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.
To describe the model, let the \(k\hbox {GB}(g)\) denote the \(g\)th bond of the \(k\)th state at time \(t\) for analysis, where \(k=\hbox {D}\), F, I, S, Gr, and \(g=1,\ldots ,G_t^k \). Let \(P_{gt}^k \) denote the \(k\hbox {GB}(g)\) price at \(t\), where \(G^{k}\) is the sample size of the \(k\)GB prices at \(t\). In the sequel, time \(t\) is fixed and the suffix is omitted from each variable. Let \(s_{g1}^k < s_{g2}^k < \cdots <s_{gM^{k}(g)}^k \) denote the future times measured in years from \(t\) at which the CFs of the \(k\hbox {GB}(g)\) are generated, and \(s_{gM^{k}(g)}^k \) is the maturity of the \(k\hbox {GB}(g)\). Then the GB pricing model is given by
$$\begin{aligned} P_g^k =\sum _{j=1}^{M^{k}(g)} {C_g^k \left( s_{gj}^k \right) D_g^k \left( s_{gj}^k \right) }\qquad (g=1, \ldots ,G^{k}). \end{aligned}$$
(3.1)
In (3.1), \(D_g^k (s)\) is a stochastic discount function which possibly depends on bond attributes, and the realization (at \(t\)) of each price \(P_g^k \) is regarded as equivalent to the realization (at \(t\)) of the whole discount function \(\{D_g^k (s):0\le s\le s_{gM^{k}(g)}^k \}\). Here \(C_g^k (s_{gj}^k )^{\prime }s (j=1,2,\ldots ,M^{k}(g)-1)\) are the coupons and \(C_g^k (s_{gM^{k}(g)}^k ) \) is the coupon plus the principal 100 (Euro) of the \(k\hbox {GB}(g)\). For example, in the case of the IGB, coupons are paid semi-annually and so with \(k=\hbox {I}\)
$$\begin{aligned} C_g^k \left( s_{gj}^k \right) =0.5c^{k}\hbox { for }j=1,2,\ldots ,M^{k}(g)-1\hbox { and }C_g^k \left( s_{gM^{k}(g)}^k \right) =0.5c^{k} +100. \end{aligned}$$
But in the case of the other GBs, coupons are paid annually.
The stochastic discount function is assumed to be decomposed into the attribute-independent mean discount function \(\overline{{D}}^{k}(s)\) and the attribute-dependent stochastic part \({\Delta }_g^k (s)\);
$$\begin{aligned} D_g^k (s)=\overline{{D}}^{k}(s)+{\Delta }_g^k (s). \end{aligned}$$
(3.2)
While in KWWDY (2012) \(\overline{{D}}^{k}(s)\) is also assumed to be attribute-dependent to look into the pricing behaviors of bond investors, here it is assumed to be attribute-independent to compare GB market prices in different states. Inserting (3.2) into the model in (3.1) yields
$$\begin{aligned} P_g^k =\sum _{j=1}^{M^{k}(g)} {C_g^k \left( s_{gj}^k \right) \overline{{D}}^{k}\left( s_{gj}^k \right) } +{\eta }_g^k\,\, with\,\, {\eta }_g^k =\sum _{j=1}^{M^{k}(g)} {C_g^k \left( s_{gj}^k \right) } {\Delta }_g^k \left( s_{gj}^k \right) . \end{aligned}$$
(3.3)
We approximate the mean discount function \(\overline{{D}}^{k}(s)\) by a polynomial of the \(p\)th order;
$$\begin{aligned} \overline{{D}}^{k}(s)=1+{\delta }_1^k s+{\delta }_2^k s^{2}+\cdots +{\delta }_p^k s^{p}, \end{aligned}$$
(3.4)
since any continuous function on a closed interval can be uniformly approximated by a polynomial.
