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Does value-at-risk encourage diversification when losses follow tempered stable or more general Lévy processes?


In this paper, we address the question of when portfolio selection based on Value-at-risk encourages diversification [in the sense of Ibragimov (Quant Financ 9(5):565–580, 2009)]. Specifically, we give sufficient conditions for the case when losses follow a Lévy process. When the process has finite variation, these conditions are also necessary. We then specialize our results to the case when losses have tempered stable distributions.

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The author wishes to thank Dr. Weidong Tian for the benefit of several discussions and the anonymous referee whose suggestions lead to a great improvement in the presentation of this paper.

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Correspondence to Michael Grabchak.



The proof of Theorem 1 is based on the following result, which is a specialization of Theorems 2.1 and 3.2 in Samorodnitsky and Taqqu (1994) [see also Theorem 2.2 in Samorodnitsky and Taqqu (1993)].

Lemma 1

Let \(\{X_t:t\ge 0\}\) and \(\{Y_t:t\ge 0\}\) be Lévy processes such that \(X_1\sim ID_0(0,M_X,0)\) and \(Y_1\sim ID_0(0,M_Y,0)\) with \(M_X((-\infty ,0])=M_Y((-\infty ,0])=0\). We have

$$\begin{aligned} M_X((r,\infty )) \le M_Y((r,\infty )) \quad \text{ for } \text{ every } r>0 \end{aligned}$$

if and only if

$$\begin{aligned} P(X_t>r) \le P(Y_t>r) \quad \text{ for } \text{ very } r>0, t>0. \end{aligned}$$

Proof (Proof of Theorem 1)

[Proof of Theorem 1] We only prove results related to Schur-convexity of \(g_r(\cdot )\) as proofs of the other results are similar.

We begin with the case when \(\mu \in \fancyscript{P}\). By positivity, we have \(\mathrm{VaR}_\gamma (X_{\varvec{u},t})\le \mathrm{VaR}_\gamma (X_{\varvec{v},t})\) for all \(t>0\) and all \(\gamma \in (0,1)\) if and only if \(P(X_{\varvec{u},t}>r)\le P(X_{\varvec{v},t}>r)\) for all \(t>0\) and all \(r>0\). By Lemma 1, this holds if and only if \(g_r(\varvec{u})=M_{\varvec{u}}((r,\infty )) \le M_{\varvec{v}}((r,\infty ))=g_r(\varvec{v})\) for all \(r>0\). This holds for all \(\varvec{u},\varvec{v}\in \mathbb R^n_+\) with \(\varvec{u}\prec \varvec{v}\) if and only if \(g_r(\cdot )\) is Schur-convex for all \(r>0\).

Now assume that \(\mu \in \fancyscript{S}\). By symmetry, \(\mathrm{VaR}_\gamma (X_{\varvec{u},t})\le \mathrm{VaR}_\gamma (X_{\varvec{v},t})\) for all \(t>0\) and all \(\gamma \in (.5,1)\) if and only if \(P(X_{\varvec{u},t}>r)\le P(X_{\varvec{v},t}>r)\) for all \(t>0\) and all \(r>0\).

First, consider the case when \(\int _{\mathbb R}\left( |x|\wedge 1\right) M(\mathrm{d}x)<\infty \). For any \(\varvec{w}\in \mathbb R^n_+\) and any \(t>0\), the distribution of \(X_{\varvec{w},t}\) is symmetric around zero and thus \(X_{\varvec{w},t} \mathop {=}\limits ^{d}X_{\varvec{w},t}^{(1)}-X^{(2)}_{\varvec{w}, t}\) where \(\{X_{\varvec{w},t}^{(1)}:t\ge 0\}\) and \(\{X_{\varvec{w},t}^{(2)}:t\ge 0\}\) are independent Lévy processes with \(X^{(1)}_{\varvec{w},1},X^{(2)}_{\varvec{w},1}\mathop {\sim }\limits ^{\mathrm{iid}}ID_0(0,M'_{\varvec{w}},0)\) and

$$\begin{aligned} M'_{\varvec{w}}(A) = \int \limits _{[x>0]} 1_A(x) M_{\varvec{w}}(\mathrm{d}x), \quad A\in \mathfrak B(\mathbb R), \end{aligned}$$

where \(M_{\varvec{w}}\) is as in (7). Note that \(|X_{\varvec{w},t}|\mathop {=}\limits ^{d}X^{(1)}_{\varvec{w},t}+X^{(2)}_{\varvec{w},t}\sim ID_0(0,2M'_{\varvec{w}},0)\). If \(r>0\) then, by symmetry,

$$\begin{aligned} P(X^{(1)}_{\varvec{w},t}+X^{(2)}_{\varvec{w},t}>r) = P(|X_{\varvec{w},t}|>r)= 2P(X_{\varvec{w},t} >r). \end{aligned}$$


