The Dutch default rate and the macroeconomy Ferdinand Rolwes 19 may 2007

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The Dutch default rate and the macroeconomy

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Doctoral Thesis Economics

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The Dutch default rate and the macroeconomy

Ferdinand Rolwes (s1008242)

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1 Introduction

1.1 Background and objectives

This research is an internship assignment from De Nederlandsche Bank (DNB). DNB is respon-sible for financial stability in the Netherlands and, in particular, for the financial healthiness of Dutch financial institutions. To be able to assess the risk of Dutch financial institutions getting into problems because of their corporate clients defaulting on loans, DNB is interested in the default behaviour of Dutch firms.

There are two main questions to be answered.

1. What macroeconomic variables are related to default behaviour of Dutch firms?

2. What is the default behaviour of Dutch firms given a two quarter zero GDP growth and a 2.5% worst case scenario?

We use a linear model for the logit default rate to answer both these questions.

The relation between default behaviour and macroeconomic variables is of interest for two reasons. First, a rising number of defaults is mostly more serious if it coincides with other economic misery. So we would like to know, for example, wether rising defaults occur at the same time as rising oil prices or a falling stock market. Second, estimating the effect of certain adverse macroeconomic scenario’s on default behaviour gives an indication of the quantitative risk. This is a popular way to assess risk because it allows people to think in terms of certain concrete scenarios. However, risk can be assessed more reliably by estimating, for several fractions α, what fraction of firms we can expect to default in the α% worst possible case. Therefore, at the end of this research, we compare both aproaches by examining a macroeconomic and a worst case scenario.

It is well known that firm specific variables like size, age and solvability have important explanatory power for defaults. See for example the studies from Altman and Sabato [4] or Altman and Rijken [2] and [3]. Ideally, we would examine the interaction between macroeconomic and firm specific variables. Unfortunately, data on Dutch firms containing firm specific variables was only available to us from 1994 (REACH dataset from Bureau van Dijk) but we could find data without firm specific variables from 1983 (Statistics Netherlands CBS). Because a long data history is important when estimating relations with macroeconomic variables, we choose to use the data with a long history and therefore ignore firm specific variables in this research.

1.2 Data sources

We have a dataset containing per sector the number of Dutch bankruptcies and the average num-ber of Dutch firms per quarter during the period 1983.1-2006.2 (94 quarters). The dataset is based on data from Statistics Netherlands CBS. Although, our data contains number of bankruptcies, in this report we will use the term defaults. This term is more common.

Data per sector aggregated to a yearly frequency is as good as exact except for the sectors Financial services and Rental and corporate services. For these two sectors data on a yearly frequency and for all sectors data on a quarterly frequency is approximate for the period 1983-1992. Detailed information on the dataset can be found in Rolwes[13].

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1.3 Overview of the report

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2 The data

In this section we explore our default and macroeconomic variables. We note some properties that will be important in this research. Finally, in subsection 2.4 we comment on the time lag between financial distress and the moment a subsequent default is recorded in our dataset.

2.1 Exploring the default rate

We are interested in the fraction of firms that defaults. Therefore we define some variables representing these default fraction. Let pdt,0be the fraction (proportion) of all firms that defaults during quarter t

pdt,0=

Number of defaults in all sectors during quarter t Average number of firms in all sectors during quarter t and pdt,i the fraction (proportion) of firms in sector i that defaults during quarter t

pdt,i =

Number of defaults in sector i during quarter t Average number of firms in sector i during quarter t

We will call pdt,0 and pdt,i respectively the economy and sector i default rate. In table 1 we present some general information on the sectors and table 2 summarizes some information on the size distribution of the firms per sector. The total number of firms in the economy is between 408665 (1983.1) and 652367 (2006.2). We see that most firms are rather small.

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1 Industry and mining (40115 - 46525)

Activities: Resource extraction (e.g. gas, sand), production and processing (e.g. food, clothing, textile, wood, building materials, machinery, furniture, vehicles, medical equipment, printing)

2 Construction (40600 - 80198)

Activities: Construction of buildings and infrastructure, installation of machinery, maintenance

3 Trade, repair consumer products (158760-165528)

Activities: Wholesale and retail (e.g. food, clothing, textile, wood, building mate-rials, machinery, furniture, vehicles, medical equipment, printing), repair consumer products (e.g. clothing, household electronics)

4 Catering (35405-37130)

Activities: hotel, restaurant and caf´e services

5 Transport, storage and communication (19086-27901)

Activities: Transport (by road, railway, water, air), travel mediation, storage, postal services, telecommunication

6 Financial services (8805-14817)

Activities: banking, investment, insurance, financial mediation

7 Rental and corporate services (48590-158905)

Activities: Rental (e.g. real estate, vehicles, machinery), real estate brokerage, con-sultancy, research

8 Other (57304-121363)

Activities: Government, education, health, non-profit activities, media, art, sports

Table 1: The sectors

Explanation: For each sector the number of firms in 1983.1 (left number) and 2006.2 (right number) is given between parentheses. For future reference, the sectors are numbered.

0 − 9 10 − 99 > 99

employees employees employees

Ind., min. 76% 20% 3%

Construction 86% 13% 1%

Trade, rep. cons. 92% 8% 0%

Catering 94% 5% 0%

Trans., stor., com. 84% 14% 1%

Financial 92% 7% 1%

Rental, corp. 92% 7% 1%

Other 89% 9% 2%

Economy 89% 9% 1%

Table 2: Size distribution of the firms

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Figure 1: The economy default rate

Mean level S.d. level S.d. first difference

×100 ×100 ×100

Ind., min. .33 .08 .04

Construction .32 .11 .04

Trade, rep. cons. .22 .05 .02

Catering .23 .10 .04

Trans., stor., com. .29 .08 .05

Financial .59 .32 .08

Rental, corp. .22 .06 .03

Other .07 .02 .01

Economy .23 .06 .02

Average .28 .10 .04

(P-value equal moments .0000 .0000 .0000)

Table 3: Some descriptive statistics of the default rates

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Figure 2: The sector default rates

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2.2 Exploring the macroeconomic variables

In this research we examine the following macroeconomic variables: gross domestic product (GDP), short and long interest rates, exchange rate, stock market return and volatility and oil price. Table 4 summarizes some details of the variables used. All variables are measured on a quarterly frequency. In this research we will use the logarithm of the exchange rate and oil price (to focus on percentage instead of absolute changes) and the growth in GDP. We define GDP growth as the (continuously compounded) growth between quarters t and t − 4:

∆%GDPt= ln

GDP in quarter t

GDP in quarter t − 4. (1)

This way we avoid seasonal effects.

Figure 3 shows plots of the macroeconomic variables against time and in table 5 we present some descriptive statistics. We can discern some important economic developments. Interest rates have gradually fallen, the guilder/euro has appreciated (mainly against the dollar), in 1988 and 2001 stock markets crashed and recently the oil price started rising again.

Gross domestic product

Gross domestic product is measured by the expenditure approach and against constant prices.

Short and long interest rates

In case of the short rate 3 month AIBOR is used and in case of the long rate the yield to maturity on (approximately) 10 year Dutch government bonds.

Exchange rate

We use the real and effective exchange rate expressed as foreign currency per unit of domestic currency (guilders until 2002 and euros afterwards). (See remark 1.) Stock market return and volatility

From daily returns on the AEX DC index we calculate (continuously compounded) quarterly returns and daily volatilities (sample standard deviations). (See remark 2.) Oil price

This is the euro oil price of one barrel UK Brent oil corrected for consumer inflation.

