Timing signal implied volatility - Can traders benefit from the currency carry trade during cri

Table (B.16) Scenario analysis of the exit strategy based on VIX These tables report the mean return andp-value of the 1-month rebalance exit

strategies based on VIXt−1 in the sample period between 28-05-1999 and 29-05-2020. The p-values are reported in parenthesis below the mean return. The lowest and highest mean returns of the five strategies are marked with a coloured cell. The abbreviations used to denote the strategies are described in Appendix

A.1.

(a) Exit1M1C

Standard deviation

Windowsize

1 1.5 2

1 4.09% 3.97% 4.01%

(0.001) (0.003) (0.003) 1.5 3.99% 4.15% 3.95%

(0.001) (0.001) (0.003) 2 4.28% 4.36% 3.96%

(0.000) (0.000) (0.003)

(b) Exit1M2C

Standard deviation

Windowsize

1 1.5 2

1 3.89% 3.72% 3.81%

(0.000) (0.000) (0.000) 1.5 3.93% 3.87% 3.59%

(0.000) (0.000) (0.001) 2 3.82% 4.00% 3.65%

(0.000) (0.000) (0.000)

Standard deviation

Windowsize

1 1.5 2

1 3.81% 3.81% 3.91%

(0.000) (0.000) (0.000) 1.5 3.98% 3.83% 3.73%

(0.000) (0.000) (0.000) 2 3.69% 3.92% 3.77%

(0.000) (0.000) (0.000)

(d) Exit1M4C

Standard deviation

Windowsize(year) 1 1.5 2

1 3.99% 3.90% 4.02%

(0.000) (0.000) (0.000) 1.5 4.24% 3.98% 3.82%

(0.000) (0.000) (0.000) 2 3.80% 4.05% 3.85%

(0.000) (0.000) (0.000)

(e) Exit1M5C

Standard deviation

Windowsize

1 1.5 2

1 3.94% 3.90% 3.95%

(0.000) (0.000) (0.000) 1.5 4.16% 3.92% 3.79%

(0.000) (0.000) (0.000) 2 3.72% 3.93% 3.84%

(0.000) (0.000) (0.000)

Table (B.17) Scenario analysis of the exit strategy based on VIX This table reports the mean return, Sharpe ratio, nominalp-value of mean return and the fraction of negative returns of the five 1-month rebalance exit- and static carry trade strategies. The results for the exit strategies are based on the VIX signal that occurred contemporaneously with the rebalance days, as well as

one-and two days lagged over the full sample period from 30-06-1999 to 29-05-2020.

The highest mean return for each of the strategies is marked by a coloured cell.

The abbreviations used to denote the strategies are described in Appendix A.1.

Carry1M Reverse1M

Signal VIXt−2 VIXt−1 VIX_t

Mean return 2.84% 3.63% 4.15% 3.73%

Sharpe ratio 0.417 0.616 0.727 0.650 nominalp-value 0.057 0.005 0.001 0.003 fraction negative 38.5% 32.5% 32.9% 33.3%

Mean return 2.60% 3.53% 3.87% 3.54%

Sharpe ratio 0.498 0.759 0.839 0.761 nominalp-value 0.023 0.001 0.000 0.001 fraction negative 38.9% 32.5% 32.1% 32.1%

Mean return 2.67% 3.78% 3.83% 3.63%

Sharpe ratio 0.546 0.886 0.893 0.843 nominalp-value 0.013 0.000 0.000 0.000 fraction negative 40.9% 33.7% 34.1% 33.3%

Mean return 2.82% 3.99% 3.98% 3.80%

Sharpe ratio 0.581 0.937 0.925 0.881 nominalp-value 0.008 0.000 0.000 0.000 fraction negative 39.3% 32.1% 32.1% 32.1%

Mean return 2.84% 3.96% 3.92% 3.74%

Sharpe ratio 0.577 0.906 0.881 0.839 nominalp-value 0.009 0.000 0.000 0.000 fraction negative 40.48% 33.3% 33.3% 33.3%

Table (B.18) Scenario analysis of the exit strategy based on VXY These tables report the mean return andp-value of the 1-month rebalance exit

strategies based on VXYt−1 in the sample period between 28-05-1999 and 29-05-2020. The p-values are reported in parenthesis below the mean return. The lowest and highest mean returns of the five strategies are marked with a coloured cell. The abbreviations used to denote the strategies are described in Appendix

A.1.

(a) Exit1M1C

Standard deviation

Windowsize

1 1.5 2

1 5.21% 5.03% 4.72%

(0.000) (0.000) (0.000) 1.5 5.35% 5.03% 4.62%

(0.000) (0.000) (0.000) 2 4.73% 4.75% 4.22%

(0.000) (0.000) (0.000)

(b) Exit1M2C

Standard deviation

Windowsize

1 1.5 2

1 4.44% 4.34% 3.94%

(0.000) (0.000) (0.000) 1.5 4.39% 4.22% 3.84%

(0.000) (0.000) (0.000) 2 3.89% 4.00% 3.60%

(0.000) (0.000) (0.001)

Standard deviation

Windowsize

1 1.5 2

1 4.24% 4.18% 3.93%

(0.000) (0.000) (0.000) 1.5 4.07% 4.14% 3.53%

(0.000) (0.000) (0.000) 2 3.71% 3.83% 3.43%

(0.000) (0.000) (0.001)

(d) Exit1M4C

Standard deviation

Windowsize

1 1.5 2

1 4.21% 4.12% 3.97%

(0.000) (0.000) (0.000) 1.5 4.04% 4.11% 3.52%

(0.000) (0.000) (0.000) 2 3.74% 3.80% 3.42%

(0.000) (0.000) (0.001)

(e) Exit1M5C

Standard deviation

Windowsize

1 1.5 2

1 4.09% 3.98% 3.86%

(0.000) (0.000) (0.000) 1.5 3.90% 3.96% 3.29%

(0.000) (0.000) (0.000) 2 3.63% 3.63% 3.28%

(0.000) (0.000) (0.002)

Table (B.19) Scenario analysis of the exit strategy based on VXY This table reports the mean return, Sharpe ratio, nominalp-value of mean return and the fraction of negative returns of the five 1-month rebalance exit- and static carry trade strategy. The results for the reverse strategies are based on the VXY signal that occurred contemporaneously with the rebalance days, as well as

one-and two days lagged over the full sample period from 30-06-1999 to 29-05-2020.

The highest mean return for each of the strategies is marked by a coloured cell.

The abbreviations used to denote the strategies are described in Appendix A.1.

Carry1M Reverse1M

Signal VXYt−2 VXYt−1 VXY_t

Mean return 2.84% 4.94% 5.03% 4.96%

Sharpe ratio 0.417 0.886 0.903 0.886 nominal p-value 0.057 0.000 0.000 0.000 fraction negative 38.5% 32.1% 31.7% 32.1%

Mean return 2.60% 4.12% 4.22% 4.18%

Sharpe ratio 0.498 0.903 0.924 0.915 nominal p-value 0.023 0.000 0.000 0.000 fraction negative 38.9% 32.1% 31.7% 32.1%

Mean return 2.67% 4.08% 4.14% 4.08%

Sharpe ratio 0.546 0.962 0.979 0.960 nominal p-value 0.013 0.000 0.000 0.000 fraction negative 40.9% 33.7% 33.3% 33.7%

Mean return 2.82% 4.11% 4.11% 4.02%

Sharpe ratio 0.581 0.961 0.967 0.940 nominal p-value 0.008 0.000 0.000 0.000 fraction negative 39.3% 32.9% 32.5% 32.9%

Mean return 2.84% 3.97% 3.96% 3.86%

Sharpe ratio 0.577 0.916 0.917 0.890 nominal p-value 0.009 0.000 0.000 0.000 fraction negative 40.48% 33.7% 33.7% 34.1%

B.8 Sub sample test of the dynamic strategies based on the change in VIX

Table (B.20) Performance matrix of reverse strategies based on VIX over sub-periods

This table reports the mean return, Sharpe ratio, nominalp-value of mean return and the fraction of negative returns of the five 1-month rebalance reverse strategies. The results are based on the level and change VIXt−1 signal over the three sub-sample periods: 1999-2006, 2006-2013 and 2013-2020. The mean returns

are marked by a 3-colour scale, using the 50th percentile. The abbreviations used to denote the strategies are described in Appendix A.1

VIXt−1

Sub periods 1999-2006 2006-2013 2013-2020 Level Change Level Change Level Change Reverse1M1C

Mean Return 3.39% 5.39% 8.92% 6.96% 3.40% 1.83%

Sharpe Ratio 0.554 0.898 1.111 0.850 0.602 0.321 Nominal p-value 0.144 0.019 0.004 0.026 0.115 0.399 Fraction negative 37.6% 34.1% 35.3% 35.3% 41.7% 41.7%

Reverse1M2C

Mean Return 4.40% 4.82% 7.79% 5.96% 2.68% 2.17%

Sharpe Ratio 0.976 1.078 1.237 0.921 0.655 0.527 Nominal p-value 0.011 0.005 0.001 0.016 0.087 0.167 Fraction negative 34.1% 32.9% 31.8% 36.5% 42.9% 42.9%

Reverse1M3C

Mean Return 5.75% 5.81% 6.76% 4.75% 1.94% 2.51%

Sharpe Ratio 1.266 1.281 1.184 0.809 0.519 0.675 Nominal p-value 0.001 0.001 0.002 0.034 0.174 0.078 Fraction negative 31.8% 30.6% 35.3% 40.0% 47.6% 45.2%

Reverse1M4C

Mean Return 6.36% 6.10% 6.62% 4.43% 1.97% 2.20%

Sharpe Ratio 1.367 1.302 1.215 0.787 0.522 0.583 Nominal p-value 0.000 0.001 0.002 0.039 0.171 0.127 Fraction negative 31.8% 30.6% 31.8% 36.5% 44.0% 41.7%

Reverse1M5C

Mean Return 6.70% 6.44% 6.05% 3.75% 1.83% 2.29%

Sharpe Ratio 1.386 1.325 1.084 0.654 0.494 0.622 Nominal p-value 0.000 0.001 0.005 0.086 0.195 0.104 Fraction negative 32.9% 31.8% 32.9% 37.6% 45.2% 42.9%

Table (B.21) Performance matrix of exit strategies based on VIX over sub periods

are marked by a 3-colour scale, using the 50th percentile. The abbreviations used to denote the strategies are described in Appendix A.1

VIXt−1

Sub periods 1999-2006 2006-2013 2013-2020 Level Change Level Change Level Change Exit1M1C

Mean Return 4.47% 5.46% 5.16% 4.18% 2.19% 1.41%

Sharpe Ratio 0.762 0.913 0.781 0.588 0.477 0.260 Nominal p-value 0.046 0.017 0.041 0.121 0.210 0.494 Fraction negative 31.8% 30.6% 31.8% 34.1% 36.9% 38.1%

Exit1M2C

Mean Return 4.33% 4.54% 4.74% 3.83% 2.04% 1.78%

Sharpe Ratio 0.996 1.042 0.834 0.645 0.563 0.471 Nominal p-value 0.010 0.007 0.029 0.090 0.140 0.217 Fraction negative 30.6% 30.6% 30.6% 35.3% 36.9% 38.1%

Exit1M3C

Mean Return 5.49% 5.52% 3.86% 2.86% 1.68% 1.96%

Sharpe Ratio 1.261 1.277 0.767 0.536 0.504 0.594 Nominal p-value 0.001 0.001 0.044 0.157 0.186 0.120 Fraction negative 28.2% 28.2% 34.1% 38.8% 41.7% 41.7%

Exit1M4C

Mean Return 6.09% 5.96% 3.92% 2.83% 1.50% 1.61%

Sharpe Ratio 1.344 1.328 0.802 0.536 0.454 0.480 Nominal p-value 0.001 0.001 0.036 0.157 0.233 0.207 Fraction negative 28.2% 28.2% 31.8% 36.5% 38.1% 38.1%

Exit1M15

Mean Return 6.44% 6.32% 3.47% 2.33% 1.45% 1.68%

Sharpe Ratio 1.362 1.343 0.691 0.427 0.427 0.505 Nominal p-value 0.000 0.001 0.070 0.258 0.262 0.185 Fraction negative 29.4% 29.4% 32.9% 37.6% 39.3% 39.3%

B.9 Sub-sample test of the dynamic strategies based on the change in VXY

Table (B.22) Performance matrix of reverse strategies based on VXY over sub periods

This table reports the mean return, Sharpe ratio, nominalp-value of mean return and the fraction of negative returns of the five 1-month rebalance exit strategies.

The results are based on the level and change VXYt−1 signal over the three sub-sample periods: 1999-2006, 2006-2013 and 2013-2020. The mean returns are marked by a 3-colour scale, using the 50th percentile. The abbreviations used to

denote the strategies are described in Appendix A.1 VXYt−1

Sub periods 1999-2006 2006-2013 2013-2020 Level Change Level Change Level Change Reverse1M1C

Mean Return 6.18% 5.35% 11.42% 7.06% 3.49% 2.15%

Sharpe Ratio 1.040 0.890 1.466 0.860 0.628 0.384 Nominal p-value 0.007 0.020 0.000 0.025 0.100 0.312 Fraction negative 32.9% 31.8% 31.8% 37.6% 42.9% 41.7%

Reverse1M2C

Mean Return 5.33% 4.66% 9.45% 6.91% 2.42% 2.16%

Sharpe Ratio 1.204 1.039 1.546 1.081 0.602 0.538 Nominal p-value 0.002 0.007 0.000 0.005 0.115 0.159 Fraction negative 29.4% 32.9% 31.8% 35.3% 45.2% 44.0%

Reverse1M3C

Mean Return 5.82% 5.40% 8.15% 5.66% 2.63% 2.16%

Sharpe Ratio 1.282 1.179 1.467 0.974 0.725 0.593 Nominal p-value 0.001 0.002 0.000 0.011 0.059 0.121 Fraction negative 31.8% 30.6% 34.1% 37.6% 44.0% 47.6%

Reverse1M4C

Mean Return 6.24% 5.47% 7.50% 4.70% 2.30% 1.80%

Sharpe Ratio 1.338 1.151 1.401 0.837 0.624 0.484 Nominal p-value 0.001 0.003 0.000 0.029 0.103 0.204 Fraction negative 31.8% 32.9% 34.1% 37.6% 42.9% 44.0%

Reverse1M5C

Mean Return 6.41% 5.47% 6.59% 4.24% 2.11% 2.02%

Sharpe Ratio 1.318 1.103 1.193 0.742 0.583 0.557 Nominal p-value 0.001 0.004 0.002 0.052 0.127 0.144 Fraction negative 32.9% 34.1% 35.3% 36.5% 44.0% 45.2%

Table (B.23) Performance matrix of exit strategies based on VXY over sub periods

This table reports the mean return, Sharpe ratio, nominalp-value of mean return and the fraction of negative returns of the five 1-month rebalance exit strategies.

denote the strategies are described in Appendix A.1 . VXYt−1

Sub periods 1999-2006 2006-2013 2013-2020 Level Change Level Change Level Change Exit1M1C

Mean Return 5.86% 5.44% 6.31% 4.13% 2.40% 1.73%

Sharpe Ratio 1.000 0.914 1.052 0.596 0.499 0.311 Nominal p-value 0.009 0.017 0.006 0.117 0.191 0.414 Fraction negative 29.4% 30.6% 28.2% 34.1% 39.3% 38.1%

Exit1M2C

Mean Return 4.79% 4.46% 5.52% 4.26% 2.07% 1.94%

Sharpe Ratio 1.102 0.995 1.032 0.748 0.542 0.490 Nominal p-value 0.004 0.010 0.007 0.050 0.155 0.198 Fraction negative 28.2% 31.8% 29.4% 34.1% 39.3% 38.1%

Exit1M3C

Mean Return 5.52% 5.31% 4.55% 3.30% 2.17% 1.94%

Sharpe Ratio 1.243 1.165 0.995 0.633 0.608 0.552 Nominal p-value 0.001 0.003 0.010 0.096 0.111 0.148 Fraction negative 28.2% 29.4% 31.8% 36.5% 41.7% 42.9%

Exit1M4C

Mean Return 6.03% 5.64% 4.37% 2.97% 1.80% 1.54%

Sharpe Ratio 1.324 1.212 0.969 0.571 0.510 0.440 Nominal p-value 0.001 0.002 0.012 0.133 0.181 0.248 Fraction negative 28.2% 30.6% 30.6% 35.3% 39.3% 39.3%

Exit1M5C

Mean Return 6.30% 5.83% 3.79% 2.61% 1.71% 1.67%

Sharpe Ratio 1.343 1.220 0.836 0.487 0.488 0.485 Nominal p-value 0.001 0.002 0.029 0.199 0.200 0.203 Fraction negative 29.4% 31.8% 31.8% 35.3% 40.5% 40.5%

Figures

C.1 Autocorrelation function plots

Figure (C.1) ACF plots for the returns of 3-month rebalance carry trade strategies

This figure presents the autocorrelation function plots of the monthly natural log-returns and squared natural log-returns up to 30 lags for the five 1-month rebalance strategies along with a 95%-confidence band. The abbreviations used for

each of the five strategies are described in Appendix A.1

Figure (C.2) ACF plots for the returns of 6-month rebalance carry trade strategies

This figure presents the autocorrelation function plots of the monthly natural log-returns and squared natural log-returns up to 30 lags for the five 6-month rebalance strategies along with a 95%-confidence band. The abbreviations used for

each of the five strategies are described in Appendix A.1

Figure (C.3) ACF plots for the returns of 12-month rebalance carry trade strategies

This figure presents the autocorrelation function plots of the monthly natural log-returns and squared natural log-returns up to 30 lags for the five 12-month rebalance strategies along with a 95%-confidence band. The abbreviations used for

each of the five strategies are described in Appendix A.1

C.2 Cumulative returns

Figure (C.4) Cumulative returns of 3-month rebalance strategies This figure presents the cumulative returns generated from the 3-month rebalancing carry trade strategies in the sample period between 30-06-1999 and 29-05-2020. The abbreviations used for each of the five strategies are described in

Appendix A.1

Figure (C.5) Cumulative returns of 6-month rebalance strategies This figure presents the cumulative returns generated from the 6-month rebalancing carry trade strategies in the sample period between 30-06-1999 and 29-05-2020. The abbreviations used for each of the five strategies are described in

Appendix A.1

C.3 Plot of the performance measures using different timing signals of VIX for differ-ent standard deviations

Figure (C.6) Exit1M1C

This figure presents the mean returns of Exit1M1C obtained from using the VIX signal that occurred contemporaneously with the rebalance days, as well as

one-and two day lagged.

