# The Yhat Blog

machine learning, data science, engineering

# An Introduction to Stock Market Data Analysis with Python (Part 2)

### Before we get to the good stuff

This post originally appeared on Curtis Miller's blog and was republished here on the Yhat blog with his permission. Kudos and thanks, Curtis! :)

This post is the second in a two-part series on stock data analysis using Python, based on a lecture I gave on the subject for MATH 3900 (Data Science) at the University of Utah. Be sure to read part 1 before the post below if you haven't already! In these posts, I will discuss basics such as obtaining the data from Yahoo! Finance using pandas, visualizing stock data, moving averages, developing a moving-average crossover strategy, backtesting, and benchmarking. This second post discusses topics including divising a moving average crossover strategy, backtesting, and benchmarking, along with practice problems for readers to ponder.

Yhat note: If you're following along with or recreating Curtis' analysis, check out our interactive data science environment Rodeo, which is particularly well suited to the task. As of last week's Rodeo release you can pop out, compare and save the plots you'll create below. This is really handy if you're working on multiple monitors. We've also made viewing dataframes a lovely experience, rather than a hassle. This post is already quite long so we havne't included screenshots of what it would look like inside of Rodeo, but check out last week's post (part 1 of this analysis) if you'd like to get a sneak peak at the Rodeo UI.

### This is not financial advice

NOTE: The information in this post is of a general nature containing information and opinions from the author’s perspective. None of the content of this post should be considered financial advice. Furthermore, any code written here is provided without any form of guarantee. Individuals who choose to use it do so at their own risk.

Call an open position a trade that will be terminated in the future when a condition is met. A long position is one in which a profit is made if the financial instrument traded increases in value, and a short position is on in which a profit is made if the financial asset being traded decreases in value. When trading stocks directly, all long positions are bullish and all short position are bearish. That said, a bullish attitude need not be accompanied by a long position, and a bearish attitude need not be accompanied by a short position (this is particularly true when trading stock options).

Here is an example. Let’s say you buy a stock with the expectation that the stock will increase in value, with a plan to sell the stock at a higher price. This is a long position: you are holding a financial asset for which you will profit if the asset increases in value. Your potential profit is unlimited, and your potential losses are limited by the price of the stock since stock prices never go below zero. On the other hand, if you expect a stock to decrease in value, you may borrow the stock from a brokerage firm and sell it, with the expectation of buying the stock back later at a lower price, thus earning you a profit. This is called shorting a stock, and is a short position, since you will earn a profit if the stock drops in value. The potential profit from shorting a stock is limited by the price of the stock (the best you can do is have the stock become worth nothing; you buy it back for free), while the losses are unlimited, since you could potentially spend an arbitrarily large amount of money to buy the stock back. Thus, a broker will expect an investor to be in a very good financial position before allowing the investor to short a stock.

We will call a plan that includes trading signals for prompting trades, a rule for deciding how much of the portfolio to risk on any particular strategy, and a complete exit strategy for any trade an overall trading strategy. Our concern now is to design and evaluate trading strategies.

We will suppose that the amount of money in the portfolio involved in any particular trade is a fixed proportion; 10% seems like a good number. We will also say that for any trade, if losses exceed 20% of the value of the trade, we will exit the position. Now we need a means for deciding when to enter position and when to exit for a profit.

Here, I will be demonstrating a moving average crossover strategy. We will use two moving averages, one we consider “fast”, and the other “slow”. The strategy is:

• Trade the asset when the fast moving average crosses over the slow moving average.
• Exit the trade when the fast moving average crosses over the slow moving average again.

A long trade will be prompted when the fast moving average crosses from below to above the slow moving average, and the trade will be exited when the fast moving average crosses below the slow moving average later. A short trade will be prompted when the fast moving average crosses below the slow moving average, and the trade will be exited when the fast moving average later crosses above the slow moving average.

We now have a complete strategy. But before we decide we want to use it, we should try to evaluate the quality of the strategy first. The usual means for doing so is backtesting, which is looking at how profitable the strategy is on historical data. For example, looking at the above chart’s performance on Apple stock, if the 20-day moving average is the fast moving average and the 50-day moving average the slow, this strategy does not appear to be very profitable, at least not if you are always taking long positions.

Let’s see if we can automate the backtesting task. We first identify when the 20-day average is below the 50-day average, and vice versa.

apple['20d-50d'] = apple['20d'] - apple['50d']
apple.tail()

Open High Low Close Volume Adj Close 20d 50d 200d 20d-50d
Date
2016-08-26 107.410004 107.949997 106.309998 106.940002 27766300 106.940002 107.87 101.51 102.73 6.36
2016-08-29 106.620003 107.440002 106.290001 106.820000 24970300 106.820000 107.91 101.74 102.68 6.17
2016-08-30 105.800003 106.500000 105.500000 106.000000 24863900 106.000000 107.98 101.96 102.63 6.02
2016-08-31 105.660004 106.570000 105.639999 106.099998 29662400 106.099998 108.00 102.16 102.60 5.84
2016-09-01 106.139999 106.800003 105.620003 106.730003 26643600 106.730003 108.04 102.39 102.56 5.65

We will refer to the sign of this difference as the regime; that is, if the fast moving average is above the slow moving average, this is a bullish regime (the bulls rule), and a bearish regime (the bears rule) holds when the fast moving average is below the slow moving average. I identify regimes with the following code.

# np.where() is a vectorized if-else function, where a condition is checked for each component of a vector, and the first argument passed is used when the condition holds, and the other passed if it does not
apple["Regime"] = np.where(apple['20d-50d'] > 0, 1, 0)
# We have 1's for bullish regimes and 0's for everything else. Below I replace bearish regimes's values with -1, and to maintain the rest of the vector, the second argument is apple["Regime"]
apple["Regime"] = np.where(apple['20d-50d'] < 0, -1, apple["Regime"])
apple.loc['2016-01-01':'2016-08-07',"Regime"].plot(ylim = (-2,2)).axhline(y = 0, color = "black", lw = 2)


apple["Regime"].plot(ylim = (-2,2)).axhline(y = 0, color = "black", lw = 2)


apple["Regime"].value_counts()


 1    966
-1    663
0     50
Name: Regime, dtype: int64


The last line above indicates that for 1005 days the market was bearish on Apple, while for 600 days the market was bullish, and it was neutral for 54 days.

Trading signals appear at regime changes. When a bullish regime begins, a buy signal is triggered, and when it ends, a sell signal is triggered. Likewise, when a bearish regime begins, a sell signal is triggered, and when the regime ends, a buy signal is triggered (this is of interest only if you ever will short the stock, or use some derivative like a stock option to bet against the market).

