Insanely high mean annual returns for volatile stocks in historical data

mean_vs_vol.py

import pandas as pd
import matplotlib.pyplot as plt
import numpy as np
from matplotlib.ticker import FuncFormatter, LogLocator

# Data:
#   lr_rf_1y_t   : float  — risk-free log return at time t (e.g., log(1.03) for 3%)
#   lr_t2        : float  — stock actual log return at time t2 = t + period_d
#   ema_var_d_t  : float  — daily variance as log(return)^2, current estimate at time t as EMA(span=365/3)
#   h_var_d      : float  — daily variance as log(return)^2, historical estimate over whole stock history
#   period_d     : int    — always 365, can be ignored
#   symbol       : str    — stock symbol (e.g., 'AAPL')
#   t            : str    — date at time t in 'YYYY-MM-DD' format
df = pd.read_csv('/storage/data/alien/experiments/historical_predictor/returns_period365_step365.tsv', sep='\t')

df['nlr_t2'] = df['lr_t2'] - df['lr_rf_1y_t'] # Risk-adjusted log return
df['r_t2'] = np.exp(df['lr_t2'])              # Multiplicative stock return
df['nr_t2'] = np.exp(df['nlr_t2'])            # Multiplicative risk-adjusted stock return

def plot_mean_vs_vol(window: int = 500):
  df_sorted = df.sort_values('ema_var_d_t').reset_index(drop=True)

  result_x = []
  result_y = []

  for i in range(len(df_sorted)):
    left = i - window
    right = i + window

    if left < 0 or right >= len(df_sorted):
      continue  # Not enough samples, skip

    avg_exp_nlr_t2 = np.mean(df_sorted['nr_t2'].iloc[left:right+1])

    result_x.append(df_sorted['ema_var_d_t'].iloc[i])
    result_y.append(avg_exp_nlr_t2)

  # Plot
  plt.figure(figsize=(12, 6))
  plt.plot(result_x, result_y, linewidth=2)

  plt.title(f'Average of 1y Stock Returns vs its current Volatility')
  plt.xlabel('Volatility at t0 as EWMA[(log return)^2, span=365/3]')
  plt.xscale('log')
  plt.ylabel('mean[S_365/S_0/RF_0] as Moving Window')
  plt.grid(True)
  plt.tight_layout()
  plt.gca().xaxis.set_major_locator(LogLocator(base=10.0, subs=[1.0, 2.0, 5.0], numticks=10))
  plt.gca().xaxis.set_major_formatter(FuncFormatter(lambda x, _: f'{x:.4f}'))

  plt.show()
plot_mean_vs_vol(window=500)

This simulation may explain it... generating random brownian walks and dropping 0.3% bancruptsies, results are the same, very high mean of high vol stocks...

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

np.random.seed(0)

def simulate_gbm(volatility, T=1.0, n_sim=10000):
  # GBM with mu = 0, sigma = volatility
  Z = np.random.normal(0, 1, n_sim)
  S_T_S_0 = np.exp(-0.5 * volatility**2 * T + volatility * np.sqrt(T) * Z)
  return S_T_S_0

vols = [0.1, 0.2, 0.3, 0.5, 0.7, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0]

results = []
for vol in vols:
  returns = simulate_gbm(vol)
  # Drop worst 0.3% percentile
  threshold = np.percentile(returns, 0.3)
  returns_filtered = returns[returns >= threshold]
  results.append({
    'volatility': vol,
    'mean_S_T_S_0': np.mean(returns_filtered)
  })

df_results = pd.DataFrame(results)
print(df_results.round(4))

# Plot
plt.figure(figsize=(10, 6))
plt.plot(df_results['volatility'], df_results['mean_S_T_S_0'], label='Mean S_T / S_0', marker='o')
plt.axhline(1.0, color='gray', linestyle='--', label='Expected Value')
plt.xlabel('Volatility (σ)')
plt.ylabel('S_T / S_0')
plt.title('Effect of Volatility on mean(S_T/S_0) vs median(S_T/S_0)')
plt.legend()
plt.grid(True)
plt.tight_layout()
plt.show()