Modeling credit risk in the presence of central bank and government intervention

Bernd Engelmann

1 Introduction

Credit risk modeling is an integral part of a lending institution’s internal and external risk management. From a statistical point of view, credit modeling is particularly challenging because of infrequent observation of information (monthly, quarterly or, in some segments, only annually), while prediction horizons in many risk applications are the remaining lifetimes of loans, which could be up to 30 years. This makes approaches driven entirely by data infeasible, and credit modeling is therefore usually done by combining data analytics with expert judgment. Experts develop a view of the future (for example, in the form of a macroeconomic scenario) and data analytics then uses past information to translate the future scenario into a credit risk parameter such as a default probability or a loss estimate.

This modeling approach is used in central bank stress testing, in which regulators such as the US Federal Reserve Bank (Fed), the European Central Bank (ECB) or the European Banking Authority (EBA) provide a set of scenarios and ask banks in their jurisdictions to evaluate the scenarios’ impacts on their portfolios. The stress test results are published and allow market participants and researchers to gain insights into banks’ resilience to economic downturns, the risk methodologies used in different banks and the reaction of market participants to the publication of stress testing outcomes (Ahnert et al 2020; Georgescu et al 2017; Sahin et al 2020).

An additional important risk application requiring scenarios is the modeling of loan loss provisions under the International Financial Reporting Standard 9 (IFRS 9) (International Accounting Standards Board 2014). An explicit requirement of this accounting standard is the calculation of the expected loss for loan portfolios under different probability-weighted scenarios. The choice of scenarios and probability weights is in general difficult to obtain from data analytics and has to be developed by expert judgment (Bellini 2019; Deloitte 2019).

Since mid-2020, challenges in credit modeling have increased considerably. Due to interventions by governments and central banks following the Covid-19 outbreak around the globe, relationships between credit risk measures and macroeconomic drivers that had remained stable for decades have become disconnected. We will illustrate this using macroeconomic data published by the Fed.

Section 2 demonstrates how macroeconomic models fail after the introduction of Covid-19 measures and shows that there is no easy fix within the framework of standard time series modeling. Section 3 outlines how outside intervention can be included in a credit model, allowing analysts to build intervention scenarios. Section 4 applies the concepts of Section 3 to the data, and analyzes the potential side effects of intervention, such as increased inflation. Section 5 states our conclusions.

2 Failure of credit models under government and central bank intervention

Table 1: Macroeconomic variables in the Fed’s CCAR 2021 data set.

Variable	Description
RGDPgr	Real gross domestic product growth
NGDPgr	Nominal gross domestic product growth
RINCOMEgr	Real disposable income growth
NINCOMEgr	Nominal disposable income growth
UR	Unemployment rate
CPIgr	Consumer Price Index (CPI) inflation rate
IR_3M	Three-month Treasury rate
IR_5Y	Five-year Treasury yield
IR_10Y	Ten-year Treasury yield
BBB_Yield	BBB corporate yield
MR	Mortgage rate
PR	Prime rate
Dow	Dow Jones Total Stock Market Index (level)
HPI	House Price Index (HPI) (level)
CRE	Commercial Real Estate Price Index (level)
VOL	Market Volatility Index (level)

To illustrate the modeling challenges that arose in mid-2020, data that was published by the Fed for the Comprehensive Capital Analysis and Review 2021 (CCAR 2021) stress test is used.¹¹ 1 The data was retrieved from http://www.federalreserve.gov/supervisionreg/ccar-2021.htm. This data is supplemented with quarterly data on mortgage loan default rates published by the Federal Reserve Bank of New York.²² 2 Annualized quarterly default rates of US residential mortgage loans (page 25 in the Excel sheet “HHD_C_Report_2021Q2.xlsx”) available from 2000 Q1 until the end of 2021 Q2 were downloaded from https://nyfed.org/3LENVeO. The CCAR 2021 data set contains quarterly data from 1976 Q1 to 2020 Q4, while the mortgage loan default rate data set contains quarterly data from 2000 Q1 to 2021 Q2. The macroeconomic variables included in the former data set are displayed in Table 1. The House Price Index (HPI) has been transformed into a year-on-year growth variable (HPIgr) for modeling purposes. Compared with quarterly HPI growth, year-on-year HPI growth is less volatile and is, therefore, preferred. Economic reasoning suggests that RINCOMEgr, UR, CPIgr, MR, PR and HPIgr should be promising candidates in a model explaining mortgage default rates. Together with the historic macroeconomic data, the Fed provides a baseline scenario reflecting its expectation of future macroeconomic development and an adverse scenario showing its view on macroeconomic data when a crisis (such as a house price crash or an economic recession) materializes. Both scenarios are provided from 2021 Q1 to 2024 Q1.