Figure (C.7) Exit1M2C

This figure presents the mean returns of Exit1M2C obtained from using the VIX signal that occurred contemporaneously with the rebalance days, as well as

one-and two day lagged.

Figure (C.8) Exit1M3C

This figure presents the mean returns of Exit1M3C obtained from using the VIX signal that occurred contemporaneously with the rebalance days, as well as

one-and two day lagged.

Figure (C.9) Exit1M4C

This figure presents the mean returns of Exit1M4C obtained from using the VIX signal that occurred contemporaneously with the rebalance days, as well as

one-and two day lagged.

Figure (C.10) Exit1M5C

This figure presents the mean returns of Exit1M3C obtained from using the VIX signal that occurred contemporaneously with the rebalance days, as well as

one-and two day lagged.

Figure (C.11) Reverse1M1C

This figure presents the mean returns of Reverse1M1C obtained from using the VIX signal that occurred contemporaneously with the rebalance days, as well as

one- and two day lagged.

Figure (C.12) Reverse1M2C

This figure presents the mean returns of Reverse1M2C obtained from using the VIX signal that occurred contemporaneously with the rebalance days, as well as

one- and two day lagged.

Figure (C.13) Reverse1M3C

This figure presents the mean returns of Reverse1M3C obtained from using the VIX signal that occurred contemporaneously with the rebalance days, as well as

one- and two day lagged.

Figure (C.14) Reverse1M4C

This figure presents the mean returns of Reverse1M4C obtained from using the VIX signal that occurred contemporaneously with the rebalance days, as well as

one- and two day lagged.

Figure (C.15) Reverse1M5C

This figure presents the mean returns of Reverse1M5C obtained from using the VIX signal that occurred contemporaneously with the rebalance days, as well as

one- and two day lagged.

C.4 Plot of the performance measures using different timing signals of VXY for differ-ent standard deviations

Figure (C.16) Exit1M1C

This figure presents the mean returns of Exit1M1C obtained from using the VXY signal that occurred contemporaneously with the rebalance days, as well as

one-and two day lagged.

Figure (C.17) Exit1M2C

This figure presents the mean returns of Exit1M2C obtained from using the VXY signal that occurred contemporaneously with the rebalance days, as well as

one-and two day lagged.

Figure (C.18) Exit1M3C

This figure presents the mean returns of Exit1M3C obtained from using the VXY signal that occurred contemporaneously with the rebalance days, as well as

one-and two day lagged.

Figure (C.19) Exit1M4C

This figure presents the mean returns of Exit1M4C obtained from using the VXY signal that occurred contemporaneously with the rebalance days, as well as

one-and two day lagged.

Figure (C.20) Exit1M5C

This figure presents the mean returns of Exit1M5C obtained from using the VXY signal that occurred contemporaneously with the rebalance days, as well as

one-and two day lagged.

Figure (C.21) Reverse1M2C

This figure presents the mean returns of Reverse1M2C obtained from using the VXY signal that occurred contemporaneously with the rebalance days, as well as

one- and two day lagged.

Figure (C.22) Reverse1M3C

This figure presents the mean returns of Reverse1M3C obtained from using the VXY signal that occurred contemporaneously with the rebalance days, as well as

one- and two day lagged.

Figure (C.23) Reverse1M4C

This figure presents the mean returns of Reverse1M4C obtained from using the VXY signal that occurred contemporaneously with the rebalance days, as well as

one- and two day lagged.

C.5 Histogram of bootstrapped 3-, 6- and 12 month rebalance carry strategies

Figure (C.24) Probability density function

This figure shows the distribution of Z-scores of the mean returns for the 3-, 6-and 12 month rebalance static carry trade strategies, in the colour grey. The distribution of the bootstrapped Z-scores are shown with the blue lines. The red lines indicate where the x-axes are equal to 1.96 and -1.96. The abbreviations used

to denote the strategies are described in Appendix A.1

Python codes

This appendix contain a selection of the total code for this thesis. The aim is to give the reader the opportunity to understand the underlying code for some of the key elements of the methodology. In Appendix D.1 the python code of static carry trade construction is shown. All calculations are based on transaction cost. Appendix D.2 shows the construction and return calculation of the exit- and reverse strategy based on VIX. The last part, Appendix D.3 shows the bootstrap resample of the static carry trade. The same procedure is used for the exit- and reverse strategy.

D.1 Portfolio construction and carry trade re-turn calculations based on 1-month rebal-ance

1 #! / u s r / b i n / env python

2 # c o d i n g : u t f−8

4 # I n [ 3 ] :

5 6

7 i m p o r t pandas a s pd # pandas f o r w o r k i n g w i t h d a t a

8 i m p o r t m a t p l o t l i b . p y p l o t a s p l t # M a t p l o t l i b f o r p l o t t i n g

9 i m p o r t numpy a s np #Numpy i s a l i b r a r y f o r w o r k i n g w i t h n u m e r i c a l d a t a

10 i m p o r t s c i p y . s t a t s

11 i m p o r t s e a b o r n a s s n s

12 i m p o r t o s

13 i m p o r t s y s

14 i m p o r t m a t p l o t l i b . p y p l o t a s p l t

15 from m a t p l o t l i b . t i c k e r i m p o r t P e r c e n t F o r m a t t e r

17 p l t . r c (’ x t i c k ’, l a b e l s i z e =13)

18 p l t . r c (’ y t i c k ’, l a b e l s i z e =13)

19 p l t . r c (’ l e g e n d ’, f o n t s i z e =14)

20 p l t . r c (’ a x e s ’, t i t l e s i z e =12)

21 p l t . r c (’ a x e s ’, l a b e l s i z e =14)

22 23

24 # I n [ 2 ] :

25 26

27 o s . c h d i r ( o s . getcwd ( ) )

29 F o r w a r d r a t e s = pd . r e a d e x c e l (”Fwd r a t e s USD . x l s x ”)

30 S p o t r a t e s = pd . r e a d e x c e l (” Spot r a t e s USD . x l s x ”)

31 Date = pd . r e a d e x c e l (” Vol i n d i c e s . x l s x ”, u s e c o l s= [ 0 ] )

33 # Only u s e d a t e s t h a t a r e i n a v a l i a b l e f o r i n d i c e s

34 S p o t r a t e s = S p o t r a t e s . merge ( Date , on=’ Date ’, how=’ r i g h t ’) . dropna ( ) . s e t i n d e x (’ Date ’)

35 F o r w a r d r a t e s = F o r w a r d r a t e s . merge ( Date , on=’ Date ’, how=’ r i g h t ’) . dropna ( ) . s e t i n d e x (’ Date ’)

37 # Only u s e max v a l u e o f e a c h month ”M”

38 s p o t r a t e s 1 M = pd . DataFrame ( )

39 s p o t r a t e s 1 M =( S p o t r a t e s . l o c [ S p o t r a t e s . groupby ( S p o t r a t e s . i n d e x . t o p e r i o d (’M’) ) .a p p l y(lambda x : x . i n d e x .max( ) ) ] )

40 s p o t r a t e s 1 M . r e s e t i n d e x ( i n p l a c e=True )

41 Forward rates1M = pd . DataFrame ( )

42 Forward rates1M =( F o r w a r d r a t e s . l o c [ F o r w a r d r a t e s . groupby (

F o r w a r d r a t e s . i n d e x . t o p e r i o d (’M’) ) .a p p l y(lambda x : x . i n d e x .max ( ) ) ] )

43 Forward rates1M . r e s e t i n d e x ( i n p l a c e=True )

44 45

46 # I n [ 3 ] :

47 48

49 # c a l c u l a t e l o g r e t u r n

51 c o l s F 1 = Forward rates1M . columns [ 1 : ]

53 f o r column i n c o l s F 1 :

54 Forward rates1M [ column ] = np . l o g ( Forward rates1M [ column ] )

56 c o l s S 1 = s p o t r a t e s 1 M . columns [ 1 : ]

58 f o r column i n c o l s S 1 :

59 s p o t r a t e s 1 M [ column ] = np . l o g ( s p o t r a t e s 1 M [ column ] )

60 61

62 # I n [ 4 ] :

63 64

65 # o n l y t a k i n g t h e mid p r i c e : I t i s u s e d f o r t h e f o r w a r d d i s c o u n t r a t e

66 M i d c o l s = [ c o l f o r c o l i n Forward rates1M . columns i f ’ MID ’ i n c o l ]

68 Fwd MID1M=pd . DataFrame ( )

70 f o r column i n M i d c o l s :

71 Fwd MID1M [ column ] = Forward rates1M [ column ]

73 M i d c o l s s p o t = [ c o l f o r c o l i n s p o t r a t e s 1 M . columns i f ’ mid ’ i n c o l ]

75 spot MID1M=pd . DataFrame ( )

77 f o r column i n M i d c o l s s p o t :

78 spot MID1M [ column ] = s p o t r a t e s 1 M [ column ]

79 80

81 # I n [ 5 ] :

82 83

84 Fwd disc 1M=pd . DataFrame ( )

86 f o r c o l i n spot MID1M . columns :

87 c u r r = c o l . s p l i t (’ ’) [ 0 ]

88 c u r r c o l s = [ c o l f o r c o l i n Fwd MID1M . columns i f c u r r i n c o l ]

89 f o r c o l 1 i n c u r r c o l s :

90 Fwd disc 1M [ c o l 1 ] = Fwd MID1M [ c o l 1 ] . v a l u e s−spot MID1M [ c o l ] . v a l u e s

91 92

93 # I n [ 6 ] :

94 95

96 c o l = [ c o l f o r c o l i n Fwd disc 1M . columns i f ’ 1M’ i n c o l ]

98 Fwd d1M1M=pd . DataFrame ( )

100 f o r column i n c o l :

101 Fwd d1M1M [ column ] = Fwd disc 1M [ column ]

102 103

104 date1M=Forward rates1M [” Date ”] . t o f r a m e ( )

105

106 Fwd dis 1M = pd . c o n c a t ( [ date1M , Fwd d1M1M ] , a x i s =1, s o r t=F a l s e )

107 108

109 # I n [ 8 ] :

110 111

112 ########P o r t f o l i o c o n s t r u c t i o n 1M r e b a l a n c e 5 c u r r e n c i e s#######

113 # 5 c u r r e n c y i n t h e h i g h and low p o r t f o l i o

114 h i g h l i s t = [ ]

115 l o w l i s t = [ ]

116

117 c o l n a m e s = Fwd dis 1M . columns [ 1 : ]

118 f o r d a t e i n Fwd dis 1M [’ Date ’ ] :

119 temp = Fwd dis 1M [ Fwd dis 1M [’ Date ’]== d a t e ] . v a l u e s [ 0 ] [ 1 : ]

120 temp1 = temp<0

121 temp2 = temp>0

122 low5 = c o l n a m e s [ temp1 ]

123 t o p 5 = c o l n a m e s [ temp2 ]

124 l i s t c r e a t e r = [ d a t e ]

125 i f l e n( t o p 5 )<5:

126 f o r v a l i n t o p 5 :

127 l i s t c r e a t e r . append ( v a l )

128 n u m z e r o s = 5−l e n( t o p 5 )

129 f o r i i n r a n g e( n u m z e r o s ) :

130 l i s t c r e a t e r . append ( 0 )

131 e l i f l e n( t o p 5 )>5:

132 o r d e r i n g = np . a r g s o r t ( temp [ temp2 ] )

133 f o r v a l i n t o p 5 [ o r d e r i n g [−5 : ] ] :

134 l i s t c r e a t e r . append ( v a l )

135 e l s e:

136 f o r v a l i n t o p 5 :

137 l i s t c r e a t e r . append ( v a l )

138

139 l i s t c r e a t e r 1 = [ d a t e ]

140 i f l e n( low5 )<5:

141 f o r v a l i n low5 :

142 l i s t c r e a t e r 1 . append ( v a l )

143 n u m z e r o s = 5−l e n( low5 )

144 f o r i i n r a n g e( n u m z e r o s ) :

145 l i s t c r e a t e r 1 . append ( 0 )

146 e l i f l e n( low5 )>5:

147 o r d e r i n g = np . a r g s o r t ( temp [ temp1 ] )

148 f o r v a l i n low5 [ o r d e r i n g [ : 5 ] ] :

149 l i s t c r e a t e r 1 . append ( v a l )

150 e l s e:

151 f o r v a l i n low5 :

152 l i s t c r e a t e r 1 . append ( v a l )

153

154 h i g h l i s t . append ( l i s t c r e a t e r )

155 l o w l i s t . append ( l i s t c r e a t e r 1 )

156

157 highDF = pd . DataFrame ( h i g h l i s t )

158 lowDF = pd . DataFrame ( l o w l i s t )

159 160

161 # I n [ 1 1 ] :

162 163

164 ########################Carry t r a d e############################

165 #C a l c u l a t e r e t u r n o f t h e h i g h e s t p o r t f o l i o

166 R l o n g = [ ]

167 f o r i n d e x , row i n highDF . i t e r r o w s ( ) :

168 d a t e l i s t = [ ]

169 i f i n d e x==l e n( highDF )−1:

170 b r e a k

171

172 f o r i i n r a n g e( 1 ,l e n( row ) ) :

173 i f row [ i ]==0:

174 d a t e l i s t . append ( np . nan )

175 c o n t i n u e

176 temp = row [ i ] . s p l i t (’ ’) [ 0 ]

177 temp1 = temp+’ BID 1M ’

178 f b i d = Forward rates1M [ temp1 ] [ i n d e x ]

179

180 i f row [ i ] i n highDF . i l o c [ i n d e x + 1 ] . v a l u e s :

181 temp2 = temp+’ mid ’

182 s m i d = s p o t r a t e s 1 M [ temp2 ] [ i n d e x +1]

183 d a t e l i s t . append ( f b i d−s m i d )

184 e l s e:

185 temp2 = temp+’ a s k ’

186 s a s k = s p o t r a t e s 1 M [ temp2 ] [ i n d e x +1]

187 d a t e l i s t . append ( f b i d−s a s k )

188 R l o n g . append ( d a t e l i s t )

189

190

191 # I n [ 1 2 ] :

192 193

194 #C a l c u l a t e r e t u r n o f t h e l o w e s t p o r t f o l i o

195 R s h o r t = [ ]

196 f o r i n d e x , row i n lowDF . i t e r r o w s ( ) :

197 d a t e l i s t = [ ]

198 i f i n d e x==l e n( lowDF )−1:

199 b r e a k

200 #p r i n t ( i n d e x )

201

202 f o r i i n r a n g e( 1 ,l e n( row ) ) :

203 i f row [ i ]==0:

204 d a t e l i s t . append ( np . nan )

205 c o n t i n u e

206 temp = row [ i ] . s p l i t (’ ’) [ 0 ]

207 temp1 = temp+’ b i d ’

208 s b i d = s p o t r a t e s 1 M [ temp1 ] [ i n d e x +1]

209

210 i f row [ i ] i n lowDF . i l o c [ i n d e x + 1 ] . v a l u e s :

