitted one (or more precisely, no alternative plan can be known). Therefore, the "expected" probability of a borrower's default is equivalent to the probability of failure of his implementation plan to date; this is the best forecast of his future financial performance at the moment. Obviously, this assumption leads to a disadvantage in estimating the probability of default (PD), which often risks being overestimated even significantly compared to the actual value. Fortunately, this revaluation is lower for companies already in financial difficulties; which clearly have fewer paths to default.

Forecasting default

The above definition of default needs to be put into practice. Specifically, excluding the assumption of paragraph (c), it is necessary to check the manifestation of two other conditions (a) and (b). With regard to condition (a), the goal is to model a way to reduce the financial health of the company under the influence of the future evolution of its business plan. It is necessary to simulate the possible occurrence and deterioration of a cash shortage situation year after year28 and identify various trajectories of difficulties. To do this, by performing the monitoring function of the principal in a long—term contractual relationship ^[10], the financial analyst/operator reviews and verifies the assumptions on the basis of which the plan is based. They assess the future availability of credit and interest rates (credit risk), market trends and their vulnerability (market risk), as well as the potential capabilities of the company in response to these forecasts (idiosyncratic risk). To do this, analysts use fundamental analysis tools.

This review aims to define:

(1) Offset levels (accuracy analysis). Prospective financial communication to the market is subject to classic moral hazard problems; there is a risk that it may be characterized by a positive bias. Assumptions are reviewed based on confidence levels in the plan and its financial projections.

(2) Degrees of uncertainty (analysis of variance). The uncertainty of estimates is quantified by the levels of variability that the analyst assigns to the probability distribution of checking all significant values (assumptions) based on publicly available information; confidential information (also depending on the volume and duration of credit relations); analytical tools used; skills and experience.

This review is carried out through the construction and modeling of various hypothetical scenarios, on the basis of which a consensus of analysts and risk assessments are formed using a variety of historical analysis tools (correctly based on a frequency approach that gives the analysis greater objectivity) as well as soft more or less proprietary information.

According to the definition of default (lack of financial resources to repay debts), modeling is usually carried out on operating and investment cash flows (free cash flow) that the company will generate in the future along with related financial obligations (debt service). The difference between operating cash flow and debt service is reflected – positively or negatively – in the net short-term financial position.

As for the above-mentioned condition (b), the credit institution loses interest in financing the company when it believes that it can no longer extract the minimum benefit. This happens when the creditor believes that the debtor has fallen into irreversible difficulties and is no longer able, even in the distant future, to produce sufficient residual cash flow to pay at least fair interest on the debt. In fact, in this case, interest is added to the debt, increasing losses on credit relations over time (increasing debt in default). Therefore, default occurs when the bank believes that short-term instruments are growing irreversibly.

There remains the problem of predicting the moment when the difficulty becomes irreversible. Each business plan consists of an initial analytical forecast for a period of time (the period necessary for the effects of certain changes to manifest themselves explicitly in the assumptions), as well as subsequent stabilization of the situation. An event becomes irreversible when it can no longer change its state.

Thus, an assessment of the irreversibility of difficulties can only be carried out during a steady state. During the initial period, prolonged increasing moments of lack of funds do not necessarily mean irreversible difficulties, but can give rise to subsequent moments of solvency. Until the forecast really reaches a stable state, it is impossible to draw final conclusions (recall the development of successful giants such as Tesla, which had to go through years of growing financial needs and debts). An important conclusion is that the difficulty becomes irreversible when short-term credit instruments are constantly increasing in a steady state. This means that the company, when implementing its business decisions at full capacity, is still unable to repay its financial obligations, and, consequently, the plan fails, demonstrating the failure of the investments made and the associated financing capital.

At the time of providing updates or monitoring of short-term credit lines, the situation turns out to be more difficult than described above. At these moments, the probability of default is assessed by the credit institution and converted into an acceptable interest rate that will be requested/applied in the future ^[4]. The problem is that this rate changes the very assessment of the probability of default, more or less noticeably affecting future cash outflows for debt servicing, thus causing the company to decline more or less rapidly to a situation of irreversible growth of short-term credit instruments. By changing the estimate of the probability of default, the pricing of the loan also changes, and starting the cycle again. This activates two alternative circles: a vicious circle, when the bank's assessment turns out to be unfavorable (when the assessment of the probability of default raises rates, which in turn increase the probability of default), draining liquidity to pay rising interest; or a virtuous circle, when the assessment turns out to be favorable, because it allows the borrower to save financial costs.