Quarterly US mortgage default rate, unemployment rate, year-on-year HPI growth and growth in real disposable income from 2000 Q1 to 2020 Q4 together with the Fed's adverse scenario until 2024 Q1. — Figure 1: Quarterly US mortgage default rate, unemployment rate, year-on-year HPI growth and growth in real disposable income from 2000 Q1 to 2020 Q4 together with the Fed’s adverse scenario until 2024 Q1.

Figure 1 shows the observed US mortgage default rate, unemployment rate, year-on-year HPI growth and growth in real disposable income over the last two decades together with the Fed’s adverse scenario until 2024 Q1. It can be seen that default and unemployment rates moved together, while HPI growth and the default rate were negatively correlated. In 2020 Q3, however, the unemployment rate spiked while the default rate was still declining – an effect that was not observable in data for previous periods. The key driver was an intervention by the US government, which first drove up unemployment through strict lockdowns following the Covid-19 outbreak and then paid aid to its citizens. This allowed borrowers to service their debt despite reduced or lost income following business closures during lockdown periods.

2.1 Credit models excluding the past default rate

Table 2: Diagnostics (coefficients, standard errors, adjusted

R^{\text{2}}

) of DR model 1 (2.1). [Standard errors are given in parentheses.

{}^{***}p<\text{0.001}

;

{}^{**}p<\text{0.01}

;

{}^{*}p<\text{0.05}

.]

Intercept	$-$ 2.491 ${}^{***}$
	$-$ (0.032)
$\text{UR}_{i-\text{1}}$	$-$ 0.095 ${}^{***}$
	$-$ (0.005)
$\text{HPIgr}_{i-\text{1}}$	$-$ 0.021 ${}^{***}$
	$-$ (0.001)
Adjusted $R^{\text{2}}$	$-$ 0.9335

Figure 2: Fit and residuals for DR model 1.

To illustrate why government interventions pose a big challenge to credit models, a simple autoregressive model of order zero ( $\operatorname{AR}(0)$ ) to explain probit-transformed default rates by the unemployment rate and HPI growth is estimated. The independent variables enter the regression with a lag of one quarter. Utilizing data from 2000 Q1 to 2019 Q4 results in the default rate model (DR model)

\varPhi^{-1}(\mathrm{DR}_{i})=-2.4905+0.0949\mathrm{UR}_{i-1}-0.0212\mathrm{% HPIgr}_{i-1}+\varepsilon_{i},

(2.1)

where $\mathrm{DR}_{i}$ is the annualized default rate of quarter $i$ , $\mathrm{UR}_{i-1}$ the unemployment rate observed in quarter $i-1$ and $\mathrm{HPIgr}_{i-1}$ the year-on-year growth in HPI observed in quarter $i-1$ . The error terms $\varepsilon_{i}$ are assumed to be independent and identically and normally distributed. More detailed diagnostics are presented in Table 2. A plot of the model fit and the residuals, shown in Figure 2, further confirms that the model explains mortgage default rates reasonably well.

Table 3: Out-of-sample predicted default rates (in percent) from model (2.1) versus realized default rates (in percent) for the four quarters in 2020.

	Q1	Q2	Q3	Q4
Estimated default rates	1.2890	1.3276	8.9060	3.7031
Observed default rates	1.1713	1.0813	0.9612	0.6488

Next, the model is applied on data for the calendar year 2020. Default probabilities are estimated and compared with realized default rates. The outcome is displayed in Table 3. While the model works quite well in the first two quarters, its predictions are highly inaccurate in the second half of 2020. The reason for this is that a sharp increase in the unemployment rate did not directly lead to defaults, owing to government interventions such as monthly Covid-19 relief checks, which were supported by financing from the central bank.

A natural question is whether a different macroeconomic model could have done a better job. Table 1 contains RINCOMEgr, a variable measuring the growth of real disposable income. In Figure 1 it can be seen that this variable spikes together with the unemployment rate. Therefore, the growth in real disposable income might offset the increasing unemployment rate in 2020 and improve default rate predictions. However, Figure 1 also shows that real disposable income growth is quite volatile. Estimating a macroeconomic model using UR and RINCOMEgr as independent variables leads to a model with a small, statistically insignificant coefficient of RINCOMEgr.

Therefore, smoothing RINCOMEgr by using a moving average should improve the model. Using an averaging window of two quarters results in the model displayed in Table 4 and Figure 3.³³ 3 Using a different averaging window such as three or four quarters leads to similar results. Therefore, only the results for averaging over two quarters are reported. Predicting default rates using this second model leads to the outcome reported in Table 5. Compared with Table 3, the numbers have improved, but not considerably. Estimation errors are still too high.

