Reina Hara

Dr Chahinaz Khiar

Nitin S Dange

Pierre Bouillac

Prof. Dr Frédéric Robert

Yannick Landais
email: yannick.landais@u-bordeaux.fr

Dr M P Bouillac

Boronic-acid Mediated Carbo-cyanation of Olefins and Vinylation of Alkyl Iodides

Phenylboronic acid was shown to mediate the free-radical multicomponent carbo-cyanation of olefins and the addition of alkyl iodides onto vinylsulfones. Reaction of the boronic acid with di-tertbutyl hyponitrite generates an aryl radical, which was able to selectively abstract the iodine atom from alkyl iodide precursors to form the key C-centered radical precursor. Arylboronic acids may thus be considered as efficient tin surrogates in these free-radical processes.

Introduction

While transition-metal catalyzed processes [1] constitute a powerful method to build C-C bonds, free-radical strategies offer an attractive alternative, allowing consecutive C-C bond formations through cascade and multicomponent reactions. [2] The addition of alkyl radicals onto C=C bonds provides an effective mean to construct C-C bonds as it is energetically favorable. [START_REF] Subramanian | Comprehensive Organic synthesis[END_REF] This addition generates a new radical species, which may then react further with a radical trap, or be oxidized or reduced respectively into carbocation or carbanion species. [START_REF] Renaud | Radicals in Organic Synthesis[END_REF] Unsaturated sulfone derivatives (vinyl-, alkynyl, cyano,…) are well-known efficient radical traps for alkyl radicals, affording, through an addition-elimination process, a new unsaturated system, along with a sulfonyl residue (Figure 1, eq. 1). [5] The power of this strategy has been illustrated earlier with simple addition of alkyl halides to vinylsulfones, [6] and more recently in three-component olefin carbo-alkynylation, alkenylation and cyanation, using the corresponding arylsulfonyl precursors (Figure 1, eq. 2). [5b,7] Depending on the nature of the substituent on the sulfonyl group, the reaction however requires that a tin reagent is present in the medium to sustain the radical chain (Figure 1, (3)). The concomitant need for tin reagents, having perceived toxicity, still hamper the development of C-C bond forming processes using arylsulfonyl derivatives. While several solutions to this problem have been reported in recent time, [8] including the use of alkylsulfonyl analogues, [5a,d][9] there is still a need for environmentally benign alternative methods to promote radical processes using arylsulfonyl derivatives. Moreover, the synthesis of the latter has been recently studied extensively, making them readily available reagents. [10] Bu 3 Sn We report here on the development of a "tin-free" strategy, using phenylboronic acids to mediate the three-component carbocyanation of olefins and the vinylation of alkyl halides under radical conditions. It is shown that alkyl radicals may be selectively generated through the fast abstraction (k ~ 10 9 M -1 s - 1 ) [11] of iodine atom from alkyl iodides by aryl radicals, issued from the fragmentation of arylboronic acids under oxidative conditions. [12] This fast abstraction outperforms all other competitive processes which might occur in the medium, including the addition to sulfonyl derivatives or the olefin partners. [START_REF]Model reactions involving phenylboronic acids and DTBHN in the presence of p-TsCN, vinylsulfones or olefins did not provide addition products[END_REF] Arylboronic acids are thus stable aryl radical precursors, which may be regarded as useful tin surrogates in above radical processes.

Bu 3 SnY X SO 2 Ph X R 2 EWG R 1 R 1 R 2 PhSO 2 -SnBu 3 EWG Y EWG EWG R 1 R 2 I II PhSO 2 t-BuO R SO 2 Ar R R ' + R X In R R ' (Bu3 Sn) 2 (Bu3 Sn) 2 O PhO I (Bu3 Sn) 2 DTBHN 1,2-DCE 65°C PhO 2 S SO 2 Ph PhO 2 C SO 2 Ph 64% OTBDMS OTBDMS (1) (2) (3)

Results and Discussion

Our studies started with the search for an oxidant able to initiate the formation of the desired aryl radical.Preliminary studies were thus carried out using α-iodothioester 2a as a radical precursor, 2-ethylbutene 3a as model olefin and p-TsCN as a cyanide source in the presence of PhB(OH)2 1 in benzene as a solvent. Several oxidants were tested as summarized in Table 1. We first investigated the use of di-tert-butyl hyponitrite (DTBHN), as recent studies by Hayashi and co-workers showed that boronic acids reacted cleanly with DTBHN to produce the corresponding aryl radical. [START_REF] Uchiyama | [END_REF] Using conditions for the 3-component carbocyanation of olefins reported earlier, we were pleased to observe that using DTBHN and replacing (Bu3Sn)2 with 1 eq. of PhB(OH)2 1 led to the expected 4a in 44% yield (Table 1, entry 1). The absence of 1 led to no reaction showing the importance of the boronic acid in the transformation (entry 2). Increasing the amount of 1 furnished 4a in 60% yield (entry 3), while a further increase led to little improvement (entry 4). The amount of DTBHN was also varied and surprisingly, an increase of the hyponitrite did not raise the yield (entries 5-6), while decreasing the quantity to 0.6 eq. had a dramatic effect on the isolated yield of 4a (entry 7). It is important noticing that in contrast with previous work using (Bu3Sn)2 reagent [7] (see Figure 1, ( 3)), the PhSO2 radical does not react with the boronic acid. [15] The process is therefore not a chain and requires at least a stoichiometric amount of DTBHN. Although conditions described in entry 3 seemed optimal, a rapid screening of other peroxides was performed as DTBHN is quite expensive.

Results summarized in entries 8-12 (Table 1) indicate that only PhCO2t-Bu in the presence of a catalytic amount of Cu(II) [16] was able to afford acceptable yields of 4a (entry 11), albeit inferior to those observed with DTBHN. Surprisingly, a poor yield was obtained with t-BuOOt-Bu, while as in DTBHN a t-BuO radical is generated upon heating at 130°C. Mn(OAc)3 in 1,2dichloroethane (DCE) was also tested (not shown), as it is known to react efficiently with boronic acids, [17] but it failed to afford traces of 4a. Using DTBHN as an oxidant, we also varied the amount of 2a (entry 13), p-TsCN (entry 14) and 3a (entry 15), showing that the olefin and the cyanide source must be used in excess. Finally, variation of the nature of the solvent was also looked at, using for instance DCE or t-BuOH, which afforded respectively 28% and 3% of 4a, while CH3CN and acetone did not afford the required compound. The nature of the boron derivative was also varied. For instance, using the phenylpinacolborate instead of 1 led to 4a in only 19% yield. Modifying the nature of the aryl radical using 4-MeOPhB(OH)2 instead of 1, led to 50% yield. Amongst the byproducts that were formed, we were able to isolate a small quantity of i (Table 1), which results from the carbo-cyanation of isobutene, itself likely issued from the decomposition of t-BuOB(OH)2. Entry 2a (eq.) 3a (eq.) 1 (eq.) Oxidant (eq.) 13 [c] 2 1 1 DTBHN (1.2) 2 11 [d] 14 [e] 1

Time (h) Yield [%] [a] 1 1 2 1 DTBHN (1.2) 2 44 2 1 2 0 DTBHN (1.2) 2 0 3 1 2 2 DTBHN (1.2) 2
2 1 DTBHN (1.2) 2 16 15 1 3 2 DTBHN (1.2) 2 57 16 1 2 2 [f] DTBHN (1.2) 2 50
[a] Yield of isolated products. [b] 130°C. [c] 1:1 CH2Cl2/water was used as a solvent. [d] Trace amount of compound i below was also isolated. [e] 1.2 eq. of p-TsCN was used. [f] 4-MeOPhB(OH)2 was used instead of 1.

With the optimal conditions set up in entry 3 in hands, we then studied the scope of the reaction, varying first the nature of the olefin (Scheme 1). The process proved to be compatible with a range of functional groups, including ketones, nitrile, phosphonates, but also bromides and carbamates. Yields range from moderate to good. The radical nature of the process is indicated by the formation of nitrile 4j, generated from carbocyanation of β-pinene. [18] PhSOC CN i These preliminary studies were then extended to radical addition varying the nature of the radical precursor 5a-d (Scheme 2). Several iodides bearing an ester (5a), α,α-difluoroester (5b), nitrile (5c) or sulfone groups (5d) were reactive under the above conditions, affording the desired nitriles 6a-g in moderate to excellent yields. Difference of reactivity between iodides 5a and 5b is worth noticing, [19] the former generally leading to higher yields. This observation had already been made previously in carbo-alkenylation reactions. [7e] Several contrasting reports have appeared recently concerning role of fluorine substituents in C-centered radicals. [20] While the strong electronegativity of the fluorine atom should increase the electrophilicity of radical intermediate I (Figure 1 (3)) and thus lead to improved reactivity toward electron-rich olefins, this is not reflected in terms of yield of 3-component adducts (compare 6b, 6d with non-fluorinated analogues 6a, 6c). A recent report by Dolbier and co-workers [21] showed that difluoroalkyl radicals (RCF2 . ) reacted faster with the most electron-deficient olefins, as a result of a polarized transition state. Charge transfer from RCF2 . to acrylonitrile for instance was shown through computational studies to lead to the development of a positive charge at the RCF2 carbon center stabilized by the electron-donation of fluorine atoms. Although 5b also bears an additional ester group, the resulting radical species may not be so electrophilic, explaining its moderate reactivity toward electronrich olefins. Similar observations were done during previous studies with related polyfluoroalkyl radicals. [7e,f] Finally, reaction with electron-rich olefins such as vinyl pivalate led to the desired nitrile 6g-a, along with the sulfonylcyanation product 6g-b. The reaction was also applied to dienes 7, 9 and 11, leading to the desired 3-component adducts albeit in moderate yields, but generally good diastereoselectivity (Scheme 3). The stereochemistry of 8, 10 and 12 were in good agreement with those of related carbo-alkenylation products prepared earlier. The cyanation using sulfonylcyanide being efficient, we also extended our investigation to the development of a carboalkenylation and vinylation of alkyl halide. [6,8] We first studied the addition of simple alkyl iodides to vinyl disulfone 14, an efficient radical trap. [6n,7e] Under the same conditions than above, the expected vinylated products were formed in satisfying yields (Scheme 4). Primary, secondary and tertiary alkyl radicals added efficiently onto 14. Interestingly, formation of vinylsulfone 15f occurred with high stereocontrol and inversion of configuration of the neomenthyl iodide precursor 13f. Finally, other vinylation agents, including diethyl-2-(chloromethylene)malonate), [7e] 1,1,2trichloroethylene, [22] or (E)-ethyl-3-chloro-2-(phenylsulfonyl)acrylate were also tested but led to disappointing results with yields not exceeding 20%. Our efforts to extend the above vinylation to a three-component carbo-alkenylation process [7e] unfortunately met with little success, affording moderate yields of products, as exemplified below with the carbo-alkenylation of vinyl pivalate 3i, which led to the 3component adduct 16 in 36% yield (Scheme 5). A final experiment was carried out in order to identify the presence of an aryl iodide and thus the formation of an aryl radical as the initiator of the radical process. Carbo-cyanation of n-octene 3f was thus carried out as described above, using functionalized boronic acid 17 instead of 1 to allow for the isolation of the corresponding aryl iodide (Scheme 6). Under these conditions, the carbo-cyanation product 4f was formed as the major compound, along with the sulfonyl-cyanation product 18 and the expected aryl iodide 19. As mentioned above, this radical process is not a chain reaction, and the electrophilic sulfonyl radical, formed upon fragmentation of p-TsCN, may thus react with excess olefin to generate 18. Based on these experimental evidences, the mechanism of the above radical processes may be summarized as in Figure 2. Addition of the alkoxy radical (resulting from the thermal decomposition of DTBHN) onto boronic acid 1, generates the phenyl radical (eq. 1). [START_REF] Uchiyama | [END_REF] The latter then abstracts the iodine atom from the alkyl iodide (eq. 2), to form radical I, which then reacts with the electron-rich olefin to generate a new nucleophilic radical II (eq. 3), being finally trapped by the sulfonyl derivative, leading to the addition product, along with the sulfonyl radical (eq. 4). PhSO2 radical does not react with 1 and thus is not able to propagate the radical chain. [15] This has two consequences: (1) equimolar amount of oxidant is required; (2) the sulfonyl radical accumulates in the medium during the reaction and may, with electron-richer olefins, provides sulfonyl-cyanation products such as 18. [7f] Fortunately, addition of sulfonyl radicals onto olefins is a reversible (fast) process, [START_REF] Chatgilialoglu | Rate constant for sulfonyl radical β-elimination[END_REF] so that sulfonyl-cyanation generally does not impede the desired carbo-cyanation pathway.

