Xavier Boulanger

Laurent Castanet

Ka and Q band Propagation Experiments in Toulouse using ASTRA 3B and ALPHASAT Satellites

Keywords: Alphasat, ASTRA 3B, Attenuation, Frequency Scaling, Propagation, Satellite Communications

ONERA operated in Toulouse, France, a beacon receiver able to collect the 19.7 GHz beacon signal of the HotBird 6 satellite. In March 2011, the RF chain was modified to be able to receive the 20.2 GHz ASTRA 3B beacon to benefit from a higher EIRP of the satellite. Since June 2015, a second beacon receiver has been installed and is able to record the 39.4 GHz beacon signal of the Alphasat satellite. The objective of this paper is to present the results of the first two years of measurements (June 2015-May 2017) of the Q-band Alphasat propagation experiment in Toulouse, as well as concurrent measurements collected at Ka-band with ASTRA 3B. First of all, the propagation experiments are briefly described. Secondly, the results of the statistical analysis are highlighted. Finally, a deep analysis of the performances of the two frequency scaling prediction methods recommended in ITU-R P.618-12 is performed.

INTRODUCTION

The Alphasat satellite was launched in July 2013. Two beacons, one at Ka-band and one at Q-band, constitute the Alphasat Aldo Paraboni scientific payload [START_REF] Koudelka | Q/V-band communications and propagation experiments using ALPHASAT[END_REF], [START_REF] Rossi | Satellite Communication and Propagation Experiments Through the Alphasat Q/V Band Aldo Paraboni Technology Demonstration Payload[END_REF]. Both signals are used by the propagation community around Europe for collecting propagation statistics [START_REF] Riva | The Aldo Paraboni Scientific Experiment: the preparation and plans for a European measurement campaign[END_REF]. The collaborative Group of the AlphaSat Aldo Paraboni propagation Experimenters (ASAPE) ensures the collaboration between all research organizations for the execution of the campaigns, data processing and analysis to test propagation models.

Over the past decade, ONERA has been involved into several Earth-space propagation experiments. From January 2008 to March 2011, ONERA operated in Toulouse, France, a beacon receiver able to collect the 19.7 GHz beacon signal of the HotBird 6 satellite [START_REF] Boulanger | Four Years of Total Attenuation Statistics of Earth-Space Propagation Experiments at Ka-Band in Toulouse[END_REF]. In March 2011, the RF chain was modified to be able to receive the 20.2 GHz ASTRA 3B beacon [START_REF] Boulanger | Four Years of Total Attenuation Statistics of Earth-Space Propagation Experiments at Ka-Band in Toulouse[END_REF], [START_REF] Boulanger | Small and large scale site diversity experiment at Ka-band in the south of France[END_REF]. From mid-2013 to end of 2014, ONERA and CNES deployed four other Ka-band beacon receivers in the South of France in order to assess the performances of Site Diversity techniques [START_REF] Boulanger | Small and large scale site diversity experiment at Ka-band in the south of France[END_REF]. Since June 2015, a Q-band beacon receiver has been installed in Toulouse (co-located to the first ASTRA 3B beacon receiver) and is able to record the 39.4 GHz beacon signal of the Alphasat satellite [START_REF] Boulanger | Alphasat Propagation Experiment in Toulouse: Two years of measurements[END_REF]. Section 2 of this paper is dedicated to the description of the Ka and Q band propagation experiments. The link parameters as well as the experimental set-up are introduced. The results of the statistical analysis, in particular the monthly and annual fluctuations at Q-band, are highlighted in Section 3. Finally, as concurrent measurements at Ka-band are available (using the co-located ASTRA 3B beacon receiver), a deep analysis of the performances of the two frequency scaling prediction methods recommended in ITU-R P.618-12 [START_REF]Propagation data and prediction methods required for the design of Earth-space telecommunication systems[END_REF] is performed. The results are shown in Section 4.

DESCRIPTION OF THE KA AND Q BAND PROPAGATION EXPERIMENTS

Main characteristics

The Ka and Q band propagation experiments are performed on the rooftop of the Electromagnetism and Radar Department building of the ONERA Toulouse Centre, France, located at 43.5710°N of latitude and 1.4716°E of longitude (Figure 1). Situated in the South-West of France, Toulouse is subject to both Atlantic Ocean and Mediterranean Sea influences. The climate normals, computed over the period 1981-2010, are the following [8]: the mean annual temperature is 13.8°C, the mean annual minimum temperature is 9.1°C and the mean annual maximum temperature is 18.5°C. The mean number of rainy days is 95.7 days/year and the mean rainfall amount is 638.3 mm/year. The precipitations are not homogeneously distributed all over the months and remain between 37.7 mm in July (the driest month) and 74 mm in May (the wettest month).

At the end of March 2011, the Ka band beacon receiver previously used for the HotBird 6 experiment was upgraded to benefit from the better EIRP of the SES ASTRA 3B Ka band beacon [START_REF] Boulanger | Four Years of Total Attenuation Statistics of Earth-Space Propagation Experiments at Ka-Band in Toulouse[END_REF], [START_REF] Boulanger | Small and large scale site diversity experiment at Ka-band in the south of France[END_REF]. ASTRA 3B is located on a geostationary orbit at 23.5°E of longitude. The signal source is the 20.199 GHz beacon (vertical polarization) with a beacon EIRP of 28.8 dBW over Toulouse. The azimuth and elevation angles from Toulouse are respectively 149.57° and 35.11º. The signal level is recorded at a 10 Hz sampling rate and the dynamic range of the beacon receiver is around 37 dB for a 50 Hz receiver bandwidth.

Since June 2015, a Q-band beacon receiver has been installed in Toulouse (colocated to the first ASTRA 3B beacon receiver) and is able to record the 39.402 GHz (linear tilted 45° polarization) beacon signal of the Alphasat satellite [START_REF] Boulanger | Alphasat Propagation Experiment in Toulouse: Two years of measurements[END_REF]. Alphasat is located on an inclined geostationary orbit (maximum ± 3°) at 25°E of longitude. The Alphasat Q band beacon EIRP is about 26 dBW over Toulouse. The mean azimuth and elevation angles from Toulouse are respectively 147.70° and 34.49º. The signal level is also recorded at a 10 Hz sampling rate and the dynamic range of the beacon receiver is around 30 dB for a 50 Hz receiver bandwidth.

