Master Markowitz Portfolio Optimization (Efficient Frontier) in Python using Pandas

What is Markowitz Portfolios Optimization (Efficient Frontier)?

The Efficient Frontier takes a portfolio of investments and optimizes the expected return in regards to the risk. That is to find the optimal return for a risk.

According to investopedia.org the return is based on the expected Compound Annual Growth Rate (CAGR) and risk metric is the standard deviation of the return.

But what does all that mean? We will learn that in this tutorial.

Step 1: Get the time series of your stock portfolio

We will use the following portfolio of 4 stocks of Apple (AAPL), Microsoft (MSFT), IBM (IBM) and Nvidia (NVDA).

To get the time series we will use the Yahoo! Finance API through the Pandas-datareader.

We will look 5 years back.

import pandas_datareader as pdr
import pandas as pd
import datetime as dt
from dateutil.relativedelta import relativedelta
years = 5
end_date = dt.datetime.now()
start_date = end_date - relativedelta(years=years)
close_price = pd.DataFrame()
tickers = ['AAPL','MSFT','IBM','NVDA']
for ticker in tickers:
  tmp = pdr.get_data_yahoo(ticker, start_date, end_date)
  close_price[ticker] = tmp['Close']
print(close_price)

Resulting in the following output (or the first few lines).

                  AAPL        MSFT         IBM        NVDA
Date                                                      
2015-08-25  103.739998   40.470001  140.960007   20.280001
2015-08-26  109.690002   42.709999  146.699997   21.809999
2015-08-27  112.919998   43.900002  148.539993   22.629999
2015-08-28  113.290001   43.930000  147.979996   22.730000
2015-08-31  112.760002   43.520000  147.889999   22.480000

It will contain all the date time series for the last 5 years from current date.

Step 2: Calculate the CAGR, returns, and covariance

To calculate the expected return, we use the Compound Average Growth Rate (CAGR) based on the last 5 years. The CAGR is used as investopedia suggest. An alternative that also is being used is the mean of the returns. The key thing is to have some common measure of the return.

The CAGR is calculated as follows.

CAGR = (end-price/start-price)^(1/years) – 1

We will also calculate the covariance as we will use that the calculate the variance of a weighted portfolio. Remember that the standard deviation is given by the following.

sigma = sqrt(variance)

A portfolio is a vector w with the balances of each stock. For example, given w = [0.2, 0.3, 0.4, 0.1], will say that we have 20% in the first stock, 30% in the second, 40% in the third, and 10% in the final stock. It all sums up to 100%.

Given a weight w of the portfolio, you can calculate the variance of the stocks by using the covariance matrix.

variance = w^T Cov w

Where Cov is the covariance matrix.

This results in the following pre-computations.

returns = close_price/close_price.shift(1)
cagr = (close_price.iloc[-1]/close_price.iloc[0])**(1/years) - 1
cov = returns.cov()
print(cagr)
print(cov)

Where you can see the output here.

# CACR:
AAPL    0.371509
MSFT    0.394859
IBM    -0.022686
NVDA    0.905011
dtype: float64
# Covariance
          AAPL      MSFT       IBM      NVDA
AAPL  0.000340  0.000227  0.000152  0.000297
MSFT  0.000227  0.000303  0.000164  0.000306
IBM   0.000152  0.000164  0.000260  0.000210
NVDA  0.000297  0.000306  0.000210  0.000879

Step 3: Plot the return and risk

This is where the power of computing comes into the picture. The idea is to just try a random portfolio and see how it rates with regards to expected return and risk.

It is that simple. Make a random weighted distribution of your portfolio and plot the point of expected return (based on our CAGR) and the risk based on the standard deviation calculated by the covariance.

import matplotlib.pyplot as plt
import numpy as np
def random_weights(n):
    k = np.random.rand(n)
    return k / sum(k)
exp_return = []
sigma = []
for _ in range(20000):
  w = random_weights(len(tickers))
  exp_return.append(np.dot(w, cagr.T))
  sigma.append(np.sqrt(np.dot(np.dot(w.T, cov), w)))
plt.plot(sigma, exp_return, 'ro', alpha=0.1) 
plt.show()

We introduce a helper function random_weights, which returns a weighted portfolio. That is, it returns a vector with entries that sum up to one. This will give a way to distribute our portfolio of stocks.

