Practical and Numerical Computation on Financial Portfolio Optimization Using Python
1 Lesson Objectives
By the end of this lesson, students should be able to:
• Understand the fundamentals of financial portfolio optimization.
• Implement portfolio optimization techniques using Python.
• Apply numerical methods to compute the minimum variance portfolio (MVP), mean-variance optimization (MVO), and constrained portfolios.
• Use Python libraries such as NumPy, Pandas, SciPy, and CVXPY for optimization.
2 Introduction to Portfolio Optimization
2.1 What is Portfolio Optimization?
Portfolio optimization is the process of selecting the best portfolio (asset alloca- tion) according to an objective, typically maximizing returns while minimizing risk. Investors aim to balance risk and return based on their preferences.
2.2 Key Concepts
• Expected Return (μ): The weighted average of asset returns, repre- senting the expected profit from a portfolio:
• Risk (Variance & Standard Deviation): Measures portfolio volatility and uncertainty:
• Covariance & Correlation: Measures how assets move relative to each other:
• Sharpe Ratio: Return-to-risk ratio used for optimal portfolio selection:
• Efficient Frontier: The set of portfolios that provides the highest ex- pected return for a given level of risk.
2.3 Types of Portfolio Optimization
1. Minimum Variance Portfolio (MVP): Minimizes portfolio risk by choosing weights that result in the lowest possible variance.
2. Mean-Variance Optimization (MVO): Maximizes return for a given risk level, following Markowitz’s modern portfolio theory.
3. Constrained Portfolio Optimization: Introduces constraints such as no short-selling, maximum investment limits, or risk bounds.
3 Practical Example: Portfolio Optimization with Analysis
3.1 Problem Statement
Consider an investor who wants to optimize a portfolio composed of five tech- nology stocks: Apple (AAPL), Google (GOOGL), Microsoft (MSFT), Amazon (AMZN), and Tesla (TSLA). The objective is to construct an efficient portfolio by minimizing risk while achieving a target return.
3.2 Step 1: Data Collection and Preprocessing
We first fetch historical stock prices and compute daily returns:
import yfinance as yf
import numpy as np
import pandas as pd
stocks = [ ’AAPL’ , ’GOOGL’ , ’MSFT’ , ’AMZN’ , ’TSLA’ ]
data = yf . download ( stocks , start=’2020−01−01 ’ , end=’2023−01−01 ’ ) [ ’Adj – Close ’ ]
returns = data . pct change ( ) . dropna ()
mean returns = returns . mean()
cov matrix = returns . cov ()
3.3 Step 2: Minimum Variance Portfolio Optimization
Using quadratic programming, we find the portfolio that minimizes risk:
def min variance portfolio ( cov matrix ) :
num assets = len ( cov matrix )
w = cp . Variable ( num assets )
objective = cp . Minimize ( cp . quad form. (w, cov matrix ))
constraints = [ cp . sum(w) == 1 , w >= 0]
prob = cp . Problem( objective , constraints )
prob . solve ()
return w. value
mvp weights = min variance portfolio ( cov matrix )
print (”Minimum – Variance – Portfolio – Weights : ” , mvp weights )
3.4 Step 3: Portfolio Performance Analysis
To evaluate the portfolio, we compute the expected return and risk:
def portfolio performance ( weights , mean returns , cov matrix ) :
port return = np . dot ( weights , mean returns )
port volatility = np . sqrt (np . dot ( weights .T, np . dot ( cov matrix , weights )))
return port return , port volatility
mvp return , mvp vol = portfolio performance ( mvp weights , mean returns , cov matrix )
print ( f ”MVP – Expected – Return : – {mvp return : . 4 f } , –MVP – Risk : – {mvp vol : . 4 f }”)
3.5 Analysis and Interpretation
From the results:
• The Minimum Variance Portfolio (MVP) provides the lowest risk but may not offer the highest return.
• The Efficient Frontier shows a range of optimal portfolios balancing risk and return.
• Investors can select portfolios based on their risk tolerance.
4 Working with a fixed dataset
Today, we will work with a fixed dataset and roundly generated data due to the rate limit on YahooFinance.
