Backtesting Investment Strategies

import pandas as pd
pd.options.display.float_format = '{:,.4f}'.format

data = pd.read_csv("dj_data.csv", parse_dates = ["Date"], index_col = "Date")

data.head(5)

	Open	High	Low	Close	Adj Close	Volume
Date
2007-01-03	12,459.5400	12,580.3496	12,404.8203	12,474.5195	12,474.5195	327200000
2007-01-04	12,473.1602	12,510.4102	12,403.8604	12,480.6904	12,480.6904	259060000
2007-01-05	12,480.0498	12,480.1299	12,365.4102	12,398.0098	12,398.0098	235220000
2007-01-08	12,392.0098	12,445.9199	12,337.3701	12,423.4902	12,423.4902	223500000
2007-01-09	12,424.7695	12,466.4297	12,369.1699	12,416.5996	12,416.5996	225190000

data.info()

<class 'pandas.core.frame.DataFrame'>
DatetimeIndex: 3333 entries, 2007-01-03 to 2020-03-30
Data columns (total 6 columns):
 #   Column     Non-Null Count  Dtype  
---  ------     --------------  -----  
 0   Open       3333 non-null   float64
 1   High       3333 non-null   float64
 2   Low        3333 non-null   float64
 3   Close      3333 non-null   float64
 4   Adj Close  3333 non-null   float64
 5   Volume     3333 non-null   int64  
dtypes: float64(5), int64(1)
memory usage: 182.3 KB

Data for 10 years, 2010-2020

• Contains one business cycle
• 2010 - recovering from the 2008 crash
• 2020 - effects of the beginning of the covid crisis

dow = data.loc['2010-01-01': '2020-03-31', "Close"].to_frame()

dow

	Close
Date
2010-01-04	10,583.9600
2010-01-05	10,572.0195
2010-01-06	10,573.6797
2010-01-07	10,606.8604
2010-01-08	10,618.1904
...	...
2020-03-24	20,704.9102
2020-03-25	21,200.5508
2020-03-26	22,552.1699
2020-03-27	21,636.7793
2020-03-30	22,327.4805

2577 rows × 1 columns

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Data visualization and returns

• analyzing and visualizing
• calculating the daily return data

import matplotlib.pyplot as plt
import numpy as np
import seaborn as sns
plt.style.use('ggplot')

dow.describe()

	Close
count	2,577.0000
mean	17,820.3425
std	5,292.2609
min	9,686.4805
25%	13,034.4902
50%	17,113.1504
75%	21,892.4297
max	29,551.4199

dow.plot(figsize = (10, 8), fontsize = 13, color='purple')
plt.legend(fontsize = 13)
plt.title("Dow Jones Closing Data: 1-3-2010 to 3-31-2020", fontsize = 20, pad = 20)
plt.show()

df.pct_change() gets the rate of change from one row to the next

dow['Return'] = dow.pct_change()

dow.dropna(inplace = True)
dow.tail(5)

	Close	Return
Date
2020-03-24	20,704.9102	0.1137
2020-03-25	21,200.5508	0.0239
2020-03-26	22,552.1699	0.0638
2020-03-27	21,636.7793	-0.0406
2020-03-30	22,327.4805	0.0319

Plot of Close and Return Together

• Primary y axis is on right with prices
• Secondary y for the return is on left

dow.plot(figsize = (10, 8), secondary_y = "Return", 
               mark_right = True, fontsize = 13)
plt.show()

dow.rename(columns = {"Close": "DJI_Close", "Return": "DJI_Return"},
                inplace = True)

dow.head(5)

	DJI_Close	DJI_Return
Date
2010-01-05	10,572.0195	-0.0011
2010-01-06	10,573.6797	0.0002
2010-01-07	10,606.8604	0.0031
2010-01-08	10,618.1904	0.0011
2010-01-11	10,663.9902	0.0043

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Backtesting a Simple Momentum Strategy

Positions:

• +1: Investing in DJI (long position) - prices go up, you win
  
• -1: Short Selling DJI (short position) - prices go down, you win
  
• 0: No position (neutral)
  

Strategies:

• Buy and Hold (Basic Strategy - passive): Initially Investing into DJI and do nothing (Position: +1 on any given day)
  
• Simple Momentum (active Strategy to be tested) taking actions based on positions:
  

a) Investing (+1) into DJI tomorrow if today´s return was positive
b) Short selling (-1) DJI tomorrow if today´s return was negative

• np.sign() - Returns an element-wise indication of the sign of a number.