211 temp2 = temp+’ MID 1M ’

212 f m i d = Forward rates1M [ temp2 ] [ i n d e x ]

213 d a t e l i s t . append(−f m i d+s b i d )

214 e l s e:

215 temp2 = temp+’ ASK 1M ’

216 f a s k = Forward rates1M [ temp2 ] [ i n d e x ]

217 d a t e l i s t . append(−f a s k+s b i d )

218 R s h o r t . append ( d a t e l i s t )

219 220

221 # I n [ 1 3 ] :

222 223

224 o u t 1 = pd . DataFrame ( R s h o r t )

225 o u t 2 = pd . DataFrame ( R l o n g )

226 r e s u l t = pd . c o n c a t ( [ o u t 1 , o u t 2 ] , a x i s =1, s o r t=F a l s e )

227 carry1M = pd . DataFrame ( )

228 carry1M [’ Date ’] = highDF [ 0 ] [ 1 : ] . v a l u e s

229 carry1M [’ carry1M5C ’]= r e s u l t . mean ( a x i s =1, s k i p n a=True )

230 231

232 # I n [ 1 5 ] :

233 234

235 ########P o r t f o l i o c o n s t r u c t i o n 1M r e b a l a n c e 4 c u r r e n c i e s#######

236 # 4 c u r r e n c y i n t h e h i g h and low p o r t f o l i o

237

238 h i g h l i s t = [ ]

239 l o w l i s t = [ ]

240

241 c o l n a m e s = Fwd dis 1M . columns [ 1 : ]

242 f o r d a t e i n Fwd dis 1M [’ Date ’ ] :

243 temp = Fwd dis 1M [ Fwd dis 1M [’ Date ’]== d a t e ] . v a l u e s [ 0 ] [ 1 : ]

244 temp1 = temp<0

245 temp2 = temp>0

246 low4 = c o l n a m e s [ temp1 ]

247 t o p 4 = c o l n a m e s [ temp2 ]

248 l i s t c r e a t e r = [ d a t e ]

249 i f l e n( t o p 4 )<4:

250 f o r v a l i n t o p 4 :

251 l i s t c r e a t e r . append ( v a l )

252 n u m z e r o s = 4−l e n( t o p 4 )

253 f o r i i n r a n g e( n u m z e r o s ) :

254 l i s t c r e a t e r . append ( 0 )

255 e l i f l e n( t o p 4 )>4:

256 o r d e r i n g = np . a r g s o r t ( temp [ temp2 ] )

257 f o r v a l i n t o p 4 [ o r d e r i n g [−4 : ] ] :

258 l i s t c r e a t e r . append ( v a l )

259 e l s e:

260 f o r v a l i n t o p 4 :

261 l i s t c r e a t e r . append ( v a l )

262

263 l i s t c r e a t e r 1 = [ d a t e ]

264 i f l e n( low4 )<4:

265 f o r v a l i n low4 :

266 l i s t c r e a t e r 1 . append ( v a l )

267 n u m z e r o s = 4−l e n( low4 )

268 f o r i i n r a n g e( n u m z e r o s ) :

269 l i s t c r e a t e r 1 . append ( 0 )

270 e l i f l e n( low4 )>4:

271 o r d e r i n g = np . a r g s o r t ( temp [ temp1 ] )

272 f o r v a l i n low4 [ o r d e r i n g [ : 4 ] ] :

273 l i s t c r e a t e r 1 . append ( v a l )

274 e l s e:

275 f o r v a l i n low4 :

276 l i s t c r e a t e r 1 . append ( v a l )

277

278 h i g h l i s t . append ( l i s t c r e a t e r )

279 l o w l i s t . append ( l i s t c r e a t e r 1 )

280

281 highDF = pd . DataFrame ( h i g h l i s t )

282 lowDF = pd . DataFrame ( l o w l i s t )

283 284

285 # I n [ 1 7 ] :

286 287

288 ########################Carry t r a d e############################

289 R l o n g = [ ]

290 f o r i n d e x , row i n highDF . i t e r r o w s ( ) :

291 d a t e l i s t = [ ]

292 i f i n d e x==l e n( highDF )−1:

293 b r e a k

294

295 f o r i i n r a n g e( 1 ,l e n( row ) ) :

296 i f row [ i ]==0:

297 d a t e l i s t . append ( np . nan )

298 c o n t i n u e

299 temp = row [ i ] . s p l i t (’ ’) [ 0 ]

300 temp1 = temp+’ BID 1M ’

301 f b i d = Forward rates1M [ temp1 ] [ i n d e x ]

302

303 i f row [ i ] i n highDF . i l o c [ i n d e x + 1 ] . v a l u e s :

304 temp2 = temp+’ mid ’

305 s m i d = s p o t r a t e s 1 M [ temp2 ] [ i n d e x +1]

306 d a t e l i s t . append ( f b i d−s m i d )

307 e l s e:

308 temp2 = temp+’ a s k ’

309 s a s k = s p o t r a t e s 1 M [ temp2 ] [ i n d e x +1]

310 d a t e l i s t . append ( f b i d−s a s k )

311 R l o n g . append ( d a t e l i s t )

312 313

314 # I n [ 1 8 ] :

315 316

317 R s h o r t = [ ]

318 f o r i n d e x , row i n lowDF . i t e r r o w s ( ) :

319 d a t e l i s t = [ ]

320 i f i n d e x==l e n( lowDF )−1:

321 b r e a k

322 #p r i n t ( i n d e x )

323

324 f o r i i n r a n g e( 1 ,l e n( row ) ) :

325 i f row [ i ]==0:

326 d a t e l i s t . append ( np . nan )

327 c o n t i n u e

328 temp = row [ i ] . s p l i t (’ ’) [ 0 ]

329 temp1 = temp+’ b i d ’

330 s b i d = s p o t r a t e s 1 M [ temp1 ] [ i n d e x +1]

331

332 i f row [ i ] i n lowDF . i l o c [ i n d e x + 1 ] . v a l u e s :

333 temp2 = temp+’ MID 1M ’

334 f m i d = Forward rates1M [ temp2 ] [ i n d e x ]

335 d a t e l i s t . append(−f m i d+s b i d )

336 e l s e:

337 temp2 = temp+’ ASK 1M ’

338 f a s k = Forward rates1M [ temp2 ] [ i n d e x ]

339 d a t e l i s t . append(−f a s k+s b i d )

340 R s h o r t . append ( d a t e l i s t )

341 342

343 # I n [ 1 9 ] :

344 345

346 o u t 1 = pd . DataFrame ( R s h o r t )

347 o u t 2 = pd . DataFrame ( R l o n g )

348 r e s u l t = pd . c o n c a t ( [ o u t 1 , o u t 2 ] , a x i s =1, s o r t=F a l s e )

349 carry1M [’ carry1M4C ’]= r e s u l t . mean ( a x i s =1, s k i p n a=True )

350 351

352 # I n [ 2 2 ] :

353 354

355 ########P o r t f o l i o c o n s t r u c t i o n 1M r e b a l a n c e 3 c u r r e n c i e s#######

356 # 3 c u r r e n c y i n t h e h i g h and low p o r t f o l i o

357 h i g h l i s t = [ ]

358 l o w l i s t = [ ]

359

360 c o l n a m e s = Fwd dis 1M . columns [ 1 : ]

361 f o r d a t e i n Fwd dis 1M [’ Date ’ ] :

362 temp = Fwd dis 1M [ Fwd dis 1M [’ Date ’]== d a t e ] . v a l u e s [ 0 ] [ 1 : ]

363 temp1 = temp<0

364 temp2 = temp>0

365 low3 = c o l n a m e s [ temp1 ]

366 t o p 3 = c o l n a m e s [ temp2 ]

367 l i s t c r e a t e r = [ d a t e ]

368 i f l e n( t o p 3 )<3:

369 f o r v a l i n t o p 3 :

370 l i s t c r e a t e r . append ( v a l )

371 n u m z e r o s = 3−l e n( t o p 3 )

372 f o r i i n r a n g e( n u m z e r o s ) :

373 l i s t c r e a t e r . append ( 0 )

374 e l i f l e n( t o p 3 )>3:

375 o r d e r i n g = np . a r g s o r t ( temp [ temp2 ] )

376 f o r v a l i n t o p 3 [ o r d e r i n g [−3 : ] ] :

377 l i s t c r e a t e r . append ( v a l )

378 e l s e:

379 f o r v a l i n t o p 3 :

380 l i s t c r e a t e r . append ( v a l )

381

382 l i s t c r e a t e r 1 = [ d a t e ]

383 i f l e n( low3 )<3:

384 f o r v a l i n low3 :

385 l i s t c r e a t e r 1 . append ( v a l )

386 n u m z e r o s = 3−l e n( low3 )

387 f o r i i n r a n g e( n u m z e r o s ) :

388 l i s t c r e a t e r 1 . append ( 0 )

389 e l i f l e n( low3 )>3:

390 o r d e r i n g = np . a r g s o r t ( temp [ temp1 ] )

391 f o r v a l i n low3 [ o r d e r i n g [ : 3 ] ] :

392 l i s t c r e a t e r 1 . append ( v a l )

393 e l s e:

394 f o r v a l i n low3 :

395 l i s t c r e a t e r 1 . append ( v a l )

396

397 h i g h l i s t . append ( l i s t c r e a t e r )

398 l o w l i s t . append ( l i s t c r e a t e r 1 )

399

400 highDF = pd . DataFrame ( h i g h l i s t )

401 lowDF = pd . DataFrame ( l o w l i s t )

402 403

404 # I n [ 2 4 ] :

405 406

407 ########################Carry t r a d e############################

408 R l o n g = [ ]

409 f o r i n d e x , row i n highDF . i t e r r o w s ( ) :

410 d a t e l i s t = [ ]

411 i f i n d e x==l e n( highDF )−1:

412 b r e a k

413

414 f o r i i n r a n g e( 1 ,l e n( row ) ) :

415 i f row [ i ]==0:

416 d a t e l i s t . append ( np . nan )

417 c o n t i n u e

418 temp = row [ i ] . s p l i t (’ ’) [ 0 ]

419 temp1 = temp+’ BID 1M ’

420 f b i d = Forward rates1M [ temp1 ] [ i n d e x ]

421

422 i f row [ i ] i n highDF . i l o c [ i n d e x + 1 ] . v a l u e s :

423 temp2 = temp+’ mid ’

424 s m i d = s p o t r a t e s 1 M [ temp2 ] [ i n d e x +1]

425 d a t e l i s t . append ( f b i d−s m i d )

426 e l s e:

427 temp2 = temp+’ a s k ’

428 s a s k = s p o t r a t e s 1 M [ temp2 ] [ i n d e x +1]

429 d a t e l i s t . append ( f b i d−s a s k )

430 R l o n g . append ( d a t e l i s t )

431 432

433 # I n [ 2 5 ] :

434 435

436 R s h o r t = [ ]

437 f o r i n d e x , row i n lowDF . i t e r r o w s ( ) :

438 d a t e l i s t = [ ]

439 i f i n d e x==l e n( lowDF )−1:

440 b r e a k

441 #p r i n t ( i n d e x )

442

443 f o r i i n r a n g e( 1 ,l e n( row ) ) :

444 i f row [ i ]==0:

445 d a t e l i s t . append ( np . nan )

446 c o n t i n u e

447 temp = row [ i ] . s p l i t (’ ’) [ 0 ]

448 temp1 = temp+’ b i d ’

449 s b i d = s p o t r a t e s 1 M [ temp1 ] [ i n d e x +1]

450

451 i f row [ i ] i n lowDF . i l o c [ i n d e x + 1 ] . v a l u e s :

452 temp2 = temp+’ MID 1M ’

453 f m i d = Forward rates1M [ temp2 ] [ i n d e x ]

454 d a t e l i s t . append(−f m i d+s b i d )

455 e l s e:

456 temp2 = temp+’ ASK 1M ’

457 f a s k = Forward rates1M [ temp2 ] [ i n d e x ]

458 d a t e l i s t . append(−f a s k+s b i d )

459 R s h o r t . append ( d a t e l i s t )

460 461

462 # I n [ 2 6 ] :

463 464

465 o u t 1 = pd . DataFrame ( R s h o r t )

466 o u t 2 = pd . DataFrame ( R l o n g )

467 r e s u l t = pd . c o n c a t ( [ o u t 1 , o u t 2 ] , a x i s =1, s o r t=F a l s e )

468 carry1M [’ carry1M3C ’]= r e s u l t . mean ( a x i s =1, s k i p n a=True )

469 470

471 # I n [ 2 8 ] :

472 473

474 ########P o r t f o l i o c o n s t r u c t i o n 1M r e b a l a n c e 2 c u r r e n c i e s#######

475 # 2 c u r r e n c y i n t h e h i g h and low p o r t f o l i o

476 h i g h l i s t = [ ]

477 l o w l i s t = [ ]

478

479 c o l n a m e s = Fwd dis 1M . columns [ 1 : ]

480 f o r d a t e i n Fwd dis 1M [’ Date ’ ] :

481 temp = Fwd dis 1M [ Fwd dis 1M [’ Date ’]== d a t e ] . v a l u e s [ 0 ] [ 1 : ]

482 temp1 = temp<0

483 temp2 = temp>0

484 low2 = c o l n a m e s [ temp1 ]

485 t o p 2 = c o l n a m e s [ temp2 ]

486 l i s t c r e a t e r = [ d a t e ]

487 i f l e n( t o p 2 )<2:

488 f o r v a l i n t o p 2 :

489 l i s t c r e a t e r . append ( v a l )

490 n u m z e r o s = 2−l e n( t o p 2 )

491 f o r i i n r a n g e( n u m z e r o s ) :

492 l i s t c r e a t e r . append ( 0 )

493 e l i f l e n( t o p 2 )>2:

494 o r d e r i n g = np . a r g s o r t ( temp [ temp2 ] )

495 f o r v a l i n t o p 2 [ o r d e r i n g [−2 : ] ] :

496 l i s t c r e a t e r . append ( v a l )

497 e l s e:

498 f o r v a l i n t o p 2 :

499 l i s t c r e a t e r . append ( v a l )

500

501 l i s t c r e a t e r 1 = [ d a t e ]

502 i f l e n( low2 )<2:

503 f o r v a l i n low2 :

504 l i s t c r e a t e r 1 . append ( v a l )

505 n u m z e r o s = 2−l e n( low2 )

506 f o r i i n r a n g e( n u m z e r o s ) :

507 l i s t c r e a t e r 1 . append ( 0 )

508 e l i f l e n( low2 )>2:

509 o r d e r i n g = np . a r g s o r t ( temp [ temp1 ] )

510 f o r v a l i n low2 [ o r d e r i n g [ : 2 ] ] :

511 l i s t c r e a t e r 1 . append ( v a l )

512 e l s e:

513 f o r v a l i n low2 :

514 l i s t c r e a t e r 1 . append ( v a l )

515

516 h i g h l i s t . append ( l i s t c r e a t e r )

517 l o w l i s t . append ( l i s t c r e a t e r 1 )

518

519 highDF = pd . DataFrame ( h i g h l i s t )

520 lowDF = pd . DataFrame ( l o w l i s t )

521 522

523 # I n [ 3 0 ] :

524 525

526 ########################Carry t r a d e############################

527 R l o n g = [ ]

528 f o r i n d e x , row i n highDF . i t e r r o w s ( ) :

529 d a t e l i s t = [ ]

530 i f i n d e x==l e n( highDF )−1:

531 b r e a k

532

533 f o r i i n r a n g e( 1 ,l e n( row ) ) :

534 i f row [ i ]==0:

535 d a t e l i s t . append ( np . nan )

536 c o n t i n u e

537 temp = row [ i ] . s p l i t (’ ’) [ 0 ]

538 temp1 = temp+’ BID 1M ’

539 f b i d = Forward rates1M [ temp1 ] [ i n d e x ]

540

541 i f row [ i ] i n highDF . i l o c [ i n d e x + 1 ] . v a l u e s :

542 temp2 = temp+’ mid ’

543 s m i d = s p o t r a t e s 1 M [ temp2 ] [ i n d e x +1]

544 d a t e l i s t . append ( f b i d−s m i d )

545 e l s e:

546 temp2 = temp+’ a s k ’

547 s a s k = s p o t r a t e s 1 M [ temp2 ] [ i n d e x +1]

548 d a t e l i s t . append ( f b i d−s a s k )

549 R l o n g . append ( d a t e l i s t )

550 551

552 # I n [ 3 1 ] :

553 554

555 R s h o r t = [ ]

556 f o r i n d e x , row i n lowDF . i t e r r o w s ( ) :

557 d a t e l i s t = [ ]

558 i f i n d e x==l e n( lowDF )−1:

559 b r e a k

560 #p r i n t ( i n d e x )

