### European Safe Bonds (ESBies) - Appendix

**APPENDIX: NUMERICAL SIMULATIONS OF THE ESBies SAFETY**

We conduct two numerical simulations to gauge the safety of the ESBies and to determine the tranching threshold that separates them from the junior tranche. In one we take a pessimistic scenario where there is a high probability of simultaneous defaults in several countries of the euro-zone. Another maybe more realistic scenario uses the default probabilities and recovery rates in the historical data in the last 20 years. We present each in turn. In both cases, we set the weight of each sovereign in the collateral pool, the asset side of the structure, equal to its average share of euro-zone (EZ) GDP between 2006 and 2010. These weights are listed in the second column of Table 1 below.

**Pessimistic Benchmark Scenario**

Table 1 orders the countries in terms of their current sovereign credit ratings (assessed by Moody’s and S&P).

We assume that there are 3 aggregate states of the world that describe the health of the EZ economy. In state 1, a catastrophe unfolds (a great depression) and default risk is very high for all countries, but more so for periphery than for core countries. The idiosyncratic default probability in this state is labeled probdef1 and is listed in the second column of Table 1. It refers to a cumulative default rate over a 5-year period. State 2 is a bad state of the world (a recession) with elevated idiosyncratic default risk in all countries (probdef2). State 3 is the good state (expansion) with low idiosyncratic default risk for all countries except Greece (probdef3). We assume that the economy is in the catastrophic state 5% of the time, in the bad state 25% of the time, and in the good state 70% of the time. The random variable that governs defaults has a fat-tailed distribution (Student-t with 4 degrees of freedom).

Second, we assume that losses given default vary by country and depend on the aggregate state of the economy; they are higher in the worse aggregate states and higher for Greece than Germany in every aggregate state of the world. They are listed in columns 4 through 6 for states 1 through 3, respectively.

Third, we assume that country defaults are idiosyncratic events within each aggregate state. Because idiosyncratic default rates are much higher in state 1 and in state 3 for all countries, default intensities are correlated. In addition, we overlay the following assumptions on default behavior that further increase the cross-country correlation between defaults:

- Whenever there is a Spanish default, the EZ countries with better credit ratings than Spain (listed in the first seven rows of Table 1) default with probability 15% and the remaining 9 countries (in the last 9 rows of table 1) default with probability 65%.
- Whenever there is an Italian default, the top 7-rated countries default with probability 25% and the remaining 9 countries default with probability 65%.
- Whenever there is either a French or a German default, all other countries default with probability 80%.

One way to express the commonality in defaults across countries is to ask what fraction of the covariation of default events can be explained with the first (first three) principal components of the default covariance matrix of the 17 EZ countries. For the parameters listed in Table 1, the first (three) principal component of defaults explains (explain) 41% (73%) of the variation between default rates. This large covariation substantially reduces the gains from diversification. We do not repeat the mistake made in the structuring of mortgage-related products where correlations between the default risk of underlying mortgage loans where routinely assumed to be below 15%.

Based on a simulation of 2,000 periods, in each of which we consider 5,000 draws of the default process (for a total of 10 million iterations), we calculated the default rate for each country and we calculate how often the portfolio of sovereign bonds (based on the GDP weights) makes losses above 10%, 20%, and 30%. The last column of Table 1 reports the unconditional default rates, measuring a cumulative rate over a 5-year period, for each country under our assumptions. They combine the information on the idiosyncratic default rates in each aggregate state with the assumptions on commonality of default when one of the four large countries defaults and with the probabilities of each of the aggregate states. For example, the unconditional default rate of Germany is 1.11%. This number results from a marginal probability of default of 5.5% conditional on being in state 1, a marginal default probability of 1.9% in state 2, and of 0.6% in state 3. The idiosyncratic probabilities of default reported in columns 3-5 only average to a default rate of 0.50% (=5%*0.5%+25%*0.1%+70%*0%) so that the remaining 0.61% (=1.11%-0.50%) arises from our auxiliary assumptions on the default of Spain, Italy, or France. Similar logic extends to the other countries. These default rates are very conservative (they are high), in light of the current credit ratings of the sovereign bonds.

Based on this simulation, we find that the portfolio of sovereign bonds has a loss distribution with a 90th percentile of 4.25%. That is, losses on the portfolio exceed 10% in 4.25% of the 10 million draws. The 80th percentile of the loss distribution is 2.71%. Finally, the 70th percentile of the loss distribution is 0.80%. This means that if we create ESBies that represent 70% of the value of the underlying collateral pool, they will be affected by losses in only 0.80% of the periods or once in every 125 5-year periods. Since our assumptions on default rates, losses given default, default correlations were all conservative, the 70% tranche is likely even safer than indicated by its 0.80% expected default rate. Note that the ESBies are safer than German bonds in this simulation.