Creating the column of strategy choices based on the above rules

dow['Position'] = np.sign(dow['DJI_Return'])
dow.sample(5)

	DJI_Close	DJI_Return	Position
Date
2019-07-22	27,171.9004	0.0007	1.0000
2011-10-28	12,231.1104	0.0018	1.0000
2020-01-14	28,939.6699	0.0011	1.0000
2018-10-25	24,984.5508	0.0163	1.0000
2018-02-02	25,520.9609	-0.0254	-1.0000

Creating the daily return of the momentum strategy

• shifting the position by one timestamp
• multiplyng times the return of the Dow
• here we can compare the DJI return, positive or negative
• which determines our position, positive or negative
• resulting in our strategy return of either positive or negative

dow['Strategy_Return'] = dow.Position.shift() * dow.DJI_Return
dow.head(10)

	DJI_Close	DJI_Return	Position	Strategy_Return
Date
2010-01-05	10,572.0195	-0.0011	-1.0000	NaN
2010-01-06	10,573.6797	0.0002	1.0000	-0.0002
2010-01-07	10,606.8604	0.0031	1.0000	0.0031
2010-01-08	10,618.1904	0.0011	1.0000	0.0011
2010-01-11	10,663.9902	0.0043	1.0000	0.0043
2010-01-12	10,627.2598	-0.0034	-1.0000	-0.0034
2010-01-13	10,680.7695	0.0050	1.0000	-0.0050
2010-01-14	10,710.5498	0.0028	1.0000	0.0028
2010-01-15	10,609.6504	-0.0094	-1.0000	-0.0094
2010-01-19	10,725.4297	0.0109	1.0000	-0.0109

Calculating the accumulative investment multiple

• add 1 to every strategy_return
• if there is no return, fill value is 0
• calculate the cummulative product, cumprod
• these values can then be translated into prices
• multiply times the original DJI_Close

dow.Strategy_Return.add(1, fill_value = 0).cumprod()

Date
2010-01-05   1.0000
2010-01-06   0.9998
2010-01-07   1.0030
2010-01-08   1.0041
2010-01-11   1.0084
              ...  
2020-03-24   0.4726
2020-03-25   0.4839
2020-03-26   0.5147
2020-03-27   0.4938
2020-03-30   0.4781
Name: Strategy_Return, Length: 2576, dtype: float64

dow['Strategy'] = dow.Strategy_Return.add(1, fill_value = 0).cumprod() * dow.iloc[0,0]
display(dow.head(5), dow.tail(5))

	DJI_Close	DJI_Return	Position	Strategy_Return	Strategy
Date
2010-01-05	10,572.0195	-0.0011	-1.0000	NaN	10,572.0195
2010-01-06	10,573.6797	0.0002	1.0000	-0.0002	10,570.3594
2010-01-07	10,606.8604	0.0031	1.0000	0.0031	10,603.5296
2010-01-08	10,618.1904	0.0011	1.0000	0.0011	10,614.8561
2010-01-11	10,663.9902	0.0043	1.0000	0.0043	10,660.6416

	DJI_Close	DJI_Return	Position	Strategy_Return	Strategy
Date
2020-03-24	20,704.9102	0.1137	1.0000	-0.1137	4,995.9025
2020-03-25	21,200.5508	0.0239	1.0000	0.0239	5,115.4960
2020-03-26	22,552.1699	0.0638	1.0000	0.0638	5,441.6291
2020-03-27	21,636.7793	-0.0406	-1.0000	-0.0406	5,220.7539
2020-03-30	22,327.4805	0.0319	1.0000	-0.0319	5,054.0941

Strategy vs Buy and Hold Visualized:

dow[['DJI_Close', 'Strategy']].plot(figsize = (12, 7), fontsize = 13)
plt.title('Simple Momentum Strategy', fontsize = 20)
plt.legend(fontsize = 13)
plt.show()

Annualizing the risk and return data for both strategies:

• aggregate and calculate the mean daily return and the standard deviation of returns
• multiply the mean daily return with 252, the number of trading days in a year
• risk is multiplied by the square root of trading days

def summary_annualize(returns):
    summary = returns.agg(['mean', 'std']).T
    summary['Return'] = summary["mean"] * 252
    summary['Risk'] = summary['std'] * np.sqrt(252)
    summary.drop(columns = ['mean', 'std'], inplace = True)
    return summary

dow.columns

Index(['DJI_Close', 'DJI_Return', 'Position', 'Strategy_Return', 'Strategy'], dtype='object')

Momentum strategy underperforms the buy-and-hold strategy in the longterm

summary_annualize(dow[["DJI_Return", "Strategy_Return"]])

	Return	Risk
DJI_Return	0.0872	0.1679
Strategy_Return	-0.0580	0.1679

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Backtesting a Simple Contrarian Strategy

• Based on the idea that trends do not hold true over longer periods of time
• Idea that positive returns today will likely be followed by negative returns tomorrow
• And negative returns today will likely be followed by positive returns tomorrow

Strategies:

• Buy and Hold (Basic Strategy): Initially Investing into DJI and do nothing (Position: +1 on all days)
• Simple Contrarian (Strategy to be tested):
  a) Short Selling (-1) DJI tomorrow if today´s return was positive
  b) Investing (+1) into DJI tomorrow if today´s return was negative

Reinstating the data

data = pd.read_csv("dj_data.csv", parse_dates = ["Date"], index_col = "Date")
dow = data.loc['2010-01-01': '2020-03-31', "Close"].to_frame()
dow['Return'] = dow.pct_change()
dow.rename(columns = {"Close": "DJI_Close", "Return": "DJI_Return"},
                inplace = True)
dow.dropna(inplace = True)
display(dow.head(5))
display(dow.tail(5))

	DJI_Close	DJI_Return
Date
2010-01-05	10,572.0195	-0.0011
2010-01-06	10,573.6797	0.0002
2010-01-07	10,606.8604	0.0031
2010-01-08	10,618.1904	0.0011
2010-01-11	10,663.9902	0.0043

	DJI_Close	DJI_Return
Date
2020-03-24	20,704.9102	0.1137
2020-03-25	21,200.5508	0.0239
2020-03-26	22,552.1699	0.0638
2020-03-27	21,636.7793	-0.0406
2020-03-30	22,327.4805	0.0319

dow["Position"] = -np.sign(dow['DJI_Return'])
dow["Strategy_Return"] = dow["Position"].shift() * dow['DJI_Return']
dow['Strategy'] = dow.Strategy_Return.add(1, fill_value = 0).cumprod() * dow.iloc[0, 0]

display(dow.head(5))
display(dow.tail(5))

	DJI_Close	DJI_Return	Position	Strategy_Return	Strategy
Date
2010-01-05	10,572.0195	-0.0011	1.0000	NaN	10,572.0195
2010-01-06	10,573.6797	0.0002	-1.0000	0.0002	10,573.6797
2010-01-07	10,606.8604	0.0031	-1.0000	-0.0031	10,540.4990
2010-01-08	10,618.1904	0.0011	-1.0000	-0.0011	10,529.2398
2010-01-11	10,663.9902	0.0043	-1.0000	-0.0043	10,483.8237

	DJI_Close	DJI_Return	Position	Strategy_Return	Strategy
Date
2020-03-24	20,704.9102	0.1137	-1.0000	0.1137	16,885.3904
2020-03-25	21,200.5508	0.0239	-1.0000	-0.0239	16,481.1826
2020-03-26	22,552.1699	0.0638	-1.0000	-0.0638	15,430.4419
2020-03-27	21,636.7793	-0.0406	1.0000	0.0406	16,056.7622
2020-03-30	22,327.4805	0.0319	-1.0000	0.0319	16,569.3350

dow[['DJI_Close', 'Strategy']].plot(figsize = (12, 9), fontsize = 13, title = "Contrarian Strategy");

summary_annualize(dow[['DJI_Return', 'Strategy_Return']])

	Return	Risk
DJI_Return	0.0872	0.1679
Strategy_Return	0.0580	0.1679

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More complex strategies: Backtesting vs Fitting

