Expected Shortfall in Python

[Post is also available at quaintitative.com]

Google VAR and you will find lots of criticisms on VAR as a measure of market risk. And you will inevitably see Expected Shortfall (ES) being put forward as an alternative.

What is the difference between the two?

Say we are trying to assess our VAR (or to put it simply, potential losses) at a confidence level of 99%, that means we will have a range of loss outcomes (or scenarios) in the 1% tail, and -

• VAR answers this question — What is the minimum loss over the whole range of outcomes in the 1% tail?
• ES answers this question — What is the average loss over the whole range of outcomes in the 1% tail?

Let’s try to compute the two measures in Python to see the difference.

First, VAR.

h = 10. # horizon of 10 days
mu_h = 0.1 # this is the mean of % returns over 10 days - 10%
sig = 0.3 # this is the vol of returns over a year - 30%
sig_h = 0.3 * np.sqrt(h/252) # this is the vol over the horizon
alpha = 0.01

VaR_n = norm.ppf(1-alpha)*sig_h - mu_h 

print("99% VaR is", round(VaR_n*100,2))
Out:
99% VaR is 3.9

And next for ES.

# with the same parameters as above
CVaR_n = alpha**-1 * norm.pdf(norm.ppf(alpha))*sig_h - mu_h

print("99% CVaR/ES is", round(CVaR_n*100,2))
Out:
99% CVaR/ES is 5.93

We don’t have to assume a normal distribution. We can also assume a t-distribution.

from scipy.stats import t
nu = 5 # degree of freedom, the larger, the closer to normal distribution
xanu = t.ppf(alpha, nu)

VaR_t = np.sqrt(h/252 * (nu-2)/nu) * t.ppf(1-alpha, nu)*sig - mu_h

print("99% VaR (Student-t with v=5) is", round(VaR_t*100,2))
Out:
99% VaR (Student-t with v=5) is 5.58
CVaR_t = -1/alpha * (1-nu)**(-1) * (nu-2+xanu**2) * t.pdf(xanu, nu)*sig_h - mu_h
print("99% CVaR (Student-t with v=5) is", round(CVaR_t*100,2))
Out:
99% CVaR (Student-t with v=5) is 13.35

The larger the degree of freedom, the closer to a normal distribution.

# to verify that the normal and Student-t VAR will be the same for big v
nu = 10000000 # degree of freedom, the larger, the closer to normal distribution
xanu = t.ppf(alpha, nu)

VaR_t = np.sqrt(h/252 * (nu-2)/nu) * t.ppf(1-alpha, nu)*sig - mu_h
print("99% VaR (Student-t with with v->infinity) is", round(VaR_t*100,2))
Out:
99% VaR (Student-t with with v->infinity) is 3.9
CVaR_t = -1/alpha * (1-nu)**(-1) * (nu-2+xanu**2) * t.pdf(xanu, nu)*sig_h - mu_h
print("99% CVaR (Student-t with with v->infinity) is", round(CVaR_t*100,2))
Out:
99% CVaR (Student-t with with v->infinity) is 5.93

We can compute something similar with actual market data. First we fit the data to normal and t-distributions.

mu_norm, sig_norm = norm.fit(returns)
nu, mu_t, sig_t = t.fit(returns)

And the respective VAR and ES can be computed quite easily.

h = 1
alpha = 0.01
xanu = t.ppf(alpha, nu)

CVaR_n = alpha**-1 * norm.pdf(norm.ppf(alpha))*sig_norm - mu_norm
VaR_n = norm.ppf(1-alpha)*sig_norm - mu_norm
 
VaR_t = np.sqrt((nu-2)/nu) * t.ppf(1-alpha, nu)*sig_norm  - h*mu_norm
CVaR_t = -1/alpha * (1-nu)**(-1) * (nu-2+xanu**2) * t.pdf(xanu, nu)*sig_norm  - h*mu_norm

And a diagram which will probably provide a good idea of the difference between VAR and ES is as follows.

The full code can be found at the notebook here.

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