On the other hand, the specification of the stochastic part of \(D_g^k (s)\) is made by
$$\begin{aligned} Cov\left( D_g^k \left( s_{gj}^k \right) ,D_h^k \left( s_{hm}^k \right) \right) =Cov\left( {\Delta }_g^k \left( s_{gj}^k \right) ,{\Delta }_h^k \left( s_{hm}^k \right) \right) =({\sigma }^{k})^{2}{\lambda }_{gh}^k \end{aligned}$$
(3.5)
for all \(j\) and \(m\), where
$$\begin{aligned} {\lambda }_{gh}^k =\left\{ {{\begin{array}{l@{\quad }l} 1&{} (g=h) \\ {\rho }^{k}e_{gh}^k &{}(g\ne h) \\ \end{array} }} \right. \,\, with\,\, e_{gh}^k =\exp \left( -\xi ^{k}\left| {s_{gM^{k}(g)}^k -s_{hM^{k}(h)}^k } \right| \right) . \end{aligned}$$
(3.6)
This specification implies that
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(1)
the longer the maturity of each bond is, the larger the variance of each price is, and
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(2)
the larger the difference of maturities of two bonds, the smaller the covariance is.
It is noted that the covariance in (3.5) is independent of the individual CF time points of two bonds except for the maturities. In this speculation the covariance of \(P_g^k \) and \(P_h^k \) becomes
$$\begin{aligned} Cov(P_g^k ,P_h^k )=({\sigma }^{k})^{2}{\lambda }_{gh}^k {\varphi }_{gh}^k \hbox { with } {\varphi }_{gh}^k =\sum _{j=1}^{M^{k}(g)} {\sum _{m=1}^{M^{k}(h)} {C_g^k \left( s_{gj}^k \right) C_h^k \left( s_{hm}^k \right) } } . \end{aligned}$$
(3.7)
Now our GB-pricing model for state \(k\) is reduced to a regression model;
$$\begin{aligned} y^{k}&= X^{k}{\beta }^{k}+ {\eta }^{k} \hbox { with } {\beta }^{k}=\left( {\delta }_1^k ,{\delta }_2^k ,\ldots ,{\delta }_p^k \right) ^{\prime } :p\times 1,\nonumber \\ y^{k}&= \left( y_1^k ,y_2^k ,\ldots ,y_{G^{k}}^k \right) ^{\prime }:G^{k}\times 1, y_g^k =P_g^k - a_g^k ,\, a_g^k =\sum \nolimits _{j=1}^{M^{k}(g)} {C_g^k \left( s_{gj}^k \right) } , \end{aligned}$$
(3.8)
and the covariance matrix;
$$\begin{aligned} Cov({\eta }^{k})=(Cov({\eta }_g^k ,\eta _h^k ))=(Cov(P_g^k ,P_h^k ))=({\sigma }^{k})^{2}({\lambda }_{gh}^k {\varphi }_{gh}^k )\equiv ({\sigma }^{k})^{2}\Phi ({\rho }^{k},\xi ^{k}). \end{aligned}$$
(3.9)
This specification naturally introduces not only a heteroscedasticity and correlation structure into the model but also a bond duration effect into the variances and covariances of the prices. The parameter \({\beta }^{k}\) in the mean discount function \(\overline{{D}}^{k}(s)\) are estimated via the GLS (generalized least squares) method, which minimizes the objective function
$$\begin{aligned} \psi ({\beta }^{k},{\rho }^{k},{\xi }^{k})= \left[ y^{k}-X^{k}{\beta } ^{k}\right] ^{\prime }\left[ \Phi ({\rho }^{k},{\xi }^{k})\right] ^{-1}\left[ y^{k}-X^{k}{\beta }^{k}\right] \end{aligned}$$
(3.10)
with respect to the unknown parameters (see Kariya and Kurata 2004 for the effectiveness of GLS).
Once \(\overline{{D}}^{k} (s)\) is estimated, the TSIR of the \(k\)GB at time \(t\) is estimated by
$$\begin{aligned} r^{k}(s)=-[\log \overline{{D}}^{k} (s)]/s\quad \textit{with}\quad 0<s\le 10. \end{aligned}$$
(3.11)
Before we apply the model to data, we look into the business cycles of the Five States and a brief history of events. In Fig. 2 the ESIs (Economic Sentiment Indicators) of EEMU states, published by the European Commission, are graphed, where the ESI is based on economic surveys and summarized as diffusion index, and the average of ESIs over 1990–2011 is set equal to 100.