• looking backward in the data to make decision on future trading can be a dangerous method
• a strategy that is "fitting" to past data cannot be relied on for future data
• strategies must be tested on future data to see how well they generalize
• forward testing is similar to patterns followed in machine learning

data = pd.read_csv("dj_data.csv", parse_dates = ["Date"], index_col = "Date")
dow = data.loc['2010-01-01': '2020-03-31', "Close"].to_frame()
dow['DJI_Return'] = dow.pct_change()
dow.columns = ["DJI_Close", "DJI_Return"]
dow.dropna(inplace = True)

Forward-Testing

• if DJI_Return, the return for the day, is positive, tomorrow will be a -1 position
• otherwise, the position is 1
• this matches the Contrarian approach

dow["Position"] = np.where(dow['DJI_Return'] > 0.0, -1, 1)

dow["Strategy_Return"] = dow["Position"].shift() * dow['DJI_Return']
dow['Strategy'] = dow.Strategy_Return.add(1, fill_value = 0).cumprod() * dow.iloc[0, 0]
dow[['DJI_Close', 'Strategy']].plot(figsize = (12, 9), fontsize = 13, title="Backtesting, Identical Results to Contrarian");

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Short-selling is not possible

• as opposed to the Contrarian approach, we take a neutral position if the previous day's return is greater than zero

dow["Position"] = np.where(dow['DJI_Return'] > 0.0, 0, 1)

dow["Strategy_Return"] = dow["Position"].shift() * dow['DJI_Return']
dow['Strategy'] = dow.Strategy_Return.add(1, fill_value = 0).cumprod() * dow.iloc[0, 0]

dow[['DJI_Close', 'Strategy']].plot(figsize = (12, 9), fontsize = 13, title="If Return > 0, No Change");

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Tweaking parameters to find profitable strategy

• if a day's returns are greater than 1%, often sellers will take the profit and sell
• so the next day will give a negative return
• selling in this case is a strategy that outperforms buying and holding

dow["Position"] = np.where(dow['DJI_Return'] > 0.01, -1, 1)

dow["Strategy_Return"] = dow["Position"].shift() * dow['DJI_Return']
dow['Strategy'] = dow.Strategy_Return.add(1, fill_value = 0).cumprod() * dow.iloc[0, 0]

dow[['DJI_Close', 'Strategy']].plot(figsize = (12, 9), fontsize = 13, title="If Return > 1% - Sell | If Not - Hold");

summary_annualize(dow[["DJI_Return", "Strategy_Return"]])

	Return	Risk
DJI_Return	0.0872	0.1679
Strategy_Return	0.1743	0.1677

Even if a Strategy seems to outperform the basic Strategy, the following issues need to be considered/analyzed as well:

• Backtesting vs. Fitting -> Was the Strategy fitted and optimized on historical data? Forward Testing is required!
  
• Transaction Costs -> Changing Positions trigger Cost. Include Costs!
  
• Tax effects -> Changing Positions can trigger (earlier) Tax Payments. Include Taxes!

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Simple Moving Averages

• also called cross-overs
• popular strategy for technical traders
• rationale = price trends persist over longer time periods
• essentially a momentum strategy, except it takes into account longer time periods and price trends

dow = data.loc['2010-01-01': '2020-03-31', "Close"].to_frame()
dow['DJI_Return'] = dow.pct_change()
dow.columns = ["DJI_Close", "DJI_Return"]
dow.dropna(inplace = True)

Calculating the average price for the immediately preceeding 50 days

• 50 trading days is a popular simple moving average window

rolling_50 = dow.DJI_Close.rolling(window = 50).mean()
print("Past 50 day averages for first examples where it can be calculated:")
display(rolling_50.iloc[50:56])
print("")
print("Past 50 day averages for last 5 days in the data:")
display(rolling_50.tail(5))

Past 50 day averages for first examples where it can be calculated:

Date
2010-03-18   10,397.0342
2010-03-19   10,400.4002
2010-03-22   10,403.9808
2010-03-23   10,409.3936
2010-03-24   10,412.8368
2010-03-25   10,417.1158
Name: DJI_Close, dtype: float64

Past 50 day averages for last 5 days in the data:

Date
2020-03-24   26,708.5479
2020-03-25   26,554.4179
2020-03-26   26,426.6679
2020-03-27   26,278.7991
2020-03-30   26,139.3959
Name: DJI_Close, dtype: float64

dow["SMA50"] = dow.DJI_Close.rolling(window = 50).mean()
dow[['DJI_Close', "SMA50"]].plot(figsize=(12,6), fontsize = 13, title="Simple Moving Average Against Daily Prices")
plt.legend(fontsize = 13)
plt.show()

SMA200

dow["SMA200"] = dow.DJI_Close.rolling(window = 200).mean()
dow[["SMA50", "SMA200"]].plot(figsize=(12,6), fontsize = 13, title="Simple Moving Averages: 50 vs 200")
plt.legend(fontsize = 13)
plt.show()

dow.dropna(inplace = True)

Strategies:

• Buy and Hold (Basic Strategy): Initially Investing into DJI and do nothing (Position: +1 on all days)
• -SMA Crossover (Momentum) (Strategy to be tested):
  a) Investing (+1): SMA50 > SMA200
  b) Short Selling (-1): SMA50 < SMA200

dow.head(5)

	DJI_Close	DJI_Return	SMA50	SMA200
Date
2010-10-19	10,978.6201	-0.0148	10,582.0898	10,507.0982
2010-10-20	11,107.9697	0.0118	10,591.3642	10,509.7780
2010-10-21	11,146.5703	0.0035	10,606.7190	10,512.6424
2010-10-22	11,132.5596	-0.0013	10,622.9712	10,515.2709
2010-10-25	11,164.0498	0.0028	10,640.1892	10,518.0002

SMA Strategy - Defining Positions

• Whenever the SMA50 crosses a longer running average, like the SMA200, it is a signal to buy or go long, because the most recent trend, rising prices, will continue into the future.
• Likewise, whenever it is the other way around, it is a signal to sell.

dow['position'] = np.sign(dow.SMA50.sub(dow.SMA200))

display(dow.head(5))
print('')
display(dow.tail(5))

	DJI_Close	DJI_Return	SMA50	SMA200	position
Date
2010-10-19	10,978.6201	-0.0148	10,582.0898	10,507.0982	1.0000
2010-10-20	11,107.9697	0.0118	10,591.3642	10,509.7780	1.0000
2010-10-21	11,146.5703	0.0035	10,606.7190	10,512.6424	1.0000
2010-10-22	11,132.5596	-0.0013	10,622.9712	10,515.2709	1.0000
2010-10-25	11,164.0498	0.0028	10,640.1892	10,518.0002	1.0000

	DJI_Close	DJI_Return	SMA50	SMA200	position
Date
2020-03-24	20,704.9102	0.1137	26,708.5479	27,010.9029	-1.0000
2020-03-25	21,200.5508	0.0239	26,554.4179	26,986.5923	-1.0000
2020-03-26	22,552.1699	0.0638	26,426.6679	26,969.1106	-1.0000
2020-03-27	21,636.7793	-0.0406	26,278.7991	26,947.2703	-1.0000
2020-03-30	22,327.4805	0.0319	26,139.3959	26,928.3739	-1.0000

dow[["SMA50", "SMA200", "position"]].plot(figsize=(12,6), secondary_y = 'position', fontsize = 13, title="SMA50, SMA200, and Position")
plt.show()

dow['Strategy_Return'] = dow['position'].shift() * dow['DJI_Return']
dow['Strategy'] = dow.Strategy_Return.add(1, fill_value = 0).cumprod() * dow.iloc[0, 0]

display(dow.head(5))
print('')
display(dow.tail(5))

	DJI_Close	DJI_Return	SMA50	SMA200	position	Strategy_Return	Strategy
Date
2010-10-19	10,978.6201	-0.0148	10,582.0898	10,507.0982	1.0000	NaN	10,978.6201
2010-10-20	11,107.9697	0.0118	10,591.3642	10,509.7780	1.0000	0.0118	11,107.9697
2010-10-21	11,146.5703	0.0035	10,606.7190	10,512.6424	1.0000	0.0035	11,146.5703
2010-10-22	11,132.5596	-0.0013	10,622.9712	10,515.2709	1.0000	-0.0013	11,132.5596
2010-10-25	11,164.0498	0.0028	10,640.1892	10,518.0002	1.0000	0.0028	11,164.0498

