Risk Measures for Time Series¶
%matplotlib inline
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import PortfolioAnalytics as PortfolioAnalytics
import PortfolioLoadData as PortfolioLoadData
import PortfolioRisk as PortfolioRisk
import PortfolioStatistics as PortfolioStatistics
%load_ext autoreload
%autoreload 2
Downside Measures: SemiDeviation, VaR and cVaR¶
We're going to look at a few measures of downside risk. We've already seen how to compute drawdowns, but we're going to look at 3 popular measures.
The first measure is the simplest, which is the semideviation, which is nothing more than the volatility of the subset of returns that are negative or below a certain threshold.
hfi = PortfolioLoadData.get_hfi_returns()
PortfolioRisk.semideviation(hfi).sort_values()
VaR and cVaR¶
We'll look at four different ways to compute Value At Risk
- Historic VaR
- Parametric Gaussian VaR
- Modified (Cornish-Fisher) VaR
- Conditional VaR
To compute the historic VaR at a certain level, say 5%, all we have to do is to find the number such that 5% of the returns fall below that number and 95% of the returns fall above that number. In other words, we want the 5 percentile return.
Note that for reporting purposes, it is common to invert the sign so we report a positive number to represent the loss i.e. the amount that is at risk.
Historical VaR¶
PortfolioRisk.var_historic(hfi, level=1).sort_values()
Conditional VaR a.k.a. Beyond VaR¶
Now that we have the VaR, the CVaR is very easy. All we need is to find the mean of the numbers that fell below the VaR!
PortfolioRisk.cvar_historic(hfi, level=1).sort_values()
Parametric Gaussian VaR¶
The idea behind this is very simple. If a set of returns is normally distributed, we know, for instance, that 50% of the returns are below the mean and 50% are above.
We also know that approx two thirds of the returns lie within 1 standard deviation. That means one third lie beyond one standard deviation from the mean. Since the normal distribution is symmetric, approximately one sixth (approx 16%) lie below one standard deviation away from the mean. Therefore, if we know the mean and standard deviation and if we assume that the returns are normally distributed, the 16% VaR would be the mean minus one standard deviation.
In general we can always convert a percentile point to a z-score (which is the number of standard deviations away from the mean that a number is). Therefore, if we can convert the VaR level (such as 1% or 5%) to a z-score, we can calculate the return level where that percent of returns lie below it.
scipy.stat.norm contains a function ppf() which does exactly that. It takes a percentile such as 0.05 or 0.01 and gives you the z-score corresponding to that in the normal distribution.
Therefore, all we need to do to estimate the VaR using this method is to find the z-score corresponding to percentile level, and then add that many standard deviations to the mean, to obtain the VaR.
PortfolioRisk.var_gaussian(hfi).sort_values()
Cornish-Fisher Modification¶
The Cornish-Fisher modification is an elegant and simple adjustment.
The z-score tells us how many standard deviations away from the mean we need to go to find the VaR. If the returns arent normal, we know that z-score will give us an inaccurate number. The basic idea is that since we can observe the skewness and kurtosis of the data, we can adjust the z-score up or down to come up with a modifed z-score. e.g. intuitively, all other things being equal, if the skewness is negative, we'll decrease the z-score further down, and if the skewness is positive, we'll push it up.
The adjusted z-score which we'll call $z_{cornishfisher}$ given by:
$$ z_{cornishfisher} = z +\frac{1}{6}(z^2-1)S + \frac{1}{24}(z^3-3z)(K-3)-\frac{1}{36}(2z^3-5z)S^2 $$We can modify the previous function by adding a "modified" parameter with a default value of True as follows. If True then the following piece of code is executed, which modifes z:
PortfolioRisk.var_gaussian(hfi, modified=True).sort_values()
var_table = [PortfolioRisk.var_historic(hfi),
PortfolioRisk.cvar_historic(hfi),
PortfolioRisk.var_gaussian(hfi, modified=False),
PortfolioRisk.var_gaussian(hfi, modified=True), ]
comparison = pd.concat(var_table, axis=1)
comparison.columns=['Historic VaR', 'cVaR', 'Gaussian', 'Cornish-Fisher']
comparison.plot.bar(title="Hedge Fund Indices: VaR at 5%",figsize=(12,6))