Consider that you are working for a firm that manages 5 assets given in the dataset asset returns .xlsx.
5 Conclusion
• The efficient frontier provides valuable insights into optimal asset allo- cation.
• Portfolio optimization techniques can be extended to factor investing, risk parity, and machine learning-based asset selection.
• Future work could involve dynamic rebalancing and robust opti- mization models to handle market changes.
1. Importing Libraries
The following Python libraries are imported:
• NumPy: For numerical operations such as matrix multiplication.
• Pandas: For handling datasets.
• Matplotlib: For plotting the efficient frontier.
• SciPy: For portfolio optimization using the minimize function.
Listing 1: Importing Required Libraries
import numpy as np import pandas as pd
import matplotlib . pyplot as plt
from scipy . optimize import minimize
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2. Loading Dataset
The asset returns dataset is loaded from an Excel file:
Listing 2: Loading Asset Returns Dataset
file path = ”/mnt/data/ asset returns . xlsx ”
returns df = pd . read excel ( file path )
—
3. Calculating Key Metrics
We calculate the mean returns and the covariance matrix:
mean returns = returns df . mean()
cov matrix = returns df . cov ()
num assets = len ( mean returns )
—
4. Defining Portfolio Variance Function
def portfolio variance ( weights , cov matrix ) :
return np . dot ( weights .T, np . dot ( cov matrix , weights ))
—
5. Defining Constraints and Bounds
The optimization problem has the following constraints:
• The sum of portfolio weights must equal 1:
• Each weight must be between 0 and 1 (no short selling):
constraints = ({ ’ type ’ : ’eq ’ , ’ fun ’ : lambda weights : np . sum( weights ) − 1}) bounds = tuple ((0 , 1) for asset in range ( num assets ))
init guess = np . ones ( num assets ) / num assets
—
6. Portfolio Optimization
The optimization problem minimizes the portfolio variance using the Sequential Least Squares Programming (SLSQP) method:
subject to:
opt results = minimize ( portfolio variance , init guess , args=(cov matrix ,) ,
method=’SLSQP’ , bounds=bounds , constraints=constraints )
7. Extracting Optimization Results
The optimal weights, return, and risk of the minimum variance portfolio are extracted:
min var weights = opt results . x
min var return = np . dot ( min var weights , mean returns )
min var risk = np . sqrt ( opt results . fun )
8. Efficient Frontier Simulation
We generate 5000 random portfolios to plot the efficient frontier. For each portfolio:
• Portfolio return:
• Portfolio risk (standard deviation):
• Sharpe ratio:
num portfolios = 5000
results = np . zeros ((3 , num portfolios ))
for i in range ( num portfolios ) :
weights = np . random . random( num assets ) weights /= np . sum( weights )
port return = np . dot ( weights , mean returns )
port risk = np . sqrt (np . dot ( weights .T, np . dot ( cov matrix , weights )))
results [0 , i ] = port risk
results [1 , i ] = port return
results [2 , i ] = port return / port risk
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9. Plotting the Efficient Frontier
We visualize the efficient frontier along with the minimum variance portfolio.
plt . figure ( fig size =(10, 6))
plt . scatter ( results [0 , : ] , results [1 , : ] , c=results [2 , : ] , cmap=’ viridis ’ ,
alpha =0.7)
plt . colorbar ( label=”Sharpe Ratio ”)
plt . scatter ( min var risk , min var return , color =’red ’ , marker= ’ * ’ ,
s=200, label=”Minimum Variance Portfolio ”) plt . title (” Efficient Frontier ”)
plt . xlabel (” Risk ( Standard Deviation )”) plt . ylabel (” Return ”)
plt . legend () plt . grid ()
plt . show ()
10. Displaying Results
Finally, the optimal portfolio details are displayed:
Listing 3: Displaying Results print (”Minimum– Variance – Portfolio : ”)
print ( f ”Risk : – { min var risk : . 4 f }”)
print ( f ”Return : – {min var return : . 4 f }”) print ( f ”Weights : – {min var weights}”)
Conclusion
This approach identifies the optimal portfolio that minimizes risk while main- taining the desired level of return. The efficient frontier represents the set of portfolios that offer the best possible return for a given level of risk.