Table 4: Diagnostics (coefficients, standard errors, adjusted

R^{\text{2}}

) of DR model 2, using UR and average RINCOMEgr as independent variables. [Standard errors are shown in parentheses.

{}^{***}p<\text{0.001}

;

{}^{**}p<\text{0.01}

;

{}^{*}p<\text{0.05}

.]

Intercept	$-$ 2.684 ${}^{***}$
	$-$ (0.074)
$\text{UR}_{i-\text{1}}$	$-$ 0.122 ${}^{***}$
	$-$ (0.011)
$\text{avRINgr}_{i-\text{1}}$	$-$ 0.021 ${}^{*}$
	$-$ (0.009)
Adjusted $R^{\text{2}}$	$-$ 0.6666

Figure 3: Fit and residuals for DR model 2.

Table 5: Out-of-time predicted default rates (in percent) from the model of Table 4 versus realized default rates (in percent) for the four quarters in 2020.

	Q1	Q2	Q3	Q4
Estimated default rates	1.1085	1.1655	5.2751	2.5807
Observed default rates	1.1713	1.0813	0.9612	0.6488

Estimated default rates under the Fed's baseline and adverse scenarios using model (...) calibrated until 2019 Q4 and predicting from 2020 Q1 to 2024 Q1. — Figure 4: Estimated default rates under the Fed’s baseline and adverse scenarios using model (2.1) calibrated until 2019 Q4 and predicting from 2020 Q1 to 2024 Q1.

Finally, model (2.1) is applied to the two scenarios provided in the data. The resulting default probability curves under the baseline and the adverse scenarios are displayed in Figure 4. Note that in this graph the model predictions, rather than the realized values, are displayed for the year 2020. The shape of the resulting default probability curves is quite peculiar and clearly beyond our expectations. The main reason for this is that unemployment was driven not by the economy but by government intervention, resulting in an atypical time series of unemployment rates. This shape of $\mathrm{UR}_{i}$ translates into the spike in the default probability curve.

Note that predictions in Figure 4 are calculated by simply inverting the predicted probit-transformed default rates without applying any bias correction because the model bias is rather small when computed in-sample, with an average realized default rate of 2.64% compared with an average estimated model default rate of 2.61%. A bias of 0.03% is much smaller than the accuracy that can be expected from these kinds of models, which is why correcting it does not increase prediction accuracy.

2.2 Credit models including the past default rate

After showing how $\operatorname{AR}(0)$ models suffer from intervention, it will be demonstrated that the same is true for $\operatorname{AR}(1)$ models. The best $\operatorname{AR}(1)$ model in terms of model fit and economic plausibility, using data until 2019 Q4, is

\varPhi^{-1}(\mathrm{DR}_{i})=-0.1877+0.8972\mathrm{DR}_{i-1}-0.0047\mathrm{% HPIgr}_{i-1}+\varepsilon_{i},

(2.2)

where HPIgr and DR enter with a lag of one quarter. The model diagnostics are displayed in Table 6, and the fit and residuals are plotted in Figure 5. It is impossible to combine UR with DR, because all possible model combinations, including those with different time lags, lead to a negative coefficient for UR.

Table 6: Diagnostics (coefficients, standard errors, adjusted

R^{\text{2}}

) of DR model 3, using DR and HPIgr as independent variables. [Standard errors are shown in parentheses.

{}^{***}p<\text{0.001}

;

{}^{**}p<\text{0.01}

;

{}^{*}p<\text{0.05}

.]

Intercept	$-$ 0.188 ${}^{***}$
	$-$ (0.032)
$\text{DR}_{i-1}$	$-$ 0.897 ${}^{***}$
	$-$ (0.017)
$\text{HPIgr}_{i-\text{1}}$	$-$ 0.005 ${}^{***}$
	$-$ (0.001)
Adjusted $R^{\text{2}}$	$-$ 0.990

Figure 5: Fit and residuals for DR model 3.

Table 7: Out-of-time predicted default rates (in percent) from model (2.2) versus realized default rates (in percent) for the four quarters in 2020.

	Q1	Q2	Q3	Q4
Estimated default rates	1.1870	1.2527	1.1592	1.0249
Observed default rates	1.1713	1.0813	0.9612	0.6488

Default probabilities under the Fed's baseline and adverse scenarios using model (...) estimated until 2019 Q4 using realized default rates until 2021 Q2 and predicting until 2024 Q1. — Figure 6: Default probabilities under the Fed’s baseline and adverse scenarios using model (2.2) estimated until 2019 Q4 using realized default rates until 2021 Q2 and predicting until 2024 Q1.

