**Appendix B. Robustness**

### *Appendix B.1. Comparing with Futures as Hedge*

The comparison of hedge effectiveness using SBBS versus BTP and/or Bund futures can be considered by reference to Table A1. The first notable outcome is that the senior SBBS is better than the Bund future even in the case of hedging German 10-year bond risk. The returns on a portfolio of Bund and an optimal hedge position in Bund futures has a standard deviation of around 67% and 64% respectively of the unhedged position in the SDC period and the recovery, respectively. The portfolio involving the senior SBBS by comparison has a relative standard deviation of 32% and 27% in these periods (the relative VaR in both periods is also much lower when the SBBS is used to hedge). Interestingly, the BTP future is a better hedge for the Italian bond than any of the SBBS individually, but a combination of SBBS achieves similar hedge effectiveness.

Overall, the Bund futures contract is a weaker hedge for most sovereign bond positions compared with SBBS. Where SBBS are ineffective as hedges for smaller sovereigns (such as IE and PT), the futures are also very ineffective. These sovereigns have a lot of idiosyncratic unhedgable risk, but their risks are significantly reduced through diversification.


**Table A1.** Futures-based hedge effectiveness SDC and Recovery.

Note: Using two different metrics, this table shows the percentage change in risk exposures achieved by hedging a position in a particular sovereign bond using a position in futures on Bunds, BTPs and a combination of both. Issuers of the bonds are indicated at the beginning of each row as follows; AT (Austria), BE (Belgium), DE (Germany), ES (Spain), FI (Finland), FR (France), GR (Greece), IE (Ireland), IT (Italy), NL (Netherlands) and PT (Portugal). The first row for each case contains the percentage change in risk due to hedging where risk is measured as *standard deviation*. The second row for each case contains the percentage change in risk achieved through hedging where risk is measured as the range between the 5th and 95th quantiles of the returns distribution (implying VaR bounds). Columns of results permit a comparison of the risk reductions for the different hedges across two sub-sample periods.

### *Appendix B.2. Hedge Effectiveness at Other Maturities*

Table A2 presents a comparison of hedge effectiveness at the 10-year term-to-maturity with those at five and two years to maturity (for conciseness only the results for hedging with all three SBBS are shown). In the pre-crisis period hedge effectiveness tends to decline as term shortens. The standard deviation of returns on the hedged positions relative to that of the unhedged positions (Rows (i)) increases on average from 0.28 at the 10 year term to 0.36 and 0.49 at the 5- and 2-year terms respectively. The relative value-at-risk comparison (Rows (ii)) increase on average less dramatically than the relative standard deviations (from 0.20 to 0.25 and 0.36, respectively). The average increase in these ratios in the pre-crisis period is a reasonably good indication of what happens at the individual sovereign level (the case of Finland is somewhat of an outlier).

The middle segmen<sup>t</sup> of Table A2 depicts the hedge effectiveness measures across terms-to-maturity for the period of the Sovereign Debt Crisis. Here, it is not so clear that any significant change occurs across the different terms on average, but there is more heterogeneity across sovereigns. The average of the ratio of standard deviations of returns on the hedged position relative to the unhedged, shown in the second last row of the table, move from 0.71 to 0.67 and then to 0.72. Relative VaRs also remain quite flat, moving from 0.67 at the 10-year term to 0.68 and 0.73 at the 5- and 2-year terms, respectively. The declines in these ratios tend to be more acute for ES, IT and PT. Significant increases occur (particularly for the 2-year term) for BE, DE, FI and NL. The hedge effectiveness results for the Recovery period show similar levels on average of the two main risk ratios between the 5- and 10-year maturities. A slightly sharper rise occurs in these ratios for the 2-year term and this is mainly explained by the FR case and two of the smaller sovereign markets (BE and FI).


**Table A2.** Hedge effectiveness: 10-, 5- & 2-year terms-to-maturity

Note: Using two different metrics, this table shows the percentage change in risk exposures achieved by hedging a position in a particular sovereign bond using a position in one or more tranches of a 70:20:10 Sovereign Bond Backed Securitisation (three maturities are considered; 2-year, 5-year and 10-year). Issuers of the bonds are indicated at the beginning of each row as follows; AT (Austria), BE (Belgium), DE (Germany), ES (Spain), FI (Finland), FR (France), GR (Greece), IE (Ireland), IT (Italy), NL (Netherlands) and PT (Portugal). The first row for each case contains the percentage change in risk due to hedging where risk is measured as *standard deviation*. The second row for each case contains the percentage change in risk achieved through hedging where risk is measured as the range between the 5th and 95th quantiles of the returns distribution (implying VaR bounds). Columns of results permit a comparison of the risk reduction for three bond maturities and acrossthreesub-sampleperiods.

### *Appendix B.3. Hedge Effectiveness under Higher Incidence of Extreme Losses*

Table A3 compares hedge effectiveness using a t-copula rather than a Gaussian copula to trigger defaults. There is almost no difference for any sovereign (or on average) in the hedge effectiveness measures over the pre-crisis sample. The largest increase in the ratio of risks occurs for the case of GR (showing a very minor 4-point rise). For the Sovereign Debt Crisis and Recovery periods, a similar small change is apparent for all sovereign bonds except those of Germany (and Finland for the SDC period).


**Table A3.** Hedge effectiveness: Gaussian versus T-copula default trigger.

Note: Using two different metrics, this table shows the percentage change in risk exposures achieved by hedging a position in a particular sovereign bond using a position in one or more tranches of a 70:20:10 Sovereign Bond Backed Securitisation where the tranche yields have been estimated with a Gaussian and a t-Copula simulator. Issuers of the bonds are indicated at the beginning of each row as follows; AT (Austria), BE (Belgium), DE (Germany), ES (Spain), FI (Finland), FR (France), GR (Greece), IE (Ireland), IT (Italy), NL (Netherlands) and PT (Portugal). The first row for each case contains the percentage change in risk due to hedging where risk is measured as *standard deviation*. The second row for each case contains the percentage change in risk achieved through hedging where risk is measured as the range between the 5th and 95th quantiles of the returns distribution (implying VaR bounds). Columns of results permit a comparison of the riskreductionusingtwowaysofestimatingthetrancheyieldsandacrossthreesub-sampleperiods.