Ph PhI Ph B(OH)2 A SO 2 Ph A R 2 EWG R 1 R 1 R 2 PhSO 2 -Ph EWG I EWG EWG R 1 R 2 I II PhSO 2 t-BuO Ph B(OH)2 X Ph + + EWG I EWG R 1 R 2 II (1)
(2)

(3) (4)

Conclusions

As a summary, we have described a "tin-free" carbo-cyanation of olefins and vinylation of alkyl iodides using phenyl boronic acids as radical mediators. The reaction of the t-BuO radical, resulting from the decomposition of DTBHN, with the boronic acid generates an aryl radical, which then abstracts the iodine atom from alkyl iodides to form a C-centered radical, finally adding onto the olefin to form a new C-C bond. Moderate to excellent yields of 2-and 3-component adducts are finally obtained, through a methodology, which is operationally simple, using readily available PhB(OH)2.

Experimental Section

General Information: All reactions were carried out under an argon atmosphere with dry solvents under anhydrous conditions. Yields refer to chromatographically and spectroscopically ( 1 H NMR) homogeneous materials. Commercial reagents were used without purification, unless otherwise stated. Silica gel 60M (230-400 mesh ASTM) was used for flash chromatography. In some cases, silica gel was preliminary deactivated by mixing with 5% (v/v) of triethylamine. CH2Cl2 was dried over activated alumina columns on MBraun Solvent Purification System (SPS-800). DCE and MeCN were distilled from CaH2 and benzene was distilled from Na/Benzophenone. 1 H-NMR and 13 C-NMR were recorded on a Brüker Avance-300 FT ( 1 H: 300 MHz, 13 C: 75.5 MHz), a Brüker DPX-400 FT ( 1 H: 400 MHz, 13 C: 100.2 MHz) and a Brüker Avance 600 ( 1 H: 600 MHz, 13 C: 150.9 MHz) apparatus using CDCl3 as internal reference unless otherwise indicated. The chemical shifts (δ) and coupling constants (J) are expressed in ppm and Hz respectively. FT-IR spectra were recorded on a Perkin-Elmer Spectrum 100 using a diamond ATR accessory. HRMS were recorded with a Waters Q-TOF 2 spectrometer in the electrospray ionization (ESI) mode.

Boronic-acid mediated carbo-cyanation -General procedure. To a solution of iodide precursor 2a (1 eq.) in dry benzene (1 M), were added olefin 3a-m (2 eq.), p-toluenesulfonyl cyanide (2 eq.) and phenyl boronic acid 1 (2 eq.), and the reaction mixture was degassed for 10 min. Then, the process was initiated by adding slowly DTBHN (1.2 eq) at room temperature. The mixture was then stirred at 65°C and the reaction progress monitored by TLC. After the reaction was completed, the mixture was concentrated under reduced pressure and the residue purified by chromatography on silica gel.

Boronic-acid mediated vinylation of alkyl iodides -General procedure. To a solution of alkyl iodide 13a-g (1 eq.) in dry benzene (1.0 M), was added (E)-1,2-bis(phenyl-sulfonyl)ethene 14 (2 eq.) and phenyl boronic acid 1 (2 eq.). The reaction mixture was then degassed for 10 min under argon, and the process initiated by adding DTBHN (1.2 eq.). The mixture was stirred at 65°C and the reaction progress monitored by TLC.

After the reaction was complete, the solvent was concentrated under reduced pressure and the residue purified by chromatography on silica gel.

Figure 1 .

 1 Figure 1. Tin-mediated C-C bond formation using addition of alkyl halides to unsaturated sulfonyl derivatives.

  Scheme 1. PhB(OH)2-mediated carbo-cyanation of olefins. Variation of the nature of the olefin.

  Scheme 2. PhB(OH)2-mediated carbo-cyanation of olefins. Variation of the nature of the radical precursor.

Scheme 3 .

 3 Scheme 3. PhB(OH)2-mediated carbo-cyanation of dienes.

Figure 2 .

 2 Figure 2. Phenylboronic acid mediated three-component carbofunctionalization of olefins.

Table 1 .

 1 Phenylboronic acid-mediated carbo-cyanation of olefin. Search for an oxidant.

		O
	PhS	+	I	2a	p-TsCN (2 eq.) Ph B(OH)2 1
					Oxidant	PhSOC	CN
				3a		4a

Acknowledgements

RH and CK thanks respectively the Yamaguchi University Faculty of Engineering (YUFE) and the Algerian PNE program for financial support. NSD thanks EU for a postdoctoral grant through H2020 Marie-Curie IF program (N°655527). The CNRS and the University of Bordeaux are thanked for financial help.

Keywords: Radicals • C-C coupling • boronic acids • tin-free methodology • sulfones


Marcel Bräutigam

Michel Dacorogna

Marie Kratz

PREDICTING RISK WITH RISK MEASURES: AN EMPIRICAL STUDY RESEARCH CENTER

Keywords: 60G70, 62M10, 62P05, 91B30, 91B70 JEL classification: C13, C22, C52, C53, G01, G32 risk measure, sample quantile process, stochastic model, VaR, volatility

published or not. The documents may come

Introduction

The introduction of risk based solvency regulations has brought the need for financial institutions to evaluate their risk on the basis of probabilistic models. Since the riskmetrics attempt by JP Morgan, the capital needed by companies to cover their risk is often identified to the quantile at a certain threshold α of the return distribution of the portfolio (see [START_REF] Zangari | RiskMetrics Technical Document[END_REF]). They called this risk measure Value-at-Risk (VaR), a name that has been adopted by the financial community. The question of the appropriateness of the risk measure to use for evaluating the risk of financial institutions has been heavily debated since the financial crisis of 2008/2009. For a review of the arguments on this subject, we refer e.g. to [START_REF] Emmer | What is the best risk measure in practice? A comparison of standard risk measures[END_REF]. Apart from the choice of adequate risk measures, there is an accepted idea that risk measurements are pro-cyclical: in times of crisis, they overestimate the future risk, while they underestimate it in quiet times. The general issue of procyclicality has been analyzed with respect to the implications for banking regulation. The discussions have mostly focused on credit risk measurement; see e.g. [START_REF] Kashyap | Cyclical implications of the Basel II capital standards[END_REF][START_REF] Gordy | Procyclicality in Basel II: Can we treat the disease without killing the patient?[END_REF][START_REF] Repullo | The Procyclical Effects of Basel II[END_REF] for contributions with respect to Basel II or, with respect to Basel III, [START_REF] Repullo | The countercyclical capital buffer of Basel III: A critical assessment[END_REF][START_REF] Athanasoglou | Bank procyclicality and output: Issues and policies[END_REF]. Consequently, this issue has been adressed explicitly in Basel III by creating a so called 'counter-cyclical' capital buffer (see [START_REF]Basel III: A global regulatory framework for more resilient banks and banking systems[END_REF]). Still, as in Basel III, most of the analyses focus on a macro-economic perspective. There have been little attempts to study the direct relation between risk measure estimates and procyclicality empirically (see e.g. [START_REF] Bec | Term Structure and Cyclicity of Value-at-Risk: Consequences for the Solvency Capital Requirement[END_REF]), and even less to quantify it. Some studies looked at the effect of crisis on the VaR estimation (see [START_REF] Norling | An empirical evaluation of Value-at-Risk during the financial crisis[END_REF]) or on new ways of estimating VaR itself (see [START_REF] Samanta | How Good is the Transformation-Based Approach to Estimate Value at Risk? Simulation and Empirical Results[END_REF]). A full quantification of procyclicality requires a dynamic reformulation of the risk measurement departing from a pure static approach estimating quantiles of an unconditional distribution. To do so, in this paper, we use an approach which generalizes in a simple way the static 'regulatory' risk measure VaR to a dynamic one. We consider the measurement itself as a random process, introducing the sample quantile process (SQP) first proposed in [START_REF] Muria | A note on look-back options based on order statistics[END_REF] and developed in [START_REF] Akahori | Some Formulae for a New Type of Path-Dependent Option[END_REF][START_REF] Embrechts | Sample Quantiles of heavy tailed stochastic processes[END_REF] as a way to measure risk. Using the SQP, we explore its dynamic behavior and its ability of predicting correctly the future risk. This allows us to precisely quantify empirically the procyclicality through a new way of studying risk measurements. Its understanding is central to an effective risk management and to the discussion which risk measures to use. We conduct our study on 11 stock indices (SI) of major economies to analyze the common behaviors and use them as realizations of the SQP process that we are exploring. When integrating with respect to the Lebesgue measure in the definition of SQP, it collapses to the evaluation of the sample VaR, with rolling windows. We explore then various SQPs based on a random measure chosen as the absolute value of the log-return taken at a power p. We assume that the sample measurement is an estimation of the future risk of the time series, as commonly done in Solvency regulations. We define a look-forward ratio which quantifies the difference between the historically predicted risk (via the SQP) and the estimated realized future risk (measured a posteriori by the VaR realized one year later). We use the ratio to quantify the accuracy of risk estimation. We run the empirical analysis over the different stock indices and find that the ratio is on average above 1 but with a high (between .4 and .5!) Root Mean Square Error (RMSE), which means large fluctuations above and below 1. Since the volatility is high in crisis time and low when there is little information hitting the market, it can be used as a proxy for the market state. That is why we suggest an analysis of the look-forward SQP ratio conditioned to the realized volatility, as done in [START_REF] Dacorogna | An Introduction to High-Frequency Finance[END_REF] for the autocorrelation of the log-returns. We find a relatively strong (between 49% and 58% for p = 0, 0.5) negative linear correlation between the log-ratio and the volatility, as well as a negative rank correlation (between 45% and 50% for the Spearman ρ, again for p = 0, 0.5) We find that in times of high volatility, the ratio is significantly below 1, while it can go even over 3 in times of low volatility signaling a strong underestimation of future risk. Looking for explanations of this effect, we model the stock indices log-returns by GARCH(1,1), a process that models the volatility clustering present in the data. Note that we use on purpose a simple GARCH to isolate the volatility clustering effect, and do not claim that this process describes well financial asset returns; indeed we do not account well for the fatness of the tails, taking a normal innovation instead of a Student one. The calibrated GARCHs are stationary and reproduce very well the average volatility. Using Monte Carlo simulations, we show that GARCH presents a very similar behavior and gives also a strong negative correlation between the log-ratio and the volatility. This negative correlation is stronger than that obtained on the data (the strength might be a bit oversized due to the normal innovation). We conclude that the procyclicality is related to the clustering and the return to the mean of the volatility. The paper is organized as follows. In Section 2, we formally introduce the Sample Quantile Processes, discuss their construction and explore their empirical behavior with 11 stock indices. In Section 3, we explore the ability of the SQP's to correctly predict the risk for the future year. By conditioning them on the annual volatility in Section 4, we show that they underestimate the risk in times of low volatility, while overestimating it in times of high volatility, confirming the pro-cyclicality of risk measurements. Introducing the GARCH(1,1) process in Section 5, we show that it can reproduce these effects well. In Section 6, we summarize the results and discuss their consequences and future developments.

