Our previous results show that the number of immunized portfolios that beat the expected rate of return (given by the initial spot rate for the planning horizon)—that is the number of positive returns obtained from equation (9)—is maximized when using a stochastic duration measure instead of its deterministic counterpart. This happens for all immunization periods, sub-samples, portfolio formation strategies, and model dimensions tested.
[Please insert Figure 2 about here]
To evaluate the robustness of the previous empirical findings, we perform a stochastic dominance analysis to all the combinations tested, i.e. to the one-, two-, and three-factor HJM parametrizations implemented using both deterministic and stochastic duration mea-sures. Based on the cumulative distribution of the percentage returns (instead of average realized returns), this analysis provides additional evidence that immunization strategies based on stochastic duration measures lead to a superior performance. The results are plotted in Figure 2. Figure 2.A shows the cumulative probability frequencies for the one-year immunization period, using one-, two-, and three-factor HJM parametrizations of the yield curve. Figures 2.B and 2.C present the same frequencies for the three-, and five-year immunization periods, respectively.
Figure 2 shows that the probability of achieving a better immunization result is always greater with the stochastic rather than with the Fisher-Weil duration measure. For the three- and five-year immunization periods, the use of a stochastic HJM three-factor dura-tion guarantees first-order stochastic dominance over the use of the Fisher-Weil duradura-tion measure. For the one-year holding period, no clear first-order stochastic dominance is noted.
Nevertheless, second-order stochastic dominance in favor of the stochastic duration approach is observed for this immunization period. Therefore, this results confirm that the superiority of the stochastic duration performance grows with the length of the immunization period considered.
[Please insert Table 5 about here]
In opposition with Agca (2005), we have tested different immunization rules using real market data instead of simulated bond prices. Therefore, the empirical evidence found against the use of deterministic duration measures cannot be attributed to its inconsistency with the interest rate HJM model dynamics adopted. However, it is then possible to argue that the yield curve estimation errors reported in Table 1 might be large enough to affect the performance of the risk measures tested.
In order to test if the outcomes of Tables 3 and 4 depend on the misfit of the spot yield curve, we estimate the following regression model for different investment horizons, different risk measures and different model dimensions:
M ean|ER (t)| = α + β × MAP E (t) + ε (t) , (11) where M ean|ER (t)| is the mean absolute excess return obtained from all immunization strategies starting at the end of month t, M AP E (t) is the corresponding end-of-month t mean absolute percentage pricing error associated to the yield curve estimation, and ε (t) is an error regression term. The results are reported in Table 5, and show that even at a 10 percent significance level, we cannot reject the null hypothesis H0: β = 0 for all immunization periods, for all duration measures, and under any model dimension. Consequently, we may conclude that our results are robust against the noise associated to the estimation of the spot interest rates.10
7 Conclusions
The duration measures of Macaulay and Fisher-Weil are still widely used in practice, even though new risk measures have emerged through the rapid development of stochastic term structure models. Moreover, the empirical evidence on the performance and efficiency of these two groups of risk measures is mixed, with some empirical studies finding that tradi-tional deterministic duration measures perform at least as well as the stochastic risk measures derived from no-arbitrage term structure models.
The stochastic term structure model used in this study belongs to the popular arbitrage-free framework proposed by Heath et al. (1992). The hypothesis under study is that the risk measures implied by stochastic term structure models are more appropriate for immunization
10Following D´ıaz et al. (2008), we have also used the changes in the target rate of return (end-of-month
one-, three-, and five-year zero coupon yields) instead of MAPE as the explanatory variable, and the changes in the mean absolute excess returns as the explained variable, but the results are broadly similar to the ones presented in Table 5. To save space, those results are not reported here but are available upon request.
purposes than their traditional counterparts. For this purpose, this study compares the immunization performance of the Fisher-Weil duration with the stochastic risk measures derived from an yield curve parametrization that is consistent with a Gaussian and multi-factor HJM term structure model, as suggested by Bj¨ork and Christensen (1999). A duration matching strategy is considered, and we run a total of 38,370 portfolios among random, bullet and barbell strategies, during a sample period that ranges from January 2000 to December 2010.
