Backtesting a Simple Mean Reversion Strategy

Linear Mean-Reversion Equity strategy

In this article we will:

• Construct a mean-reverting portfolio

• Backtest a simple mean-reverting strategy

Download and Visualise Data

data = yf.download(['AAPL', 'GOOG'])['Adj Close'].dropna()["2012-01-01":]
data.plot(logy=True)

Fit a Linear Model on the Data

x = data['GOOG']
y = data['AAPL']

y = sm.add_constant(y)

model = sm.OLS(x, y).fit()

print(model.summary())

intercept = model.params[0]
slope = model.params[1]

print("Intercept:", intercept)
print("Slope:", slope)

Visualise the Linear Model Fit

plt.scatter(data['AAPL'], data['GOOG'], color='blue', label='Data', )

x_pred = sm.add_constant(data['AAPL'])
y_pred = model.predict(x_pred)
plt.plot(data['AAPL'], y_pred, color='red', label='Linear Regression')

plt.xlabel('AAPL')
plt.ylabel('GOOG')
plt.legend()

plt.show()

Construct a Mean-Reverting Portfolio using the Linear Regression fit

stationary_train = data['GOOG'] - slope * data['AAPL']
stationary_train.plot()

---

Backtest: Simple Mean-Reversion strategy

Rolling Linear Regression Hedge Ratio

#add constant column to regress with intercept
data['const'] = 1

#fit
model = RollingOLS(
    endog=data['AAPL'].values , 
    exog=data[['const','GOOG']], 
    window=30
)
rres = model.fit()

First, we fit a linear regression model on a rolling window of 30 days worth of data.

Mean Reverting Portfolio Construction

df = rres.params
df['slope'] = df['GOOG']
df['AAPL'] = data['AAPL']
df['GOOG'] = data['GOOG']

df['portfolio'] = df['GOOG'] - df['slope'] * df['AAPL']

df['portfolio daily return'] = np.log(df['portfolio'] / df['portfolio'].shift(1))

Next, we construct a mean-reverting portfolio. Note that we are now using a rolling hedge ratio.

Trading Signal:

\(p_i = - \frac{x_i-\mu_i}{\sigma_i}\)

We use the z-score as our signal. The z-score is a measure of the number of standard deviations our portfolio is away from the mean.

df['position'] = - (df['portfolio'] - df['portfolio'].rolling(30).mean()) / df['portfolio'].rolling(30).std()

Backtest Results

df['daily pnl'] = df['portfolio daily return'] * df['position'].shift(1)
df['cum pnl'] = df['daily pnl'].cumsum()
df['cum pnl'].plot()

The backtest shows a good performance from 2012 to 2020 with consistent positive pnl.

However, the strategy has been flat since 2020 which seems to imply that the alpha has decayed from 2020 onwards.