_{1}

^{*}

We study the asymptotic behavior of the difference
*X* and
*Y* are non-negative random variables whose tail probability functions are regularly varying. The case where is the value-at-risk (VaR) at
α, is treated in [
1]. This paper investigates the case where
*Y* to the portfolio
*X*+
*Y* . Similarly to [
1], our results depend primarily on the relative magnitudes of the thicknesses of the tails of
*X* and
*Y*. Especially, we find that
*Y* if the tail of
*Y* is sufficiently thinner than that of X. Moreover, we obtain the asymptotic relationship
*X* and
*Y*. We also conducted a numerical experiment, finding that when the tail of
*X* is sufficiently thicker than that of
*Y*,

The purpose of this paper is to investigate the asymptotic behavior of the difference

Δ ρ α X , Y : = ρ α ( X + Y ) − ρ α ( X ) (1.1)

as α → 1 , where X and Y are fat-tailed random variables (loss variables) and ( ρ α ) 0 < α < 1 is a family of risk measures. The case where ρ α is an a-percentile value-at-risk (VaR), has been treated in [

ρ α ( Z ) → α → 1 ess sup ω Z ( ω ) (1.2)

for each loss random variable Z in some sense. Our result does not require any specific form for ρ α , implying that this property is robust. Furthermore, assuming some technical conditions for the probability density functions of X and Y, we study the asymptotic behavior of the Euler contribution, defined as

ρ α Euler ( Y | X + Y ) = ∂ ∂ h ρ α ( X + h Y ) | h = 1 (1.3)

(see Remark 17.1 in [

We now briefly review the financial background for this study. In quantitative financial risk management, it is important to capture tail loss events by using adequate risk measures. One of the most standard risk measures is the VaR. The Basel Accords, which provide a set of recommendations for regulations in the banking industry, essentially recommend using VaR as a measure of risk capital for banks. VaRs are indeed simple, useful, and their values are easy to interpret. For instance, a yearly 99.9% VaR calculated as x 0 means that the probability of a risk event with a realized loss larger than x 0 is 0.1%. In other words, an amount

The expected shortfall (ES) has been proposed as an alternative risk measure that is coherent (in particular, subadditive) and tractable, with the risk amount at least that of the corresponding VaR. Note that there are various versions of ES, such as the conditional value-at-risk (CVaR), the average value-at-risk (AVaR), the tail conditional expectation (TCE), and the worst conditional expectation (WCE). These are all equivalent under some natural assumptions (see [

A spectral risk measure (SRM) has been proposed as a generalization of ESs, in [

VaRs and ESs as risk measures depend on a confidence level parameter

In this paper, we consider a family

The rest of this paper is organized as follows. In Section 2, we prepare the basic settings and introduce the definitions for SRMs based on confidence level. In Section 3, we give our main results. We numerically verify our results in Section 4. Finally, Section 5 summarizes our studies. Throughout the main part of this paper, we assume that X and Y are independent. The more general case where X and Y are not independent is studied in Appendix 1. All proofs are given in Appendix 2.

Let

Note that

We now introduce the definition of SRMs.

Definition 1

1) A Borel measurable function

2) A risk measure

Remark 1 SRMs are law-invariant, comonotonic, and coherent risk measures. However, as shown in [

for each

Moreover, such a

Here, it is easy to see that

meaning that

Next, we introduce a family

Definition 2 Let

where

Condition (2.5) formally implies (1.2). Indeed, if

where we recognize

Lemma 1 Relation (2.5) is equivalent to

We now give some examples of CLBSRMs.

Example 1. Expected Shortfalls

It is easy to see that (2.5) does hold. Indeed, for any bounded continuous function f defined on

due to the bounded convergence theorem. Equivalently, we can also check that

(see [

Example 2. Exponential/Power SRMs

An admissible spectrum f corresponding to an SRM

The exponential utility function is a typical example of tractable utility functions

where p denotes the profit-and-loss (

Note that the theoretical validity of the above method is still unclear. Other methods to adequately construct SRMs from exponential utility functions have been discussed in [

Similarly to the above, an SRM

We can also verify that

We now introduce some notations and definitions used in asymptotic analysis and extreme value theory.

Let f and g be positive functions defined on

Our main purpose is to investigate the property of (1.1) for a CLBSRM

for some

In [

Let

for each

for each

where

Our main purpose in this section is to investigate in detail the asymptotic behavior of

[C1] X and Y are independent.

[C2] There is some

density function

[C3] The function

Let us adopt the notation

for

Theorem 1 Assuming [C1]-[C3],

Formally, assertions (i)-(iii) of Theorem 4.1 in [

Theorem 1 justifies the following relation:

Note that condition [C3] is not required for Theorem 1 when

Theorem 2 Assume [C1] and [C3]. Moreover, assume that

[C4] X and Y have positive, continuous, and ultimately decreasing density functions

Under these assumptions,

Theorems 1 and 2 together imply that if X and Y are independent, and if

Note that

Note that

where the last equality in the above relation is obtained from (5.12) in [

and

due to the dominated convergence theorem. Therefore, if

In Section 4, we numerically verify the above relation. Note that we can also verify a version of Assumption (S) under [C4].