$$\begin{aligned} P(X_{\varvec{u},t} >r)\le P(X_{\varvec{v},t} >r), \quad \text{ for } \text{ all } t>0, r>0, \varvec{u}\prec \varvec{v} \end{aligned}$$

is equivalent to

$$\begin{aligned} P(X^{(1)}_{\varvec{u},t}+X^{(2)}_{\varvec{u},t}>r)\le P(X^{(1)}_{\varvec{v},t}+X^{(2)}_{\varvec{v},t}>r), \quad \text{ for } \text{ all } t>0, r>0, \varvec{u}\prec \varvec{v}. \end{aligned}$$

By Lemma 1 this is equivalent to

$$\begin{aligned} 2M'_{\varvec{u}}((r,\infty )) \le 2M'_{\varvec{v}}((r,\infty )), \quad \text{ for } \text{ all } r>0, \varvec{u}\prec \varvec{v}. \end{aligned}$$

Since \(M'_{\varvec{u}}((r,\infty )) = g_{r}(\varvec{u})\) and \(M'_{\varvec{v}}((r,\infty )) =g_{r}(\varvec{v})\), this is equivalent to

$$\begin{aligned} g_{r}(\varvec{u}) \le g_{r}(\varvec{v}), \quad \text{ for } \text{ all } r>0,\varvec{u}\prec \varvec{v}, \end{aligned}$$

which is equivalent to \(g_r(\cdot )\) being Schur-convex for each \(r>0\).

Now, consider the case \(\int _{\mathbb R}\left( |x|\wedge 1\right) M(\mathrm{d}x)=\infty \). Assume that \(g_r(\cdot )\) is Schur-convex for each \(r>0\). Here the Lévy measure is not well behaved near zero, so we begin by truncating it. For a vector \(\varvec{w}\in \mathbb R_+^n\) define

$$\begin{aligned} M_{\varvec{w}}^{(m)}(A) = \int \limits _{[|x|>1/m]}1_A(x) M_{\varvec{w}}(\mathrm{d}x), \quad A\in \mathfrak B(\mathbb R). \end{aligned}$$

Let \(\{X^{(m)}_{\varvec{w},t}:t\ge 0\}\) be a Lévy process where \(X^{(m)}_{\varvec{w},1}\sim ID(0,M^{(m)}_{\varvec{w}},0)\). Since \(M^{(m)}_{\varvec{w}}\) is symmetric, \(\int _{\mathbb R} \left( |x|\wedge 1\right) M^{(m)}_{\varvec{w}}(\mathrm{d}x)<\infty \), and \(M_{\varvec{w}}^{(m)}((r,\infty ))=g_{r\vee (1/m)}(\varvec{w})\) is Schur-convex for all \(r>0\), we can use arguments as in the previous case to show that

$$\begin{aligned} P(X^{(m)}_{\varvec{u},t} >r)\le P(X^{(m)}_{\varvec{v},t} >r), \quad \text{ for } \text{ all } t>0,r>0,\varvec{u}\prec \varvec{v}. \end{aligned}$$

We now remove the truncation by taking limits. Using standard results for convergence of infinitely divisible distributions (see e.g. Theorem 15.14 in Kallenberg (2002)) it is straightforward to show that \(X^{(m)}_{\varvec{u},t}\mathop {\rightarrow }\limits ^{d}X_{\varvec{u},t}\) and \(X^{(m)}_{\varvec{v},t}\mathop {\rightarrow }\limits ^{d}X_{\varvec{v},t}\) as \(m\rightarrow \infty \). Further, so long as \(\varvec{u},\varvec{v}\ne \varvec{0}\), we have \(M_{\varvec{u}}(\mathbb R)=M_{\varvec{v}}(\mathbb R)=\infty \) and thus, by Theorem 27.4 in Sato (1999), both \(X_{\varvec{u},t}\) and \(X_{\varvec{v},t}\) have continuous distributions. Hence, for every \(t>0\) and \(r>0\),

$$\begin{aligned} P(X_{\varvec{u},t} >r)=\lim _{m\rightarrow \infty }P(X^{(m)}_{\varvec{u},t} >r)\le \lim _{m\rightarrow \infty } P(X^{(m)}_{\varvec{v},t} >r)=P(X_{\varvec{v},t} >r). \end{aligned}$$

If \(\varvec{u}=\varvec{0}\) or \(\varvec{v}=\varvec{0}\) then \(\varvec{u}=\varvec{v}=\varvec{0}\) and the result is immediate.

Before proving Corollary 1, we give the following fact, which follows from the proof of Proposition A.6 in Ibragimov (2009).

