# Credit Risk Analysis on Euro Government Bonds-Term Structures of Default Probabilities

## Abstract

In this paper, we make a comprehensive credit risk analysis on government bonds (GBs) of Germany, France, Italy, Spain and Greece over the period 2007.4–2012.3, where interest rate (IR) differential, GB price differential, default probability (DP) and credit default swap (CDS) are considered. First, applying the GB-pricing model in Kariya (Quantitative methods for portfolio analysis: MTV approach. Springer, Berlin, 1993) to these GB prices, we derive the term structures of interest rates (TSIRs) and discuss on the Maastricht convergence condition for the IR-differentials among these states relative to the German TSIRs and make some observations on some divergent tendencies. The results are associated with the business cycles and budgetary condition of each state. In the second part, to substantiate this viewpoint, we first make credit risk price spread analysis on price differentials and derive the term structures of default probabilities (TSDPs) of the French, Italian, Spanish and Greek GBs relative to the German GBs, where the corporate bond (CB) model proposed in Kariya (Advances in modern statistical theory and applications: a Festschrift for Professor Morris L. Eaton. Institute of Mathematical Statistics, Beachwood, 2013) is used in the derivation. Then it is empirically shown that the TSDPs show a significant divergent movement at the end of 2011, affected by the Euro Crisis. In addition, the TSDPs of these GBs are empirically shown to be almost linear functions of the differences of the TSIRs, which enables us to state the Maastricht condition in terms of DP. Thirdly the effectiveness of our TSDPs is empirically verified by comparing them with the corresponding CDSs against US dollars.

### Keywords

European Economic and Monetary Union Maastricht treaty Government bond pricing model Credit risk price spread Term structures of interest rates and default probability Credit default swap## 1 Introduction

The Financial Crisis in 2008–2009 and the European Crisis thereafter made many European states (countries) in the the European Economic and Monetary Union (EEMU) confronting severe budgetary and unemployment problems, as reflected in Greek economy. As of May 2013, there are 27 countries in the European Union (EU) among which 17 states form the EEMU. The European problem has been affecting global economies through trade relations and financial markets, and naturally the world concerns about their future movements, because a collapse of the EEMU would make a significantly serious impact on the world economy. As a matter of fact, the 2012 GDPs of Germany, France, Italy and Spain in the EEMU are respectively ranked the 4th, 5th, 9th, and 13th in the world and the total GDP of these four states is greater than the GDP of China, implying the global importance of these countries in view of the world trade system and financial system.

Though the currency has been integrated in the EEMU, the sovereignties of making finance for fiscal policy by issuing government bond (GB) have not yet been integrated in the system. Since these GBs are of common currency unit “euro”, the bonds issued by the EEMU states are substitutable from investors’ viewpoint and the price differentials observed in the GB markets basically show the substitutable rates or equivalently credit risks in euros, which is made by alert and sensitive investors. Hence the GB prices of the same attributes (coupon and maturity) in the EEMU states will naturally exhibit or reflect credit quality in their price differentials, where we only treat GBs with fixed coupons. In other words, the price differentials with bond attributes adjusted will directly exhibit market evaluations of the DPs (default probabilities) projected by forward-looking investors. Here it is noted that the investors who form prices in the market will have certain views and perspectives on risks of possible defaults of the GB issuers over some time horizons, given the past information on microeconomic and macroeconomic movements, business cycles, and government budgetary conditions, etc.

In this paper, via interest rate differential (IR-differential), price differential (P-differential), default probability (DP) and credit default swap(CDS), we will make a comprehensive credit risk analysis on the GB price data of the Five States; Germany, France, Italy, Spain and Greece over 2007.4–2012.3. The names of the Five States are often respectively represented by the symbols D, F, I, S, and Gr below. For example, their GBs are abbreviated as DGB, FGB, IGB, SGB and GrGB respectively. The analysis includes the derivations and comparisons of the term structure of interest rates (TSIR) implied by those GB prices, and it is associated with credit risk analysis on FGB, IGB, SGB and GrGB relative to DGB, where the price differentials are analyzed and the term structures of default probabilities (TSDPs) are derived and compared. In our terminology the credit risk of the \(k\)th state (issuer) at current time \(t\) is the TSDP \(\{p_t^k (s)\equiv P(\tau _t^k \le s) : 0<s\le s_M^k \}\) that the \(k\)th government (state) cannot timely pay the coupons nor fully redeems the principals of its issued GBs at future time \(t+s\), where \(k\)\(=\) F,I,S,Gr and \(\tau _t^k \) is the default time random variable of the \(k\)th state. Our future time horizon for analysis is commonly fixed as the 10-year term \((0,10]\) from \(t\). Throughout this paper the time \(t\) of analysis is dropped from our notation and the TSDP is simply denoted by \(\{p^{k}(s)\}\). By its definition the TSDP curve of \(p^{k}(s)\) is increasing and unconditional at each time and we will estimate \(p^{k}(s)\) with monthly cross-sectional data of the GB prices for each state \(k\).