Experimental set-up

The experimental set-up of the Ka and Q band propagation experiments in Toulouse is the following:

• An indoor Ka-band beacon receiver,

• An indoor Q-band beacon receiver,

• A profiling radiometer for total attenuation calibration,

• A tipping-bucket rain gauge for rainfall rate measurements,

• A GNSS receiver for water vapour retrieval algorithm testing (not described in this paper as out of the scope).

They are briefly described in the following sections.

Ka and Q band beacon receivers

At the end of the 90's, ONERA designed and built two indoor beacon receivers (20.7 and 41.4 GHz) for the Stentor propagation experiment [START_REF] Lemorton | A new opportunity for EHF propagation experiments: the EXPRESS campaign with the satellite STENTOR[END_REF]. After the launch failure and the loss of the satellite, the 20.7 GHz beacon receiver was modified to receive firstly the 19.7 GHz Hotbird 6 beacon [START_REF] Boulanger | Four Years of Total Attenuation Statistics of Earth-Space Propagation Experiments at Ka-Band in Toulouse[END_REF] and secondly the 20.2 GHz ASTRA 3B beacon [START_REF] Boulanger | Four Years of Total Attenuation Statistics of Earth-Space Propagation Experiments at Ka-Band in Toulouse[END_REF], [START_REF] Boulanger | Small and large scale site diversity experiment at Ka-band in the south of France[END_REF]. The Alphasat mission has given the opportunity to work at higher frequency with the Qband beacon of the Alphasat Aldo Paraboni Scientific payload. Consequently, the second beacon receiver has been slightly upgraded to receive the 39.4 GHz Alphasat beacon signal [START_REF] Boulanger | Alphasat Propagation Experiment in Toulouse: Two years of measurements[END_REF].

The main characteristics of the two indoor beacon receivers are very similar. Basically, they consist of a Cassegrain parabolic antenna of 1.20 m diameter with the RF chain immediately placed at the output of the antenna to avoid feeder loss. Three successive frequency down-conversions are performed:

• a first one from 20.2 GHz to 110 MHz or from 39.4 GHz to 220 MHz for the Ka and Q band beacon receivers respectively;

• a second one from 110 to 10 MHz or from 220 to 20 MHz for the Ka and Q band beacon receivers respectively, • a third one from 10 MHz to 2 kHz or from 20 MHz to 2.5 kHz for the Ka and Q band beacon receivers respectively.

The Alphasat beacon receiver is equipped with a dynamic tracking system to counteract orbit inclination. It is based on the Orbit Ephemeris Messages files distributed by the ASAPE Group. The resolutions of the azimuth and elevation actuators are respectively 0.003° and 0.002°. The reachable ranges are 11° for the azimuth actuator and 6.2° for the elevation actuator. The positions of both actuators are updated every minute. The ASTRA 3B beacon receiver has no dynamic tracking system (the antenna is only mechanically pointed once) as the satellite is relatively stable on its orbit. Figure 2 shows a picture of the co-located Ka and Q band beacon receivers deployed at Toulouse ONERA Centre premises. An example of raw (and uncalibrated) signal level recorded by both beacon receivers over one week (from 8 th to 14 th June 2015) is shown in Figure 3. A daily oscillation, mainly due to the slight satellite motion, can be observed on the ASTRA 3B beacon signal level. The peak-to-peak variation depends on the time of the initial pointing manoeuvre. The clear stability of the Alphasat beacon signal level proves that the tracking system works properly.

Profiling microwave radiometer

A microwave radiometer is often used in propagation experiments to calibrate the reference level of total attenuation in absorbing conditions [START_REF] Barbaliscia | Volume 3: reference book on radiometry and meteorological measurements[END_REF], [START_REF] Riva | COST IC0802 Handbook on measurements and products[END_REF]. The one deployed at Toulouse ONERA Centre premises is a TP/WVP-3000 multi-frequency profiling radiometer built by Radiometrics (USA) [START_REF] Solheim | Radiometric Profiling of Temperature, Water Vapor, and Cloud Liquid Water using Various Inversion Methods[END_REF], [START_REF] Solheim | Microwave Radiometer for Passively and Remotely Measuring Atmospheric Temperature, Water Vapor and Cloud Liquid Water Profiles[END_REF] which measures sky brightness temperatures at 12 frequency channels [START_REF] Boulanger | Four Years of Total Attenuation Statistics of Earth-Space Propagation Experiments at Ka-Band in Toulouse[END_REF]:

• 5 in Ka-band between 22 and 30 GHz around the water vapour absorption lines.

• 7 in V-band between 52 and 59 GHz corresponding to the oxygen absorption lines.

In the Ka and Q bands experiments, the radiometer is mainly used to infer total attenuation, Integrated Water Vapour Content (IWVC), and Integrated Liquid Water Content (ILWC) in absorbing conditions on the slant path. The computation of the reference level of total attenuation in absorbing conditions relies on the Radiative Transfer Equation and on the estimation of the mean radiating (or effective) temperature T mr [START_REF] Ulaby | Microwave Remote Sensing Active and Passive[END_REF]. For each radiometer frequency rad i f and a given elevation angle θ , the radiometric attenuation ( ) θ , rad i rad f A can be retrieved from [START_REF] Ulaby | Microwave Remote Sensing Active and Passive[END_REF]:

( ) ( ) ( ) ( )       - - ⋅ = θ θ θ θ , , , log 10 , 10 rad i b rad i mr c rad i mr rad i rad f T f T T f T f A (1)
where T c is the temperature of the cosmic microwave background equal to 2.73K [START_REF] Fixsen | The Temperature of the Cosmic Microwave Background[END_REF] and

( ) θ , rad i b f T
is the brightness temperature. Nevertheless, the beacon frequency f b generally does not match any of the radiometer frequencies. So, the reference level of total attenuation in absorbing conditions can be approximated from the following linear regression [START_REF] Barbaliscia | Volume 3: reference book on radiometry and meteorological measurements[END_REF], [START_REF] Riva | COST IC0802 Handbook on measurements and products[END_REF]:

( ) (
)