It’s simple to obtain signals. Let $r_t$ indicate the regime at time $t$, and $s_t$ the signal at time $t$. Then:

$$s_t = \text{sign}(r_t - r_{t - 1})$$ $$s_t \in {-1, 0, 1}$$, with -1 indicating “sell”, 1 indicating “buy”, and 0 no action. We can obtain signals like so:

# To ensure that all trades close out, I temporarily change the regime of the last row to 0
regime_orig = apple.ix[-1, "Regime"]
apple.ix[-1, "Regime"] = 0
apple["Signal"] = np.sign(apple["Regime"] - apple["Regime"].shift(1))
# Restore original regime data
apple.ix[-1, "Regime"] = regime_orig
apple.tail()

Open High Low Close Volume Adj Close 20d 50d 200d 20d-50d Regime Signal
Date
2016-08-26 107.410004 107.949997 106.309998 106.940002 27766300 106.940002 107.87 101.51 102.73 6.36 1.0 0.0
2016-08-29 106.620003 107.440002 106.290001 106.820000 24970300 106.820000 107.91 101.74 102.68 6.17 1.0 0.0
2016-08-30 105.800003 106.500000 105.500000 106.000000 24863900 106.000000 107.98 101.96 102.63 6.02 1.0 0.0
2016-08-31 105.660004 106.570000 105.639999 106.099998 29662400 106.099998 108.00 102.16 102.60 5.84 1.0 0.0
2016-09-01 106.139999 106.800003 105.620003 106.730003 26643600 106.730003 108.04 102.39 102.56 5.65 1.0 -1.0
apple["Signal"].plot(ylim = (-2, 2))


 0.0    1637
-1.0      21
1.0      20
Name: Signal, dtype: int64


We would buy Apple stock 23 times and sell Apple stock 23 times. If we only go long on Apple stock, only 23 trades will be engaged in over the 6-year period, while if we pivot from a long to a short position every time a long position is terminated, we would engage in 23 trades total. (Bear in mind that trading more frequently isn’t necessarily good; trades are never free.)

You may notice that the system as it currently stands isn’t very robust, since even a fleeting moment when the fast moving average is above the slow moving average triggers a trade, resulting in trades that end immediately (which is bad if not simply because realistically every trade is accompanied by a fee that can quickly erode earnings). Additionally, every bullish regime immediately transitions into a bearish regime, and if you were constructing trading systems that allow both bullish and bearish bets, this would lead to the end of one trade immediately triggering a new trade that bets on the market in the opposite direction, which again seems finnicky. A better system would require more evidence that the market is moving in some particular direction. But we will not concern ourselves with these details for now.

Let’s now try to identify what the prices of the stock is at every buy and every sell.

apple.loc[apple["Signal"] == 1, "Close"]


Date
2010-03-16    224.449997
2010-06-18    274.070011
2010-09-20    283.230007
2011-05-12    346.569988
2011-07-14    357.770004
2011-12-28    402.640003
2012-06-25    570.770020
2013-05-17    433.260010
2013-07-31    452.529984
2013-10-16    501.110001
2014-03-26    539.779991
2014-04-25    571.939980
2014-08-18     99.160004
2014-10-28    106.739998
2015-02-05    119.940002
2015-04-28    130.559998
2015-10-27    114.550003
2016-03-11    102.260002
2016-07-01     95.889999
2016-07-25     97.339996
Name: Close, dtype: float64


apple.loc[apple["Signal"] == -1, "Close"]

Date
2010-06-11    253.509995
2010-07-22    259.020000
2011-03-30    348.630009
2011-03-31    348.510006
2011-05-27    337.409992
2011-11-17    377.410000
2012-05-09    569.180023
2012-10-17    644.610001
2013-06-26    398.069992
2013-10-03    483.409996
2014-01-28    506.499977
2014-04-22    531.700020
2014-06-11     93.860001
2014-10-17     97.669998
2015-01-05    106.250000
2015-04-16    126.169998
2015-06-25    127.500000
2015-12-18    106.029999
2016-05-05     93.239998
2016-07-08     96.680000
2016-09-01    106.730003
Name: Close, dtype: float64


# Create a DataFrame with trades, including the price at the trade and the regime under which the trade is made.
apple_signals = pd.concat([
pd.DataFrame({"Price": apple.loc[apple["Signal"] == 1, "Close"],
"Regime": apple.loc[apple["Signal"] == 1, "Regime"],
pd.DataFrame({"Price": apple.loc[apple["Signal"] == -1, "Close"],
"Regime": apple.loc[apple["Signal"] == -1, "Regime"],
"Signal": "Sell"}),
])
apple_signals.sort_index(inplace = True)
apple_signals

Price Regime Signal
Date
2010-06-11 253.509995 -1.0 Sell
2010-07-22 259.020000 -1.0 Sell
2011-03-30 348.630009 0.0 Sell
2011-03-31 348.510006 -1.0 Sell
2011-05-27 337.409992 -1.0 Sell
2011-11-17 377.410000 -1.0 Sell
2012-05-09 569.180023 -1.0 Sell
2012-10-17 644.610001 -1.0 Sell
2013-06-26 398.069992 -1.0 Sell
2013-10-03 483.409996 -1.0 Sell
2014-01-28 506.499977 -1.0 Sell
2014-04-22 531.700020 -1.0 Sell
2014-06-11 93.860001 -1.0 Sell
2014-10-17 97.669998 -1.0 Sell
2015-01-05 106.250000 -1.0 Sell
2015-04-16 126.169998 -1.0 Sell
2015-06-25 127.500000 -1.0 Sell
2015-12-18 106.029999 -1.0 Sell
2016-05-05 93.239998 -1.0 Sell
2016-07-08 96.680000 -1.0 Sell
2016-09-01 106.730003 1.0 Sell
# Let's see the profitability of long trades
apple_long_profits = pd.DataFrame({
apple_signals["Regime"] == 1, "Price"],
"Profit": pd.Series(apple_signals["Price"] - apple_signals["Price"].shift(1)).loc[
apple_signals.loc[(apple_signals["Signal"].shift(1) == "Buy") & (apple_signals["Regime"].shift(1) == 1)].index
].tolist(),
"End Date": apple_signals["Price"].loc[
apple_signals.loc[(apple_signals["Signal"].shift(1) == "Buy") & (apple_signals["Regime"].shift(1) == 1)].index
].index
})
apple_long_profits