It is noted that in interest rate analysis and credit risk analysis, its data source naturally makes a great impact on the effectiveness of modeling and analysis. In particular when it is aimed to derive a DP (default probability) from a set of data, we distinguish the two approaches: (1) Backward-looking approach and (2) Forward-looking approach. In Backward-looking approach microeconomic and macroeconomic time series data over a past period is used for modeling and analysis and the data on defaulted firms and non-defaulted firms in the past is associated with economic, business and financial data. But these data are often generated under different environments with possibly different economic regimes. Typically, statistical or econometric models which use time series data on defaults and non-defaults belong to this approach. Examples are intensity model, survival model, classification model, rating transition model, logit-probit model, etc. In Forward-looking approach, current (cross-sectional) market data such as GB or CB (corporate bond) prices, interest rates, swap rates, stock prices, credit default swap (CDS) etc. is used to look forward over a future term with concept of DP for those firms that have not defaulted. These current market prices are supposed to reflect and include investors’ views, projection and perspectives on future economic and financial movements or budgetary conditions of firms or states over a future term, given past time series information. A typical example is that a current cross-sectional set of GB prices of different maturities will give a TSIR over a future term, which is nothing but investors’ views at current time on future interest rates.

We remark on liquidity risk. In our view liquidity will be inseparable from credit quality. In fact, liquidity risk is tightly related with credit risk and generally speaking, when economy is not in financial crisis, the lower the credit quality is, the less the liquidity is, which is in fact implied by the fact that the lower the credit quality is, the smaller the size of investors’ funds is in the market (see, e.g., Friewald et al. 2012). Also the volume of the issued GBs standing in the market (depth of the market) matters in view of liquidity because the larger it is, the more easily the investors sell or buy bonds without market impact.

In our analysis, to derive TSIRs, the forward-looking GB-pricing model proposed in Kariya (1993) and applied in Kariya et al. (2012) (shortened KWWDY 2012) is applied to each monthly cross-sectional set of GB prices. And to derive TSDPs relatively to DGB, the forward-looking CB (Corporate Bond) pricing model proposed in Kariya (2013) is also applied to each monthly cross-sectional set of FGB, IGB, SGB, and GrGB, which are regarded as CBs in the model, while DGBs are viewed as non-defaultable reference GBs.

In derivations of TSIRs, P-differentials and TSDPs, we use each monthly cross-sectional set of GB price data, where price data is observed at the last business day of each month and the period of analysis is 2007.4–2012.3. Our arguments are sometimes made associated with the conditions in the Maastricht Treaty, which we will discuss in Sect. 2 and show a legitimacy of regarding DGB as a reference GB.

In Sect. 3, first our cross-sectional GB pricing model is reviewed, and then the TSIRs of the Five States are derived cross-sectionally in each month over the period 2007.4–2012.3. The model contains the heteroscedasticity and correlation structure of prices in the stochastic discount function, which naturally introduces a bond duration structure. Then from a viewpoint of the Maastricht convergence problem on the IR-differentials in Sect. 2 we associate time series paths of their TSIRs with business cycles of the Five States (German, France, Italy, Spain and Greece), where the business cycles are measured by the ESI (Economic Sentiment Indicator). Note that the IR-differential itself is a measure of credit risk and the relationships between our TSIRs and TSDPs are considered in Sect. 6.

In Sect. 4, we look into the P-differentials of GBs relative to DGB because they provide direct credit risk measures on FGB, IGB, SGB and GrGB in terms of euros. In fact, we propose the credit risk price spread (CRiPS) as a market credit risk measure that evaluates the credit P-differentials in euros of each GB of the Four States (France, Italy, Spain and Greece) relative to the corresponding DGB-equivalent bond of the same maturity and the same coupon rate, where the prices are adjusted for the bond attributes of maturity and coupon. It in fact measures a default likelihood of each individual bond in euro, which is similar to the default distance measure that uses current stock price, and it can be used in risk management of GB portfolios. And we make some empirical observations concerning the cross-sectional and time series differences of the CRiPSs of the Four States along business cycles together with the budgetary conditions of each state and historical events.

In Sect. 5 with monthly GB price data of the Five States for the period 2007.4–2012.3, the TSDPs of the Four States are estimated via our CB pricing model, where the mean discount function derived from the DGB pricing model is used for discounting future defaultable CFs of GBs in the Four States. Here the recovery rates are assumed to be 0 because it is difficult to set specific nonzero rates for comparison. The time series movements of the TSDPs of the Four States are compared in view of business cycles, their budgetary problems and events revealed by the Greek Shocks in 2009.11, the Euro Crisis in 2010.4 and Financial Crisis in 2008.9–2009.3. The 10-year DPs of Italy and Spain are shown to be dramatically increasing over 30 % in the Euro Crisis.

In Sect. 6 the Maastricht condition on the IR-differentials are considered in terms of DPs. The DPs are regressed on the IR-differentials in time series and shown to have an almost perfect linear relationship with the IR-differentials. The result is not surprising because it is in fact due to a structural relation between our GB-pricing model and CB-pricing model, and so it can be used as a conversion formula from IR-differentials to DPs and vice versa.

In Sect. 7 to show the effectiveness of our DP measure and analysis in this paper, for each state we associate each DP \(p_k(s)\) up to term \(s\) with the CDS (credit default swap) of the same maturity \(s\). Here a CDS is a credit derivative which pays its holder the GB principal 100 euros in US dollars when the issuer gets defaulted. In our analysis we regress CDS prices (premiums) on the levels of DPs and slopes of TSDPs, the best model is selected for each maturity \(s\) and show that CDS prices are well explained by our TSDPs.