Risk Measure as a Stochastic Process

Since the risk is dynamic, changing with time, we choose a risk measure that is a stochastic process. We use the simplest dynamic extension of the VaR, namely the Sample Quantile Process, first introduced in [START_REF] Muria | A note on look-back options based on order statistics[END_REF], then studied in [START_REF] Akahori | Some Formulae for a New Type of Path-Dependent Option[END_REF][START_REF] Embrechts | Sample Quantiles of heavy tailed stochastic processes[END_REF].

Definitions

First, let us quickly recall the definition of the VaR. For a loss random variable L having a continuous distribution function F L , the VaR at level α of L is simply the quantile of order α of L:

V aR α (L) = inf x :

P (L > x) ≤ 1 -α = F -1 L (α) (2.1)
where F -1 L denotes the generalized inverse function of F L . The VaR is generally estimated on historical data, using the empirical quantile F -1 n (α) associated to a n-loss sample (L 1 , . . . , L n ) with α ∈ (0, 1) defined by:

F -1 n;L (α) = inf x : 1 n n i=1 1I (L i ≤x) ≥ α . (2.2)
Note that our choice of definition (2.2) is such that the VaR equals an order statistic of the data. It would also be possible to obtain the VaR by linear interpolation between two order statistics. This might provide a slightly different value: the smaller the data set and/or the higher the threshold, the more the two quantile definitions usually differ. By construction and given our choice of definition of a quantile, the empirical quantile will then be larger or equal to that obtained by linear interpolation. Let us turn to the Sample Quantile Process (SQP).

The loss is now represented by a stochastic process L = (L t ) t . The SQP (Q µ,α,T,t (L)) t≥0 of L at threshold α, with respect to a random measure µ defined on R + , is defined, for 0 ≤ T < t, by (see e.g. [START_REF] Akahori | Some Formulae for a New Type of Path-Dependent Option[END_REF][START_REF] Embrechts | Sample Quantiles of heavy tailed stochastic processes[END_REF]):

Q µ,α,T,t (L) = inf x : 1 t t-T µ(s)ds t t-T 1I (Ls≤x) µ(s)ds ≥ α . (2.3)
The SQP, denoted also by (Q µ,α,T (t)) t≥0 = (Q µ,α,T,t ) t≥0 when no confusion is possible, is clearly a dynamic generalization of the VaR. Indeed, choosing µ in (2.3) as the Lebesgue measure, corresponds to a rolling window VaR denoted by

Q 0,α,T (t) = inf x : 1 T t t-T 1I (Ls≤x) ds ≥ α . (2.4)
This can be read as the continuous version of the VaR defined in (2.2) on the interval [t-T, t). Nevertheless, in practice, we estimate Q 0,α,T (t) for discrete observations L i of L, which comes back to considering a 'rolling window VaR'.

Our approach is to explore the SQP as a predictor of the risk, considering µ as a random measure. We focus on the particular case where µ is defined by µ

(s) = µ p (s) = |L s | p , with p ∈ N : Q p,α,T,t = Q p,α,T (t) = inf x : 1 t t-T |L s | p ds t t-T 1I (Ls≤x) |L s | p ds ≥ α . (2.5)
This choice of µ is arbitrary, but it is one of the simplest formulations that includes the loss, and gives back the Lebesgue measure when taking p = 0 (i.e. the VaR process with a rolling window). It is also a way to study the impact of extreme movements: the higher p, the more emphasis we put on the extremes. Looking at (2.5), we can see that positive values of p will let the SQP put more weight on the tail of the distribution. And, correspondingly, negative values of p will let the SQP put more weight on the center of the distribution. We concentrate on non-negative values of p as we are interested in capturing and modelling the tail risk.

Throughout the paper, we consider the empirical estimator ( Q p,α,T,t ) t≥0 of the SQP defined by

Q p,α,T,t = inf    x : 1 i∈[t-T,t) |L i | p i∈[t-T,t) 1I (L i ≤x) |L i | p ≥ α    . (2.6)

Empirical exploration of the SQP

Given this formalism, we are now going to empirically build realized time series of SQP's computed over various stock indices, which we consider as various sample paths of the process. Since our goal is to look at the appropriateness of the VaR calculations from financial institutions, it seems natural to use stock indices and see how our various SQP's perform on this data.

Data

We consider data from 11 different stock indices. The data used are the daily closing prices from Friday, January 2, 1987 to Friday, December 30, 2016. In Table 1 you can find detailed information about the countries and indices used. The reason for such a large set of data is the need to see if the process has similar behavior over most of the developed economies. Moreover, we want a relatively long dataset, which is offered by all these stock indices. As commonly done, we refer to the S&P 500 index as our main example and focus on this specific index whenever showing figures. All the results for the other indices are given in the appendix and, if not commented on otherwise, show the same characteristics. For any time series, let us denote the closing price at time t by S(t) and by ∆t the interval between two consecutive spot prices. We focus on daily intervals. We then define the logreturn X ∆t (t) by:

X ∆t (t) = ln S(t) S(t -∆t) (2.7)
Throughout this paper we work with log-returns. If not specified otherwise, by abuse of notation, we will refer to X i = X ∆t (t i ) as the daily log-return (with ∆t = 1 day) at time t i .

Also, in some situations it might be of interest to work with the log-return losses instead of the log-returns, simply defined as: L i = -X i , the daily log-return loss at time t i .

Empirical Study and Discussion

At first, we focus on a rather qualitative evaluation, illustrated on the S&P 500. In Figure 1, we plot the sample paths of SQP at level α at monthly frequency and the VaR benchmark on the overall sample at the same level α. We look at it for different parameter values p of our random measure µ = |L S | p : p = 0, 0.5, 1, 2. The sample size, T , is taken to be one year and we consider thresholds α = 95% and 99%, respectively. When p = 0.5, the values of the SQP are more volatile (i.e. with a wider range) than for p = 0. They are not only more volatile but also smaller than the SQP for p = 0 (i.e. the rolling-window VaR), in particular with extreme values that seem to have decreased notably. Hence, the SQP, with p = 0.5, is a measure that enables a larger discrimination between the losses. For p = 1, the properties found for p = 0.5 are also present and accentuated. The volatility of the values of the SQP has once again increased for α = 0.95. The range of the SQP is also bigger when p = 1 than when p is equal to 0 and 0.5. As for p = 0.5, the SQP for p = 1 is a measure that enables a bigger discrimination between the losses. But we can also see that the differences between the SQP for the two thresholds has clearly diminished for extreme values. To conclude, the SQP for p = 1 is close to the SQP for p = 0.5, apart for the most extreme values at threshold α = 0.95, and continues the trend seen for the latter.

In addition, the SQP with p = 2 is very close to the previous SQP with p = 1. However, the differences between the SQP with the different thresholds are extremely small and almost vanished. Hence, it is difficult, if not impossible, to distinguish between the 95% and 99% thresholds. The 95% threshold collapses in fact on to the 99% threshold. As can be seen on the last graph, both SQPs hover around the VaR(99%). Moreover, for the 99% threshold, the SQP with p = 2 is equal the SQP with p = 1, which does not make it interesting anymore.

We then conclude that the choice of p = 1, 2 to define the SQP may not be the best one to assess the intensity of losses. In summary, we can observe the following. For a fixed threshold α, the SQP decreases with higher powers of p. Also, the higher the parameter p, the more volatile the SQP gets. At the same time the difference of the SQPs at level α = 95% and 99% diminishes with increasing p -we could observe that for α = 99% the SQP for p=1 and p=2 coincide. From the figures, p = 1/2 seems optimal for tail discrimination. These observations can also be confirmed when looking at it more quantitatively as done in Table 2 and its figure for the S&P 500, as well as in Table 3. In Table 2 we compare the average values achieved by the various monthly rolling SQP's with sample size T equal to one year as a function of the parameter p and the threshold α for the S&P 500. We notice from the figure again that for the highest thresholds (above and including 99%), the values for p = 1 and p = 2 are the same for all indices. We see in Table 3 that there is no marked difference between the various indices. The behavior is the same, the higher the value of p, the larger the average (in absolute terms). In the same table, the ratio between the two α is the largest on average for p = 0.5 confirming our remarks above about tail discrimination. This is not true for all indices particularly the European ones: FRA, DEU, NLD, GBR do not follow this behavior. Table 2: Average over the whole sample of rolling-window SQPs with sample size T = 1y (S&P 500) as a function of the power p for various thresholds α.

Table 3: Average over the whole sample of the rolling-window SQPs with sample size T = 1y for each of the 11 stock indices (in %) and associated ratio of the SQPs between two thresholds (95 and 99%) -using historical data. In the last column, we present the average over all indices ± the standard deviation. 3 Predictive Power for the Risk using the SQP After having introduced and analyzed the SQP we now want to focus on our main goal to test the SQP's ability to assess the estimated future risk correctly -or what we call predictive power. Let us briefly point out why this differs from the classical backtesting, which designates a statistical procedure that compares realizations with forecasts. First recall that, for the VaR, some of the backtesting methods used by European banks and the Basel committee are based on the so-called violation process counting how many times the estimated/forecast VaR(α) has been violated by realized losses during the following year; those VaR 'violations' should form a sequence of iid Bernoulli variables with probability (1 -α) (see e.g. [START_REF] Christoffersen | Evaluating interval forecasts[END_REF][START_REF] Christoffersen | Backtesting Value-at-Risk: a duration-based approach[END_REF]). We want to assess the risk, thus the quality of estimation. While the violation ratio in [START_REF] Danielsson | The Emperor has no clothes: Limits to risk modelling[END_REF] gives a measure of the quality of the estimation, here we do not question at all if the violation process is in line with the acceptable number of exceptions. Our goal is rather, when considering the risk estimation as a stochastic process, to quantify how well or wrong our VaR estimate performed, compared to the realized future value of the VaR. It is an alternative and new way of looking at risk, which, subsequently, leads also to another way to assess the quality of the risk estimation. Moreover, this way of risk quantification can be applied to any other risk measure besides the VaR (as for instance to the Expected Shortfall).