The results for predicting default rates in the year 2020 analogously to Table 3 are displayed in Table 7. The results look much better compared with the $\operatorname{AR}(0)$ models of the previous subsection. However, when we run the model on the two scenarios provided by the Fed, the outcome, displayed in Figure 6, does not look economically plausible.

Recall from Figure 1 that, compared with the 2007–9 global financial crisis, the adverse scenario includes a comparable decline in house prices and even higher unemployment rates. Yet the best $\operatorname{AR}(1)$ model predicts default rates of little more than 3% in this scenario, which is only slightly higher than the sample average of 2.64%. Estimating the model with data until 2020 Q4 would even result in predicted values below the sample average. The reason is that the $\operatorname{AR}(1)$ model is largely driven by past default rates. The historically low default rates in 2020, which are not the result of an economic process but merely the effect of government intervention, distort the model and keep model predictions unreasonably low. Therefore, $\operatorname{AR}(1)$ models might still have some value for short-term predictions so long as intervention persists. Should government intervention be discontinued abruptly, the models would fail to predict the expected increase in default rates. Moreover, they are in any case useless for long-term projections, as the severity of the model error is expected to increase with increasing suppression of the default rates by intervention.

Table 8: Results of the Kwiatkowski et al (1992) stationarity test for selected variables for the null hypothesis of level stationarity together with the number of differences required to achieve stationarity at a type I error of 5%.

Variable	$𝒑$ -value	differences
		No. of
DR	0.0301	2
CPIgr	0.1	0
HPIgr	0.1	0
MR	0.01	1
PR	0.01	1
RGDPgr	0.1	0
RINCOMEgr	0.1	0
UR	0.0509	0

The above analysis was focused on predictive accuracy but was not very clean from a statistical point of view. The high $R^{2}$ in Table 6 is likely due to the nonstationarity of the time series for default rates. To gain further insights, the Kwiatkowski et al (1992) stationarity test is performed for selected macro variables and its results are displayed in Table 8.

It can be seen that MR and PR are nonstationary, that UR is a close call when a type I error of 5% is used and that the dependent variable DR is nonstationary. When analyzing the residuals of DR model 1 and DR model 3, it can be seen that they are stationary for the former and nonstationary for the latter. To make all variables stationary, one difference is required for MR and PR, while DR has to be differenced twice. However, when trying to build a model for the twice differenced $\mathrm{DR}_{\mathrm{d}2}$ , it turns out that it is impossible to include either UR or HPIgr or both. Estimated models, regardless of the time lag used, show a positive sign for HPIgr and a negative sign for UR, contrary to economic intuition. Further, DR, UR and HPIgr are not cointegrated, which prevents the application of more advanced approaches such as the error-correction model (Engle and Granger 1987).

The discussion in this subsection and the previous one might discourage the application of credit models, since the impact of central bank and government interventions distorts their predictive power and their forecasts look unreliable. The main purpose of this paper is to propose an extension to $\operatorname{AR}(0)$ credit models by adjusting them for outside interventions. This makes model predictions more accurate in the short term and allows risk managers to include intervention scenarios in their modeling to gain a view on the likely impact of no, moderate or strong interventions by governments and central banks. In contrast to $\operatorname{AR}(1)$ models, $\operatorname{AR}(0)$ models can be “repaired” by intervention adjustments.

3 Credit risk modeling with intervention

The concept of default probability is ambiguous in credit risk management. Depending on the particular purpose (Basel regulation, IFRS 9, stress testing), different default probabilities are required. For Basel regulation, a long-term average probability of default (PD) is typically applied, which is also known as a through-the-cycle (TTC) PD, ie, a prediction of the average default rate over an economic cycle (Basel Committee on Banking Supervision 2006). In contrast, impairment models for accounting purposes require forward-looking PDs, ie, default probabilities that are predictions of default rates over the next year, which are also known as point-in-time (PIT) PDs (International Accounting Standards Board 2014). Finally, stress tests require PDs under a specific scenario, usually an economic recession, leading to a downturn PD.

A framework for transforming PIT PDs into TTC PDs and vice versa based on the simple one-factor credit risk model underlying Basel regulation has been developed in Aguais et al (2007) and Carlehed and Petrov (2012). A detailed explanation of this framework’s adaptation to Basel II/III, IFRS 9 and stress testing (which is beyond the scope of this paper) can be found in Engelmann (2021).