If the vicious circle is not interrupted at the stage that keeps the company in balance (i.e., when a deterioration in creditworthiness is equivalent to a more than proportional increase in the rate), this means that there is no such rate at which the business plan is viable, while simultaneously satisfying the requirements of the lender. Thus, it can be more accurately indicated that default occurs when the operator believes that there is no interest rate capable of covering the estimated costs of the probability of irreversible growth of short-term credit instruments. The model proposed in this study operationalizes this theory.

The internal heterogeneity and high entropy of this research context make it difficult, if not impossible, to implement a single objective analytical approach. The most popular approaches in the literature ignore the complexity of this problem in favor of increasingly complex retrospective models of analysis. On the other hand, in order to develop a successful prospective model, we consider it necessary to include a partial lack of knowledge about the system itself as a characteristic of the model itself. This ontological concept of probability corresponds to the Bayesian probabilistic approach, which answers the following question: what is the probability that the company in question will fail predicting trends in environmental competitive conditions?

Developing this theory, from this point of view, the article makes an important theoretical discovery: the probability that the plan will fail should be a single numerical estimate, since it refers to a single temporarily identified event (an increase in short-term debt at a steady state). This assumption contradicts the literature and practice based on the frequency approach to probability, which we have already criticized earlier. If the probability of non-fulfillment of the plan is unique, it should also remain the same for each part of the time during which the plan is implemented until a steady state is reached. It is not claimed that there is no probability of non-fulfillment of the plan in 1 year; rather, the probability remains the same as that of the best forecast, since any alternative calculation of probability requires additional information not available at that time. Obviously, each time the information is updated, the probability can be recalculated.

If this is true, then there is another important theoretical conclusion: with a certain capital structure planned according to the specific assumptions of the plan, the pricing of each form of debt, regardless of the year of the loan or maturity, is based on the same probability of non-fulfillment of the plan. In other words, if the rates depended solely on the probability of non-fulfillment of the plan, then all current loans (with the exception of fixed rates) and also all future loans (in any year they would have been concluded) would have one rate. It is not claimed that pricing is independent of maturity or that the different composition of funding sources does not affect pricing in any way. In fact, it is obvious that a different maturity date brings contractual characteristics and risk factors that lead to different rates. It is also obvious that the different balance between equity, short—term and long—term debt affects the probability of default (PD) - sometimes even significantly - because it generates and distributes cash outflows in different ways over time, facilitating or worsening the borrower's financial situation.

In this context, term loans simply represent the assumption of the plan, which, like others, generates its residual effect on short-term credit lines. In accordance with the above theoretical statements, the validation of these assumptions is carried out using the same probability of default, although the pricing of each term loan can be adjusted taking into account many factors (for example, a different LGD – loss given default is the proportion of non–refundable losses after default (in the recovery process)).

Finally, in addition to their own calculations, each operator also takes into account the possible solutions of other operators who are invited to support the plan. In practice, each financial operator tries to predict the solvency analyses conducted by other operators interested in the company, reflecting on the information and skills that they expect to possess (for example, a participating bank behaves differently than a new bank). All the work of the financial operator is translated into the forecast of interest rates that will be applied to its own credit lines and term loans, as well as to loans from other lenders interested in supporting the borrowed company.

The conceptual model

The purpose of the study is to mathematically operationalize the theory of forecasting default, as previously stated, in order to build a model capable of checking – given certain input variables and their bias and uncertainty – if and at what rates virtuous circles can arise (in accordance with the definition of equilibrium as an interest rate that is stable for both a business plan, and satisfactory to creditors of capital) or vicious circles (a state of default) in the future. Specifically, in this scenario modeling, the interest rate will be recalculated until the probability of default changes and vice versa. We model long-term credit relations with asymmetry, uncertainty, signaling and dynamic learning ^[10]. Moreover, it is assumed that there are competitive interdependent incentive systems between the principal and the agent.

For simplification, the financial situation is considered to be financed only by banks with the same interest rate functions, without restrictions or preferences in granting loans and without any obligations. In other words, it is as if only one bank financed the business.