Table 4: The macroeconomic variables

Remark 1: Real means the exchange rate is corrected for (consumer) inflation in both countries. Effective means the exchange rate is computed as a weighted average (weights based on trade volume) of exchange rates.

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Figure 3: The macroeconomic variables

Mean level S.d. level S.d. first difference

×100 ×100 ×100 GDP growth 2.33 1.74 1.30 Short rate 5.15 2.24 .51 Long rate 6.37 1.76 .39 ln(Exchange rate) 449.06 7.51 1.37 Return 3.54 10.03 14.31 Volatility 1.06 .55 .51 ln(Oil price) 412.87 46.19 14.51

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2.3 Some important properties

In this subsection we note some properties of the data that will be important in this research. From figures 1 and 2 we observe that the default rate is persistent. We examine this in more detail for the economy default rate by estimating the following autoregressive model of order 1 (AR(1)).

pdt,0= .00 + .85pdt−1,0+ υt (2)

(The model is estimated by Ordinary Least Squares (OLS) ignoring the equation corresponding to t = 1.) The high coefficient of the first lag confirms the observed persistency. In figure 4 we show the correlogram of pdt,0 and ˆυt. We see that the AR(1) model captures most of the serial correlation.

Figure 4: Correlograms economy default rate and residuals model (2)

Explanation: Approximate standard error of a sample correlation between unrelated series is 1/√n where n is the number of observations. The vertical dashed lines indicate two standard errors bands.

The patterns in figures 1 and 2 also suggest a negative relation with the business cycle. In order to examine this, the upper panel of figure 5 plots the economy default rate and ∆%GDPt−1 against time. The relation is not very clear because GDP growth fluctuates a lot while the default rate is persistent. However, adding ∆%GDPt−1 to (2) and estimating again we have

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(The intercept and the coefficient of GDP growth are rounded to .00 but they do deviate signif-icantly from zero.) So we see that the persistency actually implies the default rate is related to a weighted sum of lags of GDP growth. Therefore the lower panel of figure 5 plots the economy default rate andP19

j=0.82

j_∆%GDP

t−1−j against time. Now there is a nice (negative) relation.

Figure 5: Economy default rate and GDP growth

Explanation: The upper panel plots pdt,0 and ∆%GDPt−1 against time. The lower panel plots pdt,0 and

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j=0.82j∆%GDPt−1−jagainst time. All series are standardized to have zero mean and unit variance.

Now we look at the sector default rates from figure 2 and make another important observation: default rates between sectors appear to be positively related. Showing the entire correlation matrix would confront the reader with quite a lot of numbers, but table 6 presents correlations with the economy default rate. They are all significant at the 1% level. Later in this research we will see that a lot of correlation remains after conditioning on macroeconomic variables.

Finally, we check for seasonal effects. We do this by estimating the following simple model by OLS. pdt,i= 4 X j=1 δj1t∈quarter j(t) + υt (3)

(The function 1A is the indicator function for the event A.) Next we test the hypothesis δ1 = δ2= δ3= δ4. Results are summarized in table 7. No indication of seasonal effects is found.

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Correlation Correlation levels first differences

Ind., min. .91∗∗∗ .53∗∗∗

Construction .89∗∗∗ .75∗∗∗

Trade, rep. cons. .96∗∗∗ .84∗∗∗

Catering .82∗∗∗ .69∗∗∗

Trans., stor., com. .64∗∗∗ .52∗∗∗

Financial .50∗∗∗ .71∗∗∗

Rental, corp. .90∗∗∗ .79∗∗∗

Other .83∗∗∗ _.47∗∗∗

Table 6: Cross correlations between sector and economy default rates

Remark: Approximate standard error of a sample correlation between unrelated series is 1/√n where n is the number of observations. In this case n = 93. Significance is tested by comparing the sample correlation divided by its approximate standard error to the standardnormal distribution. Significance at 1%, 5% and 10% level is denoted by respectively∗∗∗,∗∗and∗. ˆ δ1/P 4 j=1δˆj δˆ2/P 4 j=1δˆj ˆδ3/P 4 j=1ˆδj ˆδ4/P 4 j=1ˆδj P-value Ind., min. 26% 25% 25% 24% .86 Construction 27% 25% 24% 24% .56

Trade, rep. cons. 26% 25% 25% 24% .66

Catering 28% 25% 24% 23% .54

Trans., stor., com. 27% 24% 24% 25% .31

Financial 26% 25% 24% 25% .99

Rental, corp. 26% 24% 25% 25% .93

Other 25% 25% 26% 24% .91

Economy 26% 25% 25% 24% .72

Table 7: Testing for seasonal effects

Explanation: This table summarizes the estimation results from model (3) and tests the hypothesis δ1 = δ2 =

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2.4 Timing of financial distress and bankruptcy

Defaults are recorded in the dataset in the quarter bankruptcy is pronounced by a court of justice. Naturally, there is some delay between the time firms start having problems meeting their financial obligations and this formal bankruptcy sentence. Therefore we assume the financial problems occur in the quarter preceding the bankruptcy. Although setting the delay at one quarter is quite arbitrary, the following data provide some support for this choice. For the period 1986-1997 on average 8.6% of the firms that went bankrupt had requested delay on their payment obligations prior to going bankrupt. Figure 6 shows a histogram of the time that passes between the request for delayed payment and the bankruptcy. This is on average approximately 3.9 months. (The average is approximate because the data on elapsed time is classified into classes of 0-3, 3-6, 6-12, 12-18, 18-24, 24-36, 36-48 and 48-60 months. We calculated the average based on the mean of these classes.) Though we do not know wether this result is also valid for firms that did not request delay on their payment prior to going bankrupt, it nicely confirms our choice of one quarter. Because of this one quarter delay we lag in this research all explanatory macroeconomic variables by one quarter.

Figure 6: Time between request for delayed payment and bankruptcy

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3 Default modeling

First we discuss some possible approaches to default modeling. Next we introduce the general model used in this research (section 3.2.1) and discuss how shocks in explanatory variables affect default behaviour through time (section 3.2.2).

3.1 An overview of possible models

In this subsection we discuss two possible approaches.

1. Individual default modeling: which firms (of a certain group) default?

2. Aggregate default modeling: what fraction of all firms (of a certain group) defaults? Both approaches are discussed in some detail below. For individual default modeling we distin-guish discrete and continuous time models.

3.1.1 Individual default modeling: discrete time

Discrete time default models model a binary random variable dt,iindicating wether firm i defaults (dt,i= 1) or not (dt,i = 0) during period t. Let λt,i be a latent variable, called the credit score, summarizing the creditworthiness. The following rule determines wether a firm defaults:

dt,i=

1 if λt,i< 0 0 if λt,i≥ 0

.

The credit score is usually modeled as a function h(.) of information relevant for the credit-worthiness and an additive disturbance. Let θ be a parameter vector, zt,i a vector containing the information relevant for the creditworthiness and υt,i a disturbance.

λt,i = h(θ, zt,i) + υt,i

If the υt,iare independent then default events are independent given fixed zt,i. This is usually not realistic since it would require all variables affecting multiple firms to be included in zt,i. (In section 4.2 we estimate, for example, a model where defaults in multiple sectors are affected by four macroeconomic variables and a latent systematic factor. It turns out that the latent systematic factor has more explanatory power than the four macroeconomic variables together! This illustrates how hard it is to capture all systematic effects in the zt,i.) The model can be extended to allow for correlation between the residuals. Letting ψt be a systematic and t,i an idiosyncratic disturbance, we can do so by specifying

υt,i= ψt+ t,i. (4)

The model can be estimated by maximizing the likelihood. Unfortunately, introducing cor-relation makes estimation a lot more complicated1_.

