Portfolio Diversification under Local and Moderate Deviations from Power Laws.

This paper analyzes portfolio diversiﬁcation for nonlinear transformations of heavy-tailed risks. It is shown that diversiﬁcation of a portfolio of convex functions of heavy-tailed risks increases the portfolio’s riskiness, if expectations of these risks are inﬁnite. On the contrary, for concave functions of heavy-tailed risks with ﬁnite expectations, the stylized fact that diversiﬁcation is preferable continues to hold. The framework of transformations of heavy-tailed risks includes many models with Pareto-type distributions that exhibit local or moderate deviations from power tails in the form of additional slowly varying or exponential factors. The class of distributions under study is therefore extended beyond the stable class.


Introduction
In the recent four decades, we have witnessed a rapid expansion of the study of heavy-tailedness and the extreme outliers phenomena in economics and finance.Beginning with Mandelbrot (1963) and Fama (1965b), numerous studies have documented that time series encountered in many fields in economics and finance are typically heavy-tailed and have infinite moments of order p ≥ α for certain α > 0 (see the discussion in Loretan & Phillips 1994, Gabaix, Gopikrishnan, Plerou & Stanley 2003, Ibragimov 2004a,b, 2005, Rachev, Menn & Fabozzi 2005, Ibragimov & Walden 2007, and references therein).
In models involving a heavy-tailed cdf F with infinite moments of order greater than or equal to α, it is typically assumed that F has Pareto (power) tails: or, more generally, that F is of Pareto-type, so that x α l(x), x → +∞. (2) Here, c 1 , c 2 are some positive constants and l(x) is a slowly varying function at infinity: l(λx)/l(x) → 1, as x → +∞, for all λ > 0, and α is the so-called tail index.Well-known examples of distributions satisfying (1) are stable laws with α ∈ (0, 2), that is, distributions that are closed under portfolio formation (see Section 2).
We mention a sample of estimates of the tail index α for returns on various stocks and stock indices: 3 < α < 5 (Jansen & de Vries 1991), 2 < α < 4 (Loretan & Phillips 1994), 1.5 < α < 2 (McCulloch 1997), 0.9 < α < 2 (Rachev & Mittnik 2000), α ≈ 3 (Gabaix et al. 2003).As discussed by Nešlehova, Embrechts & Chavez-Demoulin (2006), tail indices less than one are observed for empirical loss distributions of a number of operational risks.Furthermore, Scherer, Harhoff & Kukies (2000) and Silverberg & Verspagen (2004) report the tail indices α to be considerably less than one for financial returns from technological innovations.Rachev et al. (2005) discuss and review the vast literature that supports heavy-tailedness and Pareto distributions for equity and bond returns.2 As was shown in Ibragimov (2004aIbragimov ( ,b, 2005) ) in a general context based on majorization theory and arbitrary portfolio weights comparisons, diversification may be inferior for convolutions of stable extremely heavy-tailed and possibly dependent risks whose cdf's F satisfy power law (1) with α < 1.According to Ibragimov (2004aIbragimov ( ,b, 2005)), diversification is typically preferable for convolutions of stable heavy-tailed risks that follow (1) with α > 1.Recently, Ibragimov & Walden (2007) showed that, with bounded risks concentrated on a sufficiently large interval, diversification may be suboptimal up to a certain number of risks and then become optimal.The above findings generalize the results in Fama (1965a), Samuelson (1967) and Ross (1976) on portfolio choice in the stable framework and riskiness analysis for portfolios of stable risks with equal weights.Several examples that illustrate the phenomenon that diversification is not always preferable are presented in Kaas, Goovarets & Tang (2004).
Clearly, understanding the conditions under which diversification is preferable is core to fields such as finance, banking and insurance.Stable distributions form a small subclass of the class of Paretotype laws and the previous examples of heavy-tailed distributions may not belong to this subclass.A natural question is therefore whether the stylized facts about riskiness and portfolio diversification can be extended beyond the subclass of stable distributions.The main objective of this paper is to carry out such an extension.We analyze portfolio diversification for nonlinear transformations of heavy-tailed risks, to understand under what distributional assumptions failure of diversification occurs.We use the framework of value at risk as a measure of portfolio riskiness.
Our contribution to the literature is two-fold.First, we show that diversification of a portfolio of convex functions of heavy-tailed risks increases its riskiness if expectations of these risks are infinite (Theorem 2).However, the stylized fact that diversification is always preferable continues to hold for concave (on R + ) functions of heavy-tailed risks with finite expectations (Theorem 1).
Second, we use the results on nonlinear transformations to model a large class of Pareto-type distributions.The class of nonlinear transformations of heavy-tailed r.v.'s considered in this paper provides a natural framework for modeling risks with distributions exhibiting departures from power laws.Specifically, let us define recursively the iterations of a logarithm by ln 0 (x) = x, ln k (x) = ln ln k−1 (x) , k ≥ 1, for all large positive x.Let m ≥ 0 and let γ 1 , ..., γ m ∈ R be some constants.The stochastic framework considered in the paper covers risks Y whose tails behave as (here and throughout the paper, g(x) h(x) as x → ∞ denotes that there are constants, c and C such that 0 < c ≤ g(x)/h(x) ≤ C < ∞ for large x > 0).In particular, the choice γ k = −1, γ s = 0, 1 ≤ s ≤ m, s = k, produces deviations from power law (1) of the form P |Y | > x 1 x α ln k (x) .Similarly, the choice γ k = 1, γ s = 0, 1 ≤ s ≤ m, s = k corresponds to the deviations from power law (1) of the form x α .Also, when α = 1 the above relations correspond to deviations from the Zipf law (1) of the form x .Reminiscent of the terminology in the time series unit root literature (see Phillips 1988, Phillips & Magdalinos 2004), it is natural to refer to distributions whose tails satisfy one of relation (3) as exhibiting "local" or "moderate" deviations from power laws.Corollaries 1 and 2 provide results for when diversification will and will not be preferred for such deviations.We also show that the results above can not be generalized to the whole class of Pareto-type distributions (see Examples 2 and 1 in In Subsection 3.1).In order to highlight the main ideas and concepts discussed, the results in this paper are formulated in the framework of independent risks, which is the case where diversification works best.One can obtain analogous results for wide classes of dependent risks, including those with α−symmetric distributions and convolutions of models with common shocks.The extension to dependent risks is quite straightforward, and is similar to the extensions in Ibragimov (2004aIbragimov ( ,b, 2005)), Ibragimov & Walden (2007).
The paper is organized as follows.Section 2 introduces classes of distributions we are dealing with throughout the paper and discusses their main properties.Section 3.1 presents our main results on diversification and its effects on the value at risk of portfolios of nonlinear transformations of heavytailed risks.Section 3.2 discusses implications of the results for local and moderate deviations from power laws in form (3). Finally, Section 4 makes some concluding remarks.All proofs are left to the Appendix.

