explained return predictions model

experiments/utils.py

@@ -1,8 +1,22 @@
 import numpy as np
 
 
-def synthetic_returns(prices, sigma_r=0.02236, sigma_eps=0.14142):
+def synthetic_returns(prices, var_r=0.0005, var_eps=0.02):
+    """
+    param prices: a DataFrame of prices
+    param var_r: the Gaussian variance of the returns
+    param var_eps: the Gaussian variance of the noise term
+
+    returns: a DataFrame of "synthetic return predictions" computed as
+    alpha*(returns+noise), where alpha=var_r / (var_r + var_eps); this is the
+    coefficient that minimize the variance of the prediction error under the
+    above model.
+
+    var_r = 0.0005 and var_eps = 0.02 correspond to an information ratio
+    sqrt(alpha) of about 0.15.
+    """
     returns = prices.pct_change()
 
-    alpha = sigma_r**2 / (sigma_r**2 + sigma_eps**2)
+    alpha = var_r / (var_r + var_eps)
+    sigma_eps = np.sqrt(var_eps)
     return alpha * (returns + np.random.normal(size=returns.shape) * sigma_eps)