Look-forward SQP Ratio

For this, we consider SQP estimators ( Q p,α,T (t)) t , computed according to (2.6) during a period (of length T years) that ends at time t; for ease of notation, we may omit the hat in the following.

We measure the quality of the risk estimation by comparing it to the VaR estimated one year later (p = 0, T = 1), i.e. Q 0,α,1 (t + 1y). It means to check if our estimate Q p,α,T (t) is a good estimate at time t of the future and unknown value Q 0,α,1 (t + 1y). Hereby, it is important to note that the reference value, the VaR one year later Q 0,α,1 (t + 1y), will always be computed over a period of one year (T = 1) as it should represent the realized risk during that period of time. Note also that we compare all versions of the SQPs to the VaR as this is the reference risk measure asked by many regulators. This leads us to introducing a new random variable, the ratio of SQPs denoted by R p,α,T (t), that quantifies the difference between the historically predicted risk Q p,α,T (t) and the estimated realized future risk Q 0,α,1 (t + 1y) (considered at the time t plus one year later) computed using the period of time [t, t + 1y): Consequently, if we look at the values of the ratio R p,α,T (t), a value near 1 means the prediction Q p,α,T (t) correctly assesses the future risk estimated by Q 0,α,1 (t + 1y). Similarly, if the ratio is above one, we can say that we have an under-estimation of the future risk, i.e. the observed future risk. And to the contrary, if it is below one, we have an over-estimation of the future risk, which would lead to an over-evaluation of the needed capital.

R p,α,T (t) = Q 0,α,1 (t + 1y) Q p,α,T (t) . (3.1)

Assessing the average SQP ratio

Taking a monthly rolling window for the ratios R p,α,T (t) (i.e. updating t every month), we compute the average SQP ratio over the whole sample. Here we focus on looking at the overall averages (over the 11 different indices) of the monthly SQP ratios. Results for the 11 different indices on their own are reported in Table 4. We see, in Table 4, that all the ratios are clearly unequal to one, telling us that the future risk is, on average, not well predicted. Furthermore, we clearly see for both choices of α that the average ratio is above 1 for p = 0 -even after two standard deviations. We find back the well known result that the historical estimation of VaR underestimates the risk. For p > 0, we see that the values are below 1 and clearly decrease with p for both α = 95% and 99% (apart from p = 2 for α = 99%). This implies that our choice of parameter p puts too much weight in the tails on average, causing a clear overestimation of the actual risk. When comparing the average ratio for our two thresholds α, we see that it does not change a lot for p = 0, 0.5, while the ratio increases with α for p = 1, 2. This increase is not surprising as for those values of p, more weight is added in the tail, so the larger the threshold α, the smaller the difference to the actual future risk, and thus the smaller the overestimation. Clearly, only considering the average does not tell the whole story as it does not account for the dynamics of the rolling window -after all, we are computing the ratios on a monthly basis. Therefore, in a second step we choose to look at the Root Mean Square (RMS) distance and compute the associated Root Mean Square Error (RMSE), i.e. the average RMS distance of our ratio from a perfect prediction which equals a ratio of 1:

RM SE := 1 N N t=1 (R p,α,T (t) -1) 2 (3.2)
where N is the number of overall time points the ratio is evaluated at (as we are working with a monthly rolling window, N equals 12 times the amount of years on which the ratio is tested on). This RMSE measures the "error" of the forecasted risk for a certain sample path. For the S&P 500 index, we see in Figure 2 that the error increases with the threshold: it is smaller for 95% than for 99%, except for the SQP measure with p=2. For the 99% threshold, the error increases slightly from p = 0 to p = 0.5 where it has its maximum, and levels slightly off to its minimum for p = 1, 2. We observe a different, almost opposite pattern in the case of 95%: it decreases from p = 0 to its minimum for p = 0.5, then increases with p. All the values are clearly above 0, suggesting that the various measures do not forecast well the future risk.

As before, when considering the average SQP Ratio, we observe a similar -although slightly different -behavior for the average of all indices as with the S&P 500 index (see the last two columns in Table 5). On average, the error increases with the threshold (except for p = 1, 2), and -as with the S&P 500 -for α = 0.95 the error is smallest for p = 0.5 and increases afterwards again. For α = 0.99 its maximum is reached for p = 0, then the error levels off, being constant for p = 1, 2. Again, the average RMSE in all cases is clearly above 0 reaching 50% of the right value 1. This means that there are high fluctuations of the ratio, for the weak underestimation (of around 10%) for p = 0, as well as for a stronger overestimation for p > 0 (recall the last column of Table 4). This analysis does not give us yet an explanation for this behavior. We want to understand why and in which circumstances we over-or underestimate the risk measured by the SQP throughout the sample. It is what we tackle in the next sections.

4 Volatility Impact on the Measured Risk

Realized Volatility as an Indicator of Market States

To better understand the dynamic behavior of the ratios, we condition them on past volatility. This is inspired by the empirical study on Foreign Exchange done by Dacorogna et al. in [START_REF] Dacorogna | An Introduction to High-Frequency Finance[END_REF], where the authors conditioned the returns on past volatility, showing that, on one hand, after a period of high volatility, the returns tend to be negatively correlated. It means that the volatility should diminish in the future, with a distribution more concentrated around its mean; thus the quantiles should be smaller. It implies that the future risk would be overestimated. On the other hand, in times of low volatility, consecutive returns tend to be positively auto-correlated; thus the volatility should increase on the long term, with the consequence in the future of quantiles shifted to the right, meaning that our estimation would underestimate the future risk.

To be able to condition on volatility, we define an empirically measured volatility, σ(t -1), also called in the literature 'realized volatility', taken over a rolling sample of size T = 1 year (or 3 years) up to but not including a time t, and we annualize it by considering σ(t -1) × √ 252, obtaining the annual realized volatility v(t) at time t defined as:

v(t) = σ(t -1) × √ 252 where σ(t -1) := 1 N -1 t-1 i=t-N X i - 1 N t-1 i=t-N X i 2 (4.1)
N being the number of days in a sample of size T years, and the daily log-returns X i being defined in (2.7). Note that the notation is chosen such that the annual realized volatility v(t) at time t (January 1, 2014 for instance) is computed on the previous year until t -1 included (i.e. until December 31, 2013, in our previous example). We evaluate v(t) on a rolling basis every month. Hereby, we obtain a time series of annual realized volatility (v(t)) t , which can be used as a benchmark of the current market state. In Figure 3, we present (v(t)) t for the S&P 500 (log-)returns between 1987 and 2016 for both a rolling sample of 1 year and of 3 years. We also indicate the average annual realized volatility of the index calculated over the whole sample (represented by the dashed line). Its value is 18.46%. We can see that the volatilities above this benchmark mostly stand for periods of high instability or crisis (and not only in the USA). The high volatility of the period 1987-1989 is for instance explained by the New York Stock Exchange crash in October 1987. In 1997, Asia was hit by a crisis, as well as Russia in 1998 and Argentina in 1999-2000. In 2001, the United States experienced the bursting of the internet bubble. Following the Lehman Brother's bankruptcy, the period of 2008-2009 was a period of very high volatility. And finally, the sovereign debt crisis in Europe also impacted the S&P 500 Index in 2011-2012. This is an illustration of the increased dependence between the markets during times of crisis. We conclude from this behavior that our measure of realized volatility is a good measure that reacts well and quickly to various market states. It is thus a reasonable proxy to qualify the times of high risks and to use it to discriminate between quiet periods and periods of crisis. We use it to condition our statistics (the SQP ratio) and see if we can detect different behaviors of the price process (which underlies the risk measure) during these periods in comparison to quiet times, as it was the case for returns in [START_REF] Dacorogna | An Introduction to High-Frequency Finance[END_REF]. It is a way to look at the ability of our SQP risk measures to correctly predict the future risks and to capture the dynamics of the market.

SQP Predictive Power as a Function of Realized Volatility

In the case of the S&P 500 we show in Figure 4 both the annualized volatility and the realized ratios of SQP's for both thresholds 95% and 99%, and for p = 0 (rolling VaR). One can easily observe on these graphs a strong opposite behavior for SQP ratios computed on a 1-year sample (in blue) and for volatility (in red). Similar observations hold when using a 3-year sample to predict the estimated future risk (the figures for the analysis of the S&P 500 using a 3-year sample can be found in the appendix). In Figure 4 the dynamic behavior appears clearly. The realized ratios can differ from 1 either on the upper side or on the lower side. Looking at the dependence with respect to the annualized volatility, it can be noticed that for both graphs, the realized SQP ratios are negatively correlated with the annualized volatility. This particular feature is of importance in our study because it means that when the volatility in year t is high in the market, the realized SQP ratio in year t is quite low. Recalling the definition of the ratio, this situation means that the risks have been over-evaluated by the SQP calculated in year t (in comparison to the realized risk in year t+1y). Conversely, when the volatility is low in year t, the realized SQP ratio is often much higher than 1 in this year, which means that the risks for the year t + 1 have been under-evaluated by the calculation of the SQP in year t.

Another way of visualizing the negative correlation is to look at the various realized SQP ratios as a function of the annual realized volatility, as in Figure 5. Such figures highlight better the existing negative dependence between these two quantities.

When looking at the first row of Figure 5, the negative correlation between the SQP ratios with the annual realized volatility is clearly noticeable. It is displayed by the negative slope of the simple linear regression line. As it appears also clearly that the (negative) dependence is non linear, we consider the log-ratios instead (see the plots on the second row of Figure 5): the more volatility there is in year t, the lower are the ratios R p,α,T (t). When the log-ratios are negative, it means that losses in year t + 1 have been overestimated with the measures calculated in year t. In the original scale we can better quantify the magnitude of the effect: When looking at the case of α = 0.99, with respect to the overestimation, we can observe ratios much below 0.5, i.e. the risk computed at the height of the crisis is more than twice the size of the risk measured a year later. On the other hand, we see that the underestimation can be very big since the realized ratios for the SQP can be even larger than 3; in other words, the risk next year is more than 3 times the risk measured during the current year for this sample. We also observe that the overestimation is very systematic for high volatility, whereas it is not for the underestimation at low volatility. Indeed, in this latter case, we see also values below 1, while we do not see any value above one when the volatility is very high (above 35%). Although we show here only the S&P 500, very similar behaviors are exhibited by all the other indices (see the appendix).

It is of interest to find out which of the various measures µ p produces the strongest countercyclical behavior. To make this more quantifiable, given the highly non-linear behavior of this dependence as illustrated in Figure 5, we choose to look at the linear correlation ρ(log(R p,α,T (t), v(t)) between the logarithm of the SQP ratio (instead of the SQP ratio) and the volatility. In Table 6, we present this quantity for all indices and as average over all indices.

