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Comparing CVaR and VaR

The conditional value at risk (CVaR), or expected shortfall (ES), asks what the average loss will be, conditional upon losses exceeding some threshold at a certain confidence level. It uses VaR as a point of departure, but contains more information because it takes into consideration the tail of the loss distribution.

You'll first compute the 95% VaR for a Normal distribution of portfolio losses, with the same mean and standard deviation as the 2005-2010 investment bank portfolio_losses. You'll then use the VaR to compute the 95% CVaR, and plot both against the Normal distribution.

The portfolio_losses are available in your workspace, as well as the norm Normal distribution from scipy.stats.

This is a part of the course

“Quantitative Risk Management in Python”

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Exercise instructions

  • Compute the mean and standard deviation of portfolio_losses and assign them to pm and ps, respectively.
  • Find the 95% VaR using norm's .ppf() method--this takes arguments loc for the mean and scale for the standard deviation.
  • Use the 95% VaR and norm's .expect() method to find the tail_loss, and use it to compute the CVaR at the same level of confidence.
  • Add vertical lines showing the VaR (in red) and the CVaR (in green) to a histogram plot of the Normal distribution.

Hands-on interactive exercise

Have a go at this exercise by completing this sample code.

# Compute the mean and standard deviation of the portfolio returns
pm = portfolio_losses.____
ps = portfolio_losses.____

# Compute the 95% VaR using the .ppf()
VaR_95 = norm.ppf(0.95, loc = ____, scale = ____)
# Compute the expected tail loss and the CVaR in the worst 5% of cases
tail_loss = norm.____(lambda x: x, loc = ____, scale = ____, lb = VaR_95)
CVaR_95 = (1 / (1 - 0.95)) * ____

# Plot the normal distribution histogram and add lines for the VaR and CVaR
plt.hist(norm.rvs(size = 100000, loc = pm, scale = ____), bins = 100)
plt.axvline(x = VaR_95, c='r', label = "VaR, 95% confidence level")
plt.axvline(x = ____, c='g', label = "CVaR, worst 5% of outcomes")
plt.legend(); plt.show()
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