A compound option approach to model the interrelation between banking crises and country defaults: The case of Hungary 2008

Journal of Banking & Finance 34 (2010) 3025–3036

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A compound option approach to model the interrelation between banking crises and country defaults: The case of Hungary 2008 Dominik Maltritz * University of Erfurt, Faculty of Economics, Law and Social Sciences, Germany

a r t i c l e

i n f o

Article history: Received 15 July 2009 Accepted 1 July 2010 Available online 6 July 2010 JEL classification: F34 G13 G12

a b s t r a c t We analyze the Hungarian financial crisis of 2008 in a stochastic framework that advances structural credit risk models for country defaults: by applying compound option theory we consider payments for bailing-out the banking sector together with debt service payments in a joint crisis model. We estimate the model parameters by applying the time series maximum-likelihood approach of Duan (1994) on yield spreads of Hungarian Bonds. We find that difficulties in acquiring funds for debt servicing in combination with high outstanding debt triggered the crisis, rather than problems in the domestic banking sector. The estimated crisis probabilities dramatically rise during 2008. Ó 2010 Elsevier B.V. All rights reserved.

Keywords: Banking crises Country defaults Structural credit risk model Crises dependencies Compound option

1. Introduction The recent global financial crisis and its huge impact on economies around the world very clearly demonstrate the relevance of a thorough investigation of the determinants and causes of financial crises. In the present paper we focus on the influence of the indebtedness of countries and the interrelation between country defaults and financial crises. For many countries, in particular, small economies and developing or transition countries, this relation seems to be an important issue, since financial crises and defaults frequently occur simultaneously. Recently, e.g., turmoil in the banking sector has been observed simultaneously with (quasi-) defaults1 in several countries, such as Hungary, Iceland and Latvia. We analyze the dependency between financial crisis and sovereign debt using a stochastic framework and a compound option approach. The present paper is inspired by a strand of the

* Tel.: +49 361 737 4621; fax: +49 361 737 4629. E-mail addresses: [email protected], [email protected] 1 Quasi default means that the country alone did not have enough funds to make its required debt service payments (and support the domestic) financial sector. Rather the country needed international help to avoid a de-jure default. 0378-4266/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jbankfin.2010.07.003

literature where structural credit risk models were successfully applied to analyze and estimate country default risk. Examples include Clark (1991), Claessens and van Wijnbergen (1993), Claessens and Pennacchi (1996), Keswani (2000), and Huschens et al. (2007). The structural approach relies on the idea that a country defaults if the amount of funds the country is able and willing to spend for debt servicing is below the country’s debt service obligations. These papers generally use a stochastic framework to incorporate the fact that the amount of funds a country is able and willing to spend for debt servicing is uncertain and unobservable. The stochastic approach applied in these papers enables estimating a country’s default probability. What is more, it provides insights into the causes and determinants of defaults. In particular, the structural approach makes it possible to estimate the funds a country is able and willing to spend for debt servicing based on observable market data, such as government bond prices. Structural credit risk models have also been applied to analyze crisis risks in the banking sector. The first contribution in this context is Merton (1977) who analyzes bank deposit insurances. More recent contributions that analyze the risk of bank failures and value bank deposits include Duan et al. (1995), Gunther et al. (2001), Duan and Simonato (2002), Gropp et al. (2002), Chan-Lau et al.

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(2004), and Liao et al. (2009).2 Banking crises can be forecasted by applying this approach not only to a single bank but to all (or the majority of) banks of a country. The papers mentioned so far consider either problems in the banking sector or country defaults. Since banking crises and country defaults, however, often occur together, we aim to capture them together in one model. This requires expanding the single payment framework firstly proposed by Merton (1974), which is mostly applied in the literature. This is because we have to consider payment requirements for debt servicing, on the one hand, and payments required to bail-out the banking sector, on the other. To model the country’s payment requirements at different dates, i.e. for debt servicing and bailing-out its banking industry, we employ a compound option approach, firstly proposed by Geske (1977, 1979).3 We apply this framework to the case of the financial crisis in Hungary (2008) to analyze the essential causes of this crisis. We estimate the model parameters, i.e. the main determinants of crises occurrence, using data derived from secondary market prices of Hungarian government bonds. In particular, we consider sovereign yields and yield spreads given in the Emerging Market Bond Index by JP Morgan. The estimation of unknown quantities relies on the time series maximum-likelihood approach firstly proposed by Duan (1994). This estimation approach was firstly applied in the field of country default risk by Huschens et al. (2007). Huschens et al. (2007) also point out that it would be possible to capture multiple debt service payments of countries at different dates rather than a single payment (as in the Merton model) by using the Geske (1977) model. However, they only consider one payment date in the empirical application, i.e. they use the Merton model. Inspired by Huschens et al. (2007) we estimate Hungary’s (debt and banking) crisis risk by applying the more complex compound option model of Geske. This improves the modeling of crisis risk in two ways: first we are able to distinguish between shortterm and long-term debt service payments, and, what is more, we capture bail-out payments for the banking sector in addition to debt service payments. Based on time series of observed bond market data we estimate the development of funds Hungary is able and willing to spend for avoiding a financial crisis and a default in the course of the crisis year 2008. The estimated amount of these funds drops from approximately USD 182 billion at the beginning of 2008 to USD 108 billion at the end of 2008. Our results indicate that difficulties in acquiring funds for debt servicing (in particular by raising new credits) in combination with high outstanding debt are the main reasons for the financial crisis, rather than problems in the domestic banking sector. Based on the estimated model parameters we also can derive crisis probabilities for the crisis year. Our approach enables us to distinguish between short-term crisis risk and longterm default risk. Both types of risk considerably increase during 2008. The increase in short-term risk is, however, much stronger (reaching values of about 50% in early October 2008), which indicates that a crises was about to occur sooner rather than later. The remainder is organized as follows. In the next section we sketch the model (and discuss the influence of the determinants).

2 Another strand of the literature focuses on accounting data to forecast or explain bank distress. Recent examples include Arena (2008) and Männasoo and Mayes (2009). In comparing both approaches Agarwal and Taffler (2008) find no statistical significant differences between accounting-based and market-based approaches. Also Bongini et al. (2002) find that stock market information and accounting data have a similar ability to assess bank fragility although stock prices respond more quickly to changing conditions than ratings or balance sheet information. 3 A similar approach is proposed in Maltritz (2008) to model the interrelation between currency crisis and country defaults. Besides the different topic the present paper differs from Maltritz (2008) since we apply the model empirically to analyze the crisis in Hungary, whereas Maltritz (2008) is a purely theoretical contribution.

The third section describes the estimation of the model parameters. In the fourth section we discuss the application to Hungary and its results. On the one hand we describe the development of crisis probabilities in the pre-crisis period. On the other, we discuss the main determinants and causes of crisis. Section 5 concludes. 2. The model 2.1. The joint crisis model To evaluate the Hungarian crisis, we adapt a structural model based on the compound option approach derived by Geske (1977). This model extends the basic structural model of Merton (1974) to include multiple debt service payments instead of just a single payment. Structural credit risk models concerned with country default risk rely on the idea that the government is forced to spend money for the required debt service payments in order to avoid a default. Thus, a default occurs if the available funds are lower than the required debt service obligations. Since the amount of funds available for debt servicing at the payment date is not known with certainty, it is modeled by a stochastic process which is typically given by:

dW ¼ lW Wdt þ rW WdZ;

ð1Þ

where lW and rW are constant, and Z follows a standard Wiener process. It follows that the growth rates of the state variable for equidistant time intervals, Dt, are independent and identically normally distributed:

wt;Dt ¼ lnðW t =W tDt Þ  i:i:n:



lW 

r2W 2

 pffiffiffiffiffiffi Dt; rW Dt :

ð2Þ

We employ Geske’s (1977) framework with multiple payments required at different dates in order to consider the payments a country is required to make for bailing-out its banking industry in addition to its debt service payments.4 The government has to spend money to avoid a financial crisis regardless of whether it is to avoid a banking crisis or to avoid a default on debt service payments. Thereby, the government can spend funds only once; funds spent for bailing-out the banking sector are not available for debt servicing and vice versa. Of course, other measures exist to help the banking system, which do not require payments by the government, for example a ‘capital ratio guarantee’. Such a guarantee may help to restore confidence in the banking system and help to avoid capital withdrawals, even without any payment required by the government. Our model does not consider such measures to avoid a crisis. It focuses instead on payments actually required for the bail-out of the banking system, for example, because a government may be forced to spend money to fulfill its guarantee. These payments may be seen as a last resort after guarantees were given, but turned out to be unsuccessful. In the application, we consider the payments only that were actually required to save the Hungarian banking system. Geske (1977) has derived the value of a debt contract that securitizes two payments, B1 and B2, at T1 and T2 (T1 < T2), respectively, where at both payment dates it is possible that required payments are not made, i.e. a default occurs:

  pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi F t ¼ W t  W t N2 d1 þ rW T 1  t; . . . ; d2 þ rW T 2  t; fqg þ B2 ers ðT 2 tÞ N2 ðd1 ; d2 ; fqgÞ þ B1 ers ðT 1 tÞ N1 ðd1 Þ;

ð3Þ

4 We focus on a model with only two payments required for crisis avoidance, which we use in the empirical application on Hungary. Based on the ideas explained for the two-payment case, Geske (1977) derives the respective formulas that can be used in a situation where more than two payments are required to avoid a crisis.