End Date Price Profit
Date
2010-03-16 2010-06-11 224.449997 29.059998
2010-06-18 2010-07-22 274.070011 -15.050011
2010-09-20 2011-03-30 283.230007 65.400002
2011-05-12 2011-05-27 346.569988 -9.159996
2011-07-14 2011-11-17 357.770004 19.639996
2011-12-28 2012-05-09 402.640003 166.540020
2012-06-25 2012-10-17 570.770020 73.839981
2013-05-17 2013-06-26 433.260010 -35.190018
2013-07-31 2013-10-03 452.529984 30.880012
2013-10-16 2014-01-28 501.110001 5.389976
2014-03-26 2014-04-22 539.779991 -8.079971
2014-04-25 2014-06-11 571.939980 -478.079979
2014-08-18 2014-10-17 99.160004 -1.490006
2014-10-28 2015-01-05 106.739998 -0.489998
2015-02-05 2015-04-16 119.940002 6.229996
2015-04-28 2015-06-25 130.559998 -3.059998
2015-10-27 2015-12-18 114.550003 -8.520004
2016-03-11 2016-05-05 102.260002 -9.020004
2016-07-01 2016-07-08 95.889999 0.790001
2016-07-25 2016-09-01 97.339996 9.390007

Above, we can see that on May 17th, 2013, there was a massive drop in the price of Apple stock, and it looks like our trading system would do badly. But this price drop is not because of a massive shock to Apple, but simply due to a stock split. And while dividend payments are not as obvious as a stock split, they may be affecting the performance of our system.

# Let's see the result over the whole period for which we have Apple data
pandas_candlestick_ohlc(apple, stick = 45, otherseries = ["20d", "50d", "200d"])


We don’t want our trading system to be behaving poorly because of stock splits and dividend payments. How should we handle this? One approach would be to obtain historical stock split and dividend payment data and design a trading system for handling these. This would most realistically represent the behavior of the stock and could be considered the best solution, but it is more complicated. Another solution would be to adjust the prices to account for stock splits and dividend payments.

Yahoo! Finance only provides the adjusted closing price of a stock, but this is all we need to get adjusted opening, high, and low prices. The adjusted close is computed like so:

$$\text{price}^{\text{adj}}_t = m_t \times \text{price}_t$$

where $m_t$ is the multiplier used for the adjustment. Solving for $m_t$ requires only division and thus we can use the closing price and the adjusted closing price to adjust all prices in the series.

def ohlc_adj(dat):
"""
:param dat: pandas DataFrame with stock data, including "Open", "High", "Low", "Close", and "Adj Close", with "Adj Close" containing adjusted closing prices

:return: pandas DataFrame with adjusted stock data

This function adjusts stock data for splits, dividends, etc., returning a data frame with
"Open", "High", "Low" and "Close" columns. The input DataFrame is similar to that returned
by pandas Yahoo! Finance API.
"""
return pd.DataFrame({"Open": dat["Open"] * dat["Adj Close"] / dat["Close"],
"High": dat["High"] * dat["Adj Close"] / dat["Close"],
"Low": dat["Low"] * dat["Adj Close"] / dat["Close"],

# This next code repeats all the earlier analysis we did on the adjusted data

# np.where() is a vectorized if-else function, where a condition is checked for each component of a vector, and the first argument passed is used when the condition holds, and the other passed if it does not
# We have 1's for bullish regimes and 0's for everything else. Below I replace bearish regimes's values with -1, and to maintain the rest of the vector, the second argument is apple["Regime"]
# To ensure that all trades close out, I temporarily change the regime of the last row to 0
# Restore original regime data

"Signal": "Sell"}),
])
].tolist(),
].index
})

pandas_candlestick_ohlc(apple_adj, stick = 45, otherseries = ["20d", "50d", "200d"])


apple_adj_long_profits

End Date Price Profit
Date
2010-03-16 2010-06-10 29.355667 3.408371
2010-06-18 2010-07-22 35.845436 -1.968381
2010-09-20 2011-03-30 37.043466 8.553623
2011-05-12 2011-05-27 45.327660 -1.198030
2011-07-14 2011-11-17 46.792503 2.568702
2011-12-28 2012-05-09 52.661020 21.781659
2012-06-25 2012-10-17 74.650634 10.019459
2013-05-17 2013-06-26 57.882798 -4.701326
2013-07-31 2013-10-04 60.457234 4.500835
2013-10-16 2014-01-28 67.389473 1.122523
2014-03-11 2014-03-17 72.948554 -1.272298
2014-03-24 2014-04-22 73.370393 -1.019203
2014-04-25 2014-10-17 77.826851 16.191371
2014-10-28 2015-01-05 102.749105 -0.028185
2015-02-05 2015-04-16 116.413846 6.046838
2015-04-28 2015-06-26 126.721620 -3.184117
2015-10-27 2015-12-18 112.152083 -7.897288
2016-03-10 2016-05-05 100.015950 -7.278331
2016-06-23 2016-06-27 95.582210 -4.038123
2016-06-30 2016-07-11 95.084904 1.372569
2016-07-25 2016-09-01 96.815526 9.914477

As you can see, adjusting for dividends and stock splits makes a big difference. We will use this data from now on.

