BUG: fix annualization of alpha (#60)

Author: Vikram Narayan

empyrical/stats.py

@@ -1283,10 +1283,18 @@ def alpha_aligned(returns,
     adj_factor_returns = _adjust_returns(factor_returns, risk_free)
     alpha_series = adj_returns - (_beta * adj_factor_returns)
 
-    out = np.multiply(
-        nanmean(alpha_series, axis=0, out=out),
-        ann_factor,
-        out=out,
+    out = np.subtract(
+        np.power(
+            np.add(
+                nanmean(alpha_series, axis=0, out=out),
+                1,
+                out=out
+            ),
+            ann_factor,
+            out=out
+        ),
+        1,
+        out=out
     )
 
     if allocated_output and isinstance(returns, pd.DataFrame):

empyrical/tests/test_stats.py

@@ -733,7 +733,7 @@ def test_excess_sharpe_trans(self, returns, add_noise, translation):
     @parameterized.expand([
         (empty_returns, simple_benchmark, (np.nan, np.nan)),
         (one_return, one_return, (np.nan, np.nan)),
-        (mixed_returns, negative_returns[1:], (-8.306666666666668,
+        (mixed_returns, negative_returns[1:], (-0.9997853834885004,
                                                -0.71296296296296313)),
         (mixed_returns, mixed_returns, (0.0, 1.0)),
         (mixed_returns, -mixed_returns, (0.0, -1.0)),
@@ -801,6 +801,7 @@ def test_alpha_beta_translation(self, mean_returns, translation):
         benchmark = pd.Series(
             bench,
             index=pd.date_range('2000-1-30', periods=1000, freq='D'))
+
         # Translate returns and generate alphas and betas.
         returns_depressed = returns-translation
         returns_raised = returns+translation
@@ -814,11 +815,13 @@ def test_alpha_beta_translation(self, mean_returns, translation):
         # Alpha should change proportionally to how much returns were
         # translated.
         assert_almost_equal(
-            (alpha_standard - alpha_depressed)/252,
+            ((alpha_standard + 1) ** (1/252)) -
+            ((alpha_depressed + 1) ** (1/252)),
             translation,
             DECIMAL_PLACES)
         assert_almost_equal(
-            (alpha_raised - alpha_standard)/252,
+            ((alpha_raised + 1) ** (1/252)) -
+            ((alpha_standard + 1) ** (1/252)),
             translation,
             DECIMAL_PLACES)
         # Beta remains constant.
@@ -1108,8 +1111,8 @@ def test_down_capture(self, returns, factor_returns, expected):
          [(np.nan, np.nan)] * len(simple_benchmark)),
         (one_return, one_return, 1, [(np.nan, np.nan)]),
         (mixed_returns, negative_returns,
-         6, [(-3.81286957, -0.7826087), (-4.03558719, -0.76156584),
-             (-2.66915888, -0.61682243), (-7.8987541, -0.41311475)]),
+         6, [(-0.97854954, -0.7826087), (-0.9828927, -0.76156584),
+             (-0.93166924, -0.61682243), (-0.99967288, -0.41311475)]),
         (mixed_returns, mixed_returns,
          6, [(0.0, 1.0), (0.0, 1.0), (0.0, 1.0), (0.0, 1.0)]),
         (mixed_returns, -mixed_returns,
@@ -1209,7 +1212,7 @@ def test_roll_up_capture(self, returns, factor_returns, window, expected):
         (empty_returns, simple_benchmark, (np.nan, np.nan)),
         (one_return, one_return, (np.nan, np.nan)),
         (mixed_returns[1:], negative_returns[1:],
-         (-8.306666666666668, -0.71296296296296313)),
+         (-0.9997853834885004, -0.71296296296296313)),
         (mixed_returns, mixed_returns, (0.0, 1.0)),
         (mixed_returns, -mixed_returns, (0.0, -1.0))
     ])
@@ -1231,7 +1234,7 @@ def test_down_alpha_beta(self, returns, benchmark, expected):
         (empty_returns, simple_benchmark, (np.nan, np.nan)),
         (one_return, one_return, (np.nan, np.nan)),
         (mixed_returns[1:], positive_returns[1:],
-         (0.3599999999999995, 0.4285714285)),
+         (0.432961242076658, 0.4285714285)),
         (mixed_returns, mixed_returns, (0.0, 1.0)),
         (mixed_returns, -mixed_returns, (0.0, -1.0))
     ])