diff --git a/experiments/backtest.py b/experiments/backtest.py
index 3a3017e..1ba5858 100644
--- a/experiments/backtest.py
+++ b/experiments/backtest.py
@@ -6,6 +6,7 @@
 from typing import Callable
 import numpy as np
 import pandas as pd
+from utils import synthetic_returns
 
 # hack to allow importing from parent directory without having a package
 sys.path.append(str(Path(__file__).parent.parent))
@@ -61,7 +62,7 @@ def run_backtest(
     post_trade_cash = []
     post_trade_quantities = []
 
-    returns = prices.pct_change().dropna()
+    returns = synthetic_returns(prices).dropna()
     means = returns.ewm(halflife=125).mean()
     covariance_df = returns.ewm(halflife=125).cov()
     days = returns.index
diff --git a/experiments/taming.py b/experiments/taming.py
index 44e09e1..7cf5cdf 100644
--- a/experiments/taming.py
+++ b/experiments/taming.py
@@ -7,21 +7,28 @@
 import matplotlib.pyplot as plt
 
 
-def unconstrained_markowitz(inputs: OptimizationInput) -> np.ndarray:
+def unconstrained_markowitz(
+    inputs: OptimizationInput, long_only: bool = False
+) -> np.ndarray:
     """Compute the unconstrained Markowitz portfolio weights."""
     n_assets = inputs.prices.shape[1]
 
     mu, Sigma = inputs.mean.values, inputs.covariance.values
 
-    w = cp.Variable(n_assets)
-    c = cp.Variable()
+    if long_only:
+        w = cp.Variable(n_assets, nonneg=True)
+        c = cp.Variable(nonneg=True)
+    else:
+        w = cp.Variable(n_assets)
+        c = cp.Variable()
     objective = mu @ w
 
     chol = np.linalg.cholesky(Sigma)
     constraints = [
         cp.sum(w) + c == 1,
-        cp.norm2(chol @ w) <= inputs.risk_target,
+        cp.norm2(chol.T @ w) <= inputs.risk_target,
     ]
+
     problem = cp.Problem(cp.Maximize(objective), constraints)
     problem.solve(get_solver())
     assert problem.status in {cp.OPTIMAL, cp.OPTIMAL_INACCURATE}
@@ -30,21 +37,7 @@ def unconstrained_markowitz(inputs: OptimizationInput) -> np.ndarray:
 
 def long_only_markowitz(inputs: OptimizationInput) -> np.ndarray:
     """Compute the long-only Markowitz portfolio weights."""
-    n_assets = inputs.prices.shape[1]
-
-    mu, Sigma = inputs.mean.values, inputs.covariance.values
-
-    w = cp.Variable(n_assets, nonneg=True)
-    c = cp.Variable(nonneg=True)
-    objective = mu @ w
-    constraints = [
-        cp.sum(w) + c == 1,
-        cp.quad_form(w, Sigma, assume_PSD=True) <= inputs.risk_target**2,
-    ]
-    problem = cp.Problem(cp.Maximize(objective), constraints)
-    problem.solve(get_solver())
-    assert problem.status in {cp.OPTIMAL, cp.OPTIMAL_INACCURATE}
-    return w.value, c.value
+    return unconstrained_markowitz(inputs, long_only=True)
 
 
 def equal_weights(inputs: OptimizationInput) -> np.ndarray:
diff --git a/experiments/utils.py b/experiments/utils.py
new file mode 100644
index 0000000..a62ab64
--- /dev/null
+++ b/experiments/utils.py
@@ -0,0 +1,8 @@
+import numpy as np
+
+
+def synthetic_returns(prices, sigma_r=0.02236, sigma_eps=0.14142):
+    returns = prices.pct_change()
+
+    alpha = sigma_r**2 / (sigma_r**2 + sigma_eps**2)
+    return alpha * (returns + np.random.normal(size=returns.shape) * sigma_eps)