561

562 f o r i i n r a n g e( 1 ,l e n( row ) ) :

563 i f row [ i ]==0:

564 d a t e l i s t . append ( np . nan )

565 c o n t i n u e

566 temp = row [ i ] . s p l i t (’ ’) [ 0 ]

567 temp1 = temp+’ b i d ’

568 s b i d = s p o t r a t e s 1 M [ temp1 ] [ i n d e x +1]

569

570 i f row [ i ] i n lowDF . i l o c [ i n d e x + 1 ] . v a l u e s :

571 temp2 = temp+’ MID 1M ’

572 f m i d = Forward rates1M [ temp2 ] [ i n d e x ]

573 d a t e l i s t . append(−f m i d+s b i d )

574 e l s e:

575 temp2 = temp+’ ASK 1M ’

576 f a s k = Forward rates1M [ temp2 ] [ i n d e x ]

577 d a t e l i s t . append(−f a s k+s b i d )

578 R s h o r t . append ( d a t e l i s t )

579 580

581 # I n [ 3 2 ] :

582 583

584 o u t 1 = pd . DataFrame ( R s h o r t )

585 o u t 2 = pd . DataFrame ( R l o n g )

586 r e s u l t = pd . c o n c a t ( [ o u t 1 , o u t 2 ] , a x i s =1, s o r t=F a l s e )

587 carry1M [’ carry1M2C ’]= r e s u l t . mean ( a x i s =1, s k i p n a=True )

588 589

590 # I n [ 3 4 ] :

591 592

593 ########P o r t f o l i o c o n s t r u c t i o n 1M r e b a l a n c e 1 c u r r e n c i e s#######

594 # 1 c u r r e n c y i n t h e h i g h and low p o r t f o l i o

595 h i g h l i s t = [ ]

596 l o w l i s t = [ ]

597

598 c o l n a m e s = Fwd dis 1M . columns [ 1 : ]

599 f o r d a t e i n Fwd dis 1M [’ Date ’ ] :

600 temp = Fwd dis 1M [ Fwd dis 1M [’ Date ’]== d a t e ] . v a l u e s [ 0 ] [ 1 : ]

601 temp1 = temp<0

602 temp2 = temp>0

603 low1 = c o l n a m e s [ temp1 ]

604 t o p 1 = c o l n a m e s [ temp2 ]

605 l i s t c r e a t e r = [ d a t e ]

606 i f l e n( t o p 1 )<1:

607 f o r v a l i n t o p 1 :

608 l i s t c r e a t e r . append ( v a l )

609 n u m z e r o s = 1−l e n( t o p 1 )

610 f o r i i n r a n g e( n u m z e r o s ) :

611 l i s t c r e a t e r . append ( 0 )

612 e l i f l e n( t o p 1 )>1:

613 o r d e r i n g = np . a r g s o r t ( temp [ temp2 ] )

614 f o r v a l i n t o p 1 [ o r d e r i n g [−1 : ] ] :

615 l i s t c r e a t e r . append ( v a l )

616 e l s e:

617 f o r v a l i n t o p 1 :

618 l i s t c r e a t e r . append ( v a l )

619

620 l i s t c r e a t e r 1 = [ d a t e ]

621 i f l e n( low1 )<1:

622 f o r v a l i n low5 :

623 l i s t c r e a t e r 1 . append ( v a l )

624 n u m z e r o s = 1−l e n( low1 )

625 f o r i i n r a n g e( n u m z e r o s ) :

626 l i s t c r e a t e r 1 . append ( 0 )

627 e l i f l e n( low1 )>1:

628 o r d e r i n g = np . a r g s o r t ( temp [ temp1 ] )

629 f o r v a l i n low1 [ o r d e r i n g [ : 1 ] ] :

630 l i s t c r e a t e r 1 . append ( v a l )

631 e l s e:

632 f o r v a l i n low1 :

633 l i s t c r e a t e r 1 . append ( v a l )

634

635 h i g h l i s t . append ( l i s t c r e a t e r )

636 l o w l i s t . append ( l i s t c r e a t e r 1 )

637

638 highDF = pd . DataFrame ( h i g h l i s t )

639 lowDF = pd . DataFrame ( l o w l i s t )

640 641

642 # I n [ 3 6 ] :

643 644

645 ########################Carry t r a d e############################

646 R l o n g = [ ]

647 f o r i n d e x , row i n highDF . i t e r r o w s ( ) :

648 d a t e l i s t = [ ]

649 i f i n d e x==l e n( highDF )−1:

650 b r e a k

651

652 f o r i i n r a n g e( 1 ,l e n( row ) ) :

653 i f row [ i ]==0:

654 d a t e l i s t . append ( np . nan )

655 c o n t i n u e

656 temp = row [ i ] . s p l i t (’ ’) [ 0 ]

657 temp1 = temp+’ BID 1M ’

658 f b i d = Forward rates1M [ temp1 ] [ i n d e x ]

659

660 i f row [ i ] i n highDF . i l o c [ i n d e x + 1 ] . v a l u e s :

661 temp2 = temp+’ mid ’

662 s m i d = s p o t r a t e s 1 M [ temp2 ] [ i n d e x +1]

663 d a t e l i s t . append ( f b i d−s m i d )

664 e l s e:

665 temp2 = temp+’ a s k ’

666 s a s k = s p o t r a t e s 1 M [ temp2 ] [ i n d e x +1]

667 d a t e l i s t . append ( f b i d−s a s k )

668 R l o n g . append ( d a t e l i s t )

669 670

671 # I n [ 3 7 ] :

672 673

674 R s h o r t = [ ]

675 f o r i n d e x , row i n lowDF . i t e r r o w s ( ) :

676 d a t e l i s t = [ ]

677 i f i n d e x==l e n( lowDF )−1:

678 b r e a k

679 #p r i n t ( i n d e x )

680

681 f o r i i n r a n g e( 1 ,l e n( row ) ) :

682 i f row [ i ]==0:

683 d a t e l i s t . append ( np . nan )

684 c o n t i n u e

685 temp = row [ i ] . s p l i t (’ ’) [ 0 ]

686 temp1 = temp+’ b i d ’

687 s b i d = s p o t r a t e s 1 M [ temp1 ] [ i n d e x +1]

688

689 i f row [ i ] i n lowDF . i l o c [ i n d e x + 1 ] . v a l u e s :

690 temp2 = temp+’ MID 1M ’

691 f m i d = Forward rates1M [ temp2 ] [ i n d e x ]

692 d a t e l i s t . append(−f m i d+s b i d )

693 e l s e:

694 temp2 = temp+’ ASK 1M ’

695 f a s k = Forward rates1M [ temp2 ] [ i n d e x ]

696 d a t e l i s t . append(−f a s k+s b i d )

697 R s h o r t . append ( d a t e l i s t )

698 699

700 # I n [ 3 8 ] :

701 702

703 o u t 1 = pd . DataFrame ( R s h o r t )

704 o u t 2 = pd . DataFrame ( R l o n g )

705 r e s u l t = pd . c o n c a t ( [ o u t 1 , o u t 2 ] , a x i s =1, s o r t=F a l s e )

706 carry1M [’ carry1M1C ’]= r e s u l t . mean ( a x i s =1, s k i p n a=True )

707 708

709 # I n [ ] :

710 711

712 ################c u m u l a t i v e c a r r y t r a d e####################

713 carry1M [” cum return1M5C ”] = np . exp ( ( carry1M [’ carry1M5C ’] ) . cumsum ( ) )∗100

714 carry1M [” cum return1M4C ”] = np . exp ( ( carry1M [’ carry1M4C ’] ) . cumsum ( ) )∗100

715 carry1M [” cum return1M3C ”] = np . exp ( ( carry1M [’ carry1M3C ’] ) . cumsum ( ) )∗100

716 carry1M [” cum return1M2C ”] = np . exp ( ( carry1M [’ carry1M2C ’] ) . cumsum ( ) )∗100

717 carry1M [” cum return1M1C ”] = np . exp ( ( carry1M [’ carry1M1C ’] ) . cumsum ( ) )∗100

718 719

720 # I n [ 3 9 ] :

721 722

723 ##############Mean and Sharpe r a t i o ###########################

724 p r i n t( carry1M . mean ( )∗1 2 )

725 p r i n t( ( 1 2∗carry1M . mean ( ) ) / ( carry1M . s t d ( )∗np . s q r t ( 1 2 ) ) )

726 727

728 # I n [ 5 0 ] :

729 730

731 #############Norminal p−v a l u e from z−s c o r e ###################

732 p r i n t( s c i p y . s t a t s . norm . s f (abs( carry1M [’ carry1M1C ’] . mean ( )∗1 2 / ( ( 1 2∗ carry1M [’ carry1M1C ’] . s t d ( ) ) /np . s q r t (l e n( carry1M [’ carry1M1C ’] ) ) ) ) )∗2 )

733 p r i n t( s c i p y . s t a t s . norm . s f (abs( carry1M [’ carry1M2C ’] . mean ( )∗1 2 / ( ( 1 2∗ carry1M [’ carry1M2C ’] . s t d ( ) ) /np . s q r t (l e n( carry1M [’ carry1M2C ’] ) ) )

) )∗2 )

734 p r i n t( s c i p y . s t a t s . norm . s f (abs( carry1M [’ carry1M3C ’] . mean ( )∗1 2 / ( ( 1 2∗ carry1M [’ carry1M3C ’] . s t d ( ) ) /np . s q r t (l e n( carry1M [’ carry1M3C ’] ) ) ) ) )∗2 )

735 p r i n t( s c i p y . s t a t s . norm . s f (abs( carry1M [’ carry1M4C ’] . mean ( )∗1 2 / ( ( 1 2∗ carry1M [’ carry1M4C ’] . s t d ( ) ) /np . s q r t (l e n( carry1M [’ carry1M4C ’] ) ) ) ) )∗2 )

736 p r i n t( s c i p y . s t a t s . norm . s f (abs( carry1M [’ carry1M5C ’] . mean ( )∗1 2 / ( ( 1 2∗ carry1M [’ carry1M5C ’] . s t d ( ) ) /np . s q r t (l e n( carry1M [’ carry1M5C ’] ) ) ) ) )∗2 )

D.2 Exit- and Reverse Strategy 1-month re-balance based on VIX

1 #! / u s r / b i n / env python

2 # c o d i n g : u t f−8

4 # I n [ 1 ] :

5 6

7 i m p o r t pandas a s pd # pandas f o r w o r k i n g w i t h d a t a

8 i m p o r t m a t p l o t l i b . p y p l o t a s p l t # M a t p l o t l i b f o r p l o t t i n g

9 i m p o r t numpy a s np #Numpy i s a l i b r a r y f o r w o r k i n g w i t h n u m e r i c a l d a t a

10 i m p o r t o s

11 i m p o r t s c i p y . s t a t s

12 i m p o r t s y s

14 o s . c h d i r ( o s . getcwd ( ) )

15 carry1M = pd . r e a d e x c e l (” carry1M MID . x l s x ”, i n d e x c o l =0)

16 carry1M [’ Date ’]= carry1M [’ Date ’ ] . a s t y p e ( np . d a t e t i m e 6 4 )

17 18

19 # I n [ 2 ] :

20 21

22 #####V o l a t i l i t y i n d i c a t o r E x i t s t r a t e g y#####

23 v o l i n d e x = pd . r e a d e x c e l (” Vol i n d i c e s . x l s x ”)

25 s t d f r o m m e a n = 1 . 5

26 window = round( 2 5 3∗1 . 5 )

28 d e f g e t t h r e s h o l d ( v o l i n d e x , window , s t d f r o m m e a n ) :

29 r o l l i n g m e a n = v o l i n d e x [’ VIX I n d e x ’ ] . r o l l i n g ( window = window ) . mean ( )

30 r o l l i n g s t d = v o l i n d e x [’ VIX I n d e x ’ ] . r o l l i n g ( window = window ) . s t d ( )

31 t h r e s h o l d = ( s t d f r o m m e a n ∗ r o l l i n g s t d ) + r o l l i n g m e a n

33 r e t u r n t h r e s h o l d

35 v o l i n d e x [” T h r e s h o l d ”] = g e t t h r e s h o l d ( v o l i n d e x , window , s t d f r o m m e a n )

36 37

38 d e f g e t p o s i t i o n ( v o l i n d e x ) :

39 i n d e x a b o v e s t d = v o l i n d e x [’ VIX I n d e x ’] > v o l i n d e x [’ T h r e s h o l d ’]

40 i n d e x b e l o v s t d = v o l i n d e x [’ VIX I n d e x ’] < v o l i n d e x [’ T h r e s h o l d ’]

41 v o l i n d e x [’ pos ’] = ”NA”

42 v o l i n d e x [’ pos ’] = v o l i n d e x [’ pos ’ ] . where ( ˜ i n d e x a b o v e s t d , o t h e r = 0 )

43 v o l i n d e x [’ pos ’] = v o l i n d e x [’ pos ’ ] . where ( ˜ i n d e x b e l o v s t d , o t h e r = 1 )

45 r e t u r n v o l i n d e x [’ pos ’]

47 v o l i n d e x [’ pos ’] = g e t p o s i t i o n ( v o l i n d e x )

48 49

50 v o l i n d e x [’ D a t e s i g n a l ’] = v o l i n d e x [’ Date ’]

51 v o l i n d e x [’ Date ’] = v o l i n d e x [’ Date ’ ] . s h i f t (−1)

52 53

54 s t r a t e g y 1 M = carry1M

55 s t r a t e g y 1 M = v o l i n d e x . merge ( s t r a t e g y 1 M , on=’ Date ’, how=’ r i g h t ’)

56 57

58 # I n [ 3 ] :

59 60

61 ###########E x i t s t r a t e g y##########

62 s t r a t e g y 1 = [ ]

63 s t r a t e g y 2 = [ ]

64 s t r a t e g y 3 = [ ]

65 s t r a t e g y 4 = [ ]

66 s t r a t e g y 5 = [ ]

67 p o s i t i o n s = s t r a t e g y 1 M [’ pos ’ ] . v a l u e s

68 p o s i t i o n s d a t e = s t r a t e g y 1 M [’ Date ’ ] . v a l u e s

69 f o r i i n r a n g e( 0 ,l e n( p o s i t i o n s ) ) :

70 i f p o s i t i o n s [ i ]==1:

71 s t r a t e g y 1 . append ( 1∗s t r a t e g y 1 M [’ carry1M1C ’] . v a l u e s [ i ] )

72 s t r a t e g y 2 . append ( 1∗s t r a t e g y 1 M [’ carry1M2C ’] . v a l u e s [ i ] )

73 s t r a t e g y 3 . append ( 1∗s t r a t e g y 1 M [’ carry1M3C ’] . v a l u e s [ i ] )

74 s t r a t e g y 4 . append ( 1∗s t r a t e g y 1 M [’ carry1M4C ’] . v a l u e s [ i ] )

75 s t r a t e g y 5 . append ( 1∗s t r a t e g y 1 M [’ carry1M5C ’] . v a l u e s [ i ] )

76 e l s e:

77 s t r a t e g y 1 . append ( 0∗s t r a t e g y 1 M [’ carry1M1C ’] . v a l u e s [ i ] )

78 s t r a t e g y 2 . append ( 0∗s t r a t e g y 1 M [’ carry1M2C ’] . v a l u e s [ i ] )

79 s t r a t e g y 3 . append ( 0∗s t r a t e g y 1 M [’ carry1M3C ’] . v a l u e s [ i ] )

80 s t r a t e g y 4 . append ( 0∗s t r a t e g y 1 M [’ carry1M4C ’] . v a l u e s [ i ] )

81 s t r a t e g y 5 . append ( 0∗s t r a t e g y 1 M [’ carry1M5C ’] . v a l u e s [ i ] )

83 s t r a t e g y 1 M [’ E x i t strategy1M1C ’] = s t r a t e g y 1

84 s t r a t e g y 1 M [’ E x i t strategy1M2C ’] = s t r a t e g y 2

85 s t r a t e g y 1 M [’ E x i t strategy1M3C ’] = s t r a t e g y 3

86 s t r a t e g y 1 M [’ E x i t strategy1M4C ’] = s t r a t e g y 4

87 s t r a t e g y 1 M [’ E x i t strategy1M5C ’] = s t r a t e g y 5

88 89

90 # I n [ 4 ] :

91 92

93 #P e r f o r m a n c e m e a s u r e s : F u l l sample p e r i o d 1999−2020

94 p r i n t( s t r a t e g y 1 M . mean ( )∗1 2 )

95 p r i n t( ( 1 2∗s t r a t e g y 1 M . mean ( ) ) / ( s t r a t e g y 1 M . s t d ( )∗np . s q r t ( 1 2 ) ) )

96 97

98 # I n [ 5 ] :

99 100

101 #Nominal p−v a l u e

102 p r i n t(’ 1M1C’)

