MarketMaker.cc Team
クオンツ・リサーチ＆戦略
Building optimal crypto portfolios with Python - because YOLO isn't a strategy
Markowitz Portfolio Theory: Mathematical optimization applied to digital assets to maximize returns for a given risk level.
Hey crypto degens! 👋
Remember that time you threw your entire stack into DOGE because Elon tweeted? Or when you panic-sold everything during the last crash? Yeah, we've all been there. Today we're going to talk about something that might save your portfolio (and your sanity): Markowitz Portfolio Theory.
Harry Markowitz literally won a Nobel Prize for this stuff back in 1990. The basic idea? You can mathematically optimize your portfolio to get the best possible returns for any given level of risk. It's like having a GPS for your investments instead of driving blindfolded.
Before we dive into code, let's understand what we're dealing with:
The magic happens when you combine assets that don't move in perfect sync. When Bitcoin crashes, maybe some DeFi tokens hold up better. That's diversification working for you.
Risk vs. Return: Balancing volatile high-yield assets with a stable foundation to achieve the optimal geometric mean return.
First things first - let's get our tools ready:
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns
from scipy.optimize import minimize
import yfinance as yf
import warnings
warnings.filterwarnings('ignore')
plt.style.use('dark_background')
sns.set_palette("husl")
Let's start with the basics. We'll calculate returns and risk for a simple 2-asset portfolio.
def get_crypto_data(symbols, period="1y"):
"""
Fetch crypto data from Yahoo Finance
symbols: list of crypto symbols (e.g., ['BTC-USD', 'ETH-USD'])
period: time period for data
"""
data = yf.download(symbols, period=period)['Adj Close']
return data
crypto_symbols = ['BTC-USD', 'ETH-USD']
prices = get_crypto_data(crypto_symbols)
returns = prices.pct_change().dropna()
print("Daily Returns Preview:")
print(returns.head())
Now let's calculate some basic portfolio metrics:
def portfolio_performance(weights, returns):
"""
Calculate portfolio return and volatility
weights: array of portfolio weights
returns: dataframe of asset returns
"""
portfolio_return = np.sum(returns.mean() * weights) * 252
portfolio_vol = np.sqrt(np.dot(weights.T, np.dot(returns.cov() * 252, weights)))
return portfolio_return, portfolio_vol
weights_5050 = np.array([0.5, 0.5])
ret_5050, vol_5050 = portfolio_performance(weights_5050, returns)
print(f"50/50 Portfolio:")
print(f"Expected Annual Return: {ret_5050:.2%}")
print(f"Annual Volatility: {vol_5050:.2%}")
print(f"Sharpe Ratio: {ret_5050/vol_5050:.3f}")
Now we're cooking! The efficient frontier shows us all possible optimal portfolios. Each point represents the best possible return for a given level of risk.
def generate_random_portfolios(returns, num_portfolios=10000):
"""
Generate random portfolio combinations
"""
num_assets = len(returns.columns)
results = np.zeros((4, num_portfolios))
for i in range(num_portfolios):
weights = np.random.random(num_assets)
weights /= np.sum(weights) # Normalize to sum to 1
portfolio_return, portfolio_vol = portfolio_performance(weights, returns)
sharpe_ratio = portfolio_return / portfolio_vol
results[0,i] = portfolio_return
results[1,i] = portfolio_vol
results[2,i] = sharpe_ratio
results[3,i:] = weights
return results
results = generate_random_portfolios(returns)
portfolio_results = pd.DataFrame({
'Returns': results[0],
'Volatility': results[1],
'Sharpe_Ratio': results[2]
})
plt.figure(figsize=(12, 8))
scatter = plt.scatter(portfolio_results['Volatility'],
portfolio_results['Returns'],
c=portfolio_results['Sharpe_Ratio'],
cmap='viridis', alpha=0.6)
plt.colorbar(scatter, label='Sharpe Ratio')
plt.xlabel('Volatility (Risk)')
plt.ylabel('Expected Return')
plt.title('Efficient Frontier - Random Portfolios')
plt.show()
The Efficient Frontier: The curve representing the maximum possible expected return for a given level of risk.