Let’s now create a simulated portfolio of $1,000,000, and see how it would behave, according to the rules we have established. This includes: -Investing only 10% of the portfolio in any trade -Exiting the position if losses exceed 20% of the value of the trade. When simulating, bear in mind that: • Trades are done in batches of 100 stocks. • Our stop-loss rule involves placing an order to sell the stock the moment the price drops below the specified level. Thus we need to check whether the lows during this period ever go low enough to trigger the stop-loss. Realistically, unless we buy a put option, we cannot guarantee that we will sell the stock at the price we set at the stop-loss, but we will use this as the selling price anyway for the sake of simplicity. • Every trade is accompanied by a commission to the broker, which should be accounted for. I do not do so here. Here’s how a backtest may look: # We need to get the low of the price during each trade. tradeperiods = pd.DataFrame({"Start": apple_adj_long_profits.index, "End": apple_adj_long_profits["End Date"]}) apple_adj_long_profits["Low"] = tradeperiods.apply(lambda x: min(apple_adj.loc[x["Start"]:x["End"], "Low"]), axis = 1) apple_adj_long_profits  End Date Price Profit Low Date 2010-03-16 2010-06-10 29.355667 3.408371 26.059775 2010-06-18 2010-07-22 35.845436 -1.968381 31.337127 2010-09-20 2011-03-30 37.043466 8.553623 35.967068 2011-05-12 2011-05-27 45.327660 -1.198030 43.084626 2011-07-14 2011-11-17 46.792503 2.568702 46.171251 2011-12-28 2012-05-09 52.661020 21.781659 52.382438 2012-06-25 2012-10-17 74.650634 10.019459 73.975759 2013-05-17 2013-06-26 57.882798 -4.701326 52.859502 2013-07-31 2013-10-04 60.457234 4.500835 60.043080 2013-10-16 2014-01-28 67.389473 1.122523 67.136651 2014-03-11 2014-03-17 72.948554 -1.272298 71.167335 2014-03-24 2014-04-22 73.370393 -1.019203 69.579335 2014-04-25 2014-10-17 77.826851 16.191371 76.740971 2014-10-28 2015-01-05 102.749105 -0.028185 101.411076 2015-02-05 2015-04-16 116.413846 6.046838 114.948237 2015-04-28 2015-06-26 126.721620 -3.184117 119.733299 2015-10-27 2015-12-18 112.152083 -7.897288 104.038477 2016-03-10 2016-05-05 100.015950 -7.278331 91.345994 2016-06-23 2016-06-27 95.582210 -4.038123 91.006996 2016-06-30 2016-07-11 95.084904 1.372569 93.791913 2016-07-25 2016-09-01 96.815526 9.914477 95.900485 # Now we have all the information needed to simulate this strategy in apple_adj_long_profits cash = 1000000 apple_backtest = pd.DataFrame({"Start Port. Value": [], "End Port. Value": [], "End Date": [], "Shares": [], "Share Price": [], "Trade Value": [], "Profit per Share": [], "Total Profit": [], "Stop-Loss Triggered": []}) port_value = .1 # Max proportion of portfolio bet on any trade batch = 100 # Number of shares bought per batch stoploss = .2 # % of trade loss that would trigger a stoploss for index, row in apple_adj_long_profits.iterrows(): batches = np.floor(cash * port_value) // np.ceil(batch * row["Price"]) # Maximum number of batches of stocks invested in trade_val = batches * batch * row["Price"] # How much money is put on the line with each trade if row["Low"] < (1 - stoploss) * row["Price"]: # Account for the stop-loss share_profit = np.round((1 - stoploss) * row["Price"], 2) stop_trig = True else: share_profit = row["Profit"] stop_trig = False profit = share_profit * batches * batch # Compute profits # Add a row to the backtest data frame containing the results of the trade apple_backtest = apple_backtest.append(pd.DataFrame({ "Start Port. Value": cash, "End Port. Value": cash + profit, "End Date": row["End Date"], "Shares": batch * batches, "Share Price": row["Price"], "Trade Value": trade_val, "Profit per Share": share_profit, "Total Profit": profit, "Stop-Loss Triggered": stop_trig }, index = [index])) cash = max(0, cash + profit) apple_backtest  End Date End Port. Value Profit per Share Share Price Shares Start Port. Value Stop-Loss Triggered Total Profit Trade Value 2010-03-16 2010-06-10 1.011588e+06 3.408371 29.355667 3400.0 1.000000e+06 0.0 11588.4614 99809.2678 2010-06-18 2010-07-22 1.006077e+06 -1.968381 35.845436 2800.0 1.011588e+06 0.0 -5511.4668 100367.2208 2010-09-20 2011-03-30 1.029172e+06 8.553623 37.043466 2700.0 1.006077e+06 0.0 23094.7821 100017.3582 2011-05-12 2011-05-27 1.026536e+06 -1.198030 45.327660 2200.0 1.029172e+06 0.0 -2635.6660 99720.8520 2011-07-14 2011-11-17 1.031930e+06 2.568702 46.792503 2100.0 1.026536e+06 0.0 5394.2742 98264.2563 2011-12-28 2012-05-09 1.073316e+06 21.781659 52.661020 1900.0 1.031930e+06 0.0 41385.1521 100055.9380 2012-06-25 2012-10-17 1.087343e+06 10.019459 74.650634 1400.0 1.073316e+06 0.0 14027.2426 104510.8876 2013-05-17 2013-06-26 1.078880e+06 -4.701326 57.882798 1800.0 1.087343e+06 0.0 -8462.3868 104189.0364 2013-07-31 2013-10-04 1.086532e+06 4.500835 60.457234 1700.0 1.078880e+06 0.0 7651.4195 102777.2978 2013-10-16 2014-01-28 1.088328e+06 1.122523 67.389473 1600.0 1.086532e+06 0.0 1796.0368 107823.1568 2014-03-11 2014-03-17 1.086547e+06 -1.272298 72.948554 1400.0 1.088328e+06 0.0 -1781.2172 102127.9756 2014-03-24 2014-04-22 1.085120e+06 -1.019203 73.370393 1400.0 1.086547e+06 0.0 -1426.8842 102718.5502 2014-04-25 2014-10-17 1.106169e+06 16.191371 77.826851 1300.0 1.085120e+06 0.0 21048.7823 101174.9063 2014-10-28 2015-01-05 1.106140e+06 -0.028185 102.749105 1000.0 1.106169e+06 0.0 -28.1850 102749.1050 2015-02-05 2015-04-16 1.111582e+06 6.046838 116.413846 900.0 1.106140e+06 0.0 5442.1542 104772.4614 2015-04-28 2015-06-26 1.109035e+06 -3.184117 126.721620 800.0 1.111582e+06 0.0 -2547.2936 101377.2960 2015-10-27 2015-12-18 1.101928e+06 -7.897288 112.152083 900.0 1.109035e+06 0.0 -7107.5592 100936.8747 2016-03-10 2016-05-05 1.093921e+06 -7.278331 100.015950 1100.0 1.101928e+06 0.0 -8006.1641 110017.5450 2016-06-23 2016-06-27 1.089480e+06 -4.038123 95.582210 1100.0 1.093921e+06 0.0 -4441.9353 105140.4310 2016-06-30 2016-07-11 1.090989e+06 1.372569 95.084904 1100.0 1.089480e+06 0.0 1509.8259 104593.3944 2016-07-25 2016-09-01 1.101895e+06 9.914477 96.815526 1100.0 1.090989e+06 0.0 10905.9247 106497.0786 apple_backtest["End Port. Value"].plot()  Our portfolio’s value grew by 10% in about six years. Considering that only 10% of the portfolio was ever involved in any single trade, this is not bad performance. Notice that this strategy never lead to our stop-loss order being triggered. Does this mean we don’t need stop-loss orders? There is no simple answer to this. After all, if we had chosen a different level at which a stop-loss would be triggered, we