Remark 2

1) If

where

Note that we have

where

and therefore

Until the end of Remark 2, we assume that

2) We can relax the independence condition [C1] so that X may weakly depend on Y within the negligible joint tail condition (see Remark A.1 in [

Indeed, our proof in Appendix 2 also works by applying Theorem A.1 in [

(see Proposition 3 in Appendix 2).

3) As mentioned in Appendix A.1 of [

and then (by the same proof as Theorem 1 with (3.13))

under some assumptions. Here,

because

Note that if

which is known as the component CVaR (also known as the CVaR contribution) and widely used, particularly in the practice of credit portfolio risk management (see for instance [

In this section, we numerically investigate the behavior of

(see (5.2) in [

We numerically compute

Case 1)

We set

holds because

Indeed, we have the following result.

Proposition 1 If there is a unique solution

Note that unlike the case of SRMs,

Case 2)

with

By contrast, the convergence speed of

Case 3)

Finally, we look at the case

Similarly to Case 2), the convergence speed of

In this paper, we have studied the asymptotic behavior of the difference between

of the thicknesses of the tails of X and Y. In particular, for

Our numerical results in the case

Our results essentially depend on the assumption that X and Y are independent. However, the dependence structure of the loss variables X and Y plays an essential role in financial risk management. The case of dependent X and Y for

Kato, T. (2018) Asymptotic Analysis for Spectral Risk Measures Parameterized by Confidence Level. Journal of Mathematical Finance, 8, 197-226. https://doi.org/10.4236/jmf.2018.81015

Here, we briefly investigate the asymptotic behavior of

Note that (A.1) holds for general SRM

We consider the case where X and Y are comonotone. In other words, they are perfectly positively dependent (see Definition 4.82 of [

Proposition 2 If X and Y are comonotone, then

This proposition implies that when

Similarly to Section 4, we assume that

for a copula

1) The Gaussian copula

2) The Gumbel copula

3) The countermonotonic copula

where

normal distribution (for more details on the copulas, see, for instance, Chapter 5 of [

Note that the above findings are consistent with the comonotonic case (Proposition 2).

We describe the following conditions.

[C5] For each

[C6] There is a

for each

[C7] It holds that

for some

Conditions [C5]-[C7] strongly correspond to conditions [A5]-[A6] in [

Using a similar argument as in the proof of the uniform convergence theorem (Theorem 1.2.1 in [

for each compact set

We now introduce the following result.

Theorem 3 Assume [C5]-[C7] and (3.12). If

This theorem claims that both (3.11) and (3.14) are true under some conditions, even when X and Y are dependent.

\Proof of Lemma 1. Assume (2.5). Fix any

Because

Combining (B.1) with (B.2), we have

Conversely, if we assume (2.6), then Prokhorov’s theorem implies that for

each increasing sequence

weakly converges to

This immediately leads us to

Proof of Proposition 1. Let

where

Proof of Proposition 2. Because

Here, we see that

and thus

Similarly, because

and

We first state some propositions and prove them. For this, let

Proposition 3

Proof. If

where

Proposition 4

Proof. If

because of

Corollary 1

Proof. This follows from (3.2) and Proposition 4.

Proof of Theorem 1. Let

by virtue of Theorem 4.1(i)-(iii) in [

Furthermore, it holds that

hence

Temporarily fix any

Similarly, we have

Additionally, we have

where

By (B.8) and (B.9), we have

Combining this with (B.6) and Proposition 3, we arrive at

Because

Let

Lemma 2

Proof. Continuity and positivity are obvious. By [C4] and Theorem 1.1 in [

Let

Proposition 5 It holds that

Proof. For each

which implies our assertion.

Note that (B.11) and Proposition 5 lead to

Proposition 6 If

Proof. Let

Then, we see that

Therefore, we need to show that

First, we show that

Using (B.10), Lemmas A.1 and A.3 in [

Furthermore, we observe that

and that the function

Now, (B.18) is obvious.

Next, we observe that

Because

for some

Proposition 7 If

Proof. Let

due to [C3], (B.10), Proposition A3.8 in [

Proposition 8 If

Proof. Let

Proposition 9 If

Proof. Similarly to the proof of Proposition 8, we need to show only that

where

by [C3], (B.10), Proposition A3.8 in [

and thus we obtain (B.21) by applying the dominated convergence theorem.

Proof of Theorem 2. We can verify that the random vector

where we denote

where

First, note that condition [C5] immediately implies [C2] with

Second, note that by [C6], Proposition 3.1(i) in [

To prove Theorem 3, we give the following three propositions.

Proposition 10

Proposition 10 is obtained by an argument similar to the proof of Lemma 5.3 in [

Proposition 11 The function

Proof. Fix any

and therefore, using [C5], we arrive at

Proposition 12

Proof. Fix any

where we denote

By [C7] and the Chebyshev inequality, we get

for some

Moreover, we see that

Here, we observe that

for each

for some

Now we arrive at

by using (A.5) and (B.23). Because

Proof of Theorem 3. First, note that Proposition 10 guarantees that

where

Then, fix any

Thus, there is an

Therefore, we have

by virtue of (3.12). Because