Olivier P Faugeras
email: olivier.faugeras@tse-fr.eu

Gilles Pagès
email: gilles.pages@sorbonne-universite.fr

Risk Quantization by Magnitude and Propensity

Keywords: magnitude-propensity, risk measure, mass transportation, optimal quantization

We propose a novel approach in the assessment of a random risk variable X by introducing magnitude-propensity risk measures (mX , pX ). This bivariate measure intends to account for the dual aspect of risk, where the magnitudes x of X tell how high are the losses incurred, whereas the probabilities P (X = x) reveal how often one has to expect to suffer such losses. The basic idea is to simultaneously quantify both the severity mX and the propensity pX of the real-valued risk X. This is to be contrasted with traditional univariate risk measures, like VaR or CVaR, which typically conflate both effects. In its simplest form, (mX , pX ) is obtained by mass transportation in Wasserstein metric of the law of X to a twopoints {0, mX } discrete distribution with mass pX at mX . The approach can also be formulated as a constrained optimal quantization problem. This allows for an informative comparison of risks on both the magnitude and propensity scales. Several examples illustrate the usefulness of the proposed approach. Some variants, extensions and applications are also considered.

Introduction and outline

The evaluation and comparison of risks are basic tasks of risk analysis. The usual view in Insurance mathematics is to evaluate an univariate risk 1 X by a risk measure ρ(X), which can be thought of as a one-sided deterministic univariate summary of the random variable X. Risks X, Y are then compared through their respective risk measures ρ(X), ρ(Y ).

The starting point of this paper is the basic realization that risk, as a random variable, is intrinsically a bivariate phenomenon: magnitudes (loss amounts) occurs with given propensities (or probabilities). Hence, it appears inescapable that a risk measure, as a single univariate quantity on the magnitude scale, will conflate both effects, thus giving a somehow blurred representation of the risk borne by the random variable X. It would then be of interest to quantify risk on both the magnitude and propensity scales. The purpose of this paper is to propose such a simultaneous quantification.

The proposed approach is based on the following idea: as mentioned above, a risk measure ρ(X) can be viewed as a deterministic proxy of the random risk X. Distributionally speaking, it can be thought of as a Dirac measure δ ρ(X) (.) at ρ(X): this distribution gives full propensity one at the magnitude level ρ(X). Therefore, in order to quantify the risk with a varying propensity pX , it makes sense to look for an approximate of the distribution of X by a mixture of two Dirac, a Dirac at location zero with weight 1 -pX , and a Dirac at location mX , with weight pX . This proxy distribution thus encodes the magnitude and propensity effect of the risk borne by X, through the pair (pX , mX ). Mathematically, this problem of approximating distributions is carried out by mass transportation in Wasserstein metric. Optimal transportation to discrete measure also corresponds to the problem of optimal quantization, well-known in the Engineering and Signal Processing literature. Hence, the proposed approach to quantify risk on both the magnitude and propensity scales amounts to a special, constrained, optimal quantization problem.

The outline of the paper is as follows: In Section 2, we motivate the magnitudepropensity approach to risk measures. As a new paradigm to risk evaluation, the proposed approach needs a careful and detailed exposition of its main idea. We first argue and give some evidence of this magnitude-propensity duality of risk, using (intentionally) simplistic examples to make our point clear. We then show how some classical risk measures typically mixes both effects and explain why it would be desirable to quantify risk on both the magnitude and propensity scales. A risk measures can usually be derived as a M-functional, i.e. as a statistical parameter which is a solution of a problem of minimization of some expected loss. We next show how these expected loss minimization problems can be embedded into (degenerate) optimal transportation problems, i.e. mass transportation towards a Dirac distribution. Eventually, we propose our definition of the magnitude-propensity pair (pX , mX ), as hinted above, and make the connection to optimal quantization.

Section 3 is a theoretical study of the basic idea. We show how to compute the optimal (mX , pX ), both as direct minimization problem and as an optimal quantization problem. To that purpose we briefly recall the main facts about quantization theory. We provide existence, characterization and (partial) uniqueness results. We discuss some basic properties of the obtained magnitude-propensity pairs and study some example distributions.

In Section 4, we give some numerical illustrations of the basic idea. We explain how the optimal (mX , pX ) gives rise to magnitude-propensity plots, which allow for an informative comparison of risks on both the magnitude and propensity scale. We show how to do such comparisons for distributions with or without explicit formulas for (mX , pX ). Empirically, the optimal magnitude-propensity pair can be computed by a fixed point algorithm, which is akin to Lloyd's algorithm in optimal quantization. The methodology is illustrated on a real data set of insurance losses.

Eventually, Section 5 broadens the perspective on the proposed approach by sketching some variants, extensions and possible applications to Insurance. In particular, we extend the risk quantization to several points to account for the "tail risk" of a distribution, in addition to its "moderate risk". We next discuss several possible variants to accommodate for real-valued risks, thus giving a magnitude-propensity quantization of both the losses and the gains of a distribution. Then, multivariate generalizations are studied empirically on a set of distributions and show the influence of the dependence structure and the marginals on the multivariate magnitude-propensity measures. Next, we give an application to portfolio optimization and show how the proposed magnitude-propensity measures can be used to define acceptable risk regions and select the best or worst portfolio. At last, we mention possible extensions using covariates. We conclude in Section 6.

2 The magnitude-propensity approach to risk measures

2.1 Risk is intrinsically a bivariate propensity-magnitude random phenomenon

As a primary concept of Insurance Mathematics, it is somehow difficult to give a precise, deductive definition of risk in terms of more primitive concepts. This explains why, e.g. [START_REF] Novak | Extreme value methods with applications to finance[END_REF] p. 224, states that "There is currently no consensus concerning the meaning of the word risk". Hence, risk is often given an implicit or semantic definition in textbooks, as in [START_REF] Novak | Extreme value methods with applications to finance[END_REF] p. 191: "Risk is a possibility of an undesirable event. Though such an event is rare, its magnitude can be devastating."

The latter description of risk encompasses the duality magnitude-propensity of random variables: intuitively, a claim can be "risky" because losses may happen often, i.e. with a "high" propensity, possibly with (relatively) "small" magnitudes, or because a catastrophe of very large magnitude may happen, albeit with a (relatively) "low" propensity. We mention that this dual nature of risk is explicit in the Engineering literature (see e.g. [START_REF] Bedford | Probabilistic Risk Analysis: Foundations and Methods[END_REF]), where it is summarized by the semantic formula "risk=uncertainty+damage" in [START_REF] Kaplan | On the quantitative definition of risk[END_REF].

This duality of risk manifests itself on several levels:

1. Intrinsic magnitude-propensity aspect for a single random risk When one considers risk as a property of a single r.v., i.e. as being intrinsic, a claim X can be deemed risky due to high magnitudes, or due to high propensities, as explained above. It is easy to construct examples of two risky claims X1, X2, with same average risk E X1 = E X2, yet with X1 having high magnitude risk, while X2 having high propensity risk.

2. Relative magnitude-propensity effect for a pair of risks One can also consider that risk is not intrinsic but a relative property of one r.v. X1 w.r.t. another r.v. X2, hence a property of the pair (X1, X2), see [START_REF] Faugeras | Risk excess measures induced by hemi-metrics[END_REF] for the notion of risk excess measures. The magnitudepropensity duality then manifests itself in the comparison of risks: a claim X2 can be riskier than X1, but is it so due to a difference in magnitudes, or due to a difference in propensities (or both)? Yet, the risk excess measures of [START_REF] Faugeras | Risk excess measures induced by hemi-metrics[END_REF] typically conflates both effects.

Magnitude-propensity effect in the stochastic order

In turn, the magnitude-propensity duality encountered in comparing risks translates in the ordering structure of probability measures. We recall that the usual stochastic order ≺st (a.k.a. first-order stochastic dominance) is defined as,

X1 ≺st X2 ⇐⇒ FX 1 (x) ≥ FX 2 (x), ∀x ∈ R, (1)
⇐⇒ QX 1 (t) ≤ QX 2 (t), ∀t ∈ (0, 1), (2)
where FX , resp. QX = F -1 X , stands for the c.d.f., resp. quantile function (generalized inverse), of X. It can be described either as a comparison of magnitudes, as expressed via the quantile functions in (2), or as a comparison of propensities, as expressed via the c.d.f.s in (1). In the following toy example (intentionally simplistic to illustrate the point), we show that the stochastic order can conflate both types of differences: Toy example 1. (a) Let X1 s.t. P (X1 = 0) = 0.9, P (X1 = 100) = 0.1, while X2 is s.t. P (X2 = 0) = 0.9, P (X2 = 1000) = 0.1. Then, X1 and X2 have same propensities, but X2 is more risky than X1, solely due to a difference in magnitudes. (b) Let X1 s.t. P (X1 = 0) = 0.95, P (X1 = 1000) = 0.05, while X2 is s.t.

P (X2 = 0) = 0.5, P (X2 = 1000) = 0.5. Then, X1 and X2 have same magnitudes, but X2 is more risky than X1, solely due to a difference in propensities, In both cases, (a) and (b), one has X1 ≺st X2, as is shown on their c.d.f.s in Figure 1. Here, the stochastic comparison does not distinguish between a horizontal shift of c.d.f.s due to a difference in magnitudes (Figure 1 f.s of X 1 (blue), X 2 (red). Left: case (a), difference of magnitudes (horizontal shift). Right: case (b), differences of propensities (vertical shift). In both cases, X 1 ≺ st X 2 .

Risk measures as univariate deterministic proxies for a random risk

The main paradigm to the assessment of risk in Insurance and Financial Mathematics is based on the use of risk measures

ρ : X → R + , ( 3
)
where X is a space of non-negative measurable random variables modeling an insurance claim. These risk measures quantify risk on a magnitude scale, by converting a random loss X into a deterministic certainty equivalent ρ(X). The latter can then be used for ordering different risks, and for decision making purposes, like setting the premium for covering the risk X, see e.g. [START_REF] Rüschendorf | Mathematical Risk Analysis[END_REF]. The resulting risk measure should typically satisfy some desirable properties (coherent risk measure), like translation invariance, monotonicity, etc., see e.g. [START_REF] Artzner | Coherent measures of risk[END_REF], [START_REF] Föllmer | Stochastic Finance[END_REF].

The last decades have seen a multiplication of risk measures. Among the numerous approaches encountered in the literature, let us mention the classical premium calculation principles based on probabilistic models (see e.g. [START_REF] Mikosch | Non-life insurance mathematics[END_REF], [START_REF] Asmussen | Ruin probabilities[END_REF], [START_REF] Bühlmann | Grundlehren der Mathematischen Wissenschaften[END_REF]), the axiomatic approach where premium principles are subject to a set of desirable properties (see [START_REF] Artzner | Coherent measures of risk[END_REF]), the abstract/functional analytic approach where risk measures are derived from an acceptance set and a set of scenario measures (see [START_REF] Föllmer | Stochastic Finance[END_REF]), distortionbased measures [START_REF] Wang | Premium Calculation by Transforming the Layer Premium Density[END_REF], and eventually the approach to risk excess measures induced by hemi-metrics (see [START_REF] Faugeras | Risk excess measures induced by hemi-metrics[END_REF]). These numerous approaches have lead to considerable debate on the pros and cons of the risk measures available in the literature, see e.g. [START_REF] Embrechts | An Academic Response to Basel 3.5[END_REF].

From the magnitude-propensity point of view, the duality of risk is reflected in its measurement, i.e. in the risk measures themselves. Some risk measures focus more on the propensity aspect of risk, while some others focus more on the magnitude one, and most risk measures mix both. Let us illustrate this point with the following comparison of three well-known risk measures:

• The Value-At-Risk, defined as the left α-quantile of X, V aRα(X) := inf{x : P (X ≤ x) ≥ α}, 0 < α < 1 (4) encodes a propensity into a magnitude, by setting the V aRα(X) as the utmostleft α-quantile. (Note that some authors define VaR as the utmost-right quantile). Value-at-Risk only controls the probability of a loss, it does not capture the size of such a loss if it occurs.

• The opposite extreme is, for X an essentially bounded random variable, the essential supremum, ρ∞(X) := ess sup X, which quantifies the maximum magnitude of the loss, but gives no information on the probabilities.