In the literature there are many theoretical and empirical researches on credit risk. But to our knowledge, there seems no such paper as this paper treating credit risk unconditionally in view of forward-looking modeling. Most theoretical researches in the area of mathematical finance take a time-continuous setting in view of no-arbitrage concept. The books by Duffie and Singleton (2003) and Lando (2004) are well known. A common feature of these time-continuous theories is that all the stochastic processes are Markovian including processes of interest rate and credit risk intensity. Unfortunately speaking from a viewpoint of empirical analysis, actual interest rates and credit processes will not be Markovian since business cycles are not.

In his book, Duffie (2011) comprehensively describes his default intensity approach to corporate credit risk modeling together with time series empirical analysis, where most results are based on his previous papers such as Duffie et al. (2007, 2009), which consider frailty correlated defaults. The model is a doubly stochastic Poisson intensity model, which is extended as predictive intensity model in Duan et al. (2011).

On the other hand, a traditional approach to bond pricing will be represented by the econometric approach in Nelson and Siegel (1987), among others. They assume a specific form of non-stochastic term structure of interest rates including level, steepness, curvature and scale parameters. There are many papers associated with this model and its estimation and forecast procedure (e.g., Diebold and Li 2006). In credit risk analysis the yield curve approach often uses yield spreads between YTMs (yield-to-maturity) or par-yields derived for GBs and CBs. On the other hand, we take a price approach to measuring TSIRs, CRIPSs and TSDPs through GB price model, where TSIR and TSDP are approximated by polynomials.

## 2 The Maastricht Treaty and Our Problem

- 1.
Its inflation rate shall be less than the average of the 3 lowest rates plus 2 %.

- 2.
Its annual government budget deficit over GDP shall be less than 3 %.

- 3.
Its government debt-to-GDP ratio shall be less than 60 %.

- 4.
Its exchange rates in the past year shall be stable.

- 5.
Its long-term (10 year) interest yield average in the past year of a state that seeks currency integration shall be no more than 2.0 % higher than the unweighted arithmetic average of the similar 10-year government bond yields in the 3 EU member states with the lowest inflation.

The budget problem is described in Fig. 1, where the vertical axis and the horizontal axis respectively represent the debt-to-GDP ratio and the annual government budget deficit over GDP in 2011. It shows that though the budgetary condition of Japan is exceptionally bad, the debt-to-GDP ratios of the Five States are all larger than 60 % in 2011 against (3) and the annual government budget deficit over GDP is larger than 3 % against (2) except for Germany. In these two measures Germany is the best state among the five and hence we take DGB as our reference state for comparisons in interest analysis and credit analysis though the budgetary condition does not fully explain credit risk.

## 3 Interest Rate Differentials and Convergence Criterion

In this section we apply the GB pricing model in KWWDY (2012) to DGBs, FGBs, GrGBs, IGBs, and SGBs to derive their TSIRs and compare them to discuss on some differences of the TSIRs in association with business cycles. In Sect. 6 the results will be used to convert IR-differentials into DPs and the Maastricht convergence condition will be stated in DPs.

- (1)
the longer the maturity of each bond is, the larger the variance of each price is, and

- (2)
the larger the difference of maturities of two bonds, the smaller the covariance is.

Though the ESI represents the economic sentiments of industries in private sector, it describes the business cycles of each state, which affect the levels of IRs (at time \(t\)) and even the budgetary condition of each government. The graph was originally made by JETRO and modified by the authors. The average of ESIs over 1990–2011 is set equal to 100.

- (1)
The ESIs of the Fives hit a peak around 2007.6 almost simultaneously though the Spanish ESI was the lowest among the Fives from the peak through the downward slope up to 2009.1.

- (2)
From the peak to the points in 2008.7 the ESIs gradually decreased and then in the financial crisis period of 2008.8–2009.3 they dramatically dropped down about 30 % to the bottom (trough) in 2009.3. The bottom was common to all the ESIs.

- (3)
Then the ESIs of the Fives moved upward simultaneously up to the points in 2009.10, one month before the announcement of a huge budget deficit by Greece Government (Greek Crisis) in 2009.11.

- (4)
Since then, the ESIs separated each other and those of Italy, Spain and Greece did not go up much relative to those of Germany and France. From 2009.12 concerns about the budgetary conditions of Italy and Spain in addition to Greece spread globally and the credit rating agencies downgraded the GBs of Greece in 2009.12.

- (5)
In 2010.4 Greece requested the IMF and the EU Government a bailout and in 2010.5 the IMF and the EU Government created an emergent bailout system.

- (6)
In 2011, the ESIs of the Fives hit another peak around 2011.3 and then went down. Along this downward movement the IGBs and SGBs were sold off and their yields (interest rates) went up significantly, as will be shown later.

- (7)
Except those of Germany and France, the ESIs of Italy, Spain and Greece never reached the level in 2007.6 around 2011.3.

- (1)
From 2007.4 to 2008.6, the German economy was good enough to keep the 10 year IR about 4.5 % and the differences of short term and long term IRs were small, implying that the economy was close to the peak. It hit the peak with almost flat TSIR \(r^{D}(1)\approx r^{D}(2)\approx \cdots \approx r^{D}(10)=0.05\) in 2008.06.

- (2)
From 2008.7 to 2009.3, the interest rates dropped down rapidly and significantly along the subprime shock up and its aftermath, and the TSIRs became upward where the longer the terms are, the larger the rates. The Statistical Bureau of the European Commission identified 2009.3 as the bottom of the business cycles of European economy as in Fig. 1. The \(r^{D}(10)\) rates dropped from 4.6 % in 2008.6 to 3.28 in 2009.3 and further to 2.25 % in 2010.8, which corresponds to the business cycles of Germany in Fig. 1.

