We test for the joint significance of the variables (Wald test61). At the 5 percent level we reject the null hypothesis that the variables are jointly equal to zero, for all markets except the Portuguese market.
In conclusion, it is important to reinforce that, considering the risk of data mining, we have limited ourselves to a reduced set of information variables. There might be others which could lead to better results.62 Our objective was not to determine the best model for each country. Instead, we concentrate only on a few variables which, due to economic reasons, should represent useful information in predicting bond returns.
conditions. The objective is to show if variables like term spread, IRW and others could have been used to enhance returns.
As a passive strategy we consider the strategy that involves always holding the market (stay in bonds). Another alternative strategy could involve always staying in cash and obtain a zero excess return (because this strategy is equivalent to rolling over the one-month risk-free rate). Instead, the dynamic strategy involves holding one unit of the bond (market) if its predicted premium is positive and none of the bond if its predicted premium is negative. This position is rebalanced monthly. As it is based on the sign of the predicted excess bond return, and not in its magnitude, this strategy involves a relatively low frequency of trading and no short-selling is allowed.63
Given that our sample period is not much long, we consider both in-sample and out-of-sample estimates of predicted excess bond returns (or risk premiums). In-sample estimation assumes that investors know in advance the statistical relation between the predetermined information variables and subsequent excess bond returns. The estimates are the fitted values from the regressions of excess bond returns on the information variables for the whole sample period. This approach might exaggerate the potential profitability of the dynamic strategy. A more reasonable approach is to forecast the next month’s excess bond return using only available historical data and to update the forecast each month using new data.64 We define, as our initial sample, the period February 1994 to December 1997. Thus, the estimates are one-step ahead forecasts
63 Another alternative would be a scaled strategy in which the size of the position in the bond market depends on the magnitude of the predicted excess returns. This is a more transaction-intensive strategy and it assumes that Government bonds may be shorted with the full use of proceeds and, consequently, the results would be less reliable (see Ilmanen, 1995).
64 Similarly to what we have done to calculate the out-of-sample R2(adj.) presented in Table 4.5.
based on rolling regressions that begin in January 1998 and use all the available historical information since February 1994.
We concentrate our analysis on the multiple regression model (as reported at the bottom of each panel in Table 4.5): the model with term spread, IRW and a January dummy. This model is chosen because its variables were found to be the most significant. The introduction of the two other variables did not seem to add much in terms of explanatory power, as well as presenting some problems of high correlations.
In Figure 4.2 we compare the annualized mean excess returns for the passive trading strategy with those of the dynamic strategy considering the in-sample estimation for the period February 1994 to December 2000.65 Only with the exception of the Portuguese market (for the all maturities index), with a very weak underperformance of about 0.035 percent, the excess returns of the dynamic strategy are superior to those of the passive strategy. The outperformance varies between 0.18 percent (for the German 1-3 maturity subsector) and 3.57 percent (for the Spanish 5+ maturity subsector). The longer maturity subsector has the higher outperformance.
65 As the passive strategy involves always holding the market, the reported means are equal to the annualized monthly values of Table 4.1.
Figure 4.2 - Annualized mean of excess bond returns (in percentage) over the period February 1994 to December 2000: passive versus dynamic strategy
1.11 1.81
0.92 1.42
0.82 1.22 1.13 1.30
0.22 0.77 0.0
2.0 4.0 6.0 8.0
Spain Italy France Germany UK
SBWGBI 1-3
Stay in bonds Dynamic Strategy
2.05 3.53
2.18 3.71
1.48
2.63 1.89 2.92
0.45 1.70
0.0 2.0 4.0 6.0 8.0
Spain Italy France Germany UK
SBWGBI 3-5
Stay in bonds Dynamic Strategy
3.43 7.00
3.65 5.93
2.43 4.51
2.81 4.56
1.97 4.86
0.0 2.0 4.0 6.0 8.0
Spain Italy France Germany UK
SBWGBI 5+
Stay in bonds Dynamic Strategy
4.41 4.38
2.43 4.81
2.31 3.78
1.95 3.42
2.11 3.23
1.47 3.81
0.0 2.0 4.0 6.0 8.0
Portugal Spain Italy France Germany UK
SBWGBI all
Stay in bonds Dynamic Strategy
To evaluate if this outperformance is consistent over the sample period, we also examined the cumulative performance of both strategies. Figure 4.3 to Figure 4.6 show how the profits from the strategies grow over time for the bond market indices. The dark line shows the cumulative excess returns of the dynamic strategy. As it also shows the relative performance of the dynamic strategy versus the cash market, whenever this line declines it means that the dynamic strategy underperforms the cash market. The other solid line shows the cumulative excess return of the passive “stay in bonds”
strategy. The dashed line shows the relative performance of the dynamic versus the passive strategy. The line representing a perfect foresight strategy should always be rising or flat (if the performance of the two strategies is equal). The relative performance of the dynamic strategy over the cash market increases gradually through most of the sample period. Most of the outperformance of the dynamic strategy relative to bonds occurred during the year of 1994, as bonds were avoided during a bear market.