• The Conditional Value-at-Risk, see e.g. [START_REF] Uryasev | Conditional value-at-risk: optimization approach[END_REF] or [START_REF] Kaas | Modern actuarial risk theory: using R[END_REF], is defined as

CV aRα(X) := 1 1 -α 1 α V aRα(t)dt (5)
which reduces for a continuous distribution to the Conditional Tail Expectation (CTE), CV aRα(X) = CT Eα(X) CVaR thus clearly mixes the two aspects magnitude-propensity of the risk by computing a weighted average over a threshold.

:= E[X|X > V aRα(X)], (6)
In general, one wants to know both when/how often a catastrophe may occur, and also what is the size/extent of the loss one has to face. This suggests that reducing risk X to a single proxy on the magnitude scale as a univariate risk measure ρ(X) is somehow inadequate to account for the dual nature of risk. This idea that one numerical quantity cannot hedge against risk has already been evoked beforehand in the literature, see e.g. [START_REF] Rootzén | A single number can't hedge against economic catastrophes[END_REF]. Let us also remark that in their criticism of the VaR measure, the authors of the academic response to the Basel 3.5 framework are implicitly interested in these dual propensity and magnitude aspects of risk (see [START_REF] Embrechts | An Academic Response to Basel 3.5[END_REF] p.27 "Question W1: VaR says nothing concerning the what-if question: Given we encounter a high loss, what can be said about its magnitude?"). It thus would be of considerable practical interest to quantify risk on both the magnitude and propensity scales. This is the purpose of this paper.

Remark 1 (On elicitability). It has been argued in the literature that elicitability is also a desirable property for risk measures (see [START_REF] Embrechts | An Academic Response to Basel 3.5[END_REF]). Elicitability is a concept derived from point forecasting. Roughly speaking, in order that a point forecast, derived from a statistical functional, be consistent with their evaluation by averaging over the past, the statistical functional must be written as a (strict) Mfunctional, see [START_REF] Gneiting | Making and evaluating point forecasts[END_REF] for a precise definition. It is known that VaR is not a coherent risk measure, but is elicitable, while the Conditional Value-at-Risk (CVaR) is a coherent law invariant risk measure, but is not elicitable. CVaR is jointly elicitable with VaR, see [START_REF] Fissler | Higher order elicitability and Osband's principle[END_REF]. This consideration of joint elicitability and coherence of (CVaR,VaR), i.e. of combining magnitude and propensity type univariate risk measures, is another supplementary motivation for arguing that one should switch to bivariate risk measures for the study of univariate risks.

Remark 2 (On parametrized risk measures). For risk measures depending on a parameter, like α in V aRα(X) or CV aRα(X), arise the practical issue of choosing the "right" parameter value α for correctly representing the risk by a single numerical quantity. Common practice is to hedge against a "rare event", i.e. to take, say, α = 0.9, 0.95 or 0.99.

V aR and CV aR give in fact curves, α → V aRα(X) and α → CV aRα(X). These curves, under mild conditions, determine the distribution of X. Hence, they give the same information as the c.d.f. FX . They are just another possible analytical characterization of the distribution of X. A complete assessment of the magnitude and propensity effects of a risk X can be visualised by plotting its c.d.f. FX , or more conveniently its survival function F X , possibly on a log-log scale. This is the classical approach in the Engineering literature, see [START_REF] Kaplan | On the quantitative definition of risk[END_REF], where it is argued that "a single number is not a big enough concept to communicate risk. It takes a whole curve".

However, it becomes difficult to compare entire curves. Therefore, it is natural to look for a summary of this distribution-determining curve to as few as possible numerical quantities. This is the path favoured in Insurance and Financial Mathematics with risk measures. The magnitude-propensity measure to be introduced below, can thus be viewed as a middle-ground between univariate risk measures of Insurance Mathematics and the full curve approach in Engineering.

M-statistical functionals can be obtained from mass transportation to a degenerate distribution

The following discussion gives the key insight for defining a magnitude-propensity risk measure as an optimal transportation problem. Let us recall that the Monge-Kantorovich optimal transportation problem aims at finding a joint measure P (X,Y ) on, say, the product measurable space (R×R, B(R 2 )), with prescribed marginals (P X , P Y ), which is the solution of the optimisation problem:

Tc(P X , P Y ) := inf P (X,Y ) ∈P(P X ,P Y ) c(x, y)P (X,Y ) (dx, dy), (7)
where c : R × R → R + is a cost function and the infimum is on the set P(P X , P Y ) of joint distributions P (X,Y ) with given marginals P X , P Y . Informally, mass at x of P X is transported to y, according to the conditional distribution P (dy|x) of the optimal transportation plan P (X,Y ) , in order to recover P Y while minimising the average cost of transportation c(x, y)P (X,Y ) (dx, dy). Under regularity conditions, the optimal transportation plan is induced by a (Monge) mapping T , viz. P (X,Y ) = P (X,T (X)) . See [START_REF] Rachev | Mass transportation problems[END_REF], [START_REF] Villani | Topics in Optimal Transportation[END_REF], [START_REF] Villani | Grundlehren der Mathematischen Wissenschaften[END_REF], [START_REF] Santambrogio | Optimal transport for applied mathematicians[END_REF] for book-length treatment on the subject. When P Y is degenerate, i.e. P Y = δm with m ∈ R, then P(P X , δm) reduces to the singleton product measure P(P X , δm) = {P X (dx) × δm(dy)}. (7) thus simplifies as the expected cost between X ∼ P X and a fixed point m,

Tc(P X , δm) = c(x, m)P X (dx) = E c(X, m). (8)
Therefore, minimizing the transportation cost (7) over the set D := {δm(dy), m ∈ R} of Dirac measures is equivalent to minimizing the expected cost (8) over m ∈ R,

Tc(P X , D) := inf P Y ∈D Tc(P X , P Y ) = inf m∈R E c(X, m). (9)
In particular,

• for c(x, y) = (x -y) 2 , ( 7) is the squared L2-Wasserstein metric W2 and ( 9) writes as the variance,

W 2 2 (P X , D) = inf P Y ∈D W 2 2 (P X , P Y ) = inf m∈R E(X -m) 2 = V ar(X),
and is obviously minimised for the mean m = EX. In addition, when P X is replaced by the conditional law of X given X > V aRα(X), one obtains the CVaR (5).

• For the L1 distance, c(x, y) = |x -y|, one gets the median. [START_REF] Koenker | Econometric Society Monographs[END_REF]), that is to say the Value-At-Risk (4).

• For the asymmetric cost c(x, y) = (x -y)α1 x-y≥0 + (y -x)(1 -α)1y-x>0, with 0 < α < 1, one obtains the (left)-α-quantile, m = qα(X) (see e.g.
• For c(x, y) = y + (x-y)1 x≥y 1-α
, one gets simultaneously the Conditional Valueat-Risk and the Value-at-Risk, when E[X] < ∞: the optimal value in ( 9) is the Conditional Value-at-Risk while the argmin gives the Value-At-Risk, see [START_REF] Uryasev | Conditional value-at-risk: optimization approach[END_REF], [START_REF] Rockafellar | Conditional value-atrisk for general loss distributions[END_REF].

• And so on for other statistical functionals. In particular, see [START_REF] Faugeras | Risk excess measures induced by hemi-metrics[END_REF] for risk excess measures obtained by optimal transportation with cost functions which are hemi-metrics encoding an order.

Summarizing the above discussion, statistical functionals and risk measures, which can be expressed as an M-estimator solving (9) for a suitable cost function, can be regarded as being obtained from a special mass transportation problem towards a family of degenerate Dirac distribution δm.

2.4

The magnitude-propensity (m X , p X ) approach to measuring risk

The optimal transportation view on risk measures of Section 2.3 suggests that the limitations of the risk measures of Section 2.2 come from the fact that these univariate functionals can be viewed as being obtained by mass transportation from the P X measure to a degenerate Dirac δm measure: the latter distribution only bears a magnitude m with full propensity. Hence, the magnitude and propensity aspects of P X are mixed and encoded in the sole magnitude m of the Dirac destination measure.

It therefore becomes natural to suggest a mass transportation approach to risk measures, as in ( 9), but with the target Dirac distribution δm replaced by a two-points distribution P Y ,

P Y = (1 -p)δ0 + pδm. (10)
The latter distribution encodes both the magnitude and propensity aspects of risk: a loss of magnitude m occurs with probability p (and no loss occurs with probability 1 -p). We can therefore define of the basic idea of the paper as follows:

Definition 2.1. Let A0 := {P Y = (1 -p)δ0 + pδm, p ∈ (0, 1), m ∈ R + } be the set of such two-point distributions (10). For X ∼ P X with E[X 2 ] < ∞, the bivariate magnitude-propensity risk measure (mX , pX ) is obtained by minimizing the Wasserstein W2 distance from P X to A0, (mX , pX ) = arg inf

P Y ∈A 0 W2(P X , P Y ). (11)
In other words, the risk X is approximated in Wasserstein metric by a proxy Y ∼ B0, which bears a loss of magnitude mX , occurring with a probability pX , and no loss with probability 1 -pX . We exclude the values p = 0, p = 1 and m = 0 in the family A0, in order to obtain non degenerate measures.

This approach of approximating a distribution by a discrete one corresponds to the well-known problem of optimal quantization, see [START_REF] Graf | Foundations of quantization for probability distributions[END_REF]. The latter is itself related to k-means clustering, see [START_REF] Pollard | Quantization and the method of k-means[END_REF]. Here, the main difference is that one location of the discrete distribution is constrained to be 0. This point mass at zero encodes the absence of loss and thus the point mass pX at mX > 0 encodes loss. P Y ∈ A0 is characterized by the two parameters (m, p), and quantization to A0 gives the minimal way to represent the magnitude and propensity effect borne by X.

Theoretical analysis

The theoretical study of the optimal (mX , pX ) magnitude-propensity pair in (11) can be done by two main approaches: either as an optimal mass transportation problem or as an optimal quantization problem. The more refined results are obtained by the latter approach. However, we first make use of the former to quickly obtain a characterisation of the optimal magnitude-propensity pair.

3.1 Computation of (m X , p X ) by the optimal transportation approach

A direct optimization approach can be used to characterize the (mX , pX ) by using the explicit form of the Wasserstein metric in dimension one: denote by QX the quantile function of P X , QX (t) := inf{x : FX (x) ≥ t}, 0 < t < 1. It is then well-known (see e.g. [START_REF] Rachev | Mass transportation problems[END_REF]) that

W 2 2 (P X , P Y ) = 1 0 (QX (t) -QY (t)) 2 dt, (12)
for univariate P X , P Y with finite variance. For Y ∈ A0, its quantile function writes QY (t) = m1 1-p<t≤1 . Plugging the later in ( 12) and optimizing gives the following characterization result.

Characterization by direct optimisation

In the remainder of the paper, we simplify notations and will denote simply by F , Q, f the c.d.f., quantile function and density (if it exists) of X. 11), then it satisfies the following system of equations,

Proposition 3.1. Let X s.t. E[X 2 ] < ∞. i) Necessary conditions: if (mX , pX ) is a local minimum of (
mX = 2Q(1 -pX ), ( 13
) mX = 1 1-p X Q(t)dt pX . ( 14
)
ii) Sufficiency condition: if P X is absolutely continuous with continuous density f , then (mX , pX

) is optimal if f (Q(1 -pX )) > 0 and 2pX f (Q(1 -pX )) -Q(1 -pX ) > 0. (15)
Proof.

i) The squared Wasserstein distance between P X and P Y ∈ A0 writes

W 2 2 (P X , P Y ) = 1-p 0 (Q(t)) 2 dt + 1 1-p (Q(t) -m) 2 dt = E[X 2 ] + m 2 p -2m 1 1-p Q(t)dt := ψ(m, p)
ψ is differentiable and any optimal magnitude-propensity (mX , pX ) solving (11) must satisfy the first order conditions

∂ψ(m X ,p X ) ∂m = 0 ∂ψ(m X ,p X ) ∂p = 0 ⇔ -2 1 1-p X Q(t) + 2mX pX = 0 -2mX Q(1 -pX ) + m 2 X = 0.
For mX = 0 and pX = 0, one gets ( 13) and ( 14).

ii

) if P X has density f such that f (Q(p)) > 0, then Q is differentiable with deriva- tive the quantile-density qX (p) = Q(p) = 1 f (Q(p))
. Then ψ is twice differentiable with Hessian matrix

∂ 2 ψ ∂m 2 ∂ 2 ψ ∂m∂p ∂ 2 ψ ∂m∂p ∂ 2 ψ ∂p 2 = a b b c with a = 2p, b = 2m -Q(1 -p), c = 2m f (Q(1-p))
. The Hessian is positive-definite at the critical point (mX , pX ) if a > 0 and ac -b 2 > 0. The latter condition writes

ac -b 2 = 4pX mX f (Q(1 -pX )) -4 (m -Q(1 -pX )) 2 > 0
With (mX , pX ) satisfying ( 13), ( 14), the condition writes

4Q(1 -pX ) 2pX f (Q(1 -pX )) -Q(1 -pX ) > 0, which is (15). Remark 3. 1. If 1 -pX is in the range of F (in particular if F is continuous), then, (13) writes F (mX /2) = 1 -pX ,
and by Exercise 3.3 in [START_REF] Shorack | Probability for statisticians[END_REF] p. 113, ( 14) writes as

mX = E[X1 X>m X /2 ] 1 -F (mX /2) = E[X|X > mX /2].
This equation will be derived in the general case from the optimal quantization viewpoint, see Theorem 3.5 and Remark 4 below.