- (3)
At the bottom of the business cycles in 2009.3 the D-IRs did not drop much though the gaps between shorter rates and longer rates were widened and the upward slope of the TSIR became steep.

- (4)
After 2009.3 on, the German ESI moved up till 2011.3 in which the business cycles hit another peak, though the Greek budgetary crisis appeared in 2009.11. However, \(r^{D}(10)\) stayed at levels of more than 3 % after 2009.3 till 2010.3, and then suddenly dropped down to 2.4 % in 2010.8, which is not consistent with the movement of the German ESI.

- (5)
The period of 2010.4 through 2010.8 was reported to be the period in which Germany had to make some financial contribution to keep the EEMU system for Greece and other states.

- (6)
After that, the \(r^{D}(10)\) moved up to the level of 3.4 % in 2011.3, where 2011.3 corresponds to the peak month of the ESI. Thereafter \(r^{D}(10)\) went down along the downward movement of the ESI.

As in Fig. 5, the Greek TSIRs are divergent. In particular, from the starting point 2010.4 in Euro crisis they go over 10% and reach 20 % in 2011.4. Then at high space they increase and in 2011.10 the 10 year IR reaches 50 %, implying a state of bankruptcy in GrGB. In fact, as will be shown in Fig. 9, the 10 year DP of GrGB is more than 60 % in 2011.10. For this reason, we sometimes omit the case of GrGB from our analysis in the sequel.

- (a)
2007.4–2008.6, (b) 2008.7–2010.3,

- (c)
2010.4–2011.11, (d) 2011.12–2012.3.

In the period (a) the 4 IRs stick together without credit differentiation, representing a unification of the EEMU. This was also shown in the ESIs in Fig. 2 when their economies were good. The sub-period (b) includes the period of the subprime financial crisis and there the I-IRs and S-IRs start to separate themselves from the D-IRs and F-IRs, though the gaps are less than 2 %, as is required in the Maastricht condition (5) for participation in the EEMU.

However, in the period (c) after Greece requested a bailout to the IMF and the ECB (European Central Bank), investors concerned about the budgetary situations of Italy and Spain governments and sold IGBs and SGBs relative to DGBs and FGBs. In that period, the budget problems in fact turned out to be serious and, as is shown in Fig. 1, the annual deficit/GDP ratios of France, Italy and Spain go over 3 %, where that ratio of Greece is 7.3 %. In addition, the debt/GDP ratio of Italy at the end of 2011 is about 135 % and close to that of Greece, while the debt/GDP ratio of Spain is about 75 % and a bit less than that of Germany. The unemployment rates of these states get worse in this period. In (d), after the ECB, German Government and the IMF responded to the bailout request, the IR differentials get smaller though the gaps between Italy and Germany and between Spain and Germany are far larger than 2 %. It is noted that the F-IRs almost stick to the D-IRs until 2011.10 but thereafter the gaps widened.

No doubt, the credit risk of Italy and Spain gets worse especially in the sub-period (c) and it may be measured as IR-differential. In Sect. 5 we will make correspondence between the IR-differential and DP as our measure of credit risk after we derive TSDPs of the Four States in Sect. 4. In addition, the CDS prices of the Fours are shown to be well explained by the TSDPs.

## 4 Price Differentials (P-Differentials) and CRiPS Analysis

In this section, we first define the CRiPS (credit risk price spread) measure that shows P-differentials (in euros) of each GB of the Four States (France, Italy, Spain, Greece) relative to the corresponding DGB-equivalent bond of the same maturity and coupon rate. Secondly from the CRiPS analysis we draw some empirical observations concerning the differences of CRiPS of the Four States.

*kg*GB included in the portfolio. If the risk of DGB per unit is zero up to 10 years, this is the risk volume of the portfolio in euros relatively to DGB. If the risk of the \(g\)th DGB of maturity \(s_{gM^{D}(g)}^D \) is represented by its TSDP \(p^{D}(s)\), then the risk volume can be measured by

Sample sizes of each cross-sectional data set of GBs for each state over 2007.4–2012.3

07-4 | 07-5 | 07-6 | 07-7 | 07-8 | 07-9 | 07-10 | 07-11 | 07-12 | 08-1 | 08-2 | 08-3 | 08-4 | 08-5 | 08-6 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Germany | 35 | 36 | 36 | 35 | 35 | 36 | 35 | 36 | 36 | 36 | 36 | 37 | 36 | 37 | 37 |

France | 36 | 37 | 38 | 37 | 37 | 37 | 36 | 36 | 35 | 36 | 36 | 36 | 36 | 37 | 37 |

Italy | 30 | 29 | 28 | 29 | 30 | 29 | 30 | 31 | 31 | 31 | 31 | 31 | 32 | 32 | 31 |

Spain | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 15 | 15 | 16 | 16 | 15 | 15 | 15 | 15 |

Greece | 16 | 17 | 17 | 17 | 17 | 17 | 17 | 16 | 16 | 17 | 17 | 18 | 17 | 18 | 18 |

08-7 | 08-8 | 08-9 | 08-10 | 08-11 | 08-12 | 09-1 | 09-2 | 09-3 | 09-4 | 09-5 | 09-6 | 09-7 | 09-8 | 09-9 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Germany | 36 | 36 | 37 | 36 | 37 | 37 | 36 | 36 | 37 | 36 | 38 | 37 | 36 | 36 | 37 |