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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B1 ¼ W Q N1 ðd þ rW T 2  T 1 Þ  Bers ðT 2 T 1 Þ N1 ðdÞ;

where:

lnðW t =W Q Þ þ ðrs  r  tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; rW T 1  t lnðW t =B2 Þ þ ðr s  r2W =2ÞðT 2  tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi ; d2 ¼ rW T 2  t 2 W =2ÞðT 1

d1 ¼

with : d ¼

and

sffiffiffiffiffiffiffiffiffiffiffiffiffi T1  t q¼ : T2  t In the Geske model the borrower owns a compound option at any date t before the first payment date T1, which can be valued using the compound option formula for a call option (see Geske, 1979):

Y t ¼ W t N2 ðd1 þ rW

ð4Þ

where d1, d2 and q are calculated as described in Eq. (3). N1(  ) and N2(  ) describe the value of the one-dimensional and the twodimensional cumulative standard normal distribution for the value in parentheses, respectively, rs is the risk-less interest rate and Wt is the value of the state variable in t. In our case of a joint banking and debt crisis, B1 describes the payment for bailing-out the banking system and B2 is the required debt service payment. WQ in (3) and (4) is the crisis threshold valid at the first payment date. A crisis will occur if the state variable describing the payment capacity, i.e. the funds available for crisis avoidance, is below the threshold W T 1 < W Q . To determine this threshold valid in T1, Geske considers that at any date t before the second payment date, T1 6 t < T2, the borrower owns a (simple) call option for which W is the underlying and B2 is the strike price.5 In T1 the borrower has the opportunity to buy the option that expires in T2. The borrower is, however, not forced to buy this option. He has the option to buy this option or to refuse to do so. Thus, he owns an option to buy another option, i.e. a compound option for which T1 is the first expiry date and T2 is the second expiry date. Applied in our case, this means that by spending the funds B1 required to bail-out the banking sector at T1, the government fends off a banking crisis at T1. In addition, it gains the option, Y, which makes it possible to avoid a crisis at all: a banking crisis at T1 as well as the potential debt crises in T2. The latter can be achieved by spending the required funds for debt servicing, B2, at T2. The government buys this option at T1 only if the value of the option, Y T 1 , in T1 is greater than or equal to the required payment, B1. If the value is smaller, the country would be worse off if it spends the required funds, since the value of the option to avoid a debt crisis at T2 (and thus a crisis at all) is lower than the value of this possibility. Thus, we can specify the condition for avoiding a crisis at T1:

Y T 1  B1 :

ð5Þ

In the Merton–Geske framework, the option value, Yt, at any date t before the expiration date of the option, T2, can be priced using the Black–Scholes formula for a call option (see Black and Scholes, 1973, p. 644). We can derive a formula for the threshold by inserting the Black-Scholes formula in formula (5) for Y T 1 . The threshold, WQ, is the value of the payment capacity W T 1 for which the resulting formula becomes an equation:

5

 T1Þ

ð6Þ

:

If the state variable describing the capacity of payments for crisis avoidance, W T 1 , is greater than or equal to the threshold, WQ, the option value is higher than its price, i.e. the required payment, B1. The government avoids a crisis by spending the required funds. If by contrast the state variable is lower than this threshold, W T 1 < W Q , the option’s value is lower than its price. The government does not buy the option by spending the required funds, and a (banking) crisis occurs. 2.2. Crises probabilities

pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi T 1  t ; . . . ; d2 þ rW T 2  t ; fqgÞ

 B2 ers ðT 2 tÞ N2 ðd1 ; d2 ; fqgÞ  B1 ers ðT 1 tÞ N1 ðd1 Þ;

lnðW Q =B2 Þ þ ðr s  r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rW T 2  T 1

2 W =2ÞðT 2

Here, Geske can make use of Merton’s (1974) results. Since after T2 no further payments are required, the situation is similar to the Merton case. If the country’s payment capacity in T2 is equal to or higher than the required debt service payment, the government makes the payment and avoids a default. The country opts for default, by contrast, if the payment capacity is lower than the payment necessary to avoid the default.

The occurrence of a crisis at T1 or T2 is an uncertain event since the payment capacity, i.e. the amount of funds the country is able and willing to spend for avoiding a crisis, is not known with certainty. We model this uncertainty by assuming the state variable describing these funds is a stochastic variable. Using our framework, we can derive closed-form solutions for the crisis probabilities. As explained in the last section, the crisis probability is the probability that the state variable describing the payment capacity is below the respective crisis threshold WQ at T1 or B2 at T2. We start with discussing the short-term probability of a banking crisis in T1. At any point in time t (
1 mean wmin zfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflffl{ zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{ BlnðW =W Þ  ðl  r2 =2ÞðT  tÞC C B Q t 1 pWffiffiffiffiffiffiffiffiffiffiffiffiffiW ¼ N1 B C: A @ r T1  t W |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} 0

PT 1 ;t

ð7Þ

standard deviation

Eq. (7) is easily to explain, if we formulate the condition that a crisis occurs if the state variable is below the threshold in terms of growth rates of the state variable, wt,T = ln(WT/Wt). Given the current value of the state variable Wt and the threshold WQ the minimum growth rate necessary to avoid a crisis is wmin = ln(WQ/Wt). The crisis probability equals the probability that the realized growth rate is less than this minimum growth rate. The growth rates are independent and identically normally distributed (see Eq. (2)). Hence, the probability that the realized growth rate over T  t is less than wmin – and, thus, that WT is less than WQ – can be estimated by standardizing wmin with mean and standard deviation (as it is shown in Eq. (7)) and calculating the value of the (one-dimensional) cumulative standard normal distribution, N1(  ), for the resulting standardized growth rate. The probability that no banking crisis occurs at T1 can be calculated as the complementary probability. The probability that a crisis occurs neither at T1 nor at T2 equals the probability that the state variable is higher than the respective thresholds at the possible crises dates, i.e., higher than WQ at T1 and higher than B2 at T2. This probability can be estimated by N2(m0, m1; q) in Eq. (8). The overall crisis probability, i.e., the probability of a banking crisis at T1 or a debt crisis at T2 can be calculated as the complementary probability by:

PT 1 ;T 2 ;t ¼ 1  N2 ðm1 ; m2 ; qÞ; lnðW t =W Q Þ þ ðlW  r2W =2ÞðT 1  tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi where : m1 ¼ ; rW T 1  t lnðW t =B2 Þ þ ðlW  r2W =2ÞðT 2  tÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi m2 ¼ rW T 2  t

ð8Þ

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and

170

sffiffiffiffiffiffiffiffiffiffiffiffiffi T1  t q¼ : T2  t

160 150 140

With respect to the long-term crisis probability, meaning the probability of a debt crisis at T2, we have to distinguish between conditional and unconditional probability. At any date t (before T1), the conditional probability that a (debt) crisis will occur at T2, given that no (banking) crisis has previously occurred, can be determined by:

PT 2 ;t ¼ 1 

1  PT 1 ;T 2 ;t : 1  PT 1 ;t

ð9Þ

Short-term Threshold

130

Line O Line A Line B Line C Line X

120 110 100 90 80 70

Hungary's actual short-term payment requirements in 2008: USD 58 billion

60 50 40 30 20

This can be explained as follows: the probability that a crisis occurs neither at T1 nor at T2 equals 1  PT 1 ;T 2 ;t (see Eq. (8)). This probability that no crisis occurs is also given by the probability that no crisis occurs in the short-run times the probability that no crisis occurs in the long-run: 1  PT 1 ;T 2 ;t ¼ ð1  PT 1 ;t Þ  ð1  PT 2 ;t Þ (see Delianedis and Geske, 1998). Rearranging shows that we can calculate the probability that no crisis occurs in the long-run, ð1  P T 2 ;t Þ, by the quotient on the right-hand side of Eq. (9): ð1  P T 1 ;T 2 ;t Þ=ð1  PT 1 ;t Þ. The complementary probability gives the long-term crisis probability. We can also specify the unconditional probability that the state variable falls below the long-term threshold, i.e. the debt service payments, at the second date, estimated at date t before T1 with the values observable at t:

  2 W  rW =2ÞðT 2  tÞ e T ;t ¼ N1 lnðB2 =W t Þ  ðlp ffiffiffiffiffiffiffiffiffiffiffiffiffi P : 2 rW T 2  t

ð10Þ

This formula is also valid for calculating the crisis probability at any date between T1 and T2. 2.3. A short discussion of crisis determinants according to the adopted model Based on the derived formulas for crisis probabilities, we can now discuss the influence of certain determinants on crisis risk. Of course, an increase of the payment capacity, Wt, i.e. of the amount of funds the government is able and willing to use to avoid a crisis, lowers the crisis risk, regardless of whether we consider the short-term or the long-term crisis probability. The crisis probabilities are also lowered by a higher (positive) drift, lW, which describes the expected future development of the state variable, i.e. the prospects of a country’s ability to acquire funds for crisis avoidance. Higher indebtedness, B2, increases the probability of a debt crisis. Higher payments required to bail-out the banking sector, B1, increase the probability of a banking crisis, since they raise the threshold value, WQ.6 From the properties of Eq. (6) follows that the increase in the threshold is higher than the increase in the short-term payment, i.e. the partial derivative, o WQ/ o B1, is above one. This is displayed in Fig. 1: The slope of the curves that relates the threshold to the short-term bail-out payments is clearly above one, especially for constellations where the short-term payments are lower than the long-term payments. One of the most important results is that indebtedness also has an impact on the banking crisis risk: A higher amount of debt raises the banking crisis probability since the threshold, WQ, for a banking crisis at T1 depends not only on the current payments, B1, but also on later payments, B2. Eq. (6) shows that higher later payments increase the threshold: Since in (6) the relationship between debt servicing payments, B2, and repayments necessary to avoid a banking crisis, B1, is negative (and between B1 and the threshold WQ it is positive), a higher indebtedness raises WQ for a given B1. This is 6

Because of the positive relationship between B1 and WQ in (6), WQ rises with B1.

10 0 0

10

20

30

40

50

60

70

80

90 100 110 120 130 140 150 160 170

Short-term Payment Requirements

Fig. 1. Short-term crisis threshold (WQ) in relation to short-term payment requirements for different constellations of parameters (Line O is calculated using the constellation of parameters that approximates the situation in Hungary in 2008: rW = 0.29, B2 = 83 (billion of USD), rf = 0.03, T2  T1 = 5 years. The latter matches the observed time T2  t of about 6 years. The actual short-term payment requirements were about 58 billion, which lead to a threshold of about 110. However, the figure shows the thresholds that would be obtained when values of B1 are changed). For the other lines we changed the constellation of the remaining parameters: Line A: We assumed long-term payments of 138, which would be the case if all Hungarian debt would be long-term. All other parameters are the same as above. It can be seen that the increased long-term payments increases the short-term threshold considerably. Line B: Here all parameters are equal to that used to calculate Line O, except the time to maturity of the (long-term) debt, which is decreased by 1.5 years (see the discussion in Section 4.4). This slightly increases the threshold for given short-term payments. A shift in the same direction would follow for a lower volatility. Line C: Here all parameters are equal to that used to calculate Line O, except the time to maturity, which is increased by 1.5 years (see the discussion in Section 4.4). This slightly decreases the threshold for given short-term payments. A shift in the same direction would follow for a higher volatility. Line X: The 45°-line shows the increase of the threshold that would result from the short-term payments only, i.e. if the influence of long-term payments is neglected.

shown in Fig. 1: for Line A the long-term payments are higher as for the other lines, which leads to higher threshold. From our considerations follows that a lower debt burden not only reduces the default risk, but also the banking crisis risk. Besides the amount of outstanding debt, also its (time to) maturity, T2, influences crisis risk. Here we can derive another important result of our model (see Line B and C in Fig. 1): since banking crisis risk depends on the outstanding debt service payments, B2, the maturity of debt influences not only the risk of a debt crisis, but also the risk of a banking crisis. This is because the threshold for a banking crisis depends on the time span T2–T1 and, thus, on the debt’s time to maturity. Here a longer time span decreases the threshold and, hence, the banking crisis risk. This means a longer maturity of debt decreases banking crisis risk. Also the debt crisis risk depends on the debt’s maturity. For constellations of other parameters typically observed in reality, where the expected value of the state variable is above the payment requirements a longer time to maturity increases debt crisis risk. Also the uncertainty about a country’s payment capacity influences crisis risk. This uncertainty is modeled by the volatility of the state variable rW. The volatility can influence the short-term banking crisis risk as well as the long-term debt crisis risk and, thus, the overall crisis risk in both directions, positively as well as negatively. For constellations of input-parameters typically observed in reality, where the crisis risk is relatively low, a higher volatility implies a higher crisis risk. Only in crisis situations, where the value of the payment capacity expected for the payment date is below the crisis thresholds, may a higher volatility decrease the risk of a crisis. In rather tranquil times, lower uncertainty about the country’s payment capacity – resulting from stability,

D. Maltritz / Journal of Banking & Finance 34 (2010) 3025–3036

sustainability and credibility of the government’s economic policy – decreases the probability of (both types of) financial crises. 3. The estimation of the state variable and its parameters based on market data 3.1. Estimating the state variable based on the market value of a compound option Applying the framework presented in the last section to estimate crises probabilities and discuss the determinants of the Hungarian crisis requires the quantification of the model parameters. Here, in particular, the stochastic state variable describing the amount of funds a country is able and willing to spend to avoid a crisis (i.e., a banking crisis, on the one hand, or a default (debt crisis) on the other) is challenging. This payment capacity is not observable and there is a high degree of uncertainty about the amount of these funds. Our model accounts for this fact by assuming the payment capacity to be a stochastic variable. It is even hard to approximate the payment capacity based on observable (macro) economic variables, (among others) because it is problematic to assess which part of its revenues or the GDP the government can make available to avoid a crisis. Even if this quantity would be known, there remains the problem of assessing the amount of the payments the government is actually willing to make. Instead of estimating the amount of funds provided for crisis avoidance by considering macroeconomic variables we use observable market data on prices of Hungarian government bonds. Since the value of these securities depends on the country’s capacity to avoid a crisis and the resulting crisis risk, their prices – determined on secondary markets for the country’s government bonds by market participants through their supply and demand of these securities – reflect an assessment of crisis risk. Thus, bond market data can be used to back out market participant’s perception of crisis risk and its underlying causes, here, in particular, the amount of funds available for crisis avoidance. We can use this bond market data to value the – hypothetical – call option, Yt, the government owns at any date t before T1, at which the (next) payments are required to avoid a (banking) crisis, as well as the current value of crisis risky payments, Ft.7 In the twopayment case this compound option could be valued by using Eq. (4) and the crisis risky payment requirements by using Eq. (3),8 if all parameters on the right-hand side would be known. We do not know, however, some of the parameters on the right-hand side. As explained, the state variable, Wt, is unobservable. In the application we, thus, employ the pricing equation(s) in the reverse direction: We estimate the value of the compound option, Yt, or the payment requirements, Ft, at t from observable market data and insert it on the left-hand side of Eqs. (3) and (4), respectively. Then we solve for the unknown quantities, i.e. the unobservable state variable describing Hungary’s funds for crisis avoidance and its volatility (as well as its drift). Thereby, we assume that the amount of payments required at T1 and T2 to avoid a crisis, B1 and B2, and also the risk-less interest rate, rs, are observable and known. The option value a payment date can never be negative. If no crisis occurs, the option has a positive value that equals the difference between the funds the government is able and willing to spend for crisis avoidance and the required payments. In the crisis case its value is zero. If a crisis occurs, i.e. if the funds available for crisis avoidance are below the required funds, the holders of claims 7 In fact, we consider the value of the crisis risky payment requirements in the application instead of the compound option, as explained below. 8 As explained in the next section, we use the two-payment model because of data availability. Otherwise, one could use formulas for multiple payments as shown in Geske (1977).