The methodology used in Carlehed and Petrov (2012) can be extended for modeling government intervention. The starting point is a one-factor credit risk model:

r=\sqrt{\rho}X+\sqrt{1-\rho}\xi,

(3.1)

where $r$ is the log return on the assets of a borrower, $\rho$ the asset correlation, $\xi$ the borrower-specific risk factor and $X$ the systemic risk factor. Both $X$ and $\xi$ are assumed to follow a standard normal distribution and, further, both random variables are assumed to be independent. A borrower defaults if $r$ is below a threshold $\theta$ . Therefore, the unconditional default probability of a borrower is given as $P(r<\theta)=\varPhi(\theta)$ . Conditional on a realization $x$ of $X$ , the borrower default probability is given by

P(r<\theta\mid x)=\varPhi\bigg{(}\frac{\theta-\sqrt{\rho}x}{\sqrt{1-\rho}}% \bigg{)}=\varPhi\bigg{(}\frac{\varPhi^{-1}(P(r<\theta))-\sqrt{\rho}x}{\sqrt{1-% \rho}}\bigg{)}.

(3.2)

In (3.2) $x$ can be interpreted as the state of the economy. If $x$ is negative, the economy is in a recession; if it is positive, the economy is booming and default probabilities are low. Equation (3.2) provides a transformation between an unconditional (or TTC) default probability and a conditional (or PIT) default probability. To use this transformation, the parameters $\rho$ and $\theta$ have to be estimated and a link between macroeconomic factors and the abstract state of the economy $x$ has to be established.

From a time series of realized default rates $\mathrm{DR}_{i}$ in a particular loan segment (such as the mortgage loan default rates in the example above), the parameters $\theta$ and $\rho$ for this loan segment can be estimated. This could be done by either maximum likelihood (Demey et al 2004) or the simpler method of moments (Kupiec 2009). In the latter case $\theta$ and $\rho$ can be determined by simple formulas. Denote by $m$ the mean of the probit-transformed default rates $\varPhi^{-1}(\mathrm{DR}_{i})$ , and denote their standard deviation by $\sigma$ . Then, $\theta$ and $\rho$ are estimated as

	$\displaystyle\theta$	$\displaystyle=\frac{m}{\sqrt{1+\sigma^{2}}},$		(3.3)
	$\displaystyle\rho$	$\displaystyle=\frac{\sigma^{2}}{1+\sigma^{2}}.$		(3.4)

Once $\theta$ and $\rho$ are calculated, the state of the economy $x_{i}$ in each period $i$ can be determined from (3.2) together with (3.3) and (3.4), resulting in

x_{i}=\frac{m-\varPhi^{-1}(\mathrm{DR}_{i})}{s}.

(3.5)

Using this framework in forecasting or stress testing is now straightforward. A link between historic default rates $\mathrm{DR}_{i}$ and macroeconomic factors $K_{j}$ , $j=1,\dots,l$ , can be established by a time series model:

\varPhi^{-1}(\mathrm{DR}_{i})=\beta_{0}+\sum_{j=1}^{l}\beta_{j}K_{j,i-1}+% \varepsilon_{i},

(3.6)

where the use of a time lag of 1 is assumed.

Using (3.6), a macroeconomic scenario such as that provided by the Fed for the CCAR 2021 stress test can be translated into a scenario for $\varPhi^{-1}(\mathrm{DR}_{i})$ , which can be translated into future states of the economy $x_{i}$ using (3.5). These future states of the economy can then be used for projecting internal risk parameters of a bank under the $K_{j,i}$ scenario and measuring the scenario’s impact on a bank’s portfolios. This projection is explained with detailed examples in Engelmann (2021).

A key requirement for this approach to work is that the link between default risk and the state of the economy that is established in (3.5) holds in the future. If it is distorted by interventions from central banks and governments, the future state of the economy $x_{i}$ estimated from an econometric model does not tell us much about the number of defaults that is likely to be observed in a bank’s portfolio. This means that a model correction for government intervention is needed to improve the accuracy of default rate predictions.

Denote by $x_{\mathrm{econ},i}$ the expected economic systemic factor predicted from (3.6):

x_{\mathrm{econ},i}=\frac{m-\beta_{0}-\sum_{j=1}^{l}\beta_{j}K_{j,i-1}}{s}.

(3.7)

To improve model predictions under government intervention, an additional systemic factor $x_{\mathrm{inter},i}$ is introduced and the equation for predicting default rates is changed to

\mathrm{PD}_{i}=\varPhi\bigg{(}\frac{\theta-\sqrt{\rho}(x_{\mathrm{econ},i}+x_% {\mathrm{inter},i})}{\sqrt{1-\rho}}\bigg{)}.

