Computing the distribution of the sum of dependent random variables via overlapping hypercubes

Abstract

The original motivation of this work comes from a classic problem in finance and insurance: that of computing the Value-at-Risk (VaR) of a portfolio of dependent risky positions, i.e., the quantile at a certain level of confidence of the loss distribution. In fact, it is difficult to overestimate the importance of the concept of VaR in modern finance and insurance. It has been recommended, although with several warnings, as a measure of risk and the basis for capital requirement determination by both the guidelines of international committees (such as Basel 2 and 3 and Solvency 2) and the internal models adopted by major banks and insurance companies. However, the actual computation of the VaR of a portfolio constituted by several dependent risky assets is often a hard practical and theoretical task. To this purpose, here we prove the convergence of a geometric algorithm (alternative to Monte Carlo and quasi-Monte Carlo methods) for computing the Value-at-Risk of a portfolio of any dimension, i.e., the distribution of the sum of its components, which can exhibit any dependence structure. Moreover, although the original motivation is financial, our result has a relevant measure-theoretical meaning. What we prove, in fact, is that the H-measure of a d-dimensional simplex (for any $d\ge 2$ and any absolutely continuous with respect to Lebesgue measure H) can be approximated by convergent algebraic sums of H-measures of hypercubes (obtained through a self-similar construction).

Appendix

Proposition 2

With the notations of Sect. 2.2.4, for any absolutely continuous distribution $H$, $g_{n}^{H}\left( \alpha \right) =0$ $\ \forall$ $n\ge 1$ and $\alpha \in \left[ \frac{1}{d},\frac{2}{d+1}\right] .$

Proof

Thanks to self-similarity, it suffices to prove that for any absolutely continuous distribution $H$,

$$\begin{aligned} f_{n}^{H}\left( \alpha \right)= & {} \overset{\widehat{\rho }\left( n\right) }{ \underset{k=1}{\sum }}\sigma _{k}^{n}v_{H}\left( \widehat{Q}_{k}^{n}\cap S_{n}\left( \alpha \right) \right) -v_{H}\left( S_{n}\left( \alpha \right) \right) =0 \end{aligned}$$

where $\widehat{Q }_{k}^{n}$ are the hypercubes of the extrapolation defined in Sect. 2.2.3 (in fact, we can assume $n\ge 3$, as the cases $n=1,2$ are trivial).

To this end, we start by considering, in the hypercube $\left[ 0,1\right] ^{d}$, the trapezoid $T$ defined by

$$\begin{aligned} T=\left\{ \mathbf {x}^{\prime }=\left( x_{1},\ldots ,x_{d-1}\right) \in \left[ 0,1 \right] ^{d-1},\,0\le x_{d}\le 1-\lambda \left( \mathbf {x}^{\prime }\right) \right\} \end{aligned}$$

Then the theorem on the absolute continuity of the Lebesgue integral [see Kolmogorov and Fomin (1977)] implies that we can find $\delta >0$ such that:

• letting $T_{\delta }=\left\{ \mathbf {x}^{\prime }=\left( x_{1},\ldots ,x_{d-1}\right) \in \left[ 0,1\right] ^{d-1},\,0\le x_{d}\le 1-\lambda \left( \mathbf {x}^{\prime }\right) -\delta \right\}$, $v_{H}\left( T\right) -v_{H}\left( T_{\delta }\right) <\frac{\varepsilon }{16 \widehat{\rho }\left( n\right) }$;

• we can tile $\left[ 0,1\right] ^{d-1}$ by hypercubes of side length $\frac{1}{m}$, $m$ being sufficiently high, in such a way that for each tile $Q_{t}$, $1\le t\le m^{d}$, we can consider a rectangular hyperprism $R_{t}$ having basis $Q _{t}$ and height $\ h_{t}=1-\delta -\underset{}{\min \lambda \left( \mathbf {x}\right) \le }1-\frac{\delta }{2}- \underset{Q_{t}}{\max }\lambda \left( \mathbf {x}\right)$: hence $v_{H}\left( T_{\delta }\right) \le v_{H}\left( \overset{m^{d}}{\underset{t=1 }{\cup }}R_{t}\right) \le v_{H}\left( T\right)$.

Now, fixed $n$ and $\alpha \in \left[ \frac{1}{d},\frac{2}{d+1}\right]$, by the mentioned theorem the above construction can be reproduced (i.e., rescaled) for any trapezoid $\widehat{Q}_{k}^{n}\cap S_{n}$, replacing $1$ by $1-\left( 1-\alpha \right) ^{n}$ and $1-\delta$ by $\left( 1-\left( 1-\alpha \right) ^{n}\right) \left( 1-\delta \right)$. Moreover, set $\widehat{Q}_{k}^{n}=Q\left( \mathbf {b},l\right)$, $l>0$, and define $Q^{\prime }=Q\left( \mathbf {b},l\right) \cap \left\{ x_{d}=b_{d}\right\}$. Then

$$\begin{aligned} \widehat{Q}_{k}^{n}\cap S_{n}\!=\!\left\{ \mathbf {x}^{\prime }\!=\!\left( x_{1},\ldots ,x_{d-1}\right) \in Q^{\prime },b_{d}\le x_{d}\!\le \! \left( 1\!-\!\left( 1\!-\!\alpha \right) ^{n}\right) \left( 1-\frac{\delta }{2}\right) -\lambda \left( \mathbf {x}^{\prime }\right) \right\} \end{aligned}$$

We can also choose $\delta$ so small that if $\widehat{Q}_{k}^{n}$ has an intersection of positive volume with $S_{n}$, then it has an intersection of positive volume also with $S_{n}^{\delta }=\left\{ 0\le \lambda \left( \mathbf {x}\right) \le 1-\left( 1-\alpha \right) ^{n}-\delta ,x_{1},\ldots ,x_{d}\ge 0\right\}$.

Now we can consider an analytic distribution $H^{\prime }$ such that in

$$\begin{aligned} T_{\frac{\delta }{2}}\!=\!\left\{ \mathbf {x}^{\prime }\!=\!\left( x_{1},\ldots ,x_{d-1}\right) \!\in \! \left[ 0,1\right] ^{d-1},\,0\!\le \! x_{d}\le \left( 1\!-\!\left( 1-\alpha \right) ^{n}\right) \left( 1-\frac{\delta }{2}\right) -\lambda \left( \mathbf {x}^{\prime }\right) \right\} \end{aligned}$$

$\left| H^{\prime }\left( \mathbf {x}\right) -H\left( \mathbf {x}\right) \right| <\frac{\varepsilon }{16\widehat{\rho }\left( n\right) \left( 2m\right) ^{d}}$, while in $T_{0}-T_{\left( \frac{\delta }{2}-\sigma \right) }$, for any arbitrarily small $\sigma >0$, the density of $H^{\prime }$ can be chosen as small as we want. Moreover, by Proposition 1 of Sect. 1.4, $S_{n}$ has a cover of hypercubes, from which we can extract a cover of $p\le$ $\widehat{\rho }\left( n\right)$ not overlapping rectangular hyperprisms, to which the above construction can be analogously applied.

Then, through straightforward steps, it follows that for any arbitrarily small $\varepsilon >0$,

$$\begin{aligned} \left| f_{n}^{H}\left( \alpha \right) \right| <\varepsilon \end{aligned}$$

Hence, $f_{n}^{H}\left( \alpha \right) =0.$ $\square$