Lemma 2

Let \(X_1,X_2,\ldots ,X_n\) and \(Y_1,Y_2,\ldots ,Y_n\) be two sets of independent random variables all having symmetric unimodal densities. If \(\mathrm{VaR}_\gamma (X_i)\le \mathrm{VaR}_\gamma (Y_i)\) for all \(=1,2,\ldots ,n\) and all \(\gamma \in (.5,1)\) then \(\mathrm{VaR}_\gamma (\sum _{i=1}^nX_i)\le \mathrm{VaR}_\gamma (\sum _{i=1}^nY_i)\) for all \(\gamma \in (.5,1)\).

Proof (Proof of Corollary 1)

We focus on the case \(t=1\) as the other cases are similar. If \(a=0\) the result follows by Theorem 1. Now assume that \(a\ne 0\). Let \(Z_{1},\ldots , Z_{n}\mathop {\sim }\limits ^{\mathrm{iid}}N(0,1)\) and let \(Y_{1},\ldots ,Y_{n}\mathop {\sim }\limits ^{\mathrm{iid}}ID(0,M,0)\) be independent sequences of random variables. If \(X_{1}, \ldots ,X_{n}\mathop {\sim }\limits ^{\mathrm{iid}}\mu \) then \((X_1,\ldots ,X_n)\mathop {=}\limits ^{d}(\sqrt{a}Z_1+Y_1, \ldots ,\sqrt{a}Z_n+Y_n)\). For any \(\varvec{w}\in \mathbb R^n_+\) we have \(X_{\varvec{w}} \mathop {=}\limits ^{d}\sqrt{a} Z_{\varvec{w}} + Y_{\varvec{w}}\). If \(\varvec{u}\prec \varvec{v}\) then Theorem 1 implies that \(\mathrm{VaR}_\gamma (Y_{\varvec{u}})\le \mathrm{VaR}_\gamma (Y_{\varvec{v}})\) and Proposition A.7 in Ibragimov (2009) implies that \(\mathrm{VaR}_\gamma (\sqrt{a}Z_{\varvec{u}})\le \mathrm{VaR}_\gamma (\sqrt{a}Z_{\varvec{v}})\). By Proposition A.5 in Ibragimov (2009) both \(Z_{\varvec{u}}\) and \(Y_{\varvec{u}}\) have symmetric and unimodal densities. Thus the result follows by Lemma 2.

Proof (Proof of Proposition 1.)

[Proof of Proposition 1.] If \(f_r(\cdot )\) is convex then \(g_r(\cdot )\) is Schur-convex by Proposition C.1 on page 92 in Marshall et al. (2011). Now assume that \(M\) has a continuous Lebesgue density \(m\). This implies that \(f_r(s)\) is a continuous function of \(s\) and thus it is convex if and only if \(g_r(\cdot )\) is Schur-convex by Proposition C.1 on page 92 and Proposition C.1.c on page 95 of Marshall et al. (2011). By Leibniz’s rule \(f_r(s)\) is differentiable in \(s\) and

$$\begin{aligned} \frac{\partial }{\partial s}f_r(s) = \frac{\partial }{\partial s}\int \limits _{r/s}^\infty m(x) \mathrm{d}x = rs^{-2}m(r/s) = r^{-1}(r/s)^{2}m(r/s). \end{aligned}$$

From here, the fact that a differentiable function is convex if and only if its derivative is non-decreasing gives the equivalence between the convexity of \(f_r(\cdot )\) and the non-increasing of \(x^2m(x)\). Conditions for \(g_r(\cdot )\) to be Schur-concave can be shown in a similar way using the corresponding properties of concave and Schur-concave functions.

Proof (Proof of Corollary 3)

[Proof of Corollary 3] By Theorem 30.1 in Sato (1999), \(W_Y\sim ID(0,M^*,0)\) where \(M^*\) has a continuous density \(m^*(x)\) given by

$$\begin{aligned} m^*(x)= \int \limits _{0}^\infty \frac{1}{\sqrt{2\pi s}}e^{-x^2/(2s)}m(s)\mathrm{d}s =\frac{|x|}{\sqrt{2\pi }} \int \limits _{0}^\infty e^{-u/2}m(x^2/u)u^{-3/2}\mathrm{d}u. \end{aligned}$$


$$\begin{aligned} x^2m^*(x)= \frac{1}{\sqrt{2\pi }} \int \limits _{0}^\infty e^{-u/2} (|x|/\sqrt{u})^3 m((|x|/\sqrt{u})^2)\mathrm{d}u. \end{aligned}$$

From here the result follows by Corollary 2.

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Grabchak, M. Does value-at-risk encourage diversification when losses follow tempered stable or more general Lévy processes?. Ann Finance 10, 553–568 (2014).

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