1_{One way to estimate the model is by using theory on state space models. A general state space model consists}

of two parts: (1) a probability density function for a vector dependent variables yt given an explanatory matrix

Ztand a latent vector αt

p(yt|Zt, αt)

and (2) a dynamic process for the latent vector, depending on parameter matrices Tt and Rt and a disturbance

vector ηt

αt+1= Ttαt+ Rtηt.

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The question arises how bad it is to ignore correlation between the disturbances. This depends on the purpose of our analysis. If we are interested in estimating the parameter vector θ, ignoring the correlation is probably not a serious problem. But if we are interested in the distribution of the fraction defaulting firms, modeling the correlation is important. We illustrate this with a simple example. Suppose we have a set of 500000 firms and h(θ, z1,i) = 2.81 and υ1,i∼ N 0, 12 so P (d1,i= 1) = P (υ1,i< −2.81) ≈ .0025. (These numbers are in line with the data used in this research. See sections 2.1 and 2.1.) Let σψ be a nonnegative parameter. We draw υ1,i from (4) with ψ1 ∼ N (0, σ2ψ) and 1,i

iid

∼ N0,q1 − σ2 ψ

(so υ1,i ∼ N 0, 12). Next we compute the fraction defaulting firmsP

id1,i/500000. Histograms of 10000 simulated fractions defaulting firms are shown in figure 7 for two cases. In the uncorrelated case σψ = 0. Note that in this case there is hardly any uncertainty in the fraction defaulting firms. On the other hand, in the correlated case σψ= .06. Now the distribution has a lot more uncertainty which makes it look more realistic.

Figure 7: Simulated fractions defaulting firms

Explanation: Both histograms are based on 10000 fractions defaulting firms simulated from the discrete time model. The number of firms is 500000, h(θ, z1,i) = 2.81 and υ1,i∼ N 0, 12. The υ1,iare drawn from (4) with

ψ1∼ N (0, σ2_ψ) and 1,i iid

∼ N0,q1 − σ2 ψ

. In the uncorrelated case σψ= 0 and in the correlated case σψ= .06.

The discrete time default model is an example of a discrete choice model. Discrete choice models are discussed theoretically in Johnston[8]. Applications can be found in Hamerle et.al.[1] and Jakubic[7].

3.1.2 Individual default modeling: continuous time

Continuous time default models model the time until firm i defaults: τi. Let Fi(t) be the continuous density function (cdf) of the time until default and fi(t) the corresponding probability density function (pdf). We assume τiis a (purely) continuous random variable so the pdf exists everywhere (or, equivalently, the cdf does not jump). Let λi(t) be a latent variable, called the default intensity, summarizing the creditworthiness. It is defined by

λi(t) = fi(t) 1 − Fi(t)

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So λi(t)∆t is for small ∆t approximately the probability that default occurs during period [t, t + ∆t] conditional on no default before time t. (In duration literature λi(t) is called the hazard rate.) The pdf and cdf can be derived from the default intensity:

1 − Fi(t) = exp log 1 − Fi(t) − log 1 − Fi(0) = exp − Z t 0 λi(u)du (6) fi(t) = λi(t) 1 − Fi(t) = λi(t) exp − Z t 0 λi(u)du (7) (In (6) we used _dtd log 1 − Fi(t)

= −λi(t) and in (7) we used (5) and (6).) The following expression for the time to default will be useful.

τi = Fi−1(Ui) (8)

where Ui∼ U (0, 1)

The (logarithm of the) default intensity is usually modeled as a function h(.) of information relevant for the creditworthiness. Let θ be a parameter vector and zi(t) a vector containing the information relevant for the creditworthiness at time t.

λi(t) = exp h(θ, zi(t)).

If the Ui in (8) are drawn from independent uniform distributions then the times to default are independent given fixed zi(t). However, as was discussed in section 3.1.1, this is usually not realistic. We mention two possible extensions to model correlation given fixed zi(t).

1. Make the Ui correlated by drawing from a multivariate uniform distribution (known as a copula).

2. Add a systematic disturbance ψ(t) to the expression for the default intensity: λi(t) = exp h(θ, zi(t)) + ψ(t).

The model can be estimated by maximizing the likelihood. But extending the model to include correlation makes estimation again a lot more complicated2_.

Like in the case of discrete time default models we illustrate the effect of adding correlation with a simple example. Let σψ be a nonnegative parameter. Suppose we have a set of 500000 firms and that for all t ∈ [0, 1) h(θ, zi(t)) = −5.99 and ψ(t) = ψ where ψ ∼ N −1₂σ2ψ, σ2ψ so3

2_{In case of the second extension, if the systematic disturbance ψ(t) is a stepwise stochastic process theory on}

state space models can be used (see footnote 1).

3_{To derive P τ} i∈ [0, 1) = .0025 we may write P τi∈ [0, 1) = E [E [ Fi(1)| ψ]] = E E 1 − exp − Z 1 0 λi(u)du ψ ≈ E E Z1 0 λi(u) ψ = E [exp (−5.99 + ψ)] ≈ .0025.

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P τi ∈ [0, 1) ≈ .0025. (These numbers are in line with the data used in this research. See section 2.1.) Letting di equal one if τi< 1 and zero otherwise, we compute the fraction of firms defaulting in period [0, 1)P

id1,i/500000. The two extensions are considered one at a time. 1. Suppose the Ui are drawn from a Gaussian copula. So, letting Φ(.) be the cdf of the

standardnormal distribution and Z1, . . . , Z500000 random variables from a multivariate standardnormal distribution, we have Ui = Φ−1(Zi). We set σψ = 0. Now note that the following three events are equivalent: τi < 1, Ui < Fi(1), Zi < Φ−1(Fi(1)). So we can simulate defaults by drawing from a multivariate standardnormal distribu-tion and look which drawings are below Φ−1(Fi(1)). We choose to compare the case of no correlation between the Zi with the case of .06 correlation. However, noting that Φ−1(Fi(1)) ≈ Φ−1(.0025) ≈ −2.81, we are following exactly the same procedure as in the discrete case4_{. Therefore, for this extension, we refer to figure 7.}

2. We include a systematic disturbance and let the Ui be independent. Histograms of 10000 simulated fractions defaulting firms are shown in figure 8 for two cases. In the uncorrelated case σψ = 0 and in the correlated case σψ= .9. Note that in the uncorrelated case we can predict the number of defaulting firms very accurately given the zt,i, while in the correlated case there is considerable uncertainty.

Figure 8: Simulated fractions defaulting firms

Explanation: Both histograms are based on 10000 fractions defaulting firms simulated from the continuous time model. The number of firms is 500000, the Ui are independent and, for all t ∈ [0, 1), h(θ, zi(t)) = −5.99 and

ψ(t) = ψ where ψ ∼ N −1₂σ2 ψ, σ

2

ψ. In the uncorrelated case σψ= 0 and in the correlated case σψ= .06.

We conclude that including correlation given the zt,i makes the distribution of the fraction defaulting firms more realistic.

The continuous time default model is an example of a duration model. Duration models are discussed theoretically in Kiefer[11]. Applications can be found in Courderc and Renault[6], Carling et.al.[9], Koopman et.al.[14] and Kavvathas[10].