France | 37 | 37 | 36 | 36 | 36 | 36 | 36 | 36 | 36 | 35 | 36 | 38 | 37 | 37 | 36 |

Italy | 31 | 32 | 32 | 33 | 32 | 32 | 32 | 33 | 32 | 33 | 33 | 34 | 35 | 34 | 35 |

Spain | 14 | 14 | 15 | 16 | 16 | 16 | 17 | 17 | 17 | 17 | 18 | 18 | 19 | 19 | 19 |

Greece | 18 | 18 | 18 | 18 | 18 | 18 | 19 | 21 | 21 | 21 | 20 | 20 | 20 | 20 | 20 |

09-10 | 09-11 | 09-12 | 10-1 | 10-2 | 10-3 | 10-4 | 10-5 | 10-6 | 10-7 | 10-8 | 10-9 | 10-10 | 10-11 | 10-12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Germany | 36 | 38 | 37 | 38 | 39 | 38 | 39 | 39 | 38 | 37 | 39 | 39 | 38 | 40 | 39 |

France | 35 | 34 | 35 | 35 | 36 | 36 | 27 | 29 | 30 | 28 | 28 | 27 | 26 | 26 | 26 |

Italy | 35 | 35 | 36 | 37 | 36 | 36 | 36 | 37 | 38 | 38 | 38 | 37 | 38 | 39 | 39 |

Spain | 20 | 20 | 20 | 21 | 22 | 23 | 23 | 22 | 23 | 24 | 23 | 23 | 23 | 23 | 23 |

Greece | 20 | 21 | 21 | 22 | 22 | 23 | 23 | 21 | 21 | 22 | 21 | 21 | 21 | 21 | 21 |

11-1 | 11-2 | 11-3 | 11-4 | 11-5 | 11-6 | 11-7 | 11-8 | 11-9 | 11-10 | 11-11 | 11-12 | 12-1 | 12-2 | 12-3 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Germany | 40 | 41 | 40 | 41 | 42 | 42 | 40 | 42 | 41 | 41 | 42 | 41 | 41 | 41 | 40 |

France | 26 | 26 | 26 | 25 | 26 | 27 | 26 | 26 | 25 | 24 | 24 | 23 | 22 | 23 | 23 |

Italy | 39 | 39 | 39 | 39 | 40 | 40 | 40 | 41 | 41 | 40 | 41 | 40 | 41 | 41 | 42 |

Spain | 24 | 24 | 23 | 24 | 23 | 23 | 23 | 23 | 23 | 23 | 22 | 22 | 22 | 21 | 21 |

Greece | 21 | 21 | 20 | 19 | 18 | 18 | 18 | 18 | 17 | 16 | 16 | 15 | 17 | 16 | 4 |

- (1)
In each state the CRiPS graph at each time is almost linear, implying a legitimacy of the definition of CRiPS measure.

- (2)
Except for France, the almost linear CRiPS graphs are ordered from up to down in the order of 2007.4, 2008.6, 2009.3, 2010.8, and 2011.12. This means that along with these time points the slopes of the term structures of CRiPSs (TS-CRiPSs) become steeper and the credit risks of these states worsened relative to those of Germany. In case of France, the TS-CRiPS of 2009.3 is uniformly lower than that of 2010.8 and so the credit of FGB is improved from 2009.3 to 2010.8.

- (3)
In each state the slope of the TS-CRiPS of 2011.12, which is the mid of the Euro Crisis, is not only larger (in absolute value) than that of any other time but also it jumps down significantly from the slope of the TS-CRiPS of 2010.8. In 2011.12 the 8 year CRiPS of FGB is about \(-12\) euros, while the 9 year CRiPSs of Italy and Spain are respectively about \(-35\) and \(-30\) euros.

- (4)
In 2007.4, though the Italian CRiPSs are a bit negatively larger in the maturities of longer terms, all the CRiPSs are rather close to 0 even in Greece, meaning no significant credit differentiation at this time.

- (5)
In 2008.6 just before the subprime shock, the graph moves downward in each state but not much, implying that credit risks of these states are slightly larger than risk of Germany. For example, even the 10 year CRiPS of France is close to \(-2\) euros, while that of Italy is close to \(-5\) euros

- (6)
But in 2009.3 of the financial crisis, it moves downward significantly in each state. The 10 year CRiPSs of France, Italy, Spain and Greece are respectively about \(-5, -12, -7.5\) and \(-20\) euros, far away from DGB-equivalents.

In 2008.6 the TS-CRiPS of Spain is close and similar to that of France though the latter is slightly and uniformly smaller in absolute value in the longer terms. On the other hand, the TS-CRiPS of Greece is close and similar to that of Italy though the latter is slightly and uniformly smaller in absolute value in the longer terms. Clearly the CRiPSs of Italy and Greece are separately larger in absolute value than those of France and Spain. This fact is probably related to the fact that the debt/GDP ratios of Italy and Greece are larger. In 2009.3 the TS-CRiPS of Spain goes down away from that of France and gets close to that of Italy and the 8.3 year CRiPS measure of Spain becomes about \(-7\) euros. In 2010.8 the TS-CRiPS of Spain gets worse than that of Italy and the 9.3-year CRiPS measure of Spain becomes about \(-16.5\) euros, which is significantly different from that of Germany. On the other hand, the 9-year CRiPS of France is about \(-3.3\) euros, though it gets worse in 2012.2.