(3.8)

Unfortunately, during times of no or little central bank and government intervention there is no way to anticipate or predict intervention in future periods, and the models will do a poor job at the beginning of an intervention period. However, when the effect of intervention becomes visible after a few quarters, the models can be adjusted. More precisely, assume that default rates $\mathrm{DR}_{i}$ , $i=1,\dots,n_{e}$ , are observed in a period without intervention, while $\mathrm{DR}_{i}$ , $i=n_{e}+1,\dots,n_{\mathrm{cbg}}$ , are observed in a period with intervention, and also assume that predictions have to be made for periods $i=n_{\mathrm{cbg}}+1,\dots,n_{\mathrm{f}}$ . Then, the following steps have to be carried out to adjust a credit risk model for central bank and government intervention.

(1)

Estimate a credit risk model as in (3.6) on the periods $i=1,\dots,n_{e}$ and identify the most predictive macroeconomic variables $K_{1},\dots,K_{l}$ . Further, estimate the parameters $\theta$ and $\rho$ using the same data.
(2)

Apply the model on the periods $i=n_{e}+1,\dots,n_{\mathrm{cbg}}$ and compute the economic systemic factor $x_{\mathrm{econ},i}$ using (3.7). In addition, use the realized default rates $\mathrm{DR}_{i}$ , $i=n_{e}+1,\dots,n_{\mathrm{cbg}}$ , to compute the realized systemic factor $x_{\mathrm{real},i}$ using (3.5). The systemic factor that is due to central bank and government intervention is then computed as $x_{\mathrm{inter},i}=x_{\mathrm{real},i}-x_{\mathrm{econ},i}$ .
(3)

For the prediction periods $i=n_{\mathrm{cbg}}+1,\dots,n_{\mathrm{f}}$ , a scenario for $x_{\mathrm{inter},i}$ has to be defined. It is very unlikely that this can be done using econometrics; expert judgement has to be used instead. Experts have to develop an opinion of whether it is most likely that intervention will, for example, stop immediately, fade out within one year or be persistent over the forecasting period. The scenario for $x_{\mathrm{inter},i}$ may differ depending on the current scenario, ie, under an adverse scenario we might assume more central bank and government intervention than in the baseline scenario.
(4)

Once the scenario for $x_{\mathrm{inter},i}$ , $i=n_{\mathrm{cbg}}+1,\dots,n_{\mathrm{f}}$ , is defined, model predictions can be made by predicting $x_{\mathrm{econ},i}$ using (3.7) and, finally, default probabilities can be estimated from (3.8).

In the next section this procedure will be illustrated by continuing the previous analysis using the Fed’s data for residential mortgage default rates and the macroeconomic data for CCAR 2021.

4 Application to residential mortgage default scenarios

To adjust the credit risk model (2.1) for government intervention, an estimation of $x_{\mathrm{inter},i}$ is required. An idea of the magnitude of $x_{\mathrm{inter},i}$ can be developed from the data in 2020 by computing and comparing $x_{\mathrm{real},i}$ and $x_{\mathrm{econ},i}$ . The results are presented in Table 9.

Table 9: Comparison of realized systemic factors

x_{\text{real},i}

with model predicted systemic factors

x_{\text{econ},i}

for the four quarters in 2020.

	Q1	Q2	Q3	Q4
$x_{\text{real},i}$	0.8488	0.9551	$-$ 1.1096	$-$ 1.6096
$x_{\text{econ},i}$	0.7200	0.6800	$-$ 2.3607	$-$ 0.8267
$x_{\text{inter},i}$	0.1288	0.2751	$-$ 3.4703	$-$ 2.4362

When inferring $x_{\mathrm{inter},i}$ from data, typical estimation errors of the econometric model during times of little or no intervention should be taken into account in order to focus on the main effects instead of statistical fluctuations. Table 10 shows the summary statistics of $x_{\mathrm{real},i}-x_{\mathrm{econ},i}$ for the estimation of model (2.1) from 2000 Q1 to 2019 Q4.

Table 10: Summary statistics of

x_{\text{real},i}-x_{\text{econ},i}

in-sample from 2000 Q1 to 2019 Q4.

Minimum	25% quantile	Median	Mean	75% quantile	Maximum
$-$ 0.5433	$-$ 0.2158	0.0143	$-$ 0.0133	0.1790	0.4868

In addition, a backtest can be performed to develop an idea of typical estimation errors during normal times. For this purpose, DR model 1 (Table 2) is estimated from 2000 Q1 to 2009 Q4. The model is used to estimate the default rate in 2010 Q1. From the estimated and realized default rates in 2010 Q1, the difference $x_{\mathrm{real},i}-x_{\mathrm{econ},i}$ can be computed for 2010 Q1. After that, the estimation window is enlarged by one quarter and the analogous steps are repeated. The summary statistics of $x_{\mathrm{real},i}-x_{\mathrm{econ},i}$ from 2010 Q1 to 2019 Q4 are displayed in Table 11.