Figure 4.3 - Excess returns for the passive strategy versus the dynamic strategy over the sample period February 1994 to December 2000, for the 1-3 years to
maturity subsector
Spain
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Stay in bonds Dynamic Dynamic vs Stay in bonds
Italy
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Stay in bonds Dynamic Dynamic vs Stay in bonds
France
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Stay in bonds Dynamic Dynamic vs Stay in bonds
Figure 4.3 – continued.
Note: The dark line shows the cumulative excess returns of the dynamic strategy. The other solid line shows the cumulative excess returns of the passive “stay in bonds” strategy. The dashed line shows the relative performance of the dynamic versus the passive strategy.
Germ any
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Stay in bonds Dynamic Dynamic vs Stay in bonds
UK
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Stay in bonds Dynamic Dynamic vs Stay in bonds
Figure 4.4 - Excess returns for the passive strategy versus the dynamic strategy over the sample period February 1994 to December 2000, for the 3-5 years to
maturity subsector
Spain
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Stay in bonds Dynamic Dynamic vs Stay in bonds
Italy
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Stay in bonds Dynamic Dynamic vs Stay in bonds
France
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Stay in bonds Dynamic Dynamic vs Stay in bonds
Figure 4.4 – continued.
Note: The dark line shows the cumulative excess returns of the dynamic strategy. The other solid line shows the cumulative excess returns of the passive “stay in bonds” strategy. The dashed line shows the relative performance of the dynamic versus the passive strategy.
Germ any
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Stay in bonds Dynamic Dynamic vs Stay in bonds
UK
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Stay in bonds Dynamic Dynamic vs Stay in bonds
Figure 4.5 - Excess returns for the passive strategy versus the dynamic strategy over the sample period February 1994 to December 2000, for the 5+ years to
maturity subsector
Spain
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Stay in bonds Dynamic Dynamic vs Stay in bonds
Italy
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Stay in bonds Dynamic Dynamic vs Stay in bonds
France
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Stay in bonds Dynamic Dynamic vs Stay in bonds
Figure 4.5 – continued.
Note: The dark line shows the cumulative excess returns of the dynamic strategy. The other solid line shows the cumulative excess returns of the passive “stay in bonds” strategy. The dashed line shows the relative performance of the dynamic versus the passive strategy.
Germ any
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Stay in bonds Dynamic Dynamic vs Stay in bonds
UK
0.0 0.2 0.4 0.6 0.8 1.0 1.2
Stay in bonds Dynamic Dynamic vs Stay in bonds
Figure 4.6 - Excess returns for the passive strategy versus the dynamic strategy over the sample period February 1994 to December 2000, for the broad all
maturities index
Portugal
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Stay in bonds Dynamic Dynamic vs Stay in bonds
Spain
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6
Stay in bonds Dynamic Dynamic vs Stay in bonds
Italy
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Stay in bonds Dynamic Dynamic vs Stay in bonds
Figure 4.6 – continued.
Note: The dark line shows the cumulative excess returns of the dynamic strategy. The other solid line shows the cumulative excess returns of the passive “stay in bonds” strategy. The dashed line shows the relative performance of the dynamic versus the passive strategy.