4_{We remark that this does nog mean the models are equivalent. After all, one is in discrete time and the other}

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3.1.3 Aggregate default modeling

Let pdtbe the fraction (proportion) of firms that defaults in period t. A quite general aggregate model sets pdtequal to a function g(.) of relevant explanatory variables zt, a parameter vector θ and a disturbance υt.

pdt= g(θ, zt, υt)

The model can be estimated by maximizing the likelihood. The distribution of the fraction defaulting firms can be controlled easily by controling the distribution of the υt. Aggregate default models do not allow modeling firm specific explanatory variables.

3.2 The model

In this report we use an aggregate model for the default rates defined in section 2.1. The second objective of this research (see section 1.1) includes estimating a 2.5% worst case scenario which requires us to accurately model the distribution of the default rate. We could do so by using an individual model that captures correlation given explanatory variables. However, as mentioned in section 3.1, estimating such a model is quite complicated. Because in this research we only consider macroeconomic explanatory variables, there is no necessity to model defaults at an individual level. So we choose an aggregate model.

3.2.1 Description

Consider an economy with s sectors. Let ztbe a vector of variables relevant for the default rate at time t (including intercept), υt,ia disturbance and βia vector of parameters ∀i ∈ {0, 1, . . . , s}. We introduce the following aggregate model for pdt,i.

pdt,i=

exp z0_tβi+ υt,i 1 + exp z0_tβi+ υt,i

(9) Taking the logit5_{of both sides, we find}

˜

pdt,i:= logit(pdt,i) = zt0βi+ υt,i. (10)

Now we specify the disturbances υt,i. In case we are modeling the economy default rate we assume the disturbances are independent and identically distributed (iid) . Let σ2_ψ,0= var(υt,0). The economy model is

˜ pd_t,0 = z_t0β0+ υt,0 (11) where υt,0 iid ∼ (0, σ2 ψ,0).

In case of the sector default rate we split up the disturbances into a latent systematic (ξt) and an idiosyncratic (ψt,i) part. This way we capture the correlation between the sector default rates observed in section 2.3. Let σξ,iand σψ,i be nonnegative parameters ∀i ∈ {1, . . . , s}. The sector model is

˜

pd_t,i = z0_tβi+ υt,i (12)

where υt,i = σξ,iξt+ σψ,iψt,i ξt

iid

∼ (0, 1) , ψt,i iid

∼ (0, 1) .

5_{The logit transformation is given by logit(x) = ln} x 1−x

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Estimation and inference of the parameters are based on maximizing the Gaussian quasi loglikelihood.

3.2.2 Dynamic effects of shocks in explanatory variables

First, we note a nice property of our model for pdt,i. Differentiating (9) with respect to zt we find Dztpdt,i= pdt,iβi 1 + exp ˜pd_t,i 2.

Because ˜pd_t,i is in general low (always below -4.2 in our dataset), we may ignore the denominator and write for small ∆zt

∆%pdt,i≈ βi0∆zt. (13)

So the elements of βi are approximate semi-elasticities.

To capture the persistence observed in section 2.3 we will include the lagged default rate as explanatory variable. Let z_t∗denote explanatory variables other than the lagged default rate and the intercept and β_i∗ the corresponding parameter vector. We may write model (10) as

˜

pd_t,i = βi,0+ βi,1pd˜t−1,i+ β ∗ i 0 z_t−1∗ + υt,i or, equivalently, ˜ pd_t,i= βi,0 1 − βi,1 + β_i∗0 ∞ X j=0 β_i,1j z_t−1−j∗ + ∞ X j=0 βj_i,1υt−j,i. (14)

So including the lagged default rate makes the current default rate depend on all lags of the explanatory variables with coefficients declining at rate βi,1as the lag becomes higher. (This was also shown in section 2 using only GDP growth as explanatory variable.) This is understandable since wether a firm defaults or not depends not only on the previous period but on the entire history with more recent developments being more important.

Now consider a small shock ∆z∗ occuring in period t0 and persisting indefinitely through time. Applying (13) to (14) (interpretingP∞

j=0β j

i,1z∗t−1−j as explanatory vector) we have the following short and long run effects on the default rate.

Short run effect : ∆%pdt0+1,i≈ β ∗ i 0_∆   ∞ X j=0 βj_i,1z∗_t 0−j  = β∗i 0_∆z∗ ₍₁₅₎

Long run effect : lim

t→∞∆%pdt+1,i ≈ limt→∞β ∗ i 0 ∆   ∞ X j=0 βj_i,1z∗_t−j  = β∗_i0 1 − βi,1 ∆z∗ (16)

Note that at time t > t0 the effect on the default rate is

∆%pdt,i≈ β∗i 0 ∆   ∞ X j=0 β_i,1j z_t−1−j∗  = 1 − β t−t0 i,1 β ∗ i 0 1 − βi,1 ∆z∗. (17) So at time t a fraction 1 − βt−t0

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4 Estimation results

In this section we estimate the model from the previous subsection ˜

pdt,i = βi,0+ βi,1pd˜t−1,i+ βi∗ 0

zt−1∗ + υt,i (18)

with several macroeconomic variables substituted for z_t∗. In section 4.1 we specify a general z_t∗ containing all macreconomic variables that might be of interest. Section 4.2 estimates a parsimonious model. Finally, in section 4.3 we compare the results to those found in other research.

4.1 A general specification

In section 1.1 we explained that the relation between default behaviour and macroecomic vari-ables is of interest (1) to examine wether rising defaults occur at the same time as other forms of economic misery and (2) to estimate the effect of certain adverse macroeconomic scenarios as a quantitative indicator of risk. Therefore we choose macroeconomic variables for which there are in particular concerns of movements in unfavourable directions. We stress that these need not be the variables that add the most explanatory or forecasting power. The following macroeconomic variables are selected. For details on the variables we refer to table 4.

• GDP growth

(The economy is always at risk of slowing down.) • Short interest rate: level and change

(The next years, a decline in the saving rate caused by aging is likely to boost interest rates.)

• Log exchange rate: level and change

(Currently, there are concerns about a large depreciation of the US dollar.) • Stock market: return and volatility

(Stock market crashes are hard to predict but do happen occasionally.) • Log oil price: level and change

(Since the oil crises there is a permanent concern about the possibility of rising oil prices.) In this subsection we include all macroeconomic variables simultaneously in z∗

t. These vari-ables and their estimated coefficients are discussed in the following subsubsections. Results for parameters concerning the intercept, lagged default rate and disturbance term are deferred to section 4.2. This section merely identifies which macroeconomic variables are significant.

4.1.1 Gross domestic product

In section 2 we already observed a negative relation between the default rate and GDP growth. The following explanation is quite common. GDP equals aggregate demand in an economy. Therefore it is related to the sales of firms. The lower GDP growth, the harder it is for firms to generate income through sales and the more likely it is that firms cannot meet their obligations and default.

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GDP growth

Ind., min. −2.41∗∗∗

Construction −.99

Trade, rep. cons. −1.15∗

Catering −.43

Trans., stor., com. −3.45∗∗∗

Financial −3.08∗∗∗

Rental, corp. −2.93∗∗∗

Other −.51

Economy −1.33∗∗

Pooled −1.69∗∗∗

(P-value equal coefficients .0694)

Table 8: Estimated coefficients GDP growth

Explanation: Model (18) is estimated with the macroecomic variables from section 4.1 included in zt∗. The

coefficients of GDP growth are reported. Pooled results for variable j are obtained by estimation under the restriction β∗

1,j = . . . = β8,j∗ . This restriction is tested and the p-value is reported. Significance at 1%, 5% and 10% level is

denoted by respectively∗∗∗_,∗∗_and∗_.