Table 11: Summary statistics of

x_{\text{real},i}-x_{\text{econ},i}

in a backtest from 2010 Q1 to 2019 Q4.

Minimum	25% quantile	Median	Mean	75% quantile	Maximum
$-$ 0.7394	$-$ 0.2982	$-$ 0.1884	$-$ 0.1749	$-$ 0.0809	0.3360

Unlike the in-sample errors in Table 10, the errors in the backtest are not symmetric and the distribution is shifted to the negative range. This means that estimated default rates have a tendency to be overestimated. This could mean that the model is misspecified. However, including more macroeconomic variables in the regression leads to only minor improvements. Alternatively, this could indicate that mild intervention was already present between 2010 and 2019, resulting in slightly lower default rates than predicted by the model.

Comparing the values in Tables 10 and 11 shows that a difference of no greater than 0.5–0.75 between $x_{\mathrm{real},i}$ and $x_{\mathrm{econ},i}$ can be attributed to statistical estimation errors. Higher values can be attributed only to government and central bank intervention. Therefore, a reasonable calibration of $x_{\mathrm{inter},i}$ could be 0 for 2020 Q1 and 2020 Q2, 3.0 for 2020 Q3 and 2.0 for 2020 Q4. Once the calibration is done, a scenario for $X_{\mathrm{inter},i}$ is needed for making predictions, which is mainly driven by future plans of politicians and central bankers as far into the future as they can be anticipated.

To illustrate the procedure, three intervention scenarios are defined, reflecting a fast decline, a slow decline and no decline in government and central bank intervention until 2024 Q1.

Scenario 1:: intervention fades out linearly to 0 until 2021 Q1.
Scenario 2:: intervention is linearly reduced from 2.0 to 1.0 until 2024 Q1.
Scenario 3:: intervention stays high at 2.0 until 2024 Q1.

The following three subsections analyze short-term predictions, stress tests and adverse effects of intervention under the three different intervention scenarios.

4.1 Short-term prediction

Short-term model prediction can be analyzed under the scenario of no intervention and the three different intervention scenarios defined above by comparing default probabilities with observed default rates in the first two quarters of 2021. To apply model (2.1) for this purpose, macroeconomic data for 2020 Q4 and 2021 Q1 is needed. The 2020 Q4 data is contained in the data set for the CCAR 2021 stress test. In 2021 Q1 the US unemployment rate⁴⁴ 4 Retrieved from the Federal Reserve Economic Data website. URL: https://fred.stlouisfed.org/series/UNRATE. was 6.2% and year-on-year HPI growth⁵⁵ 5 Retrieved from the Federal Housing Finance Agency website. URL: https://bit.ly/3x8iG8a. was 7.2%. The predictions under different intervention scenarios are summarized in Table 12.

Table 12: Default rate predictions under different intervention scenarios and realized values in 2021 Q1 and 2021 Q2.

	2021 Q1	2021 Q2
No intervention	2.6816	2.2646
Intervention scenario 1	0.9145	1.1053
Intervention scenario 2	0.6552	0.5686
Intervention scenario 3	0.6157	0.5010
Realized value	0.4200	0.3300

4.2 Stress test under intervention scenarios

The outcomes of the base and adverse scenarios provided by the Fed under intervention scenario 1 and the projections of the adverse scenario under intervention scenarios 2 and 3 are shown in Figure 7. Compared with the predictions in Figure 2 all projections look reasonable. In all cases an increase in default rates should be expected compared with the historically low default rates in 2020. The level of increase, however, depends on the scenario that materializes and the level of government intervention that will prevail in the near future.

Default probabilities under the Fed's baseline and adverse scenarios including the intervention scenarios 1--3. — Figure 7: Default probabilities under the Fed’s baseline and adverse scenarios including the intervention scenarios 1–3.

Looking at the numbers produced so far, we might conclude that interventions are always beneficial. This might be true in the short term, but central bank and government interventions could lead to adverse long-term effects.

4.3 Adverse effects of intervention

An obvious adverse effect could be increasing inflation if intervention consists mainly of paying aid to citizens by printing money. This scenario is not included in the Fed’s data since the central bank, not too surprisingly, will not consider its own failure in controlling inflation as a realistic scenario. Market participants, however, should consider increased inflation, which reduces borrowers’ ability to pay back loans, as a potential scenario.

Figure 8: Quarterly US mortgage default rates, inflation rates and the four-quarter rolling average inflation rate from 2000 Q1 to 2020 Q4.