∑ ⋅ + = rad i f rad i rad i b rad f A a a f A θ θ , , 0 (2)
where the coefficients a i depend on the elevation, the location and the number of frequencies taken into account. Radiosounding (RAOBS) profiles representative of the sites where the beacon measurements are carried out are often used as input of a Radiative Transfer Simulator to retrieve those coefficients. In a similar way, IWVC and ILWC can be inferred from:

( ) ( )

∑ ⋅ + = rad i f rad i rad i f A b b IWVC θ θ , 0 (3) ( ) (
)

∑ ⋅ + = rad i f rad i rad i f A c c ILWC θ θ , 0 (4)
For both ASTRA 3B Ka-band and Alphasat Q-band propagation experiments, monthly slant path (35° of elevation) coefficients a i (for ASTRA 3B) and a' i (for Alphasat) have been derived using radiosounding profiles collected in Bordeaux (44.84°E, -0.58°E, 200 km away from Toulouse) and four radiometric channels (23.84, 30, 51.25, and 52.28 GHz) leading to :

A c A c A c c ILWC + + + = (7)
For the sake of conciseness, tables of coefficients will be not presented in this paper. Finally, the full methodology described in [START_REF] Boulanger | Four Years of Total Attenuation Statistics of Earth-Space Propagation Experiments at Ka-Band in Toulouse[END_REF] is then used to combine radiometric and beacon measurements to retrieve calibrated time series of rain and total attenuation. An example of calibrated time series of total attenuation is shown in Figure 4.

Tipping-bucket rain gauge

A corrosion resistant dual-chamber tipping bucket rain gauge able to record the exact instant of tips (1 second resolution) is also available. It was manufactured by FIEDLER, CZ and the main characteristics are:

• collecting area of 500 cm 2 • 0.1 mm per pulse resolution • 0.015 mm accuracy The rain gauge data are converted into rain rate measurements (in mm/h) according to the methodology described in [START_REF]On the processing of tipping bucket rain gauge data for study group 3 experimental database[END_REF]10.

STATISTICAL ANALYSIS

Two full years (June 2015 -May 2017) of data have been processed. Daily rain and total attenuation validated time series have been produced and statistical analyses have been performed. The availabilities of valid data used for the different statistical analyses are reported in Table 1.

Figure 5 shows the Complementary Cumulative Distribution Functions (CCDF) of concurrent rain and total attenuation at 20.2 GHz and 39.4 GHz. A small year-to-year variability can be noticed. As an example, the total attenuation values at 20.2 GHz exceeded for 1 % of the time are 1.9 dB and 1.6 dB for the first and second year of measurements, respectively. At 39.4 GHz, the total attenuation values exceeded for 1 % of the year are 4. Figure 6 shows CCDF of concurrent rainfall rate and rain attenuation at 39.4 GHz for the full two-year period. It is worth noting that the dynamic range of the beacon receiver is reached for 0.03 %. The rainfall rate value exceeded for 0.01 % of the average year is around 35 mm/h. Figure 7 shows the monthly CCDF of rain and total attenuation at 39.4 GHz. A high month-to-month variability can be noticed with the strongest rain attenuation (and so the less favourable propagation conditions) from May to August and the weakest rain attenuation in December and January. Up-link power control (ULPC) is one of the most effective fade mitigation techniques (FMT) to increase link availability and service quality of satellite communication systems operating at very high frequencies (Ka-band and above) [START_REF] Poiares-Baptista | Up-link power control on a high-availability satellite link operating at 30 GHz[END_REF], [START_REF] Lin | Results of an experiment to demonstrate the effectiveness of open-loop up-link power control for Ku-band satellite links[END_REF], and [START_REF] Dissanayake | Application of open-loop uplink power control in Ka-band satellite links[END_REF]. It allows the Earth station transmit power to be adjusted with respect to the propagation conditions and so severe attenuation (mainly caused by rain in these frequency bands) to be counteracted. The estimation of the attenuation on the up-link can be performed using opportunistic signals (down-link, beacons …) which often operate at lower frequencies than the up-link. The scaling from attenuation A down at frequency f down to attenuation A up at frequency f up > f down by means of frequency scaling prediction methods is then the key point in the design of ULPC techniques.

Two different rain attenuation frequency scaling prediction methods are currently recommended in ITU-R P.618-12 [START_REF]Propagation data and prediction methods required for the design of Earth-space telecommunication systems[END_REF]:

• Section 2.2.1.3.1 of ITU-R P.618-12 which provides a method of predicting the statistical variation of the rain attenuation at frequency f up conditioned on the rain attenuation at frequency f down . It is often referred to instantaneous frequency scaling (IFS) as it predicts the instantaneous uplink rain attenuation distribution based on the measured instantaneous downlink rain attenuation. • Section 2.2.1.3.2 of ITU-R P.618-12 which provides a simplified method of predicting the equiprobable rain attenuation at frequency f up conditioned to the rain attenuation at frequency f down . It is often referred to long-term frequency scaling as it predicts the uplink rain attenuation based on knowledge of the downlink rain attenuation at the same probability of exceedance.

Instantaneous frequency scaling

The model has been first proposed by [START_REF] Bertorelli | Modelling of short-term frequency scaling for rain attenuation using ITALSAT data[END_REF] and upgraded in [START_REF] Castanet | Characterisation and modelling of instantaneous frequency scaling in 20-50 GHz band[END_REF]. It relies on two strong hypotheses:

• The probability distributions of rain attenuation on the up-link A up and on the downlink A down conditioned to the presence of rain attenuation on the links are log-normal distributions with parameters sets ( ) and( ) down down σ µ , respectively for the up-link and the down-link, meaning:

up up σ µ ,
( ) ( ) ( ) ( )                - = > >         - = > >

Comparisons

The final comparisons with both long-term and instantaneous frequency scaling prediction methods are reported in Figure 9 and Figure 10, respectively. On a long-term basis, it could be observed a slight underestimation of the method described in Section 2.2.1.3.2 of ITU-R P.618-12, while a perfect agreement is reached using the method described in Section 2.2.1.3.1 of ITU-R P.618-12. On an instantaneous basis, as expected, the optimal parameterization of the method described in Section 2.2.1.3.1 of ITU-R P.618-12 performs better than the generic parameterization (i.e. using ( 14)) which, however, still exhibits an appropriate behaviour even for very low probability levels. As the availabilities of valid data are around 95%, reliable statistical analyses have been performed. A high month-to-month variability can be noticed with more than 20 dB difference of rain attenuation at 39.4 GHz from the most favourable month to the worst. Using concurrent measurements at 20.2 and 39.4 GHz, the two prediction methods of frequency scaling recommended in ITU-R P.618-12 have been tested. On a long-term basis, a slight underestimation of the method described in Section 2.2.1.3.2 of ITU-R P.618-12 can be noticed, while a perfect agreement is reached using the method described in Section 2.2.1.3.1 of ITU-R P.618-12. On an instantaneous basis, the method described in Section 2.2.1.3.1 of ITU-R P.618-12 shows very good performances. The ASTRA 3B and Alphasat propagation experiments are still on-going and the testing activity of frequency scaling prediction methods needs to be consolidated from the expected new results.