Table 6: Pearson correlation between the log of the SQP ratios and the volatility, for each index and for T = 1 year, over the whole historical sample, and for two thresholds (95 and 99%). In the last column, we present the average over all indices ± the standard deviation. We see that the Pearson correlation is always significantly negative, the average values for different thresholds and values of p are above 40%. The negative linear correlation is strongest for the lower values of p: in the case of α = 0.95 it is strongest for p = 0.5 (although p = 0 is quite similar), and in the case of α = 0.99, it is strongest for p = 0. In general one can say that the correlation is similar for both thresholds. One might also conclude that, for those observations, SQP with p = 0 or 0.5, are better suited than with higher values of p to emphasize the cyclical signal, thus for designing counter-cyclical models. Due to the non-linear nature of the dependence, we also consider the notion of rank correlation, looking for instance at the Spearman rho (the values for each index separately as well as for the Kendall tau are available in the appendix). In Table 7, we display only the average ρ over all indices as we have seen in the other tables that indices do not differ significantly from one another in their behaviors. In this table, the results confirm those obtained when considering the correlation between the logarithm of the SQP ratios and the volatility (e.g. highest value in absolute terms for low values of p for both thresholds α, decreasing in p for α = 0.95, almost constant in p for p > 0 in the case of α = 0.99), only the magnitude is a bit lower than for the linear correlation.

Table 7: Average over all indices of the Spearman-ρ between the annualized volatility and the SQP ratios computed on a 1-year sample α = 0.95 α = 0.99 p = 0 -0.47 ± 0.06 -0.50 ± 0.04 p = 0.5 -0.46 ± 0.07 -0.45 ± 0.06 p = 1 -0.42 ± 0.09 -0.46 ± 0.06 p = 2 -0.37 ± 0.11 -0.46 ± 0.06

Note that the same analysis was performed using a 3-year sample (T=3) to compute the estimated future risk Q T,α,t and the annualized realized volatility, then to look at the dependence between annualized volatility and SQP ratios. On average, effects are similar in magnitude and with mostly similar tendencies for both linear correlation and non linear correlation. The corresponding tables can be found in the appendix. We need to further explore the dependence, to better understand the behavior of the parametrized risk measure (SQP). It is what we develop in the next section, introducing a simple model to highlight the procyclical behavior we observed so far.

Thus, we consider a natural model, namely the GARCH one, as it has been developed for modelling volatility clustering.

Given the aim of this study, we choose to use the simplest version, the GARCH(1,1), to model the log-returns (X t ) t as: X t+1 = t σ t , where the variance σ 2 t of X t satisfies

σ 2 t = ω + α X 2 t + β σ 2 t-1 , with ω > 0, α > 0, β > 0,
and where the error terms ( t ) t constitute a Gaussian white noise. Here again, one could use a more sophisticated model with Student-t error terms in order to better fit the tails, but mixing fat tails with volatility clustering would not help us isolate and identify the main cause of the procyclicality, hence this choice of standard Gaussian noise. To estimate our model, we fit the parameters ω, α, β to each full sample of the 11 indices, using a robust optimization method proposed by Zumbach in [START_REF] Zumbach | The pitfalls in fitting GARCH (1,1) processes[END_REF]. Compared to the other methods, we saw that the method by Zumbach was the most precise in reproducing the average annual realized volatility. As a side note, we remark that the robustness is especially important as other optimization methods may not always fulfill the stationarity condition for the GARCH process, namely α + β < 1. From Table 8, we see that the optimization gives in all cases, parameters that fulfill the GARCH stationarity condition. The estimated parameters do not vary much from one index to the other (except for AUS, ITA and JPN). Further, we see that the annualized volatilities of the GARCH reproduce very well the historical ones on average, when estimated over 1000 replications of the GARCH process. Given those GARCH parameters, we replicate for each index 1000 simulated paths with the same sample size as the original data sample. On each of these replications, we apply the same statistical analyses as those done on the historical data in Section 4. This means we analyze the behavior of the SQP for varying parameters p and thresholds α, we look at the RMSE and the SQP-ratios; as these results are not central to our analysis, we do not present them here. Then we look closer at the correlation between annual realized volatility and the logarithm of the SQP-ratios; it is what is developed next. All in all, we observe qualitatively similar results with the GARCH simulations as with the historical data.

Dependence between SQP Ratios and Annualized Volatility in the GARCH Model

As in §4.2, we first look at the Pearson correlation between the annualized daily volatility and the logarithm of the SQP ratios computed on a 1-year sample. Those are depicted in Table 9. As for historical data, we observe a negative correlation here too. The negative Pearson correlations (again, using the logarithm of the SQP ratios) of the GARCH for the averages are higher (in absolute value) than in the data (see Table 6), the difference between the two increasing with p. Furthermore, as in the case of the historical data, the negative correlation is stronger for the lower powers p -except that, in contrast to the historical data, the behavior for α = 0.95 is different for p = 1, 2 (while it was strongly decreasing with p for historical data, here it remains at similar values).

Table 9: Average Pearson correlation between the annualized volatility and the log of SQP ratios computed on a 1-year sample, using a GARCH(1,1) model, and for two thresholds (95 and 99%) AVG±σ α = 0.95 α = 0.99 p = 0 -0.63 ± 0.07 -0.61 ± 0.07 p = 0.5 -0.66 ± 0.07 -0.61 ± 0.07 p = 1 -0.66 ± 0.07 -0.60 ± 0.07 p = 2 -0.63 ± 0.07 -0.60 ± 0.07

As before, we also illustrate the obtained results with the S&P 500 index, considering one (simulated) sample path of the fitted GARCH(1,1) model, and look at the various realized SQP ratios (as well as its logarithm) for p = 0, as a function of the annual realized volatility; see Figure 6. For a better comparison between the GARCH realization and the historical data, we use the same y-scale for both. The experiment has been replicated with many sample plaths and gives the same conclusion. We observe a similar behavior for the GARCH realization (Fig. 6) and the historical data (Fig. 5):

• We have a negative correlation in both cases and the slope of the simple linear regression lines look similar. • The more volatility there is in year t, the lower are the ratios R p,α,T,t , which means that losses in year t + 1 have been overestimated with the measures calculated in year t. • This overestimation can result in ratios below 0.5, i.e. in the above realization the risk computed at the height of the crisis is more than double the size of the risk measured a year later. • For the underestimation, we see that the realized ratios can be larger than 2; in other words, the risk next year is more than 2 times the risk measured during the current year for this sample. As can be seen from Figure 6, the linear correlation is not an optimal approximation of the dependence structure (as already observed for historical data, although slightly stronger for GARCH). Thus, we consider the log SQP ratio to take into account the non linear dependence and, additionally, compute a rank correlation -see e.g. Table 10.

Table 10: Average Spearman's ρ rank correlation between the annualized volatility and the SQP ratios computed on a 1-year sample, using a GARCH(1,1) model AVG±σ α = 0.95 α = 0.99 p = 0 -0.61 ± 0.08 -0.59 ± 0.08 p = 0.5 -0.64 ± 0.08 -0.59 ± 0.08 p = 1 -0.64 ± 0.08 -0.58 ± 0.09 p = 2 -0.61 ± 0.09 -0.58 ± 0.09

For the GARCH(1,1) model, the rank correlation is of the same order (slightly lower) than the Pearson correlation. This is similar to the historical data (there the differences were a bit more pronounced, especially for low α and low p). On average (i.e. when looking at the average values of the 11 indices considered) we see the same behavior with respect to α and p for the Spearman as well as the Pearson correlation in this case.

Overall, a simple GARCH model captures pretty well the main features we have seen in the data. Thus, this procyclicality seems related to the clustering of volatility and its return to the mean. Despite having the same tendencies, we still have seen differences between the two. Thus, up to now, we do not have evidence to claim that the procyclicality in the data is solely due to the GARCH effect.

Conclusion

Popularized by JP Morgan in 1996 (see [START_REF] Zangari | RiskMetrics Technical Document[END_REF]), the Value-at-Risk measure is still, nowadays, mostly used by banks and some insurance companies for risk management purposes. This use has been reinforced by regulators even if it has been shown that Value-at-Risk was not a coherent measure contrary to Expected Shortfall in [START_REF] Artzner | Coherent measures of risks[END_REF]. However, as a lesson learned from the 2008 financial crisis, the Basel Committee is moving towards Expected Shortfall. Nonetheless, this study remains timely since VaR is the measure used by both banks and insurances and our analysis could be generalized for Expected Shortfall at a later stage. However, the question remains if measuring risk using a risk measure on historical data is appropriate to forecast the risk financial institutions face. In this study, we show empirically that risk measures based on VaR are in fact pro-cyclical. Looking at 11 stock indices, we find that in times of low volatility, the risk is underestimated, while in times of crises the risk is overestimated.

Although pro-cyclicality of risk models has widely been assumed by market actors, it was never clearly quantified, nor shown empirically with enough evidence. Our way to quantify this effect is to model the risk measure as a stochastic process and condition it on realized volatility. We use the Sample Quantile Process (SQP), a 'dynamic generalization of VaR', that can be defined in many ways depending on the choice of measure µ that is part of its definition. After studying various choices of parameters, we conduct a study of the dynamics of risk measures as a function of realized volatility. To measure the so called predictive power, we introduce a new quantity, a ratio of risk measures to quantify the quality of risk prediction.

While we only used it for the SQP/VaR, it is a universal quantity that can be applied to any other risk measure like the Expected Shortfall without restrictions. It is a novel way of measuring the accuracy of risk estimation that does not rely on number of violations. Our study confirms the assumption presented above and allows us to quantify this effect. We observed that simple GARCH(1,1) models provide similar characteristics as in the data, i.e. we saw a negative correlation between realized volatility and the SQP-ratios with even slightly higher values than in the historical data. We thus relate this effect to the volatility clustering present in financial markets. Still, we do not have evidence to claim that the procyclicality in the data is solely due to the GARCH effect -further and more statistical investigations are in progress. As our observations are confirmed on 11 different stock indices we can conclude that the risk management imposed by regulators favors pro-cyclical behaviors of the market actors. Indeed, it does not reflect the genuine potential risks and losses. Following a pure historical estimation of risk measures, banks would be required to keep more capital than needed during a crisis (period of high volatility), while the requirements during quiet period grossly underestimate the risk, leaving the banks, or financial institutions in general, unprepared in case of crisis. Finally, having empirically quantified a pro-cyclical effect of the sample VaR, we can now take it into account to suggest an anti-cyclical risk measure. To conclude, in view of our results, the introduction of anti-cyclical risk management measures, which are at the opposite of what is currently required for financial institutions by regulators, would better prepare the financial system to cope with future crises by enhancing the capital requirements in quiet times and relaxing them during crises. Clearly, these measures should be in line with the empirical results without hampering the system or introducing rules that weaken economic valuation of liabilities as it is currently done. This is what we are developing. Studies are under way to design SQP with the right dynamical behavior that would be a good basis for anti-cyclical regulation. In each row we consider the SQP Ratios and then the logarithm of the SQP Ratios: From left to right, SQP ratio for α = 95%, 99%, then logarithm of SQP ratio for α = 95%, 99%. From top to bottom: GBR, ITA, JPN, NLD. In each row we consider the SQP Ratios and then the logarithm of the SQP Ratios: From left to right, SQP ratio for α = 95%, 99%, then logarithm of SQP ratio for α = 95%, 99%. From top to bottom: SGP, SWE, USA B.6.2 On a simulated GARCH realization In each row we consider the SQP Ratios and then the logarithm of the SQP Ratios: From left to right, SQP ratio for α = 95%, 99%, then logarithm of SQP ratio for α = 95%, 99%. From top to bottom: AUS, CAN, DEU, FRA. In each row we consider the SQP Ratios and then the logarithm of the SQP Ratios: From left to right, SQP ratio for α = 95%, 99%, then logarithm of SQP ratio for α = 95%, 99%. From top to bottom: GBR, ITA, JPN, NLD. In each row we consider the SQP Ratios and then the logarithm of the SQP Ratios: From left to right, SQP ratio for α = 95%, 99%, then logarithm of SQP ratio for α = 95%, 99%. From top to bottom: On real data, on a realization of a GARCH(1,1) simulation.

