1. Introduction
Nowadays, risk measures have been widely used in both finance and insurance, such as capital requirement calculation and insurance pricing, etc. In their seminal work, ref. [
1] axiomatically introduced and characterized coherent risk measures. Later, refs. [
2,
3] axiomatically introduced and characterized a broader class of convex risk measures. In the context of insurance, ref. [
4] introduced an axiomatic approach to insurance pricing, which is also known as distortion premium principle. For more details about (univariate) risk measures, we refer to [
5]. Mathematically, the random loss of a financial position is commonly described by a random variable defined on some measurable space
. In practice, these classical risk measures usually require the acquirement of accurate distribution functions of the random variables. This requirement is mathematically equivalent to requiring an
accurate probability measure on
, which is sometimes known as a reference probability or a scenario in the literature. For instance, value at risk (VaR) and expected shortfall (ES) are the cases, which are two risk measures popularly used in practice.
From the practical point of view, it is usually difficult to accurately acquire the
true distribution functions of the random variables, because the distribution functions are usually estimated from data (samples) or statistical hypothesis (simulation). In other words, it is usually hard to acquire an
accurate (reference) probability measure on
. Hence, the model uncertainty problem naturally arises. This kind of uncertainty could stem from the Knightian uncertainty; for instance, see [
5]. There are two natural and important approaches to deal with model uncertainty. One approach is to consider random variables on some measurable space
without assuming a reference probability (scenario) on it; for instance, see [
5]. Such models are also called
model-free models; for instance, see [
1]. Another approach is to consider multiple scenarios (reference probabilities) on
, in which the set of scenarios describes model uncertainty; for instance, see [
6,
7,
8] and the references therein. For more earlier studies on model uncertainty problem, we refer to [
9,
10,
11,
12,
13,
14], and the references therein. It should be mentioned that model uncertainty might have different disguises in different situations. For instance, ref. [
15] studied background risk measures. Ref. [
16] studied various systemic risk measures. Ref. [
17] studied various VaR- and expectile-based systemic risk measures. Ref. [
18] studied factor risk measures. Ref. [
19] established an elegant conditional distortion risk measure. Various conditional risk measures also appear in [
14,
20,
21] in different disguises, just name a few. In this paper, we also take into account the model uncertainty issue. Moreover, we adopt the way of considering
model-free models.
In the present paper, we take a different perspective to describe model uncertainty. Specifically, we use an auxiliary random variable to describe model uncertainty. For example, for each value the auxiliary random variable takes, there corresponds a scenario (reference probability) on some measurable space. Our approach is to, respectively, measure the risk of a random loss provided that the auxiliary random variable takes different values first, and then aggregate these risk evaluations into a single value. In such a procedure of risk evaluation, the auxiliary random variable serves a role like a condition or environment. Hence, we regard it as a random environment, and refer to the resulting risk measures as conditional risk measures under a random environment later.
In this paper, by axiomatic approaches, we introduce conditional coherent and convex risk measures under a random environment, respectively. After studying their properties, we then axiomatically characterize them. The axioms are newly presented in the presence of model uncertainty, although they are motivated by those classical axioms. Finally, as examples, we also discuss the connections of the introduced conditional coherent risk measures under random environments with two existing risk measures. This paper mainly gives some theoretical results, and it is expected to make a meaningful complement to the study of coherent and convex risk measures under model uncertainty.
The rest of the paper is organized as follows. In
Section 2, we state preliminaries.
Section 3 is devoted to the main results of this paper. In
Section 4, we provide two examples to discuss the connections of the proposed risk measures with the existing risk measures.
2. Preliminaries
In this paper, for any non-empty set
S, we take the power set
as the
-algebra, making
a measurable space. The power set
consists of all subsets of
S. The reason that we use the power set
as the
-algebra on
S is to avoid various tedious measurability consideration, because we will encounter integrals of various random variables with respect to non-additive measures in the sequel. For more about integral with respect to non-additive measures, we refer to [
22]. We denote by
the linear space of all bounded measurable functions (i.e., random variables) on the measurable space
.