We present the mathematical details of our model. We adopt the notational convention that uppercase letters represent random variables and lowercase letters represent fixed quantities. Time is measured in discrete intervals (for example, years). Here it indicates the (random) time when either the debt is repaid or the company is in default. Budget management modeling is carried out as follows: ()

(1)

where is the short–term net financial position (STNFP) at the time , and – indicates the change in STNFP at the time . So, this is the free cash flow from operating and investing activities, and this is the serviced debt at the moment. This is expressed as follows:

(2)

where – denotes the net change in term loans at the time (repayment of term debt less proceeds from the issue of new debt), and, respectively, represent interest expenses on outstanding term debt and STNFP. We assume that interest expense is a linear function of the outstanding debt at the beginning of each period:

(3)

where is the interest rate. To simplify , we assume that , for all where the rate is constant. Thus, the rate is only a function of the probability of default. To emphasize this, we write it down .

To model the impact of analyst revisions, we assume that is a random variable with an average value (offset) and a standard deviation. For simplicity, we accept and as independent of, but our analysis can easily be extended to time-dependent forecasts. Thus, and accordingly represent the reliability of the plan and the uncertainty of the plan according to the revisions of analysts. The key point is that according to the Bayesian probability model, they are input parameters that can be configured by the analyst. We denote as the result of a random variable , where denotes a sample. We assume that the company enters a stable state after a certain time , when it becomes permanent for .

To estimate the average value and distribution of various quantities of interest, we repeatedly simulate the main random process and then take empirical averages. Here we illustrate this point of view and explain how we estimate the probability of default. We define a default event as an increase in STNFP in a steady state. Formally, the default event is defined as:

(4)

where denotes a specific selection. The probability of default PD is approximately calculated as:

(5)

When it is large enough, the probability of default does not depend on the samples and therefore we omit it from the notation. On the other hand, the probability of default depends significantly on the interest rate , and we emphasize this in the notation as . In general, if (0,1) and represents some rate, and we set , then . This prompts us to determine the equilibrium rate. For this purpose, we define a composite function:

(6)

The equilibrium rate is a fixed point of the function , that is , it satisfies

(7)

Explicit calculation, as a rule, is not a trivial process; therefore, we turn to approximation methods. Given the recursive structure of the problem, we choose a reliable technique known as the fixed point method (Burden et al., 2015). In short, this method generates a sequence such that , and then approximates for . However, the fixed point algorithm extracts only a subset of the fixed points that we consider stable. To calculate unstable fixed points, we resort to the more complex fzero MATLAB routine. This routine uses a combination of bisection of the secant line and inverse quadratic interpolation of Brent and Forsyth.

For a deeper study of the lender's behavior, we associate the above-defined equilibrium rate with the rate that maximizes the benefit for the bank. The profitability of the bank is determined in this way:

(8)

if or

(9)

if.

The constant 1 is the discount factor, and a is the so–called default loss. We estimate the expected return as

(10)

where, again, we can omit the dependency on, if large enough. On the other hand, we emphasize the dependence on the interest rate by writing

Results

A company is a future-oriented competitive system and is considered in a state of equilibrium as long as its stakeholders trust its future. In moments of predictable crisis in this system, credit institutions become stakeholders who ultimately decide whether to maintain this balance, to what extent and to what point. Banks will continue to renew and increase the company's credit lines as long as they consider it profitable. Ultimately, it depends on the probability that they attach to the irreversible growth of the firm's credit lines in a stationary state. This probability of default is measured based on the reliability and uncertainty of the business plan and is updated based on interest rates that are stable for the plan itself. Thus, equilibrium is a situation of stable and constant meeting points between predictable supply and demand trends for a company's credit.

The case of JSC "Infa-Hotel"

To understand the potential of the model and demonstrate its specific functioning, it is applied to the case study of JSC Infa-Hotel, the leader of the hospitality industry in Moscow (Hotel Savoy, luxury segment).

In 2021-2022, the Infa-Hotel company entered into bankruptcy proceedings, finding itself under the weight of debt obligations. In April 2023, the company received court approval for a debt restructuring plan. As a result of the restructuring process, Infa-Hotel Company's regulatory value for the absolute liquidity ratio was in the range of 0.2–0.5 – the indicator was below the norm in 2021, but since 2022 the indicator has returned to normal, and continues to improve in 2023. Debt obligations decreased from 78.678 million rubles to 6.750 million rubles.

The purpose of this case is to explore the consequences arising from our model, applying it to the periods before and after default. In particular, we represent the company in three different time periods:

(a) in December 2022, just before the default, to understand the predictability of this event;

(b) in December 2018, to find out whether the default of Infa-Hotel was predictable four years before it occurred and to what extent;

(c) in December 2023 to assess the validity of the company's recovery plan.