France
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Stay in bonds Dynamic Dynamic vs Stay in bonds
Germ any
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Stay in bonds Dynamic Dynamic vs Stay in bonds
UK
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
Stay in bonds Dynamic Dynamic vs Stay in bonds
The inferences for the risk-adjusted performance are even more favourable to the dynamic strategy, as shown in Table 4.6 below. The risk is measured by volatility, shortfall frequency and frequency of “being wrong”. The volatility (measured by the standard deviation of returns) of the dynamic strategy is, as should be66, lower than that of the static strategy. Contrary to what we found for the passive strategy, the T-test for the monthly returns of the dynamic strategy allows us to reject the hypothesis of a mean equal to zero in almost all the cases.67 Thus, the Sharpe ratio is clearly higher.
We can see that the dynamic strategy had some success in avoiding negative returns. The shortfall probability, defined as the frequency of negative realized excess returns, ranges between 16 and 25 percent, while the corresponding values for the passive strategy ranges between 29 and 45 percent. Investors following such a dynamic strategy avoid the bond market for several months and earn zero.
Following the information variables and trading according the predicted excess bond returns, leads to a correct decision with a frequency between 57 (Italy 1-3 years to maturity) and 72 percent (for Portugal and Spain all maturities). For most of the countries, this frequency is superior to 60 percent.
66 Since it has, at most, the same level of volatility as bonds, and the rest of the time it is invested in cash.
67 We have also performed a T-test for the difference in returns of the two strategies and, in several cases, the differential return is statistically significant.
Table 4.6 - Performance of passive and dynamic bond trading strategies (in-sample) for the period February 1994 to December 2000
This table compares the in-sample performance of a passive/static and a dynamic bond investment strategy for each country and for each maturity subsector. Panel A reports the results of the passive strategy while Panel B presents the results of the dynamic strategy. The passive strategy involves always staying in bonds. The dynamic strategy involves buying one unit of the bond (market) if its predicted premium is positive and none of the bond if its predicted premium is negative. The expected bond risk premium is the fitted value from the regressions of realized excess bond returns on term spread, IRW and the January dummy for the whole sample period. The table reports the annualized mean (in percentage) and standard deviation of the excess returns for each trading strategy, the T-test for the hypothesis of the monthly mean=0 (for the dynamic strategy), the annualized Sharpe ratio (the ratio of the mean and standard deviation), the shortfall frequency (observed frequency of negative excess returns) and, for the dynamic strategy, the frequency of forecasts with correct sign (observed frequency of predicting positive bond risk premiums when the realized premium is positive plus observed frequency of predicting negative bond risk premiums when the realized premium is negative).
*** Statistically significant at 1% ** Statistically significant at 5% * Statistically significant at 10%
Portugal Spain Italy France Germany UK Panel A - Static Strategy
SBWGBI 1-3
Mean 1.107 0.924 0.820 1.127 0.217
Standard Deviation 1.993 2.116 1.359 1.383 1.603
T-test mean=0 1.46 1.15 1.59 2.14 ** 0.36
Sharpe Ratio 0.555 0.437 0.604 0.815 0.136
Shortfall frequency 0.42 0.41 0.45 0.39 0.45
SBWGBI 3-5
Mean 2.053 2.181 1.483 1.890 0.446
Standard Deviation 3.816 3.984 2.832 2.735 3.097
T-test mean=0 1.42 1.44 1.38 1.82 * 0.38
Sharpe Ratio 0.538 0.547 0.523 0.691 0.144
Shortfall frequency 0.37 0.39 0.43 0.41 0.39
SBWGBI 5+
Mean 3.433 3.649 2.435 2.812 1.967
Standard Deviation 6.346 6.757 5.506 4.612 6.445
T-test mean=0 1.42 1.42 1.16 1.60 0.80
Sharpe Ratio 0.541 0.540 0.442 0.610 0.305
Shortfall frequency 0.40 0.43 0.40 0.37 0.39
SBWGBI all
Mean 4.411 2.435 2.306 1.945 2.110 1.469
Standard Deviation 3.153 4.240 4.124 4.001 3.223 5.212
T-test mean=0 3.43*** 1.51 1.47 1.28 1.72 * 0.74
Sharpe Ratio 1.399 0.574 0.559 0.486 0.655 0.282
Shortfall frequency 0.29 0.40 0.41 0.40 0.37 0.40
Table 4.6 - Performance of Passive and Dynamic Bond Trading Strategies for the period February 1994 to December 2000 (continued)
*** Statistically significant at 1% ** Statistically significant at 5% * Statistically significant at 10%
Note: In the case of Portugal, the period is restricted to January 1995 to December 2000 as explained previously.
