Scaled VaR: approximation vs reality

Previous question: Understanding VaR rescaling

After understanding the usual VaR scaling formula $$\text{VaR}_{T,\alpha}=\sqrt{T}\text{VaR}_{1,\alpha}$$ I wanted to know by how much it deviates from the real (simulated) value.

That is,

1. Calculate $$\text{VaR}_{T,\alpha}=V_0\sigma\Phi^{-1}(\alpha)\sqrt{N}$$
2. Simulate $$V_T-V_0=V_0(1+R_1)(1+R_2)\dots(1+R_T)-V_0$$ with $$R_i$$ normally distributed and get the empirical/simulated VaR

I coded this in Python

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm

def VaR_Scaled(V0, alpha, sigma, T):
return V0*sigma*np.sqrt(T)*norm.ppf(1-alpha) # Var scaling formula

def VaR_Simulated(V0, alpha, sigma, T, NumSimul):
V = np.full((NumSimul, T+1), V0)
for i in range(1, T+1):
V[:,i] = np.multiply(V[:,i-1], np.full((1, NumSimul), 1) + np.random.normal(loc = 0, scale = sigma, size = (1, NumSimul)))
return -np.percentile(V[:,T]-np.full((NumSimul, 1), V0), 1-alpha)

Scaled = VaR_Scaled(100.000, 1.0, 1.0, 2)
Simulated = VaR_Simulated(100.000, 1.0, 1.0, 2, 10000)
print('The scaled VaR is ', Scaled,', the simulated VaR is ', Simulated)
print('their difference is ', np.abs(Scaled-Simulated))


and tried to plot the size of this difference against $$V_0$$, $$\sigma$$, $$T$$ and $$\alpha$$

• It increases linearly with $$V_0$$
• It increases exponentially with $$\sigma$$ (tried $$0\le\sigma\le 1$$)
• It increases with the square root of $$T$$ (as expected)
• With $$0\le\alpha\le 1000/10000$$ it decreases at first and then increases (logarithmically? Picture below)
• It doesn't depend on the number of simulations (assuming we do at least a few thousand)

In any case, the approximation doesn't seem good most of the time.

Questions:

1. Is the approximation really this bad or am I missing something? The derivation comes from using only the first-order terms in the taylor expansions of $$\log$$ and $$\exp$$ so maybe this easily explains the bad approximation
2. If the approximation is in fact bad, why do we keep using it (other than simplicity)? Do banks really use this formula?
3. Why the strange behaviour (first decrease then increase) with respect to the confidence $$\alpha$$?