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against the Hungarian government or Hungarian Banks suffer losses since there are not enough funds available to pay the contractual amount of payments to these stakeholders. Thus, the value of the state variable at a payment date equals the sum of the required payments and the option value. In absence of arbitrage opportunities this must hold true at any date t before the payment dates. Hence the value of the compound option is given by:

Y t ¼ W t  Ft :

ð11Þ

Since there is a chance that the required payments will not be made (fully), the value, Ft,i, of such crisis risky payments at any date before their maturity is lower than the current value of risk-less investments, Bt,i, which follows from discounting the contractual payments Bi by the risk-less interest rate, rs: F t;i < Bt;i ¼ Bi ers ðT i tÞ . The risk of potential losses should be reflected in a lower price and a resulting internal rate of return that exceeds the risk-less interest rate, rr > rs. Thus, the value of a default risky payment at date Ti is F t;i ¼ Bi err ðT i tÞ . The current market value all future payment requirements Bi follows from summing up the current values of all outstanding payments (with i = 1, 2 in the two-payment case):

Ft ¼

I X

Bi err;i ðT i tÞ :

ð12Þ

i¼1

By inserting the sum of these payments into Eq. (11) we obtain the market value of the compound option:

Yt ¼ Wt 

I X

Bi err;i ðT i tÞ :

ð13Þ

i¼1

In fact, yield spreads to the risk-less interest rates (risk-less interest rates are represented by the US treasury yield curve) are typically observed for developing and transition countries and widely used in the literature as an indicator for crisis risk. Several papers applying structural credit risk models (e.g., Claessens and Pennacchi, 1996; Keswani, 2000, or Huschens et al., 2007) also use such bond market data. We face the problem that the state variable Wt in Eq. (13) is unobservable. In fact, estimating this variable is one of the major aims of our approach. Thus, we directly consider the market value of outstanding payments as given by Eq. (12) instead of the market value of the compound option in the application. 3.2. The maximum likelihood estimation approach for the state variable and its parameters If the market value of the required payments, Ft, is specified from observable data on risk-adjusted interest rates, as described in the last section, we can estimate the funds a country is going to spend for avoiding a crisis, Wt, by applying the pricing equation for the risky payments (3) in reverse direction. This means we insert the market value of the debt on the left-hand side and solve it for the state variable Wt (in an iterative way). This requires that the other parameters in (3) are known. As explained, we assume that the required payments, B1 and B2, and the risk-less interest rate, rs, are observable and known. Since the state variable is not observable, also its volatility is a latent quantity. Thus, we have two unknown quantities but just one equation. Our approach to solve the problem and to estimate the state variable and its volatility (and also its drift) is explained in the following. Since we have to estimate two unknown quantities based on a single estimation we must add more structure to the problem. We apply a time series approach: Instead of considering Eq. (3) for a single date only, we consider a time series of observed market values F tn . Based on this time series we estimate both the state variable and its volatility simultaneously by using a

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D. Maltritz / Journal of Banking & Finance 34 (2010) 3025–3036

maximum-likelihood approach, as firstly proposed by Duan (1994). As explained in the following, this estimation approach is purely based on the assumptions of the model and requires no additional assumptions.9 If we knew the volatility, rW, we could calculate the values of the state variable, W tn , corresponding to an observed time series of market values, F tn , (n = 0, . . ., N) using Eq. (3). The properties of the stochastic process (see Eq. (1)) require rW to be constant over time. Thus, its value, though unknown, must be the same for the whole time series. We estimate the volatility parameter applying an iterative approach: We include an arbitrary (and constant) value for the volatility and calculate the corresponding time series of the state variable, W tn , for given values of the required payments F tn . We do this for a multitude of possible volatility values. For each volatility value and the corresponding time series of the state variable, W tn , we calculate the value of the likelihood function. We choose that value as the estimator of the volatility that yields the highest value of this corresponding likelihood function. The likelihood function relies on the assumptions of the model. Since it follows from Eq. (1) that the growth rates of the state variable for equidistant time intervals, Dt, are independent and identically normally distributed (see Eq. (2)), the likelihood function corresponding to the normal distribution should be used. The values of the state variable and their growth rates are, however, not observable directly – instead they are estimated based on the observed market values of the required payments, F tn , depending on the given volatility value. Thus, instead of the likelihood function of the growth rates, we use the likelihood function of the observable market values of payments expressed in terms of growth rates, which is given by:10

"

 pffiffiffiffiffiffiffi  1 w  l ^ w;N;Dt 2 tn ^ w;Dt þ 2pr 2 r^ w;Dt n¼1    @F tn þ ln þ ln W tn : @W tn

LLF ¼ 

N X

ln

ð14Þ

The partial derivative of Ft with respect to Wt is:

 pffiffiffiffiffiffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi @F t ¼ 1  N2 d1 þ rW T 1  t;d2 þ rW T 2  t; q : @W t

ð15Þ

^ w;Dt can be determined by The standard deviation of the time series r considering Eq. (2) and the given value of the volatility parameter of the stochastic process rW, which is used to estimate the particular time series of the state variable:

pffiffiffiffiffiffi

r^ w;Dt ¼ rW Dt:

ð16Þ

The mean of the time series can be estimated from the observed N growth rates of the state variable by:

l^ w;N;Dt ¼

N 1 X wtn : N n¼1

ð17Þ

Having specified the volatility estimator by choosing the value of the volatility that yields the highest value of the likelihood func^ W , the corresponding time series of values of the state varition, r ^ tn , gives the estimates for these quantities. The estimator ables, W 9 Duan’s approach exploits the model assumptions and avoids additional assumptions. Thus, it is less restrictive as alternative approaches, e.g. the Kalman filter, used in other contributions (see, e.g., Claessens and Pennacchi, 1996; Keswani, 2000). The Kalman filter requires a linear approximation of the non-linear pricing equation, which is a potential source of errors. In addition, the pricing errors, which arise when the pricing equation is used to calculate the latent quantities from observable quantities, are assumed to be independently identically normally distributed and independent from the error terms in the stochastic process of the latent state variable. 10 This is proofed and discussed in more detail in Duan (1994).

^ W , is derived from the corresponding for the drift parameter, l ^ w;N;Dt , of this time series and the estimated volamean estimator, l ^ W , using Eq. (2): tility, r

l^ W ¼

l^ w;N;Dt Dt

þ

r^ 2W 2

:

ð18Þ

4. The case of Hungary In the following we analyze the situation of Hungary in 2008, when the country was struck by a financial crisis in mid October. Here a de-jure default and a collapse of the banking system could only be prevented through substantial help from the IMF and the EU. By applying the estimation approach described in the last section we estimate the development of the state variable describing the payment capacity simultaneously with the parameters of the stochastic process (lW and rW) based on data from secondary markets of Hungarian government bonds. We describe the results in Section 4.2. In Section 4.3 we discuss the crises probabilities that result from these estimates. In Section 4.4 we provide some robustness tests. In Section 4.1 we start with explaining our proceeding in applying the estimation approach and, in particular, how we specified the input data for the estimation. 4.1. Quantification of the (exogenous) input data For the application of our model and the estimation approach described in the last section the following data need to be known: first, the payments required to avoid a crisis and their payment dates, second, market data on risk-adjusted interest rates of government bonds and, third, risk-less interest rates. Based on the payment requirements and risk-adjusted interest rates we can derive the market value of the outstanding payments (which is considered instead of the market value of the hypothetical compound option) using Eq. (12). We insert the value of these outstanding payments into Eq. (3) (on the left-hand side). Also, the required payments and the risk-less interest rates are inserted as exogenous variables in (3). We do this for a time series of observed market values. Based on this time series we apply the maximum likelihood time series approach explained above to solve simultaneously for the time series of the state variable and the corresponding volatility (and also the drift). Thus, we derive a time series that describes the development of the country’s payment capacity based on the observed market data. In addition, we obtain the related uncertainty expressed by the volatility and the expected future development described by the drift. To infer the market value of the required payments, F tn at date tn we use data on risk-adjusted interest rates of Hungarian bonds from the Emerging Market Bond Index global (EMBI) provided by JP Morgan. Country specific sub-indices of the EMBI are calculated by JP Morgan as the weighted average of data on several bond issues. To be included in the calculation an issue has to fulfill several liquidity criteria (see, Cavanagh and Long, 1999). Most important, the issue must have a minimum amount of USD 500 million face value outstanding. In addition, the EMBI data provides the advantage that it alleviates the impact of special features of a single bond issue – which can distort its reflection of crisis risk. For every country specific sub-index JP Morgan provides specific time series on the risk-adjusted interest rate and the corresponding spread.11 This risk-adjusted interest rate of the Hungarian EMBI is inserted into Eq. (12) for rr,i to discount the risky payment requirements. The corresponding spread, s, i.e. the difference to the risk-less benchmark 11 We consider so-called ‘‘stripped‘‘ yields and spreads, where the value of collaterals is excluded from the prices, since including these collaterals would distort the reflection of crisis risk.