Then we iterate 20.000 times (could be any value, just want to have enough to plot our graph), where we make a random weight w, then calculate the expected return by the dot-product of w and cagr-transposed. This is done by using NumPy’s dot-product function.

What a dot-product of np.dot(w, cagr.T) does is to take elements pairwise from w and cagr and multiply them and sum up. The transpose is only about the orientation of it to make it work.

The standard deviation (assigned to sigma) is calculated similar by the formula given in the last step: variance = w^T Cov w (which has dot-products between).

This results in the following graph.

Returns vs risks

This shows a graph which outlines a parabola. The optimal values lie along the upper half of the parabola line. Hence, given a risk, the optimal portfolio is one corresponding on the upper boarder of the filled parabola.

Considerations

The Efficient Frontier gives you a way to balance your portfolio. The above code can by trial an error find such a portfolio, but it still leaves out some consideratoins.

How often should you re-balance? It has a cost to do that.

The theory behind has some assumptions that may not be a reality. As investopedia points out, it assumes that asset returns follow a normal distribution, but in reality returns can be more the 3 standard deviations away. Also, the theory builds upon that investors are rational in their investment, which is by most considered a flawed assumption, as more factors play into the investments.

The full source code

Below here you find the full source code from the tutorial.

import pandas_datareader as pdr
import datetime as dt
import pandas as pd
from dateutil.relativedelta import relativedelta
import matplotlib.pyplot as plt
import numpy as np

years = 5
end_date = dt.datetime.now()
start_date = end_date - relativedelta(years=years)
close_price = pd.DataFrame()
tickers = ['AAPL', 'MSFT', 'IBM', 'NVDA']
for ticker in tickers:
    tmp = pdr.get_data_yahoo(ticker, start_date, end_date)
    close_price[ticker] = tmp['Close']
returns = close_price / close_price.shift(1)
cagr = (close_price.iloc[-1] / close_price.iloc[0]) ** (1 / years) - 1
cov = returns.cov()
def random_weights(n):
    k = np.random.rand(n)
    return k / sum(k)
exp_return = []
sigma = []
for _ in range(20000):
    w = random_weights(len(tickers))
    exp_return.append(np.dot(w, cagr.T))
    sigma.append(np.sqrt(np.dot(np.dot(w.T, cov), w)))
plt.plot(sigma, exp_return, 'ro', alpha=0.1)
plt.show()

Multiple Time Frame Analysis on a Stock using Pandas

What will we investigate in this tutorial?

A key element to success in trading is to understand the market and the trend of the stock before you buy it. In this tutorial we will not cover how to read the market, but take a top-down analysis approach to stock prices. We will use what is called Multiple Time Frame Analysis on a stock starting with a 1-month, 1-week, and 1-day perspective. Finally, we will compare that with a Simple Moving Average with a monthly view.

Step 1: Gather the data with different time frames

We will use the Pandas-datareader library to collect the time series of a stock. The library has an endpoint to read data from Yahoo! Finance, which we will use as it does not require registration and can deliver the data we need.

import pandas_datareader as pdr
import datetime as dt

ticker = "MSFT"
start = dt.datetime(2019, 1, 1)
end = dt.datetime.now()
day = pdr.get_data_yahoo(ticker, start, end, interval='d')
week = pdr.get_data_yahoo(ticker, start, end, interval='wk')
month = pdr.get_data_yahoo(ticker, start, end, interval='mo')

Where the key is to set the interval to ‘d’ (Day), ‘wk’ (Week), and ‘mo’ (Month).

This will give us 3 DataFrames, each indexed with different intervals.

Dayly.

                  High         Low  ...      Volume   Adj Close
Date                                ...                        
2019-01-02  101.750000   98.940002  ...  35329300.0   98.860214
2019-01-03  100.190002   97.199997  ...  42579100.0   95.223351
2019-01-04  102.510002   98.930000  ...  44060600.0   99.652115
2019-01-07  103.269997  100.980003  ...  35656100.0   99.779205
2019-01-08  103.970001  101.709999  ...  31514400.0  100.502670

Weekly.