Although the results seem to indicate that the dynamic strategy outperforms the passive one, will this survive after transactions costs are taken into account? In Table 4.7 we report the number of times that the dynamic strategy switches between the bond and cash markets during the period. As shown, this simple investment strategy implies few transactions: about one or two a year. So, transaction costs will not have a relevant impact on performance.
Portugal Spain Italy France Germany UK Panel B - Dynamic Strategy
SBWGBI 1-3
Mean 1.806 1.420 1.217 1.304 0.765
Standard Deviation 1.339 1.596 1.057 1.277 1.093
T-test mean=0 3.55*** 2.34** 3.03*** 2.69*** 1.84*
Sharpe Ratio 1.349 0.889 1.151 1.021 0.700
Shortfall Frequency 0.22 0.20 0.23 0.30 0.17
Frequency of forecasts
with correct sign 0.66 0.57 0.60 0.63 0.61
SBWGBI 3-5
Mean 3.528 3.712 2.627 2.915 1.700
Standard Deviation 2.471 3.199 1.985 2.209 1.992
T-test mean=0 3.76*** 3.05*** 3.48*** 3.47*** 2.24**
Sharpe Ratio 1.428 1.160 1.323 1.320 0.853
Shortfall Frequency 0.19 0.17 0.22 0.22 0.16
Frequency of forecasts
with correct sign 0.65 0.66 0.61 0.69 0.60
SBWGBI 5+
Mean 7.000 5.933 4.513 4.563 4.858
Standard Deviation 4.603 5.555 3.513 3.540 4.481
T-test mean=0 4.00*** 2.81*** 3.38*** 3.39*** 2.85***
Sharpe Ratio 1.521 1.068 1.285 1.289 1.084
Shortfall Frequency 0.20 0.24 0.17 0.18 0.20
Frequency of forecasts
with correct sign 0.71 0.63 0.67 0.71 0.60
SBWGBI all
Mean 4.376 4.814 3.776 3.415 3.228 3.811
Standard Deviation 3.107 3.045 3.387 2.607 2.548 3.593
T-test mean=0 3.45*** 4.16*** 2.93*** 3.45*** 3.33*** 2.79***
Sharpe Ratio 1.408 1.581 1.115 1.310 1.267 1.061
Shortfall Frequency 0.25 0.19 0.22 0.17 0.19 0.22
Frequency of forecasts
with correct sign 0.72 0.72 0.65 0.67 0.71 0.58
Table 4.7 – Number of transactions for the dynamic strategy over the period February 1994 to December 2000.
As mentioned previously, we also conducted an out-of-sample analysis for the period January 1998 to December 2000. The results are reported in Table 4.8 below.
One can observe that they are not much different from those of the in-sample analysis.
However, the difference between the two strategies is not as big probably due to the fact that the shorter period considered is characterised by low bond market returns.
SBWGBI 1-3 SBWGBI 3-5 SBWGBI 5+ SBWGBI all
Portugal 5
Spain 10 9 9 10
Italy 15 17 13 12
France 9 10 8 8
Germany 11 9 6 7
UK 12 14 12 14
Table 4.8 - Performance of passive and dynamic bond trading strategies (out-of-sample) for the period January 1998 to December 2000
This table compares the out-of-sample performance of a passive/static and a dynamic bond investment strategy for each country and for each maturity subsector. Panel A reports the results of the passive strategy while Panel B presents the results of the dynamic strategy. The passive strategy involves always staying in bonds. The dynamic strategy involves buying one unit of the bond (market) if its predicted premium is positive and none of the bond if its predicted premium is negative. The expected bond risk premium is the fitted value from the rolling regressions of realized excess bond returns on term spread, IRW and the January dummy (rolling one-step ahead forecasts) that begin in January 1998 and uses all the available historical information since February 1994. The table reports the annualized mean (in percentage) and standard deviation of the excess returns for each trading strategy, the T-test for the hypothesis of the monthly mean=0, the annualized Sharpe ratio (the ratio of the mean and standard deviation), the shortfall frequency (observed frequency of negative excess returns) and, for the dynamic strategy, the frequency of forecasts with correct sign (observed frequency of predicting positive bond risk premiums when the realized premium is positive plus observed frequency of predicting negative bond risk premiums when the realized premium is negative).