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.
From Fig. 2 it is observed:
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(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.
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(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.
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(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.
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(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.
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(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.
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(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.
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(7)
Except those of Germany and France, the ESIs of Italy, Spain and Greece never reached the level in 2007.6 around 2011.3.
Now we apply the model to each monthly data set of cross-sectional GB prices from 2007.4 through 2012.3 (5 years). In Table 2 the sample size of each monthly data set is given for each state over the period. The sample sizes of DGB vary from 35 to 42 and tend to increase on the average. On the other hand, the average sample size of FGB is about 37 for the period up to 2010.3 but thereafter it is reduced to about 25. In other words, the French Government did not issue many GBs after 2010.3. In the case of SGB the sample size tends to increase almost monotonically from 16 through 24, while in the case of IGB they tend to increase from 28 to 42 over time. However, Greece issued less GBs and it tends to increase from 16 to 23 up to 2010.5 and thereafter tends to decrease to 16 except for the case of 2012.3 with sample size 2. In 2012.3 it seems that Greece could not issue new bonds after the Greek Crisis.
In selecting a model in (3.4), we need to specify the order \(p\) of the polynomial. Figure 3 shows the AIC values and the averages of the 60 minimized values
‘s of the objective function in (3.8) when \(p\) changes. Of course, as \(p\) increases, the average decreases since the parameter increases. In the case of DGB and IGB there are a moderate sample size for each set of data and hence we might be able to take \(p=8\). However considering the other cases with smaller samples and taking into account the fact that \(p=6\) in the cases of USGB and JGB (Japanese GB) (see KWWDY 2012), we take \(p=6\) below, which is common to all the models of the Fives.
In Fig. 4, we plot the TSIRs of DGBs, denoted by D-TSIRs, which is estimated through (3.11).
Some observations follow:
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(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.
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(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.
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(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.
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(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.
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(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.
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(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.
In Fig. 5 the TSIRs of the Four States (France, Italy, Spain and Greece) are plotted. It is first noted that the scales of each vertical axis in these graphs are different. The reason why the graphs of French \(r^{F}(10)\) and \(r^{F}(9)\) are cut off in between the period is that no FGBs of 10 year and 9 year maturities beyond the cut-off points are available in our data set, where no extrapolation is made in estimating IRs.
It is observed from Fig. 5 that French TSIRs move more like German TSIRs over all and that the TSIRs of Italy, Spain and Greece move in completely different manner, though up to 2007.7 all the IRs including the German rates are almost same. In fact, they moved up significantly after the Greek Shock in 2010.4, implying that some potential budgetary or credit problems in the latter states are revealed. The 10-year IRs of these three states remain at more than 4 % level for almost all the months even when the business cycles are around the trough in 2009.3, while the 10-year German IR dropped to 2.4 % in 2010.8. This implies that the credit risk of the three states gets larger than that of Germany. In fact, it is often the case that credit risk is analyzed through the IR-differentials \(r^{k}(j)-r^{D}(j)\)’s between \(k-\)IR and D\(-\)IR of common maturity \(j \)years, where \(k=\hbox {F ,I, S, Gr}\) and \(k-\)IR stands for the interest rate of the \(k\)th state at each time \(t\). In Sect. 6 the IR-differentials are analyzed in association with DPs.
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.
The D-IRs, F-IRs, I-IRs and S-IRs of 7-year maturity are drawn for the Five States over the period 2007.4–2012.3 in Fig. 6. From Fig. 6 it is easily observed that I-IRs and S-IRs gradually get separated from D-IRs and F-IRs. The period is divided into 4 sub-periods;
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(a)
2007.4–2008.6, (b) 2008.7–2010.3,
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(c)
2010.4–2011.11, (d) 2011.12–2012.3.
We respectively call them (a) sub-period of no differentiation, (b) sub-period of differentiation, (c) sub-period of divergence and (d) sub-period of stabilization.
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.