103 p r i n t( s c i p y . s t a t s . norm . s f (abs( s t r a t e g y 1 M [’ E x i t strategy1M1C ’] . mean ( )∗1 2 / ( ( 1 2∗s t r a t e g y 1 M [’ E x i t strategy1M1C ’ ] . s t d ( ) ) /np . s q r t (l e n( s t r a t e g y 1 M [’ E x i t strategy1M1C ’] ) ) ) ) )∗2 )

104 p r i n t(’ 1M2C’)

105 p r i n t( s c i p y . s t a t s . norm . s f (abs( s t r a t e g y 1 M [’ E x i t strategy1M2C ’] . mean ( )∗1 2 / ( ( 1 2∗s t r a t e g y 1 M [’ E x i t strategy1M2C ’ ] . s t d ( ) ) /np . s q r t (l e n( s t r a t e g y 1 M [’ E x i t strategy1M2C ’] ) ) ) ) )∗2 )

106 p r i n t(’ 1M3C’)

107 p r i n t( s c i p y . s t a t s . norm . s f (abs( s t r a t e g y 1 M [’ E x i t strategy1M3C ’] . mean ( )∗1 2 / ( ( 1 2∗s t r a t e g y 1 M [’ E x i t strategy1M3C ’ ] . s t d ( ) ) /np . s q r t (l e n( s t r a t e g y 1 M [’ E x i t strategy1M3C ’] ) ) ) ) )∗2 )

108 p r i n t(’ 1M4C’)

109 p r i n t( s c i p y . s t a t s . norm . s f (abs( s t r a t e g y 1 M [’ E x i t strategy1M4C ’] . mean

( )∗1 2 / ( ( 1 2∗s t r a t e g y 1 M [’ E x i t strategy1M4C ’ ] . s t d ( ) ) /np . s q r t (l e n( s t r a t e g y 1 M [’ E x i t strategy1M4C ’] ) ) ) ) )∗2 )

110 p r i n t(’ 1M5C’)

111 p r i n t( s c i p y . s t a t s . norm . s f (abs( s t r a t e g y 1 M [’ E x i t strategy1M5C ’] . mean ( )∗1 2 / ( ( 1 2∗s t r a t e g y 1 M [’ E x i t strategy1M5C ’ ] . s t d ( ) ) /np . s q r t (l e n( s t r a t e g y 1 M [’ E x i t strategy1M5C ’] ) ) ) ) )∗2 )

112 113

114 # I n [ 1 2 ] :

115 116

117 #F r a c t i o n o f n e g a t i v e#

118 p r i n t(sum( s t r a t e g y 1 M [’ E x i t strategy1M1C ’]<0 ) / (sum( s t r a t e g y 1 M [’ E x i t strategy1M1C ’]>=0)+sum( s t r a t e g y 1 M [’ E x i t strategy1M1C ’]<0 ) ) )

119 p r i n t(sum( s t r a t e g y 1 M [’ E x i t strategy1M2C ’]<0 ) / (sum( s t r a t e g y 1 M [’ E x i t strategy1M2C ’]>=0)+sum( s t r a t e g y 1 M [’ E x i t strategy1M2C ’]<0 ) ) )

120 p r i n t(sum( s t r a t e g y 1 M [’ E x i t strategy1M3C ’]<0 ) / (sum( s t r a t e g y 1 M [’ E x i t strategy1M3C ’]>=0)+sum( s t r a t e g y 1 M [’ E x i t strategy1M3C ’]<0 ) ) )

121 p r i n t(sum( s t r a t e g y 1 M [’ E x i t strategy1M4C ’]<0 ) / (sum( s t r a t e g y 1 M [’ E x i t strategy1M4C ’]>=0)+sum( s t r a t e g y 1 M [’ E x i t strategy1M4C ’]<0 ) ) )

122 p r i n t(sum( s t r a t e g y 1 M [’ E x i t strategy1M5C ’]<0 ) / (sum( s t r a t e g y 1 M [’ E x i t strategy1M5C ’]>=0)+sum( s t r a t e g y 1 M [’ E x i t strategy1M5C ’]<0 ) ) )

123 124

125 # I n [ 7 ] :

126 127

128 ##############Sub sample t e s t###############

129 i n d i c e s = l i s t(r a n g e( 8 5 , 2 5 2 ) ) #p e r i o d 1999−2006

130 d r o p p e d d a t a 1 = s t r a t e g y 1 M . drop ( i n d e x=i n d i c e s )

131 #P e r f o r m a n c e m e a s u r e s

132 mean = d r o p p e d d a t a 1 . mean ( )∗12

133 s r = 12∗d r o p p e d d a t a 1 . mean ( ) / ( d r o p p e d d a t a 1 . s t d ( )∗np . s q r t ( 1 2 ) )

134 135

136 # I n [ 9 ] :

137 138

139 #P−v a l u e

140 p r i n t(’ 1M1C’)

141 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 1 [’ E x i t strategy1M1C ’] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 1 [’ E x i t strategy1M1C ’] . s t d ( ) ) /np .

s q r t (l e n( d r o p p e d d a t a 1 [’ E x i t strategy1M1C ’] ) ) ) ) )∗2 )

142 p r i n t(’ 1M2C’)

143 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 1 [’ E x i t strategy1M2C ’] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 1 [’ E x i t strategy1M2C ’] . s t d ( ) ) /np .

s q r t (l e n( d r o p p e d d a t a 1 [’ E x i t strategy1M2C ’] ) ) ) ) )∗2 )

144 p r i n t(’ 1M3C’)

145 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 1 [’ E x i t strategy1M3C ’] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 1 [’ E x i t strategy1M3C ’] . s t d ( ) ) /np .

s q r t (l e n( d r o p p e d d a t a 1 [’ E x i t strategy1M3C ’] ) ) ) ) )∗2 )

146 p r i n t(’ 1M4C’)

147 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 1 [’ E x i t strategy1M4C ’] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 1 [’ E x i t strategy1M4C ’] . s t d ( ) ) /np .

s q r t (l e n( d r o p p e d d a t a 1 [’ E x i t strategy1M4C ’] ) ) ) ) )∗2 )

148 p r i n t(’ 1M5C’)

149 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 1 [’ E x i t strategy1M5C ’] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 1 [’ E x i t strategy1M5C ’] . s t d ( ) ) /np .

s q r t (l e n( d r o p p e d d a t a 1 [’ E x i t strategy1M5C ’] ) ) ) ) )∗2 )

150 151

152 # I n [ 1 9 ] :

153 154

155 #F r a c t i o n o f n e g a t i v e#

156 p r i n t(sum( d r o p p e d d a t a 1 [’ E x i t strategy1M1C ’]<0 ) / (sum( d r o p p e d d a t a 1 [’ E x i t strategy1M1C ’]>=0)+sum( d r o p p e d d a t a 1 [’ E x i t strategy1M1C ’ ]<0 ) ) )

157 p r i n t(sum( d r o p p e d d a t a 1 [’ E x i t strategy1M2C ’]<0 ) / (sum( d r o p p e d d a t a 1 [’ E x i t strategy1M2C ’]>=0)+sum( d r o p p e d d a t a 1 [’ E x i t strategy1M2C ’ ]<0 ) ) )

158 p r i n t(sum( d r o p p e d d a t a 1 [’ E x i t strategy1M3C ’]<0 ) / (sum( d r o p p e d d a t a 1 [’ E x i t strategy1M3C ’]>=0)+sum( d r o p p e d d a t a 1 [’ E x i t strategy1M3C ’ ]<0 ) ) )

159 p r i n t(sum( d r o p p e d d a t a 1 [’ E x i t strategy1M4C ’]<0 ) / (sum( d r o p p e d d a t a 1 [’ E x i t strategy1M4C ’]>=0)+sum( d r o p p e d d a t a 1 [’ E x i t strategy1M4C ’ ]<0 ) ) )

160 p r i n t(sum( d r o p p e d d a t a 1 [’ E x i t strategy1M5C ’]<0 ) / (sum( d r o p p e d d a t a 1 [’ E x i t strategy1M5C ’]>=0)+sum( d r o p p e d d a t a 1 [’ E x i t strategy1M5C ’ ]<0 ) ) )

161 162

163 # I n [ 2 0 ] :

164 165

166 i n d i c e s = l i s t(r a n g e( 0 , 1 6 8 ) ) #p e r i o d 2013−2020

167 d r o p p e d d a t a 2 = s t r a t e g y 1 M . drop ( i n d e x=i n d i c e s )

168 #P e r f o r m a n c e m e a s u r e s

169 mean = d r o p p e d d a t a 2 . mean ( )∗12

170 s r = 12∗d r o p p e d d a t a 2 . mean ( ) / ( d r o p p e d d a t a 2 . s t d ( )∗np . s q r t ( 1 2 ) )

171 172

173 # I n [ 2 3 ] :

174 175

176 #P−v a l u e

177 p r i n t(’ 1M1C’)

178 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 2 [’ E x i t strategy1M1C ’] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 2 [’ E x i t strategy1M1C ’] . s t d ( ) ) /np .

s q r t (l e n( d r o p p e d d a t a 2 [’ E x i t strategy1M1C ’] ) ) ) ) )∗2 )

179 p r i n t(’ 1M2C’)

180 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 2 [’ E x i t strategy1M2C ’] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 2 [’ E x i t strategy1M2C ’] . s t d ( ) ) /np .

s q r t (l e n( d r o p p e d d a t a 2 [’ E x i t strategy1M2C ’] ) ) ) ) )∗2 )

181 p r i n t(’ 1M3C’)

182 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 2 [’ E x i t strategy1M3C ’] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 2 [’ E x i t strategy1M3C ’] . s t d ( ) ) /np .

s q r t (l e n( d r o p p e d d a t a 2 [’ E x i t strategy1M3C ’] ) ) ) ) )∗2 )

183 p r i n t(’ 1M4C’)

184 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 2 [’ E x i t strategy1M4C ’] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 2 [’ E x i t strategy1M4C ’] . s t d ( ) ) /np .

s q r t (l e n( d r o p p e d d a t a 2 [’ E x i t strategy1M4C ’] ) ) ) ) )∗2 )

185 p r i n t(’ 1M5C’)

186 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 2 [’ E x i t strategy1M5C ’] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 2 [’ E x i t strategy1M5C ’] . s t d ( ) ) /np .

s q r t (l e n( d r o p p e d d a t a 2 [’ E x i t strategy1M5C ’] ) ) ) ) )∗2 )

187 188

189 # I n [ 2 4 ] :

190 191

192 #F r a c t i o n o f n e g a t i v e#

193 p r i n t(sum( d r o p p e d d a t a 2 [’ E x i t strategy1M1C ’]<0 ) / (sum( d r o p p e d d a t a 2 [’ E x i t strategy1M1C ’]>=0)+sum( d r o p p e d d a t a 2 [’ E x i t strategy1M1C ’ ]<0 ) ) )

194 p r i n t(sum( d r o p p e d d a t a 2 [’ E x i t strategy1M2C ’]<0 ) / (sum( d r o p p e d d a t a 2 [’ E x i t strategy1M2C ’]>=0)+sum( d r o p p e d d a t a 2 [’ E x i t strategy1M2C ’ ]<0 ) ) )

195 p r i n t(sum( d r o p p e d d a t a 2 [’ E x i t strategy1M3C ’]<0 ) / (sum( d r o p p e d d a t a 2 [’ E x i t strategy1M3C ’]>=0)+sum( d r o p p e d d a t a 2 [’ E x i t strategy1M3C ’ ]<0 ) ) )

196 p r i n t(sum( d r o p p e d d a t a 2 [’ E x i t strategy1M4C ’]<0 ) / (sum( d r o p p e d d a t a 2 [’ E x i t strategy1M4C ’]>=0)+sum( d r o p p e d d a t a 2 [’ E x i t strategy1M4C ’ ]<0 ) ) )

197 p r i n t(sum( d r o p p e d d a t a 2 [’ E x i t strategy1M5C ’]<0 ) / (sum( d r o p p e d d a t a 2 [’ E x i t strategy1M5C ’]>=0)+sum( d r o p p e d d a t a 2 [’ E x i t strategy1M5C ’ ]<0 ) ) )

198

199

200 # I n [ 2 5 ] :

201 202

203 i n d i c e s = l i s t(r a n g e( 0 , 8 4 ) ) #p e r i o d 2006−2013

204 d r o p p e d d a t a 3 = s t r a t e g y 1 M . drop ( i n d e x=i n d i c e s )

205 i n d i c e s 1=l i s t(r a n g e( 1 6 9 , 2 5 2 ) )

206 d r o p p e d d a t a 3 = d r o p p e d d a t a 3 . drop ( i n d e x=i n d i c e s 1 )

207 #P e r f o r m a n c e m e a s u r e s

208 mean = d r o p p e d d a t a 3 . mean ( )∗12

209 s r = 12∗d r o p p e d d a t a 3 . mean ( ) / ( d r o p p e d d a t a 3 . s t d ( )∗np . s q r t ( 1 2 ) )

210 211

212 # I n [ 3 0 ] :

213 214

215 #P−v a l u e

216 p r i n t(’ 1M1C’)

217 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 3 [’ E x i t strategy1M1C ’] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 3 [’ E x i t strategy1M1C ’] . s t d ( ) ) /np .

s q r t (l e n( d r o p p e d d a t a 3 [’ E x i t strategy1M1C ’] ) ) ) ) )∗2 )

218 p r i n t(’ 1M2C’)

219 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 3 [’ E x i t strategy1M2C ’] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 3 [’ E x i t strategy1M2C ’] . s t d ( ) ) /np .

s q r t (l e n( d r o p p e d d a t a 3 [’ E x i t strategy1M2C ’] ) ) ) ) )∗2 )

220 p r i n t(’ 1M3C’)

221 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 3 [’ E x i t strategy1M3C ’] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 3 [’ E x i t strategy1M3C ’] . s t d ( ) ) /np .

s q r t (l e n( d r o p p e d d a t a 3 [’ E x i t strategy1M3C ’] ) ) ) ) )∗2 )

222 p r i n t(’ 1M4C’)

223 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 3 [’ E x i t strategy1M4C ’] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 3 [’ E x i t strategy1M4C ’] . s t d ( ) ) /np .

s q r t (l e n( d r o p p e d d a t a 3 [’ E x i t strategy1M4C ’] ) ) ) ) )∗2 )

224 p r i n t(’ 1M5C’)

225 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 3 [’ E x i t strategy1M5C ’] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 3 [’ E x i t strategy1M5C ’] . s t d ( ) ) /np .

s q r t (l e n( d r o p p e d d a t a 3 [’ E x i t strategy1M5C ’] ) ) ) ) )∗2 )

226 227

228 # I n [ 3 1 ] :

229 230

231 #F r a c t i o n o f n e g a t i v e#

232 p r i n t(sum( d r o p p e d d a t a 3 [’ E x i t strategy1M1C ’]<0 ) / (sum( d r o p p e d d a t a 3 [’ E x i t strategy1M1C ’]>=0)+sum( d r o p p e d d a t a 3 [’ E x i t strategy1M1C ’ ]<0 ) ) )

233 p r i n t(sum( d r o p p e d d a t a 3 [’ E x i t strategy1M2C ’]<0 ) / (sum( d r o p p e d d a t a 3 [’ E x i t strategy1M2C ’]>=0)+sum( d r o p p e d d a t a 3 [’ E x i t strategy1M2C ’ ]<0 ) ) )

234 p r i n t(sum( d r o p p e d d a t a 3 [’ E x i t strategy1M3C ’]<0 ) / (sum( d r o p p e d d a t a 3 [’ E x i t strategy1M3C ’]>=0)+sum( d r o p p e d d a t a 3 [’ E x i t strategy1M3C ’ ]<0 ) ) )

235 p r i n t(sum( d r o p p e d d a t a 3 [’ E x i t strategy1M4C ’]<0 ) / (sum( d r o p p e d d a t a 3 [’ E x i t strategy1M4C ’]>=0)+sum( d r o p p e d d a t a 3 [’ E x i t strategy1M4C ’ ]<0 ) ) )

236 p r i n t(sum( d r o p p e d d a t a 3 [’ E x i t strategy1M5C ’]<0 ) / (sum( d r o p p e d d a t a 3 [’ E x i t strategy1M5C ’]>=0)+sum( d r o p p e d d a t a 3 [’ E x i t strategy1M5C ’ ]<0 ) ) )

237 238

239 # I n [ 1 0 ] :

240 241

242 ########V o l a t i l i t y i n d i c a t o r R e v e r s e s t r a t e g y#########

243 v o l i n d e x = pd . r e a d e x c e l (” Vol i n d i c e s . x l s x ”)

244 245

246 s t d f r o m m e a n = 1 . 5

247 window = round( 2 5 3∗1 . 5 )