2. Setting a := Q(1 -pX ), the sufficiency condition (15) writes also

af (a) < 2(1 -F (a)).
In view of the fact that E

[X] = ∞ 0 tf (t) = ∞ 0 (1 -F (t)
)dt < ∞, the latter condition has to be understood as a condition on the tail decrease of the density. It can also be expressed as a condition on the failure rate (or hazard function), as

h(a) := f (a) 1 -F (a) < 2 a .
A general existence result will be given in Theorem 3.5, by quantization methods.

Examples

We illustrate the formulas obtained in Proposition 3.1 on the following examples:

Example 1 (Uniform distribution). For X ∼ U [0,a] , i.e. Q(p) = ap, (13) and ( 14) give pX = 2/3 and mX = 2a/3.

Example 2 (Exponential distribution). For X ∼ Exp(λ), with λ > 0, the memorylessness property yields that E[X|X > a] = a + λ -1 . Hence, the optimal threshold is aX = λ -1 = E[X] and one has pX = e -2 ≈ 0.135 and mX

= 2 λ = 2 E[X].
It is noteworthy that, in both examples, pX is fixed and does not depend on the parameter (a, resp. λ) of the distribution.

Example 3 (Pareto distribution). One considers the following one-parameter (version) of the Pareto Distribution, X ∼ P a(θ), defined by P (X > x) = (1 + x) -θ , for x ≥ 0. We assume that θ > 2, so that E[X 2 ] < ∞. One has that

E[X|X > a] = θ θ -1 (1 + a) -1.
Using the characterisation ( 24), one finds that the optimal threshold is a = 1 θ-2 . Hence, (25) gives

mX = 2 θ -2 , pX = θ -2 θ -1 θ .
For some other distributions, one may not have a closed form expression.

A review of Optimal Quantization

Additional insight is gained by viewing the basic approach (11) as a constrained optimal quantization problem. We first summarize the basic facts and terminology of (unconstrained) optimal quantization theory. The following can be regarded as a quick introduction to the field. Optimal quantization originates from the engineering and signal processing literature (see [START_REF] Lloyd | Least squares quantization in PCM[END_REF], [START_REF] Gersho | Vector quantization and signal compression Boston[END_REF]). It aims at optimally discretizing a continuous (stationary) signal in view of its transmission. It was developed originally for analog-to-digital conversion, compression, pattern recognition.

Following [START_REF] Gersho | Vector quantization and signal compression Boston[END_REF] and [START_REF] Graf | Foundations of quantization for probability distributions[END_REF], a N -vector quantizer on (R d , ||.||) is a mapping T : R d → {x1, . . . , xN }, where {x1, . . . , xN } is a codebook of size N . Thus, associated with T is a partition {Ai}, with Ai = {x ∈ R d : T (x) = xi}, of the input space so that a quantizer writes

T (x) = N 1=1 xi1A i (x), x ∈ R d
and is determined by the pairs {(xi, Ai), i = 1, . . . n}. An N -optimal quantizer for a distribution P X is a N -quantizer which minimises the mean squared error (or distortion) D(T ;

P X ) := inf T E(X -T (X)) 2 .
Equivalently, it can be shown [START_REF] Pollard | Quantization and the method of k-means[END_REF], [START_REF] Graf | Foundations of quantization for probability distributions[END_REF]) that an optimal quantizer is a Monge map minimising the Wasserstein metric W2(P X , P Y ) between P X and P Y , where P Y is a discrete measure with N points. It is known, see [START_REF] Gersho | Vector quantization and signal compression Boston[END_REF], that given the codebook, the optimal partition is given by the Voronoi partition,

Ai := {x ∈ R d : ||x -xi|| ≤ ||x -xj||, j = i}.
Conversely, given a partition, the optimal codebook is given by the centers (centroids),

yi = E(X|X ∈ Ai).
These properties are the basis of Lloyd's iterative algorithm. This also explains why optimal quantizers are sometimes defined directly by their codebook and corresponding Voronoi partition, see e.g. [START_REF] Pagès | Numerical probability. Universitext. An introduction with applications to finance[END_REF]. As a consequence, the distortion writes as a sole function of the centers, as

D(T ; P X ) = DN (x1, . . . , xN ) := E min x i ||X -xi|| 2 (16)
If the support of P X has at least N elements, existence of an N -optimal quantizer follows from the fact that (x1, . . . , xN ) → DN (x1, . . . , xN ) is 1-Lipshitz and a nontrivial compacity argument, see e.g. Lemma 8 in [START_REF] Pollard | Quantization and the method of k-means[END_REF], Theorem 5.1 in [START_REF] Pagès | Numerical probability. Universitext. An introduction with applications to finance[END_REF] or Theorem 4.12 in [START_REF] Graf | Foundations of quantization for probability distributions[END_REF]. In dimension one, a known sufficient condition for uniqueness of the N -optimal quantizer is that P X be absolutely continuous with a log-concave density f . By Proposition 6.6 in [START_REF] Pagès | Numerical probability. Universitext. An introduction with applications to finance[END_REF], if x N := (x1, . . . , xN ) has pairwise distinct component and P X (∪ 1≤i≤N ∂Ai) = 0, then

DN is continuously differentiable with gradient ∇DN = ∂D N ∂x i with ∂DN ∂xi (x N ) = E[2(xi -X)1X∈A i ]. (17)
These calculations are the basis for a stochastic gradient descent based algorithm known as Competitive Learning Vector Quantization, see [START_REF] Pagès | Numerical probability. Universitext. An introduction with applications to finance[END_REF]. In the onedimensional case, explicit expressions of the gradient can be written in terms of the c.d.f. F and of the cumulative first moment function

K(x) := E[X1 X≤x ] as ∂DN ∂xi (x N ) = 2xi F (x i+ 1 2 ) -F (x i-1 2 ) -2 K(x i+ 1 2 ) -K(x i-1 2 ) ,
where

x i+ 1 2 = x i+1 +x i 2
corresponds to the boundaries of the Voronoi cells. Additional formulas for the Hessian are available (see [START_REF] Pagès | Numerical probability. Universitext. An introduction with applications to finance[END_REF] p. 155). These formulas allow for a Newton-Raphson zero search procedure.

Existence and characterization of (m X , p X ) by constrained optimal quantization

We can now tackle the study of (mX , pX ) from the optimal quantization viewpoint. One thus introduces the constrained two-points quantizer with centers {x0, x1} := {0, m}, i.e. as a mapping T : [0, ∞) → {0, m} with T (x) = m1 x≥a , where a ≥ 0 is a threshold to determinate. Then, the optimal quantization problem with constrained knot at zero writes inf

(a,m)∈[0,∞) 2 E[(X -T (X)) 2 ].
By the results of the previous section (see ( 16)), one already knows that a = m/2, i.e. that the Voronoi regions write A0 = {x : 0 ≤ x ≤ m/2} and A1 = {x : x ≥ m/2}. As a consequence, and in view of ( 16) the distortion/objective function writes as a sole function of the magnitude m, as

L(m) := E[X 2 ∧ (X -m) 2 ]. (18)
We first study the differentiability properties of the objective function (18) in the next Section.

B-differentiability properties of the objective function

For the general N -points quantization problem, the square root of the distortion function is 1-Lipshitz, see Pagès (2018) p. 136. Hence, it has a derivative a.e. by Rademacher's Theorem. Therefore, L is differentiable a.e. In addition, since the integrand H : x → X 2 ∧ (X -x) 2 is piecewise differentiable, it is easy to show that it has directional derivatives everywhere. A convenient tool in the setting of optimisation of piecewise smooth functions is the concept of Bouligand derivative (B-derivative), see [START_REF] Robinson | Local structure of feasible sets in nonlinear programming. III. Stability and sensitivity[END_REF] and [START_REF] Scholtes | Introduction to piecewise differentiable equations[END_REF]. It drops the requirement of linearity of the differential, represents a first-order approximation and allows to have a single-valued notion of differential. We give below a simplified definition for functions of one variable.

Definition 3.2. A function f : R → R is Bouligand differentiable (B-differentiable) at x0 if there exists a positive homogeneous function ∇ B f (x0) : R → R s.t. f (x0 + v) = f (x0) + ∇ B f (x0)(v) + o(v), ∀v ∈ R. ( 19
)
For the study of the B-differentiability of the objective function L, we first give a lemma on the differentiability of its integrand.

Lemma 3.3 (B-derivative of a min function). Let H(x) = min(h1(x), h2(x)), where h1, h2 : R → R are differentiable functions. Then H is B-differentiable, with Bderivative given by i

) if x is s.t. h1(x) < h2(x), then ∇ B H(x)(v) = h 1 (x)v, and conversely, if h1(x) > h2(x), then ∇ B H(x)(v) = h 2 (x)v. ii) if x is s.t. H(x) = h1(x) = h2(x), then ∇ B H(x)(v) = min(h 1 (x)v, h 2 (x)v).
Proof. i) Assume w.l.o.g. that h1(x) < h2(x), so that H(x) = h1(x). Let us show that H(x) = h1(x) remains true on a neighborhood of x. Set d = h2(x)-h1(x) > 0. Let 0 < ε1 < d, 0 < ε2 < d -ε1. By continuity of h1 and h2 at x, there exists δ > 0 s.t. ∀|v| < δ,

h1(x + v) ≤ h1(x) + ε1 (20) h2(x + v) ≥ h2(x) -ε2 (21)
h1(x) + ε1 = h2(x) + ε1 -d, therefore (20) and (21) give

h1(x + v) ≤ h2(x) + ε1 -d < h2(x) -ε2 ≤ h2(x + v).
Hence, H(x + v) = h1(x + v) for all |v| < δ. Thus, the Bouligand derivative of H at x reduces to the classical derivative h 1 (x) of h1.

ii) One has

h1(x + v) = h1(x) + h 1 (x)v + o(v) h2(x + v) = h2(x) + h 2 (x)v + o(v) (22) Since H(x) = h1(x) = h2(x), H(x + v) -H(x) = min(h1(x + v) -h1(x), h2(x + v) -h2(x)).
We then use the inequality

| min(a, b) -min(c, d)| ≤ max(|a -c|, |b -d|), applied to a = h1(x + v) -h1(x), b = h2(x + v) -h2(x), c = h 1 (x)v, d = h 2 (x)v, to deduce that |H(x+v)-H(x)-min(h 1 (x)v, h 2 (x)v)| ≤ max(|o(v)|, |o(v)|) = |v| max(|o(1)|, |o(1)|),
viz. H is B-differentiable at x, with B-derivative

∇ B H(x)(v) = min(h 1 (x)v, h 2 (x)v),
as the latter expression is positively homogeneous.