Notations and classes of distributions
We say that a r.v.X with density p : R → R and the convex distribution support An 1998).A distribution is said to be log-concave if its density p satisfies the above inequalities.Examples of log-concave distributions include the normal, uniform, exponential and logistic distributions, the Gamma distribution Γ(α, β) with α ≥ 1, the Beta distribution B(a, b) with a ≥ 1 and b ≥ 1, and the Weibull distribution W(γ, α) with α ≥ 1.If a r.v.X is log-concavely distributed, then its density has at most an exponential tail, that is, p(x) = o(exp(−λx)) for some λ > 0, as x → ∞ and all the power moments E|X| γ , γ > 0, of the r.v.exist (see Corollary 1 in An 1998).We denote by LC the class of symmetric log-concave distributions.
Let R + = [0, ∞).Throughout the paper, M denotes the class of differentiable odd functions f : R → R such that f is concave and increasing on R + and M denotes the class of odd functions f : R → R such that f is convex and increasing on R + .Further, M (resp.M ) denotes the subclass of M (resp.M ) consisting of functions f which are strictly concave (strictly convex) on R + .
By CT SLC, we denote the class of convolutions of log-concave distributions and distributions of transforms f (Y ), f ∈ M , of symmetric stable r.v.'s Y ∼ S α (σ, 0, 0) with characteristic exponents α ∈ [1, 2] and σ > 0. In what follows, we write X ∼ LC (resp., X ∼ CT SLC) if the distribution of the r.v.X belongs to the class LC (resp., CT SLC).The class CT SLC thus consists of distributions of r.v.'s X such that, for some k ≥ 1, and independent r.v.'s Y 0 ∼ LC and where θ ∈ {0, 1}, (It will follow from our analysis that α i = 1 is a special case, for which the assumption f i ∈ M is needed for the value at risk comparisons to be strict).
3 Main results