*** Statistically significant at 1% ** Statistically significant at 5% * Statistically significant at 10%
Portugal Spain Italy France Germany UK
Panel A - Static Strategy SBWGBI 1-3
Mean 0.403 0.391 0.403 0.491 0.195
Standard Deviation 1.181 1.173 1.118 1.163 1.445
T-test mean=0 0.59 0.58 0.62 0.73 0.23
Sharpe Ratio 0.341 0.333 0.360 0.422 0.135
Shortfall frequency 0.50 0.42 0.44 0.42 0.47
SBWGBI 3-5
Mean 1.141 0.802 1.071 1.190 0.423
Standard Deviation 2.289 2.342 2.453 2.469 2.732
T-test mean=0 0.86 0.59 0.76 0.84 0.27
Sharpe Ratio 0.499 0.343 0.437 0.482 0.155
Shortfall frequency 0.39 0.39 0.44 0.44 0.42
SBWGBI 5+
Mean 2.053 1.593 2.223 2.230 2.648
Standard Deviation 4.367 4.373 4.492 4.704 5.826
T-test mean=0 0.81 0.63 0.86 0.82 0.79
Sharpe Ratio 0.470 0.364 0.495 0.474 0.454
Shortfall frequency 0.39 0.47 0.39 0.39 0.42
SBWGBI all
Mean 0.939 1.326 1.071 1.606 1.454 2.058
Standard Deviation 3.014 3.039 2.723 3.338 2.999 4.642
T-test mean=0 0.54 0.76 0.68 0.83 0.84 0.77
Sharpe Ratio 0.312 0.436 0.393 0.481 0.485 0.443
Shortfall frequency 0.39 0.39 0.42 0.39 0.39 0.42
Table 4.8 - Performance of passive and dynamic bond trading strategies (out-of-sample) for the period January 1998 to December 2000 (continued)
Portugal Spain Italy France Germany UK
Panel B - Dynamic Strategy SBWGBI 1-3
Mean 1.045 0.497 0.403 0.423 0.953
Standard Deviation 0.961 0.884 1.033 1.159 1.148
T-test mean=0 1.88* 0.97 1.35 0.63 1.44
Sharpe Ratio 1.088 0.561 0.390 0.365 0.830
Shortfall Frequency 0.31 0.25 0.31 0.42 0.19
Frequency of forecasts
with correct sign 0.67 0.47 0.67 0.53 0.72
SBWGBI 3-5
Mean 2.393 1.282 1.589 1.514 2.115
Standard Deviation 1.768 1.597 2.234 2.315 2.026
T-test mean=0 2.34** 1.39 1.23 1.13 1.81*
Sharpe Ratio 1.353 0.803 0.711 0.654 1.044
Shortfall Frequency 0.19 0.19 0.33 0.33 0.19
Frequency of forecasts
with correct sign 0.75 0.58 0.56 0.61 0.69
SBWGBI 5+
Mean 3.808 2.507 3.812 2.729 4.135
Standard Deviation 3.440 3.223 3.638 4.249 4.944
T-test mean=0 1.92* 1.35 1.81* 1.11 1.45
Sharpe Ratio 1.107 0.778 1.048 0.642 0.836
Shortfall Frequency 0.19 0.25 0.22 0.25 0.28
Frequency of forecasts
with correct sign 0.75 0.58 0.69 0.67 0.61
SBWGBI all
Mean 0.939 2.744 1.648 2.155 1.894 4.297
Standard Deviation 3.014 2.350 1.984 3.041 2.723 3.545
T-test mean=0 0.54 2.02* 1.44 1.23 1.20 2.10**
Sharpe Ratio 0.312 1.168 0.831 0.709 0.696 1.212
Shortfall Frequency 0.39 0.19 0.22 0.25 0.25 0.25
Frequency of forecasts
with correct sign 0.61 0.75 0.58 0.67 0.67 0.64
*** Statistically significant at 1% ** Statistically significant at 5% * Statistically significant at 10%
Note: In the case of Portugal, the expected bond risk premium is the fitted value from the rolling regressions of realized excess bond returns on term spread, IRW and the January dummy (rolling one-step ahead forecasts) that begin in January 1998 and use all the available historical information since January 1995.