**5. Results**

### *5.1. Effectiveness of Hedging without Diversification*

The hedge effectiveness results, in terms of the percentage change in risk (usually reduction in risk), are shown for each of three sub-sample periods in Tables 1–3 and the daily returns on the hedged and unhedged positions, for the case of hedging with just the senior and for the case of hedging with a mixture of the senior and mezzanine, are shown in Figures 3–5.<sup>11</sup>

The visual evidence indicates that hedging is generally effective in the pre-Sovereign Debt Crisis period in reducing the volatility of returns (with some isolated exceptions). Hedging is not effective for high-risk sovereigns during the height of the SDC but effectiveness returns to some extent during the recovery. In general, the combined hedge works better than the single hedge in the crises and recovery periods. Figure 3 reveals that hedging is quite consistently effective for three countries (Germany (DE), France (FR) and the Netherlands (NL)). In these cases, the composite hedge seems to eliminate the occasional blips present in the single hedge case. Similarly consistent levels of effectiveness are found for AT and FI (not displayed).

Figure 4 shows the cases of (BE, ES and IT). This clearly reveals how idiosyncratic the experiences of each of the high-risk periphery countries was during the crisis (making them difficult to hedge using SBBS). It is interesting that the composite hedge (senior and mezzanine) works better than the single hedge during the crisis and recovery (apart from one particular day). This tends to improve further with the inclusion of the junior SBBS as a hedge instrument (this more general case is not displayed in the figure but can be seen from the tabulated results discussed below).

Figure 5 shows the more volatile cases of (GR, IE and PT). Here again, there is evidence of hedge ineffectiveness during the crisis with improvement only obvious during the recovery for IE and PT. Again, the composite hedge is better than the single hedge during the recovery for these countries and is particularly good in protecting from the more extreme movements. Although hedging is least effective in reducing risk in the case of the smaller markets, their idiosyncratic risk is more amenable to control using diversification strategies (this is addressed below).

<sup>10</sup> A similar analysis employing dynamic conditional correlation methods, compiling hedge ratios using conditional variances and covariances in line with the relations presented in Appendix A, did not in fact lead to significantly different hedge effectiveness ratios. In addition, the combined use of the methods of Gibson et al. (2017) and a stochastic volatility modelling approach designed to address the presence of isolated outliers, due to Chan and Grant (2016), also failed to change the conclusions drawn from the more straightforward application of a rolling linear regression approach. These alternative methods may however lead to improvements when used at higher frequency with regular updating. Extending the analysis to such a frequency is beyond the scope or needs of the present study.

<sup>11</sup> The cases of Austria (AT) and Finland (FI) are not shown above but are quite similar to the case of NL.


**Table 1.** Hedge effectiveness: Previous to the Sovereign Debt Crisis.

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 in the pre-SDC period (this is used as an indicator of *hedge effectiveness* in the text). 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 are arranged in three broad categories. The first category concerns the use of a single SBBS tranche in the hedge. The second category concerns the use of a pair of SBBS in hedging. The last category involves the use of all three SBBS in the hedge. The most effective hedge instrument/combination within the first two categories is highlighted with colour.




**Table 2.** *Cont.*

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 during the SDC period (this is used as an indicator of *hedge effectiveness* in the text). 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 are arranged in three broad categories. The first category concerns the use of a single SBBS tranche in the hedge. The second category concerns the use of a pair of SBBS in hedging. The last category involves the use of all three SBBS in the hedge. The most effective hedge instrument/combination within the first two categories is highlighted with colour.

**Table 3.** Hedge effectiveness Post-Sov-debt-crisis.


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 in the post-SDC period (this is used as an indicator of *hedge effectiveness* in the text). 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 are arranged in three broad categories. The first category concerns the use of a single SBBS tranche in the hedge. The second category concerns the use of a pair of SBBS in hedging. The last category involves the use of all three SBBS in the hedge. The most effective hedge instrument/combination within the first two categories is highlighted with colour.

**Figure 3.** Single and Composite Hedging (DE, FR, NL). Note: This figure facilitates a comparison of the dispersion of returns on hedged long positions in German, French and Dutch sovereign bonds respectively with the dispersion of returns on the same sovereign bonds without hedging. The first column of figures concern the case of hedging the long positions using only the senior SBBS. The second column of figures concerns the case of hedging the individual bond positions with both the senior and mezzanine SBBS. The returns are measured in basis points (left axis) and the bonds considered are those with a 10 year term-to-maturity. The cases of Austria (AT) and Finland (FI) are not shown above but are quite similar to the case of NL.

**Figure 4.** Single and Composite Hedging (BE, ES, IT). Note: This figure facilitates a comparison of the dispersion of returns on hedged long positions in Belgian, Spanish and Italian sovereign bonds respectively with the dispersion of returns on the same sovereign bonds without hedging. The first column of figures concern the case of hedging the long positions using only the senior SBBS. The second column of figures concerns the case of hedging the individual bond positions with both the senior and mezzanine SBBS. The returns are measured in basis points (left axis) and the bonds considered are those with a 10 year term-to-maturity.

