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,
- Calculate $\text{VaR}_{T,\alpha}=V_0\sigma\Phi^{-1}(\alpha)\sqrt{N}$
- 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:
- 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
- If the approximation is in fact bad, why do we keep using it (other than simplicity)? Do banks really use this formula?
- Why the strange behaviour (first decrease then increase) with respect to the confidence $\alpha$?
Thanks in advance!