ACKNOWLEDGMENT

This work has been carried out under ONERA internal funding and under CNES R&T studies: R-S13/TC-0005-048-"Dual-Polarization Propagation Experiment at Ka And Q Bands" and R-S16/TC-0005-084 -"Improvement of Propagation Models at Ka and Q/V Bands over Europe".

Figure 1 .

 1 Figure 1. Overview of the Ka and Q band propagation experiments in Toulouse, France

Figure 2 .

 2 Figure 2. Co-located ASTRA 3B (right) and Alphasat (left) beacon receivers deployed at Toulouse ONERA Centre premises.

Figure 3 .

 3 Figure 3. Raw beacon signal level recorded by the ASTRA 3B and Alphasat beacon receivers from 8 th to 14 th June 2015.

Figure 4 .

 4 Figure 4. Calibrated time series of total attenuation from 13 th to 14 th June 2015

  3 dB and 3.3 dB. For 0.1 % of the time, the total attenuation values at 20.2 GHz are 5.8 dB and 4.1 dB for the first and second year of measurements, respectively. At 39.4 GHz, the total attenuation values exceeded for 0.1 % of the year are 17.6 dB and 11.4 dB. (a) (b) Figure 5. CCDF of concurrent attenuation at 20.2 and 39.4 GHz. (a) Rain attenuation. (b) Total attenuation

Figure 6 .

 6 Figure 6. CCDF of concurrent rainfall rate and rain attenuation at 39.4 GHz for the full two-year period

2 •

 2 Figure 8. Mean (a) and standard deviation (b) of the log-attenuation at 39.4 GHz conditioned to the attenuation at 20.2 GHz

Figure 9 .

 9 Figure 9. Test of long-term frequency scaling prediction methods

Table 1 .

 1 Availabilities of valid data.

	1	st Year	2	nd Year	Total
	(June 2015 -	(June 2016 -	(June 2015 -
	May 2016)	May 2017)	May 2017)

Consequently, there is a correlation coefficient ρ between the two underlying Gaussian processes G up and G down of the rain attenuation processes A up and A down , respectively. Mathematically, it leads to: where E[] is the expectation operator. So, it is straightforward that:

where n is a White Gaussian noise with zero mean and unit variance. Using [START_REF] Barbaliscia | Volume 3: reference book on radiometry and meteorological measurements[END_REF], [START_REF] Riva | COST IC0802 Handbook on measurements and products[END_REF] leads to:

( )

Parameters set ( [START_REF] Lemorton | A new opportunity for EHF propagation experiments: the EXPRESS campaign with the satellite STENTOR[END_REF] can then be retrieved from [START_REF] Solheim | Radiometric Profiling of Temperature, Water Vapor, and Cloud Liquid Water using Various Inversion Methods[END_REF]: Finally, based on data collected in Spino d'Adda during the ITALSAT experiment and in Louvain la Neuve, Lessive and Darmstadt during the OLYMPUS experiment, [START_REF]Information document about frequency scaling in[END_REF] has proposed a generic modelling of ρ:

which is the current parameterization of Section 2.2.1.3.1 of ITU-R P.618-12 recommended for frequency scaling up to 55 GHz.

Long-term frequency scaling

The long-term rain attenuation frequency scaling prediction method described in Section 2.2.1.3.2 of ITU-R P.618-12 is an empirical model. It gives an attenuation ratio directly as a function of frequency (in the range 7 to 55 GHz) and attenuation:

( )

where:

( )

A down and A up are the equiprobable values of the rain attenuation at frequencies f down and f up (GHz), respectively.

Tests of the prediction methods

The data presented in Section III are used to test the two above models. As a reminder, ASTRA 3B and Alphasat satellites are respectively located at 23.5° and 25° of longitude giving almost the same slant path (0.6° difference in elevation, and 1.90 difference in azimuth). The frequency scaling analysis it then appropriate.

Parameterization

According to [START_REF] Solheim | Microwave Radiometer for Passively and Remotely Measuring Atmospheric Temperature, Water Vapor and Cloud Liquid Water Profiles[END_REF], it is interesting to have a look on the experimental mean and standard deviation of the log-attenuation at 39.4 GHz conditioned to the attenuation at 20.2 GHz. The model can be considered as valid if:

• there is a linear relationship between the mean of the log-attenuation at 39.4 GHz and the log-attenuation at 20.2 GHz, • the standard deviation of the log-attenuation at 39.4 GHz is independent of the attenuation at 20.2 GHz. This is observed on the results of Figure 8, thus validating the theoretical framework of the prediction method described in Section 4.1. The attenuation values at 20.2 GHz have been voluntarily truncated in the range 1 to 7 dB because, first, calibration issues can impact the results at very low attenuation levels and, second, the attenuation at 39.4 GHz could be higher than the dynamic range of the Q-band beacon receiver (~30 dB) when attenuation at 20.2 GHz is higher than 7 dB. The optimal experimental value of the correlation coefficient ρ is estimated using the standard deviation of the log-attenuation at 39.4 GHz (see (13b)). This extracted value of ρ is then used to compute the mean of the log-attenuation at 39.4 GHz conditioned to the attenuation at 20.2 GHz (see (13a)). It experimentally leads to


Diana Dorobantu
email: diana.dorobantu@univ-lyon1.fr

Yahia Salhi
email: yahia.salhi@univ-lyon1.fr

Pierre-E Thérond

Modelling net carrying amount of shares for market consistent valuation of life insurance liabilities *

Keywords: Insurance, Best Estimate Technical Provision, Impairment Losses, Correlated Brownian Motions, Joint Density

The attractiveness of insurance saving products is driven, among others, by dividends payments to policyholders and participation in profits. These are mainly constrained by regulatory measures on profit-sharing on the basis of statutory accounts. Moreover, since both prudential and financial reporting regulation require market consistent best estimate measurement of insurance liabilities, cash-flows projection models have to be used for such a purpose in order to derive the underlying financial incomes. Such models are based on Monte-Carlo techniques. The latter should simulate future accounting profit and losses needed for profit-sharing mechanisms.