4.1.2 Interest rate

Firms often finance their activities partly by debt. Therefore the costs of firms are positively related to interest rates. So if interest rates are higher, firms have more cost and are more likely to default.

Table 9 shows the estimated coefficients of the level and change in the short rate. We see that only for Construction a significant relation is found with the level of the short rate. Possibly, results in this sector are different from those in other sectors because construction firms are substantially affected by interest rates through another channel than cost of debt. Because in particular private households find it easier to finance construction work on their homes when interest rates are low, we expect demand for construction work to be negatively related to interest rates. This should especially affect small construction firms and about 86% of construction firms has less than 10 employees. The strong rejection of the hypothesis of equal coefficients supports the view that the Construction sector is an exception. It is remarkable that the level and not the change of the interest rate is significant. For one would expect firms to get used to the level and only react to changes.

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Short rate ∆ Short rate

Ind., min. 1.11∗ −3.84

Construction 1.88∗∗∗ −2.69

Trade, rep. cons. .76 −.71

Catering −.47 1.04

Trans., stor., com. −.44 .88

Financial .08 2.79

Rental, corp. 1.10 −1.47

Other −.11 −5.54

Economy .79 .66

Pooled .86 −1.60

(P-value equal coefficients .0025 .2829)

Table 9: Estimated coefficients short rate

coefficients of the level and the change of the short rate are reported. Pooled results for variable j are obtained by estimation under the restriction β∗

1,j = . . . = β∗8,j. These restrictions are tested and the p-values are reported.

Significance at 1%, 5% and 10% level is denoted by respectively∗∗∗_,∗∗_and∗_.

4.1.3 Exchange rate

The exchange rate is expressed as the price of domestic currency in terms of foreign currency. Firms in sectors doing a lot of international business are expected to be affected by exchange rates. However, the sign of the relation is ambiguous. Business conditions of importing firms depend positively on the exchange rate because imports become cheaper if the exchange is high. Business conditions of exporting firms depend negatively on the exchange rate because exports become more expensive, and therefore demand for them drops, if the exchange rate is high.

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ln(ER) ∆ ln(ER)

Ind., min. .45∗ .14

Construction .15 .75

Trade, rep. cons. .63∗∗ .13

Catering .67∗ .59

Trans., stor., com. 1.75∗∗∗ −.72

Financial .99∗∗ −.03

Rental, corp. .61∗∗ .15

Other −.26 .65

Economy .42∗ .12

Pooled .48∗∗ .21

Table 10: Estimated coefficients exchange rate

coefficients of the logarithm and the change of the logarithm of the exchange rate are reported. Pooled results for variable j are obtained by estimation under the restriction β∗

1,j= . . . = β∗8,j. These restrictions are tested and the

p-values are reported. Significance at 1%, 5% and 10% level is denoted by respectively∗∗∗_,∗∗ _and∗_.

4.1.4 Stock market return and volatility

Merton’s theory predicts that the probability of default is negatively related to stock return and positively to volatility6_{. However, since the vast majority of firms examined in this research is} not listed on a stock exchange, it is doubtful wether the stock market sufficiently reflects their financial healthiness to observe these relations. But, because stock market return (and, thereby also, volatility) are popular for scenario analysis we do check for a possible relation.

Table 11 shows the estimated coefficients of stock market return and volatility. Both in case of return and volatility none of the coefficients deviate significantly from zero and the signs are mixed. Clearly, we may conclude that the default rate is unrelated to the stock market.

6_{This theory is based on the rule that a default occurs when the value of assets is lower than that of debt.}

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Return Volatility

Ind., min. −.01 1.19

Construction .27 2.05

Trade, rep. cons. .16 .95

Catering .04 −4.70

Trans., stor., com. .07 1.87

Financial −.15 3.31

Rental, corp. −.23 −.51

Other −.15 −3.73

Economy −.03 1.23

Pooled .02 .97

Table 11: Estimated coefficients stock market return and volatility

coefficients of return and volatility are reported. Pooled results for variable j are obtained by estimation under the restriction β∗

1,j = . . . = β∗8,j. These restrictions are tested and the p-values are reported. Significance at 1%, 5%

and 10% level is denoted by respectively∗∗∗_,∗∗ _and∗_.

4.1.5 Oil price

The oil price affects the price of a lot of products that are used in firm’s production processes. Therefore, the cost of firms and thus their probability of defaulting are positively related to the oil price.

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ln(Oil price) ∆ ln(Oil price)

Ind., min. .10∗∗ −.09

Construction .07 −.05

Trade, rep. cons. .11∗∗∗ −.15

Catering .19∗∗∗ −.22∗

Trans., stor., com. .21∗∗∗ −.08

Financial .15∗∗ −.04

Rental, corp. .10∗∗ −.06

Other .08 −.02

Economy .07∗ −.06

Pooled .10∗∗∗ −.10

Table 12: Estimated coefficients oil price

coefficients of the logarithm and the change of the logarithm of the oil price are reported. Pooled results for variable j are obtained by estimation under the restriction β∗

1,j= . . . = β∗8,j. These restrictions are tested and the p-values

are reported. Significance at 1%, 5% and 10% level is denoted by respectively∗∗∗_,∗∗ _and∗_.

4.2 A parsimonious specification

In this subsection we estimate model (18) without the variables that were found to be insignificant in the previous subsection. This time we include the following variables in z_t∗.

• GDP growth

• Short interest rate (level) • Exchange rate (level) • Oil price (level)

The results are shown in table 12. The results have not changed much compared to the general model from the previous subsection. Most importantly, for the exchange rate and the oil price coefficient estimates decreased somewhat. Furthermore, for the short rate the sector Rental and corporate services is now significant. Significance of the GDP growth coefficients hardly changed.

Now we look more closely at the model estimated in table 12. Subsequently, we summarize shortly the effects of some macroeconomic shocks, examine the residuals of the model, analyze the explained variance and check wether parameter estimates are robust with respect to the data period used.

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shock between about 60% and 95% of the long run effect is realized. Although GDP growth shocks will be examined more closely in the next section, we remark already that the effects of GDP growth shocks are somewhat low. (A persistent 3% decrease in GDP growth raises the long run default rate only by about 25% while figure 1 shows fluctuations much larger through the business cycle.)

Lower bound Mean effect Upper bound

GDP growth: +.01 (Economy)

Short run −2.38% −1.40% −.43%

Long run −14.91% −9.01% −3.11%

Short rate: +.01 (Construction)

Short run .32% 1.52% 2.73%

Long run .96% 6.60% 12.24%

Exchange rate: +1% (Trans.,stor.,com.)

Short run .75% 1.48% 2.22%

Long run 1.39% 2.95% 4.51%

Oil price: +10% (Trans.,stor.,com.)

Short run .52% 1.79% 3.05%

Long run −6.46% 3.55% 13.57%

Table 13: Short and long run effects of macroeconomic shocks

Explanation: Short and long run effects (percentage changes) on the default rate of the economy or a certain sector are computed using respectively (15) and (16) based on estimation results from table 12. Upper and lower bounds of a 95% confidence interval are reported as well.

Consider the residuals ˆυt,i from (18). Table 14 presents some descriptive statistics. We see that, in general, the residuals are somewhat leptokurtic and skewed to the right, but skewness and kurtosis mostly do not deviate significantly from those of a normal distribution. The Jarque-Bera test confirms this. (We remark that deviations from the normal distribution also usually do not affect consistency of the parameter estimators. It would only make them less efficient.) An important assumption of the model is independency of the residuals. To examine this assumption we show in figures 9 and 10 correlograms of respectively the economy and the sector residuals. There is an unfortunate spike of negative correlation at the first lag of economy residuals. But because this spike is mostly insignificant for the sector residuals and negative serial correlation in quarterly residuals is hard to explain, we further ignore it. For the rest, the correlograms give no reason to doubt the assumption of independent residuals.

