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1 INTRO TO PORTFOLIO RISK MANAGEMENT IN PYTHON Portfolio Composition Dakota Wixom Quantitative Analyst QuantCourse.com
2 Calculating Portfolio Returns PORTFOLIO RETURN FORMULA: R : Portfolio return R w p a n a n : Return for asset n : Weight for asset n R = R w + R w R w p a 1 a 1 a 2 a 2 a n a 1
3 Calculating Portfolio Returns in Python Assuming StockReturns is a pandas DataFrame of stock returns, you can calculate the portfolio return for a set of portfolio weights as follows: In [1]: import numpy as np In [2]: portfolio_weights = np.array([0.25, 0.35, 0.10, 0.20, 0.10]) In [3]: port_ret = StockReturns.mul(portfolio_weights, axis=1).sum(axis=1) In [4]: port_ret Out [4]: Date In [5]: StockReturns["Portfolio"] = port_ret
4 Equally Weighted Portfolios in Python Assuming StockReturns is a pandas DataFrame of stock returns, you can calculate the portfolio return for an equally weighted portfolio as follows: In [1]: import numpy as np In [2]: numstocks = 5 In [3]: portfolio_weights_ew = np.repeat(1/numstocks, numstocks) In [4]: StockReturns.iloc[:,0:numstocks].mul(portfolio_weights_ew, axis=1).sum(ax Out [4]: Date
![Plotting Portfolio Returns in Python To plot the daily returns in Python: In [1]: StockPrices["Returns"] =](/docs-images/87/95838613/images/5-0.jpg "StockPrices[\"Adj Close\"].pct_change() In [2]: StockReturns = StockPrices[\"Returns\"] In [3]: StockReturns.")
5 Plotting Portfolio Returns in Python To plot the daily returns in Python: In [1]: StockPrices["Returns"] = StockPrices["Adj Close"].pct_change() In [2]: StockReturns = StockPrices["Returns"] In [3]: StockReturns.plot()
6 Plotting Portfolio Cumulative Returns In order to plot the cumulative returns of multiple portfolios: In [1]: import matplotlib.pyplot as plt In [2]: CumulativeReturns = ((1+StockReturns).cumprod()-1) In [3]: CumulativeReturns[["Portfolio","Portfolio_EW"]].plot() Out [3]:
7 Market Capitalization
8 Market Capitalization Market Capitalization: The value of a company's publically traded shares. Also referred to as Market Cap.
9 Market-Cap Weighted Portfolios In order to calculate the market cap weight of a given stock n: w = mcap n mcap n n mcap i=1 i
10 Market-Cap Weights in Python To calculate market cap weights in python, assuming you have data on the market caps of each company: In [1]: import numpy as np In [2]: market_capitalizations = np.array([100, 200, 100, 100]) In [3]: mcap_weights = market_capitalizations/sum(market_capitalizations) In [4]: mcap_weights Out [4]: array([0.2, 0.4, 0.2, 0.2])
11 INTRO TO PORTFOLIO RISK MANAGEMENT IN PYTHON Let's practice!
12 INTRO TO PORTFOLIO RISK MANAGEMENT IN PYTHON Correlation and Co- Variance Dakota Wixom Quantitative Analyst QuantCourse.com
13 Pearson Correlation EXAMPLES OF DIFFERENT CORRELATIONS BETWEEN TWO RANDOM VARIABLES:
14 Pearson Correlation A HEATMAP OF A CORRELATION MATRIX:
15 Correlation Matrix in Python Assuming StockReturns is a pandas DataFrame of stock returns, you can calculate the correlation matrix as follows: In [1]: correlation_matrix = StockReturns.corr() In [2]: print(correlation_matrix) Out [2]:
16 Portfolio Standard Deviation Portfolio standard deviation for a two asset portfolio: σ : Portfolio standard deviation p w: Asset weight σ: Asset volatility σ p = w σ + w σ + 2w w ρ σ σ ,2 1 2 ρ : Correlation between assets 1 and 2 1,2
17 The Co-Variance Matrix To calculate the co-variance matrix (Σ) of returns X:
18 The Co-Variance Matrix in Python Assuming StockReturns is a pandas DataFrame of stock returns, you can calculate the covariance matrix as follows: In [1]: cov_mat = StockReturns.cov() In [2]: cov_mat Out [2]:
19 Annualizing the Covariance Matrix To annualize the covariance matrix: In [2]: cov_mat_annual = cov_mat*252
20 Portfolio Standard Deviation using Covariance The formula for portfolio volatility is: σ P ortfolio = wt Σ w σ P ortfolio : Portfolio volatility Σ: Covariance matrix of returns w: Portfolio weights (w is transposed portfolio weights) The dot-multiplication operator T
21 Matrix Transpose Examples of matrix transpose operations:
22 Dot Product The dot product operation of two vectors a and b:
23 Portfolio Standard Deviation using Python To calculate portfolio volatility assumy a weights array and a covariance matrix: In [1]: import numpy as np In [2]: port_vol = np.sqrt(np.dot(weights.t, np.dot(cov_mat, weights))) In [3]: port_vol Out [3]: 0.035
24 INTRO TO PORTFOLIO RISK MANAGEMENT IN PYTHON Let's practice!
25 INTRO TO PORTFOLIO RISK MANAGEMENT IN PYTHON Markowitz Portfolios Dakota Wixom Quantitative Analyst QuantCourse.com
26 100,000 Randomly Generated Portfolios
27 Sharpe Ratio The Sharpe ratio is a measure of risk-adjusted return. To calculate the 1966 version of the Sharpe ratio: S = Ra rf σ a S: Sharpe Ratio R : Asset return r : Risk-free rate of return f a σ : Asset volatility a
28 The Efficient Frontier
29 The Markowitz Portfolios Any point on the efficient frontier is an optimium portfolio. These two common points are called Markowitz Portfolios: MSR: Max Sharpe Ratio portfolio GMV: Global Minimum Volatility portfolio
30 Choosing a Portfolio How do you choose the best Portfolio? Try to pick a portfolio on the bounding edge of the efficient frontier Higher return is available if you can stomach higher risk
31 Selecting the MSR in Python Assuming a DataFrame df of random portfolios with Volatility and Returns columns: In [1]: numstocks = 5 In [2]: risk_free = 0 In [3]: df["sharpe"] = (df["returns"]-risk_free)/df["volatility"] In [4]: MSR = df.sort_values(by=['sharpe'], ascending=false) In [5]: MSR_weights = MSR.iloc[0,0:numstocks] In [6]: np.array(msr_weights) Out [6]: array([0.15, 0.35, 0.10, 0.15, 0.25])
32 Past Performance is Not a Guarantee of Future Returns Even though a Max Sharpe Ratio portfolio might sound nice, in practice, returns are extremely difficult to predict.
33 Selecting the GMV in Python Assuming a DataFrame df of random portfolios with Volatility and Returns columns: In [1]: numstocks = 5 In [2]: GMV = df.sort_values(by=['volatility'], ascending=true) In [3]: GMV_weights = GMV.iloc[0,0:numstocks] In [4]: np.array(gmv_weights) Out [4]: array([0.25, 0.15, 0.35, 0.15, 0.10])
34 INTRO TO PORTFOLIO RISK MANAGEMENT IN PYTHON Let's practice!
Financial Returns. Dakota Wixom Quantitative Analyst QuantCourse.com INTRO TO PORTFOLIO RISK MANAGEMENT IN PYTHON
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