In this paper we deal with impairment losses on equity securities for financial portfolios which rely on instrument-by-instrument assessment (when projection models consider groups of shares). Our motivation is to describe the joint distribution of market value and impairment provision of a book of equity securities, with regard to the French accounting rules for depreciation. The results we obtain enable to improve the ability of projection models to represent such an asymmetric mechanism. Formally, an impairment loss is recognized for an equity instrument if there has been a significant and prolonged decline in its market value below the carrying cost (acquisition value). Such constraints are formalized using an assumption on the dynamics of the equity, and leads to a complex option-like pay-off.

Using this formulation, we propose analytical formulas for some quantitative measurements related the impairments losses of a book of financial equities. These are derived on a general framework and some tractable example are illustrated. We also investigate the operational implementation of these formulas and compare their computational time to a basic simulation approach.

Introduction

Over the past decade, market consistent best estimate measurement became the new standard when it comes to life insurance valuation. This principle was first generalized by the Market Consistent Embedded Value (MCEV) principles published by the CFO Forum in 2009 and was then considered by the European Union for prudential purposes (Solvency II) and by the International Accounting Standard-Setter (IASB) for financial reporting purposes, see [START_REF] Thérond | About Market Consistent Valuation in Insurance[END_REF], [START_REF] Vedani | Market inconsistencies of market-consistent European life insurance economic valuations: pitfalls and practical solutions[END_REF] and [START_REF] Wüthrich | Market-Consistent Actuarial Valuation[END_REF].

Due to the complexity of life insurance mechanisms, in most cases, this kind of measurement requires to use some cash-flows projection models in order to estimate the market consistent best estimate using Monte Carlo simulations. As a matter of fact, projecting cash-flows of life insurers requires modelling future guarantees revaluation and policyholders behaviours (especially regarding lapses) which both are dependent on future economic scenarios. These economic scenarios have to be drawn consistently with the current observed prices on the market (see [START_REF] Thérond | About Market Consistent Valuation in Insurance[END_REF] for more details on market consistency principles). In order to comply with the market consistency, risk neutral approaches appear to be the present standard even if some other approaches would meet the requirement (e.g. real world economic scenarios coupled with stochastic deflators when it comes to discount future cash-flows). Each generated scenario should take into account the behaviours of insurer (through asset management, revaluation policy, etc.) but also of policyholders (mortality, partial or total lapses, etc.).

Moreover, in some jurisdictions (e.g. Italy, Germany or France), participation mechanisms are constrained by regulation. For example, in France, life insurers have to dedicate, at least, 85% of financial incomes to re-evaluate policyholder's annuities or savings. Here, financial incomes has to be understood as local Generally Accepted Accounting Principles (GAAP) financial revenues. As a consequence, and in order to apprehend this kind of constraints, local GAAP financial statements have to be projected, too.

Modelling local GAAP financial statements appears to be quite a challenge due to the complexity of some accounting mechanisms and the granularity at which they have to be considered. It is especially the case for projecting financial assets due to the impairment provisions which are recognized on an instrument-by-instrument basis and may have a significant impact on local GAAP financial revenue and thus on the constraints an insurer has to deal with when considering revaluation decisions.

As a matter of fact, life insurers have to deal with guaranteed rate and participation features. Cashflows projection models have to be able to incorporate management rules such as the asset-and-liability management (ALM), e.g. asset management policy regarding the entity's objectives (giving good bonuses to policyholders in order to avoid lapses, keeping some unrealized gains to prevent the consequences of potential future market drops, containing P&L's volatility, etc.) Equity securities management play a major role in this ALM, since selling shares with unrealized gains and losses enables to modify the local GAAP financial result. As a consequence, it is of major importance to be able to precisely model the correlated behaviour of the market value and net carrying amounts of such financial assets.

Such a problem is the main motivation of this work. In fact, our main goal is to derive some analytical formulas in order to be able to simulate the joint carrying amount and market value of financial equity portfolios, regarding French impairment rules. This subject is of paramount importance, from an operational perspective, since cash-flows projecting models generally do not consider individual financial assets but groups of financial assets (model points). The main challenge consists in deriving this joint distribution for a book of n assets without computing the impairment mechanism for each individual asset.

In this paper, we propose closed form formulas of impairment provisions for a book of n equities with different assumptions on their dependency. First, we consider the case of comonotonic shares driven by the same Brownian motion. In fact, the valuation principle associated with the acquisition cost measurement leads to distinctly recognize several lines of the same title acquired at different times, thus at different costs. So we may have, if any, different levels of impairment reserves for the same security but acquired at different dates. Recently [START_REF] Azzaz | Some characteristics of an equity security next-year impairment[END_REF] considered a similar case but with a different impairment rule. Indeed, they investigated a criteria based on a separated events, e.g. decline and prolonged events. However, in that paper the authors only considered a single equity. Therefore, in the current work we intend to extend the results of [START_REF] Azzaz | Some characteristics of an equity security next-year impairment[END_REF] to a more general framework but with a joint realisation of the above mentioned impairment conditions. We should note that one can also derive the quantities of interest in the framework of [START_REF] Azzaz | Some characteristics of an equity security next-year impairment[END_REF] using the result developed in Section 4, e.g. next-period expected impairment and its cumulative distribution function (c.d.f.). Secondly, we are tackling the case of linearly dependent shares. In a similar way, we are able to develop analytical of the distributional specificities of next-period impairment. This is mainly based on the existence of the joint law of a multidimensional drifted Brownian motion with a vector of its element-wise supremum over a given period. We should note that in our case we did use the joint density to characterize the quantities of interest. This density is known for n = 2 but its analytical form for n ≥ 3 does only exist for some particular correlation matrices, see [START_REF] Escobar | A note on the distribution of multivariate brownian extrema[END_REF]. We thus propose the computation of the next period impairment c.d.f. of some particular cases where the aforementioned density does exist based on the method of images, see [START_REF] Keller | The scope of the image method[END_REF].

