Parametric VaR

A formula instead of a history

Historical VaR reads the loss from past returns. Parametric VaR assumes returns follow a normal distribution and needs only two numbers, the mean and the standard deviation. The loss exceeded with probability 1 - confidence is then z * sigma - mu, where z is the standard normal quantile (1.645 at 95%, 2.326 at 99%). It is fast, smooth, and extends easily to a whole portfolio, which is why it was the industry standard for decades.

from statistics import NormalDist

def normal_var(mu, sigma, confidence):
    z = NormalDist().inv_cdf(confidence)
    return z * sigma - mu

print(round(NormalDist().inv_cdf(0.95), 4), round(NormalDist().inv_cdf(0.99), 4))
print(round(normal_var(0.0005, 0.012, 0.95), 5))        # a 95% one-day loss limit
print(round(normal_var(0.0005, 0.012, 0.99), 5))        # a higher confidence, a larger limit

# The multiplier z for common confidence levels
from statistics import NormalDist

for confidence in (0.90, 0.95, 0.975, 0.99, 0.999):
    print(confidence, round(NormalDist().inv_cdf(confidence), 3))       # 1.282, 1.645, 1.96, 2.326, 3.09

From one asset to a portfolio, and to ten days

For a portfolio, the mean is weights @ mu and the standard deviation is sqrt(weights @ cov @ weights), the quantities from the portfolio-variance topic. Diversification shows up directly: low correlations shrink the standard deviation and the VaR. For a longer horizon, the mean grows with the number of days and the standard deviation with its square root, so a 10-day VaR is z * sigma * sqrt(10) - mu * 10.

import numpy as np
from statistics import NormalDist

def normal_var(mu, sigma, confidence):
    return NormalDist().inv_cdf(confidence) * sigma - mu

weights = np.array([0.6, 0.4])
mu = np.array([0.0005, 0.0003])
cov = np.array([[0.00016, 0.00004], [0.00004, 0.00009]])             # volatilities 1.26% and 0.95%, correlation 0.33

mean, std = float(weights @ mu), float(np.sqrt(weights @ cov @ weights))
print(round(mean, 5), round(std, 5))
print(round(normal_var(mean, std, 0.99), 5))
print(round(normal_var(mean * 10, std * np.sqrt(10), 0.99), 5))      # the 10-day figure

Where the normal assumption fails

Real returns have fat tails: extreme days are much more frequent than a normal distribution allows. The parametric VaR then looks too small, most of all at high confidence. Comparing it with the historical VaR on the same data shows the gap: positive means the normal model understates the risk. At 95% the two are often close, at 99% they can differ a lot.

# Fat-tailed returns: the normal model understates the 99% loss
import numpy as np
from statistics import NormalDist

rng = np.random.default_rng(2)
returns = rng.standard_t(3, size=5000) * 0.008           # a fat-tailed distribution (3 degrees of freedom)

for confidence in (0.95, 0.99):
    historical = -np.quantile(returns, 1 - confidence)
    parametric = NormalDist().inv_cdf(confidence) * returns.std(ddof=1) - returns.mean()
    print(confidence, round(float(historical), 5), round(float(parametric), 5), round(float(historical - parametric), 5))

• Parametric VaR = z * sigma - mu with z = NormalDist().inv_cdf(confidence): only the mean and the standard deviation are needed.
• A portfolio uses weights @ mu and sqrt(weights @ cov @ weights); a horizon scales the mean by days and the volatility by sqrt(days).
• Real returns have fat tails, so compare with the historical VaR: a positive gap means the normal model understates the risk.

Next up: expected shortfall, the average loss on the days when the VaR is exceeded.

Lab · what's graded

• Turn two numbers into a limit with normal_var25 pts
• Cover a whole portfolio with portfolio_normal_var25 pts
• Look further ahead with horizon_var25 pts
• Measure the fat tails with normality_gap25 pts