Corollary 3.4 (B-Differentiability of the objective function). L is B-differentiable on x ≥ 0, with B-derivative given by

∇ B L(0)(v) = -2 E[X]v1v>0,
and, for x > 0,

∇ B L(x)(v) = 2v E[(x -X)1 X>x/2 ] + P (X = x/2) min(0, xv). ( 23
)
Proof. Applying Lemma 3.3 to the integrand function H(x) = X 2 ∧ (X -x) 2 , i.e. with h1(x) = X, h2(x) = (X -x) 2 , gives the B-derivative of the integrand function: almost surely, for v ∈ R,

∇ B H(x)(v) =                0 x < 0 min(0, -2Xv) x = 0 2(x -X)v 0 < x < 2X min(0, 2Xv) x = 2X 0 x > 2X
In addition, the second order terms in the Taylor expansions ( 22) of h1 and h2 (which are exact) do not depend on X, so that reasoning as in 3.3, it is easy to see that one gets a remainder term in ( 19) for H which does not depend on X, with at most linear growth:

H(x + v) = H(x) + ∇ B H(x)(v) + v (v), a.s., with | (v)| ≤ |v|. Therefore, if one sets ∇ B L(x)(v) := E[∇ B H(x)(v)],
then, by linearity,

|L(x + v) -L(x) -∇ B L(x)(v)| |v| ≤ |v| → 0, as |v| → 0. By integration, ∇ B L(0)(v) = -2 E[X]v v > 0 0 v ≤ 0 and for x > 0, ∇ B L(x)(v) = (2(x -t)10<x<2t + min(0, 2tv)1x=2t) P X (dt) = E[2(x -X)1 X>x/2 ]v + P (X = x/2) min(0, xv). v → ∇ B L(x)(v) is positively homogeneous, for x ≥ 0. Hence, L is B-differentiable.

Main existence and characterization result

With these tools, we can now give the main existence and characterization result of the optimal magnitude-propensity pair.

Theorem 3.5. i) If E[X 2
] < ∞ and the support of P X contains at least two points, then there exists a magnitude-propensity pair (pX , mX ) minimizing (11).

ii) An optimal magnitude-propensity pair (pX , mX ) is characterized by solving for a the equation

2a = E[X|X > a], a > 0, (24)
and then setting mX = 2a, pX = P (X > a).

(25)

In addition, P X ({a}) = 0.

Proof. i) By the results of the Section 3.2, the objective function L of ( 18) is locally Lipshitz hence continuous and its lower level sets are compact, see e.g. Lemma 8 in [START_REF] Pollard | Quantization and the method of k-means[END_REF]. Hence, the general existence result follows from Weierstrass theorem. See also Exercise 3 p. 139 in [START_REF] Pagès | Numerical probability. Universitext. An introduction with applications to finance[END_REF].

ii) Since ∇ B L(.)(.) gives a first order approximation of L, a necessary condition for mX to be a minimizer is that

∇ B L(mX )(v) ≥ 0, ∀v ∈ R. For mX = 0, ∇ B L(0)(v) = -2 E[X]
< 0 for v > 0, therefore 0 cannot be a critical point. For mX > 0 and v > 0, the first order condition writes

E[(mX -X)1 X>m X /2 ] = 0. (26)
For mX > 0, v < 0, it reduces to

2 E[(mX -X)1 X>m X /2 ] + mX P (X = mX /2) = 0. (27)
Plugging ( 26) into (27) implies that P (X = mX /2) = 0 and yields (24), upon reparametrizing by a = mX /2.

Remark 4.

i) We hopefully obtain the same formula between (24) of Theorem 3.5 and ( 13), ( 14) of Proposition 3.1. The quantization approach allows to obtain a general existence result.

ii) P X does not charge the optimal threshold a = mX /2, even if P X is not absolutely continuous. By ( 23), this imply that ∇ B L(mX )(v) is linear in v, that is to say, L is differentiable (in the classical sense) at the optimal mX , with

L (mX ) = 2 E[(mX -X)1 X>m X /2 ] = 2 E[(mX -X)1 X≥m X /2 ].

Uniqueness

The matter of uniqueness is notoriously a difficult topic in optimal quantization, see [START_REF] Graf | Foundations of quantization for probability distributions[END_REF], [START_REF] Pagès | Numerical probability. Universitext. An introduction with applications to finance[END_REF]. We provide below in Theorem 3.6 a set of sufficiency conditions for the uniqueness of the magnitude-propensity pair. The main condition is the log-concavity of x → x 3 f (x), which is fulfilled when the density f is itself log-concave. It is a condition similar to the classical condition of [START_REF] Trushkin | Sufficient conditions for uniqueness of a locally optimal quantizer for a class of convex error weighting functions[END_REF] and [START_REF] Kieffer | Uniqueness of locally optimal quantizer for log-concave density and convex error weighting function[END_REF] for the uniqueness in (unconstrained) optimal quantization.

Theorem 3.6. Let P X be an absolutely continuous distribution on R+, with density f , 0 ∈ supp(P X ) and

E[X 2 ] < ∞. Assume i) f : x → x 3 f (x) is strictly log-concave on R+; ii) limx→0 xf (x) = 0.
Then, the optimal magnitude-propensity pair is unique.

Proof. Prior to the proof, let us make the following elementary remarks. Being logconcave, the function f : x → x 3 f (x) has a limit in [0, ∞) as x → 0. Thus, we may set f (0) = limx→0 x 3 f (x) = 0, by assumption ii). Consequently, I := {x > 0 : f (x) > 0} is an interval of R+ of the form (0, b] or (0, b), with b ∈ (0, ∞], since 0 ∈ supp(P X ), and f (x) = f (x)/x 3 is continuous on (0, b). Moreover, still by log-concavity, f has a left-handed limit f (b-) at b. Finally, f is continuous on (0, b), with left-handed limit f (b-) ≥ 0.

The optimal magnitude mX is s.t. y * = mX /2 is a zero of the function

Φ(y) = 2yF (y) -K(y), (28)
Φ (0) = 2F (0) -limy→0 yf (y) = 2 > 0. Case ia) if b = ∞, then limy→∞ Φ (y) = -limy→∞ yf (y) = 0 since E[X] < ∞.
Thus Φ > 0 and Φ is increasing on (0, ∞). But since Φ(0) = -E[X] < 0 and limy→∞ Φ(y) = 0, Φ cannot have a zero in (0, ∞), which contradicts the general existence Theorem 3.5. Thus, necessarily, b < ∞.

Case ib) if b < ∞, then lim y→b Φ (y) = -bf (b-) ≤ 0. If f (b-) = 0, then Φ > 0 on (0, b), so that Φ is increasing on [0, b) from -E[X] up to 0,
* = mX /2 ∈ (0, b) s.t. Φ(y * ) = 0. Yet, limy→∞ Φ(y) = 0 since E[X] < ∞,
which is a contradiction. Therefore, Φ (y0) < 0. With the fact that Φ (0) = 2 > 0 and Φ is decreasing on (0, y0), this implies that there exists a unique y1 ∈ (0, y0) s.t. Φ (y1) = 0. On (y0, b), Φ cannot have any zeros, since Φ is increasing, Φ (y0) < 0 and either

Φ (b-) = -bf (b-) < 0 if b < ∞, or limy→∞ Φ (y) = 0 if b = ∞, since E[X] < ∞.
So, in both cases, Φ is increasing on (0, y1) and decreasing on (y1, b). Moreover Φ(0) = -E[X] < 0 and lim y→b Φ(y) = 0, and the general existence Theorem 3.5 ensures that Φ has at least a zero in (0, b/2). Hence, necessarily Φ(y1) ≥ 0 and Φ has at most one zero, located in (0, y1).

In particular, if f is itself log-concave, then the conditions of Theorem 3.6 are fulfilled, as shown in the next corollary.

Corollary 3.7. If f is a log-concave density on (0, b), then the assumptions i) ii) of Theorem 3.6 hold.

Proof.

i) x → 3/x is decreasing and f r /f is non-increasing, therefore ζ is decreasing, i.e. f : x → x 3 f (x) is strictly log-concave.

ii) Let y0 > 0 s.t. f (y0) > 0. For 0 < y < y0 < b, one has, by log-concavity, that

ln f y + y0 2 ≥ 1 2 ln f (y) + 1 2 ln f (y0) ⇐⇒ ln f (y) ≤ 2 ln f y + y0 2 -ln f (y0) ⇐⇒ 0 ≤ f (y) ≤ f 2 y + y0 2 /f (y0) ⇐⇒ 0 ≤ yf (y) ≤ yf 2 y + y0 2 /f (y0),
which implies limy→0 yf (y) = 0 by continuity. (Note that the statement is trivial if limy→0 f (y) < ∞.)

Remark 5. Examples of distributions satisfying the hypotheses of Theorem 3.6, but not those of Corollary 3.7 (in particular, whose density is not log-concave) are the power distributions

f (x) = (a + 1)x a 1 (0,1) (x), -1 < a < 0,
and the Gamma distributions

f (x) = x a-1 Γ(a) e -x 1x>0, 0 < a < 1.
Remark 6. If, in addition to being log-concave, the density is non-increasing, a simpler proof of Theorem 3.6 can be given as follows: Let X be the symmetrized version of X, with density f (x) = 1 2 f (|x|). Let λ ∈ [0, 1], x, y ∈ R. Since ln f is non-increasing and concave,

ln f (|λx + (1 -λ)y|) ≥ ln f (λ|x| + (1 -λ)|y|) ≥ λ ln f (|x|) + (1 -λ)f (|y|),
viz. f is log-concave. Therefore, by the existence and uniqueness results of [START_REF] Trushkin | Sufficient conditions for uniqueness of a locally optimal quantizer for a class of convex error weighting functions[END_REF], [START_REF] Kieffer | Uniqueness of locally optimal quantizer for log-concave density and convex error weighting function[END_REF], there exists a unique three-points stationary quantizer of X.

The optimal magnitude mX is obtained by deriving the optimal two-points quantizer of X with a point constrained to be zero. mX must verify the stationary condition

E[(mX -X)1 X≥m X /2 ] = 0. ( 32
)
On the other hand, a three-points (unconstrained) quantizer (x1, x2, x3) for X must satisfy the stationary condition E[(x-X)1A i ( X] = 0. Let us show that that (-mX , 0, mX ) is a three-points stationary quantizer for X. Indeed, for x1 = -mX , the stationary condition writes (-mX -t)1 t≤m X /2 f (t)dt = 0, which gives (32) by the change of variable x = -t and the symmetry of f . For x3 = mX , the stationary condition is also (32) since mX > 0. For x2 = 0, it writes

m X /2
-m X /2 x f (x)dx = 0, which is automatically satisfied since f is even. Therefore, (-mX , 0, mX ) is the unique three points stationary quantizer for X. Thus mX is unique.

Discussion and properties

The optimal magnitude-propensity pair (mX , pX ) obtained by ( 25) or ( 13), ( 14) has several interesting characteristics. First, from (13), resp. ( 14), it is clear that the magnitude mX can be interpreted either as (twice) a Value-at-Risk V aRα(X), resp. as a Conditional Value-at-Risk CV aRα(X), for a special value of α: mX = CV aRp X (X) = 2V aR1-p X (X).

(33)

It is thus comforting that one obtains, for the magnitude effect, a quantity akin to the classical and well-used univariate risk measures. In addition, such an interpretation gives an answer to the problem mentioned in Remark 2 about the "right" choice of α for parametrized risk measures like V aRα(X) or CV aRα(X): the corresponding α is determined by pX , hence by the distribution P X . It is no longer a subjective choice dependent on the user. Second, the magnitude-propensity paradigm mX , pX , by quantifying risk on a bivariate scale, is fundamentally different from the traditional approach to measuring risk by a univariate coherent risk measure ρ. Hence, it may not make sense to look for properties similar to those of coherent risk measures. However, one has the following noteworthy properties for the magnitude mX . Proposition 3.8 (positive homogeneity/scaling of the magnitude). For a > 0, maX = amX , and paX = pX Proof. From the expression (16) of the distortion as a function of the centers,

W 2 2 (P aX , P Y ) = E[min{aX 2 , (aX -m) 2 }] = a 2 E[min{X 2 , (X -m/a) 2 }].
Therefore m aX a = mX , and paX = pX .

Recall the definition of the convex order: X ≤cx Y if E φ(X) ≤ E φ(Y ), for all integrable, convex functions φ, see e.g. [START_REF] Rüschendorf | Mathematical Risk Analysis[END_REF]. One has: Proposition 3.9 (Monotonicity of the magnitude). If X ≤cx Y , then mX ≤ mY .