Finally, we derive the explicit form of the joint c.d.f. of the aggregate carrying amounts and the impairment losses both for the comonotonic and linearly dependent shares. This c.d.f. is of paramount importance in practice as it should help the determination of a representative share of a book of equities when a model point approach is considered in the internal model of the insurer. In fact, due to constraints on computation times and implementation costs but also due to the complexity of a share-by-share approach, insurers generally rely on a model point that represents a book of equity, see [START_REF] Bennemann | Impairment estimates of equity portfolios represented by model points[END_REF]. This leads to the application of the accounting depreciation rules to the representative equity instead of applying it individually to each security. The results obtained in [START_REF] Azzaz | Some characteristics of an equity security next-year impairment[END_REF] (with a slightly different impairment criterion, that of IAS 39) have shown the inadequacy of the representation of impairments using the only significant loss criterion. Hence, using this joint c.d.f. should enhance the construction of the model point.

The remainder is organized as follows. In Section 2 we introduce the accounting standards that French insurers have to apply for accounting non-depreciable financial assets (e.g. funds and stocks). This gives arise to a mathematical formulation for the impairment rule as a complex option-like cash-flow. In Section 3, we formulate this problem using a Black and Scholes framework, where the equities follow a geometric Brownian motion. The impairment is formulated and its distribution and expectation are investigated for a single share and give the basis for the computations for a book of n equities that follow in Section 4. In this section, we introduce the main result of the paper while distinguishing comonotonic of linearly dependent shares. Finally, in Section 5 we investigate the computational performance of the analytical results and analyse the bias estimation stemming from a basic simulation-based approach.

2 Accounting measurement of equity securities 2.1 General Measurement. In France, the revaluation of life insurance policies is constrained by regulatory measures on profit-sharing on the basis of statutory accounts. Therefore, it is necessary to describe the specific modalities that apply to the measurement and recognition of the financial assets. Precisely, the accounting standards for equity instruments are governed by Art. R343-10 of the French Insurance Code and sections 123-1 to 123-19 of the French Insurance Accounting Regulation 1 for their depreciation methods2 . On an individual basis, these rules can be summarized as follows: (i) at initial recognition, an equity security is measured and recognized at its acquisition cost; (ii) the subsequent measurement of that kind of financial instruments remains at their acquisition cost; (iii) under some conditions (described below), an impairment provision may be recognized.

From this accounting scheme, at each annual reporting date, the net carrying amount of an equity security is equal to its acquisition cost less the impairment provision. The differences between the opening and closing values of the impairment provision are recognized through profits-and-losses (P&L). When this equity security is sold during the reporting period, the difference between its net carrying amount (at the opening) and the sold price is recognized through P&L.

Impairment losses and provision.

French accounting regulations state that provisions for impairment have to be accounted on an individual basis, i.e. unrealised gains from some shares do not compensate unrealised losses from other for the purpose of determining such provisions. Considering an individual financial asset, an impairment provision has to be recognized if, at reporting date: (i) the market value is inferior to the acquisition value; and (ii) the unrealized loss presents a durable nature.

For non-amortizable financial assets, several cases have to be considered regarding the durable or permanent condition. So, an unrealized loss of an equity security is presumed durable if: (i) an impairment provision was recognized in the opening balance sheet; (ii) it exists some objective hints that the unrealized loss presents a durable nature; (iii) the unrealized loss has been significant and prolonged.

In the following, we will focus on the last criterion which is a quantitative one. Unlike the IAS 39 standards (see [START_REF] Azzaz | Some characteristics of an equity security next-year impairment[END_REF]), the French standard defines how to consider a significant and prolonged impairment : when the unrealized loss was higher than 20 % of the acquisition cost during the latest 6-months. Beyond the impairment trigger, two other disposals are quite different from IAS 39 impairment mechanism: (i) the impairment provision is re-evaluated each year (with the possibility of reversal), (ii) if the insurance company has the will and the ability to hold the equity security over the next years, the impairment provision could be lesser than the unrealized loss3 . In the following, without loss of generality, we will not consider this latter possibility.

2.3 Notation. In this section, we consider the impairment of a single asset i, denoted S i , according to the accounting principles discussed above. Suppose that we are at the beginning of the period t and we seek to quantify the (t + 1)-period impairment. At the end of this period, the equity is impaired whenever its recoverable amount falls under the acquisition value S i t i = a i > 0 (t i denotes the acquisition date). This rule must be considered jointly with the constraint on the prolonged effect of the drop-down. Indeed, the impairment is triggered when the observed market value over the period [t + 1/2, t + 1] remained below a given level α of the acquisition value, where α ∈ [0, 1]. These together induce a recognition of a loss (a i -S i t+1 ). More formally, given the information acquired up to time t > t i and denoting S i t+1 = sup u∈[t+1/2,t+1] S i u , the future impairment PDD i t+1 is given as follows:

PDD i t+1 = (a i -S i t+1 ) + 1 S i t+1 ≤αa i .
(2.1)

As a consequence, the net accounting value of this title is given by NAV i t+1 = a i -P DD i t+1 .

(2.2)

Model assumptions

The issue we want to deal with consists in the projection of both net accounting and market values of equity securities belonging to a life insurer's financial assets portfolio.

As mentioned previously, both new prudential and financial reporting standards require market consistent valuation of insurance liabilities. Concerning with-profits life insurance contracts, in most case, such evaluations are conducted using Monte Carlo simulations based on future cash-flows realisations drawn from a projection model. Such a model has to represent asset-and-liability management, especially local GAAP P&L consequences of selling (or not) financial assets. Moreover, since Solvency II and IFRS 17 require market consistent measurement, in most cases, risk-neutral economic scenario generators are considered. More specifically, the stock dynamics are evolving under the risk neutral probability measure. This practice roots in the use of the quantified provisions to assess the market consistent of the liability best estimate under the regulatory requirement. As noted before, the latter should take into account the financial returns that are accounted for in a profit sharing mechanism.