	DJI_Close	DJI_Return	SMA50	SMA200	position	Strategy_Return	Strategy
Date
2020-03-24	20,704.9102	0.1137	26,708.5479	27,010.9029	-1.0000	-0.1137	8,400.9733
2020-03-25	21,200.5508	0.0239	26,554.4179	26,986.5923	-1.0000	-0.0239	8,199.8681
2020-03-26	22,552.1699	0.0638	26,426.6679	26,969.1106	-1.0000	-0.0638	7,677.0940
2020-03-27	21,636.7793	-0.0406	26,278.7991	26,947.2703	-1.0000	0.0406	7,988.7066
2020-03-30	22,327.4805	0.0319	26,139.3959	26,928.3739	-1.0000	-0.0319	7,733.6867

dow[['DJI_Close', 'Strategy']].plot(figsize = (12, 9), fontsize = 13, title="SMA Cross-Over Momentum Stratgy");

summary_annualize(dow[['DJI_Return', 'Strategy_Return']])

	Return	Risk
DJI_Return	0.0878	0.1676
Strategy_Return	-0.0229	0.1677

Contrarian: Trying the opposite strategy from momentum

• Whenever the SMA50 crosses a longer running average, like the SMA200, take the short position
• and vice versa

dow['position'] = -np.sign(dow.SMA50.sub(dow.SMA200))
dow[['SMA50', 'SMA200', 'position']].plot(figsize = (12, 9), fontsize = 13, secondary_y = 'position', title="SMA Cross-Over Momentum Stratgy");

dow['Strategy_Return'] = dow['position'].shift() * dow['DJI_Return']
dow['Strategy'] = dow.Strategy_Return.add(1, fill_value = 0).cumprod() * dow.iloc[0, 0]

dow[['DJI_Close', 'Strategy']].plot(figsize = (12, 9), fontsize = 13, title="SMA Cross-Over Momentum Stratgy");

summary_annualize(dow[['DJI_Return', 'Strategy_Return']])

	Return	Risk
DJI_Return	0.0878	0.1676
Strategy_Return	0.0229	0.1677

---

---

Backtesting the "Perfect Strategy"

• illustration of what it would look like to know what the stock will do beforehand and act accordingly

dow = data.loc['2010-01-01': '2020-03-31', "Close"].to_frame()
dow['DJI_Return'] = dow.pct_change()
dow.columns = ["DJI_Close", "DJI_Return"]
dow.dropna(inplace = True)
dow['position'] = np.sign(dow['DJI_Return'])
dow['Strategy_Return'] = dow['position'] * dow['DJI_Return']
dow['Strategy'] = dow.Strategy_Return.add(1, fill_value = 0).cumprod() * dow.iloc[0, 0]

display(dow.head(5))
display(dow.tail(5))

	DJI_Close	DJI_Return	position	Strategy_Return	Strategy
Date
2010-01-05	10,572.0195	-0.0011	-1.0000	0.0011	10,583.9465
2010-01-06	10,573.6797	0.0002	1.0000	0.0002	10,585.6085
2010-01-07	10,606.8604	0.0031	1.0000	0.0031	10,618.8266
2010-01-08	10,618.1904	0.0011	1.0000	0.0011	10,630.1695
2010-01-11	10,663.9902	0.0043	1.0000	0.0043	10,676.0210

	DJI_Close	DJI_Return	position	Strategy_Return	Strategy
Date
2020-03-24	20,704.9102	0.1137	1.0000	0.1137	185,617,768,656.5365
2020-03-25	21,200.5508	0.0239	1.0000	0.0239	190,061,144,946.2066
2020-03-26	22,552.1699	0.0638	1.0000	0.0638	202,178,296,243.3131
2020-03-27	21,636.7793	-0.0406	-1.0000	0.0406	210,384,695,787.1586
2020-03-30	22,327.4805	0.0319	1.0000	0.0319	217,100,711,786.1638

dow[['DJI_Close', 'Strategy']].plot(figsize = (12, 9), fontsize = 13, title="The Perfect Strategy", logy = True);

summary_annualize(dow[['DJI_Return', 'Strategy_Return']])

	Return	Risk
DJI_Return	0.0872	0.1679
Strategy_Return	1.6609	0.1314