Proof. By e.g. Proposition 1 in [START_REF] Puccetti | Sharp bounds on the expected shortfall for a sum of dependent random variables[END_REF]

, X ≤cx Y if and only if E[X|X > a] ≤ E[Y |Y > a], for all a ≥ 0. Hence, the curve a → E[X|X > a] is below the curve a → E[Y |Y > a]
. Thus, the optimal threshold aX for X, satisfying (24), is lower than the corresponding optimal threshold aY for Y . Therefore, mX ≤ mY .

Eventually, the magnitude effect will be larger than the average of X, which seems a desirable property from the point of view of the classical premium calculation principles [START_REF] Bühlmann | Grundlehren der Mathematischen Wissenschaften[END_REF]:

Proposition 3.10. If P X is absolutely continuous, then mX ≥ EX. Proof. Let F (x) = P (X ≥ x), K(x) = E[X1 X≥x ] = (x,∞) tf (t)dt and β(x) = K(x)/F (x). Then, β (x) = -xf (x)F (x) + f (x)K(x) F 2 (x) = f (x) F (x) K(x) F (x) -x ≥ 0 as K(x) ≥ xF (x). Therefore, β is non-decreasing, hence β(x) ≥ β(0) = E[X].
Therefore, the solution mX of the equation

x = β(x/2) satisfy mX ≥ E[X].
4 Magnitude-propensity risk comparison: numerical illustrations and empirical aspects

Risk comparison with magnitude-propensity plots

A fundamental objective of risk analysis is the comparison of risks. The proposed approach allows to compare risks on both the magnitude and propensity scales, by displaying a point of coordinates (mX , pX ) on magnitude and propensity axes. Figure 2 illustrates the resulting magnitude-propensity plot, for the uniform, exponential and Pareto distributions of Examples 1-3. Both the uniform and exponential distribution have constant propensity along the parameter, and a magnitude linearly increasing with the mean of the distribution. The blue dots show the (mX , pX ) points for a distribution with mean E[X] = 1, 2, 5, and 10: the uniform distribution has a higher propensity, with a lesser magnitude effect. For the Pareto distribution, the red circles show the (mX , pX ) points with parameter value θ = 2.1, 2.5, 5, 10: the tail effect is reflected in the behavior of the magnitude-propensity pair. For a heavy-tailed distribution (θ close to 2), one has a large magnitude with a very small propensity, while for a short-tailed distribution (θ large), the magnitude effect remain limited but with a larger propensity. This behaviour is in accordance with what could be expected from intuition.

4.2 Distributions with no closed form expressions for (m X , p X )

For some more complicated distributions, one may not have a closed form expression of the (mX , pX ) as in Examples 1-3. However, one can use the computation capabilities of, e.g., Mathematica Wolfram Research, Inc. ( 2020) to obtain a closed-form expression of the E[X|X > a] and then find numerically the root of (24). We illustrate the procedure for the Gamma and Weibull distributions, two distributions frequently encountered in insurance mathematics.

Example 4 (Gamma distribution). For X ∼ Γ(α, β), its density writes f (x) = x α-1 e -(x/β) . One has

E[X|X > a] = β Γ(1 + α, a/β) Γ(α, a/β)
where Γ is the incomplete Gamma function. We take a family of Γ(α, 2) distribution, with shape parameter α varying from 0.1 to 2.9 by stepsize of 0.2. We solve (24) numerically with initial point E[X]. The resulting magnitude-propensity plot is displayed in Figure 4.

2.9 2.7 2.5 2.3 2.1 1.9 1.7 1.5 1.3 1.1 0.9 0.7 0.5

0.3 0.1 0 1 2 3 4 m X 0.0 0.2 0.4 0.6 0.8 1.0 p X Figure 3: Magnitude-propensity plot for Gamma distribution X ∼ Γ(α, 2), for α varying from 0.1 to 2.9. The corresponding mean E[X] is indicated near the circled (m X , p X ) points.
The coordinates of the circled points indicate the magnitude-propensity (mX , pX ) values and are labeled with the expectation E[X]. Here, it is interesting that both mX and pX are increasing with the mean parameter, contrary to the Pareto case of Figure 2. This corresponds to intuition: let us recall that as the shape parameter α increase, the density f (x) is shifted to the right and shrinks toward zero for small x values, see Figure 4. Hence, both the magnitude and propensity coding the risk borne by X will increase. Example 5 (Weibull distribution). For X ∼ W (α, β), its density writes f (x) = x α-1 e -(x/β) α . One has

E[X|X > a] = βe a α β -α Γ(1 + 1/α, a α β -α )
where Γ is the incomplete Gamma function. The effect of shifting the parameters is illustrated in Figure 5 below.

Empirical computations and illustrations

The characterization (24) shows that the optimal threshold is a fixed point of the function

a → E[X|X > a] 2 . ( 34
)
For empirical data, i.e. when the distribution of X is unknown but one has a sample of realizations of X, one can replace the unknown expectation in (34) by a sample estimator. For the empirical measure, one obtains an estimate of (mX , pX ) as a fixed point of

a → n i=1 Xi1X i >a 2 n i=1 1X i >a
.

A Mathematica code is given in the supplementary appendix. The latter would correspond to Lloyd's algorithm in the unconstrained quantization problem. See [START_REF] Pagès | Numerical probability. Universitext. An introduction with applications to finance[END_REF] for discussion of numerical optimal quantization. Alternatively, one could seek directly for a (global) minimizer of the objective function L of (18), using a generic minimizer program (e.g. the command "NMinimize" in Wolfram Research, Inc. ( 2020)). These approaches are illustrated on a synthetic and a real dataset. For the synthetic data set, we simulated a sample of 100 i.i.d. Uniform on [0, 1] random variables. From Example 1, we known that we should obtain (mX , pX ) = (2/3, 2/3). LLoyd's algorithm on the sample gives (mX , pX ) ≈ (0.67, 0.65). We standardized the original data by dividing each entry by 10 6 . There is one clear outlier at 15500 (the famed "Andrew" hurricane). It is noteworthy that Lloyd's algorithm on the full dataset give (mX , pX ) ≈ (15500, 0.0333), i.e. the outlier value as magnitude, with propensity one over the number of sample points (1/30): the outlier has a dwarfing effect on the whole data set and this is reflected on the magnitude-propensity measure. If the outlier value is removed, one gets (mX , pX ) ≈ (2401.67, 0.206897), which better reflects the magnitude and propensity effects of the data set.

Perspective: towards general risk quantization

In order to make the presentation clear, the discussion so far has been directed towards i) univariate risks X ∈ [0, ∞[;

ii) using the simplest way to quantify risk on the magnitude and propensity scale, i.e. on the basic idea (11) with a constrained two-points quantizer {0, mX }.

As a constrained optimization problem, the proposed approach naturally suggests to consider several variants and extensions. To stimulate further research on the topic, we conclude the paper with a brief sketch below of some of these possible variants and extensions.

Variant: three-points quantification

One can refine the quantized proxy by looking for a quantization to a discrete distribution with more than two points. For example, one can refine one's measure of risk into a classification between a "moderate" and a "large" risk (or, say, "low" risk and "tail" risk), by using a three points discrete measure, P Y = (1 -p1 -p2)δ0 + p1δm 1 + p2δm 2 , with m1 < m2. With a such three points discrete measure, one can encode and quantify in the magnitude and propensity scale both moderate, resp. large risk, with (m1, p1), resp. (m2, p2). Such a simultaneous quantization of the "mild" and "extreme" magnitude-propensity parts of an insurance risk X could reveal to be very valuable in applications. For example, in Reinsurance (see e.g. [START_REF] Albrecher | Reinsurance: actuarial and statistical aspects[END_REF]), the insurer and reinsurer have to agree on an optimal risk sharing policy, with the insurer usually ceding to the reinsurer the "extreme" part of the risk (the one with high magnitude and low propensity), while keeping and managing the "mild" part. The proposed 2-levels risk quantization approach seems to be particularly well-suited for identifying and quantifying these parts. Also, for capital adequacy, the "tail" magnitude-propensity risk measure (m2, p2) could serve as a basis from which one can compute a sufficient safety margin against adverse extreme risks.

Eventually, let us remark that one can of course increase the number of points in the quantization proxy to more than three points. From an approximation viewpoint, the more points, the better is the quantization of the distribution. However, the more points one uses, the more complicated the framework becomes, as the number of magnitudes and propensities measures increase, rendering the task of comparing and summarizing risks less straightforward. Hence, there is a balance to reach between simplicity of the framework and accuracy of approximation.

Extension to real-valued risk

So far, we have restricted on non-negative risk, mainly for clarity and simplicity of presentation. There are several variants which can be proposed to accommodate for a real-valued risk X ∈ R (so that, in the insurance convention where X stands for a loss, X < 0 stands for a gain).

1. First, one can apply the previous framework to the conditional distribution of the positive part X|X ≥ 0 or to X + = max(X, 0). This makes sense if the gains do not offset the losses in the application considered, so that one can restrict the measure of risk to the losses only.

2. Second, one can look for three-points risk quantization to account for the negative part of the distribution. For example, one can apply the framework of ( 11) with

P Y (.) = p-δm -(.) + (1 -p+ -p-)δ0(.) + p+δm + (.), (35)
with m-≤ 0 ≤ m+. The latter constraint allows for an easy interpretation of (m+, p+), resp. (m-, p-), as the magnitude-propensity measure of the loss, resp. gain, part of the distribution. Another possibility is to impose a quantification point at the mean, instead of zero, i.e. to use as a target distribution

P Y (.) = p-δm -(.) + (1 -p+ -p-)δ E[X] (.) + p+δm + (.), (36)
with m-≤ E[X] ≤ m+. Here, the interpretation is also straightforward: the proxy Y takes a "large" value m+ with likelihood p+, a "small" value m-with likelihood p-, and a "moderate" value corresponding to the mean of X, with the remaining propensity (1 -p+ -p-). Depending on the signs of E[X], m+ and m-, positive, resp. negative, these values interprets as quantification of the magnitudes of the losses, resp. of the gains (in the Insurance convention). Notice that the (36) reduces to (35) when X is replaced by X

-E[X]. In view of the fact that W 2 2 (P X , P Y ) = W 2 2 (P X-E[X] , P Y -E[Y ] ) + (E[X] -E[Y ]) 2 ,
it would also make sense to have a discrete proxy Y which has the same mean as X.

3. A third possibility is to standardize by the mean, by simply subtracting it and then apply a two-points quantization to the centered distribution. More precisely, one could quantize X -E[X] to a two-point distributions, P Y (.) = pδm 1 (.) + (1 -p)δm 2 (.), with m1 < m2. This allows to summarize the distribution of X into a mean effect E[X], which itself decomposes into a "moderate risk" of magnitude m1 +E[X] with propensity p1, and a "large" risk of magnitude m2 + E[X] with propensity p2. This gives a quintuplet of descriptive statistics, which summarizes the characteristics of the distribution and can be thought as an alternative to Tukey's box plot. However, compared to (36), it does not provide a propensity quantification of the mean. Such a variant may thus be deemed more useful as a descriptive statistics tool than a Risk one.

In all these variants, the rationale is to have a discrete proxy which summarizes the distribution of X and has an intuitive interpretation. The question of which variant is more sensible from a risk perspective is somehow partially a subjective issue, and is thus left to the appreciation of the reader for the application at hand. What matters is that risk comparisons between several risks be performed within the same framework.

Multivariate risk

A multivariate generalisation to risk vectors X = (X1, . . . , X d ) ∈ R d is similar in spirit: one solves ( 11), with P Y a discrete two-points distribution P Y = (1 -p)δ0 + pδm. The difference with the univariate case is that the optimal (m, p) now gives one propensity measure p X and d magnitudes measures (mX 1 , . . . , mX d ) =: m X to quantify the multivariate risk X.

Table 1 below illustrates the approach in the case d = 2, for a variety of distributions. The magnitude values were obtained by direct minimization of the objective function on a sample of size N = 5000, for each distribution considered. The corresponding propensities were computed by counting the number of sample points in the Voronoi region.

