Value at risk (VaR) is a statistic used to try and quantify the level of financial risk within a firm or portfolio over a specified time frame. VaR provides an estimate of the maximum loss from a given position or portfolio over a period of time, and you can calculate it across various confidence levels.
Estimating the risk of a portfolio is important to long-term capital growth and risk management, particularly within larger firms or institutions. VaR is typically framed as something like this:
In this post I'll walk you through the steps to calculate this metric across a portfolio of stocks.
There are two main ways to calculate VaR:
In this post, we'll focus on using method (2) (variance-covariance). In short, the variance-covariance method looks at historical price movements (standard deviation, mean price) of a given equity or portfolio of equities over a specified lookback period, and then uses probability theory to calculate the maximum loss within your specified confidence interval. You can read more detail here, but we'll calculate it step by step below using Python.
Before we get started, note that the standard VaR calculation assumes the following:
In order to calculate the VaR of a portfolio, you can follow the steps below:
import pandas as pd
from pandas_datareader import data as pdr
import fix_yahoo_finance as yf
import numpy as np
import datetime as dt
# Create our portfolio of equities
tickers = ['AAPL','FB', 'C', 'DIS']
# Set the investment weights (I arbitrarily picked for example)
weights = np.array([.25, .3, .15, .3])
# Set an initial investment level
initial_investment = 1000000
# Download closing prices
data = pdr.get_data_yahoo(tickers, start="2018-01-01", end=dt.date.today())['Close']
#From the closing prices, calculate periodic returns
returns = data.pct_change()
returns.tail()
# Generate Var-Cov matrix
cov_matrix = returns.cov()
cov_matrix
This will allow us to calculate the standard deviation and mean of returns across the entire portfolio.
# Calculate mean returns for each stock
avg_rets = returns.mean()
# Calculate mean returns for portfolio overall,
# using dot product to
# normalize individual means against investment weights
# https://en.wikipedia.org/wiki/Dot_product#:~:targetText=In%20mathematics%2C%20the%20dot%20product,and%20returns%20a%20single%20number.
port_mean = avg_rets.dot(weights)
# Calculate portfolio standard deviation
port_stdev = np.sqrt(weights.T.dot(cov_matrix).dot(weights))
# Calculate mean of investment
mean_investment = (1+port_mean) * initial_investment
# Calculate standard deviation of investmnet
stdev_investment = initial_investment * port_stdev
Next, we can plug these variables into our percentage point function (PPF) below.
# Select our confidence interval (I'll choose 95% here)
conf_level1 = 0.05
# Using SciPy ppf method to generate values for the
# inverse cumulative distribution function to a normal distribution
# Plugging in the mean, standard deviation of our portfolio
# as calculated above
# https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.norm.html
from scipy.stats import norm
cutoff1 = norm.ppf(conf_level1, mean_investment, stdev_investment)
#Finally, we can calculate the VaR at our confidence interval
var_1d1 = initial_investment - cutoff1
var_1d1
#output
#22347.7792230231
Here we are saying with 95% confidence that our portfolio of 1M USD will not exceed losses greater than 22.3k USD over a one day period.
What if we wanted to calculate this over a larger window of time? Below we can easily do that by just taking our 1 day VaR and multiplying it by the square root of the time period (this is due to the fact that the standard deviation of stock returns tends to increase with the square root of time).
# Calculate n Day VaR
var_array = []
num_days = int(15)
for x in range(1, num_days+1):
var_array.append(np.round(var_1d1 * np.sqrt(x),2))
print(str(x) + " day VaR @ 95% confidence: " + str(np.round(var_1d1 * np.sqrt(x),2)))
# Build plot
plt.xlabel("Day #")
plt.ylabel("Max portfolio loss (USD)")
plt.title("Max portfolio loss (VaR) over 15-day period")
plt.plot(var_array, "r")
1 day VaR @ 95% confidence: 20695.24
2 day VaR @ 95% confidence: 29267.49
3 day VaR @ 95% confidence: 35845.21
4 day VaR @ 95% confidence: 41390.49
5 day VaR @ 95% confidence: 46275.97
6 day VaR @ 95% confidence: 50692.79
7 day VaR @ 95% confidence: 54754.47
8 day VaR @ 95% confidence: 58534.99
9 day VaR @ 95% confidence: 62085.73
10 day VaR @ 95% confidence: 65444.11
11 day VaR @ 95% confidence: 68638.36
12 day VaR @ 95% confidence: 71690.43
13 day VaR @ 95% confidence: 74617.76
14 day VaR @ 95% confidence: 77434.51
15 day VaR @ 95% confidence: 80152.33
As mentioned in the calculation section, we are assuming that the returns of the equities in our portfolio are normally distributed when calculating VaR. Of course, we can't predict that moving forward, but we can at least check how the historical returns have been distributed to help us assess whether VaR is suitable to use for our portfolio.
import matplotlib.mlab as mlab
import matplotlib.pyplot as plt
# Repeat for each equity in portfolio
returns['AAPL'].hist(bins=40, normed=True,histtype="stepfilled",alpha=0.5)
x = np.linspace(port_mean - 3*port_stdev, port_mean+3*port_stdev,100)
plt.plot(x, scipy.stats.norm.pdf(x, port_mean, port_stdev), "r")
plt.title("AAPL returns (binned) vs. normal distribution")
plt.show()
AAPl returns vs. normal distribution
FB returns vs. normal distribution
C returns vs. normal distribution
DIS returns vs. normal distribution
From the above we can see the returns have all been fairly normally distributed for our chosen stocks since 2018.
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