In the next section these observations are substantiated by introducing CB-pricing model and CRiPS measures are transformed into default probabilities of \(k\hbox {GB}(g)\hbox {s}\).

## 5 Term Structures of Default Probabilities (TSDPs) of FGB, IGB, SGB and GrGB

In this section we derive the (relative) TSDPs of each state up to 10 years via Kariya (2013) model, and then compare their time series movements of the TSDPs over the period 2007.4–2012.3. In Sect. 6 we will associate the TSDPs with the IR-differentials and relate them to the Maastricht condition (5) in Sect. 2.

- (1)
Though the levels of DPs \(\{p^{k}(s)\}\) for each state are different where \(s=1,2,\ldots ,10\), the patterns of time path of each \(p^{k}(s)\) over the sample period are similar. All the DPs of the Four States are getting larger relatively to their past DPs over the period.

- (2)
This implies that only Germany has a sound credit position over the period, which was observed in terms of IR-differentials in Sect. 3 as well as P-differentials in Sect. 4.

- (3)
Overall the 8-year DP \(p^{F}(8)\) of France takes 10.3 % at its maximum in 2012.1. In 2012.1 S&P downgraded the rating of 9 Euro GBs including FGB. It is noted that FGBs of 10-year maturity disappear after 2010.3 and the FRBs of 9-year maturity disappear after 2011.3 because their GBs are not available in the market.

- (4)
On the other hand, the 10-year DP \(p^{I}(10)\) of Italy takes 37 % at its maximum in 2011.12, the 10-year DP \(p^{S}(10)\) of Spain takes 30 % at its maximum in 2011.9 but the Greek case is as catastrophically large as 80 %.

- (5)
Around the financial crisis period 2008.9–2009.3, the TSDPs \(\{p^{k}(s):s=1,2,\ldots ,10\}\) shift above for each state \(k\). But the effect is different among the states and the local maximum of 10-year DPs \(p^{k}(10)^{\prime }s\) around that period is about 6 % in France, about 13 % in Italy, about 10 % in Spain and about 20 % in Greece, the last of which is already the largest.

- (6)
In the sub-period 2007.4–2008.6, the 10-year DPs \(p^{I}(10)\)’s of Italy are larger than those of France and Spain, which will probably reflect the high debt/GDP ratios of Italy even in the sub-period.

In the next section we relate these results on DPs to the results on the IR-differentials and discuss on the Maastricht condition (5) in terms of DPs.

## 6 Default Probabilities (DPs), IR-Differentials and Maastricht Condition (5)

In Fig. 11 the time series graphs of the IR differentials and DPs of Italy and Spain among others are plotted for \(s=3,5,7\). It is observed that the IR differentials and the DPs fluctuate together in every detail, though the fluctuations of the IR-differentials may not be easily seen in details.

Results of regression analysis between the DPs \(\{p^{k}(s)\}\) and the IR-differentials \(\{{\delta }^{k}(s)\equiv r^{k}(s) - r^D (s) \}\) for 2004.4–2012.3

\(\beta \) | (t-value) | \(\alpha \) | (t-value) | adj R\(^2\) | ||
---|---|---|---|---|---|---|

France | 2yrs | 2.001 | (122.06) | 0.00003 | (0.76) | 0.996 |

3yrs | 2.936 | (92.99) | 0.00014 | (1.43) | 0.993 | |

5yrs | 4.830 | (74.19) | 0.00045 | (1.75) | 0.989 | |

7yrs | 6.642 | (73.77) | 0.00083 | (1.99) | 0.989 | |

9yrs | 8.440 | (21.60) | 0.00137 | (1.34) | 0.919 | |

Spain | 2yrs | 1.908 | (339.37) | 0.00049 | (5.72) | 0.999 |

3yrs | 2.784 | (225.66) | 0.00103 | (5.33) | 0.999 | |

5yrs | 4.675 | (225.65) | 0.00107 | (3.16) | 0.999 | |

7yrs | 6.199 | (158.86) | 0.00359 | (5.57) | 0.998 | |

9yrs | 7.747 | (136.45) | 0.00491 | (5.24) | 0.997 | |

Italy | 2yrs | 1.904 | (297.66) | 0.00041 | (3.70) | 0.999 |

3yrs | 2.733 | (168.47) | 0.00268 | (9.18) | 0.998 | |

5yrs | 4.440 | (181.03) | 0.00292 | (6.56) | 0.998 | |

7yrs | 6.089 | (142.95) | 0.00289 | (3.85) | 0.997 | |

9yrs | 7.206 | (85.37) | 0.01311 | (8.92) | 0.992 | |

Greece | 2yrs | 1.009 | (35.24) | 0.03599 | (4.68) | 0.955 |

3yrs | 1.475 | (29.82) | 0.04785 | (4.93) | 0.938 | |

5yrs | 2.636 | (33.80) | 0.04906 | (5.24) | 0.951 | |

7yrs | 3.248 | (25.82) | 0.07019 | (5.45) | 0.919 | |

9yrs | 5.204 | (41.15) | 0.04591 | (5.75) | 0.969 |

It is remarked that the linear relationship does not accidentally hold but can be shown to hold structurally in the relation of the bond pricing model in Sect. 3 and the credit risk model in Sect. 5 though we do not show it here. In other words, the credit risk measures of the DP and the IR-differential are almost equivalent.