Next we compare the variance explained by the macroeconomic variables, the latent sys-tematic (ξt) and the idiosyncratic (ψt,i) disturbances. Note from table 12 that the systematic standard deviations are all significantly different from zero but not from each other. Recalling that υt,i = σξ,iξt+ σψ,iψt,i we rewrite (18) into

˜

pd_t,i− βi,0− βi,1pd˜t−1,i= βi∗ 0_z∗

t−1+ σξ,iξt+ σψ,iψt,i.

Because the systematic and idiosyncratic disturbances are independent from everything we have the following variance decomposition:

var ˜pdt,i− βi,1pd˜t−1,i

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Skewness Kurtosis Jarque-Bera

Ind., min. .59∗∗ 3.10 5.52∗

Construction .34 3.06 1.79

Trade, rep. cons. .30 3.54 2.49

Catering .22 5.69∗∗∗ 28.81∗∗∗

Trans., stor., com. .07 3.75 2.27

Financial .08 3.28 .40

Rental, corp. .09 2.46 1.25

Other .05 3.11 .09

Economy .39 3.59 3.75

Table 14: Some descriptive statistics of the estimated residuals

Explanation: The residuals are the υt,ifrom the models estimated in table 12. The hypotheses that skewness

equals 0 and kurtosis 3 are tested assuming the sample skewness and kurtosis have the same distribution as in the case of a normal distribution. We also report the Jarque-Bera test statistic. Significance at 1%, 5% and 10% level is denoted by respectively∗∗∗,∗∗ and∗.

Figure 9: Correlogram economy residuals

Explanation: The residuals are the υt,i from the economy model estimated in table 12. Approximate standard

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Figure 10: Correlogram sector residuals

Explanation: The residuals are the υt,ifrom the sector model estimated in table 12. Approximate standard error

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or, equivalently,

β∗_i0var z_t−1∗ β_i∗ var ˜pd_t,i− βi,1pd˜t−1,i

+

σ2_ξ,i

var ˜pd_t,i− βi,1pd˜t−1,i +

σ_ψ,i2

var ˜pd_t,i− βi,1pd˜t−1,i

= 1. (19)

(The first lag is excluded form the decomposition because it explains by far most of the vari-ance (from table 12 well over 50%) and because it is not independent from the macroeconomic variables.) We call these three fractions respectively the macroeconomic, latent systematic and idiosyncratic part. For the model estimated in table 12, they are shown in table 15. We see that more variance is explained by the latent systematic than by the macroeconomic part. Although the macroeconomic variables were not selected because of their explanatory power, this does illustrate the difficulty of finding all relevant systematic variables. The macroeconomic variables have the most explanatory power for the sectors Industry and mining, Transport, storage and communication, Financial services and Rental and corporate services.

Macroeconomic Latent systematic Idiosyncartic

part part part

Ind., min. 22% 20% 58%

Catering 10% 29% 60%

Trans., stor., com. 33% 23% 44%

Financial 25% 27% 48%

Other 8% 13% 79%

Average 19% 30% 51%

Table 15: Variance decomposition

Explanation: This table shows the variance decomposition (19) based on the model estimated in table 12.

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of the 99% confidence interval. This suggests that the parameter does not depend on the data period used and thus that the relation is stable through time. We see that that for the economy model most relations are stable. Remarkably, for the sector model most relations with the oil price are stable but those with GDP growth are not.

In terce p t Fir st lag GDP gro wth Sh ort rat e ln (ER) ln (Oil p rice ) Sy stematic s.d . Idi os yn c ratic s.d . Ind., min. X X X X X Construction X X X

Trade, rep. cons. X X X

Catering X X

Trans., stor., com. X X X X

Financial X X X

Rental, corp. X

Other X X

Economy X X X X X X X

Table 16: Robustness of parameter estimates with respect to the data period used

Explanation: This table indicates which of the parameters estimated in table 12 are always between the 99% confidence bounds in plots like those in figure 11.

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Figure 11: Robustness of parameter estimates with respect to data period used

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4.3 Results from other research

We read seven other researches on the relation between the default rate and the macroeconomy. We split them in two groups.

The first group of researches considers a broad selection of mostly nontraded firms. Koopman and Lucas[12], Hamerle et.al.[1] and Jakub´ık[7] investigate defaults of respectively US, German and Finnish firms while Carling et.al.[9] look at defaults on bank loans to Swedish firms. In all these researches GDP growth (or a related variable measuring the economy’s output like new business orders, producer or consumer confidence, unemployment or the output gap) are found to be significantly negatively related to defaults. Particularly interesting is the research from Koopman and Lucas who use data over a 64 year period (1933-1997)! The other researches use data periods between 9 and 25 years. Hamerle et.al. and Carling et.al. include firm spe-cific explanatory variables in their research. Jakub´ık[7] also examines the interest rate and the exchange rate. He finds a significant positive relation with the interest rate for the aggregate economy and the sectors Construction, Manufacturing and Trade and with the exchange rate for the aggregate economy and (only) the sector Trade. The other researches do not examine the interest and exchange rate.

The second group concerns firms traded on stock exchanges. These firms are mostly larger and therefore might differ from the firms examined in this research and those from the first group. Koopman et.al.[14] and Kavvathas[10] model rating transitions while Couderc and Renault[6] only look at default events. The data period varies between 17 and 25 years. These researches also confirm the negative relation between defaults and GDP growth (except Kavvathas who does not consider GDP growth). Also, they look at stock market variables and find defaults to be significantly positively related to (S&P 500) volatility and significantly negatively to (S&P 500) returns (which is consistent with Merton’s theory, see footnote 6). Finally, Kavvathas reports a significant positive relation between defaults and interest rates.

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5 Scenario analysis

In this section we examine the quantitative risk of the default rate. We do this by examining the default behaviour given an unfavourable macroeconomic scenario of two quarters zero GDP growth and a 2.5% worst case scenario. Both scenarios are compared to a certain base scenario. First we discuss the models used and then we present the scenario analysis results.

5.1 The scenario analysis model

In this section we use model (18) with only GDP growth included in z_t∗. This allows us to examine a GDP growth scenario without the need to make assumptions on the other macroeconomic variables. The model is given by

˜

pd_t,i = βi,0+ βi,1pd˜t−1,i+ βi,2∆%GDPt−1+ υt,i (20)

υt,i iid

∼ 0, σ2_ξ,i+ σ2_ψ,i

and the estimated parameters are shown in table 17. Compared to the results in table 12, the coefficients of GDP growth are somewhat closer to zero. This is consistent with macroeconomic theory which states that an increase in GDP (due to a spontaneous increase in aggregate demand) lowers the default rate but also leads to higher interest rates and an appreciating exchange rate and this has an increasing effect on the default rate.

Intercept First lag GDP growth Systematic Idiosyncratic

s.d. s.d.