D. Maltritz / Journal of Banking & Finance 34 (2010) 3025–3036 600 500 400 300 200 100

02.01.2009

02.01.2008

02.01.2007

01.01.2006

01.01.2005

02.01.2004

01.01.2003

01.01.2002

31.12.2000

01.01.2000

01.01.1999

0

Fig. 2. Sovereign yield spreads (in basis points) for Hungary (JP Morgan’s EMBI global).

curve, s = rs  rr is displayed in Fig. 2, which gives a first glance on the assessment of Hungary’s crisis risk by market participants. As it is usually done in the literature, we use yields on US treasuries as a risk-less benchmark, rs.12 Since the risk-adjusted interest rate, rr, is the sum of the risk-less rate, rs, and the spread, s, that reflects the crisis risk, the risk-less interest rate can be calculated directly from the JP Morgan data by subtracting the yield spread from the risk-adjusted yield: rs = rr  s. To specify the payments required to avoid a crisis we have to obtain information on the payments required to bail-out the banking system, on the one hand, and the required debt service payments, on the other. With respect to future debt service payments, detailed timely information describing all payment dates and the required payments in detail is not available. The available sources only provide data that distinguish between short-term and long-term payments. We consider current data provided by DataStreamÒ (the original source is Economist Intelligence UnitÒ (EIU)). Our compound option model enables us to consider separately several debt service obligations due at different dates. By distinguishing between short-term and long-term payments, we, thus, advance the existing literature concerned with the application of structural credit risk models in the context of sovereign default risk, which typically relies on single payment or threshold models that neglect the time structure of payments. Findings in the empirical literature (see Detragiache and Spilimbergo, 2001), however, indicate that not only the whole amount of debt, but also the composition of debt influence crisis risk significantly, i.e. a higher ratio of short-term to long-term debt is related to a higher crisis risk. Thus, taking short-term and long-term debt service payments separately into account seems to be important. The question arises as to which kinds of debt are to be considered. Is it better to focus on the domestic debt or, rather, is the external debt important? We consider the country’s external debt since providing foreign exchange for external debt servicing is the bottleneck for developing and transition countries mostly. As can be seen in many examples, such countries try to avoid a default by financing domestic debt through printing money (and accept the resulting inflation). This means domestic debt is a rather soft constraint compared to external debt. Thus, it is common in the literature on country default risk for emerging and transition countries to consider external debt rather than domestic debt. 12 Of course, any country can default in principle. However, even in the major financial crisis of 2008 the interest rates of US treasuries are comparatively low, which implies that market participants figure the default risk negligible although the United States were additionally able to support the domestic banking sector with large amounts of funds.

3031

According to EIU, Hungary’s total outstanding external debt at the beginning of 2008 was 138 billion of US Dollars. Fifty-five billion of the debt was short-term with a maturity up to 1 year.13 The remaining 83 billion was long-term debt. Thus, the ratio of shortterm debt to total debt is on a high level of about 40%. Since we have no information about the maturity of the long-term debt we approximate the long-term maturity by the average maturity of the Hungarian government bonds included in the calculation of the EMBI global. Data on the average maturity is provided by JP Morgan. This approach implies the assumption that the average maturity of the bonds is a good proxy for the maturity of the whole debt.14 The average maturity in the observation period varies between 6 and 7 years.15 The second type of payments required to avoid a crisis are the payments needed to bail-out the banking system. One could argue that helping the banks would be easy for the Hungarian government since it can provide the funds in Hungarian Forint, which the central bank can easily print. This implies first of all that the government and the society must accept the resulting inflation. However, Hungary faced the additional problem that the Hungarian private sector and especially the banking sector were largely indebted in foreign currency and the debt was mostly to foreign creditors. If the government provides rescue funds in Forint, these funds will be exchanged into foreign currency to a large part in order to fulfill payment obligations in foreign currency. Providing funds in Forint without any backing by a foreign currency would hardly solve the problems of the banking system. Instead, it would increase the problems of the exchange markets where Hungary faced a rapid loss of the Forint’s value. The exchange of these funds into international currency (to fulfill the foreign payment obligations) would result in a decrease of their value that leads to a (further) devaluation of the exchange rate. This negatively effects Hungary’s private enterprises and its citizens, e.g., because it exacerbates servicing of their high debt burden in foreign currency and because of the increasing prices of imports. The governmental sector also faces negative effects such as the exchange rate’s adjusted value of taxes declines, which make external debt servicing even harder. In addition, a drop in the exchange rate would hamper the intended joining of the European Monetary Union.16 For these reasons, we made the assumption that the government needed funds with ‘‘real value”, i.e. foreign currency, to bail-out the banks and could not just print the required money. Our approach ultimately is determined by actual events. In reality, Hungary finally used and required funds from the international rescue package to help its banking system.17 The Hungarian government provided a rescue package for the banking system amounting to approximately 3 billion USD (or Euro 2,3 billon or HuF 600 billion),18 which was not increased later on. These funds amounting to USD 3 billion are, thus, assumed as the amount of payments required to bail-out the banking system.

13 The short-term debt comprises on the one hand debt with an original time to maturity up to 1 year and, on the other, long-term debt payments (i.e. payments with an original time to maturity longer than 1 year) that become due within 1 year. 14 Since JP Morgan considers only bonds with a remaining time to maturity longer than 1 year in the calculation of the EMBI (see, Cavanagh and Long, 1999) there is no systematical bias towards a short maturity in this approximation. 15 We performed several robustness checks with different maturities. The results are discussed in Section 4.4. 16 To join the EMU, new members have to fulfill the exchange rate stability criterion meaning that the currency has respected the normal fluctuation margins of the Exchange Rate Mechanism (ERM II) (15%) without severe tensions for at least 2 years. 17 According to www.usatoday.com Hungarian authorities said: ‘‘The aid package for ‘‘Hungarian banks of systemic importance” comes as part of the $25.1 billion standby loan for Hungary announced last month by the International Monetary Fund, the European Union and the World Bank.” 18 See, e.g., www.reuters.com.

D. Maltritz / Journal of Banking & Finance 34 (2010) 3025–3036 190 170 150 130 110 90 70

State Variable (Payment Capacity)

30-Dec-08

15-Dec-08

30-Nov-08

31-Oct-08

15-Nov-08

15-Oct-08

30-Sep-08

15-Sep-08

31-Aug-08

31-Jul-08

16-Aug-08

1-Jul-08

16-Jul-08

1-Jun-08

16-Jun-08

1-May-08

16-May-08

1-Apr-08

16-Apr-08

1-Mar-08

17-Mar-08

31-Jan-08

15-Feb-08

50 1-Jan-08

As in the case of debt service payments, we do not have precise information as to the exact date the help was, in fact, required. But we assume that it was rather short-term help. As for the shortterm debt service payments we assume an average maturity for the bail-out payments of 1 year. Since we do not have precise information about the exact dates when short-term payments become due – neither for the debt service payments nor for the bail-out payments – and we, thus, assume a similar average maturity (of 1 year) we can sum up both types of payments to the cumulated amount of short-term payments, B1. The long-term debt service payments constitute the payments B2. We apply our model in the crisis year 2008. Since we can observe the market data that provides the information on crisis risk, i.e. risky interest rates and spreads of Hungarian government bonds, in daily frequency, we apply the estimation approach with daily time series. In employing the time series estimation approach described in Section 3.2 we only consider data observable at the considered estimation date and before this date. Thus, we obtain the same estimates of crisis risk that would have occurred by applying the approach at this date in the past. The results are not biased by including market data that became observable at later dates. For January 1, 2008, for example, we consider the data observed at this date. Since we need a time series of observations we include data from prior dates in the estimation. For January 2, 2008, we additionally include data observable at this date (as well as the data used in the estimation for January 1) and so on for the following days of the year. In running the estimation the time series should be as long as possible since our estimation approach implies that the estimators of the unknown parameters are assumed to converge towards the true values with an increasing number of observations, if there are no structural breaks in the time series. We include observations from March 1, 2005, onwards in the estimation. We have chosen this date since it seems to be a turning point in the development of Hungarian crisis risk. We observe a considerable jump in the spreads at this specific day. Between February 28 and March 1, 2005 the spreads more than doubled from 26 to 57 basis points (see Fig. 2). A structural break in the time series at this date is confirmed using the Chow breakpoint test.19 In the time before we observe a downward trend and very low spreads and crisis risk, whereas from March 2005 onwards spreads and crisis risk increased. Since the estimation approach implies a continuous stochastic process without jumps and structural breaks, including data from dates prior March 2005 would be problematic.