                  High         Low  ...       Volume   Adj Close
Date                                ...                         
2019-01-01  103.269997   97.199997  ...  157625100.0   99.779205
2019-01-08  104.879997  101.260002  ...  150614100.0   99.769432
2019-01-15  107.900002  101.879997  ...  127262100.0  105.302940
2019-01-22  107.879997  104.660004  ...  142112700.0  102.731720
2019-01-29  106.379997  102.169998  ...  203449600.0  103.376968

Monthly.

                  High         Low  ...        Volume   Adj Close
Date                                ...                          
2019-01-01  107.900002   97.199997  ...  7.142128e+08  102.096245
2019-02-01  113.239998  102.349998  ...  4.690959e+08  109.526405
2019-03-01  120.820000  108.800003  ...  5.890958e+08  115.796768
2019-04-01  131.369995  118.099998  ...  4.331577e+08  128.226700
2019-05-01  130.649994  123.040001  ...  5.472188e+08  121.432449
2019-06-01  138.399994  119.010002  ...  5.083165e+08  132.012497

Step 2: Combine data and interpolate missing points

The challenge to connect the DataFrames is that they have different index entries. If we add the data points from Daily with Weekly, there will be a lot of missing entries that Daily has, but Weekly does not have.

                   day        week
Date                              
2019-01-02  101.120003         NaN
2019-01-03   97.400002         NaN
2019-01-04  101.930000         NaN
2019-01-07  102.059998         NaN
2019-01-08  102.800003  102.050003
...                ...         ...
2020-08-13  208.699997         NaN
2020-08-14  208.899994         NaN
2020-08-17  210.279999         NaN
2020-08-18  211.490005  209.699997
2020-08-19  209.699997  209.699997

To deal with that we can choose to interpolate by using the DataFrame interpolate function.

import pandas_datareader as pdr
import datetime as dt
import pandas as pd

ticker = "MSFT"
start = dt.datetime(2019, 1, 1)
end = dt.datetime.now()
day = pdr.get_data_yahoo(ticker, start, end, interval='d')
week = pdr.get_data_yahoo(ticker, start, end, interval='wk')
month = pdr.get_data_yahoo(ticker, start, end, interval='mo')
data = pd.DataFrame()
data['day'] = day['Close']
data['week'] = week['Close']
data['week'] = data['week'].interpolate(method='linear')
print(data)

Which results in the following output.

                   day        week
Date                              
2019-01-02  101.120003         NaN
2019-01-03   97.400002         NaN
2019-01-04  101.930000         NaN
2019-01-07  102.059998         NaN
2019-01-08  102.800003  102.050003
...                ...         ...
2020-08-13  208.699997  210.047998
2020-08-14  208.899994  209.931998
2020-08-17  210.279999  209.815997
2020-08-18  211.490005  209.699997
2020-08-19  209.699997  209.699997

Where the missing points (except the first entry) will be linearly put between. This can be done for months as well, but we need to be more careful because of three things. First, some dates (1st of the month) do not exist in the data DataFrame. To solve that we use an outer-join, which will include them. Second, this introduces some extra dates, which are not trading dates. Hence, we need to delete them afterwards, which we can do by deleting the column (drop) and removing rows with NA value (dropna). Thirdly, we also need to understand that the monthly view looks backwards. Hence, the 1st of January is first finalized the last day of January. Therefore we shift it back in the join.

import pandas_datareader as pdr
import datetime as dt
import pandas as pd

ticker = "MSFT"
start = dt.datetime(2019, 1, 1)
end = dt.datetime.now()
day = pdr.get_data_yahoo(ticker, start, end, interval='d')
week = pdr.get_data_yahoo(ticker, start, end, interval='wk')
month = pdr.get_data_yahoo(ticker, start, end, interval='mo')

data = pd.DataFrame()
data['day'] = day['Close']
data['week'] = week['Close']
data['week'] = data['week'].interpolate(method='index')
data = data.join(month['Close'].shift(), how='outer')
data['month'] = data['Close'].interpolate(method='index')
data = data.drop(columns=['Close']).dropna()
data['SMA20'] = data['day'].rolling(20).mean()

Step 3: Visualize the output and take a look at it

To visualize it is straight forward by using matplotlib.