Ind., min. −1.49∗∗∗ _.73∗∗∗ _−2.43∗∗∗ _.05∗∗∗ _.10∗∗∗

Construction −.86∗∗∗ _.85∗∗∗ _−.83 _.07∗∗∗ _.10∗∗∗

Trade, rep. cons. −.92∗∗∗ _.85∗∗∗ _−1.23∗∗ _.07∗∗∗ _.07∗∗∗

Catering −.54∗∗∗ _.91∗∗∗ _−1.16 _.09∗∗∗ _.13∗∗∗

Trans., stor., com. −1.69∗∗∗ _.70∗∗∗ _−2.93∗∗∗ _.08∗∗∗ _.13∗∗∗

Financial −.50∗∗∗ _.89∗∗∗ _−2.67∗∗∗ _.09∗∗∗ _.09∗∗∗

Rental, corp. −1.44∗∗∗ .76∗∗∗ −2.64∗∗∗ .09∗∗∗ .06∗∗∗

Other −2.32∗∗∗ _.68∗∗∗ _−1.14 _.05∗∗∗ _.16∗∗∗

Economy −.84∗∗∗ _.86∗∗∗ _−1.46∗∗∗ _NA _.08∗∗∗

Pooled −.75∗∗∗ _.86∗∗∗ _−1.47∗∗∗ _.07∗∗∗ _.11∗∗∗

(P-value equal coefficients .0000 .0000 .2619 .0315 .0000)

Table 17: Estimated parameters model (18) with GDP growth only

Explanation: Model (18) is estimated with only GDP growth included in zt∗. All estimated parameters are

reported. Pooled results are obtained by estimating the sector model under the restriction β1,j = . . . = β8,j or

σj,1= . . . = σj,8for certain j while allowing the other parameters to differ per sector. These restrictions are tested

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We also require a model for the explanatory variables which is in this case only GDP growth. It turns out that an AR(1) model fits GDP growth quite well. Let γ be a parameter vector and σν a nonnegative parameter.

∆%GDPt = γ0+ γ1∆%GDPt−1+ νt (21)

νt iid

∼ 0, σ_ν2

The model is estimate by OLS using data over the period 1978.1-2006.2 (114 observations). (Equations corresponding to t = 1 are ignored.) Table 18 summarizes the estimation results. Note that all parameters are significantly different from zero at the 1% level.

Intercept .01∗∗∗

First lag .61∗∗∗

Standard deviation .02∗∗∗

Table 18: Estimation results model (21).

Explanation: The parameters of model (21) are estimated. Significance at 1%, 5% and 10% level is denoted by respectively∗∗∗_,∗∗_and∗_.

In the next subsection we need to draw realizations of the disturbances υt,i in (20) and νt in (21). Although in table 14 excess kurtosis of the residuals was not significant, we do take it into account because kurtosis is important when estimating worst case scenarios. We assume the disturbances follow after standardizing a standardized t-distribution with df degrees of freedom. The pdf of a standardized t-distribution evaluated at a real number x is given by

Γdf +1₂ Γdf₂p(df − 2)π 1 + x 2 df − 2 df +12 . The kurtosis is 3df − 6 df − 4

and we set the degrees of freedom such that this equals the sample kurtoses. Table 19 shows the sample kurtoses of the disturbances and the degrees of freedom of the fitted t-distributions. For the sector model we pool the kurtoses because the kurtoses differ quite a lot between the sectors.

Sample kurtosis Degrees of freedom

Sector model 4.31 8.60

Economy model 4.12 9.38

GDP model 5.49 6.41

Table 19: Sample kurtosis and degrees of freedom of disturbances in (20) and (21).

Explanation: The sector and economy model refer to respectively υt,i∀i ∈ {1, . . . , 8} and υt,0in (20); the GDP

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5.2 Scenario analysis results

To analyze scenarios we generate 200000 paths of the logit default rate using (20) and (21) given certain starting values for ˜pdt,i and ∆%GDPt. Disturbances υt,i and νt are generated by mul-tiplying draws from the t-distributions discussed above by their respective standard deviations q

σ2 ξ,i+ σ

2

ψ,iand σν. Finally, we invert (10) to find the default rate:

pdt,i=

exp ˜pd_t,i 1 + exp ˜pdt,i

.

(See footnote 5.) In this section we are interested in the average 2007 default rate. Define

¯ pd_2007,i=1 4 4 X t=1 pd2007.t,i.

(We skip the quarters 2006.3 and 2006.4 because it takes some time before the default rate reacts to what we assume for the period after 2006.2.)

5.2.1 Base scenario

First we compute the expected average 2007 default rate without assuming making assumptions on what happens after 2006.2. We do so by setting the starting values for ˜pd_t,i and ∆%GDPt equal to the known values from 2006.2, generating the logit default rates and computing ¯pd2007,i. Next we compute the average of all generated ¯pd_2007,i.

Table 20 shows the average generated ¯pd_2007,i. The results in this table only serve as a benchmark for the results from the following two subsections.

The average generated economy default rate is plotted against time in figure 12 at the end of this section. It remains approximately constant because both the default rate and GDP growth were in 2006.2 already close to their long run averages.

Default rate

Ind., min. .31%

Construction .26%

Trade, rep. cons. .22%

Catering .31%

Trans., stor., com. .29%

Financial .83%

Rental, corp. .20%

Other .06%

Economy .22%

Table 20: Base scenario

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5.2.2 GDP growth scenario

In this subsubsection we assume zero GDP growth in quarters 2006.3 and 2006.4. (In 2006.2 GDP growth was 2.8%.) After 2006.4 GDP growth evolves according to (21). Logit default rates evolve according to (20) with the ˜pdt,i from quarter 2006.2 as starting value. We generate the logit default rates and compute ¯pd_2007,i. Next we compute the average of all generated ¯pd_2007,i. Table 21 compares the average generated ¯pd_2007,i given the zero GDP growth scenario to the one given the base scenario. The percentage difference and its 95% confidence interval are reported. (The confidence interval captures uncertainty in the percentage difference caused by uncertainty in the estimated parameters of models (20) and (21). So it refers to the expected effect and not the true effect.) The effects are surprisingly small even if we look at the upper bounds. We can compare this to historical events by looking at the upper panel of figure 5. During the period 1983-1991 there were about three brief sharp drops in GDP growth. In these cases the default rate did not visibly react. However, during the more lengthy GDP growth slowdowns of 1991-1993 and 2000-2003 the default rate doubled approximately. So, it appears that the default rate only reacts substantially to long lasting GDP growth developments (see the lower panel of figure 5). But the long run effects that can be computed from table 17 using (16) (10% higher economy default rate for each persistent percentage point drop in GDP growth) are also quite small. Possibly, our specification of the default rate depending on higher lags of GDP growth only through the lagged default rate is too simple. Directly including more lags might enable the model to better distinguish between short drops in GDP growth and more lengthy recessions. Note that, in accordance with the estimation results from table 17, the sectors Industry and mining, Transport storage and communication, Financial services and Rental and corporate services are affected most by the zero GDP growth scenario.

In figure 12 at the end of this section the average generated economy default rate (given the zero GDP growth scenario) is plotted against time.

Lower bound % Difference default rate Upper bound

Ind., min. 5% 10% 16%

Construction −1% 4% 10%

Catering 1% 7% 12%

Trans., stor., com. 7% 12% 18%

Other −1% 4% 10%

Economy 2% 8% 13%

Table 21: Zero GDP growth scenario

Explanation: This table shows the percentage difference between the average generated ¯pd2007,i given the zero

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5.2.3 Worst case scenario

Finally, we look at the 2.5% worst case scenario for the average 2007 default rate without making assumptions on what happens after 2006.2. Like in the case of the base scenario, we do so by setting the starting values for ˜pdt,i and ∆%GDPt equal to the known values of quarter 2006.2, generating the logit default rates and computing ¯pd_2007,i. Next, from all generated ¯pd_2007,i we pick the .975th percentile.