Let
be a fixed measurable space, where the sample space
represents all possible states of the real world. In this paper, we will work on
rather than the space of all essentially bounded random variables on certain probability space, so that we can deal with model uncertainty; for instance, see [
5,
6,
7]. The random loss of a financial position is described by an element in
. The space
will also serve as random environments. For the terminology convenience, for any random environment
, each value that
Z takes is called a
state of
Z.
Given a non-empty set
S, a mapping
is called a set function on
, if
A set function
on
is called normalized, if
; and called monotone, if
for any
with
A normalized monotone set function
on
is called a finitely additive probability measure on
, if
for any disjoint
. Note that a finitely additive probability measure on
is a probability measure whenever
S is a finite set (i.e., the number of elements of
S is finite.). We denote by
the set of all finitely additive probability measures on
. Simply, we write
for
. Given a
for any measurable function (i.e., random variable)
V on
, denote by
the expectation of
V with respect to
. For more about the expectation under finitely additive probability measures, we refer to [
5].
We denote by the real numbers, and the non-negative real numbers. For a non-empty set A, stands for the indicator function of A. By convention, For any , Ran(X) stands for the range of X.
Give an environment
, for any state
and any random loss
, we denote by
the restriction of
X to the subset
of
, that is,
,
Notice that the domains of
and
are different in general. If the random environment
Z is degenerate, then
and
X are the same.
Given a non-empty set
S, in general, a risk measure on
can be defined as any functional
. In particular, for any random variable
, the quantity
is called the risk measure of
X. A risk measure
is called
monotone if
for
with
,
translation-invariant if
for
and
,
positively homogeneous if
for
and
,
subadditive if
for
. A risk measure is called
coherent if it is monotone, translation-invariant, positively homogeneous and subadditive,
convex if it is monotone and translation-invariant, and satisfies that
for
and
. For a coherent risk measure
, both positive homogeneity and translation invariance imply that
for
. For more details about risk measures, we refer to [
5].
We end this section by introducing the definitions of conditional risk measures (CRMs) under a fixed state and a random environment, respectively.
Definition 1. Given a random environment and a state , a CRM under a fixed state z is defined as any functional . In particular, for any random loss , the quantity is called the risk measure of X under a fixed state z.
Definition 2. A CRM under a random environment is defined as any functional . In particular, for any random loss and any random environment , the quantity is called the risk measure of X under a random environment Z.
Note that there are two input arguments for a CRM under a random environment. The first input argument represents the random loss of a financial position. Compared with the first input argument, the second input argument serves only as a role of a kind of condition or environment. Different elements of represent different random environments. In the majority, we specify a condition (environment) first, and then measure the risk of random losses . Nevertheless, considering that the random environment Z can be any element of , we think of the CRM under a random environment as a bivariate functional on .
3. Main Results
In this section, we present the main results of this paper. By axiomatic approaches, we introduce and characterize conditional coherent and convex risk measures under a fixed state and random environment, respectively. Specifically, in the first subsection, we introduce and characterize conditional coherent risk measures under a fixed state and random environment, respectively. In the second subsection, we introduce and characterize conditional convex risk measures under a fixed state and random environment, respectively.
3.1. Conditional Coherent Risk Measures Under Random Environment
In this subsection, we list some axioms for CRMs under a random environment first. Then we construct conditional coherent risk measures under random environments and study their properties. Finally, we axiomatically characterize it.
Similar to the definition of classical coherent risk measures, we start with the definition of conditional coherent risk measures (CCRMs) under a fixed state.
Definition 3. Given, arbitrarily, a random environment and a state , a CRM under a fixed state z is called coherent if it satisfies the following properties (axioms):
A1 Local indifference:
For , if , then .
A2 Local monotonicity:
For , if , then .
A3 Local translation invariance:
for any and any .
A4 Local positive homogeneity:
for any and any .
A5 Local sub-additivity:
for any .