B.3 RMSE of SQP Ratios

B.4 Annualized Volatility

Figure 1 : 2 .

 12 Figure 1: Sample Quantile Processes (SQP) rolling every month with T = 1 year, and thresholds α = 95% and 99%, respectively. From left to right and from top to bottom: p = 0, p = 0.5, p = 1 and p = 2. Dashed lines correspond to VaR(α) computed over the whole sample. The y-scale is the same in the four graphs to allow comparison between the different measures.

Figure 2 :

 2 Figure 2: RMSE (defined in (3.2)) of the SQP Ratios R p,α,1 (t) taking p = 0, 0.5, 1, 2 and two thresholds α = 95% (in blue), 99% (in red), on the S&P 500

Figure 3

 3 Figure 3 Annual realized volatility (defined in (4.1)) of the S&P 500 returns over a rolling sample of size T = 1, 3 year(s). The dashed line represents the sample average of the volatility

Figure 4 :

 4 Figure 4: SQP Ratios and annualized volatility time series computed over 1 year (T = 1y) for α = 95% (left) and α = 99% (right)

Figure 5 :

 5 Figure 5: SQP Ratios as a function of annualized volatility for p = 0: On the left α = 95% and on the right α = 99%; first row SQP-ratios, second row logarithm of SQP-ratios

Figure 6 :

 6 Figure 6: SQP Ratios as a function of annualized volatility for one simulated sample path of the GARCH(1,1) (that fits the S&P500) for p = 0: On the left α = 95% and on the right α = 99%; first row SQP-ratios, second row logarithm of SQP-ratios

Figure A. 2 :

 2 Figure A.2: SQP rolling every month with T = 1 year, and thresholds α = 95% and 99% for different indices and values of p. From left to right: p = 0, p = 0.5, p = 1 and p = 2. From top to bottom: GBR, ITA, JPN, NLD. Dashed lines correspond to VaR(α) computed over the whole sample.

Figure A. 3 :Figure A. 4 :

 34 Figure A.3: SQP rolling every month with T = 1 year, and thresholds α = 95% and 99% for different indices and values of p. From left to right: p = 0, p = 0.5, p = 1 and p = 2. From top to bottom: SGP, SWE, USA. Dashed lines correspond to VaR(α) computed over the whole sample.

Figure A. 5 :

 5 Figure A.5: RMSE (defined in (3.2)) of the SQP Ratios R p,α,1 (t) taking p = 0, 0.5, 1, 2 and two thresholds α = 95% (in blue), 99% (in red). Each figure is for one index, shown in alphabetical order: AUS, CAN, DEU, FRA, GBR, ITA, JPN, NLD, SGP, SWE, USA.

Figure A. 6 :

 6 Figure A.6: Annual realized volatility (defined in (4.1)) over a rolling sample of size T = 1, 3 year(s). The dashed line represents the sample average of the volatility. Each figure is for one index, shown in alphabetical order: AUS, CAN, DEU, FRA, GBR, ITA, JPN, NLD, SGP, SWE, USA.

Figure A. 7 :

 7 Figure A.7: SQP Ratios and annualized volatility time series computed over 1 year (T = 1y) for α = 95% and α = 99%. The two figures corresponding to an index are shown next to each other, i.e. we have two indices per row. The indices are shown in alphabetical order: AUS, CAN, DEU, FRA,GBR, ITA.

Figure A. 8 :

 8 Figure A.8: SQP Ratios and annualized volatility time series computed over 1 year (T = 1y) for α = 95% and α = 99%. The two figures corresponding to an index are shown next to each other, i.e. we have two indices per row. The indices are shown in alphabetical order: JPN, NLD, SGP, SWE, USA

Figure A. 9 :

 9 Figure A.9: SQP Ratios as a function of annualized volatility for p = 0 for α = 95%, 99%. In each row we consider the SQP Ratios and then the logarithm of the SQP Ratios: From left to right, SQP ratio for α = 95%, 99%, then logarithm of SQP ratio for α = 95%, 99%. From top to bottom: AUS, CAN, DEU, FRA.

Figure A. 10 :

 10 Figure A.10: SQP Ratios as a function of annualized volatility for p = 0 for α = 95%, 99%.In each row we consider the SQP Ratios and then the logarithm of the SQP Ratios: From left to right, SQP ratio for α = 95%, 99%, then logarithm of SQP ratio for α = 95%, 99%. From top to bottom: GBR, ITA, JPN, NLD.

Figure A. 11 :

 11 Figure A.11: SQP Ratios as a function of annualized volatility for p = 0 for α = 95%, 99%.In each row we consider the SQP Ratios and then the logarithm of the SQP Ratios: From left to right, SQP ratio for α = 95%, 99%, then logarithm of SQP ratio for α = 95%, 99%. From top to bottom: SGP, SWE, USA

Figure A. 12 :

 12 Figure A.12: SQP Ratios as a function of annualized volatility for one simulated sample path of the GARCH(1,1) -which is fitted to each index separetely -for p = 0 for α = 95%, 99%.In each row we consider the SQP Ratios and then the logarithm of the SQP Ratios: From left to right, SQP ratio for α = 95%, 99%, then logarithm of SQP ratio for α = 95%, 99%. From top to bottom: AUS, CAN, DEU, FRA.

Figure A. 13 :

 13 Figure A.13: SQP Ratios as a function of annualized volatility for one simulated sample path of the GARCH(1,1) -which is fitted to each index separetely -for p = 0 for α = 95%, 99%.In each row we consider the SQP Ratios and then the logarithm of the SQP Ratios: From left to right, SQP ratio for α = 95%, 99%, then logarithm of SQP ratio for α = 95%, 99%. From top to bottom: GBR, ITA, JPN, NLD.

Figure A. 14 :

 14 Figure A.14: SQP Ratios as a function of annualized volatility for one simulated sample path of the GARCH(1,1) -which is fitted to each index separetely -for p = 0 for α = 95%, 99%.In each row we consider the SQP Ratios and then the logarithm of the SQP Ratios: From left to right, SQP ratio for α = 95%, 99%, then logarithm of SQP ratio for α = 95%, 99%. From top to bottom: SGP, SWE, USA

  500

Figure A. 15 :

 15 Figure A.15: SQP rolling every month with T = 3 year, and thresholds α = 95% and 99% and different values of p using the S&P 500 index. From left to right: p = 0, p = 0.5, p = 1 and p = 2. Dashed lines correspond to VaR(α) computed over the whole sample.

Figure A. 16

 16 Figure A.16 Average over the whole sample of rolling-window SQPs with sample size T = 3y (S&P 500) as a function of the power p for various thresholds α.

Figure A. 17

 17 Figure A.17RMSE (defined in (3.2)) of the SQP Ratios R p,α,3 (t) (i.e. on a 3-year sample) taking p = 0, 0.5, 1, 2 and two thresholds α = 95% (in blue), 99% (in red), on the S&P 500

Figure A. 18 :Figure A. 19 :

 1819 Figure A.17RMSE (defined in (3.2)) of the SQP Ratios R p,α,3 (t) (i.e. on a 3-year sample) taking p = 0, 0.5, 1, 2 and two thresholds α = 95% (in blue), 99% (in red), on the S&P 500

Table 1 :

 1 Information about the different indices used

	Country	Country Code Index	Description
	Australia	AUS	AORDD	The All Ordinaries index (AOI) is made up of the 500
				largest companies listed on the Australian Securities Ex-
				change (ASI)
	Canada	CAN	GSPTSED	The Canada S&P/TSX 300 Composite comprises ap-
				proximately 71% of market capitalization for Canadian-
				based, Toronto Stock Exchange (TSX) listed companies
	France	FRA	CACTD	The CAC All-Tradable Index represents the 250 largest
				stocks, by capitalisation, traded on Euronext Paris
	Germany	DEU	CDAXD	Germany CDAX Total Return Index includes the shares
				of all companies listed on Frankfurt Stock Exchange
	Great Britain GBR	FTASD	UK FTSE All-Share Index is capitalisation-weighted and
				comprises more than 600 (out of 2,000+) companies
				traded on the London Stock Exchange
	Italy	ITA	BCIPRD	Italy BCI Global Return Index includes over 325 compa-
				nies listed on the Milan Stock Exchange
	Japan	JPN	TOPXD	The Tokyo Price Index (TOPIX) is a capitalization-
				weighted price index of all First Section stocks (more
				than 1600 companies)
	Netherlands	NLD	AEX	The AEX index is a stock market index composed of a
				maximum of 25 of the most actively traded securities of
				Dutch companies on Euronext Amsterdam
	Singapore	SGP	STI	FTSE Straits Times Index (STI) is capitalisation-
				weighted and tracks the performance of the top 30 com-
				panies listed on the Singapore Exchange
	Sweden	SWE	OMXAFGX Sweden OMX Affärsvärldens General Index includes all
				the shares of companies registered on the Stockholm
				Stock Exchange
	United States
	of America	USA	SPXD	S&P 500 Composite Price Index represents about 75% of
				the stock market's capitalization

Table 4 :

 4 SQP ratios (defined in (3.1)) average over the whole historical sample, for 1 year, for each index and for two different thresholds (95 and 99 %). In the last column, we present the average over all indices ± the standard deviation.

	Mean for	AUS	CAN	FRA	DEU	ITA	JPN	NLD	SGP	SWE	GBR	USA	AVG (± σ)
						SQP ratio for 1y (α = 95%)
	p = 0	1.04	1.09	1.08	1.11	1.10	1.11	1.10	1.06	1.10	1.08	1.06	1.08 (0.02)
	p = 0.5	0.81	0.83	0.81	0.85	0.84	0.85	0.83	0.78	0.84	0.83	0.82	0.83 (0.02)
	p = 1	0.64	0.65	0.65	0.64	0.66	0.68	0.66	0.60	0.68	0.68	0.67	0.66 (0.02)
	p = 2	0.50	0.47	0.50	0.51	0.49	0.50	0.50	0.44	0.50	0.51	0.50	0.49 (0.02)
						SQP ratio for 1y (α = 99%)
	p = 0	1.04	1.10	1.06	1.10	1.11	1.14	1.11	1.12	1.09	1.08	1.08	1.09 (0.03)
	p = 0.5	0.80	0.81	0.85	0.88	0.83	0.85	0.88	0.82	0.83	0.88	0.84	0.84 (0.03)
	p = 1	0.74	0.77	0.79	0.80	0.78	0.79	0.80	0.74	0.76	0.81	0.79	0.78 (0.02)
	p = 2	0.74	0.77	0.79	0.80	0.78	0.79	0.80	0.74	0.76	0.81	0.79	0.78 (0.02)

To be clear, let us look at a simple example. Choosing α = 0.95, t = January 2014, and T = 1 year, the denominator Q p,0.95,1 of the ratio R p,0.95,1 (t) corresponds then to the empirical estimate of the SQP computed on a 1 year sample from January 2013 (t -T ) to December 2013 (i.e. the month before t, as we are considering the interval [t -T, t)), which acts as an estimate for our future risk in 2014 (if T would be 3 years, Q p,0.95,3 would be computed from January 2011 up to December 2013). We define the future risk Q 0,α,1 (t + 1y) as the VaR estimated on the one year sample from January 2014 (t) until December 2014 (last month before t + 1y). Finally, the ratio of the two quantities is built.

