Marginal and Component Value-at-Risk: A Python Example

Value-at-risk (VaR), despite its drawbacks, is a solid basis to understand the risk characteristics of the portfolio. There are many approaches to calculate VaR (historical simulation, variance-covariance, simulation). Marginal VaR is defined as the additional risk that a new position adds to the portfolio. The practical reading of the marginal VaR, as very nicely stated in Quant at risk http://www.quantatrisk.com/2015/01/18/applied-portfolio-value-at-risk-decomposition-1-marginal-and-component-var/ is: the highest the marginal VaR, the exposure to the respective asset should be reduced to lower the portfolio VaR. The component VaR shows the reduction of the portfolio value-at-risk resulting from removal of a position. The sum of component VaR of the shares in the portfolio equals the portfolio diversified VaR, while the sum of the individual VaR presents the undiversified portfolio VaR, assuming perfect correlation between assets in the portfolio.

Also it is worth noting that based on simple mathematics, the equations can be changed, i.e. there are several ways to calculate the component VaR.

We take the following steps in Python to come up with the marginal VaR of a 3-asset portfolio:

(This works for Python 2.7 for Python 3.4 -> change in print statements in the codes. Additionally, to some of the lines some formatting can be applied - for instance returns are not presented in percentages.)

import numpy as np

import pandas as pd

import pandas.io.data as web

from scipy.stats import norm

Value=1e6 # $1,000,000

CI=0.99 # set the confidence interval

tickers =['AAPL', 'MSFT', 'YHOO']

numbers=len(tickers)

data=pd.DataFrame()

for share in tickers:

    data[share]=web.DataReader(share, data_source='yahoo', start='2011-01-01', end='2015-05-15')['Adj Close']

data.columns=tickers

ret=data/data.shift(1)-1 # calculate the simple returns

ret.mean()*252 #annualize the returns

covariances=ret.cov()*252 #gives the annualized covariance of returns

variances=np.diag(covariances) #extracts variances of the individual shares from covariance matrix

volatility=np.sqrt(variances) #gives standard deviation

weights=np.random.random(numbers)

weights/=np.sum(weights) # simulating random percentage of exposure of each share that sum up to 1; if we want to plug in our own weights use: weights=np.array([xx,xx,xx])

Pf_ret=np.sum(ret.mean()*weights)*252 #Portfolio return

Pf_variance=np.dot(weights.T,np.dot(ret.cov()*252,weights)) #Portfolio variance

Pf_volatility=np.sqrt(Pf_variance) #Portfolio standard deviation

USDvariance=np.square(Value)*Pf_variance

USDvolatility=np.sqrt(USDvariance)

covariance_asset_portfolio=np.dot(weights.T,covariances)

covUSD=np.multiply(covariance_asset_portfolio,Value)

beta=np.divide(covariance_asset_portfolio,Pf_variance)

def VaR():

    # this code calculates Portfolio Value-at-risk (Pf_VaR) in USD-terms and Individual Value-at-risk (IndividualVaR) of shares in portfolio.  

    Pf_VaR=norm.ppf(CI)*USDvolatility

    IndividualVaR=np.multiply(volatility,Value*weights)*norm.ppf(CI)

    IndividualVaR = [ '$%.2f' % elem for elem in IndividualVaR ]

    print 'Portfolio VaR: ', '$%0.2f' %Pf_VaR

    print 'Individual VaR: ',[[tickers[i], IndividualVaR[i]] for i in range (min(len(tickers), len(IndividualVaR)))]

VaR() #call the function to get portfolio VaR and Individual VaR in USD-terms

def marginal_component_VaR():

     # this code calculates Marginal Value-at-risk in USD-terms and Component Value-at-risk of shares in portfolio. 

    marginalVaR=np.divide(covUSD,USDvolatility)*norm.ppf(CI)

    componentVaR=np.multiply(weights,beta)*USDvolatility*norm.ppf(CI)

    marginalVaR = [ '%.3f' % elem for elem in marginalVaR ]

    componentVaR=[ '$%.2f' % elem for elem in componentVaR ]

    print 'Marginal VaR:', [[tickers[i], marginalVaR[i]] for i in range (min(len(tickers), len(marginalVaR)))]

    print 'Component VaR: ', [[tickers[i], componentVaR[i]] for i in range (min(len(tickers), len(componentVaR)))]

marginal_component_VaR(): #call the function