Since the state of a given environment is pre-specified, the financial meanings of the above Axioms A1–A5 can be interpreted totally similarly to the classic setting; for instance, see [
1,
4,
5,
6,
23].
Note that Axiom A2 implies Axiom A1. Now, we turn to introduce the definition of conditional coherent risk measures (CCRMs) under a random environment.
Definition 4. A CRM under a random environment is called coherent, if for any random environment and each state , there is a CCRM under a fixed state z such that the following properties (axioms) hold:
B1 Outcome indifference:
For any , if for any , then .
B2 Global monotonicity:
For any , if for any , then .
B3 Risk consistency:
For any and , if for any , then .
B4 Multiplication consistency:
For any and , if for any , then .
B5 Global sub-additivity:
For any , if for any , then .
Axioms B1–B5 can be interpreted in the context of finance as follows. Note first that exactly represents the risk measure of random loss X under the condition that the environment Z takes the state z, as will be seen in the sequel. Axiom B1 means that for two random losses and , if their state-wise riskiness and have the same distribution under the environment’s probability distribution, then their overall riskiness should be the same. This characteristic has some similarity to the law invariance in the classic setting. B2 says that if random loss is less risky than another random loss in an almost-all-state-wise sense, then the overall riskiness of should not exceed that of . B3 says that for two random losses X and Y, if the riskiness of Y is the sum of that of X and a fixed amount of c in an almost-all-state-wise sense, then the overall riskiness of Y should also be the sum of the overall riskiness of X and the fixed amount c. This characteristic has some similarity to the translation invariance in the classic setting. B4 says that for two random losses X and Y, if the riskiness of Y is a multiple of that of X in an almost-all-state-wise sense, then the overall riskiness of Y should also be the same multiple of the overall riskiness of X. This characteristic has some similarity to the positive homogeneity in the classic setting. B5 says that for three random losses X, Y, and W, if the riskiness of W is the sum of that of X and Y in an almost-all-state-wise sense, then the overall riskiness of W should not exceed the sum of the overall riskiness of X and Y. This characteristic has some similarity to the subadditivity in the classic setting.
Remark 1. - (i)
Note that when arbitrarily given a random environment , the existence of such a family of CCRMs under a fixed state as in Definition 2.1 will be illustrated later; see (1) and Proposition 3.1 below. Given a family of CCRMs under a fixed state such that B1–B5 hold, then we also say that the CRM under a random environment is the corresponding CCRM under a random environment. - (ii)
B4 implies that for any random environment . Both B4 and B3 imply that for . Each of Axioms B2–B5 can imply Axiom B1.
Now, we turn to construct CCRMs under a random environment. Given a random environment
and a state
, let
be a subset of
, and
be a subset of
. Then, we define CRMs under a fixed state and random environment
and
, respectively, as follows:
and
where
is given by (
1).
Remark 2. - (i)
For each , it could be viewed as a plausible probability measure on , that is, the possible probability distribution of X under the state z. When dealing with financial data, one should first confirm the random environment Z, for example, rate of interest, exchange rate, or the population mean, or so on, and then under each state of the random environment, use historical data in different time window or re-sampling to simulate the possible probability distributions of random loss X.
- (ii)
For each , it could be regarded as a plausible probability distribution for Z. In practice, once the random environment Z is confirmed, one could use historical data in different time window or re-sampling to simulate the possible probability distributions of random environment Z.
The next proposition shows the properties of the CRMs defined by (
1) and (
2).
Proposition 1. - (1)
The CRM under a fixed state defined by (1) is coherent, that is, it satisfies Axioms A1–A5. - (2)
The CRM under a random environment ρ defined by (2) is coherent, that is, it satisfies Axioms B1–B5.
Proof. A1 and B1 are clear. A2–A5 and B2–B5 are straightforward since the expectation under the finitely additive probability measure is also linear with respect to the random variable. The proof is completed. ☐
Next, we proceed to characterize the CCRM under a random environment. The next theorem will show that any CCRM under a random environment is of a form as in (
2), which is one of the main results of this paper.