The results obtained clearly suggest that stochastic risk measures outperform the tradi-tional Fisher-Weil duration for immunization purposes. Furthermore, we also conclude that the superiority of the stochastic duration measures is better captured under a multi-factor term structure model and for longer immunization periods. Note that these conclusions are based not only on the analysis of average excess returns and downside risks for each duration approach, but are also justified by a stochastic dominance analysis. We have also compared the influence of portfolio formation techniques on the average final return of the bond port-folio, and found that bullet portfolios tend to provide a better immunization performance than barbell strategies.
Our conclusions are in line with, for instance, Fooladi and Roberts (1992), Soto (2001) or Agca (2005) in respect to the prevalence of additional returns due to portfolio formation techniques, but diverge from Wu (2000) and Agca (2005) when comparing the performance of stochastic and deterministic duration measures. This divergence arises because Wu (2000) and Agca (2005) only use a single-factor term structure model while we test multi-factor HJM specifications.
Note that our analysis avoids any model bias that might favor HJM risk measures be-cause we compare all risk measures using real market data. Moreover, we also show that the superior performance of the stochastic duration measures is not driven by yield curve estimation errors.
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Table1:Yieldcurveestimationerrors. January2000 to December2010 January2000 to July2007 August2007 to December2010 One factor
Two factors Three factors One factor Two factors Three factors One factor Two factors
Three factors Average12.59.36.19.97.73.411.38.78.2 St.Deviation9.78.05.710.29.72.17.66.36.3 Median12.911.69.512.07.55.812.69.89.7 Minimum2.41.30.62.41.30.69.64.44.3 Maximum21.317.416.415.113.311.221.317.416.4 Table1showsthesummarystatisticsoftheend-of-monthabsolutepercentagepricingerrors(expressedinbasispoints)associatedtotheestimationof theGermangovernmentzero-couponyieldcurve.TheGermanspotyieldcurveisestimatedbetweenJanuary2000andDecember2010,andthrough theconsistentHJMparametrizationproposedinequation(5),whichisimplementedusingone,two,andthreefactors.
Table2:Descriptivestatisticsfortheone-,three-,andfive-yearzero-couponbondyields. OnefactorTwofactorsThreefactors 1year3years5years1year3years5years1year3years5years PanelA:Fullperiod(January2000-December2010) Average3.09%3.65%4.22%2.97%3.52%4.03%2.78%3.17%3.51% StandardDeviation1.29%0.97%0.77%1.21%0.98%0.75%1.18%0.97%0.73% Median3.16%3.64%4.15%3.03%3.58%4.05%2.99%3.36%3.84% Minimum0.57%1.55%2.51%0.51%1.20%2.11%0.25%0.88%1.58% Maximum5.23%5.31%5.68%5.20%5.31%5.54%5.03%5.18%5.20% PanelB:Before-crisisperiod(January2000-July2007) Average3.37%3.87%4.39%3.36%3.85%4.30%3.18%3.53%3.83% StandardDeviation0.92%0.76%0.69%0.91%0.76%0.63%0.87%0.72%0.62% Median3.41%3.72%4.25%3.37%3.70%4.19%3.14%3.48%3.66% Minimum1.96%2.53%3.13%2.02%2.48%3.11%1.88%2.16%2.52% Maximum5.23%5.31%5.68%5.20%5.31%5.54%5.03%5.18%5.20% PanelC:Crisisperiod(August2007-December2010) Average2.52%3.19%3.85%2.16%2.81%3.46%1.95%2.39%2.83% StandardDeviation1.44%0.95%0.56%1.39%0.79%0.56%1.27%0.68%0.43% Median1.88%2.81%3.77%1.35%2.47%3.27%0.94%1.91%2.51% Minimum0.57%1.55%2.51%0.51%1.20%2.11%0.25%0.88%1.58% Maximum4.73%4.78%4.82%4.59%4.63%4.62%4.55%4.49%4.49% Table2showsthesummarystatisticsoftheend-of-monthone-,three-,andfive-yearGermanspotratesestimated,betweenJanuary2000and December2010,throughtheconsistentHJMparametrizationproposedinequation(5)thatisimplementedusingone,two,andthreefactors.
