Introduction

In financial institutions good estimates of credit risk are important both for pricing individual loans and for managing risk at the aggregate level. The authorities, central banks, and supervisors are concerned with the stability of banks and the financial system make assessments of credit risk at the aggregate level. Since credit risk is not directly observable, it is usually estimated by using statistical models. In this paper we present such a model linking macroeconomic variables to an aggregate measure of credit risk on loans made to corporate borrowers, where the risk is measured at the industry level. The alternative to this direct approach is to estimate credit risk for each firm and then aggregate the risk for all firms in the industry. The advantage with a direct modeling is that it is easier to link the development in macroeconomic variables to the development in credit risk. Firm-specific risk typically relies on financial ratios from the firms' financial statements and it is challenging to link macroeconomic variables to individual firms.

Credit risk is the risk of incurring a loss on a loan. The expected loss on a loan is determined by the probability of default and the loss provided that default has occurred. Default is here defined as the event when payments are not made according to the loan agreement. For many financial institutions credit risk, and in particular credit risk on loans to corporate borrowers, is the major source of risk. Other sources of risk are risks related to investments in securities (market risk), securing future financing (funding risk), and possible losses due to operational failures. Banks' exposures to credit risk in different industries vary. It is therefore important to both estimate the risk at the industry level and to take into account the exposure to different firms. We therefore use the debt-weighted probability of default (DWPD) per industry as our aggregated risk measure. When computing this measure we use the firm-specific estimates of probabilities of default (PDs) from Norges Bank's SEBRA model. The SEBRA model, see Eklund et al. (2001) and Bernhardsen and Larsen (2007), estimates the default risk for each firm based primarily on financial ratios computed from the firms' financial statements. The PDs are then aggregated by weighting each firm's PD with its debt and then taking the sum over all firms. (2)

Modeling of aggregate credit risk has become increasingly important when analysing the economic development in countries. Norges Bank monitors the stability of the financial system in Norway and follows closely the development in credit risk on corporate loans. Norges Bank's assessment of credit risk is also included in the Financial Stability report, which is published twice a year. One method for uncovering and identifying potential risks to the financial system or to individual financial institutions is to perform stress tests.3 One type of stress test is a prediction of financial results and balance sheet items of banks or a group of banks based on a set of assumptions about future economic developments. In a macro stress test future economic developments are typically represented by key macroeconomic variables such as GDP growth, interest rates, and exchange rates. The term stress test means that the set of assumptions (which may be termed a scenario) are chosen to represent a very negative development in the economy. Norges Bank uses several models when performing stress tests. The main models are a macro model and micro models covering the risks in the household and corporate sectors as well as a model for banks, see Figure 1. Andersen et al. (2008) presents this model framework in detail. Based on an assumed negative macroeconomic development, a model called the Small Macro Model is used to predict developments in future macroeconomic variables. These variables are then combined with micro information and separate micro models for households and firms. The output from the micro models, debt at risk, is then combined with the macroeconomic variables to predict banks' future income and capital adequacy.

The model presented in this paper may be used as a part of a stress test for estimating the development in corporate credit risk within the framework shown in Figure 1. Direct modeling of aggregate credit risk may be a supplement to the current micro approach. In the current approach estimated future macro variables are used to predict firms' future financial statements. In a second step, the default risk is then estimated for each firm based on these financial statements. This approach is described in detail in Bernhardsen and Syversten (2009).

We proceed as follows: We first present the data underlying the model in section two. Section three presents the model. In order to evaluate the model performance, we perform backtests at the industry level. These backtests are presented in section four. It is also important to make assessment of risk for portfolios that do not contain all firms in an industry. Section five therefore presents an analysis of the errors made when using aggregate estimates of risk on smaller loan portfolios.

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Data

The debt-weighted default probability for an industry at time t is

[DWPD.sup.t] = [SIGMA][[jPD.sup.i].sub.t] [[w.sup.i].sub.t] (1)

where [[PD.sup.i].sub.t] is the probability of default for firm i as estimated at time t. [[PD.sup.i].sub.t] is estimated by using Norges Bank's default prediction model SEBRA Basic, see Bernhardsen and Larsen (2007). Each firm's weight is equal to the ratio of the firm's debt ([[D.sup.i].sub.t]) to aggregated portfolio debt,

[[w.sup.i].sup.t]= [[D.sup.i].sup.t]/[[SIGMA].sup.i] [[D.sup.i].sup.t]

Debt-weighted probabilities of default may be interpreted as the average expected fraction of 1 krone of loan in the portfolio that defaults next year. By using each firm's debt in the weight, we explicitly take into account the loan exposure of different firms. The risk-weighted debt (RWD) for the portfolio is the expected amount of debt that is expected to default the next year,

[RWD.sup.t] = [DWPD.sup.t] x [D.sup.t] (3)

where [D.sup.t] is the total amount of debt in the portfolio. In other words, risk-weighted debt is simply the debt-weighted probability of default scaled by the level of debt.

Table 1 reports selected descriptive statistics for the sample of firms for the years 1988-2008. The sample consists of Norwegian joint stock firms. Statistics are reported for 14 industries and for all industries aggregated. We also report statistics for all firms when we exclude oil-related firms (Oil services and Oil and gas). The industry classification is based on NACE Rev. 1.1. (4) The most important industries as measured by their share of bank debt are Commercial property, Shipping, and Manufacturing and mining with, respectively, 39.8, 13.8, and 12.7 of total bank debt at the end of 2008. Bank debt is here measured according to the information about the firms' debt in their balance sheets. The number of firms has been increasing during the sample period and the number of firms varies between the industries. Trade and retail has the highest average number of firms with above 27 000. Manufacturing and mining, Commercial property, and Business services have all average number of firms of above 10 000.

Table 1 also reports the descriptive statistics per industry for the yearly computed DWPDs and the yearly mean of the probabilities of default (MPDs). The average MPD was about 4.5 percent for all firms excluding oil-related firms while the average DWPD was about 2.5 percent. This highlights the importance of taking into account exposure when...