Theorem 1. Let be an arbitrarily given random environment. For any state , let be a CCRM under a fixed state z, and be the corresponding CCRM under a random environment. Then,
- (1)
can be represented by where .
- (2)
can be represented by where is given by (3), and
Proof. - (1)
For any
, denote by
the natural extension of
X from
to
, that is,
is defined by
Clearly,
. Moreover, we define a risk measure
by
First, we claim that the risk measure
defined by (
5) is coherent. In fact, For any
with
, we know that
. Hence by Axiom A2,
. Thus, monotonicity of
follows from (
5). Similarly, translation invariance of
follows from Axiom A3 and (
5). Positive homogeneity of
follows from Axiom A4 and (
5). Subadditivity of
follows from Axiom A5 and (
5).
Therefore, by the representation theorem for coherent risk measures (e.g., see [
5]), we know that the risk measure
can be represented by
where
where the first equality is due to (
5), and the second equality is due to Axiom A1.
Now, we proceed to show (
3). For any
, write
, that is,
. Clearly,
. Recall that
. Hence, from Axiom A1, (
5) and the definition of
, it follows that
which, together with (
6), implies (
3). The desired assertion holds.
- (2)
An arbitrarily fixed random environment
. Denote
Note that for each
, there is some
depending on
h so that
Hence, we define a risk measure
by
where
,
, with some
.
First, we claim that the risk measure
defined by (
7) is well defined. In fact, if there are two random variables
such that both
and
then
for any
. Hence, by Axiom B1 we know that
. Thus,
as in (
7) is well defined.
Now, we further conclude that
Indeed, for any
, there exists
such that
. Since
is local translation-invariant and local monotone,
for any
. Thus,
Conversely, for any
, we define a random variable
by
Clearly,
. Moreover, for each
,
whenever
. Hence, by Axiom A1, we have that for any
,
which yields that
Therefore, (
8) follows from (
9) and (
12).
☐
Next, we proceed to show that the risk measure
on
defined by (
7) is coherent. In fact, due to Axioms B2–B4, it is not hard to verify that
is monotone, translation-invariant, and positively homogenous. Hence, it is sufficient for us to show the subadditivity. Indeed, given any
, similar to (
10), we define
. By (
11), we know that
and
Denote
. Then,
. Similar to (
10), we define
, and have that
Thus, for any
,
which, together with Axiom B5, gives rise to
From (
13) and (
7), it follows that
which means that
is subadditive. Therefore,
is a coherent risk measure.
Consequently, by the representation theorem for coherent risk measures (e.g., see [
5]), we know that the risk measure
can be represented by
where
By (
8) and the definition of
as in (
7),
can be represented by
where
. Thus, (
4) is proved. Theorem 3.1 is proved.
3.2. Conditional Convex Risk Measures Under Random Environment
In this subsection, we extend the previous class of CCRMs under a random environment to a broader class of conditional convex risk measures under a random environment. We construct and axiomatically characterize conditional convex risk measures under a random environment.
Similar to the definition of classical convex risk measures, we start with the definition of conditional convex risk measures (CConRMs) under a fixed state.
Definition 5. Given, arbitrarily, a random environment and a state , a CRM under a fixed state z is called convex if it satisfies Axioms A1–A3 and A6 below:
Similar to the explanation of the financial meanings of Axioms A1–A5 above, since the state of a given environment is pre-specified, the financial meanings of Axiom A6 above can be interpreted totally similarly to the classic setting; for instance, see [
2,
3,
5].
Now, we turn to introduce the definition of conditional convex risk measures (CConRMs) under a random environment.
Definition 6. A CRM under a random environment is called convex If, for any random environment and each state , there is a CConRM under a fixed state z such that Axioms B1–B3, B0 and B6 below hold:
B0 Standard:
.
B6 Global convexity:
For any and , if for any , then .