may have seen it triggered. Stop-loss orders are automatically triggered and ask no question as to why the order was triggered. This means that both a genuine change in trend or a momentary fluctuation can trigger a stop-loss, with the latter being the more concerning reason since not only do you have to pay for the order, there is no guarantee that you will sell the stock at the price you set, which could make your losses worse. Meanwhile, the trend on which you based your trade still holds, and had the stop-loss not been triggered, you may have made a profit. That said, a stop-loss can help you protect against your own emotions, staying wedded to a trade even though it has lost its value. They’re also good to have if you cannot monitor or quickly access your portfolio, like when you are on vacation. I have provided links both for and “against” the use of stop-loss orders, but from now on I’m not going to require our backtesting system to account for them. While less realistic (and I do believe an industrial-strength system should account for a stop-loss rule), this simplifies the backtesting task. A more realistic portfolio would not be betting 10% of its value on only one stock. A more realistic one would consider investing in multiple stocks. Multiple trades may be ongoing at any given time involving multiple companies, and most of the portfolio will be in stocks, not cash. Now that we will be investing in multiple stops and exiting only when moving averages cross (not because of a stop-loss), we will need to change our approach to backtesting. For example, we will be using one pandas DataFrame to contain all buy and sell orders for all stocks being considered, and our loop above will have to track more information. I have written functions for creating order data for multiple stocks, and a function for performing the backtesting. def ma_crossover_orders(stocks, fast, slow): """ :param stocks: A list of tuples, the first argument in each tuple being a string containing the ticker symbol of each stock (or however you want the stock represented, so long as it's unique), and the second being a pandas DataFrame containing the stocks, with a "Close" column and indexing by date (like the data frames returned by the Yahoo! Finance API) :param fast: Integer for the number of days used in the fast moving average :param slow: Integer for the number of days used in the slow moving average :return: pandas DataFrame containing stock orders This function takes a list of stocks and determines when each stock would be bought or sold depending on a moving average crossover strategy, returning a data frame with information about when the stocks in the portfolio are bought or sold according to the strategy """ fast_str = str(fast) + 'd' slow_str = str(slow) + 'd' ma_diff_str = fast_str + '-' + slow_str trades = pd.DataFrame({"Price": [], "Regime": [], "Signal": []}) for s in stocks: # Get the moving averages, both fast and slow, along with the difference in the moving averages s[1][fast_str] = np.round(s[1]["Close"].rolling(window = fast, center = False).mean(), 2) s[1][slow_str] = np.round(s[1]["Close"].rolling(window = slow, center = False).mean(), 2) s[1][ma_diff_str] = s[1][fast_str] - s[1][slow_str] # np.where() is a vectorized if-else function, where a condition is checked for each component of a vector, and the first argument passed is used when the condition holds, and the other passed if it does not s[1]["Regime"] = np.where(s[1][ma_diff_str] > 0, 1, 0) # We have 1's for bullish regimes and 0's for everything else. Below I replace bearish regimes's values with -1, and to maintain the rest of the vector, the second argument is apple["Regime"] s[1]["Regime"] = np.where(s[1][ma_diff_str] < 0, -1, s[1]["Regime"]) # To ensure that all trades close out, I temporarily change the regime of the last row to 0 regime_orig = s[1].ix[-1, "Regime"] s[1].ix[-1, "Regime"] = 0 s[1]["Signal"] = np.sign(s[1]["Regime"] - s[1]["Regime"].shift(1)) # Restore original regime data s[1].ix[-1, "Regime"] = regime_orig # Get signals signals = pd.concat([ pd.DataFrame({"Price": s[1].loc[s[1]["Signal"] == 1, "Close"], "Regime": s[1].loc[s[1]["Signal"] == 1, "Regime"], "Signal": "Buy"}), pd.DataFrame({"Price": s[1].loc[s[1]["Signal"] == -1, "Close"], "Regime": s[1].loc[s[1]["Signal"] == -1, "Regime"], "Signal": "Sell"}), ]) signals.index = pd.MultiIndex.from_product([signals.index, [s[0]]], names = ["Date", "Symbol"]) trades = trades.append(signals) trades.sort_index(inplace = True) trades.index = pd.MultiIndex.from_tuples(trades.index, names = ["Date", "Symbol"]) return trades def backtest(signals, cash, port_value = .1, batch = 100): """ :param signals: pandas DataFrame containing buy and sell signals with stock prices and symbols, like that returned by ma_crossover_orders :param cash: integer for starting cash value :param port_value: maximum proportion of portfolio to risk on any single trade :param batch: Trading batch sizes :return: pandas DataFrame with backtesting results This function backtests strategies, with the signals generated by the strategies being passed in the signals DataFrame. A fictitious portfolio is simulated and the returns generated by this portfolio are reported. """ SYMBOL = 1 # Constant for which element in index represents symbol portfolio = dict() # Will contain how many stocks are in the portfolio for a given symbol port_prices = dict() # Tracks old trade prices for determining profits # Dataframe that will contain backtesting report results = pd.DataFrame({"Start Cash": [], "End Cash": [], "Portfolio Value": [], "Type": [], "Shares": [], "Share Price": [], "Trade Value": [], "Profit per Share": [], "Total Profit": []}) for index, row in signals.iterrows(): # These first few lines are done for any trade shares = portfolio.setdefault(index[SYMBOL], 0) trade_val = 0 batches = 0 cash_change = row["Price"] * shares # Shares could potentially be a positive or negative number (cash_change will be added in the end; negative shares indicate a short) portfolio[index[SYMBOL]] = 0 # For a given symbol, a position is effectively cleared old_price = port_prices.setdefault(index[SYMBOL], row["Price"]) portfolio_val = 0 for key, val in portfolio.items(): portfolio_val += val * port_prices[key] if row["Signal"] == "Buy" and row["Regime"] == 1: # Entering a long position batches = np.floor((portfolio_val + cash) * port_value) // np.ceil(batch * row["Price"]) # Maximum number of batches of stocks invested in trade_val = batches * batch * row["Price"] # How much money is put on the line with each trade cash_change -= trade_val # We are buying shares so cash will go down portfolio[index[SYMBOL]] = batches * batch # Recording how many shares are currently invested in the stock port_prices[index[SYMBOL]] = row["Price"] # Record price old_price = row["Price"] elif row["Signal"] == "Sell" and row["Regime"] == -1: # Entering a short pass # Do nothing; can we provide a method for shorting the market? #else: #raise ValueError("I don't know what to do with signal " + row["Signal"]) pprofit = row["Price"] - old_price # Compute profit per share; old_price is set in such a way that entering a position results in a profit of zero # Update report results = results.append(pd.DataFrame({ "Start Cash": cash, "End Cash": cash + cash_change, "Portfolio Value": cash + cash_change + portfolio_val + trade_val, "Type": row["Signal"], "Shares": batch * batches, "Share Price": row["Price"], "Trade Value": abs(cash_change), "Profit per Share": pprofit, "Total Profit": batches * batch * pprofit }, index = [index])) cash += cash_change # Final change to cash balance results.sort_index(inplace = True) results.index = pd.MultiIndex.from_tuples(results.index, names = ["Date", "Symbol"]) return results # Get more stocks microsoft = web.DataReader("MSFT", "yahoo", start, end) google = web.DataReader("GOOG", "yahoo", start, end) facebook = web.DataReader("FB", "yahoo", start, end) twitter = web.DataReader("TWTR", "yahoo", start, end) netflix = web.DataReader("NFLX", "yahoo", start, end) amazon = web.DataReader("AMZN", "yahoo", start, end) yahoo = web.DataReader("YHOO", "yahoo", start, end) sony = web.DataReader("SNY", "yahoo", start, end) nintendo = web.DataReader("NTDOY", "yahoo", start, end) ibm = web.DataReader("IBM", "yahoo", start, end) hp = web.DataReader("HPQ", "yahoo", start, end)  signals = ma_crossover_orders([("AAPL", ohlc_adj(apple)), ("MSFT", ohlc_adj(microsoft)), ("GOOG", ohlc_adj(google)), ("FB", ohlc_adj(facebook)), ("TWTR", ohlc_adj(twitter)), ("NFLX", ohlc_adj(netflix)), ("AMZN", ohlc_adj(amazon)), ("YHOO", ohlc_adj(yahoo)), ("SNY", ohlc_adj(yahoo)), ("NTDOY", ohlc_adj(nintendo)), ("IBM", ohlc_adj(ibm)), ("HPQ", ohlc_adj(hp))], fast = 20, slow = 50) signals  Price Regime Signal Date Symbol 2010-03-16 AAPL 29.355667 1.0 Buy AMZN 131.789993 1.0 Buy GOOG 282.318173 -1.0 Sell HPQ 20.722316 1.0 Buy IBM 110.563240 1.0 Buy MSFT 24.677580 -1.0 Sell NFLX 10.090000 1.0 Buy NTDOY 37.099998 1.0 Buy SNY 16.360001 -1.0 Sell YHOO 16.360001 -1.0 Sell 2010-03-17 SNY 16.500000 1.0 Buy YHOO 16.500000 1.0 Buy 2010-03-22 GOOG 278.472004 1.0 Buy 2010-03-23 MSFT 25.106096 1.0 Buy 2010-05-03 GOOG 265.035411 -1.0 Sell 2010-05-10 HPQ 19.435830 -1.0 Sell 2010-05-14 NTDOY 35.799999 -1.0 Sell 2010-05-17 SNY 16.270000 -1.0 Sell YHOO 16.270000 -1.0 Sell 2010-05-19 AMZN 124.589996 -1.0 Sell MSFT 23.835187 -1.0 Sell 2010-05-21 IBM 108.322991 -1.0 Sell 2010-06-10 AAPL 32.764038 0.0 Sell 2010-06-11 AAPL 33.156405 -1.0 Sell 2010-06-18 AAPL 35.845436 1.0 Buy 2010-06-28 IBM 111.397697 1.0 Buy 2010-07-01 IBM 105.861499 -1.0 Sell 2010-07-06 IBM 106.630175 1.0 Buy 2010-07-09 NTDOY 36.950001 1.0 Buy 2010-07-20 IBM 109.298956 -1.0 Sell 2016-06-23 AAPL 95.582210 1.0 Buy TWTR 17.040001 1.0 Buy 2016-06-27 AAPL 91.544087 -1.0 Sell FB 108.970001 -1.0 Sell 2016-06-28 SNY 36.040001 -1.0 Sell YHOO 36.040001 -1.0 Sell 2016-06-30 AAPL 95.084904 1.0 Buy NFLX 91.480003 0.0 Sell 2016-07-01 NFLX 96.669998 -1.0 Sell SNY 37.990002 1.0 Buy YHOO 37.990002 1.0 Buy 2016-07-11 AAPL 96.457473 -1.0 Sell NTDOY 27.700001 1.0 Buy 2016-07-14 MSFT 53.407133 1.0 Buy 2016-07-25 AAPL 96.815526 1.0 Buy FB 121.629997 1.0 Buy 2016-07-26 GOOG 738.419983 1.0 Buy 2016-08-18 NFLX 96.160004 1.0 Buy 2016-09-01 AAPL 106.730003 1.0 Sell 2016-09-02 AMZN 772.440002 1.0 Sell FB 126.510002 1.0 Sell GOOG 771.460022 1.0 Sell HPQ 14.490000 1.0 Sell IBM 159.550003 1.0 Sell MSFT 57.669998 1.0 Sell NFLX 97.379997 1.0 Sell NTDOY 28.840000 1.0 Sell SNY 43.279999 1.0 Sell TWTR 19.549999 1.0 Sell YHOO 43.279999 1.0 Sell bk = backtest(signals, 1000000) bk  End Cash Portfolio Value Profit per Share Share Price Shares Start Cash Total Profit Trade Value Type Date Symbol 2010-03-16 AAPL 9.001907e+05 1.000000e+06 0.000000 29.355667 3400.0 1.000000e+06 0.0 99809.2678 Buy AMZN 8.079377e+05 1.000000e+06 0.000000 131.789993 700.0 9.001907e+05 0.0 92252.9951 Buy GOOG 8.079377e+05 1.000000e+06 0.000000 282.318173 0.0 8.079377e+05 0.0 0.0000 Sell HPQ 7.084706e+05 1.000000e+06 0.000000 20.722316 4800.0 8.079377e+05 0.0 99467.1168 Buy IBM 6.089637e+05 1.000000e+06 0.000000 110.563240 900.0 7.084706e+05 0.0 99506.9160 Buy MSFT 6.089637e+05 1.000000e+06 0.000000 24.677580 0.0 6.089637e+05 0.0 0.0000 Sell NFLX 5.090727e+05 1.000000e+06 0.000000 10.090000 9900.0 6.089637e+05 0.0 99891.0000 Buy NTDOY 4.126127e+05 1.000000e+06 0.000000 37.099998 2600.0 5.090727e+05 0.0 96459.9948 Buy SNY 4.126127e+05 1.000000e+06 0.000000 16.360001 0.0 4.126127e+05 0.0 0.0000 Sell YHOO 4.126127e+05 1.000000e+06 0.000000 16.360001 0.0 4.126127e+05 0.0 0.0000 Sell 2010-03-17 SNY 3.136127e+05 1.000000e+06 0.000000 16.500000 6000.0 4.126127e+05 0.0 99000.0000 Buy YHOO 2.146127e+05 1.000000e+06 0.000000 16.500000 6000.0 3.136127e+05 0.0 99000.0000 Buy 2010-03-22 GOOG 1.310711e+05 1.000000e+06 0.000000 278.472004 300.0 2.146127e+05 0.0 83541.6012 Buy 2010-03-23 MSFT 3.315733e+04 1.000000e+06 0.000000 25.106096 3900.0 1.310711e+05 0.0 97913.7744 Buy 2010-05-03 GOOG 1.126680e+05 9.959690e+05 -13.436593 265.035411 0.0 3.315733e+04 -0.0 79510.6233 Sell 2010-05-10 HPQ 2.059599e+05 9.897939e+05 -1.286486 19.435830 0.0 1.126680e+05 -0.0 93291.9840 Sell 2010-05-14 NTDOY 2.990399e+05 9.864139e+05 -1.299999 35.799999 0.0 2.059599e+05 -0.0 93079.9974 Sell 2010-05-17 SNY 3.966599e+05 9.850339e+05 -0.230000 16.270000 0.0 2.990399e+05 -0.0 97620.0000 Sell YHOO 4.942799e+05 9.836539e+05 -0.230000 16.270000 0.0 3.966599e+05 -0.0 97620.0000 Sell 2010-05-19 AMZN 5.814929e+05 9.786139e+05 -7.199997 124.589996 0.0 4.942799e+05 -0.0 87212.9972 Sell MSFT 6.744502e+05 9.736573e+05 -1.270909 23.835187 0.0 5.814929e+05 -0.0 92957.2293 Sell 2010-05-21 IBM 7.719409e+05 9.716411e+05 -2.240249 108.322991 0.0 6.744502e+05 -0.0 97490.6919 Sell 2010-06-10 AAPL 8.833386e+05 9.832296e+05 3.408371 32.764038 0.0 7.719409e+05 0.0 111397.7292 Sell 2010-06-11 AAPL 8.833386e+05 9.832296e+05 3.800738 33.156405 0.0 8.833386e+05 0.0 0.0000 Sell 2010-06-18 AAPL 7.865559e+05 9.832296e+05 0.000000 35.845436 2700.0 8.833386e+05 0.0 96782.6772 