ENH: Add Alpha & Beta stats (#1221)

* added beta & alpha / resolved merge conflict

* simplified beta calculation

* remove DS_store

* move beta & alpha / use log return

* Update backtesting/_stats.py
* Update backtesting/_stats.py

* alpha & beta test

* #noqa: E501

* add space

* update docs

* Revert unrelated change
* Add comment

Author: jensnesten

README.md

@@ -68,6 +68,8 @@ CAGR [%]                                16.80
 Sharpe Ratio                             0.66
 Sortino Ratio                            1.30
 Calmar Ratio                             0.77
+Alpha [%]                              450.62
+Beta                                     0.02
 Max. Drawdown [%]                      -33.08
 Avg. Drawdown [%]                       -5.58
 Max. Drawdown Duration      688 days 00:00:00

backtesting/_stats.py

@@ -152,6 +152,13 @@ def _round_timedelta(value, _period=_data_period(index)):
         s.loc['Sortino Ratio'] = (annualized_return - risk_free_rate) / (np.sqrt(np.mean(day_returns.clip(-np.inf, 0)**2)) * np.sqrt(annual_trading_days))  # noqa: E501
     max_dd = -np.nan_to_num(dd.max())
     s.loc['Calmar Ratio'] = annualized_return / (-max_dd or np.nan)
+    equity_log_returns = np.log(equity[1:] / equity[:-1])
+    market_log_returns = np.log(c[1:] / c[:-1])
+    cov_matrix = np.cov(equity_log_returns, market_log_returns)
+    beta = cov_matrix[0, 1] / cov_matrix[1, 1]
+    # Jensen CAPM Alpha: can be strongly positive when beta is negative and B&H Return is large
+    s.loc['Alpha [%]'] = s.loc['Return [%]'] - risk_free_rate * 100 - beta * (s.loc['Buy & Hold Return [%]'] - risk_free_rate * 100)  # noqa: E501
+    s.loc['Beta'] = beta
     s.loc['Max. Drawdown [%]'] = max_dd * 100
     s.loc['Avg. Drawdown [%]'] = -dd_peaks.mean() * 100
     s.loc['Max. Drawdown Duration'] = _round_timedelta(dd_dur.max())

backtesting/backtesting.py

@@ -1262,6 +1262,8 @@ def run(self, **kwargs) -> pd.Series:
             Sharpe Ratio                          0.58038
             Sortino Ratio                         1.08479
             Calmar Ratio                          0.44144
+            Alpha [%]                           394.37391
+            Beta                                  0.03803
             Max. Drawdown [%]                   -47.98013
             Avg. Drawdown [%]                    -5.92585
             Max. Drawdown Duration      584 days 00:00:00

backtesting/test/_test.py

@@ -317,6 +317,8 @@ def test_compute_stats(self):
                 'Start': pd.Timestamp('2004-08-19 00:00:00'),
                 'Win Rate [%]': 46.96969696969697,
                 'Worst Trade [%]': -18.39887353835481,
+                'Alpha [%]': 394.37391142027462,
+                'Beta': 0.03803390709192,
         })
 
         def almost_equal(a, b):