[Paper Review] Aggregation of Risks and Asymptotic independence

Abhimanyu Mitra, Sidney I. Resnick|ArXiv.org|Jul 25, 2008

Probability and Risk Models19 references3 citations

TL;DR

This paper establishes sufficient conditions under which the tail probability of the sum of asymptotically independent risks in the Gumbel domain of attraction behaves asymptotically like a constant multiple of the tail of a single risk. It extends results to non-identical and lighter-tailed marginals, provides examples including subexponential and non-subexponential cases, and applies the findings to approximate portfolio optimization via linear combinations of risks with positive coefficients.

ABSTRACT

We study the tail behavior of the distribution of the sum of asymptotically independent risks whose marginal distributions belong to the maximal domain of attraction of the Gumbel distribution. We impose conditions on the distribution of the risks $(X,Y)$ such that $P(X + Y > x) \sim (const)P (X > x)$. With the further assumption of non-negativity of the risks, the result is extended to more than two risks. We note a sufficient condition for a distribution to belong to both the maximal domain of attraction of the Gumbel distribution and the subexponential class. We provide examples of distributions which satisfy our assumptions. The examples include cases where the marginal distributions of $X$ and $Y$ are subexponential and also cases where they are not. In addition, the asymptotic behavior of linear combinations of such risks with positive coefficients is explored leading to an approximate solution of an optimization problem which is applied to portfolio design.

Motivation & Objective

• To analyze the asymptotic tail behavior of the sum of two or more asymptotically independent risks whose marginal distributions lie in the maximal domain of attraction of the Gumbel distribution.
• To extend existing results on risk aggregation beyond independence to the case of asymptotic independence, particularly when marginal distributions are not necessarily subexponential.
• To provide sufficient conditions under which the tail of the sum $X+Y$ is asymptotically equivalent to $ (1+c)P(X>x) $, where $ c = \lim_{x\to\infty} P(Y>x)/P(X>x) $.
• To explore the asymptotic behavior of linear combinations of such risks with positive coefficients and apply the results to approximate portfolio design optimization.
• To assess the accuracy of asymptotic approximations for exceedance probabilities in numerical experiments and relate this to the quality of approximate solutions in optimization.

Proposed method

• Derives sufficient conditions on the joint distribution of $(X,Y)$ such that $ \lim_{x\to\infty} \frac{P(X+Y>x)}{P(X>x)} = 1 + c $, with $ c = \lim_{x\to\infty} \frac{P(Y>x)}{P(X>x)} $.
• Imposes regular variation and convergence conditions on the normalized tail behavior of the joint distribution, ensuring convergence to a product of Gumbel-type extreme value distributions.
• Uses properties of the maximal domain of attraction of the Gumbel distribution ($MDA(\Lambda)$) and the concept of asymptotic independence via bivariate extreme value limits.
• Extends the two-risk result to $n$-variate sums under non-negativity and the same tail equivalence condition, maintaining the asymptotic equivalence to a constant multiple of the marginal tail.
• Applies the asymptotic equivalence to linear combinations $ \tilde{a}_1X + \tilde{a}_2Y $ with positive coefficients, deriving approximate solutions for portfolio optimization problems.
• Employs Monte Carlo simulation with 10,000 samples to estimate exceedance probabilities and evaluate relative errors between approximate and true optimal solutions across different correlation levels ($\rho = -0.9, 0, 0.9$).

Experimental results

Research questions

• RQ1 Under what conditions does the sum of two asymptotically independent risks in $MDA(\Lambda)$ exhibit tail equivalence to a constant multiple of a single risk’s tail?
• RQ2 Can the asymptotic tail equivalence $ P(X+Y>x) \sim (1+c)P(X>x) $ be extended to cases where the marginal distributions are not identically distributed or where one has a lighter tail not in $MDA(\Lambda)$?
• RQ3 How accurate is the asymptotic approximation $ P(X+Y>x) \approx (1+c)P(X>x) $ in practice, especially when applied to portfolio optimization with linear combinations of risks?
• RQ4 What is the relationship between the accuracy of the asymptotic approximation and the performance of approximate solutions in portfolio design problems?
• RQ5 Are there distributions in $MDA(\Lambda)$ with asymptotic independence for which the limit $ \lim_{x\to\infty} \frac{P(X+Y>x)}{P(X>x)} $ exists and is neither 2 nor $\infty$, thus lying strictly between?

Key findings

• For risks $X,Y$ in $MDA(\Lambda)$ with $ c = \lim_{x\to\infty} P(Y>x)/P(X>x) \in (0,\infty) $, the paper establishes $ \lim_{x\to\infty} \frac{P(X+Y>x)}{P(X>x)} = 1 + c $, which includes the case $c=1$ yielding $P(X+Y>x) \sim 2P(X>x)$.
• The result holds even when one marginal distribution is lighter-tailed and not in $MDA(\Lambda)$, provided the limit $c$ exists and is finite.
• When $X$ and $Y$ are non-negative and satisfy the tail equivalence condition, the result extends to sums of more than two risks, maintaining the asymptotic equivalence to $ (1 + \sum c_i)P(X>x) $.
• Numerical experiments show that for $\rho = -0.9$ and $\rho = 0$, the relative error between the approximate and true optimal portfolio probabilities is below 10% for thresholds $x \geq 5$, indicating good approximation accuracy.
• For $\rho = 0.9$, relative errors increase (up to 81% for $x=20$), but the approximate solution remains within the same order of magnitude as the true optimal solution when both weights are positive.
• The study confirms that when the asymptotic approximation is accurate, the corresponding approximate solution to the portfolio optimization problem is also accurate, and vice versa.