**Figure 5.** Single and Composite Hedging (GR, IE, PT). Note: This figure facilitates a comparison of the dispersion of returns on hedged long positions in Greek, Irish and Portuguese sovereign bonds respectively with the dispersion of returns on the same sovereign bonds without hedging. The first column of figures concern the case of hedging the long positions using only the senior SBBS. The second column of figures concerns the case of hedging the individual bond positions with both the senior and mezzanine SBBS. The returns are measured in basis points (left axis) and the bonds considered are those with a 10 year term-to-maturity.

The tabulations of risk reductions can provide some more depth to the graphical analysis. The pre-Sovereign Debt Crisis period comparisons are in Table 1 and this is clearly a period where hedge effectiveness is high for all countries (at least for some of the seven hedges). The best hedges for each sovereign are highlighted in colour for the cases of 1-SBBS and 2-SBBS as hedges. In the case of the single hedge, it is the senior-SBBS that gives the best protection (the percentage reduction for the best single hedges—based on standard deviation of daily returns—ranges from 79% for DE to 36% for GR). In almost all cases, the 2-SBBS hedge provides some marginal improvement in hedge effectiveness over the 1-SBBS hedge case (all percentage reductions above 50%). In many cases the 3-SBBS hedge is best overall but this may not always be worthwhile from a cost and operational perspective.

Table 2 shows summary statistics for comparisons of risk changes through hedging during the sovereign debt crisis period. For the 1-SBBS hedge, only DE remains well hedged using the senior SBBS (reducing risk by 68%). Roughly half of the risk is avoided by single-SBBS hedging for the case of FI and NL while the remaining sovereigns are clearly not amenable to single-SBBS hedging in this crisis period. Moving to 2-SBBS or 3-SBBS hedging generally gives rise to some small risk reduction for most sovereigns relative to the single-SBBS case. Table 3 covers the recovery period (from Mario Draghi's London Speech until the end of the last quarter of 2016). By use of composite hedging, it is usually possible to reduce risks by half or more. The exceptions remain GR and PT.

### *5.2. Post-Hedging Diversification of Risks*

Once risks have been hedged, there is potential for dealers to diversify remaining risks by operating simultaneously across many of the sovereign markets. At each moment of a trading day (and at the end of each day), it is likely that a dealer will have a portfolio of outstanding positions in many markets. The individual net hedged-positions are likely to be much smaller than unhedged positions (e.g., if the hedge-ratio is close to 1 then the individual net hedged-positions will be minor). These net positions in individual sovereigns could be subject to further netting (i.e., if some are net-long and others are net-short). Risk reduction from this additional type of netting is not included in the following analysis (i.e., portfolios will be restricted to be hedged long-positions in the components). This most probably implies a significant underestimate of the risk reduction that could be achieved by diversification.

The upper panel of Figure 6 compares returns on a cross-country portfolio of hedged and unhedged positions for the pre-SDC period. In this case, it is assumed that the portfolio results from an equal weighting on the underlying sovereigns. Firstly, it is clear that the portfolio of hedged positions has a much smaller dispersion than the portfolio of unhedged components. Despite the equal weighting of components and the equal capital exposure assumption, diversification reduces risks much more for the portfolio with hedged positions and this is due to the fact that there are mostly idiosyncratic risks surviving in the hedged case while the unhedged case will involve considerable systematic risk.

The lower panel of Figure 6 concerns the more volatile conditions of the SDC and the recovery. In this case, there is not much benefit from the hedging because most risk is actually idiosyncratic. This means that diversification is equally effective in the case of the unhedged portfolio. In general, the risks are much lower than risks for typical single sovereigns during this period. Hedging starts to matter again during the recovery.

**Figure 6.** Portfolio returns with and without hedged components. Note: This figure facilitates a comparison of the dispersion of returns on a portfolio of hedged long positions in 11 sovereigns with the dispersion of returns on the same portfolio of long positions without hedging. The first column of figures concern the case of hedging the components of the portfolio of long positions using only the senior SBBS. The second column of figures concerns the case of hedging the individual bond positions with both the senior and mezzanine SBBS. The returns are measured in basis points (left axis) and the bonds considered are those with a 10 year term-to-maturity. The returns are measured in basis points (left axis).

A related comparison is that of the dispersion of returns on the safest single sovereign asset and those of the portfolio of hedged sovereign positions (the German 10 year Bund returns are used for this discussion, but these are almost indistinguishable from the returns on the similarly safe senior SBBS). Figure 7 displays these returns through time for the pre-SDC period and the sample from the start of the SDC through to the more recent recovery. It is very obvious that the portfolio of hedged sovereigns has an extremely low risk level when compared with the most safe sovereign (especially in the pre-SDC period). This demonstrates the effectiveness of diversification in reducing risks when risks are idiosyncratic and not too extreme. The effectiveness of diversification is compromised from the start of the SDC, but there is still some significant reduction in risk for the majority of the sample (with a small number of extreme outliers). For normal risk levels, diversification is a strong driver of risk reduction.