import pandas_datareader as pdr
import datetime as dt
import matplotlib.pyplot as plt
import pandas as pd

ticker = "MSFT"
start = dt.datetime(2019, 1, 1)
end = dt.datetime.now()
day = pdr.get_data_yahoo(ticker, start, end, interval='d')
week = pdr.get_data_yahoo(ticker, start, end, interval='wk')
month = pdr.get_data_yahoo(ticker, start, end, interval='mo')

data = pd.DataFrame()
data['day'] = day['Close']
data['week'] = week['Close']
data['week'] = data['week'].interpolate(method='index')
data = data.join(month['Close'].shift(), how='outer')
data['month'] = data['Close'].interpolate(method='index')
data = data.drop(columns=['Close']).dropna()
data.plot()
plt.show()

Which results in the following graph.

As expected the monthly price is adjusted to be the closing day-price the day before. Hence, it looks like the monthly-curve is crossing the day-curve on the 1st every month (which is almost true).

To really appreciate the Multiple Time Frames Analysis, it is better to keep the graphs separate and interpret them each isolated.

Step 4: How to use these different Multiple Time Frame Analysis

Given the picture it is a good idea to start top down. First look at the monthly picture, which shows the overall trend.

Month view of MFST.

In the case of MSFT it is a clear growing trend, with the exception of two declines. But the overall impression is a company in growth that does not seem to slow down. Even the Dow theory (see this tutorial on it) suggest that there will be secondary movements in a general bull trend.

Secondly, we will look at the weekly view.

Weekly view of MFST

Here your impression is a bit more volatile. It shows many smaller ups and downs, with a big one in March, 2020. It could also indicate a small decline in the growth right and the end. Also, the Dow theory could suggest that it will turn. Though it is not certain.

Finally, on the daily view it gives a more volatile picture, which can be used to when to enter the market.

Day view of MFST

Here you could also be a bit worried. Is this the start of a smaller bull market.

To sum up. In the month-view, we have concluded a growth. The week-view shows signs of possible change. Finally, the day-view is also showing signs of possible decline.

As an investor, and based on the above, I would not enter the market right now. If both the month-view and week-view showed growth, while the day-view decline, that would be a good indicator. You want the top level to show growth, while a day-view might show a small decline.

Finally, remember that you should not just use one way to interpret to enter the market or not.

Step 5: Is monthly the same as a Simple Moving Average?

Good question, I am glad you asked. The Simple Moving Average (SMA) can be calculated easy with DataFrames using rolling and mean function.

Best way is to just try it.

import pandas_datareader as pdr
import datetime as dt
import matplotlib.pyplot as plt
import pandas as pd

ticker = "MSFT"
start = dt.datetime(2019, 1, 1)
end = dt.datetime.now()
day = pdr.get_data_yahoo(ticker, start, end, interval='d')
week = pdr.get_data_yahoo(ticker, start, end, interval='wk')
month = pdr.get_data_yahoo(ticker, start, end, interval='mo')

data = pd.DataFrame()
data['day'] = day['Close']
data['week'] = week['Close']
data['week'] = data['week'].interpolate(method='index')
data = data.join(month['Close'].shift(), how='outer')
data['month'] = data['Close'].interpolate(method='index')
data = data.drop(columns=['Close']).dropna()
data['SMA20'] = data['day'].rolling(20).mean()
data.plot()
plt.show()

As you see, the SMA is not as reactive on the in crisis in March, 2020, as the monthly view is. This shows a difference in them. This does not exclude the one from the other, but shows a difference in how they react.

Comparing the month-view with a Simple Moving Average of a month (20 trade days)

Please remember, that the monthly view is first updated at the end of a month, while SMA is updated on a daily basis.

Other differences is that SMA is an average of the 20 last days, while the monthly is the actual value of the last day of a month (as we look at Close). This implies that the monthly view can be much more volatile than the SMA.

Conclusion

It is advised to make analysis from bigger time frames and zoom in. This way you first look at overall trends, and get a bigger picture of the market. This should eliminate not to fall into being focused on a small detail in the market, but understand it on a higher level.

12% Investment Solution

Would you like to get 12% in return of your investments?

D. A. Carter promises and shows how his simple investment strategy will deliver that in the book The 12% Solution. The book shows how to test this statement by using backtesting.

Did Carter find a strategy that will consistently beat the market?

Actually, it is not that hard to use Python to validate his calculations. But we can do better than that. If you want to work smarter than traditional investors then continue to read here.