The collateral pool has zero losses in 36% of the 10mi draws. Conditional on having a non-zero loss, the average loss is 3.3%. The unconditional average loss across all 10mi draws is 2.12%. This is the probability weighted average of a marginal loss of 11.09% in state 1, 4.14% in state 2, and 1.06% in state 3. Conditional on a portfolio loss in excess of 30% (the states of the world in which ESBies are affected), the mean loss is 44.6%. Conditional on a portfolio loss between 0 and 30% (the states of the world in which the junior tranche is affected), the mean portfolio loss is 2.85%. The mean loss on the junior tranche (including the states of the world with no losses) is 1.82%.

According to these same simulations, losses rarely exceed 30%; they do so with probability 0.80%. In our proposal, the first 30% of losses would go to the holders of the junior tranches. To further protect the holders of ESBies, a capital guarantee could be added to the protection offered by subordination. To eliminate losses that occur between the 99.2th and 99.7th percent of the loss distribution, a capital guarantee of 14.56% would be needed in our example. If 60% of EZ GDP is securitized in the form of ESBies, the long-run size of the program would be 60% of euro 9.17 trillion (2010 number) or 5.50 trillion. ESBies would represent 70% of this or they would be euro 3.85 trillion. The required capital guarantee would be 800 billion euros. Note that this capital would only be touched with a very small likelihood (0.8%).

**Table 1. Defaults and Losses on the Collateral Pool of ESBies in a Pessimistic Scenario** *Notes:* Column 1 reports the country and the Moody’s and S&P credit rating as of August 2011. The second column reports the weight of each country in the collateral pool. Column 3-5 report the idiosyncratic (not the total) default rate conditional on aggregate state 1, 2, and 3. Columns 6-8 report the loss given default conditional on default occurring in aggregate state 1, 2, and 3. Column 9 reports the unconditional expected default rate which combines the idiosyncratic probabilities of default in columns 3-5, the auxiliary assumptions on default of Spain, Italy, France, and Germany, and the probability distribution over aggregate states.

In a second exercise, we use historical expected default rates and recovery rates on sovereign bonds. In fact, the recovery rates we use are the same as in the previous exercise, and those correspond to historical average recovery rates from previous sovereign defaults.^{1} We use five-year average cumulative default rates by initial credit rating for corporate bonds. These default rates are higher than the corresponding default rates for sovereign bonds. We use the corporate rates because the sovereign default rates are exactly zero for all bonds rated A or above for the 1983-2007 sample period we use as a data source. In that sense, even the historical scenario is pessimistic. We recalibrate the probabilities of default in aggregate states 1, 2, and 3 in order to arrive at an expected default rate for each country that corresponds to the expected default rate on a company with the same credit rating as that country (as of August 2011).

Table 2 contains the parameter assumptions as well as the resulting expected default rate for each country. We note that for the AAA, AA, and A rated countries, the expected default rates still exceed the historical averages on equally-rated corporate bonds, which are 0.08%, 0.18%, and 0.50%, respectively. The rest of the simulation exercise is identical to our pessimistic benchmark case described above. The first (three) principal components of default realizations explain 50% (84%) of the common variation across countries.

The main result is that the 70% ESBies tranche now sustains losses in only 0.11% of periods, as opposed to the 0.80% of periods under our benchmark calibration. Even an 80% ESBies tranche would only sustain losses with probability 0.5% compared to 2.7% in the benchmark. The capital guarantee, which covered losses between the 99.2th and 99.7th percentiles in the benchmark case, becomes redundant in this historical case. An 11.9% capital guarantee would cover losses between the 99.88th and 99.96th percentile of the loss distribution.

**Table 2. Defaults and Losses on the Collateral Pool of ESBies in a Historical Scenario** *Notes:* Column 1 reports the country and the Moody’s and S&P credit rating as of August 2011. The second column reports the weight of each country in the collateral pool. Column 3-5 report the idiosyncratic (not the total) default rate conditional on aggregate state 1, 2, and 3. Columns 6-8 report the loss given default conditional on default occurring in aggregate state 1, 2, and 3. Column 9 reports the unconditional expected default rate which combines the idiosyncratic probabilities of default in columns 3-5, the auxiliary assumptions on default of Spain, Italy, France, and Germany, and the probability distribution over aggregate states.

**NOTES**

** ^{1}**We use data on recovery rates and 5-year cumulative default rates from a March 2008 Moody’s Global Credit Research report titled “Sovereign Default and Recovery Rates, 1983-2007,” Exhibit 8 and 9.