Buy 2010-06-28 IBM 6.974378e+05 9.832296e+05 0.000000 111.397697 800.0 7.865559e+05 0.0 89118.1576 Buy 2010-07-01 IBM 7.821270e+05 9.788006e+05 -5.536198 105.861499 0.0 6.974378e+05 -0.0 84689.1992 Sell 2010-07-06 IBM 6.861598e+05 9.788006e+05 0.000000 106.630175 900.0 7.821270e+05 0.0 95967.1575 Buy 2010-07-09 NTDOY 5.900898e+05 9.788006e+05 0.000000 36.950001 2600.0 6.861598e+05 0.0 96070.0026 Buy 2010-07-20 IBM 6.884589e+05 9.812025e+05 2.668781 109.298956 0.0 5.900898e+05 0.0 98369.0604 Sell 2016-06-23 AAPL 3.951693e+05 1.863808e+06 0.000000 95.582210 1900.0 5.767755e+05 0.0 181606.1990 Buy TWTR 2.094333e+05 1.863808e+06 0.000000 17.040001 10900.0 3.951693e+05 0.0 185736.0109 Buy 2016-06-27 AAPL 3.833670e+05 1.856135e+06 -4.038123 91.544087 0.0 2.094333e+05 -0.0 173933.7653 Sell FB 5.795130e+05 1.862921e+06 3.770004 108.970001 0.0 3.833670e+05 0.0 196146.0018 Sell 2016-06-28 SNY 7.885450e+05 1.880959e+06 3.110001 36.040001 0.0 5.795130e+05 0.0 209032.0058 Sell YHOO 9.975770e+05 1.898997e+06 3.110001 36.040001 0.0 7.885450e+05 0.0 209032.0058 Sell 2016-06-30 AAPL 8.169157e+05 1.898997e+06 0.000000 95.084904 1900.0 9.975770e+05 0.0 180661.3176 Buy NFLX 9.907277e+05 1.893981e+06 -2.640000 91.480003 0.0 8.169157e+05 -0.0 173812.0057 Sell 2016-07-01 NFLX 9.907277e+05 1.893981e+06 2.549995 96.669998 0.0 9.907277e+05 0.0 0.0000 Sell SNY 8.045767e+05 1.893981e+06 0.000000 37.990002 4900.0 9.907277e+05 0.0 186151.0098 Buy YHOO 6.184257e+05 1.893981e+06 0.000000 37.990002 4900.0 8.045767e+05 0.0 186151.0098 Buy 2016-07-11 AAPL 8.016949e+05 1.896589e+06 1.372569 96.457473 0.0 6.184257e+05 0.0 183269.1987 Sell NTDOY 6.133349e+05 1.896589e+06 0.000000 27.700001 6800.0 8.016949e+05 0.0 188360.0068 Buy 2016-07-14 MSFT 4.264099e+05 1.896589e+06 0.000000 53.407133 3500.0 6.133349e+05 0.0 186924.9655 Buy 2016-07-25 AAPL 2.424604e+05 1.896589e+06 0.000000 96.815526 1900.0 4.264099e+05 0.0 183949.4994 Buy FB 6.001543e+04 1.896589e+06 0.000000 121.629997 1500.0 2.424604e+05 0.0 182444.9955 Buy 2016-07-26 GOOG -8.766857e+04 1.896589e+06 0.000000 738.419983 200.0 6.001543e+04 0.0 147683.9966 Buy 2016-08-18 NFLX -2.703726e+05 1.896589e+06 0.000000 96.160004 1900.0 -8.766857e+04 0.0 182704.0076 Buy 2016-09-01 AAPL -6.758557e+04 1.915427e+06 9.914477 106.730003 0.0 -2.703726e+05 0.0 202787.0057 Sell 2016-09-02 AMZN 1.641464e+05 1.979327e+06 213.000000 772.440002 0.0 -6.758557e+04 0.0 231732.0006 Sell FB 3.539114e+05 1.986647e+06 4.880005 126.510002 0.0 1.641464e+05 0.0 189765.0030 Sell GOOG 5.082034e+05 1.993255e+06 33.040039 771.460022 0.0 3.539114e+05 0.0 154292.0044 Sell HPQ 7.081654e+05 2.006030e+06 0.925746 14.490000 0.0 5.082034e+05 0.0 199962.0000 Sell IBM 8.996254e+05 2.015652e+06 8.018727 159.550003 0.0 7.081654e+05 0.0 191460.0036 Sell MSFT 1.101470e+06 2.030572e+06 4.262865 57.669998 0.0 8.996254e+05 0.0 201844.9930 Sell NFLX 1.286492e+06 2.032890e+06 1.219993 97.379997 0.0 1.101470e+06 0.0 185021.9943 Sell NTDOY 1.482604e+06 2.040642e+06 1.139999 28.840000 0.0 1.286492e+06 0.0 196112.0000 Sell SNY 1.694676e+06 2.066563e+06 5.289997 43.279999 0.0 1.482604e+06 0.0 212071.9951 Sell TWTR 1.907771e+06 2.093922e+06 2.509998 19.549999 0.0 1.694676e+06 0.0 213094.9891 Sell YHOO 2.119843e+06 2.119843e+06 5.289997 43.279999 0.0 1.907771e+06 0.0 212071.9951 Sell bk["Portfolio Value"].groupby(level = 0).apply(lambda x: x[-1]).plot()  A more realistic portfolio that can invest in any in a list of twelve (tech) stocks has a final growth of about 100%. How good is this? While on the surface not bad, we will see we could have done better. ### Benchmarking Backtesting is only part of evaluating the efficacy of a trading strategy. We would like to benchmark the strategy, or compare it to other available (usually well-known) strategies in order to determine how well we have done. Whenever you evaluate a trading system, there is one strategy that you should always check, one that beats all but a handful of managed mutual funds and investment managers: buy and hold SPY. The efficient market hypothesis claims that it is all but impossible for anyone to beat the market. Thus, one should always buy an index fund that merely reflects the composition of the market. SPY is an exchange-traded fund (a mutual fund that is traded on the market like a stock) whose value effectively represents the value of the stocks in the S&P 500 stock index. By buying and holding SPY, we are effectively trying to match our returns with the market rather than beat it. I obtain data on SPY below, and look at the profits for simply buying and holding SPY. spyder = web.DataReader("SPY", "yahoo", start, end) spyder.iloc[[0,-1],:]  Open High Low Close Volume Adj Close Date 2010-01-04 112.370003 113.389999 111.510002 113.330002 118944600 99.292299 2016-09-01 217.369995 217.729996 216.029999 217.389999 93859000 217.389999 2180977.0  # We see that the buy-and-hold strategy beats the strategy we developed earlier. I would also like to see a plot. ax_bench = (spyder["Adj Close"] / spyder.ix[0, "Adj Close"]).plot(label = "SPY") ax_bench = (bk["Portfolio Value"].groupby(level = 0).apply(lambda x: x[-1]) / 1000000).plot(ax = ax_bench, label = "Portfolio") ax_bench.legend(ax_bench.get_lines(), [l.get_label() for l in ax_bench.get_lines()], loc = 'best') ax_bench  Buying and holding SPY beats our trading system, at least how we currently set it up, and we haven’t even accounted for how expensive our more complex strategy is in terms of fees. Given both the opportunity cost and the expense associated with the active strategy, we should not use it. What could we do to improve the performance of our system? For starters, we could try diversifying. All the stocks we considered were tech companies, which means that if the tech industry is doing poorly, our portfolio will reflect that. We could try developing a system that can also short stocks or bet bearishly, so we can take advantage of movement in any direction. We could seek means for forecasting how high we expect a stock to move. Whatever we do, though, must beat this benchmark; otherwise there is an opportunity cost associated with our trading system. Other benchmark strategies exist, and if our trading system beat the “buy and hold SPY” strategy, we may check against them. Some such strategies include: • Buy SPY when its closing monthly price is aboves its ten-month moving average. • Buy SPY when its ten-month momentum is positive. (Momentum is the first difference of a moving average process, or$MO^q_t = MA^q_t - MA^q_{t - 1}\$.)

