Introducción
Mi objetivo es recuperar un único Valor en Riesgo (VaR) de una N(0, H) variable aleatoria $X$ a $\alpha \in (0,1)$ de nivel de confianza donde H es un conocido d-dimensional positiva definida la matriz (con $d >3$) En otras palabras, mi objetivo es recuperar la $\alpha$ cuantil de una N(0, H) variable aleatoria $X$ , es decir,
$$\text{VaR}_{\alpha}(X) = \text{inf} \ \{x \in \mathbb{R}: F_X(x) > \alpha\}$$
I consider the $\texto{VaR}_\alpha$ for $\alpha \en \{0.1, 0.05, 0.01\}$ in the context of a portfolio comprising $d>3$ financial log returns where we assume that the portfolio log return follows a multivariate normal distribution, i.e. N(0, w^T H w). We define w as an equally weighted vector of length $d$ meaning that each asset is equally represented in our portfolio. I do not wish to consider a reduced model in which the portfolio's returns are considered as a single univariate time series, i.e. N(0, $\sigma^2$), as this ignores complicated correlation among the individual returns. In this context VaR is the maximum loss with 1-$\alpha$ probabilidad. La idea es usar el VaR se estima en un análisis de backtesting (prueba de Kupiec, Christoffersen de Prueba) con el fin de evaluar los diferentes modelos utilizados para la previsión de la matriz de covarianza H.
Enfoque
En caso de unimodal simétrica distribuciones tales como la distribución normal multivariante considerado en la pregunta, de cuantiles corresponden a Mayor Densidad De Las Regiones. Como se ha señalado aquí, la densidad más alta de la región de una N(0,H) variable aleatoria es un elipsoide centrado en su media, 0, y orientado por la matriz de covarianza H:
$$x: x^TH^{-1}x \le y$$
In order to find a single quantile that satisfies the definition of Value-at-Risk we proceed as follows: the set of solutions that satisfy the condition of the ellipsoid equation, the level set, is retrieved by randomly generating many points on this ellipsoid. Then, I evaluate the portfolio return at each solution satisfying the condition: for example, let's suppose $s = (x_1, x_2, \dots, x_d)$ is a solution/ point on the ellipsoid, then the portfolio return at this point will be $w\cdot s$, which corresponds to the mean of the solution vector because we stated that each asset is equally represented in the portfolio and $x_i \en s$ corresponds to the return of asset $i$. Consequently, the lowest return, the minimum, on the ellipsoid is the maximum loss.
Question
Now, in order to ensure that the maximum loss on the ellipsoid is the maximum loss with 1-$\alpha$ probability, hence is the Value-at-Risk of X at the $\alpha$ confidence level, I need to define the cutoff value for the ellipsoid, i.e. $s$. At this point, I am a bit confused concerning the definition of the cutoff value for the ellipsoid.
Here it is stated that the cutoff value for the ellipsoid can be determined from the Chi-square with d degrees of freedom. For instance, the highest density region capturing 0.95 probability of the N(0,H) is found with y= value such that Chi-square with d degrees of freedom 0.95. When defining the cutoff value from the chi-square with d degrees of freedom and following the procedure explained in the approach section, I would interpret the lowest return on the ellipsoid as the maximum loss with 0.95 probability, i.e. $\text{VaR}_{0.05}$. Es correcto afirmar que estos máxima pérdida de valores, a continuación, la pérdida máxima con 1-α probabilidad, por lo tanto es el Valor en Riesgo? VaR aumenta con una disminución en alfa, pero al mismo tiempo VaR parece un poco una grande para el contexto que me hace dudar de mi interpretación y la definición del valor de corte para el elipsoide determina a partir de la Chi-cuadrado con d grados de libertad y su relación con el α cuantil de la distribución (Valor en Riesgo).
Cualquier comentario que pudiera me apunte en la dirección correcta son más que bienvenidos.
Python de código para la obtención de puntos al azar sobre el elipsoide
import numpy as np
from scipy.linalg import sqrtm
from scipy.stats import ortho_group
from scipy.stats._continuous_distns import chi2
dim = 10
# Create a positive definite matrix H with shape (dim, dim).
# evals is the vector of eigenvalues of H. This can be replaced
# with any vector of length `dim` containing positive values.
evals = (np.arange(1, dim + 1)/2)**2
# Use a random orthogonal matrix to generate H.
R = ortho_group.rvs(dim)
H = R.T.dot(np.diag(evals).dot(R))
# y determines the level set to be computed.
y = chi2.ppf(q=0.95, df=dim)
# Generate random points on the ellipsoid
nsample = 100000
r = np.random.randn(H.shape[0], nsample)
# u contains random points on the hypersphere with radius 1.
u = r / np.linalg.norm(r, axis=0)
xrandom = sqrtm(H).dot(np.sqrt(y)*u)
# Compute maximum loss on the ellipsoid
xrandom_min = np.min(np.array([np.mean(x) for x in xrandom.T]))
print("Maximum loss i.e. Value-at-Risk:")
print(xrandom_min)