Distribution

m X1 m X2 p X Independence copula, Uniform marginals 0.562383 0.554198 0.8388 Minimal copula, Uniform marginals 0.50628 0.49372 1 Minimal copula, X 1 ∼ Γ(2, 3), X 2 ∼ U [0,1] 8.45238 0.296782 0.5904 Maximal copula, Uniform marginals 0.661274 0.661274 0.668 Gaussian copula ρ = 0, X 1 ∼ Γ(1.5, 2), X 2 ∼ Γ(0.5, 2) 4.61801 1.2804 0.5348 Gaussian copula ρ = 0.5, X 1 ∼ Γ(1.5, 2), X 2 ∼ Γ(0.5, 2) 4.54941 1.52264 0.5282 Gaussian copula ρ = 0.95, X 1 ∼ Γ(1.5, 2), X 2 ∼ Γ(0.5, 2) 5.11324 2.03133 0.458 Gaussian copula ρ = -0.5, X 1 ∼ Γ(1.5, 2), X 2 ∼ Γ(0.5, 2) 4.59007 0.632589 0.5384 Gaussian copula ρ = -0.95, X 1 ∼ Γ(1.5, 2), X 2 ∼ Γ(0.5, 2) 4.92603 0.165642 0.465 For the first distribution with the independence copula and same uniform marginals, one gets the same magnitude measures mX 1 ≈ mX 2 for both marginals, as expected. Compared to the univariate case of Example 1, one gets a higher propensity and lower magnitudes. With the minimal copula, an interesting phenomenon occurs: the propensity jumps to 1, whereas magnitudes decrease to one-half. As all sample points are on the diagonal line x2 = 1 -x1, all points are closer to the mid-point (1/2, 1/2) than to the origin, thus explaining the observed effect. Note that when the marginals are no longer the same, this unit propensity effect in the minimal copula case no longer occurs. With the maximal copula and same marginals, one obtains exactly the same magnitude propensity values as in the univariate case of Example 1, which is also coherent with intuition since X2 = X1 almost surely, and the vector (X1, X2) thus reduces to a uniform distribution on the unit diagonal.

For a distribution with a Gaussian copula and Γ(1.5, 2), Γ(0.5, 2) marginals, the magnitudes reflects the difference in riskiness of the marginals. They show an increase with an increase in positive correlation. With increasing negative dependence, the magnitudes usually show some decrease, but not always, as evidenced by the increase of mX 1 in the last line of Table 1.

These simulations evidence a complex pattern of the influence of the dependence structure and the marginals on the magnitude-propensity measures in the multivariate case. This calls for further research.

Aggregate risk/ Portfolio optimization

In risk management, a multivariate risk X = (X1, . . . , X d ) often occurs via the aggregate risk S := d i=1 Xi, or more generally via a portfolio risk X β := β T X, with β in the unit simplex {β ∈ R d + : βi = 1}, see e.g. McNeil, Frey, and Embrechts (2005) Chapter 6, and [START_REF] Rockafellar | Conditional value-atrisk for general loss distributions[END_REF], [START_REF] Uryasev | Conditional value-at-risk: optimization approach[END_REF]. This reduces the study of the multivariate risk to the univariate one, so that the framework (11) can be applied without further ado.

We give an application to portfolio optimization. Let X = (X1, X2) be distributed according to a Gaussian copula with unit variances and mild correlation ρ = 0.5, together with Gamma marginals X1 ∼ Γ(1.5, 2), X2 ∼ Γ(0.5, 2). The univariate portfolio risk X β = βX1 +(1-β)X2 now writes as a function of 0 ≤ β ≤ 1. It induces a propensity-magnitude risk measure (mX β , pX β ) for each value of β, as shown in Figure 5.4. The black labels over the circled points indicate the value of β = 0, 0.05, . . . , 1, so that β = 0 corresponds to X β being equal to the second marginal X2 only, while β = 1 corresponds to X β being equal to the first marginal X1 only. X1 has a fatter tail and more skewed distribution than X2, which translates here into larger magnitude and propensity measures for β = 1 than for β = 0, see Figure 5.4.

Figure 8: Magnitude-propensity plot of a portfolio X β = βX 1 + (1 -β)X 2 (Blue circled points), with Gamma marginals, X 1 ∼ Γ(1.5, 2), X 2 ∼ Γ(0.5, 2), and a Gaussian copula for (X 1 , X 2 ) with correlation ρ = 0.5, β = 0, 0.05, . . . , 1 (black labels). Expected risk curve m X β p X β = 1 (orange dotted curve).

The portfolio optimization problem deals with selecting the set of portfolios with acceptable/optimal risk profiles. The magnitude-propensity risk measure (mX β , pX β ) of the portfolio can be used to formulate the portfolio optimization problem in several ways. First, it can be used to define acceptable risk zones. One can look for portfolio with a bound on the magnitude risk, the propensity risk, or a combination thereof. For example, the risk manager may decide that he is willing to accept portfolios whose expected risk, as evaluated by mX β pX β , is lower than some constant. The orange dotted curve on Figure 5.4 show the equation mX β pX β = 1. In this example, portfolios with roughly 0 ≤ β ≤ 0.5 are deemed acceptable. Second, one can also look for the optimal/worst portfolio, in the sense that β minimizes/maximizes the expected risk mX β pX β . In this example, numerical inspection shows that the optimal portfolio is found for β = 0.1, and the worst one for β = 0.90. Here, it is interesting to note that the best, resp. worst, portfolio do not corresponds to β = 0, resp. β = 1. This reflects the diversification effect.

Here, we have briefly sketched some possibilities to illustrate how the proposed framework could be applied, and leave for further research a thorough investigation of the topic.

Covariates

Another extension is to take into account the effect of covariates X on a loss variable Y , by quantizing the risk of the conditional distribution Y |X, (or a linear approximation thereof, as in quantile regression). Note that a related, but distinct, approach occurs in Frequency-Severity models (see e.g. [START_REF] Frees | Regression modeling with actuarial and financial applications[END_REF] Chapter 16): there, the loss distribution has a large proportion of zeros, and the modeling is done in two parts, one for the frequency of zeros, and the other for the severity. Among other, models include the Tobit model [START_REF] Tobin | Estimation of relationships for limited dependent variables[END_REF], with a censored latent variable and the individual risk model, see [START_REF] Frees | Regression modeling with actuarial and financial applications[END_REF].

Conclusion

In this paper, we introduce the magnitude-propensity risk measures (mX , pX ), as a way to simultaneously quantify both the severity and the propensity of a risk X. Introduced as a particular mass transportation problem in the Wasserstein metric W2 of the law of X to a two-points {0, mX } discrete distribution with mass pX at mX , it generalizes M -functionals, in particular the traditional risk measures like VaR and CVaR. The key observation is to view the proposed approach as a constrained optimal quantization problem, with a fixed null center. This allows to parametrize the problem with a single parameter, the optimal threshold determining the Voronoi regions, and to derive it as a solution of a fixed point equation. General existence and characterization results are obtained using B-derivatives, and sufficiency conditions for uniqueness are given. The obtained magnitude mX has interesting interpretations in terms of classical risk measures, as (twice) a VaR or as an CVaR. In addition, it has noteworthy properties, like positive homogeneity, monotonicity w.r.t. convex order and being larger than the mean.

Visualization and comparison of risks can be done on magnitude-propensity plots, which allow for an informative comparison of risks. The effect of tails, shift in the density and outliers is reflected in the (mX , pX ) pair. Empirically, the pair can be estimated e.g. by using a variant of Lloyd's algorithm for optimal quantization, or by direct global minimisation. This novel paradigm of visualizing and comparing risk on the bivariate magnitude-propensity scale offers a broader perspective on risk assessment and evaluation.

Eventually, the variants and extensions considered in Section 5 broadens the perspective and scope of applications of the basic approach. Multilevel quantization can help into distinguishing between "moderate" and "tail risk", which is useful in Reinsurance. Extensions to real-valued risks allows to also quantify the "gain part" of a distribution and broadens the scope of applicability of the magnitude-propensity paradigm. Multivariate generalisation highlight some interesting phenomena, like the complex pattern of the influence of the dependence structure and the marginals on the magnitude-propensity measures in the multivariate case. A portfolio optimization problem can be formulated using the magnitude-propensity framework, and reveals an interesting diversification effect. We hope that these variants and extensions would stimulate further research on this novel paradigm of evaluating and comparing risk.

  left), and a vertical shift of c.d.f.s due to a difference of propensities (Figure 1 right).

Figure 1 :

 1 Figure1: C.d.f.s of X 1 (blue), X 2 (red). Left: case (a), difference of magnitudes (horizontal shift). Right: case (b), differences of propensities (vertical shift). In both cases, X 1 ≺ st X 2 .

  where we have set F (y) = b y f (t)dt and K(y) = b y tf (t)dt. Moreover, since f is log-concave, ln f has right and left-handed derivatives on (0, b), and, by monotonicity, these one-sided derivatives have limits as x ↓ 0 and x ↑ b. Therefore, Φ can be continuously defined on y ∈ [0, b] ∩ R + , as Φ (y) = 2F (y) -yf (y). (29) On (0, b), Φ has right and left-handed derivatives, so let us define Φ r as the righthanded derivative Φ r (y) := -3f (y) -yf r (y) = -yf (y) := 3/y + f r (y)/f (y) (31) with f r denoting the right-handed derivative of f . By assumption, x → x 3 f (x) is strictly log-concave, therefore ζ is decreasing. Let us show that there exists a unique y1 ∈ (0, b) s.t. Φ (y1) = 0. For that purpose, consider the following cases: Case i) Assume that lim y→b ζ(y) = ≥ 0. Then, by strict log-concavity, ζ(y) > ≥ 0 on I, i.e. Φ < 0 and Φ decreasing on [0, b]. By assumption ii),

  which contradicts the existence of y * = mX /2 ∈ [0, b/2) such that Φ(y * ) = 0. Therefore, f (b-) > 0, lim y→b Φ (y) < 0 and there exists a unique y1 ∈ (0, b) s.t. Φ (y1) = 0. Case ii) Assume that lim y→b ζ(y) = < 0. Case iia) If limy→0 ζ(y) = η ≤ 0. Then, since ζ is decreasing, ζ < 0 on (0, b), hence Φ r > 0, so that Φ is increasing on (0, b). But this is clearly impossible, since Φ (0) = 2 and lim y→b Φ (y) = 0, resp. = -bf (b-) ≤ 0, for b = ∞, resp. b < ∞. Thus, necessarily limy→0 ζ(y) = η > 0. Case iib) If limy→0 ζ(y) = η > 0. Then, since ζ is decreasing, there exists a unique y0 ∈ I s.t. Φ r < 0 on (0, y0), and Φ r > 0 on (y0, b). Assume that Φ (y0) ≥ 0. Then, Φ > 0 on (0, b) \ {y0} and Φ is increasing on [0, b]. By the general existence Theorem 3.5, there exists some y

Figure 2 :

 2 Figure 2: Magnitude-propensity plots for the uniform U [0,a] distribution (solid line), Exponential Exp(λ) (dotted) and Pareto P a(θ) (dashed), for varying values of the parameter a, λ, θ.

Figure 4 :

 4 Figure 4: Probability density functions f for Gamma distributions Γ(α, 2), with α = 0.1 (blue), 1 (orange), 3 (green), 7 (red).

Figure 5 :

 5 Figure 5: Magnitude-propensity plot for a Weibull W (α, β) distribution for β = 2 (blue), β = 1.5 (red), β = 3 (green). The magnitude-propensity (m X , p X ) points are labeled by the values of α = 0.5, 1, 2, 4.

Figure 6 :

 6 Figure 6: US Hurricane Losses data (Reported losses) in the U.S. 1949 --1999. Available at https://datarepository.wolframcloud.com/resources/US-Hurricane-Loss.

Figure 5 .

 5 1 illustrates the proposed variant on a simulated dataset of size 1000 sampled from a standard log-normal distribution. The theoretical density function (blue curve) together with the histogram of the data is plotted along with the three quantization points (red circles) of respective magnitudes 0, 2.165, 8.200, with corresponding propensities 0.539, 0.393, 0.068: one has a "moderate" risk (m1, p1) = (2.165, 0.393) of relatively low magnitude and high propensity, and a tail risk (m2, p2) = (8.2, 0.068) of large magnitude but low propensity. This example illustrates the duality of risk exposed in Section 2 and the relevance of the proposed paradigm in distinguishing both phenomena and quantifying them.