**Figure 7.** German 10-year returns compared with those from an 11-country portfolio of sovereigns hedged with SBBS. Note: This figure facilitates a comparison of the dispersion of returns on a portfolio of hedged long positions in 11 sovereigns with the dispersion of returns on a single sovereign bond that is widely considered to be the safest sovereign bond investment in the euro area (namely the German Bund). The long positions are hedged using the senior SBBS. The returns are measured in basis points (left axis) and the bonds considered are those with a 10 year term-to-maturity. (**a**) shows the comparison for the period previous to the Sovereign Debt Crisis while (**b**) covers the period of the Sovereign Debt Crisis and recovery. The hedged portfolio has returns with a dispersion which is much lower than the safe haven asset in the non-crisis sub-sample. Even in the crisis and recover periods, the hedged portfolio compares favourably with the safe asset investment in terms of dispersion of returns.

Table 4 broadens the analysis of the relative risks following diversification to the case of a size-weighted portfolio and to other maturities. These results pertain to the case where the senior SBBS is used as the hedge instrument.<sup>12</sup> The tabulated results for the 10-year bonds confirm what was clear from the graphical analysis. In the pre-SDC period, most risk measures indicate a reduction of around 70% in risk due to a combination of diversification and hedging relative to diversification on its own. It also confirms that risks after hedging and diversifying are less than a third of those faced by investing in the lowest risk sovereign (close to what is the maximum reduction possible when investing the same capital in 11 assets with similar idiosyncratic risks rather than 1). In this period (for the 10 year term), there is little difference between the equal and size-weighted portfolio results.

For the 10 year term, there is little benefit from hedging before diversifying during the Sovereign Debt Crisis period (the ratio of the risks on portfolio returns when components are hedged is roughly equal to the risks with unhedged components which is tantamount to saying that almost all risks are idiosyncratic). Diversification reduces risks again in the Recovery period and this is particularly true for the size-weighted portfolio. Risk is relatively high for the equally-weighted portfolio compared with the German bond for the Sovereign Debt Crisis period, but otherwise risk is similar to or less than that of the German for the SDC and Recovery, respectively.

In the case of the 5-year term, risks are reduced by around 45% in the pre-SDC period via a combination of hedging and diversification relative to just diversification (the extra risk reduction achieved in the 2 year case is similar to that of the 10 year). As with the 10 year case, hedging does not achieve significant extra benefit in terms of risk reduction for the 5-year bonds during the SDC (and this is also true of the two year term-to-maturity). There is about a ten percentage point difference in the risk reduction due to hedging during the Recovery between the 2 and 5 year terms with more

<sup>12</sup> The weights used in the analysis are related to the size of the individual sovereign float relative to that of the total of 11 sovereigns in the euro area market (weights are provided in the notes for Table 4).

reduction achieved in the 5 year category. As for comparison with the German sovereign, an equally weighted portfolio of sovereigns has risk of 1.52 and 3.52 times the volatility of the German bond (for the 5 and 2 year case, respectively) during the SDC (the associated relative VaR measures and the size-weighted comparisons show a smaller difference in these risks, but, in all cases, risks are greater than holding just the German). The Recovery period does not appear to be as positive for risk reduction through the combination of hedging and diversification for the 2 year term. However, portfolio risks remain low and are not much greater than those of the lowest risk sovereign.


**Table 4.** Diversification and hedge effectiveness.

Note: Using two different metrics, this table shows the percentage change in risk exposures achieved by hedging and diversifying (using the optimal hedge position in one or more tranches of a 70:20:10 Sovereign Bond Backed Securitisation). The portfolio of assets being hedged and diversified involves the 10 year bonds issued by; 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). The risk reduction is measured relative to the unhedged portfolio and relative to an investment in just the German Bund which is considered the safe haven asset. Columns of results permit a comparison across three sub-sample periods and for risk reductions achieved by diversifying across an equally-weighted and size-weighted portfolio, respectively.

### *5.3. Summary of Results for Extensions*

The possibility that similar hedge (and diversification) strategies might be just as good as SBBS has been ignored. The effectiveness of hedging using futures (in particular, German Bund and Italian BTP bond futures) is addressed in Appendix B. The principal finding here is that the senior SBBS is a much better hedge than Bund futures (even ignoring the fact that hedging with futures involves other additional costs and basis risks). Additionally, where SBBS are ineffective as hedges for smaller sovereigns (such as IE and PT), the futures are also very ineffective so there is no clear superiority in terms of hedge effectiveness using futures rather than SBBS.

Appendix B also contains details of the generalisation of the analysis to other maturities and to the case where SBBS yields are generated using a multivariate t-copula to trigger defaults. The outcome of these extensions is straightforward to summarise. Firstly, extending to shorter maturities leads to a general reduction in hedge effectiveness on average in the pre-crisis period and a less pervasive reduction in hedge effectiveness for the two-year maturity in the recovery period. Ultimately, this does not impact greatly on the risks that remain after diversification. Secondly, since the t-copula produces a higher incidence of extreme losses that more often spillover to mezzanine and senior tranches, the yield movements that result are less correlated with those of the German sovereign bonds than is the case with the Gaussian copula. The senior tranche therefore becomes less effective as a hedge for German bonds. Otherwise, the changes in hedge effectiveness due to use of the t-copula are insignificant.