Axioms B0 and B6 can be interpreted in the context of finance as follows. Note first that exactly represents the risk measure of random loss X under the condition that the environment Z takes the state z, as will be seen in the sequel. B0 can be viewed as a kind of technical condition. Nevertheless, B0 could be interpreted as follows: keeping global monotonicity B2 and risk consistency B3 in mind, B0 says that for any bounded random loss X, its riskiness should be uniformly finite with respect to almost all states of the random environment. B6 says that for three random losses X, Y, and W, if the riskiness of W is a certain convex combination of the riskiness of X and Y in an almost-all-state-wise sense, then the overall riskiness of W should not exceed the same combination of the overall riskiness of X and Y. This characteristic has some similarity to the convexity in the classic setting.
Throughout this subsection, for a given random environment
and any state
, let
be a functional from
to
with
Let
be a functional from
to
with
.
Now, we turn to construct CConRMs under a random environment. Given a random environment
and a state
, we define CRMs under a fixed state and random environment
and
, respectively, as follows:
and
, where
is given by (
15).
The next proposition shows the properties of the CRMs defined by (
15) and (
16).
Proposition 2. - (1)
The CRM under a fixed state defined by (15) is convex, that is, it satisfies Axioms A1–A3 and A6. - (2)
The CRM under a random environment ρ defined by (16) is convex, that is, it satisfies Axioms B0–B3 and B6.
Proof. For any
, there exists
such that
. Note that the expectation under finitely additive probability measure is monotone, not only are both A2 and B2 obvious, but also it holds that
Since the expectation under finitely additive probability measure is also translation-invariant, A3 and B3 are clear. Moreover,
Meanwhile, from
, we can obtain B0. Together with (
17), it holds that there exists
such that
Thus, by B2, B3 and
, we have that
For any
, we have that
for any
. Hence, A6 is proved. ☐
For any
, by (
18), for any
, we have that
Therefore, B6 is proved. The proposition is proved.
Next, we proceed to characterize the CConRM under a random environment. The next theorem will show that any CConRM under a random environment is of a form as in (
16), which is another one of the main results of this paper.
Theorem 2. Let be an arbitrarily given random environment. For any state , let be a CConRM under a fixed state z, and be the corresponding CConRM under a random environment. Then,
- (1)
can be represented by - (2)
can be represented by where is given by (19), and
Proof. - (1)
For any given
, we define the risk measure
as in (
5), that is,
where
is the natural extension of
X from
to
. Similar to the proof of Theorem 3.1(1), we can steadily show that
is a convex risk measure on
.
Therefore, by the representation theorem of convex risk measure (e.g., see [
5]), we know that the risk measure
can be represented as
where
Consequently, by the definition of
, we obtain the desired assertion (
19).
- (2)
Basically, the argument is quite similar to the proof of Theorem 1(2). Specifically, an arbitrarily fixed random environment
. Similarly, denote
Note that for each
, there is some
depending on
h so that
Similar to (
7), we well define a risk measure
by
where
,
, with some
.
☐
In fact, for any
, there exists
such that
. For any
, by the local monotonicity and local translation invariance of
,
which, togetrher with Axiom B0, yields that
for any
. Therefore,
Conversely, for any
, similar to (
10), we define a random variable
by
Clearly,
. Moreover, for each
,
whenever
. Hence, by Axiom A1 and the local translation invariance of
, we have that for any
,
which implies that
Therefore, (
22) follows from (
24) and (
26).
Next, we proceed to show that the risk measure
on
defined by (
21) is convex. Indeed, from Axioms B2 and B3, it follows that
is monotone and translation-invariant. Moreover, with the help of Axiom B6, by an argument similar to (
14), we can show the convexity of
. Therefore, the risk measure
is convex.
Consequently, by the representation theorem for convex risk measures (e.g., see [
5]), we know that the risk measure
can be represented by
where
By (
22) and the definition of
as in (
21),
can be represented by
where
The desired assertion (
20) is proved. The proof of Theorem 3.2 is completed.