## 7 CDSs and DPs: Effectiveness of Our DP Measures

As is well known, a CDS here is the credit derivative which pays its holder the GB principal in US dollars when the issuer of the GB gets defaulted. Note that the issuers of CDSs are investment bankers, hedge funds, insurance companies, etc. and hence they may be different from bond investors. In this section, to show the effectiveness of the DP measures and related analyses in this paper, we associate our DP measures of each maturity for each state with the CDSs of the same maturity for the same state. For this purpose we regress CDS prices (premiums) on levels of DPs and slopes of TSDPs derived in Sect. 5 and show that CDS prices are well explained by those variables.

Regression models of CDSs for France, Italy and Spain in the Base model O in (7.1)

France | \(\beta 1\) | Const | adjR2 | Italy | \(\beta 1\) | Const | adjR2 | Spain | \(\beta 1\) | Const | adjR2 |
---|---|---|---|---|---|---|---|---|---|---|---|

FCDS(1) | 1.3734 | 0.3698 | .7009 | ICDS(1) | 0.8901 | 0.4925 | .7747 | SCDS(1) | 0.4993 | 0.7620 | .8417 |

(9.49) | (8.45) | (11.47) | (3.31) | (14.25) | (11.65) | ||||||

FCDS(2) | 0.8673 | 0.3262 | .8707 | ICDS(2) | 0.4831 | 0.2668 | .8465 | SCDS(2) | 0.3237 | 0.6456 | .8629 |

(16.03) | (9.70) | (14.51) | (1.97) | (15.50) | (8.59) | ||||||

FCDS(3) | 0.5644 | 0.3137 | .8467 | ICDS(3) | 0.3357 | 0.1749 | .8884 | SCDS(3) | 0.2565 | 0.5308 | .8939 |

(14.52) | (7.42) | (17.42) | (1.46) | (17.92) | (6.78) | ||||||

FCDS(4) | 0.4273 | 0.2700 | .8620 | ICDS(4) | 0.2551 | 0.2799 | .9273 | SCDS(4) | 0.2322 | 0.4898 | .9370 |

(15.44) | (5.79) | (22.05) | (2.93) | (23.80) | (6.70) | ||||||

FCDS(5) | 0.3457 | 0.2322 | .8724 | ICDS(5) | 0.2060 | 0.4279 | .9437 | SCDS(5) | 0.2171 | 0.4480 | .9541 |

(16.15) | (4.56) | (25.25) | (5.20) | (28.13) | (6.10) | ||||||

FCDS(7) | 0.2577 | 0.1721 | .8280 | ICDS(7) | 0.1590 | 0.3575 | .9508 | SCDS(7) | 0.1625 | 0.3223 | .9542 |

(13.56) | (2.38) | (27.11) | (4.56) | (28.16) | (4.31) | ||||||

ICDS(10) | 0.1439 | \(-\)0.1366 | .9166 | ||||||||

(20.46) | (\(-\)1.09) |

- (1)
Except for the case of

*ICDS*(10) all the regression coefficients are significant and the longer the maturities are, the less the coefficients and the larger the \(t\)-values are. - (2)
Except for the cases of

*ICDS*(10) and*ICD*(3), all the constant terms are significant and decreasing along the period of maturities. - (3)
In the cases of \(ICDS(j),\,j=4,5,7,10\) and \(SCDS(j),\,j=4,5,7\), the adjusted \(R^{2}\)’s are greater than 0.9 and so in these cases each \(kDP(j)\) derived from GB prices will be in a good correspondence with the corresponding \(kCDS(j)\) in the derivative market for each time.

- (4)
In the case of the FCDSs, the adjusted \(R^{2}\) of

*FCDS*(1) is the lowest as 0.7, the worst even in all the states, and the other adjusted \(R^{2}\)’s are larger than 0.8. - (5)
In the case of the ICDSs, the adjusted \(R^{2}\)’s are increasing from \(R^{2}(1)=0.77\) to \(R^{2}(7) =0.95\) but in the case of

*ICDS*(10), \(R^{2}(10)=0.92\), the constant term is negative and its t-value is insignificant. - (6)
In the case of the SCDS, the least adjusted \(R^{2}\) is 0.84 for

*SCDS*(1) and is monotonically increasing up to 0.95 for*SCDS*(7). This case will be the best case among the three states so long as the base model is concerned.

*FCDS*(1) and

*ICDS*(1).

Best regression results for France, Italy and Spain among Models A, B and C

- (1)
In the case of France, Model A improves Model O significantly for \(FCDS(j),\,j=1,2,3\), none of the three models cannot improve it for \(FCDS(j),\, j=4,5\) and Model C improves it for

*FCDS*(7) in adjusted \(R^{2}\) with the \(t\)-value of \({\beta }_{F2} (7)\) significant. In the case of*FCDS*(1) the adjusted \(R^{2}\) changes from 0.7 to 0.9 when the slope variable is added to Model O. In Fig. 12 the improvement is visualized. - (2)
In the case of Italy, Model A improves Model O for \(ICDS(j),\,j=1,2,3,4\). But the \(t\)-values of \({\beta }_1^I (j)\)’s with \(j=1,2,3\) become less than 2, though we keep the \(DP(j)\) variables in our basic viewpoint. In Fig. 12 the improvement is visualized for

*ICDS*(1), where the adjusted \(R^{2}\) changes from0.77 to 0.87. On the other hand, none of the three models improve Model O significantly for \(ICDS(j), j=5,7,10\) in terms of \(t\)-values of \({\beta }_2^I (j)\) or adjusted \(R^{2}\). - (3)
In the case of Spain, Model A improve Model O for \(SCDS(j),\,j=1,2, 3,4,5\) but not for \(SCDS(j),\,j=7\).