Accordingly, under the risk neutral measure, the equities share have the same drift rate that is equal to the risk-free rate but have different volatilities. Hereafter, we consider that the dynamics of these stock prices can be described by a geometric Brownian motion as follows

dS i t = S i t rdt + σ i dW i t , t ≥ 0 (3.1)
where r and σ i are some constant parameters referring, respectively, to the risk-free interest rate and the volatility of the i th equity. Here, W i = (W i t ) t≥0 is a standard Brownian motion such that for each i = j we have cov(W i t , W j t ) = ρ ij t, where ρ ij ∈] -1, 1[ is the correlation coefficient. Moreover, let us introduce the following notation

X i t = µ i t + σ i W i t , where µ i = 2r -σ 2 i 2 ,
which will be used throughout the sequel.

A primal quantity of interest when dealing with impairment is the distributional behaviour of the impairment provision at a given time. To this end, the next proposition gives the c.d.f. of the impairment provision of a single title at time t, given the available information at time s. In the following proposition, the notation P t (•) (resp. E t (•)) stands for the conditional probability

P •|F i t (resp. conditional expecta- tion E(•|F i t ))
where F i t is the filtration generated by the Brownian motion W i , i.e. F i t = σ{W i s , s ≤ t}.

Proposition 1 Consider an asset i with acquisition value a i > 0, acquisition date t i and future impairment P DD i given by Equation (2.2). Let t > 1 2 . For all t i ≤ s < t -1 2 , the cumulative distribution function of PDD i t is given as follows:

P s (PDD i t ≤ x) =        0 if x < 0, p 1 (a i , S i s ) if 0 ≤ x < a i (1 -α), p 2 (a i , S i s ) if a i (1 -α) ≤ x < a i , 1 if x ≥ a i , with p 1 (a i , S i s ) = 1 -Φ w, ln( αa i S i s )-µ i δ σ i √ δ ; √ 2δ-1 √ 2δ + αa i S i s 2µ i σ 2 i Φ w + 2µ i σ i δ -1 2 , -ln( αa i S i s )-δµ i σ i √ δ ; - √ 2δ-1 √ 2δ , p 2 (a, S i s ) = 1 -Φ w, ln( a i -x S i s )-µ i δ σ i √ δ ; √ 2δ-1 √ 2δ + αa i S i s 2µ i σ 2 i Φ   w + 2µ i σ i δ -1 2 , -ln( (a i -x)S i s a 2 i α 2 )-δµ i σ i √ δ ; - √ 2δ-1 √ 2δ   , w = ln αa i S i s -µ i (δ -1 2 ) σ i δ -1 2 , δ = t -s and Φ(U, V ; θ)
is the joint c.d.f. of two correlated Gaussian r.v.s U and V with a given correlation coefficient θ.

Proof. See Appendix A.1.
If this first result enables to simulate future impairment losses, from an operational perspective, we still have to deal with two issues: (i) the joint evolution of market value and net accounting value has to be considered for applying management rules in projection models; (ii) insurer's internal models do not consider equity securities on an individual basis (even if the impairment rules apply at this level of measurement). In the following, we deal with these two issues with the objective of obtaining mathematical results which enable an operational implementation (see 5).

Main results

In this section, we derive the main quantities needed to assess the impairment of a book of n equities. These are related to the joint distribution of the next period impaired value and the underlying shares values. First, we start with a particular case on which the equities have the same source of uncertainty. This is to take into account a common practice in real world applications where the same Brownian motion is used to generate the dynamics of the stocks and thus compute the related impaired values. This model is also of interest when the insurance company holds many shares of a stock that have been purchased at different dates. Following the accounting rules this share should be split in different lines based on the acquisition date. Secondly, we consider a more general framework where the equity securities are not perfectly correlated. For both frameworks, we denote

Γ t = PDD 1 t + • • • + PDD n t , (4.1)
where PDD i t , for i = 1, • • • , n are defined in Equation (2.2). Moreover, we are also considering the aggregate dynamics Σ = (Σ t ) t≥0 defined as follows:

Σ t = S 1 t + • • • + S n t . (4.2)
From an operational perspective, our main goal is to be able to simulate some realizations of (Γ t+1 , Σ t+1 ) conditionally to (Γ t , Σ t ).

4.1 Portfolio of comonotonic securities. Our first results consist in deriving the joint distribution of the market value and the impairment reserve of shares of the same company, acquired at different times.

As a matter of fact, the valuation principle associated with the acquisition cost measurement leads to distinctly recognize several lines of the same title acquired at different times, thus at different costs. So we may have, if any, different levels of impairment reserves for the same security but acquired at different dates.

As noted before, in practice, the use of Monte-Carlo simulations based on an asset-by-asset approach often leads to large running times. Therefore, the practitioners are generally used to group the equities into homogeneous groups and to replace the group of contracts with a representative security called a model point, see [START_REF] Bennemann | Impairment estimates of equity portfolios represented by model points[END_REF]. When doing this, they advocate the use comonotonic dynamics for modelling the various securities within each model point. More formally, we consider that the stocks have the same dynamic

dS t = S t rdt + σdW t ,
where r, σ > 0 and W = (W t ) t≥0 is a standard Brownian motion. Here X t = µt + σW t where µ = (2r -σ 2 )/2. In the other hand, this case could correspond, for instance, to the presence of the same share in the asset book of the insurer with different acquisition dates and values. These are treated separately as the impairment provision does depend on the acquisition value, see Equation (2.2). Given this particular case, our aim is then to derive the joint probability P t (Γ t+1 ≤ y, Σ t+1 ≤ x), where as previously mentioned the notation

P t (•) (resp. E t (•)) stands for the conditional probability P (•|F t ) (resp. conditional expectation E(•|F t ))
where F t is in this section the filtration generated by the Brownian motion W , i.e. F t = σ{W s , s ≤ t}. First, we generalize the result in Proposition 1 and derive a closed form for c.d.f. of the aggregate loss Γ defined in Equation (4.1).

Proposition 2 Let consider a book of n shares with different acquisition dates t i and acquisition costs

a i > 0, i = 1, . . . , n. Let t > 1 2 . For all max(t 1 . . . t n ) ≤ s < t -1 2
, the next period probability of recording an impairment on this book is given by

P s (Γ t > 0) = 1 -p 1 (max(a 1 . . . a n ), S s ),
where p 1 is given in Proposition 1.