To get an impression of the impact of inflation, the time series of observed quarterly inflation rates is displayed in Figure 8. Similar to real disposable income growth, quarterly inflation is volatile, and estimating a model using raw quarterly inflation would result in a statistically insignificant model coefficient for inflation. Therefore, a moving average is applied to smooth out the fluctuations and capture the true economic effect.

Table 13: Diagnostics (coefficients, standard errors, adjusted

R^{\text{2}}

) of DR model 4, using UR and average CPIgr over a four-quarter rolling window as independent variables. [Standard errors are shown in parentheses.

{}^{***}p<\text{0.001}

;

{}^{**}p<\text{0.01}

;

{}^{*}p<\text{0.05}

.]

Intercept	$-$ 2.891 ${}^{***}$
	$-$ (0.086)
$\text{UR}_{i-\text{1}}$	$-$ 0.135 ${}^{***}$
	$-$ (0.011)
$\text{avCPIgr}_{i-\text{1}}$	$-$ 0.034 ${}^{*}$
	$-$ (0.017)
Adjusted $R^{\text{2}}$	$-$ 0.660

Figure 9: Fit and residuals for DR model 4.

Using a four-quarter rolling average results in the model displayed in Table 13. The in-sample fit in Figure 9 shows that the model calibrates reasonably well. To evaluate the impact of inflation, it is assumed that in our baseline scenario the inflation rate gradually rises to triple that of the baseline scenario provided by the Fed until 2024 Q1, leading to an inflation rate of 4.4% at the end of the projection horizon. In the adverse scenario, a tenfold inflation rate is assumed, leading to inflation rates of around 20%, as last seen in the 1980s. The model in Table 13 is applied to the scenarios in Figure 7 together with the increased inflation rate, and the results are shown in Figure 10, which illustrates that side effects such as increased inflation can negate the impact of intervention. In its adverse scenario, the Fed has assumed that economic recovery starts in 2023 and that by 2024 Q1 the first positive effects of this recovery are visible in the economic data, resulting in lower loan default rates, as displayed in Figure 7. High inflation can wipe out this recovery and keep default rates at crisis levels.

Default probabilities under the Fed's baseline and adverse scenarios including the intervention scenarios 1--3 together with an increased inflation assumption. — Figure 10: Default probabilities under the Fed’s baseline and adverse scenarios including the intervention scenarios 1–3 together with an increased inflation assumption.

In Figure 10 the crude assumption was made that inflation is independent of intervention and that it has no impact on other macroeconomic factors such as unemployment. However, the risk of increased inflation should go hand in hand with the level of government intervention, and macroeconomic factors are interdependent. Further refining the inflation scenario along these lines could result in an even stronger indication that interventions by government and central bankers can backfire, leading to worse outcomes than no or mild interventions. Analyzing these scenarios could reveal that assuming governments and central banks will properly manage a crisis might be a risky strategy and, in addition, allows a rough estimate of this risk to be obtained conditional on economic scenarios.

5 Conclusion

In this paper a simple approach for including central bank and government intervention in credit models was developed and illustrated using the Fed’s data for the CCAR 2021 stress test. It was shown how government interventions distort the economic link between default rates and macroeconomic variables, resulting in highly inaccurate predictions by macroeconomic models that were estimated on data at times of little to no intervention. The impact of intervention can be calibrated on the first 2–3 quarters once the effect of intervention becomes visible and should improve short-term predictions considerably.

This approach was motivated by central bank and government interventions in the United States and many European countries following the outbreak of the Covid-19 pandemic. However, the interventions by governments in different countries have been somewhat heterogeneous. The approach of this paper is suitable for interventions that increase the ability of borrowers to service their loans, such as government payments to citizens and corporations suffering income losses due to Covid-19 measures. Other forms of intervention have allowed borrowers to suspend their payments on loans for some time. Such interventions are substantially different in nature since they do not improve borrowers’ ability to pay but do reduce the level of information banks receive from their clients. In that case, the modeling approach of this paper is not helpful, and a purely economic model is needed to estimate the effective number of defaults remaining when the payment moratorium ends.

Government interventions may come with the risk of side effects. In the case of payments to support borrowers, one obvious risk is an increase in inflation, which might reduce the future ability of borrowers to pay back their loans if the cost of living increases much faster than salaries. This paper illustrated how this effect could be evaluated by defining inflation scenarios, revealing that the impact of rising inflation can be substantial. This result could allow risk managers to develop views on the conditions under which, and to what extent, government interventions might even cause credit portfolio quality to deteriorate in the long run.

Declaration of interest

The author reports no conflicts of interest. The author alone is responsible for the content and writing of the paper.

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