248

249 d e f g e t t h r e s h o l d ( v o l i n d e x , window , s t d f r o m m e a n ) :

250 r o l l i n g m e a n = v o l i n d e x [’ VIX I n d e x ’ ] . r o l l i n g ( window = window ) . mean ( )

251 r o l l i n g s t d = v o l i n d e x [’ VIX I n d e x ’ ] . r o l l i n g ( window = window ) . s t d ( )

252 t h r e s h o l d = ( s t d f r o m m e a n ∗ r o l l i n g s t d ) + r o l l i n g m e a n

253

254 r e t u r n t h r e s h o l d

255

256 v o l i n d e x [” T h r e s h o l d ”] = g e t t h r e s h o l d ( v o l i n d e x , window , s t d f r o m m e a n )

257 258

259 d e f g e t p o s i t i o n ( v o l i n d e x ) :

260 i n d e x a b o v e s t d = v o l i n d e x [’ VIX I n d e x ’] > v o l i n d e x [’ T h r e s h o l d ’]

261 i n d e x b e l o v s t d = v o l i n d e x [’ VIX I n d e x ’] < v o l i n d e x [’ T h r e s h o l d ’]

262 v o l i n d e x [’ pos ’] = ”NA”

263 v o l i n d e x [’ pos ’] = v o l i n d e x [’ pos ’ ] . where ( ˜ i n d e x a b o v e s t d , o t h e r = −1)

264 v o l i n d e x [’ pos ’] = v o l i n d e x [’ pos ’ ] . where ( ˜ i n d e x b e l o v s t d ,

o t h e r = 1 )

265

266 r e t u r n v o l i n d e x [’ pos ’]

267

268 v o l i n d e x [’ pos ’] = g e t p o s i t i o n ( v o l i n d e x )

269 270

271 v o l i n d e x [’ D a t e s i g n a l ’] = v o l i n d e x [’ Date ’]

272 v o l i n d e x [’ Date ’] = v o l i n d e x [’ Date ’ ] . s h i f t (−1)

273

274 s t r a t e g y 1 M = carry1M

275 s t r a t e g y 1 M = v o l i n d e x . merge ( s t r a t e g y 1 M , on=’ Date ’, how=’ r i g h t ’)

276 277

278 # I n [ 1 1 ] :

279 280

281 ###R e v e r s e s t r a t e g y####

282 s t r a t e g y 1 = [ ]

283 s t r a t e g y 2 = [ ]

284 s t r a t e g y 3 = [ ]

285 s t r a t e g y 4 = [ ]

286 s t r a t e g y 5 = [ ]

287 p o s i t i o n s = s t r a t e g y 1 M [’ pos ’ ] . v a l u e s

288 p o s i t i o n s d a t e = s t r a t e g y 1 M [’ Date ’ ] . v a l u e s

289 f o r i i n r a n g e( 0 ,l e n( p o s i t i o n s ) ) :

290 i f p o s i t i o n s [ i ]==1:

291 s t r a t e g y 1 . append ( 1∗s t r a t e g y 1 M [’ carry1M1C ’] . v a l u e s [ i ] )

292 s t r a t e g y 2 . append ( 1∗s t r a t e g y 1 M [’ carry1M2C ’] . v a l u e s [ i ] )

293 s t r a t e g y 3 . append ( 1∗s t r a t e g y 1 M [’ carry1M3C ’] . v a l u e s [ i ] )

294 s t r a t e g y 4 . append ( 1∗s t r a t e g y 1 M [’ carry1M4C ’] . v a l u e s [ i ] )

295 s t r a t e g y 5 . append ( 1∗s t r a t e g y 1 M [’ carry1M5C ’] . v a l u e s [ i ] )

296 e l s e:

297 s t r a t e g y 1 . append (−1∗s t r a t e g y 1 M [’ carry1M1C ’] . v a l u e s [ i ] )

298 s t r a t e g y 2 . append (−1∗s t r a t e g y 1 M [’ carry1M2C ’] . v a l u e s [ i ] )

299 s t r a t e g y 3 . append (−1∗s t r a t e g y 1 M [’ carry1M3C ’] . v a l u e s [ i ] )

300 s t r a t e g y 4 . append (−1∗s t r a t e g y 1 M [’ carry1M4C ’] . v a l u e s [ i ] )

301 s t r a t e g y 5 . append (−1∗s t r a t e g y 1 M [’ carry1M5C ’] . v a l u e s [ i ] )

302

303 s t r a t e g y 1 M [’ R e v e r s e strategy1M1C ’] = s t r a t e g y 1

304 s t r a t e g y 1 M [’ R e v e r s e strategy1M2C ’] = s t r a t e g y 2

305 s t r a t e g y 1 M [’ R e v e r s e strategy1M3C ’] = s t r a t e g y 3

306 s t r a t e g y 1 M [’ R e v e r s e strategy1M4C ’] = s t r a t e g y 4

307 s t r a t e g y 1 M [’ R e v e r s e strategy1M5C ’] = s t r a t e g y 5

308 309

310 # I n [ 1 3 ] :

311 312

313 #P e r f o r m a n c e m e a s u r e s : F u l l sample p e r i o d 1999−2020

314 p r i n t( s t r a t e g y 1 M . mean ( )∗1 2 )

315 p r i n t( ( 1 2∗s t r a t e g y 1 M . mean ( ) ) / ( s t r a t e g y 1 M . s t d ( )∗np . s q r t ( 1 2 ) ) )

316 317

318 # I n [ 1 6 ] :

319 320

321 #Nominal p−v a l u e

322 p r i n t(’ 1M1C’)

323 p r i n t( s c i p y . s t a t s . norm . s f (abs( s t r a t e g y 1 M [’ R e v e r s e strategy1M1C ’ ] . mean ( )∗1 2 / ( ( 1 2∗s t r a t e g y 1 M [’ R e v e r s e strategy1M1C ’ ] . s t d ( ) ) /np .

s q r t (l e n( s t r a t e g y 1 M [’ R e v e r s e strategy1M1C ’ ] ) ) ) ) )∗2 )

324 p r i n t(’ 1M2C’)

325 p r i n t( s c i p y . s t a t s . norm . s f (abs( s t r a t e g y 1 M [’ R e v e r s e strategy1M2C ’ ] . mean ( )∗1 2 / ( ( 1 2∗s t r a t e g y 1 M [’ R e v e r s e strategy1M2C ’ ] . s t d ( ) ) /np .

s q r t (l e n( s t r a t e g y 1 M [’ R e v e r s e strategy1M2C ’ ] ) ) ) ) )∗2 )

326 p r i n t(’ 1M3C’)

327 p r i n t( s c i p y . s t a t s . norm . s f (abs( s t r a t e g y 1 M [’ R e v e r s e strategy1M3C ’ ] . mean ( )∗1 2 / ( ( 1 2∗s t r a t e g y 1 M [’ R e v e r s e strategy1M3C ’ ] . s t d ( ) ) /np .

s q r t (l e n( s t r a t e g y 1 M [’ R e v e r s e strategy1M3C ’ ] ) ) ) ) )∗2 )

328 p r i n t(’ 1M4C’)

329 p r i n t( s c i p y . s t a t s . norm . s f (abs( s t r a t e g y 1 M [’ R e v e r s e strategy1M4C ’ ] . mean ( )∗1 2 / ( ( 1 2∗s t r a t e g y 1 M [’ R e v e r s e strategy1M4C ’ ] . s t d ( ) ) /np .

s q r t (l e n( s t r a t e g y 1 M [’ R e v e r s e strategy1M4C ’ ] ) ) ) ) )∗2 )

330 p r i n t(’ 1M5C’)

331 p r i n t( s c i p y . s t a t s . norm . s f (abs( s t r a t e g y 1 M [’ R e v e r s e strategy1M5C ’ ] . mean ( )∗1 2 / ( ( 1 2∗s t r a t e g y 1 M [’ R e v e r s e strategy1M5C ’ ] . s t d ( ) ) /np .

s q r t (l e n( s t r a t e g y 1 M [’ R e v e r s e strategy1M5C ’ ] ) ) ) ) )∗2 )

332 333

334 # I n [ 1 7 ] :

335 336

337 #F r a c t i o n o f n e g a t i v e

338 p r i n t(sum( s t r a t e g y 1 M [’ R e v e r s e strategy1M1C ’]<0 ) / (sum( s t r a t e g y 1 M [’ R e v e r s e strategy1M1C ’]>=0)+sum( s t r a t e g y 1 M [’ R e v e r s e strategy1M1C

’]<0 ) ) )

339 p r i n t(sum( s t r a t e g y 1 M [’ R e v e r s e strategy1M2C ’]<0 ) / (sum( s t r a t e g y 1 M [’ R e v e r s e strategy1M2C ’]>=0)+sum( s t r a t e g y 1 M [’ R e v e r s e strategy1M2C

’]<0 ) ) )

340 p r i n t(sum( s t r a t e g y 1 M [’ R e v e r s e strategy1M3C ’]<0 ) / (sum( s t r a t e g y 1 M [’ R e v e r s e strategy1M3C ’]>=0)+sum( s t r a t e g y 1 M [’ R e v e r s e strategy1M3C

’]<0 ) ) )

341 p r i n t(sum( s t r a t e g y 1 M [’ R e v e r s e strategy1M4C ’]<0 ) / (sum( s t r a t e g y 1 M [’ R e v e r s e strategy1M4C ’]>=0)+sum( s t r a t e g y 1 M [’ R e v e r s e strategy1M4C

’]<0 ) ) )

342 p r i n t(sum( s t r a t e g y 1 M [’ R e v e r s e strategy1M5C ’]<0 ) / (sum( s t r a t e g y 1 M [’ R e v e r s e strategy1M5C ’]>=0)+sum( s t r a t e g y 1 M [’ R e v e r s e strategy1M5C

’]<0 ) ) )

343 344

345 # I n [ 1 8 ] :

346 347

348 ########S p l i t i n t o 3 sample p e r i o d s##########

349 i n d i c e s = l i s t(r a n g e( 8 5 , 2 5 2 ) ) #p e r i o d 1999−2006

350 d r o p p e d d a t a 1 = s t r a t e g y 1 M . drop ( i n d e x=i n d i c e s )

351 #P e r f o r m a n c e m e a s u r e s

352 mean = d r o p p e d d a t a 1 . mean ( )∗12

353 s r = 12∗d r o p p e d d a t a 1 . mean ( ) / ( d r o p p e d d a t a 1 . s t d ( )∗np . s q r t ( 1 2 ) )

354 355

356 # I n [ 2 1 ] :

357 358

359 #Nominal p−v a l u e

360 p r i n t(’ 1M1C’)

361 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M1C ’ ] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 1 [’ R e v e r s e strategy1M1C ’ ] . s t d ( ) ) / np . s q r t (l e n( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M1C ’ ] ) ) ) ) )∗2 )

362 p r i n t(’ 1M2C’)

363 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M2C ’ ] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 1 [’ R e v e r s e strategy1M2C ’ ] . s t d ( ) ) / np . s q r t (l e n( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M2C ’ ] ) ) ) ) )∗2 )

364 p r i n t(’ 1M3C’)

365 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M3C ’ ] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 1 [’ R e v e r s e strategy1M3C ’ ] . s t d ( ) ) / np . s q r t (l e n( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M3C ’ ] ) ) ) ) )∗2 )

366 p r i n t(’ 1M4C’)

367 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M4C ’ ] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 1 [’ R e v e r s e strategy1M4C ’ ] . s t d ( ) ) / np . s q r t (l e n( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M4C ’ ] ) ) ) ) )∗2 )

368 p r i n t(’ 1M5C’)

369 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M5C ’ ] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 1 [’ R e v e r s e strategy1M5C ’ ] . s t d ( ) ) / np . s q r t (l e n( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M5C ’ ] ) ) ) ) )∗2 )

370 371

372 # I n [ 2 2 ] :

373 374

375 #F r a c t i o n o f n e g a t i v e

376 p r i n t(sum( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M1C ’]<0 ) / (sum(

d r o p p e d d a t a 1 [’ R e v e r s e strategy1M1C ’]>=0)+sum( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M1C ’]<0 ) ) )

377 p r i n t(sum( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M2C ’]<0 ) / (sum(

d r o p p e d d a t a 1 [’ R e v e r s e strategy1M2C ’]>=0)+sum( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M2C ’]<0 ) ) )

378 p r i n t(sum( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M3C ’]<0 ) / (sum(

d r o p p e d d a t a 1 [’ R e v e r s e strategy1M3C ’]>=0)+sum( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M3C ’]<0 ) ) )

379 p r i n t(sum( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M4C ’]<0 ) / (sum(

d r o p p e d d a t a 1 [’ R e v e r s e strategy1M4C ’]>=0)+sum( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M4C ’]<0 ) ) )

380 p r i n t(sum( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M5C ’]<0 ) / (sum(

d r o p p e d d a t a 1 [’ R e v e r s e strategy1M5C ’]>=0)+sum( d r o p p e d d a t a 1 [’ R e v e r s e strategy1M5C ’]<0 ) ) )

381 382

383 # I n [ 2 3 ] :

384 385

386 i n d i c e s = l i s t(r a n g e( 0 , 1 6 8 ) ) #p e r i o d 2013−2020

387 d r o p p e d d a t a 2 = s t r a t e g y 1 M . drop ( i n d e x=i n d i c e s )

388 #P e r f o r m a n c e m e a s u r e s

389 mean = d r o p p e d d a t a 2 . mean ( )∗12

390 s r = 12∗d r o p p e d d a t a 2 . mean ( ) / ( d r o p p e d d a t a 2 . s t d ( )∗np . s q r t ( 1 2 ) )

391 392

393 # I n [ 2 7 ] :

394 395

396 #Nominal p−v a l u e

397 p r i n t(’ 1M1C’)

398 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M1C ’ ] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 2 [’ R e v e r s e strategy1M1C ’ ] . s t d ( ) ) / np . s q r t (l e n( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M1C ’ ] ) ) ) ) )∗2 )

399 p r i n t(’ 1M2C’)

400 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M2C ’ ] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 2 [’ R e v e r s e strategy1M2C ’ ] . s t d ( ) ) / np . s q r t (l e n( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M2C ’ ] ) ) ) ) )∗2 )

401 p r i n t(’ 1M3C’)

402 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M3C ’ ] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 2 [’ R e v e r s e strategy1M3C ’ ] . s t d ( ) ) /

np . s q r t (l e n( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M3C ’ ] ) ) ) ) )∗2 )

403 p r i n t(’ 1M4C’)

404 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M4C ’ ] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 2 [’ R e v e r s e strategy1M4C ’ ] . s t d ( ) ) / np . s q r t (l e n( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M4C ’ ] ) ) ) ) )∗2 )

405 p r i n t(’ 1M5C’)

406 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M5C ’ ] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 2 [’ R e v e r s e strategy1M5C ’ ] . s t d ( ) ) / np . s q r t (l e n( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M5C ’ ] ) ) ) ) )∗2 )

407 408

409 # I n [ 2 8 ] :

410 411

412 #F r a c t i o n o f n e g a t i v e

413 p r i n t(sum( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M1C ’]<0 ) / (sum(

d r o p p e d d a t a 2 [’ R e v e r s e strategy1M1C ’]>=0)+sum( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M1C ’]<0 ) ) )

414 p r i n t(sum( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M2C ’]<0 ) / (sum(

d r o p p e d d a t a 2 [’ R e v e r s e strategy1M2C ’]>=0)+sum( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M2C ’]<0 ) ) )

415 p r i n t(sum( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M3C ’]<0 ) / (sum(

d r o p p e d d a t a 2 [’ R e v e r s e strategy1M3C ’]>=0)+sum( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M3C ’]<0 ) ) )

416 p r i n t(sum( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M4C ’]<0 ) / (sum(

d r o p p e d d a t a 2 [’ R e v e r s e strategy1M4C ’]>=0)+sum( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M4C ’]<0 ) ) )

417 p r i n t(sum( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M5C ’]<0 ) / (sum(

d r o p p e d d a t a 2 [’ R e v e r s e strategy1M5C ’]>=0)+sum( d r o p p e d d a t a 2 [’ R e v e r s e strategy1M5C ’]<0 ) ) )

418 419

420 # I n [ 2 9 ] :

421 422

423 i n d i c e s = l i s t(r a n g e( 0 , 8 4 ) ) #p e r i o d 2006−2013

424 d r o p p e d d a t a 3 = s t r a t e g y 1 M . drop ( i n d e x=i n d i c e s )

425 i n d i c e s 1=l i s t(r a n g e( 1 6 9 , 2 5 2 ) )

426 d r o p p e d d a t a 3 = d r o p p e d d a t a 3 . drop ( i n d e x=i n d i c e s 1 )