Table 22 compares the .975th percentile of all generated ¯pd_2007,i to the average generated ¯

pd2007,i given the base scenario. The percentage difference and its 95% confidence interval are reported. (The confidence interval captures uncertainty in the percentage difference caused by uncertainty in the estimated parameters of models (20) and (21).) We see that the 2.5% worst case scenarios are a lot worse than the zero GDP growth scenarios from the previous subsection. For most sectors the 2.5% worst case scenario is three to four times as bad as the the zero GDP growth scenario. But for the sectors Construction, Catering and Other, which are relatively insensitive to GDP growth, the worst case scenario is over nine times as bad.

The .025th and .975th percentiles of the generated economy default rates are plotted against time in figure 12 at the end of this section. The figure shows clearly that in our estimated model the 2.5% worst case scenario is a lot worse than then the zero GDP growth scenario.

Lower bound % Difference default rate Upper bound

Ind., min. 25% 33% 41%

Catering 54% 62% 70%

Trans., stor., com. 35% 43% 51%

Other 32% 40% 48%

Economy 20% 28% 36%

Table 22: 2.5% worst case scenario

Explanation: This table shows the percentage difference between the .975th percentile of all generated ¯pd2007,i

and the average generated ¯pd2007,i given the base scenario. The percentage difference is an estimate because of

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Figure 12: Forecasting the economy default rate

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6 Conclusions

In this research we modeled the (quarterly logit) Dutch default rate of the entire economy and of 8 sectors using a linear model.

First, we examined the relation with several macroeconomic variables (one quarter lagged). The following variables were considered: GDP growth, interest rate, exchange rate, stock market return and volatility and oil price. A convincing negative relation with GDP growth was found. The relation with the (logarithm of the real euro) oil price is also significant in several sectors. Furthermore, there is some indication of a positive relation with the (short) interest rate for the sector Construction and with the (logarithm of the real) exchange rate for the sectors Transport, storage and communication, Financial services and Rental and corporate services. No relation with stock market return and volatility was found. (See table 12 for an overview.) Remarkably, for the interest rate, exchange rate and oil price not the change but the level of the variables turned out to be significant. For the economy default rate the relations with the macroeconomic variables are stable through time, but for the sector default rates most relations are unstable except those with the oil price(!). The first lag of the logit default rate has a highly significant coefficient. This implies that the effect of persistent macroeconomic shocks gradually increases over time. Our macroeconomic variables explain on average about a fifth of the variance of the logit default rate corrected for the forecast from its first lag. A latent factor affecting all sectors explains about thirty percent and the rest is explained by sector specific disturbances. Other research mainly confirms the results on GDP growth and, to a limited extent, interest and exchange rate. Furthermore, the stock market is often found to be related but always for firms listed on a stock exchange.

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References

[1] T.Liebig A. Hamerle and H. Scheule. Forecasting credit portfolio risk. Deutsche Bundesbank, 2004.

[2] E. Altman and H. Rijken. How rating agencies achieve rating stability. Journal of Banking & Finance, 2004.

[3] E. Altman and H. Rijken. The effects of rating through the cycle on rating stability, rating timeliness and default prediction performance. New York University, 2005.

[4] E. Altman and G. Sabato. Modeling credit risk for SME’s: evidence from the US market. ABACUS, 2007.

[5] J. Durbin and S. Koopman. Time series analysis by state space methods. Oxford University Press, first edition, 2001.

[6] F.Couderc and O. Renault. Times-to-default: life cycle, global and industry cycle impacts. FAME, 2005.

[7] P. Jakub´ık. Does credit risk vary with economic cycles: the case of Finland. Bank of Finland, 2006.

[8] J. Johnston and J. Dinardo. Econometric Methods. McGraw-Hill, fourth edition, 1997. [9] J. Lind´e K. Carling, T. Jacobson and K. Roszbach. Capital charges under Basel II: corporate

credit risk modelling and the macro economy. Sveriges Riksbank, 2002.

[10] D. Kavvathas. Estimating credit rating transition probabilities for corporate bonds. Univer-sity of Chicago, 2001.

[11] N. Kiefer. Economic duration data and hazard functions. Journal of Economic Literature, 1988.

[12] S. Koopman and A. Lucas. Business and default cycles for credit risk. Tinbergen Institute, 2003.

[13] F. Rolwes. Explanation dataset Dutch bankruptcies and firms. De Nederlandsche Bank, 2006.

[14] A. Lucas A. Monteiro S. Koopman, R. Kr¨aussl. Credit cycles and macro fundamentals. Tinbergen Institute, 2006.

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Appendix 1: Additional results on interest rates

In this appendix we reestimate the model from section 4.1 using the long interest rate or the term spread instead of the short interest rate.

First we show in table A1 the results when the long rate is used instead of the short rate. The estimated coefficients are a bit larger than those of the short rate reported in table 9. Partly this can be explained by the lower volatility of the long rate (see table 5), but, in this specification, the relations are also just somewhat more significant. But if we would reestimate the model from section 4.2 we find the long rate has similar significance as the short rate. The relations with the change in the long rate all have the wrong sign and are mostly insignificant.

Long rate ∆ Long rate

Ind., min. 1.71∗ −6.21∗

Construction 3.30∗∗∗ −8.21∗∗

Trade, rep. cons. 1.72∗∗ −2.94

Catering .71 −1.29

Trans., stor., com. .42 −7.53

Financial −.69 −.02

Rental, corp. 2.12∗∗ −4.74

Other 1.78 −1.07

Economy 1.23∗ −3.67

Pooled 1.72∗∗ −4.36

Table A1: Estimated coefficients long rate

Explanation: Model (18) is estimated with the macroecomic variables from section 4.1 included in zt∗but with

the short rate replaced by the long rate. The coefficients of the level and the change of the long rate are reported. Pooled results for variable j are obtained by estimation under the restriction β∗

1,j= . . . = β∗8,j. These restrictions

are tested and the p-values are reported. Significance at 1%, 5% and 10% level is denoted by respectively∗∗∗_,∗∗

and∗_.

(46)

Term spread ∆ Term spread

Ind., min. −.46 −.34

Construction −.42 −3.42

Trade, rep. cons. .79 −1.84

Catering 3.32∗∗ −4.20

Trans., stor., com. 2.30 −8.29∗

Financial −1.16 −2.59

Rental, corp. .29 −2.17

Other 3.95∗∗ 3.71

Economy −.37 −3.23

Pooled .32 −1.62

Table A2: Estimated coefficients term spread

Explanation: Model (18) is estimated with the macroecomic variables from section 4.1 included in z∗

t but with

the short interest rate replaced by the term spread. The coefficients of the level and the change of the term spread are reported. Pooled results for variable j are obtained by estimation under the restriction β∗

1,j= . . . = β8,j∗ . These

restrictions are tested and the p-values are reported. Significance at 1%, 5% and 10% level is denoted by respectively

The Dutch default rate and the macroeconomy Ferdinand Rolwes 19 may 2007

The Dutch default rate and the macroeconomy

The Dutch default rate and the macroeconomy

Contents

1

Introduction

1.1

Background and objectives

1.2

Data sources

1.3

Overview of the report

2

The data

2.1

Exploring the default rate

2.2

Exploring the macroeconomic variables

2.3

Some important properties

2.4

Timing of financial distress and bankruptcy

3

Default modeling

3.1

An overview of possible models

3.2

The model

4

Estimation results

4.1

A general specification

4.2

A parsimonious specification

4.3

Results from other research

5

Scenario analysis

5.1

The scenario analysis model

5.2

Scenario analysis results

6

Conclusions

References

Appendix 1: Additional results on interest rates