Master Dow Theory with Python Pandas

What will we cover in this tutorial?

Dow theory was proposed by Charles H. Dow and is not an exact science. It is more how to identify trends in the market. In this tutorial we investigate the approach by testing it on data. Notice, that there are various ways to interpret it and often it is done by visual approximations, while we in this tutorial will make some rough assumptions to see if it beats the buy-and-hold approach of a stock.

First we will make our assumption on how to implement the Dow theory approach to make buy and sell indicators, which we will use as buy and sell markers in the market.

Step 1: Understand the Dow theory to make buy and sell indicators

The essence of Dow theory is that there are 3 types of trend in the market. The primary trend is a year or more long trend, like a bull market. Then on a secondary trend, the market can move in opposite direction for 3 weeks to 3 months. This can result in a pullback, that can seem like a bear market within the bull market. Finally, there are micro trends (less than 3 weeks) which can be considered as noise.

According to Dow theory each market has 3 phases. Our objective as an investor is to identify when a bear market turns into bull market.

Some visual example to understand the above will help a bit. A general bull market with primary and secondary trends could look like this.

Primary bull market trend with secondary bear market trends.

Where you should notice that the temporary lows are all increasing along the way.

A similar picture for a bear market could be.

Primary bear market trend with secondary bull market trends.

Here you should notice how the secondary bull markets peaks are also in a decreasing trend.

Step 2: Identify when a primary market trend changes

The key here is to identify when a primary stock trend goes from bull to bear or opposite.

Please also notice that Dow theory talks about the market and we here are looking at a stock. Hence, we have an assumption that the market and the stock have a strong enough correlation to use the same theory.

From a primary bear to a primary bull market could look like as follows.

From bear to bull market

We have added some markers in the diagram.

  • LL : Low-Low – meaning that the low is lower than previous low.
  • LH : Low-High – meaning that the high is lower than previous high.
  • HH : High-High – meaning that the high is higher than previous high.
  • HL : High-Low – meaning that the low is higher than previous low.

As you see, the bear market consists of consecutive LL and LH, while a bull market consists of consecutive HH and LH. The market changes from bear to bull when we confidently can say that we will get a HH, which we can do when we cross from the last LL over the last LH (before we reach HH).

Hence, a buy signal can be set when we reach a stock price above last LH.

Similar we can investigate the when a primary trends goes from bull to hear market.

From bull to a bear trend.

Where we have the same types of markers.

We see that the trend changes from bull to bear when we go from HL to LL. Hence, a sell indicator is when we are sure we reach a LL (that is before it is a LL).

Again, this is not an exact science and is just a way to interpret it. We will try it out on real stock data to see how it performs.

Step 3: Get some data and calculate points of lows and highs

We will use Pandas-datareader to get the time series data from Yahoo! Finance.

import pandas_datareader as pdr
import datetime as dt

ticker = pdr.get_data_yahoo("TWTR", dt.datetime(2020,1,1), dt.datetime.now())
print(ticker)

Resulting in a time series for Twitter, which has the ticker TWTR. You can find other tickers for other companies by using the Yahoo! Finance ticker lookup.

                 High        Low       Open      Close    Volume  Adj Close
Date                                                                       
2020-01-02  32.500000  31.959999  32.310001  32.299999  10721100  32.299999
2020-01-03  32.099998  31.260000  31.709999  31.520000  14429500  31.520000
2020-01-06  31.709999  31.160000  31.230000  31.639999  12582500  31.639999
2020-01-07  32.700001  31.719999  31.799999  32.540001  13712900  32.540001
2020-01-08  33.400002  32.349998  32.349998  33.049999  14632400  33.049999
...               ...        ...        ...        ...       ...        ...
2020-08-12  38.000000  36.820000  37.500000  37.439999  11013300  37.439999
2020-08-13  38.270000  37.369999  37.430000  37.820000  13259400  37.820000
2020-08-14  37.959999  37.279999  37.740002  37.900002  10377300  37.900002
2020-08-17  38.090000  37.270000  37.950001  37.970001  10188500  37.970001
2020-08-18  38.459999  37.740002  38.279999  38.009998   8548300  38.009998

First thing we need to get is to find the low and highs. First challenge here is that the stock price is going up and down during the day. To simplify our investigation we will only use the Close price.