Random sampling is fun, but we want the mathematically optimal solution. Time to bring out the big guns - scipy optimization!
def negative_sharpe_ratio(weights, returns, risk_free_rate=0.02):
"""
Calculate negative Sharpe ratio (we minimize this)
"""
portfolio_return, portfolio_vol = portfolio_performance(weights, returns)
sharpe = (portfolio_return - risk_free_rate) / portfolio_vol
return -sharpe
def minimize_volatility(weights, returns):
"""
Calculate portfolio volatility (we minimize this)
"""
_, portfolio_vol = portfolio_performance(weights, returns)
return portfolio_vol
def portfolio_return_objective(weights, returns):
"""
Calculate portfolio return (we maximize this)
"""
portfolio_return, _ = portfolio_performance(weights, returns)
return -portfolio_return # Negative because we minimize
def optimize_portfolio(returns, objective='sharpe', target_return=None):
"""
Optimize portfolio based on different objectives
"""
num_assets = len(returns.columns)
constraints = ({'type': 'eq', 'fun': lambda x: np.sum(x) - 1}) # Weights sum to 1
bounds = tuple((0, 1) for _ in range(num_assets)) # No short selling
initial_guess = num_assets * [1. / num_assets]
if objective == 'sharpe':
result = minimize(negative_sharpe_ratio, initial_guess,
args=(returns,), method='SLSQP',
bounds=bounds, constraints=constraints)
elif objective == 'min_vol':
result = minimize(minimize_volatility, initial_guess,
args=(returns,), method='SLSQP',
bounds=bounds, constraints=constraints)
elif objective == 'target_return':
constraints = ({'type': 'eq', 'fun': lambda x: np.sum(x) - 1},
{'type': 'eq', 'fun': lambda x: portfolio_performance(x, returns)[0] - target_return})
result = minimize(minimize_volatility, initial_guess,
args=(returns,), method='SLSQP',
bounds=bounds, constraints=constraints)
return result
max_sharpe = optimize_portfolio(returns, 'sharpe')
min_vol = optimize_portfolio(returns, 'min_vol')
print("🎯 Maximum Sharpe Ratio Portfolio:")
for i, symbol in enumerate(crypto_symbols):
print(f"{symbol}: {max_sharpe.x[i]:.3f}")
ret_sharpe, vol_sharpe = portfolio_performance(max_sharpe.x, returns)
print(f"Return: {ret_sharpe:.2%}, Volatility: {vol_sharpe:.2%}")
print(f"Sharpe Ratio: {ret_sharpe/vol_sharpe:.3f}\n")
print("🛡️ Minimum Volatility Portfolio:")
for i, symbol in enumerate(crypto_symbols):
print(f"{symbol}: {min_vol.x[i]:.3f}")
ret_minvol, vol_minvol = portfolio_performance(min_vol.x, returns)
print(f"Return: {ret_minvol:.2%}, Volatility: {vol_minvol:.2%}")
Let's scale this up to a proper crypto portfolio with multiple assets:
crypto_portfolio = ['BTC-USD', 'ETH-USD', 'BNB-USD', 'ADA-USD', 'SOL-USD', 'DOT-USD']
prices_multi = get_crypto_data(crypto_portfolio, period="2y")
returns_multi = prices_multi.pct_change().dropna()
correlation_matrix = returns_multi.corr()
plt.figure(figsize=(10, 8))
sns.heatmap(correlation_matrix, annot=True, cmap='RdYlBu_r', center=0)
plt.title('Crypto Asset Correlation Matrix')
plt.show()
def efficient_frontier(returns, num_portfolios=50):
"""
Calculate the efficient frontier
"""
ret_range = np.linspace(returns.mean().min()*252, returns.mean().max()*252, num_portfolios)
efficient_portfolios = []
for target_ret in ret_range:
try:
result = optimize_portfolio(returns, 'target_return', target_ret)
if result.success:
ret, vol = portfolio_performance(result.x, returns)
efficient_portfolios.append([ret, vol, result.x])
except:
continue
return np.array(efficient_portfolios)
efficient_port = efficient_frontier(returns_multi)
plt.figure(figsize=(14, 10))