Table 5 :

 5 Average RMSE for the SQP ratios (eq. 3.1, with T = 1 year) -using historical data for each index. In the last column, we present the average over all indices ± the standard deviation.

	Mean for	AUS	CAN	FRA	DEU	ITA	JPN	NLD	SGP	SWE	GBR	USA	AVG (± σ)
					RMSE of the SQP ratio for 1y (α = 95%)
	p = 0	0.36	0.46	0.44	0.50	0.49	0.54	0.52	0.46	0.47	0.44	0.42	0.46 (0.05)
	p = 0.5	0.33	0.42	0.38	0.43	0.39	0.46	0.42	0.42	0.39	0.37	0.38	0.40 (0.03)
	p = 1	0.44	0.47	0.43	0.47	0.45	0.50	0.46	0.52	0.44	0.42	0.45	0.46 (0.03)
	p = 2	0.56	0.59	0.54	0.55	0.55	0.60	0.55	0.62	0.56	0.53	0.57	0.57 (0.03)
					RMSE of the SQP ratio for 1y (α = 99%)
	p = 0	0.37	0.56	0.39	0.50	0.52	0.61	0.58	0.72	0.52	0.47	0.51	0.52 (0.10)
	p = 0.5	0.45	0.49	0.39	0.45	0.42	0.58	0.48	0.65	0.46	0.44	0.54	0.49 (0.08)
	p = 1	0.45	0.49	0.38	0.42	0.40	0.53	0.44	0.59	0.45	0.41	0.52	0.46 (0.06)
	p = 2	0.45	0.49	0.38	0.42	0.40	0.53	0.44	0.59	0.45	0.41	0.52	0.46 (0.06)

Table 8 :

 8 Descriptive Statistics of the GARCH models for the 11 indices

		AUS CAN FRA DEU ITA JPN NLD SGP SWE GBR USA
	GARCH Parameters
	ω [10 -6 ]	4.13	1.32	3.11	2.51 2.66 5.96 2.63	3.42	4.08	2.10	2.24
	α [10 -1 ]	1.66	1.08	0.98	0.95 0.83 1.48 1.12	1.33	1.12	1.13	1.07
	β [10 -1 ]	7.93	8.79	8.83	8.89 9.02 8.19 8.74	8.48	8.65	8.68	8.76
	α + β	0.96 0.99	0.98	0.98 0.99 0.97 0.99 0.98	0.98	0.98	0.98
	Fitting Results
	Likelihood [10 4 ]	2.54	2.58	2.35	2.39 2.32 2.25 2.37	2.36	2.30	2.51	2.45
	Volatility [%]	16.0	15.8	19.9	20.0 20.9 21.3 21.3	20.9	21.0	16.6	18.3
	Historical [%]	16.0	15.9	20.0	20.1 21.1 21.3 21.6	21.0	21.1	16.7	18.5

Looking for Explanations via GARCH modellingIn our empirical analysis, we observe volatility clustering as well as the return to the mean of volatility. Those are both prominent characteristics of a GARCH model. Thus, we think that part of the procyclical behavior with volatility might be explained by this clustering.

Acknowledgement: We are grateful to Prof. Jiro Akahori for his suggestion of using the SQP and for fruitful discussions on it.

Appendix A: Tables

A. [START_REF] Akahori | Some Formulae for a New Type of Path-Dependent Option[END_REF] Rank Correlation on Real Data Table A.1: Spearman rank correlation between the SQP ratios and the volatility, for each index and for T = 1 year, over the whole historical sample, and for two thresholds (95 and 99%). In the last column, we present the average over all indices ± the standard deviation.

C.1.2 On Simulated GARCH realizations

Table A.9: Pearson correlation between the logarithm of the SQP ratios and the volatility, for each index and for T = 3 years, using a GARCH(1,1) model (average over 1000 realizations), and for two thresholds (95 and 99%). In the last column, we present the average over all indices ± the standard deviation.


Pierre Franck

Luan Odorizzi

Marlice Santos

Claire Botelho Costa

Odair Aparecido Lavigne

Armin Fernandes

Bischoff

Odair Lavigne

Aparecido

Alberto Luan

Odorizzi

Marlice Botelho Costa

Claire Lavigne

Odair Aparecido Fernandes

Armin Bischoff

Impact of agro-ecological infrastructures on the control of insect pests in apple orchards

Keywords: conservation biological control, predation, parasitism, codling moth, rosy apple aphid, floral resources

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Flowering strips significantly increase the local abundance and biodiversity of flying predators and parasitoids in the agroecosystem [START_REF] Marshall | Field margins in northern Europe: their functions and interactions with agriculture[END_REF]. However, only few studies have demonstrated their role in pest control and their impact on the dynamic of arthropod populations and on trophic interaction within the crop [START_REF] Lavandero | Increasing floral diversity for selective enhancement of biological control agents: A double-edged sword?[END_REF][START_REF] Géneau | Selective flowers to enhance biological control of cabbage pests by parasitoids[END_REF].

The objective of this work was to test the effect of biodiversity in flowering strips on the biological control of two major insect pests in apple orchard, the codling moth (CM) and the rosy apple aphid (RAA).

Material and methods

Orchard experimental design

The experiment was carried out in 2014 and 2015 in three untreated apple orchards (480 to 1200 m 2 ) at INRA Montfavet, France (43°54'51.57 "N, 4°52'56.15" E). Three strip treatments (30 to 70 m 2 ) were established along the longer edge of each orchard (Figure 1): (a) a wildflower strip mixture comprising 30 vascular plant species (WS) optimized for high and long lasting productions of floral resources, (b) a grass strip mixture (GS) including Lolium perenne (28%) and Festuca arundinacea (72%), (c) an unsown strip allowing the natural development of spontaneous vegetation (SV). Grass and wildflower strips were sown in orchard edge in March 2014.

Figure 1. Schema of the experimental design. Shaded symbols indicate the selected trees to monitor apple pests and their enemies in each apple orchard.

Entomological observations

The arthropod communities involved in RAA and CM control were monitored at 18 or 27 apple trees per orchard (Figure 1). These observation trees were selected at three distances from the margin strips (1 m, 4 m and 13 m). The same trees were selected in 2014 and 2015. The number of RAA colonies per apple tree was counted at the end of the RAA growing period (19/05/14 and 03/06/15). The occurrence of predators and aphid mummies were recorded in up to six RAA colonies per tree (Table 1; [START_REF] Dib | Effect of management strategies on arthropod communities in the colonies of rosy apple aphid, Dysaphis plantaginea Passerini (Hemiptera: Aphididae) in south-eastern France[END_REF]. CM larvae were collected weekly from May to October in cardboard band traps wrapped around the tree trunks. The number of CM larvae and adult emergences were recorded to infer the parasitism of diapausing and non diapausing larvae (Table 2; [START_REF] Maalouly | Temporal dynamics of parasitoid assemblages parasitizing the codling moth[END_REF]. Furthermore, the percentages of flowering entomophilous plants and the number of predators and parasitoids in each strip treatment were recorded in May of each year to shed light on their respective impact on the control of the apple pests.

Statistical analysis

The occurrence of predators in RAA colonies and CM parasitism were analyzed using generalized linear models assuming a binomial distribution (glmer function, lme4 package, R version 3.2.4). The strip treatment, the distance to the orchard margin and their interaction were factors tested. All models also included the observation date as an additional fixed factor to account for temporal variation and a random 'orchard' effect to account for a block factor.

Results and discussion

The percentages of flowering entomophilous plants in the margin strips were significantly higher in WS than in SV and in GS both year (Figure 2). We mainly observed hoverflies and ladybirds in the margin strips. The number the hoverflies was higher in WS than SV and GS in 2015 (Figure 2). However, we did not detect significant differences in the number of ladybirds and hymenopteran parasitoids among the margin strips.

The abundances of RAA colonies and CM larvae per apple tree were not homogenously distributed within the orchards. RAA infestations were significantly higher at the closest distances from the margin, but infestations were lower close to WS than close to GS and SV strips. Inversely, the number of CM larvae were significantly lower at the lowest distances from margin strips.

The occurrence of predators and parasitoids in RAA colonies and codling moth parasitism within the orchards were not significantly different among the classes of distances to the margin and among the margin strip treatments. However, the arthropod communities involved in RAA and CM control significantly changed within and between years (Tables 1 &2). These results suggest that provisioning of additional flower resources in orchard margins had a limited impact on the abundance of pest natural enemies both within the orchards and their edges if floral resources are already present in the environment. Selection of entomophilous plants according to temporal natural enemy's requirement could additionally enhance RAA and CM controls in the apple orchards.

Figure 2 .

 2 Figure 2. Percentage of entomophilous plants in flower (upper panels) and number of overflies (lower panels) in WS (white), SV (grey) and GS (black) margin strips in May 2014 (left panels) and 2015 (right panels).

Table 1 .

 1 Mean number of RAA colonies per apple tree and mean proportions of RAA with an observed predator (hoverflies, ladybirds and earwigs) or a mummy (parasitoids) in 2014 and in 2015. Minimum and maximum values per orchard are indicated in brackets.

		2014	2015
	RAA colonies	41.9 [31.8-56.0]	22.3 [11.3-27.4]
	Hoverflies	0.1 [0.0-0.4]	0.7 [0.4-0.9]
	Ladybirds	0.6 [0.5-0.9]	0.5 [0.2-0.7]
	Earwigs	0.6 [0.5-0.9]	0.2 [0.0-0.8]
	Parasitoids	0.7 [0.5-1.0]	0.6 [0.3-0.7]

Table 2 .

 2 Mean number of CM larvae per apple tree, parasitism rates and proportion of parasitoids (Ascogaster quadridentata, Pristomerus vulnerator and Perilampus tristis) in 2014. A total of 2673 CM larvae were collected (including 927 diapausing). The diapausing samples correspond to the emergences that occurred in 2015. Minimum and maximum values per orchard are indicated in brackets.

		Non-diapausing	Diapausing
	CM larvae	27.7 [18.4-33.4]	14.7 [6.9-19.8]
	Parasitism rate	0.27 [0.15-0.36]	0.32 [0.15-0.38]
	A. quadridentata	0.59 [0.33-0.63]	0.43 [0.21-0.69]
	P. vulnerator	0.13 [0.10-0.26]	0.17 [0.00-0.23]
	P. tristis	0.27 [0.26-0.40]	0.38 [0.31-0.51]

Acknowledgements

This work was supported by the Agropolis Fondation (ANR-10-LABX-0001) and by the PEERLESS project (ANR-12-AGRO-0006).