427 #P e r f o r m a n c e m e a s u r e s

428 mean = d r o p p e d d a t a 3 . mean ( )∗12

429 s r = 12∗d r o p p e d d a t a 3 . mean ( ) / ( d r o p p e d d a t a 3 . s t d ( )∗np . s q r t ( 1 2 ) )

430 431

432 # I n [ 3 2 ] :

433

434

435 #Nominal p−v a l u e

436 p r i n t(’ 1M1C’)

437 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M1C ’ ] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 3 [’ R e v e r s e strategy1M1C ’ ] . s t d ( ) ) / np . s q r t (l e n( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M1C ’ ] ) ) ) ) )∗2 )

438 p r i n t(’ 1M2C’)

439 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M2C ’ ] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 3 [’ R e v e r s e strategy1M2C ’ ] . s t d ( ) ) / np . s q r t (l e n( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M2C ’ ] ) ) ) ) )∗2 )

440 p r i n t(’ 1M3C’)

441 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M3C ’ ] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 3 [’ R e v e r s e strategy1M3C ’ ] . s t d ( ) ) / np . s q r t (l e n( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M3C ’ ] ) ) ) ) )∗2 )

442 p r i n t(’ 1M4C’)

443 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M4C ’ ] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 3 [’ R e v e r s e strategy1M4C ’ ] . s t d ( ) ) / np . s q r t (l e n( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M4C ’ ] ) ) ) ) )∗2 )

444 p r i n t(’ 1M5C’)

445 p r i n t( s c i p y . s t a t s . norm . s f (abs( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M5C ’ ] . mean ( )∗1 2 / ( ( 1 2∗d r o p p e d d a t a 3 [’ R e v e r s e strategy1M5C ’ ] . s t d ( ) ) / np . s q r t (l e n( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M5C ’ ] ) ) ) ) )∗2 )

446 447

448 # I n [ 3 3 ] :

449 450

451 #F r a c t i o n o f n e g a t i v e

452 p r i n t(sum( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M1C ’]<0 ) / (sum(

d r o p p e d d a t a 3 [’ R e v e r s e strategy1M1C ’]>=0)+sum( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M1C ’]<0 ) ) )

453 p r i n t(sum( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M2C ’]<0 ) / (sum(

d r o p p e d d a t a 3 [’ R e v e r s e strategy1M2C ’]>=0)+sum( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M2C ’]<0 ) ) )

454 p r i n t(sum( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M3C ’]<0 ) / (sum(

d r o p p e d d a t a 3 [’ R e v e r s e strategy1M3C ’]>=0)+sum( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M3C ’]<0 ) ) )

455 p r i n t(sum( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M4C ’]<0 ) / (sum(

d r o p p e d d a t a 3 [’ R e v e r s e strategy1M4C ’]>=0)+sum( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M4C ’]<0 ) ) )

456 p r i n t(sum( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M5C ’]<0 ) / (sum(

d r o p p e d d a t a 3 [’ R e v e r s e strategy1M5C ’]>=0)+sum( d r o p p e d d a t a 3 [’ R e v e r s e strategy1M5C ’]<0 ) ) )

D.3 Bootstrap resample and confidence inter-val of static carry trade

1 #! / u s r / b i n / env python

2 # c o d i n g : u t f−8

4 # I n [ 1 ] :

5 6

7 ############B o o t s t r a p p i n g#####################

8 i m p o r t pandas a s pd # pandas f o r w o r k i n g w i t h d a t a

9 i m p o r t m a t p l o t l i b . p y p l o t a s p l t # M a t p l o t l i b f o r p l o t t i n g

10 i m p o r t numpy a s np #Numpy i s a l i b r a r y f o r w o r k i n g w i t h n u m e r i c a l d a t a

11 i m p o r t o s

12 i m p o r t s y s

13 i m p o r t s e a b o r n a s s n s

14 i m p o r t s c i p y . s t a t s

16 #Import Kevin Sheppard a r c h p a c k a g e : B o o t s t r a p p i n g

17 i m p o r t a r c h

18 from a r c h . b o o t s t r a p i m p o r t S t a t i o n a r y B o o t s t r a p #s t a t i o n a r y b o o t s t r a p P o l i t i s and Romona

19 from a r c h . b o o t s t r a p i m p o r t o p t i m a l b l o c k l e n g t h #C a l c u l a t e o p t i o m a l b l o c k l e n g h

20 o s . c h d i r ( o s . getcwd ( ) )

22 ###R e t u r n s o v e r t h e whole sample p e r i o d 1999−2020###

23 carry1M = pd . r e a d e x c e l (” carry1M MID . x l s x ”)

24 carry3M = pd . r e a d e x c e l (” carry3M MID . x l s x ”)

25 carry3M = carry3M [ :−1 ]

26 carry6M = pd . r e a d e x c e l (” carry6M MID . x l s x ”)

27 carry6M = carry6M [ :−1 ]

28 carry1Y = pd . r e a d e x c e l (” carry1Y MID . x l s x ”)

29 carry1Y = carry1Y [ :−1 ]

30 31

32 # I n [ 2 ] :

33 34

35 ################1 month r e b a l a n c e################

36 37

38 # I n [ 3 ] :

41 d e f s h a r p e r a t i o ( x ) :

42 mu , sigma = 12∗x . mean ( ) , np . s q r t ( 1 2∗x . v a r ( ) )

43 v a l u e s = np . a r r a y ( [ mu , sigma , mu / sigma ] ) . s q u e e z e ( )

44 i n d e x = [’ mu ’, ’ sigma ’, ’ SR ’]

45 r e t u r n pd . S e r i e s ( v a l u e s , i n d e x=i n d e x )

46 47

48 # I n [ 4 ] :

49 50

51 #8 7 7 7 8 8 3 8 8

52 e n t r o p y = [ 8 7 7 7 8 8 3 8 8 ]

53 a l l p a r a m s = {}

54 a l l m u=pd . DataFrame ( )

55 a l l s i g m a=pd . DataFrame ( )

56 a l l c i = np . z e r o s ( ( 2 , 3 ) )

57 f o r i , c o l i n enumerate( carry1M . columns [ 2 : 7 ] ) :

58 c a r r y 1 = carry1M [ c o l ] . to numpy ( )

59 o p t = o p t i m a l b l o c k l e n g t h ( c a r r y 1∗∗2 )

60 b l o c k l e n = o p t . l o c [ 0 ,’ s t a t i o n a r y ’]

62 r s = np . random . RandomState ( e n t r o p y )

63 bs = S t a t i o n a r y B o o t s t r a p ( b l o c k l e n , c a r r y 1 , r a n d o m s t a t e=r s )

64 r e s u l t s = bs .a p p l y( s h a r p e r a t i o , 1 0 0 0 )

65 a l l m u [ c o l ]= r e s u l t s [ : , 0 ]

66 a l l s i g m a [ c o l ]= r e s u l t s [ : , 1 ]

67 68

69 a l l c i = np . c o n c a t e n a t e ( ( a l l c i , s h a r p e r a t i o ( c a r r y 1 ) . to numpy ( ) . r e s h a p e ( 1 , 3 ) ) )

70 c i = bs . c o n f i n t ( s h a r p e r a t i o , r e p s =10000, method=’ b a s i c ’)

71 a l l c i = np . c o n c a t e n a t e ( ( a l l c i , c i ) )

72 73

74 a l l c i = a l l c i [ 2 : , : ]

75 a l l c i = pd . DataFrame ( a l l c i , i n d e x = [ [ v a l f o r s u b l i s t i n [ [’{}

params ’.f o r m a t( c o l ) ,’{} Lower ’.f o r m a t( c o l ) , ’{} Upper ’.f o r m a t( c o l ) ] f o r c o l i n carry1M . columns [ 2 : 7 ] ] f o r v a l i n s u b l i s t ] ] , columns =[’ mu ’, ’ sigma ’, ’ SR ’] )

77 p r i n t( a l l c i )

78 79

80 # I n [ 7 ] :

81 82

83 p l t . r c (’ t e x t ’, u s e t e x=F a l s e )

84 p l t . r c (’ f o n t ’, f a m i l y=’ s e r i f ’)

85 p l t . rcParams [’ a x e s . l i n e w i d t h ’] =0.9

86 p l t . rcParams [’ x t i c k . l a b e l s i z e ’]=27

87 p l t . rcParams [’ y t i c k . l a b e l s i z e ’]=27

89 f i g , a x e s = p l t . s u b p l o t s ( 1 , 5 , f i g s i z e = ( 4 5 , 1 2 ) , s h a r e y=True )

90 x1 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

91 s 1= s c i p y . s t a t s . norm . p d f ( x1 , l o c =( a l l c i . l o c [’ carry1M1C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry1M1C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

92 s n s . k d e p l o t ( ( a l l m u [’ carry1M1C ’] / 1 2 ) / ( ( a l l s i g m a [’ carry1M1C ’] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , c o l o r=”#1f 3 b 4 d ”, ax=a x e s [ 0 ] , l e g e n d=

F a l s e )

93 a x e s [ 0 ] . f i l l b e t w e e n ( x1 , s 1 , c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

94 a x e s [ 0 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

95 a x e s [ 0 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

97 x2 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

98 s 2= s c i p y . s t a t s . norm . p d f ( x2 , l o c =( a l l c i . l o c [’ carry1M2C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry1M2C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

99 s n s . k d e p l o t ( ( a l l m u [’ carry1M2C ’] / 1 2 ) / ( ( a l l s i g m a [’ carry1M2C ’] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , ax=a x e s [ 1 ] , c o l o r=”#1f 3 b 4 d ”, l e g e n d=

F a l s e )

100 a x e s [ 1 ] . f i l l b e t w e e n ( x2 , s 2 , f a c e c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

101 a x e s [ 1 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

102 a x e s [ 1 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

103 104

105 x3 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

106 s 3= s c i p y . s t a t s . norm . p d f ( x3 , l o c =( a l l c i . l o c [’ carry1M3C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry1M3C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

107 s n s . k d e p l o t ( ( a l l m u [’ carry1M3C ’] / 1 2 ) / ( ( a l l s i g m a [’ carry1M3C ’] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , c o l o r=”#1f 3 b 4 d ”, ax=a x e s [ 2 ] , l e g e n d=

F a l s e )

108 a x e s [ 2 ] . f i l l b e t w e e n ( x3 , s 3 , f a c e c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

109 a x e s [ 2 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

110 a x e s [ 2 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

111 112

113 x4 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

114 s 4= s c i p y . s t a t s . norm . p d f ( x4 , l o c =( a l l c i . l o c [’ carry1M4C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry1M4C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

115 s n s . k d e p l o t ( ( a l l m u [’ carry1M4C ’] / 1 2 ) / ( ( a l l s i g m a [’ carry1M4C ’] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , c o l o r=”#1f 3 b 4 d ”, ax=a x e s [ 3 ] , l e g e n d=

F a l s e )

116 a x e s [ 3 ] . f i l l b e t w e e n ( x4 , s 4 , f a c e c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

117 a x e s [ 3 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

118 a x e s [ 3 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

119

120 x5 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

121 s 5= s c i p y . s t a t s . norm . p d f ( x5 , l o c =( a l l c i . l o c [’ carry1M5C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry1M5C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

122 s n s . k d e p l o t ( ( a l l m u [’ carry1M5C ’] / 1 2 ) / ( ( a l l s i g m a [’ carry1M5C ’] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , c o l o r=”#1f 3 b 4 d ”, ax=a x e s [ 4 ] , l e g e n d=

F a l s e )

123 a x e s [ 4 ] . f i l l b e t w e e n ( x5 , s 5 , f a c e c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

124 a x e s [ 4 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

125 a x e s [ 4 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

126

127 a x e s [ 0 ] . s e t x l a b e l (’ Carry1M1C z−s c o r e ’, f o n t s i z e = 2 7 )

128 a x e s [ 1 ] . s e t x l a b e l (’ Carry1M2C z−s c o r e ’, f o n t s i z e = 2 7 )

129 a x e s [ 2 ] . s e t x l a b e l (’ Carry1M3C z−s c o r e ’, f o n t s i z e = 2 7 )

130 a x e s [ 3 ] . s e t x l a b e l (’ Carry1M4C z−s c o r e ’, f o n t s i z e = 2 7 )

131 a x e s [ 4 ] . s e t x l a b e l (’ Carry1M5C z−s c o r e ’, f o n t s i z e = 2 7 )

132

133 a x e s [ 0 ] . s e t y l a b e l (’ P r o b a b i l i t y D e n s i t y ’, f o n t s i z e = 2 7 )

134

135 p l t . s a v e f i g (’ pdf 1M . png ’)

136 137

138 # I n [ 6 ] :

139 140

141 ###############3 months r e b a l a n c e###############

142 e n t r o p y = [ 8 7 7 7 8 8 3 8 8 ]

143 a l l p a r a m s = {}

144 all mu3M=pd . DataFrame ( )

145 a l l s i g m a 3 M=pd . DataFrame ( )

146 a l l c i = np . z e r o s ( ( 2 , 3 ) )

147 f o r i , c o l i n enumerate( carry3M . columns [ 2 : 7 ] ) :

148 c a r r y 1 = carry3M [ c o l ] . to numpy ( )

149 o p t = o p t i m a l b l o c k l e n g t h ( c a r r y 1∗∗2 )

150 b l o c k l e n = o p t . l o c [ 0 ,’ s t a t i o n a r y ’]

151

152 r s = np . random . RandomState ( e n t r o p y )

153 bs = S t a t i o n a r y B o o t s t r a p ( b l o c k l e n , c a r r y 1 , r a n d o m s t a t e=r s )

154 r e s u l t s = bs .a p p l y( s h a r p e r a t i o , 1 0 0 0 )

155 all mu3M [ c o l ]= r e s u l t s [ : , 0 ]

156 a l l s i g m a 3 M [ c o l ]= r e s u l t s [ : , 1 ]

157

158 a l l c i = np . c o n c a t e n a t e ( ( a l l c i , s h a r p e r a t i o ( c a r r y 1 ) . to numpy ( ) . r e s h a p e ( 1 , 3 ) ) )

159 c i = bs . c o n f i n t ( s h a r p e r a t i o , r e p s =10000, method=’ b a s i c ’)

160 a l l c i = np . c o n c a t e n a t e ( ( a l l c i , c i ) )

161

162 a l l c i = a l l c i [ 2 : , : ]

163 a l l c i = pd . DataFrame ( a l l c i , i n d e x = [ [ v a l f o r s u b l i s t i n [ [’{}

params ’.f o r m a t( c o l ) ,’{} Lower ’.f o r m a t( c o l ) , ’{} Upper ’.f o r m a t( c o l ) ] f o r c o l i n carry3M . columns [ 2 : 7 ] ] f o r v a l i n s u b l i s t ] ] , columns =[’ mu ’, ’ sigma ’, ’ SR ’] )

164

165 p r i n t( a l l c i )

166 167

168 # I n [ 1 0 ] :

169 170

171 p l t . r c (’ t e x t ’, u s e t e x=F a l s e )

172 p l t . r c (’ f o n t ’, f a m i l y=’ s e r i f ’)

173 p l t . rcParams [’ a x e s . l i n e w i d t h ’] =0.9

174 p l t . rcParams [’ x t i c k . l a b e l s i z e ’]=27

175 p l t . rcParams [’ y t i c k . l a b e l s i z e ’]=27

176

177 f i g , a x e s = p l t . s u b p l o t s ( 1 , 5 , f i g s i z e = ( 4 5 , 1 2 ) , s h a r e y=True )

178 x1 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

179 s 1= s c i p y . s t a t s . norm . p d f ( x1 , l o c =( a l l c i . l o c [’ carry3M1C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry3M1C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

180 s n s . k d e p l o t ( ( all mu3M [’ carry3M1C ’] / 1 2 ) / ( ( a l l s i g m a 3 M [’ carry3M1C ’] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , c o l o r=”#1f 3 b 4 d ”, ax=a x e s [ 0 ] , l e g e n d=

F a l s e )

181 a x e s [ 0 ] . f i l l b e t w e e n ( x1 , s 1 , c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

182 a x e s [ 0 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

183 a x e s [ 0 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

184

185 x2 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

186 s 2= s c i p y . s t a t s . norm . p d f ( x2 , l o c =( a l l c i . l o c [’ carry3M2C params ’,’

mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry3M2C params ’,’ sigma ’ ] . v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

187 s n s . k d e p l o t ( ( all mu3M [’ carry3M2C ’] / 1 2 ) / ( ( a l l s i g m a 3 M [’ carry3M2C ’] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , ax=a x e s [ 1 ] , c o l o r=”#1f 3 b 4 d ”, l e g e n d=

F a l s e )

188 a x e s [ 1 ] . f i l l b e t w e e n ( x2 , s 2 , f a c e c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

189 a x e s [ 1 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

190 a x e s [ 1 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

191 192

193 x3 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

194 s 3= s c i p y . s t a t s . norm . p d f ( x3 , l o c =( a l l c i . l o c [’ carry3M3C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry3M3C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