Figure 7 :

 7 Figure 7: Three points risk quantization from a sample of size 1000 sampled from a standard log-normal distribution. Density function f X (x) (blue curve). Risk quantization points (m X , p X ) (red circles).

Table 1 :

 1 Multivariate magnitude-propensity risk. N = 5000.

Note that in the latter reference CVaR is called Expected Shortfall, and that other terms abound in the literature such as Tail Value-at-Risk (TVaR), Average Value-at-Risk (AVaR), Expected Tail Loss (ETL), Superquantile, etc.

Acknowledgments

Olivier P. Faugeras acknowledges funding from ANR under grant ANR-17-EURE-0010 (Investissements d'Avenir program). Declaration of interest: none.


Martine Vanhove

The contrastive particle na / ni in Beja (North-Cushitic)

Keywords: Beja, discourse analysis, pragmatics, discourse particle, contrast, semantic opposition, adversative, concessive, mirative, topic

Based on a corpus of semi-spontaneous data, this article is a first description and analysis of the functions of the Beja contrastive discourse particle na / ni within a functional, pragmatic and discourse organisation approach. It is shown that this particle is a marker of semantic opposition, denial of expectation, including adversativity, concessivity and mirativity, and possibly marginally a topic marker.

Introduction

Contrastive particles have given rise to a very large body of research related to information structure, discourse organisation and clause linkage, within different theoretical and methodological approaches (in particular pragmatics, formal semantics, functional typology, corpus linguistics, textual and discourse analysis, and cognitive linguistics). 1 Alexandra Vydrina herself dedicated a part of her 2017 PhD thesis to a corpus-based analysis of information structure and discourse organisation, specifically on topic and focus constructions in Kakabe (Mande, Niger-Congo), including contrastive focus, contrastive assertion and contrastive topic. She further developed her approach in a descriptive and theoretically oriented article [START_REF] Vydrina | Topicality in Sentence Focus utterances[END_REF]. In both works, she showed how useful it is to study information structure within discourse organization strategies, and that instead of analysing isolated utterances, it is preferable to look at the contribution of an utterance and specificities of grammatical marking to the discourse organization as a whole. In this line, the aim of this contribution is to provide a preliminary description and corpus-based analysis of one of the contrastive discourse particles of Beja, namely na / ni.

As defined in Cuenca et al. (2019: 5) "contrast is a negative coherence relation involving the comparison of two discourse segments that express propositional contents or states of affairs presented in an antonymic relation." It is considered as one of the basic relations expressed by discourse markers.

It is also well known that "contrast in discourse structure [is] intertwined with contrast in information structure" (Umbach 2004: 156), and numerous studies have dealt with contrastive focus and contrastive topic.

In addition, the term "contrast" is often used as a cover term for both semantic opposition and denial of expectation, i.e. the adversative function (and one of its subtypes, the concessive function) [START_REF] Malchukov | Towards a semantic typology of adversative and contrast marking[END_REF][START_REF] König | Concessive connectives and concessive sentences[END_REF]. For instance, as mentioned in [START_REF] Malchukov | Towards a semantic typology of adversative and contrast marking[END_REF], following Lakoff (1971: 141), the English particle but expresses semantic opposition in John is short but Bill is long, and denial of expectation in John is short but he is good at basketball. Moreover, " [o]nly the denial-of-expectation but allows a paraphrase employing a concessive conjunction (cf. Although John is short, he is good at basketball)" (Malchukov 2004: 180). While some languages, such as English, or Beja as we will see, use the same conjunction (but in English) for semantic opposition and denial of expectation, other languages, such as Russian, use different conjunctions (see e.g. [START_REF] Lakoff | Ifs, ands and buts about conjunction[END_REF]Malchukov 2004: 179-181).

At the same time, a similarity between adversativity and mirativity has long been recognized because of the unexpectedness feature that they share (see e.g. Podlesskaja 1993: 113 for Japanese;Malchukov 2004: 187-191 for Russian). As explained by Malchukov in his typology of adversative and contrastive connectives,

[i]n the adversative construction the event denoted by the second conjunct is unexpected in view of the first one (it is expected that the two events are not compatible), while in the mirative construction the event referred to in the second conjunct is unexpected as such, without any relation to the event in the first one. (Malchukov 2004: 187) For example, the Japanese conjunction ga is analysed as mirative in (1):

(1)

Yuriko wa sutto tachiagat-ta ga... kiregire ni it-ta

Yuriko TOP immediately stand.up-PAST CONJ indistinct talk-PAST 'Juriko immediately stood up and suddenly (lit. but) started to mumble' [START_REF] Podlesskaja | Slozhnoe predlozhenije v sovremennom japonskom jazyke. Materialy k tipologii polipredikativnosti[END_REF]: 113, cited in Malchukov 2004: 188) The paper is organized as follows. Section 2 presents brief information about the language that is relevant for this article, as well as the data used. Section 3 discusses the use of na / ni as a marker of semantic opposition, Section 4 as an adversative particle, Section 5 as a mirative particle, and Section 6 as a possible topic marker. Section 7 concludes.

General information on Beja and the data

Beja (ISO 639-1, glottolog beja1238), locally named biɖawjeːt, is the sole language of the North-Cushitic branch of the Afroasiatic phylum. It is spoken by approx. 2,000,000 speakers in Eastern Sudan, 2 where it is unwritten, and by approx. 60,000 speakers in Northern Eritrea, where a Latin script has been developed some three decades ago for school education. It seems that there are a few speakers left in Southern Egypt.

Beja has rich and complex inflectional and derivational morphology, both in the nominal and verbal domains. It is partly templatic, i.e. consisting of consonantal roots to which various patterns are applied, for verb inflection and derivation, noun formation, verb-noun derivation, 2

The country where I did my fieldwork.

adjective and plural formation. Beja is also partly agglutinative-fusional, with suffixes and enclitics (the majority), as well as prefixes and proclitics, some of which are portmanteau morphemes, such as the definite article.

Syntactically, Beja is predominantly head-final; the canonical constituent order is (X)(S)(O)V, and dependent clausematrix clause. Within a clause, constituent order is not particularly rigid and may vary for pragmatic reasons, but in complex sentences, the clause order is more rigid. There are two main types of complex sentences: deranked subordination with sequential, simultaneity, general and manner converbs, and balanced subordination 'I could see myself as younger than the man, but because I also felt that I was the strongest, I'll tease it (the leopard), he thought.' (BEJ_MV_NARR_15_leopard_047-050)

The present study is exclusively based on a 10 hour-length corpus of narrative, poetic, and procedural texts, interviews, and an 8.45 min-long dialogue that I recorded in Sudan at Sinkat, Wagar, Erkowit, and Port-Sudan between 2001 and 2011. 4 It was not possible to elicit and test data for the purpose of this article at that time, nor since then. It should also be noted that a strong hierarchy of discourse genres and restrictive rules of politeness mean that Beja speakers are reluctant to be recorded in ordinary conversations for fear of disrespecting the researcher, hence the scarcity of this genre in the corpus. It may have had an impact on the low rate of occurrences of the contrastive particle in my data: only 30 examples were found, eleven with =ni, fourteen with =na, four with na and one with na=. Nine occurrences were found in the 3

The terminology "deranked subordination" and "balanced subordination" was introduced by [START_REF] Cristofaro | Subordination[END_REF] to refer to subordinated clauses with non-finite verb forms, and those with finite verb forms, respectively.

4

The data are partly available online at https://corporan.huma-num.fr/Archives/corpus.php. They are segmented into intonation units and fully annotated. In the transcriptions, / represents a minor prosodic break, // a major prosodic break. sole dialogue, eight in reported speech, and thirteen in narrative parts. In addition to the sociolinguistic explanation, the low number of occurrences of the contrastive particle may also be related to the competition with the use of the adversative Arabic loan lakiːn 'but', which usually expresses a mere semantic opposition between two states of affairs (3), and the simultaneity converb that can express a concessive meaning when following the matrix clause (4). However, the number of tokens does not exceed much that of the contrastive particle (approx. 40 tokens for each construction in my data). Note that as opposed to na / ni, their position is strictly restricted to the juncture between two adjacent clauses. 'I failed with the fortune and the car, but I killed the leopard!' (BEJ_MV_NARR_06_foreigner_74-80)

(4)

allaːji-i=d iː-ʔiʃ adgir-a=b aka-jeː

God-GEN=DIR 1SG-let\AOR can-CVB.MNR=INDF.M.ACC become-CVB.SMLT 'I leave it to God, although I can (face it).' (BEJ_MV_NARR_01_rich_SP2_255)

Semantic opposition

One of the functions of the contrastive particle na / ni is to mark a simple semantic opposition between two States of Affairs (SoA), without any presupposition. This is the case in nine examples of the data: the particle is enclitic or proclitic to the temporal adverb bhoːb 'formerly', enclitic to two pronouns, one noun, one verb, one relative marker, and is twice a free-standing particle. The morphosyntactic configurations are illustrated below.

Example ( 5) is an excerpt from the Pear Story (Chafe 1980), which describes the moment when the boy with the bicycle and the group of three children separate, going in opposite directions.

The contrast is marked by the enclitic =ni on the demonstrative pronoun, and the two SoA compared are syntactically adjacent.

( 'While they (the group of three children) walked back, but while this one (the boy with the bicycle) was leaving, on the road, when they found the fallen hat, one (of them) took it, the one in the middle.' (BEJ_MV_NARR_29_pear2_174-182)

In (6), the contrastive particle is enclitic to a finite verb, and compares the different roles of children in a game, the "bride" and the other players, again in adjacent clauses. 'We used to make this one circle there, we used to make the other circle here, the bride stays in this circle, in the place where we are, on the other hand, the men stay, er, to look after the bride.' (BEJ_MV_NARR_55_tanduuy_028-045)

In the following example the contrastive particle cliticizes to the last element of a relative clause, the finite verb with the clitic relative marker =eːb. The speaker contrasts, in adjacent clauses, two alternative solutions to a problem: calling the police, or agreeing beforehand on a written contract.

( In example [START_REF] Umbach | On the notion of contrast in information structure and discourse structure[END_REF], the speaker compares the gloomy present state of a summer resort with its bustling activity in ancient times. Note that the two contrasted SoA are not adjacent: they are separated by 23 intonation units (from both speakers of the interview). The contrastive particle is enclitic to a temporal adverb.

( 'Now, it is not in this condition anymore. (A long description of the former activities of the summer resort follows). But, formerly, the summer resort used to work well.' (BEJ_MV_INT_01_hotel_SP2_049-071)

Example ( 9) illustrates a case where the broader linguistic context, in which it is amply discussed that an unscrupulous rich man refused to pay poor people who worked for him, knowing that he did not risk to be prosecuted, serves as the first linguistic segment to which the apodosis of the conditional sentence is compared. The contrastive particle occurs as a freestanding particle, in its own intonation unit, after a pause of 686 ms. 5

To answer a question of one of the reviewers, there was no hesitation in the speaker's utterance, according to my consultant.

i-raːkʷ i-ndi //

FUT.SG-be_afraid\INT 3SG.M-say\IPFV .

'… If there were witnesses for that, and our case became really strong, on the other hand, he too, from the beginning, would have been frightened.' (BEJ_MV_CONV_01_rich_SP2_265-280)

In ( 10), one of the contrastive segments is left implicit. It is deductible from the common ground shared by the speakers, namely that there are people who believe in the last judgement and others who do not. The connective of nominal coordination =wa 'and' has an associative value here, as it is used on only one noun. 6 The contrastive particle is enclitic to the connective.

(10) ti=gijaːma=wa=na / w=ʔiːmaːn i-bari-n da i-kati-n=eːk // DEF.F=doomsday=COORD=CONTR . DEF.SG.M=faith 3-have\IPFV-PL man\PL 3-become\IPFV-PL=if .

lʔaː-n-eː=d eː-ʔiʃ-n=eːb hiːs-ani //

after-life-L-GEN.PL=DIR 3-let\IPFV-PL=REL.SG.M think-IPFV.1SG .

'If they are men who have faith on doomsday and (all that goes with it) (as opposed to those who don't), I think they should leave it for the after-life.' (BEJ_MV_CONV_01_rich_SP2_256-260)

Adversative

The contrastive particle can also mark adversativity between two states of affairs, that is a contrast that involves counter-expectation, including presupposition and unexpectedness. It is often of the concessive subtype. There are eleven examples of this sort in the data. The contrastive marker occurs three times as a clitic to a main verb, four times as a clitic to a conditional conjunction attached to a finite verb, twice as a clitic to a manner converb used in predicative function, once to a discourse particle, and once as a free-standing particle.