Table3:Descriptivestatisticsoftheexcessreturnsbydurationapproach,modeldimension andimmunizationhorizon. OnefactorTwofactorsThreefactors FWDSDFWDSDFWDSD PanelA:One-yearimmunizationperiod(excessreturnsinbasispoints) Average14.215.015.416.517.118.8 StandardDeviation29.528.929.628.629.428.3 Median9.19.610.411.510.511.8 Minimum-474.8-237.1-475.0-219.7-213.5-168.9 Maximum473.6406.5473.9409.4474.6405.9 MeanComparison(p-value)0.17100.12220.0850 No.Obs4,8984,8864,8984,8814,8984,889 AverageBullet28.028.229.633.929.734.9 AverageBarbell7.112.77.613.67.913.9 PanelB:Three-yearimmunizationperiod(excessreturnsinbasispoints) Average19.720.019.920.320.122.2 StandardDeviation18.218.618.017.918.017.5 Median15.115.215.015.215.515.7 Minimum-15.0-15.0-17.0-17.6-16.8-10.8 Maximum113.8117.6114.7151.2116.3143.8 MeanComparison(p-value)0.10560.04810.0430 No.Obs1,2361,2321,2361,2281,2361,223 AverageBullet32.735.833.935.136.036.8 AverageBarbell13.312.915.213.114.915.3 continuesonnextpage
Table3:(continued) OnefactorTwofactorsThreefactors FWDSDFWDSDFWDSD PanelC:Five-yearimmunizationperiod(excessreturnsinbasispoints) Average22.824.822.526.124.727.7 StandardDeviation17.317.216.716.116.816.6 Median17.519.317.520.417.721.7 Minimum-16.2-13.7-16.1-17.7-14.9-14.7 Maximum69.169.473.376.675.777.6 MeanComparison(p-value)0.08360.03760.0246 No.Obs278267278264278264 AverageBullet43.445.943.547.043.949.6 AverageBarbell16.618.116.719.816.620.6 Table3summarizesthedescriptivestatisticsoftheexcessreturnsdefinedthroughequation(9),expressedinbasispoints,andgeneratedbyduration matchedportfoliosformedusingtheFisher-Weilriskmeasure(FWD)andthestochasticduration(SD)obtainedimplicitlyfromequation(7).Both riskmeasuresareimplementedundertheGaussianHJMmodel(3),usingone,two,andthreefactors.PanelsA,B,andCshowtheexcessreturn statisticsfromone-,three-andfive-yearimmunizationperiods,respectively.The“MeanComparison(p-value)”yieldsthelevelofsignificancetoreject theonetailt-testfortheindependenceofsampleaverageexcessreturnsproducedbydeterministicandstochasticdurationmeasures.Asusual,the meandifferenceequalszerounderthenullhypothesis.Theaverageexcessreturnsofbulletandbarbellportfoliosarereportedatthebottomofeach panel.Bulletandbarbellportfoliosareselectedateachstartingdate,andrefertotheportfolioswiththesmallestandlargestdurationdifferences, respectively,betweenthetwocomponentbonds.
Table 4: Percentage of negative excess returns by duration approach, model dimension and portfolio formation strategy.
One factor Two factors Three factors
FWD SD FWD SD FWD SD
Panel A: One-year immunization period Random portfolios
Full period 19.7% 12.1% 15.2% 8.6% 12.6% 8.4%
Before crisis 18.1% 11.6% 13.4% 8.2% 11.7% 8.2%
During crisis 21.4% 17.9% 16.1% 12.6% 14.8% 10.9%
Bullet 10.6% 9.6% 8.5% 7.4% 8.5% 7.4%
Barbell 12.8% 12.8% 11.7% 10.6% 11.7% 10.6%
Panel B: Three-year immunization period Random portfolios
Full period 16.0% 8.9% 13.4% 7.1% 9.7% 5.7%
Before crisis 13.3% 6.6% 11.2% 4.9% 8.5% 4.1%
During crisis 18.0% 14.2% 14.9% 11.7% 11.6% 8.6%
Bullet 8.3% 6.3% 8.3% 6.3% 8.3% 4.2%
Barbell 10.4% 8.3% 8.3% 8.3% 8.3% 6.3%
Panel C: Five-year immunization period Random portfolios
Full period 5.3% 3.6% 3.2% 2.9% 3.2% 2.4%
Before crisis 5.3% 3.6% 3.2% 2.9% 3.2% 2.4%
During crisis n.a. n.a. n.a. n.a. n.a. n.a.