Case A: 2022

The business plan is based on information gathered from the Annual Report for 2021 and corporate news published at that time. In particular, our assumptions are as follows (Table 6).

Table 1 – Input data for case simulation a: 2022

Regarding debt service, we are considering:

STNFP

Long-term debt 78,678 which must be repaid within 12 years starting from t=1 with the specified payments (Table 7).

Table 2 – Payments according to the repayment periods of case a debt: 2022

Rate: , where is a constant value, is a constant value.

Free cash flows are slightly positive on average, but turn negative in many scenarios (Figure 1).

Figure 1 – Results of free cash flow simulation (FCF) for Infa-Hotel in 2022. 50 implementations are shown in blue. The average value (in red) is calculated based on 2500 implementations

Nevertheless, the huge debt burden makes banking intervention impractical at any interest rate (Figure 2).

Figure 2 – Simulation of the results for the Infa-Hotel company in 2022

Figure 3 – Simulation of the results for the Infa-Hotel company in 2022 with a fixed rate of 6%

Please note that the probability of default at a zero rate exceeds 40%. Assuming a fixed interest rate on all amortised loans at 6% (as it really happens on average), the situation does not change. The absence of short-term debt at the beginning of the period makes PD significantly independent of the interest rate on STFP (Figure 3). The only reasonable way out at that time, it would seem, was to give up the fight altogether.

Case B: 2018

The business plan is based on information from the Annual Report for 2017 (when the complete renovation of the room stock of the Savoy Hotel was carried out) and corporate news of that time. In particular, our assumptions are as follows (Table 3).

Table 3 – Input data for the simulation of case b: 2018

Regarding debt service, we are considering:

STNFP

Long-term debt of 183.448 which must be repaid within 12 years starting from t=1 with the specified payments (Table 2).

Table 2 – Payments according to the repayment periods of case b: 2018

The trend of free cash flow on average looks higher than in 2022 (Figure 4).

Figure 4 – Results of free cash flow simulation (FCF) for the Infa-Hotel company in 2018. 50 implementations are shown in blue. The average value (in red) is calculated based on 2500 implementations

However, the situation is always moving towards a lack of equilibrium (Figure 5).

Figure 5 – Simulation of the results for the Infa-Hotel company in 2018

Nevertheless, the bank can make a profit at rates of about 13%, although this profitability does not compensate for the accepted risk (PD is constantly growing and requires a higher interest rate). Assuming a fixed interest rate on all amortised loans at 6% (as it happens on average), the model reaches an equilibrium point only theoretically, given that any provision of short-term loans (at any interest rate) will lead to losses for the bank (Figure 6).

Figure 6 – Simulation of the results for the Infa-Hotel company in 2018 with a fixed rate of 6%

In conclusion, we can say that with reasonable assumptions about the company's future activities, it should have suspended operations four years earlier with almost half a billion less debts!

Case C: 2023

The business plan is based on the approved restructuring plan and performance results for 2023. In particular, our assumptions are as follows (Table 4).

Table 4 – Input data for case simulation c: 2023

Regarding debt service, we are considering:

STNFP

Long-term debt of 6,750 that must be repaid within 5 years starting from t=1 with the specified payments (Table 5).

Table 5 – Payments according to the repayment periods of the case debt c: 2023

The trend of free cash flow is shown in Figure 7.

Figure 7 – Simulation of the results for the Infa-Hotel company in 2023 with a fixed rate of 6%

The situation is improving significantly compared to the past and is almost reaching equilibrium (Figure 7), with positive profitability for banks at rates between the minimum discount rate and about 50%.

In conclusion, it can be noted that the lack of balance will correspond to the behavior of creditors who operate under restructuring conditions. This means that at reasonable market rates, even if the associated probability of default (PD) required further rate increases, the bank did not move in this direction, nevertheless ensuring "acceptable" profitability (also taking into account losses from credit cuts). Under the conditions of a free market, the agreement would most likely not have taken place, and in fact no new bank chose to provide new credit lines to Infa-Hotel in 2023.

The following happens: an increase in the discount rate of the loan offer leads to an increase in the minimum convenient rate that the bank is obliged to apply to the counterparty. This confirms that business restructuring plans implemented at variable interest rates are under stress due to the current restrictive monetary policy of the central bank of the Russian Federation. That is, by applying this model in the archaic period before the onset of default and after debt restructuring and reorganization of the company's activities, we "kind of go back to the past" in order to make a forecast about the probability of default risk.