16-Jan-08

3032

Short-term Crisis Threshold

Fig. 3. The development of the state variable describing the market’s assessment of Hungary’s capacity to make payments for crisis avoidance (in billions of US Dollars) and the short-term crisis threshold in 2008.

The increasing spreads of Hungarian government bonds reflect increasing crisis risk. According to our model this is related to a worsening of the market participant’s assessment of Hungary’s payment capacity, i.e. the funds it is able and willing to spend for crisis avoidance. This can be seen in Fig. 3 where the development of the payment capacity is displayed. During 2008 the market participant’s assessment of the amount of funds available for making the required debt servicing payments and bailing-out the banking sector declined dramatically from USD 182 billion at the beginning of 2008 to USD 108 billion at the end of 2008. Following a continual decline in the first three months of the year from USD 182 billions to 165 billions a first dramatic drop in the payment capacity from about USD 165 billion to USD 145 billion happened in late March/early April 2008. Here, a referendum initiated by the oppo-

sitional Fidesz party forced the government to ease its strict policy of budget consolidation and lead to a government crisis. An even sharper drop of the payment capacity, which started in late September 2008, finally led to the Hungarian crisis in October 2008. Thus, our results indicate that the market participant’s assessment of Hungary’s ability to raise the required funds for crisis avoidance, i.e. debt servicing and bailing-out the banking sector, worsened dramatically in the course of 2008. This (among others) may reflect the impact of the international financial crisis on Hungary’s economy: Because of its high budget and current account deficit, Hungary heavily depended on potential lenders’ willingness to lend and on their assessment of the Hungarian economy and its prospects. Thus, the worsening in Hungary’s solvency – in the market participant’s opinion – caused massive problems in raising new capital by raising credits or issuing bonds to finance the repayment of old debt service obligations. So, the issuing of new government bonds at the beginning of October failed.20 The state variable dropped by 43 billion from USD 145 billion on September 22 to USD 102 billion on October 21. Only between October 13 and October 17 the state variable dropped by 24 billions, from USD 135 to UDS 111 billion. These numbers indicate that the drop of Hungary’s solvency finally lead to the crisis in October 2008, rather than financing needs for bailing-out the banking sector. The required bail-out payments of USD 3 billion seem rather small compared to the drop in (the assessment of) Hungary’s payment capacity of USD 43 billion. The decline in the state variable was halted by massive help from the IMF and the EU, finally amounting to about 25 billion USD. The agreement on this international help in mid October 2008 finally stopped the worsening in the markets assessment of Hungary’s solvency. So, shortly after a first help agreement with the ECB about liquidity assistance to Hungary on October 16, 2008, the state variable reached its bottom at USD 102 billion at October 21. In late 2008 the estimation of the amount of payments Hungary will be able to pay for crisis avoidance is about USD 108 billion, but gradually increased over time. As explained above, these funds are required to finance a total amount of necessary payments for crisis avoidance of about USD 141 billion. The required payments comprise outstanding external debt of 138 billion, from which 55 billion (i.e. about 40%) were due in 2008, plus bail-out payments for the banking sector of 3 billion. The total outstanding debt increased by the factor 4.5 since 2000,

19 The F-statistic is 8.647 and the p-value is 0.00330. The log-likelihood test statistic is 8.634 and the corresponding p-value is 0.00329. For both test statistics we can, thus, reject the null that there is no structural break at this date with a probability of more than 99%.

20 see www.reters.com: ‘‘BUDAPEST, Oct 10 (Reuters) – Hungary’s debt agency AKK announced plans on Friday to cut net government debt sales by 200 billion Forint ($1.09 billion) over the rest of the year, but the government debt market was at a near standstill [. . .]. A 15-year bond auction scheduled for Oct 22 was cancelled.”

4.2. Hungary’s payment capacity in 2008

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D. Maltritz / Journal of Banking & Finance 34 (2010) 3025–3036

21 Some papers use an approach putted forward by Moody’s KMV, were the shortterm debt plus 50% of long-term debt are considered in estimating default risks (focusing on a short-term maturity of 1 year). In the case of Hungary even this approach would lead to a much lower crisis risk compared to our results. Furthermore, this threshold is arbitrary and not endogenously derived by modeling the influence of later payments based on theory guided considerations.

100% 90% 80% 70% 60% 50% 40% 30% 20%

Joint PoC

Short-term PoC

30-Dec-08

30-Nov-08

15-Dec-08

31-Oct-08

15-Nov-08

15-Oct-08

30-Sep-08

15-Sep-08

31-Aug-08

16-Aug-08

16-Jul-08

31-Jul-08

01-Jul-08

16-Jun-08

01-Jun-08

16-May-08

16-Apr-08

01-Apr-08

17-Mar-08

01-Mar-08

31-Jan-08

15-Feb-08

16-Jan-08

0%

01-May-08

10% 01-Jan-08

from around USD 30 billion outstanding in 2000 to USD 138 billion outstanding at the beginning of 2008. Thereby, the ratio of shortterm to overall debt was on a high level of about 40%. One advantage of our compound option model is that it enables to distinguish between short-term debt and long-term debt and their interrelation as well as their influence on crisis risk. Thus, we can consider short-term and long-term payments separately. Hungary’s estimated payment capacity was above the total amount of required payments of about USD 141 billions during the first quarter of 2008, from early April to mid September the numbers of both quantities were quite similar. Because of the dramatic drop in the estimated payment capacity in late September and early October the payment capacity was well-below the over all payment requirements from early October onwards. Nevertheless, the payment capacity was well-above the required short-term payments of about USD 58 billion. Thus, one should not expect a crisis in the short-run – if only the short-term payments are considered. Since our model accounts for the influence of the long-term payments we obtain different results. According to our model the threshold for a crisis in the short-run is given by WQ, as explained in Section 2 (see, Eq. (6)), which is approximately twice as high as the short-term payment requirements of USD 58 billion. The numbers for WQ are shown by the dashed line in Fig. 3. Since the payment capacity described by our state variable is close or even below the short-term threshold, we found a crisis to be very likely even in the short-run. Given the numbers of the payment capacity a one-payment model of the Merton-type that only considers the short-term payments would come up with a much lower estimation of crisis risk.21 The two-payment model used here enables us to consider in a joint model payments required for bailing-out the banking system additionally to debt service payments as well as the dependencies between these payments. Considering both types of payments in a joint model gives a more accurate picture of crisis risk than neglecting either banking crisis risk or debt crisis risk, since both contribute to the short-term threshold. This holds true even if the debt service payments are very high in relation to the bailout payments, as in the case of Hungary. Including the bail-out payments gives a more accurate picture of the situation and is better than neglecting them. To be precise at this point is especially important since any amount of money required to be paid in the short-run increases the short-term threshold on an even higher amount due to the influence of the outstanding debt service payments (see Eq. (6)), as discussed in Section 2.3. How far the increase in the threshold exceeds the additional amount of short-term payment requirements depends on the constellation of parameters. Hungary in 2008 faced a situation with extraordinary high short-term debt service payments in relation to long-term payments. As can be seen in Fig. 1, for this constellation the slope of the curves is not extraordinary steep, but nevertheless clearly higher than one. Thus, the small amount of additional payments of USD 3 billion raised the threshold by roughly 4 billion, i.e. the increase of the threshold is about one third higher than the increase in the payment requirements. If we would observe a (more usual) situation, where the ratio of short-term debt to long-term debt is lower, an increase of the short-term payment requirements by the USD 3 billion would increase the threshold on much higher amounts. This can be seen by the steep slope of the curves in Fig. 1 for lower

Long-term PoC

Fig. 4. Estimated crisis probabilities for Hungary in 2008.