Bullet 8.0% 8.0% 8.0% 4.0% 8.0% 4.0%
Barbell 8.0% 8.0% 8.0% 4.0% 8.0% 4.0%
Table 4 presents the percentage of duration matched portfolios with negative excess returns—as defined in equation (9)—which are formed using the Fisher-Weil risk measure (FWD) and stochastic durations (SD).
Both risk measures are implemented under the Gaussian HJM model (3), using one, two, and three factors.
Panels A, B, and C show the negative excess returns for one-, three-, and five-year immunization periods, respectively. The full period covers January 2000 to December 2010, the before crisis period extends from January 2000 until July 2007, while the crisis period ranges from August 2007 to December 2010. Since
the crisis period on our database lasts for only 3.3 years, results are not available (n.a.) for the
five-year immunization period. Bullet and barbell portfolios are selected at each starting date, and refer to the portfolios with the smallest and largest duration differences, respectively, between the two component bonds.
Table5:Theinfluenceofyieldcurveestimationerrorsontheimmunizationperformance. OnefactorTwofactorsThreefactors FWDSDFWDSDFWDSD PanelA:One-yearimmunizationperiod No.Obs120120120120120120 Coefficient0.05590.06250.09530.08790.04670.0454 (1.0406)(1.2098)(0.8557)(0.8004)(0.6969)(0.6535) R-Squared0.0100.0140.0220.0180.0050.004 PanelB:Three-yearimmunizationperiod No.Obs969696969696 Coefficient0.05830.06480.04480.04520.14770.0068 (0.7372)(0.8027)(0.6612)(0.6591)(0.3979)(0.0180) R-Squared0.0100.0110.0080.0080.0030.000 PanelC:Five-yearimmunizationperiod No.Obs727272727272 Coefficient0.22280.00830.25790.04970.31550.1223 (1.3704)(0.0520)(0.3647)(0.3278)(0.4896)(0.1773) R-Squared0.0700.0000.1010.0040.0090.001 Table5presentstheregressions(11)using,asdependent,theaverageabsoluteexcessreturnsarisingfromtheimmunizationstrategies—implemented viaFisher-Weil(FWD)andstochastic(SD)durationmeasures—thathavestartedineachend-of-month,and,astheindependent,theend-of-month meanabsolutepercentagepricingerrorsproducedbytheestimationofthespotyieldcurvethroughtheparametrization(5)thatisconsistentwith one-,two-,andthree-factorHJMmodels.PanelsA,B,andCshowtheresultsforone-,three-,andfive-yearimmunizationperiods,respectively.The t-statisticsarepresentedinparenthesisbelowtherespectivecoefficient.
Figure1:One-,three-,andfive-yearspotinterestrates. 0%1%2%3%4%5%6% Interest Rate
Figure 1.A: One factor One yearThree yearsFive years
0%
1%
2%
3%
4%
5%
6% Interest Rate
Figure 1.B: Two factors One yearThree yearsFive years
0%
1%
2%
3%
4%
5%
6% Interest Rate
Figure 1.C: Three factors One yearThree yearsFive years Figure1plotstheevolutionoftheone-,three-,andfive-yearspotinterestratesduringthesampleperiod(January2000toDecember2010).The shadedareacorrespondstothesub-periodbetweenAugust2007,thebeginningofthefinancialcrisis,andNovember2009,whentheescalationofthe GreekcrisispronouncedthebeginningoftheEMUsovereigndebtcrisis.
Figure 2: Comparison of stochastic and deterministic duration approaches through first order stochastic dominance.
Figure 3.A: One year immunization period Figure 3.B: Three years immunization period Figure 3.C: Five years immunization period
0.0
Figure 2 shows the cumulative probability distribution of excess returns—as defined by equation (9)—for different immunization periods, and considering one-, two-, and three-factor HJM parametrizations of the yield curve. The line labeled as “FWD” corresponds to the use of the Fisher-Weil deterministic duration measure, while the solid line labeled as “SD” is generated by a stochastic duration measure.