strategy-lab/to_explore/pyquantnews/23_ConditionalValueAtRisk.ipynb

This code calculates Value at Risk (VaR) and Conditional Value at Risk (CVaR) for a portfolio of stocks. It uses historical stock data to compute returns, then scales these returns to simulate a portfolio's performance. It defines functions to calculate VaR and CVaR, both of which are measures of potential losses in a portfolio. The code also generates visualizations of these risk metrics to aid in financial risk assessment. This is practical for portfolio management and risk analysis in finance.

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import numpy as np
import pandas as pd
import yfinance as yf

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import matplotlib.pyplot as plt

Define a list of stock tickers representing the portfolio components

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oex = ['MMM','T','ABBV','ABT','ACN','ALL','GOOGL','GOOG','MO','AMZN','AXP','AIG','AMGN','AAPL','BAC',
       'BRK-B','BIIB','BLK','BA','BMY','CVS','COF','CAT','CVX','CSCO','C','KO','CL','CMCSA',
       'COP','DHR','DUK','DD','EMC','EMR','EXC','XOM','META','FDX','F','GD','GE','GM','GILD',
       'GS','HAL','HD','HON','INTC','IBM','JPM','JNJ','KMI','LLY','LMT','LOW','MA','MCD','MDT','MRK',
       'MET','MSFT','MS','NKE','NEE','OXY','ORCL','PYPL','PEP','PFE','PM','PG','QCOM',
       'SLB','SPG','SO','SBUX','TGT','TXN','BK','USB','UNP','UPS','UNH','VZ','V','WMT',
       'WBA','DIS','WFC']

Count the number of stocks in the portfolio

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num_stocks = len(oex)

Download historical stock data for the defined period

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data = yf.download(oex, start='2014-01-01', end='2016-04-04')

Calculate daily returns and de-mean the returns by subtracting the mean

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returns = data['Adj Close'].pct_change()
returns = returns - returns.mean(skipna=True)

Plot the de-meaned returns for visualization

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returns.plot(legend=None)

Define a function to scale weights so their absolute values sum to one

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def scale(x):
    return x / np.sum(np.abs(x))

Generate random weights for the portfolio and scale them

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weights = scale(np.random.random(num_stocks))
plt.bar(np.arange(num_stocks), weights)

Define a function to calculate Value at Risk (VaR)

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def value_at_risk(value_invested, returns, weights, alpha=0.95, lookback_days=500):
    """Calculates Value at Risk (VaR) for a portfolio
    
    Parameters
    ----------
    value_invested : float
        Total value of the portfolio
    returns : pd.DataFrame
        Historical returns of the portfolio components
    weights : np.ndarray
        Weights of each asset in the portfolio
    alpha : float, optional
        Confidence level for VaR, by default 0.95
    lookback_days : int, optional
        Number of days to look back for historical data, by default 500
    
    Returns
    -------
    float
        The Value at Risk (VaR) at the specified confidence level
    """
    
    # Fill missing values in returns with zero and calculate portfolio returns

    returns = returns.fillna(0.0)
    portfolio_returns = returns.iloc[-lookback_days:].dot(weights)

    # Calculate the VaR as the percentile of portfolio returns

    return np.percentile(portfolio_returns, 100 * (1 - alpha)) * value_invested

Define the total value invested in the portfolio

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value_invested = 1000000

Calculate Value at Risk (VaR) using the defined function

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value_at_risk(value_invested, returns, weights, alpha=0.95, lookback_days=520)

Define parameters for lookback days and confidence level

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lookback_days = 500
alpha = 0.95

Calculate portfolio returns using historical data and weights

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portfolio_returns = returns.fillna(0.0).iloc[-lookback_days:].dot(weights)

Calculate VaR and express it as a return rather than absolute loss

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portfolio_VaR = value_at_risk(value_invested, returns, weights, alpha=0.95)
portfolio_VaR_return = portfolio_VaR / value_invested

Plot histogram of portfolio returns and mark the VaR on the plot

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plt.hist(portfolio_returns, bins=30)
plt.axvline(portfolio_VaR_return, color='red', linestyle='solid')
plt.legend(['VaR', 'Returns'])
plt.title('Historical VaR')
plt.xlabel('Return')
plt.ylabel('Observation Frequency')

Define the number of iterations for VaR calculation and initialize an array

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N = 1000
VaRs = np.zeros((N, 1))

Iterate to calculate VaR over different lookback windows

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for i in range(N):
    VaRs[i] = value_at_risk(value_invested, returns, weights, lookback_days=i + 1)

Plot the VaR values over different lookback windows

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plt.plot(VaRs)
plt.xlabel('Lookback Window')
plt.ylabel('VaR')

Define a function to calculate Conditional Value at Risk (CVaR)

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def cvar(value_invested, returns, weights, alpha=0.95, lookback_days=500):
    """Calculates Conditional Value at Risk (CVaR) for a portfolio
    
    Parameters
    ----------
    value_invested : float
        Total value of the portfolio
    returns : pd.DataFrame
        Historical returns of the portfolio components
    weights : np.ndarray
        Weights of each asset in the portfolio
    alpha : float, optional
        Confidence level for CVaR, by default 0.95
    lookback_days : int, optional
        Number of days to look back for historical data, by default 500
    
    Returns
    -------
    float
        The Conditional Value at Risk (CVaR) at the specified confidence level
    """
    
    # Calculate VaR and portfolio returns for the specified lookback period

    var = value_at_risk(value_invested, returns, weights, alpha, lookback_days=lookback_days)
    
    returns = returns.fillna(0.0)
    portfolio_returns = returns.iloc[-lookback_days:].dot(weights)
    var_pct_loss = var / value_invested
    
    # Calculate the mean of returns below the VaR threshold

    return np.nanmean(portfolio_returns[portfolio_returns < var_pct_loss]) * value_invested

Calculate CVaR using the defined function

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cvar(value_invested, returns, weights, lookback_days=500)

Calculate VaR again for consistency

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value_at_risk(value_invested, returns, weights, lookback_days=500)

Calculate portfolio returns using historical data and weights

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lookback_days = 500

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portfolio_returns = returns.fillna(0.0).iloc[-lookback_days:].dot(weights)

Calculate VaR and CVaR and express them as returns

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portfolio_VaR = value_at_risk(value_invested, returns, weights)
portfolio_VaR_return = portfolio_VaR / value_invested

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portfolio_CVaR = cvar(value_invested, returns, weights)
portfolio_CVaR_return = portfolio_CVaR / value_invested

Plot histogram of portfolio returns, marking VaR and CVaR on the plot

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plt.hist(portfolio_returns[portfolio_returns > portfolio_VaR_return], bins=20)
plt.hist(portfolio_returns[portfolio_returns < portfolio_VaR_return], bins=10)
plt.axvline(portfolio_VaR_return, color='red', linestyle='solid')
plt.axvline(portfolio_CVaR_return, color='red', linestyle='dashed')
plt.legend(['VaR', 'CVaR', 'Returns', 'Returns < VaR'])
plt.title('Historical VaR and CVaR')
plt.xlabel('Return')
plt.ylabel('Observation Frequency')

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