(I first read of these strategies here.) The general lesson still holds: don’t use a complex trading system with lots of active trading when a simple strategy involving an index fund without frequent trading beats it. This is actually a very difficult requirement to meet.

As a final note, suppose that your trading system did manage to beat any baseline strategy thrown at it in backtesting. Does backtesting predict future performance? Not at all. Backtesting has a propensity for overfitting, so just because backtesting predicts high growth doesn’t mean that growth will hold in the future.

### Conclusion

While this lecture ends on a depressing note, keep in mind that the efficient market hypothesis has many critics. My own opinion is that as trading becomes more algorithmic, beating the market will become more difficult. That said, it may be possible to beat the market, even though mutual funds seem incapable of doing so (bear in mind, though, that part of the reason mutual funds perform so poorly is because of fees, which is not a concern for index funds).

This lecture is very brief, covering only one type of strategy: strategies based on moving averages. Many other trading signals exist and employed. Additionally, we never discussed in depth shorting stocks, currency trading, or stock options. Stock options, in particular, are a rich subject that offer many different ways to bet on the direction of a stock. You can read more about derivatives (including stock options and other derivatives) in the book Derivatives Analytics with Python: Data Analysis, Models, Simulation, Calibration and Hedging.

Another resource (which I used as a reference while writing this lecture) is the O’Reilly book Python for Finance.

Remember that it is possible (if not common) to lose money in the stock market. It’s also true, though, that it’s difficult to find returns like those found in stocks, and any investment strategy should take investing in it seriously. This lecture is intended to provide a starting point for evaluating stock trading and investment, and I hope you continue to explore these ideas.

### Problems

#### Problem 1

Devise a trading strategy as described in lecture based on moving-average crossovers (you do not need a stop-loss). Pick a list of at least 15 stocks that have existed since January 1st, 2010. Backtest your strategy with the stocks chosen and benchmark the performance of your portfolio against the performance of SPY. Are you able to beat the market?

#### Problem 2

Realistically, with every trade a commission is applied. Read about how commission works, and modify the backtest() function in the lecture to allow multiple commission structures (flat fee, percentage of portfolio, etc.) to be simulated.

Additionally, our current moving average crossover strategy results in a trading signal triggering the moment two moving averages cross. We would like to make sure signals are more robust, either by:

Triggering a trade when the moving averages differ by a fixed amount Triggering a trade when the moving averages differ by some amount of (rolling) standard deviations, which are defined by: $$SD^n_t = \sqrt{\frac{1}{n - 1} \sum_{i = 0}^{n - 1} (x_{t - 1} - MA^n_t)^2}$$

(pandas does have means for computing rolling standard deviations.) Regarding the latter, if the moving averages differ by p \times SD^n_t, a trading signal is sent. Modify the function ma_crossover_orders() so that these restrictions can be implemented. Specifically, you should have the ability to set how many days are in the window of the rolling standard deviation (it need not be the same as either the fast or slow moving average windows), and how many standard deviations the moving averages must differ by in order for a signal to be sent. (The current behavior of these functions should still be possible; in fact, it should be the default behavior.)

Once these changes have been made, repeat problem 1, including a realistic commission scheme (consider looking up one from a brokerage firm) when simulating the performance of the portfolio, and requiring the moving averages differ by some fixed number or standard deviations in order for signals to be sent.

#### Problem 3

We did not set up our trading system to allow for shorting stocks. Short selling is much trickier, since losses from short selling are unlimited (a long position, on the other hand, limits losses to the total value of the assets purchased). Read about short selling here. Then modify the function backtest() to allow for short selling. How will the function decide how to conduct short sales, including how many shares to short and how to account for shorted stocks when conducting other trades? We leave this up to you to decide. As a hint, the number of shares being shorted can be represented internally in the function by a negative number.

Once this is done, repeat Problem 1, perhaps also using features implemented in Problem 2.

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