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Introduction

In quantitative finance, risk management and volatility forecasting play a crucial role in designing trading strategies. This article explores a systematic approach to take-profit decisions using the Conditional Value at Risk (CVaR) on the right tail, also known as the Expected Upside Potential (EUP), combined with a Heterogeneous Auto-Regressive (HAR) model for volatility prediction.

We will analyze an equally weighted portfolio of the “Magnificent Seven” stocks and implement a data-driven methodology to determine dynamic sell signals.

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1️. Data and Portfolio Construction

We start by selecting seven major tech stocks: Apple (AAPL), Microsoft (MSFT), NVIDIA (NVDA), Amazon (AMZN), Tesla (TSLA), Meta (META), and Alphabet (GOOGL). Using Yahoo Finance, we fetch historical adjusted closing prices from January 1, 2020, to March 15, 2025 and compute their daily returns.

import numpy as np
import pandas as pd
import yfinance as yf
import statsmodels.api as sm
import matplotlib.pyplot as plt

plt.close('all')

tickers = ["AAPL", "MSFT", "NVDA", "AMZN", "TSLA", "META", "GOOGL"]
data = yf.download(tickers, start="2020-01-01", end="2025-03-15", auto_adjust=False)["Close"]
returns = data.pct_change().dropna()

# Equally weighted portfolio
weights = np.ones(len(tickers)) / len(tickers)
portfolio_returns = returns.dot(weights)
portfolio_prices = (1 + portfolio_returns).cumprod() * 100  # Base 100

2️. Calculating VaR (95%) and Expected Upside Potential (EUP)

To quantify the risk and potential upside, we use:

• VaR (95%): The 95th percentile of rolling 60-day returns.

• EUP (Right-Tail CVaR): The average return of observations exceeding the 95th percentile.

window_eup = 60
rolling_var = portfolio_returns.rolling(window=window_eup).quantile(0.95)

def eup_cvar(x, alpha=95):
    threshold = np.percentile(x, alpha)
    tail = x[x >= threshold]
    return tail.mean() if len(tail) > 0 else np.nan

eup_series = portfolio_returns.rolling(window=window_eup).apply(lambda x: eup_cvar(x, alpha=95), raw=True)
signal = portfolio_returns > rolling_var  # Take-profit trigger

3️. HAR Model for Volatility Prediction

The Heterogeneous Auto-Regressive (HAR) Model is useful for forecasting daily volatility based on:

• Daily volatility (2-day window)

• Weekly volatility (5-day window)

• Monthly volatility (22-day window)

vol_daily   = portfolio_returns.rolling(window=2).std()
vol_weekly  = portfolio_returns.rolling(window=5).std()
vol_monthly = portfolio_returns.rolling(window=22).std()

vol_data = pd.DataFrame({
    'vol_daily': vol_daily,
    'vol_weekly': vol_weekly,
    'vol_monthly': vol_monthly
}).dropna()

X = sm.add_constant(vol_data[['vol_daily', 'vol_weekly', 'vol_monthly']])
y = vol_data['vol_daily'].shift(-1)
data_valid = pd.concat([X, y], axis=1).dropna()
X_valid = data_valid.iloc[:, :-1]
y_valid = data_valid.iloc[:, -1]

if len(X_valid) > 0:
    model = sm.OLS(y_valid, X_valid).fit()
    vol_data.loc[X_valid.index, 'vol_forecast'] = model.predict(X_valid)
else:
    vol_data['vol_forecast'] = np.nan

4️. Calculating the Sell Ratio and Sale Percentage

We compute the sell ratio as:

The ratio is mapped to a sale percentage between 0% and 100%, based on predefined thresholds:

• Sell Ratio ≤ 1.0 → 0% Sale

• Sell Ratio ≥ 3.0 → 100% Sale

• Between 1.0 and 3.0 → Interpolated proportionally

common_index = portfolio_returns.index.intersection(vol_data['vol_forecast'].dropna().index)
eup_aligned = eup_series.loc[common_index]
signal_aligned = signal.loc[common_index]
vol_forecast_aligned = vol_data['vol_forecast'].loc[common_index]

ratio = pd.Series(np.nan, index=common_index)
for date in common_index:
    if signal_aligned.loc[date]:
        if not pd.isna(eup_aligned.loc[date]) and not pd.isna(vol_forecast_aligned.loc[date]):
            ratio.loc[date] = eup_aligned.loc[date] / vol_forecast_aligned.loc[date]

def sale_percentage(r, threshold_min=1.0, threshold_max=3.0):
    if r <= threshold_min:
        return 0.0
    elif r >= threshold_max:
        return 1.0
    else:
        return (r - threshold_min) / (threshold_max - threshold_min)

sales_percentage = ratio.dropna().apply(sale_percentage)

5️. Printing and Visualizing Sell Signals

print("Last 10 Sell Ratios (EUP / Volatility Forecast):")
print(ratio.dropna().tail(10))
print("\nLast 10 Sale Percentages:")
print(sales_percentage.tail(10))

We plot the portfolio price evolution, highlighting sell signals with red markers:

fig, ax = plt.subplots(figsize=(12, 6))
ax.plot(portfolio_prices.index, portfolio_prices, label="Portfolio Price", color="blue")

sales_dates = sales_percentage.index
if not sales_dates.empty:
    ax.scatter(sales_dates, portfolio_prices.loc[sales_dates],
               color="red", marker="v", s=100, label="Sell Signal")
    for date in sales_dates:
        perc = sales_percentage.loc[date] * 100
        ax.annotate(f"{perc:.0f}%", (date, portfolio_prices.loc[date]),
                    textcoords="offset points", xytext=(0,10), ha="center", color="red")

ax.set_title("Sell Signals Based on EUP (Right-Tail CVaR) and HAR Volatility Model")
ax.set_xlabel("Date")
ax.set_ylabel("Portfolio Price (Base 100)")
ax.legend()
ax.grid(True)
plt.show()

Conclusion

This approach integrates EUP-based take-profit signals with a HAR volatility model to adaptively determine optimal sell percentages. Unlike fixed thresholds, this data-driven strategy adjusts dynamically to market conditions, offering a more systematic way to take profits based on expected upside potential and forecasted risk.