short-term payments (or for higher long-term payments). From these considerations follows that it is important to capture the situation very precisely, which implies that a model that includes bail-out payments for banks in addition to debt service payments improves the assessment of crisis risk, even if the amount of bail-out payments is small in relation to debt service payments. The small additional contribution of the bail-out payments to the threshold and the crisis risk in the situation actually observed may be seen as a result of our framework. This enables us to conclude that the crisis was caused by Hungary’s extremely high debt burden (in relation to its payment capacity) and more particularly by the extraordinary high short-term debt and rather not by the problems in the banking sector and by the comparatively low payments needed to fix these problems, since their impact on the threshold in the given situation was quite low. Thus, we can state that problems of the banking sector may have their part in triggering the crisis, but did not cause it. 4.3. Hungary’s crisis risk in 2008 The deteriorating values of the funds Hungary was able to spend for crisis avoidance in relation to its high payment requirements, especially for debt servicing, lead to a considerable increase in crisis risk in the course of 2008. In addition to the state variable and the required payments the estimated parameters for volatility and drift influence the estimated crisis probabilities (see, Eqs. (7)– (9)). For all daily estimates we obtain the same parameter for the volatility of 0.29, which indicates that the results are very stable and the estimated value achieved convergence to the ‘‘true” volatility. The estimated values for the drift decline during the observation period. This is mainly because of the turmoil during the financial crisis and the resulting decline in the payment capacity and the average growth rate.22 In Section 4.4.2 we provide the drift estimates for different robustness checks. It turns out that differences between different scenarios are very low and that the change of the estimates of the drift within time, i.e. estimates for specific dates of the time series, are very similar to those of the original estimation. The variation in the drift over time is reflected by our estimates since we run a new estimation of the payment capacity and its parameters for every day in the observation period, as explained in Section 4.1.23 Whereas the drift does not concern the functional 22 For the beginning of 2008 the estimates are well above 10% (e.g., 12.8% for January 1). Till September 2008 the estimates slightly decrease because of the deterioration of the payment capacity. Triggered by the crisis in October 2008 we observe a drop to values of about 3% in late 2008 (e.g., 3.2% on December 12). 23 As a result, we obtain the estimation for the drift that corresponds to the market data available at this specific date, i.e. the best estimator of the drift for the data known at this date. The estimation is neither biased by data that become available at later dates, nor does it neglect data that are already available.

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D. Maltritz / Journal of Banking & Finance 34 (2010) 3025–3036

relation between observed market prices and estimates of the payment capacity (see Eqs. (3) and (4)), it does enter the formula for the crisis probabilities. To calculate these probabilities, we used the most recent drift values estimated for every day in the observation period. Fig. 4 shows the resulting crisis probabilities. Our model and the data given enable us to infer a short-term crisis probability (see Eq. (7)) and a long-term crisis probability (see Eq. (9)). Furthermore, we can estimate an overall crisis probability, i.e. the risk of suffering a crisis in the short- or in the long-run (see Eq. (8)). The points of time displayed on the x-axis of Fig. 4 display the date of prognosis when the numbers figured out by the lines are estimated. The dashed line shows the short-term default probability and the thin solid line shows the long-term crisis probability (given that no crisis has occurred in the short-run). The bold solid line displays the overall default probability (see Eq. (8)). Leading up to the crisis in early October 2008, the estimated short-term crisis probabilities increased considerably during the year 2008. The short-term crisis risk was on a fairly low level of 2% at the beginning of 2008. Here, the country’s estimated amount of payment ability of USD 182 billion was well-above the shortterm payment requirements of USD 58 billion. It also exceeds the resulting short-term threshold, which varies around USD 110 billion. The crisis risk gradually increased during the first quarter. A first jump to about 12% crisis probability resulted from the referendum initiated by Fidesz and the resulting political turmoil in late March. This jump in the crisis risk reflects the drop in the state variable during that time. After a calming in early May the crisis risk gradually increased again till late September. On September 22, the short-term crisis risk started to rise dramatically with an extreme jump from 13% to 59% between October 3 and October 21. During the whole observation period the conditional long-term crisis risk, i.e. the risk of suffering a crisis in the long-run given that no crisis has occurred in the short-run, is considerably lower than the short-term risk. This is mainly because of a positive development of Hungary’s ability to raise the required funds for debt servicing, which is reflected in our model by the positive estimation of the drift parameter discussed above. Thus, the projection of the amount of funds the country is able to raise for debt servicing in the next approximately 6 years (which is the average maturity of the long-term debt) is higher than the current estimates (shown in Fig. 4) and the resulting projection for the next year. There is a multiplicative link between the overall crisis probability and the short- and long-term crisis risk: As can be seen by (rearranging) Eq. (9): the joint probability that no crisis occurs is the product of the probability not to suffer a short-term crisis and the (conditional) probability not to suffer a long-term crisis. Thus, the overall crisis probability must be above the short- and long-term probability. During the whole observation period the overall probability was mainly driven by the short-term probability. The resulting overall crisis risk raises from 2% at the beginning of 2008 to 63% in early October, when Hungary was virtually bankrupt and needed international help to avoid a de-jure default. The probability that Hungary will obtain such help is reflected in the market data, which explains that the crisis probability is well-below 100%.

related via the pricing equation for the compound option (4) or the debt (3).24 We would, thus, obtain different estimates of the state variable for different volatility estimates. Instability of the volatility estimates would, hence, cast doubts about the trustworthiness of the estimates of the payment capacity. It is reassuring that the estimated volatility values are very stable with respect to different settings (see column 3 of Table 1). When changes in the length of the time series are made, e.g., this influences the results of the estimation only marginally. As explained above, we run a new estimation for every day in the observation period by adding the data observed at this day, whereas we do not consider the data observed at later dates. We obtain the same volatility value of 0.29 for every estimation we run (for 2008). This may be seen as an indication that the volatility estimates achieved convergence to the true volatility value. The stability of the results is further confirmed by running estimations for different sub-samples, namely the years before 2008. When we concentrate on the pre-crisis period, the results do not change significantly. Considering all dates till the end of 2007 in the estimation yields a volatility estimate of 0.29. Even if we run the estimation for the period before July 2007, including only the – supposedly – more tranquil time before the international financial crises started, the results yield only a very slight change with an estimate of 0.28. We obtain the same value when we consider data of the years 2005 and of 2006 only. Such slight changes in the volatility have only marginal influence on the estimated payment capacity. This can be seen in column 4 of Table 1. To analyze the robustness of the payment capacity, we calculate the average values of the estimated payment capacity obtained for different volatility estimators and/or sub-samples. Comparing the deviation between the estimates of the state variable obtained for a volatility of 0.29 (‘‘original” estimation – Scenario O) and for a volatility of 0.28 (Scenario A) shows rather small differences25. For a volatility of 0.28 we would obtain an average payment capacity of USD 140.26 billion, whereas the original estimation yields 143.54 billion. We obtain, thus, a rather small relative deviation between both estimates of 2.34%. We also estimate the volatility and the payment capacity in the pre-crisis year 2007 (Scenario B). The estimates of the payment capacity for 2007 are calculated using the true volatility estimates for the respective dates, i.e. 0.28 up to September 3, 2007 and 0.29 from this day onwards. In the pre-crisis time, the state variable describing the payment capacity is relatively stable and we do not observe such dramatic changes as is the case in 2008. The average of the state variable for the pre-crisis year 2007 is USD 184, which is slightly higher than the values observed in the beginning of 2008 (see Fig. 4). As explained in Section 4.1 we do not have precise information about the average time span till the long-term debt becomes due. Thus, we check the robustness of our results with respect to different assumptions about this time span. In the original estimations discussed so far, we approximate the maturity of the long-term debt by the average maturity of the bonds included in the calculation of the EMBI. We re-run our estimations using a 3 year band around the original value of the maturity, i.e. calculate new values for the maturity by adding 1.5 years on the one hand and subtracting 1.5 years on the other. The resulting 3 year band means a

4.4. Robustness tests 4.4.1. Robustness of the payment capacity One may ask how trustworthy and stable our estimates of the state variable (payment capacity), its parameters, and the crisis probabilities are. A discussion of the estimation of the payment capacity directly touches the issue of the stability of the volatility estimates, since both payment capacity and volatility are estimated simultaneously from observed market values and are directly

24 The drift, by contrast, does not enter these pricing formulas, which is a rather general result of option pricing theory (see Merton, 1974) and holds true in our model as well. Thus, we discuss at first the estimation of payment capacity and volatility and later on the estimates of crisis probability and drift. 25 Note: These are average values for the year 2008. Although the volatility of 0.28 is not the best estimator for 2008, we can specify the estimates of the state variable that correspond to this volatility estimator. These values for the state variable are shown in Table 1 (Scenario A).