195 s n s . k d e p l o t ( ( all mu3M [’ carry3M3C ’] / 1 2 ) / ( ( a l l s i g m a 3 M [’ carry3M3C ’] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , c o l o r=”#1f 3 b 4 d ”, ax=a x e s [ 2 ] , l e g e n d=

F a l s e )

196 a x e s [ 2 ] . f i l l b e t w e e n ( x3 , s 3 , f a c e c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

197 a x e s [ 2 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

198 a x e s [ 2 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

199 200

201 x4 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

202 s 4= s c i p y . s t a t s . norm . p d f ( x4 , l o c =( a l l c i . l o c [’ carry3M4C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry3M4C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

203 s n s . k d e p l o t ( ( all mu3M [’ carry3M4C ’] / 1 2 ) / ( ( a l l s i g m a 3 M [’ carry3M4C ’] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , c o l o r=”#1f 3 b 4 d ”, ax=a x e s [ 3 ] , l e g e n d=

F a l s e )

204 a x e s [ 3 ] . f i l l b e t w e e n ( x4 , s 4 , f a c e c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

205 a x e s [ 3 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

206 a x e s [ 3 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

207

208 x5 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

209 s 5= s c i p y . s t a t s . norm . p d f ( x5 , l o c =( a l l c i . l o c [’ carry3M5C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry3M5C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

210 s n s . k d e p l o t ( ( all mu3M [’ carry3M5C ’] / 1 2 ) / ( ( a l l s i g m a 3 M [’ carry3M5C ’] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , c o l o r=”#1f 3 b 4 d ”, ax=a x e s [ 4 ] , l e g e n d=

F a l s e )

211 a x e s [ 4 ] . f i l l b e t w e e n ( x5 , s 5 , f a c e c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

212 a x e s [ 4 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

213 a x e s [ 4 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

214

215 a x e s [ 0 ] . s e t x l a b e l (’ Carry3M1C z−s c o r e ’, f o n t s i z e = 2 7 )

216 a x e s [ 1 ] . s e t x l a b e l (’ Carry3M2C z−s c o r e ’, f o n t s i z e = 2 7 )

217 a x e s [ 2 ] . s e t x l a b e l (’ Carry3M3C z−s c o r e ’, f o n t s i z e = 2 7 )

218 a x e s [ 3 ] . s e t x l a b e l (’ Carry3M4C z−s c o r e ’, f o n t s i z e = 2 7 )

219 a x e s [ 4 ] . s e t x l a b e l (’ Carry3M5C z−s c o r e ’, f o n t s i z e = 2 7 )

220

221 a x e s [ 0 ] . s e t y l a b e l (’ P r o b a b i l i t y D e n s i t y ’, f o n t s i z e = 2 7 )

222

223 p l t . s a v e f i g (’ pdf 3M . png ’)

224 225

226 # I n [ 1 1 ] :

227 228

229 ##################6 months r e b a l a n c e###############

230 e n t r o p y = [ 8 7 7 7 8 8 3 8 8 ]

231 a l l p a r a m s = {}

232 all mu6M=pd . DataFrame ( )

233 a l l s i g m a 6 M=pd . DataFrame ( )

234 a l l m u=pd . DataFrame ( )

235 a l l c i = np . z e r o s ( ( 2 , 3 ) )

236 f o r i , c o l i n enumerate( carry6M . columns [ 2 : 7 ] ) :

237 c a r r y 1 = carry6M [ c o l ] . to numpy ( )

238 o p t = o p t i m a l b l o c k l e n g t h ( c a r r y 1∗∗2 )

239 b l o c k l e n = o p t . l o c [ 0 ,’ s t a t i o n a r y ’]

240

241 r s = np . random . RandomState ( e n t r o p y )

242 bs = S t a t i o n a r y B o o t s t r a p ( b l o c k l e n , c a r r y 1 , r a n d o m s t a t e=r s )

243 r e s u l t s = bs .a p p l y( s h a r p e r a t i o , 1 0 0 0 )

244 SR = pd . DataFrame ( r e s u l t s [ : , −1 : ] , columns =[’ SR ’] )

245 all mu6M [ c o l ]= r e s u l t s [ : , 0 ]

246 a l l s i g m a 6 M [ c o l ]= r e s u l t s [ : , 1 ]

247

248 a l l c i = np . c o n c a t e n a t e ( ( a l l c i , s h a r p e r a t i o ( c a r r y 1 ) . to numpy ( ) . r e s h a p e ( 1 , 3 ) ) )

249 c i = bs . c o n f i n t ( s h a r p e r a t i o , r e p s =10000, method=’ b a s i c ’)

250 a l l c i = np . c o n c a t e n a t e ( ( a l l c i , c i ) )

251

252 a l l c i = a l l c i [ 2 : , : ]

253 a l l c i = pd . DataFrame ( a l l c i , i n d e x = [ [ v a l f o r s u b l i s t i n [ [’{}

params ’.f o r m a t( c o l ) ,’{} Lower ’.f o r m a t( c o l ) , ’{} Upper ’.f o r m a t( c o l ) ] f o r c o l i n carry6M . columns [ 2 : 7 ] ] f o r v a l i n s u b l i s t ] ] , columns =[’ mu ’, ’ sigma ’, ’ SR ’] )

254

255 p r i n t( a l l c i )

256 257

258 # I n [ 1 2 ] :

259 260

261 p l t . r c (’ t e x t ’, u s e t e x=F a l s e )

262 p l t . r c (’ f o n t ’, f a m i l y=’ s e r i f ’)

263 p l t . rcParams [’ a x e s . l i n e w i d t h ’] =0.9

264 p l t . rcParams [’ x t i c k . l a b e l s i z e ’]=27

265 p l t . rcParams [’ y t i c k . l a b e l s i z e ’]=27

266

267 f i g , a x e s = p l t . s u b p l o t s ( 1 , 5 , f i g s i z e = ( 4 5 , 1 2 ) , s h a r e y=True )

268 x1 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

269 s 1= s c i p y . s t a t s . norm . p d f ( x1 , l o c =( a l l c i . l o c [’ carry6M1C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry6M1C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

270 s n s . k d e p l o t ( ( all mu6M [’ carry6M1C ’] / 1 2 ) / ( ( a l l s i g m a 6 M [’ carry6M1C ’] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , c o l o r=”#1f 3 b 4 d ”, ax=a x e s [ 0 ] , l e g e n d=

F a l s e )

271 a x e s [ 0 ] . f i l l b e t w e e n ( x1 , s 1 , c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

272 a x e s [ 0 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

273 a x e s [ 0 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

274

275 x2 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

276 s 2= s c i p y . s t a t s . norm . p d f ( x2 , l o c =( a l l c i . l o c [’ carry6M2C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry6M2C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

277 s n s . k d e p l o t ( ( all mu6M [’ carry6M2C ’] / 1 2 ) / ( ( a l l s i g m a 6 M [’ carry6M2C ’] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , ax=a x e s [ 1 ] , c o l o r=”#1f 3 b 4 d ”, l e g e n d=

F a l s e )

278 a x e s [ 1 ] . f i l l b e t w e e n ( x2 , s 2 , f a c e c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

279 a x e s [ 1 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

280 a x e s [ 1 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

281 282

283 x3 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

284 s 3= s c i p y . s t a t s . norm . p d f ( x3 , l o c =( a l l c i . l o c [’ carry6M3C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry6M3C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

285 s n s . k d e p l o t ( ( all mu6M [’ carry6M3C ’] / 1 2 ) / ( ( a l l s i g m a 6 M [’ carry6M3C ’] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , c o l o r=”#1f 3 b 4 d ”, ax=a x e s [ 2 ] , l e g e n d=

F a l s e )

286 a x e s [ 2 ] . f i l l b e t w e e n ( x3 , s 3 , f a c e c o l o r=”#d8dcd6 ”, l a b e l=”Non

b o o t s t r a p z−s c o r e ”)

287 a x e s [ 2 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

288 a x e s [ 2 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

289 290

291 x4 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

292 s 4= s c i p y . s t a t s . norm . p d f ( x4 , l o c =( a l l c i . l o c [’ carry6M4C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry6M4C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

293 s n s . k d e p l o t ( ( all mu6M [’ carry6M4C ’] / 1 2 ) / ( ( a l l s i g m a 6 M [’ carry6M4C ’] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , c o l o r=”#1f 3 b 4 d ”, ax=a x e s [ 3 ] , l e g e n d=

F a l s e )

294 a x e s [ 3 ] . f i l l b e t w e e n ( x4 , s 4 , f a c e c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

295 a x e s [ 3 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

296 a x e s [ 3 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

297

298 x5 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

299 s 5= s c i p y . s t a t s . norm . p d f ( x5 , l o c =( a l l c i . l o c [’ carry6M5C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry6M5C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

300 s n s . k d e p l o t ( ( all mu6M [’ carry6M5C ’] / 1 2 ) / ( ( a l l s i g m a 6 M [’ carry6M5C ’] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , c o l o r=”#1f 3 b 4 d ”, ax=a x e s [ 4 ] , l e g e n d=

F a l s e )

301 a x e s [ 4 ] . f i l l b e t w e e n ( x5 , s 5 , f a c e c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

302 a x e s [ 4 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

303 a x e s [ 4 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

304

305 a x e s [ 0 ] . s e t x l a b e l (’ Carry6M1C z−s c o r e ’, f o n t s i z e = 2 7 )

306 a x e s [ 1 ] . s e t x l a b e l (’ Carry6M2C z−s c o r e ’, f o n t s i z e = 2 7 )

307 a x e s [ 2 ] . s e t x l a b e l (’ Carry6M3C z−s c o r e ’, f o n t s i z e = 2 7 )

308 a x e s [ 3 ] . s e t x l a b e l (’ Carry6M4C z−s c o r e ’, f o n t s i z e = 2 7 )

309 a x e s [ 4 ] . s e t x l a b e l (’ Carry6M5C z−s c o r e ’, f o n t s i z e = 2 7 )

310

311 a x e s [ 0 ] . s e t y l a b e l (’ P r o b a b i l i t y D e n s i t y ’, f o n t s i z e = 2 7 )

312

313 p l t . s a v e f i g (’ pdf 6M . png ’)

314 315

316 # I n [ 1 3 ] :

317 318

319 #######################1 y e a r r e b a l a n c e##############

320 e n t r o p y = [ 8 7 7 7 8 8 3 8 8 ]

321 a l l p a r a m s = {}

322 all mu12M=pd . DataFrame ( )

323 a l l s i g m a 1 2 M=pd . DataFrame ( )

324 a l l c i = np . z e r o s ( ( 2 , 3 ) )

325 f o r i , c o l i n enumerate( carry1Y . columns [ 2 : 7 ] ) :

326 c a r r y 1 = carry1Y [ c o l ] . to numpy ( )

327 o p t = o p t i m a l b l o c k l e n g t h ( c a r r y 1∗∗2 )

328 b l o c k l e n = o p t . l o c [ 0 ,’ s t a t i o n a r y ’]

329

330 r s = np . random . RandomState ( e n t r o p y )

331 bs = S t a t i o n a r y B o o t s t r a p ( b l o c k l e n , c a r r y 1 , r a n d o m s t a t e=r s )

332 r e s u l t s = bs .a p p l y( s h a r p e r a t i o , 1 0 0 0 )

333 SR = pd . DataFrame ( r e s u l t s [ : , −1 : ] , columns =[’ SR ’] )

334 all mu12M [ c o l ]= r e s u l t s [ : , 0 ]

335 a l l s i g m a 1 2 M [ c o l ]= r e s u l t s [ : , 1 ]

336 337

338 a l l c i = np . c o n c a t e n a t e ( ( a l l c i , s h a r p e r a t i o ( c a r r y 1 ) . to numpy ( ) . r e s h a p e ( 1 , 3 ) ) )

339 c i = bs . c o n f i n t ( s h a r p e r a t i o , r e p s =10000, method=’ b a s i c ’)

340 a l l c i = np . c o n c a t e n a t e ( ( a l l c i , c i ) )

341

342 a l l c i = a l l c i [ 2 : , : ]

343 a l l c i = pd . DataFrame ( a l l c i , i n d e x = [ [ v a l f o r s u b l i s t i n [ [’{}

params ’.f o r m a t( c o l ) ,’{} Lower ’.f o r m a t( c o l ) , ’{} Upper ’.f o r m a t( c o l ) ] f o r c o l i n carry1Y . columns [ 2 : 7 ] ] f o r v a l i n s u b l i s t ] ] , columns =[’ mu ’, ’ sigma ’, ’ SR ’] )

344

345 p r i n t( a l l c i )

346 347

348 # I n [ 1 4 ] :

349 350

351 p l t . r c (’ t e x t ’, u s e t e x=F a l s e )

352 p l t . r c (’ f o n t ’, f a m i l y=’ s e r i f ’)

353 p l t . rcParams [’ a x e s . l i n e w i d t h ’] =0.9

354 p l t . rcParams [’ x t i c k . l a b e l s i z e ’]=27

355 p l t . rcParams [’ y t i c k . l a b e l s i z e ’]=27

356

357 f i g , a x e s = p l t . s u b p l o t s ( 1 , 5 , f i g s i z e = ( 4 5 , 1 2 ) , s h a r e y=True )

358 x1 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

359 s 1= s c i p y . s t a t s . norm . p d f ( x1 , l o c =( a l l c i . l o c [’ carry1Y1C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry1Y1C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

360 s n s . k d e p l o t ( ( all mu12M [’ carry1Y1C ’] / 1 2 ) / ( ( a l l s i g m a 1 2 M [’ carry1Y1C ’ ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , c o l o r=”#1f 3 b 4 d ”, ax=a x e s [ 0 ] , l e g e n d

=F a l s e )

361 a x e s [ 0 ] . f i l l b e t w e e n ( x1 , s 1 , c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

362 a x e s [ 0 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

363 a x e s [ 0 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

364

365 x2 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

366 s 2= s c i p y . s t a t s . norm . p d f ( x2 , l o c =( a l l c i . l o c [’ carry1Y2C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry1Y2C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

367 s n s . k d e p l o t ( ( all mu12M [’ carry1Y2C ’] / 1 2 ) / ( ( a l l s i g m a 1 2 M [’ carry1Y2C ’ ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , ax=a x e s [ 1 ] , c o l o r=”#1f 3 b 4 d ”, l e g e n d

=F a l s e )

368 a x e s [ 1 ] . f i l l b e t w e e n ( x2 , s 2 , f a c e c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

369 a x e s [ 1 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

370 a x e s [ 1 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

371 372

373 x3 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

374 s 3= s c i p y . s t a t s . norm . p d f ( x3 , l o c =( a l l c i . l o c [’ carry1Y3C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry1Y3C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

375 s n s . k d e p l o t ( ( all mu12M [’ carry1Y3C ’] / 1 2 ) / ( ( a l l s i g m a 1 2 M [’ carry1Y3C ’ ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , c o l o r=”#1f 3 b 4 d ”, ax=a x e s [ 2 ] , l e g e n d

=F a l s e )

376 a x e s [ 2 ] . f i l l b e t w e e n ( x3 , s 3 , f a c e c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

377 a x e s [ 2 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

378 a x e s [ 2 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

379 380

381 x4 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

382 s 4= s c i p y . s t a t s . norm . p d f ( x4 , l o c =( a l l c i . l o c [’ carry1Y4C params ’,’ mu ’ ] . v a l u e s [ 0 ] / 1 2 ) / ( ( a l l c i . l o c [’ carry1Y4C params ’,’ sigma ’ ] .

v a l u e s [ 0 ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , s c a l e =1)

383 s n s . k d e p l o t ( ( all mu12M [’ carry1Y4C ’] / 1 2 ) / ( ( a l l s i g m a 1 2 M [’ carry1Y4C ’ ] / np . s q r t ( 1 2 ) ) /np . s q r t ( 2 5 2 ) ) , c o l o r=”#1f 3 b 4 d ”, ax=a x e s [ 3 ] , l e g e n d

=F a l s e )

384 a x e s [ 3 ] . f i l l b e t w e e n ( x4 , s 4 , f a c e c o l o r=”#d8dcd6 ”, l a b e l=”Non b o o t s t r a p z−s c o r e ”)

385 a x e s [ 3 ] . a x v l i n e ( x = 1 . 9 6 , l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

386 a x e s [ 3 ] . a x v l i n e ( x =−1.96, l i n e s t y l e =’ dashed ’, c=” f i r e b r i c k ”)

387

388 x5 = np . l i n s p a c e (−2 , 7 , 1 0 0 0 )

389 s 5= s c i p y . s t a t s . norm . p d f ( x5 , l o c =( a l l c i . l o c [’ carry1Y5C params ’,’

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