Taking that decision might limit and not give correct results, but it surely simplifies our work.

Next up, we need to identify highs and lows. This can be done to see when a daily difference goes from positive to negative.

import pandas_datareader as pdr
import datetime as dt

ticker = pdr.get_data_yahoo("TWTR", dt.datetime(2020,1,1), dt.datetime.now())
ticker['delta'] = ticker['Close'].diff()
growth = ticker['delta'] > 0
ticker['markers'] = growth.diff().shift(-1)
print(ticker)

Please notice the shit(-1) as it moves the indicator on the day of the change.

2020-08-05  37.340000  36.410000  36.560001  36.790001   10052100  36.790001  0.440002   False
2020-08-06  37.810001  36.490002  36.849998  37.689999   10478900  37.689999  0.899998    True
2020-08-07  38.029999  36.730000  37.419998  37.139999   11335100  37.139999 -0.549999    True
2020-08-10  39.169998  37.310001  38.360001  37.439999   29298400  37.439999  0.299999    True
2020-08-11  39.000000  36.709999  37.590000  37.279999   20486000  37.279999 -0.160000    True
2020-08-12  38.000000  36.820000  37.500000  37.439999   11013300  37.439999  0.160000   False
2020-08-13  38.270000  37.369999  37.430000  37.820000   13259400  37.820000  0.380001   False
2020-08-14  37.959999  37.279999  37.740002  37.900002   10377300  37.900002  0.080002   False
2020-08-17  38.090000  37.270000  37.950001  37.970001   10188500  37.970001  0.070000   False
2020-08-18  38.459999  37.740002  38.279999  38.009998    8548300  38.009998  0.039997     NaN

Where we have output above. The True values are when we reach Highs or Lows.

Now we have identified all the potential HH, LH, LH, and LL.

Step 4: Implement a simple trial of sell and buy

We continue our example on Twitter and see how we can perform.

Our strategy will be as follows.

  • We either have bought stocks for all our money or not. That is, either we have stocks or not.
  • If we do not have stocks, we buy if stock price is above last high, meaning that a HH is coming.
  • If we do have stocks, we sell if stock price is below last low, meaning that a LL is coming.

This can mean that we enter market in the last of a bull market. If you were to follow the theory complete, it suggest to wait until a bear market changes to a bull market.

import pandas_datareader as pdr
import datetime as dt

ticker = pdr.get_data_yahoo("TWTR", dt.datetime(2020,1,1), dt.datetime.now())
ticker['delta'] = ticker['Close'].diff()
growth = ticker['delta'] > 0
ticker['markers'] = growth.diff().shift(-1)
# We want to remember the last_high and last_low
# Set to max value not to trigger false buy
last_high = ticker['Close'].max()
last_low = 0.0
# Then setup our account, we can only have stocks or not
# We have a start balance of 100000 $
has_stock = False
balance = 100000
stocks = 0
for index, row in ticker.iterrows():
  # Buy and sell orders
  if not has_stock and row['Close'] > last_high:
    has_stock = True
    stocks = balance//row['Close']
    balance -= row['Close']*stocks
  elif has_stock and row['Close'] < last_low:
    has_stock = False
    balance += row['Close']*stocks
    stocks = 0
  # Update the last_high and last_low
  if row['markers']:
    if row['delta'] > 0:
      last_high = row['Close']
    else:
      last_low = row['Close']

print("Dow returns", balance + stocks*ticker['Close'].iloc[-1])
# Compare this with a simple buy and hold approach.
buy_hold_stocks = 100000//ticker['Close'].iloc[0]
buy_hold = 100000 - buy_hold_stocks*ticker['Close'].iloc[0] + buy_hold_stocks*ticker['Close'].iloc[-1]
print("Buy-and-hold return", buy_hold)

Which results in the following results.

Dow returns 120302.0469455719
Buy-and-hold return 117672.44716644287

That looks promising, but it might be just out of luck. Hence, we want to validate with other examples. The results say a return of investment of 20.3% using our Dow theory approach, while a simple buy-and-hold strategy gave 17.7%. This is over the span of less than 8 months.

The thing you would like to achieve with a strategy is to avoid big losses and not loose out on revenue. The above testing does not justify any clarification on that.