random_results = generate_random_portfolios(returns_multi, 5000)
plt.scatter(random_results[1], random_results[0],
c=random_results[2], cmap='viridis', alpha=0.3, s=10)
if len(efficient_port) > 0:
plt.plot(efficient_port[:,1], efficient_port[:,0], 'r-', linewidth=3, label='Efficient Frontier')
max_sharpe_multi = optimize_portfolio(returns_multi, 'sharpe')
min_vol_multi = optimize_portfolio(returns_multi, 'min_vol')
ret_sharpe_multi, vol_sharpe_multi = portfolio_performance(max_sharpe_multi.x, returns_multi)
ret_minvol_multi, vol_minvol_multi = portfolio_performance(min_vol_multi.x, returns_multi)
plt.scatter(vol_sharpe_multi, ret_sharpe_multi, marker='*', color='gold', s=500, label='Max Sharpe')
plt.scatter(vol_minvol_multi, ret_minvol_multi, marker='*', color='red', s=500, label='Min Volatility')
plt.colorbar(label='Sharpe Ratio')
plt.xlabel('Volatility (Risk)')
plt.ylabel('Expected Return')
plt.title('Multi-Asset Crypto Portfolio Optimization')
plt.legend()
plt.show()
print("🚀 Optimal Multi-Asset Allocations:")
print("\nMaximum Sharpe Ratio Portfolio:")
sharpe_weights = pd.Series(max_sharpe_multi.x, index=crypto_portfolio).sort_values(ascending=False)
for asset, weight in sharpe_weights.items():
if weight > 0.01: # Only show significant allocations
print(f"{asset}: {weight:.1%}")
print(f"\nPortfolio Metrics:")
print(f"Expected Return: {ret_sharpe_multi:.1%}")
print(f"Volatility: {vol_sharpe_multi:.1%}")
print(f"Sharpe Ratio: {ret_sharpe_multi/vol_sharpe_multi:.2f}")
Multi-Asset Diversification: Constructing a robust portfolio by combining uncorrelated crypto assets into a stable geometric structure.
For the true portfolio optimization ninjas, let's implement some advanced techniques:
def risk_parity_portfolio(returns):
"""
Risk Parity Portfolio - each asset contributes equally to portfolio risk
"""
def risk_contribution(weights, cov_matrix):
portfolio_vol = np.sqrt(np.dot(weights.T, np.dot(cov_matrix, weights)))
marginal_contrib = np.dot(cov_matrix, weights) / portfolio_vol
contrib = weights * marginal_contrib
return contrib
def risk_parity_objective(weights, cov_matrix):
contrib = risk_contribution(weights, cov_matrix)
target_contrib = np.ones(len(weights)) / len(weights)
return np.sum((contrib - target_contrib)**2)
num_assets = len(returns.columns)
cov_matrix = returns.cov() * 252
constraints = ({'type': 'eq', 'fun': lambda x: np.sum(x) - 1})
bounds = tuple((0.001, 1) for _ in range(num_assets))
initial_guess = num_assets * [1. / num_assets]
result = minimize(risk_parity_objective, initial_guess,
args=(cov_matrix,), method='SLSQP',
bounds=bounds, constraints=constraints)
return result
risk_parity_result = risk_parity_portfolio(returns_multi)
print("⚖️ Risk Parity Portfolio:")
rp_weights = pd.Series(risk_parity_result.x, index=crypto_portfolio).sort_values(ascending=False)
for asset, weight in rp_weights.items():
print(f"{asset}: {weight:.1%}")
ret_rp, vol_rp = portfolio_performance(risk_parity_result.x, returns_multi)
print(f"\nRisk Parity Metrics:")
print(f"Expected Return: {ret_rp:.1%}")
print(f"Volatility: {vol_rp:.1%}")
print(f"Sharpe Ratio: {ret_rp/vol_rp:.2f}")
def backtest_portfolio(weights, prices):
"""
Simple backtest of portfolio performance
"""
returns = prices.pct_change().dropna()
portfolio_returns = (returns * weights).sum(axis=1)
cumulative_returns = (1 + portfolio_returns).cumprod()
total_return = cumulative_returns.iloc[-1] - 1
annualized_return = (1 + total_return) ** (252 / len(portfolio_returns)) - 1
annualized_vol = portfolio_returns.std() * np.sqrt(252)