***

Based on this theory, in a consistent probabilistic environment, we model the equilibrium of a loan company based on predictable conditions of its credit demand (market trends, business plan, LGD, etc.) and predictable conditions of the credit supply segment (interest rate function, availability of information and analytical tools, etc.). These forecasts mutually influence each other and consist of assessments of credit, market and idiosyncratic risks. The model quantifies the probability of default (PD) by estimating the intensity of checking for future simulated default events. This PD is a unique numerical estimate (therefore, the interest rate regime between different forms of debt depends on factors external to PD). This PD is a function of the rate applicable in the future by credit institutions, and vice versa.

In response to research questions, the model provides important results:

1) given the interest rate on the debt, it provides the probability that the company will remain viable in the future;

2) it verifies the existence of a rate capable of ensuring the financial health of the company and at the same time minimal satisfaction of creditors;

3) it verifies the existence of a rate that maximizes the profitability of creditors while ensuring the financial health of the company;

4) it evaluates the intensity of the bargaining power of the borrower and the lender;

5) it determines the maximum level of sustainable debt at rates considered satisfactory to creditors (if any);

6) It determines the impact of a certain debt structure/restructuring on financial health.

Conclusion

The default theory and its associated model can unify research and tools in various fields (corporate finance, credit risk management, financial intermediation, structured finance, project finance, corporate restructuring, etc.).

The operational result of the model is the creation of an "individualized" model for predicting default in the debtor's financial market, where market and competitive forces (market risk), the potential and vulnerability of the company (idiosyncratic risk), financial market skills, information asymmetries and future credit trends (credit market risk) dynamically interact to create a future-oriented real-time system to predict the company's PD. A model with the following characteristics appears.

Firstly, it makes extensive use of "soft" information about the company's future, which has been shown to be fundamental for assessing credit risk. This has two advantages:

(1) The model can also be applied to startups, companies in the process of restructuring or radical transformation;

(2) any predictable internal (strategic and operational) and external (competitive and loan offers) changes of the company may be subject to an assessment update with a real-time reaction. Due to this ability, the model can be used by credit institutions at the stages of provision/updating/monitoring, as well as at the pricing stage, as well as by market operators for bond valuation and pricing.

Secondly, human assessment skills are integrated into the assessment model, as the literature and regulatory authorities have long recommended. The introduction of human subjectivity into the model reproduces the complexity of the market, emphasizing the importance of the experience of operators and their freedom. This contributes to entrepreneurial innovation momentum, as well as healthy competitiveness among financial operators.

Thirdly, the assessment focuses on the evolution of the borrower's debt structure with due consideration for debt repayment periods, rate variability, collateral, etc. This allows you to quantify debt service for each year and get an accurate estimate of the probability of default (PD). Therefore, the mechanisms should facilitate the selection of firms, structuring debt based on the characteristics and duration of their financial needs, and setting the right price for each type of loan product. This should improve the functioning of the financial market by mitigating the misallocation of financial liabilities.

Fourth, the default forecast is formulated before reaching a state of stationarity. Consequently, the horizon of PD estimation is usually expanded in comparison with lag models that show myopic predictive abilities. The advantage of this is the ability to assess the real sustainability of the company's business model, especially in terms of socio-environmental sustainability - a condition that is becoming increasingly important. A joint consideration of these characteristics of the model makes it possible to introduce an assessment of idiosyncratic risk into the assessment model, taking into account the mechanisms of interaction between the specific incentive systems of each operator. This leads to the creation of a truly future-oriented valuation model, which, as such, should not be influenced by the classical stationary limitations of the most common valuation systems today, which have generated serious market inefficiencies.

Thus, our main conclusion is that credit risk measurement tools and the operators who use them must take a step back in order to move forward, re-mastering the technical aspects of fundamental analysis. It is necessary to create rating systems that "return to the future."

Exploring the prospects for future research, if the risk of default is thus included in the cost of capital, our work can outline interesting prospects in the field of corporate valuation and the study of leverage dynamics — a topic far from achieving indisputable results ^[7]. Regarding the limitations of the model, we focus on optimal betting strategies with functions that are constant over time (i.e., a constant bet). Since it is currently unclear to us how to extend fixed point analysis to dynamic betting strategies over time, this problem represents an interesting area of future research. Our model was developed under the assumption of a single bank that finances the company and assumes simplifications in building cash flows compared to reality. However, it is much more complex than existing models because it tries to replicate the financial management of a company. This implies the need for a much larger volume of input data and, accordingly, an associated increase in costs and processing time.