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D. Maltritz / Journal of Banking & Finance 34 (2010) 3025–3036 Table 1 Results for several robustness checks. Scenario O A Bc C D E F

(Time to) long-term maturity b

EMBI-numbers EMBI-numbers EMBI-numbers EMBI-numbers 1.5 years EMBI-numbers +1.5 years EMBI-numbers EMBI-numbers

Volatility

Payment capacitya (Billions of USD)

Drifta

Joint crisis probabilitya (%)

0.29 0.28 0.28/0.29 0.32 0.27 0.32 0.27

143.54 140.26 184.59 159.61 132.20 155.25 137.24

0.0715 0.0742 0.1321 0.0794 0.0639 0.0658 0.0766

18.11 18.20 1.20 15.19 20.66 17.95 18.37

a

These are average values calculated by including the values for the year 2008 (observation period) in the calculation for all scenarios, except Scenario B. According to the EMBI of JP Morgan, the maturity of long-term Hungarian (bond) debt varies between 6 and 7 years in the observation period. c In Scenario B the results for the pre-crisis years 2007 are considered. This means, we include the values observed till the end of the year 2007 and we use the volatility observed in this year: This was 0.28 up to September 3, 2007 and 0.29 from this date onwards. b

4.4.2. Robustness of the crisis probability In the next step, we analyze the robustness of the crisis probability for which the payment capacity and its parameters serve as input data. Besides payment capacity and its volatility discussed in the last subsection, the crisis probability also depends on the drift. The results for the drift obtained for different settings of other parameters discussed above are shown in column 5 in Table 1.26 The average drift does not vary much between different scenarios. As explained in the last subsections, also the volatility does not differ much between different scenarios, which implies that the time series of the payment capacity obtained for given market values are also quite similar. Since there are no large differences between the input data, we do not expect large deviations in the crisis probabilities either. Column 6 of Table 1 overviews the results for the joint crisis probabilities obtained for different scenarios. It shows the averages (calculated for the observation period) of the joint crisis probability (as determined by Eq. (8)) for several scenarios with different input data discussed above. From the values of the joint crisis probability derived in the original estimation (Scenario O – the numbers displayed by the bold solid line in Fig. 4), we obtain an average value for the joint crisis probability of 18.11%. The average crisis probability in 2007 was, of course, lower. The numbers match the results of early 2008 (see Fig. 4). For Scenario C, where the maturity is 1.5 years lower than the maturity given by the EMBI, the average crisis probability is 15.19%, which is 2.9% points lower than the average of the original estimation. For Scenario D, where the maturity is increased by 26 Note: Since the drift is calculated by formula (18) (see also Eq. (2)) from the mean estimator of the time series of growth rates of the payment capacity (by adding half the variance), the drift is above the estimated mean growth rate of the respective time series.

100% 90% 80% 70% 60% 50% 40% 30% 20% 10%

Joint PoC (Original Estimation)

1-Jan-09

2-Dec-08

17-Dec-08

2-Nov-08

17-Nov-08

2-Oct-08

17-Oct-08

2-Sep-08

Joint PoC (Scenario C)

17-Sep-08

2-Aug-08

18-Aug-08

3-Jul-08

18-Jul-08

3-Jun-08

18-Jun-08

3-May-08

18-May-08

3-Apr-08

18-Apr-08

3-Mar-08

19-Mar-08

2-Feb-08

17-Feb-08

3-Jan-08

0% 18-Jan-08

deviation of approximately 50% since the original maturity in the observation period is about 6 years. The results are shown by Scenario C and D in Table 1. The maximum likelihood estimator of the volatility changes slightly for differences in the maturity. For the reduced maturity (1.5 years) we obtain a higher volatility estimator of 0.32, for the increased maturity (+1.5 years) the estimated volatility is 0.27. From Eq. (3) follows that these changes in the volatility lead to different estimates of the payment capacity. For the reduced maturity (1.5 years) the average payment capacity in the observation period is USD 159.61 billion, for the increased maturity (+1.5 years) we obtain an average payment capacity of USD 132.2 billion. This means, the payment capacity reacts quite inelastic to changes in the maturity. With a relative deviation of +10% and 8%, respectively, i.e. a variation of 18%, the deviation of the payment capacity is comparatively low in relation to roughly 50% variation in the long-term maturity.

Joint PoC (Scenario D)

Fig. 5. Estimated crisis probabilities for Hungary in 2008 with different forecast periods (maturity of long-term debt).

1.5 years, the average of the crisis probability is 20.66, i.e. about 2.5% points higher. Fig. 5 shows the results for the crisis probability obtained for different settings at different points of time. It can be seen that especially in the period till early October, when the crisis actually occurs, the deviations are very small. Since our model and the derived crisis probabilities are designed to forecast crisis, these small differences in the pre-crisis period are particularly good news. In the following crisis period the deviation is somewhat higher, yet the major part of the differences between the average crisis probability of Scenario O and Scenario C/D of +2.5/2.9% points results from the crisis period starting in October 2008. Even in the crisis period the probability reacts rather inelastic to changes in the maturity: the relative deviation between crisis probabilities for the original estimation and those for the alternative scenarios is approximately +14%/16% for a variation in the maturity of about 50%. It is important to mention that the deviations in the crisis probability result mainly because of the change in the forecast interval for the crisis probability (which is equal to the long-term maturity) and not because of an explicit change in risk assessment, i.e. different crisis probabilities for the same forecast interval.27 Such a change in the risk assessment would result from differences in the input data (estimated for different maturities) used to calculate the crisis probability. The last two rows of Table 1 show the crisis

27 In fact, deviations in the crisis probability may occur because of two reasons. On the one hand, we would obtain different values of the crisis probability for different forecast intervals, even if the other input data (volatility, payment capacity and drift) for the calculations of the crisis probability are not changed. On the other hand, we may obtain different values for the other input data when we run estimations with different values for the maturity of the long-term payments, which in turn yields different crisis probabilities even for the same forecast interval.

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probabilities of alternative Scenarios E and F. Here, we consider the original maturity given by EMBI, but employ input data estimated for the changed maturities, i.e. a volatility of 0.27 and 0.32, respectively, (and the corresponding payment capacity and drift). It can be seen that the differences of the crisis probability for Scenario E and F to the estimates obtained in the original estimation (Scenario O) are very low. (In Fig. 5 the lines for Scenario E and F would be virtually similar to the line of Scenario O.) This means, the major part of the differences of the crisis probabilities between Scenario O and Scenario C/D discussed above, is from the change in the forecast interval; for a longer (shorter) forecast interval the estimated crisis probability is higher (lower).28 All in all, the results are quite robust if we keep in mind the considerable change in the settings. Thus, the robustness checks show that the results are rather robust, no matter whether we consider the payment capacity and its stochastic parameters or the crisis probabilities estimated based on these quantities. 5. Conclusion We consider the interrelation between country defaults and domestic banking crises in a stochastic framework. Our model is based on the literature where structural credit risk models are applied in the context of country default risk. We advance this literature by considering potential payments required for bailing-out the struggling domestic banking sector. Since these bail-out payments for the domestic banking sector in general do not become due at the same time as the debt service payments, we are forced to expand the model approach typically used in the literature for country default risk to incorporate (at least) two instead of just one payment requirement at two different dates. To incorporate these payments for bailing-out the banking sector, on the one hand, and debt service payments on the other, we apply a model that relies on compound option theory, as outlined in Geske (1977, 1979). This approach enables us, additionally, to consider short-term and long-term debt service payments separately. We estimate the unobservable model parameters by using secondary market data on sovereign yield spreads of Hungarian Bonds, as given by JP Morgan in its Emerging Markets Bond Index. By applying the time series maximum-likelihood approach of Duan (1994) we can simultaneously estimate a stochastic state variable that describes Hungary’s payment capacity, i.e. the amount of funds it is able and willing to spend for crisis avoidance, as well as the stochastic parameters, i.e. volatility and drift, of this state variable. Our results show that Hungary faced increasing problems in acquiring the funds for debt servicing and bailing-out the banking sector in the course of 2008. These problems in combination with extremely high outstanding debt finally caused the crisis. Problems in the domestic banking sector, by contrast, seem to influence the

28 Such a change in the forecast interval has a larger influence on the crisis probabilities in more risky situations than in less risky periods, as Fig. 5 shows.

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