*FCDS*(7), while the shorter slope \((DP^{k}(5)-DP^{k}(2))/3\) of TSDPs on Base Models of FCDS,ICDS and SCDS. From the significance of \(t\)-values for the slope variables and adjusted \(R^{2}\), we summarize our model selection in the south-east corner of Table 4 where the DP variables are kept in the models even if the \(t\)-values are insignificant as in

*ICDS*(\(j\)) with \(j=1,2,3\).

It is noted that CDS markets exist as derivative markets for GBs and so they are closely related. But the players who form or quote CDS prices are not necessarily investors in the GB markets and GDS markets are rather smaller. Taking into this point account, our empirical result that the variations of \(DP(j)\) variable and \(CDS(j)\) variable are synchronized will imply that our method and model of deriving TSDPs are effective to a large extent. And the results will be used for making a certain decision on pricing CDSs and including CDSs for risk management in bond portfolios.

It is remarked that the results on the high adjusted \(R^{2}\)’s in Table 4 are overall results for the total period and may not be as effective as the apparent high numbers. In fact, the fluctuations in the graphs include a big trendy mountain (Euro Crisis) that may be under a long cycle as in Fig. 12. Hence so long as the regression models pick the big movements, the adjusted \(R^{2}\)’s tend to be large. But this is a limitation of regression analysis itself and the results will still show the co-movements of the CDSs with the levels of the DPs and the slopes. In time series context the CDS, DP and Slope variables may be co-integrated to a certain degree, though we do not here explore for it. Note that the DPs and Slopes data are cross-sectionally derived data by our model.

## 8 Conclusion

We made a comprehensive credit risk analysis on FGB, IGB, SGB and GrGB via the IR-differentials, the GB price differentials, and the DPs in comparison with DGB where the sample period was 2007.4–2012.3. First after making some discussions and observations on the business cycles of the Five States and the Maastricht convergence condition of IRs, we derived the TSIRs via the bond-pricing model in KWWDY (2012) and compared them. In association with the budgetary conditions and business cycles of the Five States, we found that (a) in the sub-period 2007.4–2008.6 the TSIRs of the Five States moved together at almost same levels, meaning that no credit differentiation was found in the IR-differentials, (b) in the sub-period 2008.7–2010.3 all the IRs tended to decrease gradually as downward trend movements but their IR-differentials exhibited some credit differentiation, (c) in the sub-period 2010.4–2011.12 the IR-differentials tended to diverge and (d) after that they tended to get stabilized gradually. Though the F-TSIRs moved together with the D-TSIR before the Euro Crisis, then they deviate. This deviation would have made the role of the German Government more important for the stability of the EEMU.

Secondly we proposed what we call CRiPS measure, which measures directly credit risk via the P-differential between a given GB and DGB-equivalent in terms of euros. The CRiPS measure is model-free once the mean discount function of Germany is estimated. In our empirical analysis the 10-year CRIPSs of Italy and Spain in the mid of the Financial Crisis 2009.3 were respectively about \(-10\) euros and \(-7\) euros, but in the mid of the Euro Crisis 2011.12 they jumped down to about \(-35\) euros and \(-30\) euros. Since the CRIPS measure is additive, it can be used to measure credit risk volume (in euros) of a bond portfolio.

Thirdly the TSDPs of the Four States were derived via CB-pricing model in Kariya (2013) and compared. The model enabled us to transform the term structures of CRiPSs into TSDPs, and it turned out that the 10-year CRiPSs of Italy and Spain in 2011.12 respectively corresponded to about 35 and 30 % where the recovery rate was assumed to be zero. In addition we substantiated the observations on the credit differentiation obtained through our TSIR analysis in terms of the DPs. Also it was observed that the Financial Crisis did not affect the DPs of France, did increase the DPs of Italy, Spain and Greece to the levels of 6, 10 and 20 % respectively, but after that their DPs decreased. The time series movements of the TSDPs of the Four States were associated with business cycles, Financial Crisis and Euro Crisis.

Fourthly the time series relationships between the IR-differentials and the DPs of each maturity were shown to be strongly linear by our regression analysis, which enables us to convert the IR-differentials into the DPs for each maturity and vice versa. Consequently the Maastricht convergence condition can be stated in terms of the DPs. It was observed that Italy, Spain and Greece did not meet the required condition in the period of the Euro Crisis though they are the members of the EEMU, creating the instability of the EEMU system that was an economic concern in the global world. But an explicit solution that President Van Rompuy (2012) planned for a genuine integration of the EEMU will stabilize the EEMU.

Finally we made compared the CDS prices of each maturity to our TSDPs by term series regression and found that the CDS prices were well explained by the DP levels and the slopes of TSDPs. Since the CDS prices are formed in a different market by different players, this result will show that our approach and model to deriving the TSDPs are effective. In addition the regression model will enable us to use for trading CDSs.

Overall, our empirical model analysis on credit risks of the main states of the EEMU will be effective and the results therein will be useful for decision making in credit investment and risk management.

## Notes

### Acknowledgments

This research was supported by the Japan Society for the Promotion of Science (JSPS), Grant-in-Aid for Scientific Research (A),No.23243040.

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