4. Examples
In this section, through two examples, we discuss the connections between our axiomatic approach and some existing risk measures in the literature. The first example examines the connection with the factor-separable risk measures introduced by [
24]. The second example shows that the random environment can describe the model uncertainty discussed by [
7].
Example 1. (
Factor-separable risk measures)
. The factor-separable risk measures are introduced by [24]. Let be certain measurable space. We denote by the space of all measurable functions (i.e., random variables) on . In the construction of factor-separable risk measures, assume that a risk (random) factor Z takes finitely many values , and Ω is partitioned into (disjoint) subsets where . For the risk factor Z, a factor-separable risk measure is then defined aswhere ρ and , , are coherent risk measures. For more details, we refer to [24]. Next, for our discussion convenience, we set
and
. Compared with the random loss
, we view the risk factor
as a
random environment. Since
is coherent, by the representation for coherent risk measures (e.g., see [
5]), there is a subset
of
so that
Indeed, the set
above can be chosen as
Notice that for any
, by the linearity of expectation,
For any given random loss
, we define a random variable
by
Since
is coherent, for each
,
is a CCRM under a fixed state
. Furthermore, for any
, we define a probability measure
on
as follows:
and
We endow the random variable
a probability mass function under
as follows:
Then,
Thus, from (
27)–(
30), it follows that the factor-separable risk measure
defined by (
27) can be expressed by
where
, which is in accordance with the CCRM
under a random environment
Z as in (
2).
In the next example, we will show that the random environment can describe the model uncertainty.
Example 2. (
Scenario-based risk measures)
. Ref. [7] introduced the -mixture of ES. Let be a set of probability measures on some measurable space We denote by the -mixture of ES, which is defined byfor some with and distribution functions on , where stands for the expected shortfall (ES) of X under probability measure with probability level . For more details, we refer to [7]. Next, for our discussion convenience, we set . We will discuss the connection of the CCRMs under a random environment with the -mixture of ES.
As an important case studied by [
7], we assume that the probability measures
are mutually singular, that is, there exist finitely many
, such that
,
for any
and
,
. In this situation, we define a random environment
by
, if
for some (unique)
. Clearly,
. Moreover, for each
, we define a distortion function
by
where
is some distribution function on
. By [
5], we know that the distortion risk measure of a random loss
X with the distortion function
under the probability measure
is a weighted value at risk (WVaR), that is,
For each
, we define a risk measure
by
Then, it is not hard to verify that
is a CCRM under a fixed state
i. Furthermore, by (
2), for any random loss
, the corresponding CCRM of
X under a random environment
Z is defined by
where
is some subset of
. Hence,
Thus, by Proposition 1 we can see that a CCRM
under a random environment
Z becomes a supremum of
-mixtures of ES.
5. Empirical Study
In this section, we take data from the stock market to calculate the risk measure for daily returns with respect to six stocks: Tesla, Netflix, Apple, Microsoft, P&G, Johnson & Johnson. The random environment is taken as the risk-free interest rate. We categorize the levels of the risk-free interest rate into three levels: high, medium, and low. We use the re-sampling method to simulate different potential distributions of the risk-free interest rate. Each re-sampling of the distribution can be regarded as a
in (
2). Under each level of the risk-free interest rate, we also use the re-sampling method to calculate the potential distributions of the daily return of these six stocks and take the average of the worst 10% to obtain
in (
1). Then, by taking the maximum of all the re-samplings for
, we obtain the
as in (
1). Furthermore, by taking the maximum of the weighted sum of
with respect to the distribution of
, we finally achieve the risk measure for these six stocks.
The data for risk-free interest rate are shown in
Figure 1. The data for daily stock price is shown are
Figure 2. The data for daily returns of all the six stocks are shown in
Figure 3. One can notice that the range of the daily returns for Tesla and Netflix is the most drastic, the range for Apple and Mircosoft is less drastic, and the performance for P&G and Johnson & Johnson is much more stable. Also, from the risk measures for daily stock returns, which are listed in
Table 1, one can also observe the same results.