Step 5: Try out some other tickers to test it

A first investigation is to check how the algorithm performs on other stocks. We make one small adjustment, as the comparison to buy on day-1, might be quite unfair. If price is low, it an advantage, while if the price is high, it is a big disadvantage. The code below runs on multiple stocks and compare the first buy with a Dow approach (as outlined in this tutorial) with a buy-and-hold approach. The exit of the market might also be unfair.

import pandas_datareader as pdr
import datetime as dt
def dow_vs_hold_and_buy(ticker_name):
  ticker = pdr.get_data_yahoo(ticker_name, dt.datetime(2020,1,1), dt.datetime.now())
  ticker['delta'] = ticker['Close'].diff()
  growth = ticker['delta'] > 0
  ticker['markers'] = growth.diff().shift(-1)
  # We want to remember the last_high and last_low
  # Set to max value not to trigger false buy
  last_high = ticker['Close'].max()
  last_low = 0.0
  # Then setup our account, we can only have stocks or not
  # We have a start balance of 100000 $
  has_stock = False
  balance = 100000
  stocks = 0
  first_buy = None
  for index, row in ticker.iterrows():
    # Buy and sell orders
    if not has_stock and row['Close'] > last_high:
      has_stock = True
      stocks = balance//row['Close']
      balance -= row['Close']*stocks
      if first_buy is None:
        first_buy = index
    elif has_stock and row['Close'] < last_low:
      has_stock = False
      balance += row['Close']*stocks
      stocks = 0
    # Update the last_high and last_low
    if row['markers']:
      if row['delta'] > 0:
        last_high = row['Close']
      else:
        last_low = row['Close']
  dow_returns = balance + stocks*ticker['Close'].iloc[-1]
  # Compare this wiith a simple buy and hold approach.
  buy_hold_stocks = 100000//ticker['Close'].loc[first_buy]
  buy_hold_returns = 100000 - buy_hold_stocks*ticker['Close'].loc[first_buy] + buy_hold_stocks*ticker['Close'].iloc[-1]
  print(ticker_name, dow_returns > buy_hold_returns, round(dow_returns/1000 - 100, 1), round(buy_hold_returns/1000 - 100, 1))

tickers = ["TWTR", "AAPL", "TSLA", "BAC", "KO", "GM", "MSFT", "AMZN", "GOOG", "FB", "INTC", "T"]
for ticker in tickers:
  dow_vs_hold_and_buy(ticker)

Resulting in the following output.

TWTR   True  20.3  14.4
AAPL  False  26.4  52.3
TSLA   True 317.6 258.8
BAC    True -16.3 -27.2
KO     True  -8.2 -14.6
GM     True   8.9 -15.1
MSFT  False  26.2  32.1
AMZN  False  32.8  73.9
GOOG  False   7.1  11.0
FB     True  18.3  18.2
INTC  False -34.9 -18.4
T     False -25.3 -20.8

This paints a different picture. First, it seems more random if it outperforms the buy-and-hold approach.

The one performing best is the General Motors Company (GM), but it might be due to unlucky entering of the market. The stock was high in the beginning of the year, and then fell a lot. Hence, here the Dow helped to exit and enter the market correct.

Intel Corporation (INTC) is working a lot against us. While there is a big loss (-18.4%), it is not saved by our Dow theory algorithm. There was a big loss in stock value 24th of July with 20% from close the day before to open. The Dow cannot save you for situations like that and will sell on the far bottom.

The Apple (AAPL) is also missing a lot of gain. The stock is in a great growth in 2020, with some challenges in March and after (Corona hit). But looking and buy and sell signals, it hits sell higher than the following buy and losses out on gain.

Amazon (AMZN) seems to be the same story. Growth in general and hitting buying on higher than previous sell, and loosing out on profit.

Next steps and considerations

We have made some broad simplifications in our algorithm.

  • Only consider Close value, while a normal way to find the markers are on a OHLC candlestick diagram.
  • If we used the span of the day price, then we might limit our losses with a stop-loss order earlier.
  • This is not an exact science, and the trends might need a different way to identify them.

Hence, the above suggest it can be more adjusted to real life.

Another thing to keep in mind is that you should never make your investment decision on only one indicator or algorithm choice.