Kilian Croci
email: kilian.croci@ensam.eu

Florent Ravelet

Jean-Christophe Robinet

Amélie Danlos

Experimental Study of Cavitation in Laminar Flow 1

Keywords: laminar flows, gaseous cavitation regimes, cavitation inception, bubbles, hysteresis

An experimental set-up has been especially developed in order to observe cavitation in laminar flows. Experiments have been carried out with a silicon oil of viscosity passing through a Venturi-type geometry with 18°/8° convergent/divergent angles respectively. The range of Reynolds numbers at the inlet section is between 350 and 1000. For these flow conditions, the Venturi only experiences air releasing gaseous cavitation for which two dynamic regimes are identified. They are characterized by two critical Reynolds numbers, induced by major hydrodynamic changes in the flow, in addition to a hysteresis effect between the inception and the disappearance of cavitation.

Introduction

Several regimes can occur in turbulent cavitating flows. In studies about water flows coming across obstacles, and in particular with Venturi geometries, four main cavitation regimes can be identified:  An intermittent regime which corresponds to the cavitation inception. This regime occurs for relatively high pressures and is characterized by the intermittent inception and the implosion [START_REF] Brandner | Cavitation about a jet in crossflow[END_REF] of small vapor bubbles at the Venturi throat.  For lower pressure conditions a "closed cavity" regime, presenting relatively stable vapor/liquid mixture cavities, is observed. Re-entrant jets, induced by a recirculation flow, occur at the cavity rear inducing the mixing of liquid and vapor without cloud shedding.  A transient regime of partial cavitation [START_REF] Croci | Investigation of two mechanisms governing cloud cavitation shedding: experimental study and numerical highlight[END_REF] is obtained decreasing the pressure in the test section. This highly unsteady regime is characterized by several cloud cavitation shedding dynamics induced respectively by: the re-entrant jet and by the periodical condensations and implosions of coherent cloud cavitation structures named condensation shock wave [START_REF] Ganesh | Bubbly shock propagation as a mechanism for sheet-to-cloud transition of partial cavities[END_REF].  For really low pressure conditions, cavities can reach important sizes and a periodic regime (also called supercavitation) can be established [START_REF] Brennen | Cavitation and bubble dynamics[END_REF]. In this particular regime the condensation shock wave, which occurs periodically, is the dominant shedding dynamic in the flow.

These regimes and shedding mechanisms still remain of principal interest both experimentally and numerically. However cavitation flows have, to the authors' knowledge, been very little investigated using fluids with highviscosity [START_REF] Washio | Study on cavitation starting at the point of separation on a smooth wall in hydraulic oil flow[END_REF]. The present study focuses on the different cavitation regimes observed in laminar flows. Two gaseous cavitation regimes are highlighted and a comparison is made with the turbulent regimes previously presented. The paper is organized as follow: first the experimental set-up is described in §2. The cavitation regimes are then highlighted ( §3) and finally conclusions and perspectives are proposed in §4 and §5.

Experimental Set-up

The experiments are conducted in a closed test loop where a pump impels a silicone oil flow with a flow rate from to . The Bluesil FLD 47 V100 ® silicone oil main characteristics are given in Table 1.

Viscosity ( ) Volumic Mass ( ) Vapor Pressure ( ) Surface Tension ( ) The test section is represented in figure 1. It consists of a rectangular inlet section of width and height and a square section at the throat. The Venturi-type geometry presents convergent/divergent angles respectively. The velocity is measured with an ultrasonic flow-meter and studied for at the inlet test section corresponding to Reynolds numbers ( ) taken from to . In the meantime the pressure is measured in the test section upstream ( ) and downstream ( ) of the throat. The cavitation number, such as the Reynolds number, is defined at the beginning of the test section as follow:

⁄

. Finally, the temperature is measured with a PT100 sensor and the kinematic viscosity is reevaluated accordingly based on our own calibration. In the following parts of the study, the inception and the different cavitation regimes will be described and characterized as a function of the cavitation and the Reynolds numbers (figure 2).

Cavitation Inception

The inception of cavitation, widely studied into water, is an intricate phenomenon which depends on several parameters. As explained in the works of Brennen [START_REF] Brennen | Cavitation and bubble dynamics[END_REF], it might depend on the water quality regarding on the dissolved and the free gas content. For a saturated liquid experiencing a low pressure region, important outgassing and vaporization may occur in the flow generating the intermittent regime described in §1. Moreover when a nucleus goes through a region with a critical pressure drop in the surrounding liquid, its radius would be able to grow unboundedly. Such as in a turbulent water-flow a low pressure region can be generated in a laminar flow, inducing outgassing and nuclei explosive growths. In a Venturi geometry the velocity, as well as the pressure drop, is maximum at the center of the throat. Consequently a large air bubble might be generated in this central region which is illustrated in figure 2. After this inception, the bubble is carried away by the stream and, contrary to the water-flows, does not implode because of the high viscosity of the oil. A part of the air bubble can then be captured by the separated zone Separated flow Pressure drop zone *Corresponding Author, Kilian Croci: kilian.croci@ensam.eu downstream of the Venturi throat which is already present in the non-cavitating flow at low Reynolds number. In summary for a critical Reynolds number , an adverse pressure gradient can be generated leading to an explosive bubble growth near the separated region. The recirculation region can then captures a part of the expanded bubble and make it reach back to the Venturi throat on one wall-side. After a few repetitions of this cycle, two elongated "tadpole shape" bubbles settle on both side-wall at the Venturi throat.

As a result the cavitation inception, previously defined, occurs above a critical inlet Reynolds number and depends on the cavitation number. The inception conditions ( ) are exposed in figure 3 with the blue curve. Under this curve, a dynamic regime can be defined: the bubbly recirculation regime, characterized by bubbles' inception and recirculation to the wall-sides of the Venturi throat. For lower Reynolds numbers and higher cavitation numbers, the recirculation regime disappears but not the two tadpole bubbles at the throat which oscillate with micro-bubble sheddings until a critical desinent Reynolds number . It defines a second regime called the tadpole regime. The figure 3 shows the hysteresis between the inception and disappearance of cavitation with the two different cavitation regimes previously introduced. The two asymptotes, corresponding to the two critical Reynolds numbers characterized earlier, might be due to the recirculation flow characteristics. For Reynolds numbers higher than , the oil flow might be fast enough to generate a pressure drop zone at the throat and, in the meantime, the shear stress might be enough developed in order that a bubble generated at the throat get captured by a thick recirculation zone and go back to the Venturi throat. The desinent critical Reynolds number , which is around , might correspond to the disappearance of the recirculation zone. Near this condition, the two bubbles become less and less stable and symmetry breaks can often be observed. Regarding to the figure 3, the two different regimes can be defined depending on the flow conditions. The bubbly recirculation regime is defined by the inception curve (occurring for ) for which the cavitation number slowly decreases with the Reynolds number. Under this curve, bubbles are generated at the throat, are carried away by the flow, get captured by the recirculation zone and its negative velocities and, finally, they reach the Venturi slope to the throat where they settle to one or the other wall-side. It results in the presence of two attached bubbles at both wall-side of the Venturi throat in addition to recirculating bubbles as illustrated in figure 4. On the other hand the tadpole regime (figure 5) is the result of a hysteresis between inception and the disappearance of these two bubbles. Indeed after the settlement of the bubbles at the throat, they do not disappear immediately when the Reynolds number is decreased and the cavitation number is increased. However in these conditions the bubbly recirculation regime is stopped and only the two oscillating bubbles persist. The tadpole regime occurs for flow conditions which are below the desinent curve (red dots in figure 2) and is limited by the critical desinent Reynolds number . For Reynolds numbers , the desinent curve seems to rejoin the inception curve meaning that in high velocity conditions the hysteresis disappears. In these two regimes micro-bubble sheddings can be observed, such as in figure 4 and 5. These sheddings, that occur with high frequencies, might be due to the gas diffusion [START_REF] Gross | Diffusion-driven nucleation from surface nuclei in hydrodynamic cavitation[END_REF] of the dissolved air present in the flow at interface of bubbles. This mechanism induces a growth of the bubble until a critical size leading to the shedding.

Cavitation Regimes

Conclusions

In the present paper, a first overview about the different cavitation regimes in laminar flows ( ) has been made and the outcomes have been compared to the usual dynamical regimes observed into water-turbulent cavitating flows. The following summarizes the important conclusions.

Y (mm) a) b)

Y (mm)

*Corresponding Author, Kilian Croci: kilian.croci@ensam.eu  Two different gaseous cavitation dynamical regimes have been identified in the laminar flow, namely the bubbly recirculation and the tadpole regime.  The inception of cavitation in the oil is similar to that one observed into water with the intermittent regime except that the bubbles generated do not condensate (and implode) in higher pressure regions, due to the high viscosity of the oil.  The study of cavitation inception and desinence reveals a hysteresis which induces the tadpole regime.  The two regimes are characterized by two critical Reynolds numbers, and , which represent hydrodynamic transitions: namely the disappearance of the recirculation region and the emergence of an adverse pressure gradient at the throat.

Future Developments

In an upcoming article, the cavitation inception will be highlighted with top and side visualizations while the two regimes exposed in this paper will be characterized with the measurements of bubble sizes and shedding frequencies depending on cavitation and Reynolds numbers. In a meantime other dynamical regimes will be investigated for higher velocity inlet and thus higher Reynolds numbers. The first observations suggest the inception of a second bubbly recirculation zone at the top of the Venturi as well as a transition from gaseous to hydrodynamic cavitation for higher Reynolds numbers.

Figure 1 -

 1 Figure 1 -Venturi geometry sketch. The width of the test section is and the height at the inlet section .

Figure 2 -

 2 Figure 2 -Sketch of the flow at the inception. The flow separation occurs at relatively low Reynolds numbers in non-cavitating flows but when an adverse pressure gradient is generated near this region, the growth and capture of air bubbles become possible leading to attached air cavities and recirculating bubbles.

Figure 3 -

 3 Figure 3 -Map of the dynamic regimes as a function of Reynolds and cavitation numbers. The blue and the red curves correspond repectively to the inception and to the desinence of cavitation. The blue dotted line denotes the inception critical Reynolds number whereas the red dotted line represents the desinent critical Reynolds number . The cycle of cavitation hysteresis is defined as follow. I : Absence of cavitation; (decreasing the cavitation number) II : Inception of cavitation and bubbly recirculation regime; (decreasing the Reynolds number) III : Subcritical tadpole regime and disappearance of cavitation when the cavitation number is enough increased. The uncertainties are estimated to for the Reynolds numbers and to 2% for the cavitation numbers.

Figure 4 -

 4 Figure 4 -Top view of the bubbly recirculation regime captured at and . The oil stream is coming from the left to the right. The Venturi throat is positioned at . Dark regions are indicative for the contours of the bubbles. The two tadpole bubbles (left) are attached near the throat with micro-bubble shedding whereas recirculating bubbles (right) reach back the Venturi slope to the throat.

Figure 5 -

 5 Figure 5 -Two typical "Tadpole" bubbles observed with a top view for (a) and and for (b) and .

Table 1 -

 1 Characteristics of Bluesil FLD 47 V100 ® silicone oil at given by the constructor.

Acknowledgement

The authors wish to acknowledge the great assistance of Mme Guylaine Ducouret from ESCPI Paris for the characterization of the silicone oil used in the present study.