Diversification of nonlinear transformations of heavy-tailed risks
Let 0 < q < 1/2.Given a r.v.(risk) Z, we denote by V aR q [Z] the value at risk (VaR) of Z at level q, that is, its (1 − q)−quantile: V aR q [Z] = inf{z ∈ R : P (Z > z) ≤ q} (throughout the paper, we interpret the positive values of Z as a risk holder's losses).For n ≥ 1, the sample mean X n represents the return on the portfolio of risks X 1 , ..., X n with equal weights w n = (1/n, 1/n, ..., 1/n): 1 shows that diversification continues to be preferable for convex transformations of heavy-tailed risks with finite expectations.
Theorem 1 The following conclusions hold.
On the contrary, Theorem 2 demonstrates that the stylized fact on diversification being preferable is reversed for concave (on R + ) transformations of heavy-tailed risks with infinite expectations.
These results may seem intuitive, as α = 1 is the value at which the preferability of diversification is reversed for stable distributions (see Ibragimov 2004a,b, 2005, Ibragimov & Walden 2007).For example, in the case of i.i.d.symmetric Cauchy risks X i ∼ S 1 (σ, 0, 0), i = 1, ..., n, we have X n ∼ S 1 (σ, 0, 0), so that diversification has no effect on the portfolio VaR.However, outside the class of stable distributions the effects of diversification on portfolio riskiness are not that simple.In fact, the intuition that distributions with thinner tails than Cauchy-type power laws (equations (1) with α = 1) are always "good" to diversify and distributions with heavier tails are always "bad," may not hold.Let us give two examples: Example 1 Risks with heavier tails than those of Cauchy distributions, where diversification may be preferable: An example of dependent risks with such properties is provided by the following construction.Consider risks U i given by symmetric Cauchy r.v.'s and Z > 0 is a positive r.v.independent of Y i s.This is an example of a model with a common shock Z that affects all dependent risks U i .The risks U i have tails that are heavier than those of Cauchy distributions with α = 1 in the case of common shocks Z with infinite first moments: EZ = ∞.However, using Theorem 1 and conditioning arguments, we get that Example 2 Risks with thinner tails than those of Cauchy distributions, where diversification may not be preferable: Consider i.i.d.risks X 1 , X 2 such that X i = f (Y i ), where f ∈ M and Y i ∼ S 1 (σ, 0, 0) are i.i.d.symmetric Cauchy r.v.'s.In particular, as discussed in the introduction and in the next subsection, the above setup models risks with tails exhibiting deviations from power laws (1) in the form x .Further, let Z i , i = 1, 2, be i.i.d.symmetric stable risks with the tail index α > 1, say α = 1.1 : Z i ∼ S α (σ, 0, 0), so that the tails of their distributions decline to zero faster than those of Cauchy distributions.Given A > 0, define the risks , where I(•) stands for the indicator function.Clearly, the tails of distributions of the risks x α for large x and are thus thinner than the tails of Cauchy distributions with α = 1.
this implies that there exists a sufficiently large A such that diversification can not be said to be optimal for the risks Y (A) i with thinner tails than those of Cauchy distributions: 2 )].

Diversification and value at risk under deviations from power laws
As indicated in the introduction, the class of nonlinear transformations of heavy-tailed r.v.'s considered in this paper provides a natural framework for modeling risks with distributions exhibiting local to moderate departures from power laws in forms ( 2) and (3).
Let m ≥ 0, γ 1 , ..., γ m ∈ R. Further, let x 0 be a large positive constant.Consider the odd increasing on R function V defined by For k ∈ {1, 2, ..., m} and sufficiently large x we have that the function G(x) = x m i=k g i (x), where Let us show, using (5), that we get that g i (x) = o g i (x) x as x → ∞ for k ≤ i ≤ m, and The first term in this expression is o(1) as shown above, and, as d dx [ln j (x)] = (x j−1 k=1 ln k (x)) −1 for j ≥ 2, each of the other summands is also o(1).From (5) and the above relations it follows that (6) indeed holds and, consequently, G (x) (and, therefore, V (x)) has the same sign as g k (x) for sufficiently large x.
The results on nonlinear transformations of heavy-tailed risks imply Corollaries 1 and 2 below.These corollaries concern the analysis of portfolio diversification under local, and moderate deviations from power laws in form (3). For instance, Corollary 1 demonstrates that the stylized facts on portfolio diversification continue to hold for portfolios of nonlinear transformations of risks with the tail index α ≥ 1 whose distributions have tails that satisfy (3) with α > 1 or with α = 1, γ 1 = 0, ...., γ k−1 = 0, γ k > 0 for some k ∈ {1, 2, ..., m}.Thus, Corollary 1 implies that portfolio diversification decreases portfolio riskiness in the world of risks whose distributions are even slightly thinner than those of Cauchy r.v.'s.
Corollary 1 Let the parameters of the function V be such that γ 1 = 0, ..., γ k−1 = 0, γ k < 0 for some k ∈ {1, 2, ..., m}.Further, let x 0 in the definition of V be sufficiently large so that the function is welldefined, odd and increasing on R and is strictly concave on [0, ∞).If X 1 and X 2 are i.i.d risks such that X then the conclusions of Theorem 1 hold.However, Corollary 2 implies that diversification always leads to an increase in riskiness for portfolios of nonlinear functions of risks with α ≤ 1 whose distributions have tails satisfying (3) with α < 1 or with α = 1, γ 1 = 0, ...., γ k−1 = 0, γ k < 0 for some k ∈ {1, 2, ..., m}.Thus, Corollary 2 shows that riskiness of a portfolio increases with diversification for risks with distributions whose tails are even slightly heavier than those of Cauchy r.v.'s.
Corollary 2 Let the parameters of the function V be such that γ 1 = 0, ..., γ k−1 = 0, γ k > 0 for some k ∈ {1, 2, ..., m}.Further, let x 0 in the definition of V be sufficiently large so that the function is welldefined, odd and increasing on R and is strictly convex on [0, ∞).If X 1 and X 2 are i.i.d risks such that According to relation ( 12), the part of Theorem 1 for transformations of stable r.v.'s holds.Relation (13) shows that Theorem 2 is true.
Proof of Corollaries 1 and 2. Corollary 1 follows from Theorem 1 since, under the assumptions of the corollary, the function V belongs to the class f ∈ M .Similarly, Corollary 2 is a consequence of Theorem 2 and the fact that, under its assumptions, the function V belongs to the class f ∈ M .