Example ( 11) is extracted from the same dialogue as example (9). In his question (11a), Speaker 1 proposes that the poor men should force the rich man to pay them for their work, but Speaker 2 (11b) is against it and prefers another solution, which goes against Speaker 1's expectation, expressed in (11a). In Speaker 2's answer, the contrastive particle occurs on a finite verb.

( Plain coordination is marked on each coordinated noun.

In the story from which example ( 12) is extracted, a married woman was kidnapped by another man, by whom she had children. The legitimate husband managed to find her, and after he had asked her about her current condition, her answer shows that she inferred that he wanted her back without her children, the socially acceptable behaviour. The contrastive particle, enclitic to the conditional conjunction, underlines her denial of her husband's expectation, and could be schematized as "if p then normally not q", and paraphrased as "if one has illegitimate children, one should not keep them".

( In example ( 13), as in the two preceding examples, the contrastive particle has a general adversative meaning. In the preceding context of this example, the speaker mentions that there is no well in his village, then provides a long list of the wells and reservoirs found in the surroundings. The speaker ends his discourse by a statement about the absence of thirst. It is first introduced by the discourse particle ibakiː 'anyway', confirming that the numerous water spots prevent from being thirsty. It is immediately followed by the enclitic contrastive particle, which does not go back to the preceding list of wells, but contrasts with the mention of the absence of a well 22 intonation units before, an absence which could have led the population and cattle to lack water in an arid environment. In the desert, the absence of a well normally presupposes thirst for living beings.

( 'I could see myself as younger than the man, however, because I also felt that I was the strongest, I'll tease it (the leopard), he thought.' (BEJ_MV_NARR_15_leopard_047-050)

In ( 15), the predicative manner converb marked with =ni is included in a conditional clause (without an apodosis, a frequent rhetorical strategy), expressing a concessive relation, "although your friends are happy, you are sad". Happiness of others is not supposed to be compatible with one's own sadness.

(

Mirative

As described by Malchukov,

In the adversative construction the event denoted by the second conjunct is unexpected in view of the first one (it is expected that the two events are not compatible), while in the mirative construction the event referred to in the second conjunct is unexpected as such, without any relation to the event in the first one. (Malchukov 2004: 187) Seven examples of this sort, involving an element of surprise and unexpectedness, were found in the Beja data: four with the contrastive particle clitic to a verb, two clitic to a noun, and one clitic to a conditional conjunction.

Example ( 16) occurs in the same context as example (12), and directly precedes it. The woman begins her answer about her current condition by reminding her husband of the unexpected visit of the stranger who took the opportunity to kidnap her, while the husband was looking after his camels. The two contrastive particles are clitic to finite verbs.

( 'Gosh, we want to chat with you. Gosh, you, at Gwinib, lo and behold, there is a man with donkey legs, he is made of legs!' (BEJ_MV_NARR_34_donkeyman_53-58)

) 16
An out-of-the-blue mirative meaning is also the case in example ( 18), where the contrastive particle this time is clitic to a noun. Note that the particle also occurs in a conditional clause as in example ( 15), but here it has a reason meaning and a matrix clause preceding it. The unexpectedness of the SoA comes from the shared knowledge that a fisherman is supposed to catch fish, not donkey corpses or pieces of jar.

( 'He says, they said: God! protect me against the misfortune of today! since sometimes, lo and behold, I fish out the corpse of a donkey, and sometimes I fish out a piece of jar!' In ( 19), also with the contrastive particle on a noun, the unexpectedness comes from the fact that a man, even if very thirsty, is not supposed to drink a whole jerrycan of water, let alone go and get a second one. 'He takes a jerrycan and gets down. Lo and behold, after he finishes the jerrycan, he gets another one!' (BEJ_MV_NARR_42_hunter_bread_028-035)

In the preceding context of example (20), people show their surprise when they see two men asleep in the mosque, instead of the man and woman they were expecting to discover. They ask them why they are there. A misunderstanding follows, as in the context of Islamic rules of hospitality, it is normal for travellers and homeless people to sleep in a mosque. In their answer with the contrastive particle in a reason clause, marked with the conditional clitic, the answer underlines the unexpectedness of the question.

(20) allaːj-i gaw i-kati=eːk=ni / hinin hoːj bʔaː-naː=eːk / God-GEN house 3SG.M-become\IPFV=if=CONTR . 1PL.NOM 3DAT/ABL lie_down-PFV.1PL=if 7 .

'Lo and behold, since it is God's house, if we slept in it (it was our right)!' (BEJ_MV_NARR_84_MountainSon_371-373)

Contrast and/or topic?

There are three instances for which the exact function of the particle is ambiguous. In each of them, the noun to which the particle is attached is a repetition of the same noun in the preceding 7

Like in example (15), the apodosis is left unexpressed.

segment. On one hand, the contrast between two linguistic segments seems to be still at stake, but on the other hand, the repetition of the noun may indicate that the particle is grammaticalizing towards a marker of the topic of the second segment.

In example (21), extracted from a procedural text from an adult who was explaining the rules of a children's game, there is a thematic break between the first segment and the second. In the first segment, the speaker describes the course of the game when it is the turn of the first group of players to chase the "bride". He goes on with the description of the turn of the second group when they have won the first round. The example could thus be interpreted as either marking the topic of the second segment, or a semantic opposition between the two segments.

( 'The bride who has entered the circle, if we have killed her without us being made enter the circle, it becomes our turn, and we then choose (another) bride. We, then, as for our turn, then we play and (…) if our bride enters the circle, we are the winners.' (BEJ_MV_NARR_55_tanduuy_084-102)

In example ( 22), the second clause with the particle is simply a rephrasing of the first clause, and no thematic break is involved. =na seems to behave as a mere marker of topic continuity. However, another possible interpretation is that =na could underline the contrast between the fact that the fairies have worked, but that the shoemaker has not rewarded them yet as expected, i.e. an adversative meaning, which was unmarked in the first formulation of the question.

( 'They say: Gosh, how can we pay their wages to these fairies? As for the wages, what shall we do?' (BEJ_MV_NARR_17_shoemaker_233-244)

Conclusion

The above description has shown that the Beja contrastive particle na / ni is polysemous and multifunctional, fitting in the broad characterization of the concept of contrast, which groups together mere semantic opposition between two states of affairs, adversativity (and its concessivity subtype), and mirativity.

Morphosyntactically, what is striking is the great freedom of position of this particle within any type of clause. However, further research is needed to explain such syntactic diversity.

The scope of the contrastive particle is not limited to the adjacent word, and from the point of view of discourse organisation it can contrast two adjacent states of affairs, or jump over long stretches of discourse to compare two states of affairs, as in examples ( 8) and ( 13).

In a few instances, when it is enclitic to a noun, the particle may be undergoing a process of grammaticalization towards a topic marker, which needs to be checked on more data.

Even if not a coordinating element per se, it is striking that na / ni mirrors neatly the right-hand part of the semantic map of coordinating connectives proposed by Malchukov (2004: 178), and reproduced below. The non-relevant semantic functions for Beja are surrounded with dashed lines.

Figure 1: A semantic map for coordinating connectives (adapted from Malchukov 2004: 178) To conclude, I want to insist on the fact that this is a preliminary description and analysis based on a limited set of data, which would need to be complemented by more data and tests.

(

  )

	tʔa oːn	whaːl	oːnaːj	ki=haːj	/
	(…) masjaf	bhoːb=na	/ fagar ʃanha ʃagaːm-i	/
	summer_resort formerly=CONTR . clever work	work-AOR.3SG.M .

now DEM.SG.M.ACC DEF.SG.M=situation PROX.SG.M.GEN NEG.IPFV=be_there\PFV[3SG.M] .

well (i.e. contrary to what one would suspect), there

  is no thirst here. Example (2) above, repeated below as (14), illustrates the case where the particle is enclitic to a converb used in predicative function (here after the comparative clitic =ka), marking the contrast between the first SoA, i.e. being the youngest, with the second SoA, i.e. being the strongest. These two SoA, as is the rule for adversative relations, are not expected to be compatible. The presupposition is that normally younger persons are not stronger than older ones. It clearly entails a concessive relation, which could be paraphrased as "although I was the youngest, I was the strongest".

	(14) i=tak=iː=ka	ʔaɖami=ka=b	aka-jeː
	DEF.M=man=ABL.SG=CMPR young=CMPR=INDF.M.ACC become-CVB.SMLT
	i=nafs=i		rh-i	/ ʔakr-aː=ka=na		han
	DEF.M=soul=POSS.1SG.NOM see-AOR.1SG . be_strong-CVB.MNR=CMPR=CONTR also
	dh=i		a-tʔi=ajt	/ ʔaːlaʤ-ani	i-ni	//
	DIR=POSS.1SG.ACC 1SG-resemble\PFV=CSL . tease-IPFV.1SG 3SG.M-say\PFV .
	13) i=kʷaːkʷi	sad	areː eː-fi	/ suːr tʔawaːj niː-di	/ ibakiː=na	/
	DEF.M=Sinkat reservoir then 3.SG.M-be_there . before T'away	1PL-say\AOR . anyway=CONTR .
	mangar hoːj	ki=haːj	/
	thirst	3DAT/ABL NEG.IPFV= be_there\PFV.[3SG.M] .

'There is also Sinkat reservoir. Before, we used to call it T'away. Anyway,

  17) is another example with the contrastive particle clitic to a finite verb. Here the statement comes completely out of the blue, and the mirative meaning comes from the shared knowledge that a human being is not supposed to have donkey legs. PROX.SG.M.ACC DEF.M=Gwinib=LOC.SG . man donkey-GEN leg-PL=COP.3SG .

	(17) iraːni baruːk	wanas-naj	diː-tiːt	// iraːni baruːk
	gosh	2SG.M.NOM chat-IPFV.1PL say-CVB.SEQ . gosh	2SG.M.NOM
	oːn			i=gʷinib=iːb	/ tak meːk-i	ragad-a=i	/
	i=ragad-a-i		i-kati=ni	/ eː-fi
	DEF.M=leg-PL-GEN 3SG.M-become\IPFV=CONTR . 3SG.M-be_there\IPFV
	jaː iraːni	uːn	ani	/ geː oː=tak	/ beːtaj
	ADRF gosh	PROX.SG.M.NOM 1SG.NOM . DM	DEF.SG.M.ACC=man . that_one
	rhi-ta=ni		/ w=haːj	giːg-jaː=ni	/ firaj	hoːj
	see-PFV.2SG.M=CONTR . REL.SG.M=COM leave-PFV.3SG.M=CONTR . offspring\PL 3DAT/ABL
	a-mri	/
	1SG-find\PFV .
	'Gosh, as for me, the man you saw that day unexpectedly, who left with me unexpectedly, I got children
	from him.' (BEJ_MV_NARR_51_camel_stallion_295-301)

(

  .SG.M.ACC DEF.M=wage=POSS.3SG.ACC=CONTR . how do-IPFV.3SG.M

	22) iraːnaj eːt	/ ti=ʤaːntaːji aːn	hinin	/ kak /
	gosh	PROX.PL.F.ACC . DEF.F=jinni	PROX.PL.M.NOM	1PL.NOM . how .
	iraːnaj / oːn	i=ʤaza=joː	/ ni-kʷeːs	eː-d-na	//
	gosh	. PROX.SG.M.ACC DEF.M=wage=POSS.3SG.ACC . 1PL-reimburse\IPFV 3-say\IPFV-PL .
	oːn		i=ʤaza=joː=na	/ kak sak-iːni

PROX

Acknowledgements

My gratitude goes to my Beja consultants, in particular Ahmed Abdallah Mohamed-Tahir and his family at Sinkat, Yacine Ahmed Hamid and his family in Khartoum, and to my colleague Mohamed-Tahir Hamid Ahmed. The financial support of my research unit, Llacan, the ANR projects CorpAfroAs and CorTypo, and the Labex EFL is gratefully acknowledged. Many thanks also to the two reviewers for their insightful remarks. The usual disclaimers apply.