sharpe_ratio = annualized_return / annualized_vol
max_dd = (cumulative_returns / cumulative_returns.expanding().max() - 1).min()
return {
'total_return': total_return,
'annualized_return': annualized_return,
'annualized_volatility': annualized_vol,
'sharpe_ratio': sharpe_ratio,
'max_drawdown': max_dd,
'cumulative_returns': cumulative_returns
}
strategies = {
'Max Sharpe': max_sharpe_multi.x,
'Min Volatility': min_vol_multi.x,
'Risk Parity': risk_parity_result.x,
'Equal Weight': np.ones(len(crypto_portfolio)) / len(crypto_portfolio)
}
plt.figure(figsize=(14, 8))
for name, weights in strategies.items():
backtest_results = backtest_portfolio(weights, prices_multi)
plt.plot(backtest_results['cumulative_returns'], label=f"{name} (Sharpe: {backtest_results['sharpe_ratio']:.2f})")
plt.title('Portfolio Strategy Backtests')
plt.xlabel('Date')
plt.ylabel('Cumulative Returns')
plt.legend()
plt.yscale('log')
plt.grid(True, alpha=0.3)
plt.show()
*Algorithmic Backtesting: Simulating historical performance to validate theoretical optimization models.*
performance_summary = pd.DataFrame()
for name, weights in strategies.items():
results = backtest_portfolio(weights, prices_multi)
performance_summary[name] = [
f"{results['annualized_return']:.1%}",
f"{results['annualized_volatility']:.1%}",
f"{results['sharpe_ratio']:.2f}",
f"{results['max_drawdown']:.1%}"
]
performance_summary.index = ['Annual Return', 'Annual Volatility', 'Sharpe Ratio', 'Max Drawdown']
print("\n📊 Strategy Performance Summary:")
print(performance_summary)
Before you go all-in on mathematical optimization, here are some hard truths about crypto:
1. Past Performance ≠ Future Results The crypto market is young and chaotic. Those correlations you calculated? They might flip overnight when regulations change or when the next big hack happens.
2. Transaction Costs Matter Rebalancing your portfolio costs money. In DeFi, gas fees can eat your lunch. Factor this into your strategy.
3. Liquidity Issues Not all cryptos are equally liquid. That small-cap altcoin might look great in your optimization, but try selling it during a crash.
4. Regime Changes Crypto markets have different "regimes" - bull markets, bear markets, crab markets. What works in one might not work in another.
def practical_portfolio_rebalancing(target_weights, current_weights, threshold=0.05):
"""
Only rebalance when weights drift beyond threshold
"""
weight_diff = np.abs(target_weights - current_weights)
needs_rebalancing = np.any(weight_diff > threshold)
if needs_rebalancing:
print("🔄 Rebalancing needed!")
for i, (target, current) in enumerate(zip(target_weights, current_weights)):
if abs(target - current) > threshold:
print(f"Asset {i}: {current:.1%} → {target:.1%}")
else:
print("✅ Portfolio within tolerance, no rebalancing needed")
return needs_rebalancing
current_allocation = np.array([0.35, 0.25, 0.15, 0.10, 0.10, 0.05])
target_allocation = max_sharpe_multi.x
practical_portfolio_rebalancing(target_allocation, current_allocation)
You now have a complete toolkit for crypto portfolio optimization:
Remember: In crypto, even the best mathematical models can't predict when Elon will tweet about Dogecoin or when the next exchange will get hacked. Use portfolio theory as your foundation, but always keep some powder dry and never invest more than you can afford to lose.
Now go forth and optimize responsibly! 🚀
All the code from this tutorial is available on GitHub: https://github.com/suenot/markowitz
Happy optimizing! 📈