Proof. See Appendix A.2

Another indicator of interest is the expected loss at the end of a given period. Of course, one can use the c.d.f. in the last proposition to retrieve the expected value of impairments by integration of the survival function. However, this would be painful, and numerically complex. Therefore, in the next result we give the explicit form of the latter conditional on the information up to time s.

Proposition 3 For n shares with different acquisition values a i and different acquisition dates t i ≤ s < t -1 2 , i = 1, . . . , n and t > 1 2 , the next period expected impaired amount is given as follows:

E s (Γ t ) = n i=1 a i (1 -p 1 (a i , S s )) + n i=1 e rδ   -S s Φ   ln a i α Ss -(µ + σ 2 )(δ -1 2 ) σ δ -1 2 , ln a i α Ss -(µ + σ 2 )δ σ √ δ ; √ 2δ -1 √ 2δ   +(a i α) 2(µ+σ 2 ) σ 2 S -2r σ 2 s Φ   ln a i α Ss + (µ + σ 2 )(δ -1 2 ) σ δ -1 2 , -ln a i α Ss -(µ + σ 2 )δ σ √ δ ; - √ 2δ -1 √ 2δ     .
where p 1 is given in Proposition 1 and δ = t -s.

Proof. See Appendix A.3

The above results are of paramount importance when investigating the impairment characteristics and thus can be incorporated on a periodic assessment the impact of an impairment of a book of comonotonic equities on the profit sharing mechanism. For instance, this can be integrated on scenario based approach. However, from a risk management point of view it is important to measure the risk of any deviation in the profit or loss resulting from any impairment of such a security. Therefore, in the following result we are looking for the joint market value of an individual stock is the price at which this asset is quoted on an exchange.

Theorem 1 Let t, x, y ≥ 0. Assume, without loss of generality, that the acquisition costs a i , i = 1 . . . n of n shares are ordered, i.e. a 1 ≤ a 2 • • • ≤ a n . In the case of n shares with different acquisition values a i and different acquisition dates t i < t, i = 1 . . . n, the joint c.d.f. P t (Γ t+1 ≤ y, Σ t+1 ≤ x) can be written as follows:

√ 2φ 1 σ ln x nS t - µ σ -Φ ω n , 1 σ ln x nS t - µ σ ; 1 √ 2 +1 x n ≤anα a n α S t 2µ σ 2 e -µ 2 σ 2 Φ ω n + 2µ σ , 1 σ ln xS t n(a n α) 2 - µ σ ; - 1 √ 2 +1 x n >anα Φ ω n , 1 σ ln x nS t - µ σ ; 1 √ 2 -Φ ω n , 1 σ ln a n α S t - µ σ ; 1 √ 2 + a n α S t 2µ σ 2 e -µ 2 σ 2 Φ ω n + 2µ σ , - 1 σ ln a n α S t - µ σ ; - 1 √ 2 + n-1 j=0 1 1 n-j ( n i=1+j a i -y)≤ x n ∧a j+1 α × Φ ω j+1 , 1 σ ln x n ∧ a j+1 α S t - µ σ ; 1 √ 2 - a j+1 α S t 2µ σ 2 e -µ 2 σ 2 Φ ω j+1 + 2µ σ , 1 σ ln S t ( x n ∧ a j+1 α) (a j+1 α) 2 - µ σ ; - 1 √ 2 -Φ ω j+1 , 1 σ ln n i=j+1 a i -y (n -j)S t - µ σ ; 1 √ 2 + a j+1 α S t 2µ σ 2 e -µ 2 σ 2 Φ ω j+1 + 2µ σ , 1 σ ln S t ( n i=1+j a i -y) (n -j)(a j+1 α) 2 - µ σ ; - 1 √ 2 - n-1 j=1 1 1 n-j ( n i=1+j a i -y)≤ x n ∧a j+1 α 1 x n >a j α -1 1 n-j ( n i=1+j a i -y)<a j α Φ ω j , 1 σ ln a j α S t - µ σ ; 1 √ 2 - a j α S t 2µ σ 2 e -µ 2 σ 2 Φ ω j + 2µ σ , 1 σ ln S t a j α - µ σ ; - 1 √ 2 +1 1 n-j ( n i=1+j a i -y)≤ x n <a j α × -Φ ω j , 1 σ ln x nS t - µ σ ; 1 √ 2 + a j α S t 2µ σ 2 e -µ 2 σ 2 Φ ω j + 2µ σ , 1 σ ln xS t n(a j α) 2 - µ σ ; - 1 √ 2 +1 1 n-j ( n i=1+j a i -y)≤ x n ∧a j α Φ ω j , 1 σ ln n i=1+j a i -y (n -j)S t - µ σ ; 1 √ 2 + a j α S t 2µ σ 2 e -µ 2 σ 2 Φ ω j + 2µ σ , 1 σ ln ( n i=1+j a i -y)S t (n -j)(a j α) 2 - µ σ ; - 1 √ 2 ,
where

ω j = √ 2 σ ln (a j α/S t ) + µ/σ √ 2.
Proof. See Appendix A.4

From an operational perspective, this first result enables to deal with several lines of a same title which were acquired at different times. Moreover it is already an improvement of the main used approach in cash-flows projection models which consists to aggregate a book of stocks (averaging both market values and acquisition costs) and consider it as if it was a single equity security. Of course this main used approach leads to misrepresent impairment losses dynamic which is far to be linear.

4.2 Shares with linear dependency. In this section, we intend to expand our first result, assuming that the stocks are correlated and they have the following dynamics :

dS i t = S i t rdt + σ i dW i t , (4.3)
where r, σ i > 0 and W i , i = 1, . . . , n are n correlated Brownian motions (BM) , cov(W i t , W j t ) = ρ ij t with ρ ij ∈] -1, 1[. In this section, F t is the filtration generated by the n BMs.

Since the expected impaired amount does not depend on the correlation coefficients, we easily find the following result as in the previous section.

Proposition 4 For n correlated shares with different acquisition dates t i and acquisition costs a i , the next period expected impaired amount is given as follows for t i ≤ s < t -1 2 , i = 1 . . . n and t > 1 2 :

E s (Γ t ) = n i=1 a i (1 -p 1 (a i , S i s )) + n i=1 e rδ   -S i s Φ   ln a i α S i s -(µ i + σ 2 i )(δ -1 2 ) σ i δ -1 2 , ln a i α S i s -(µ i + σ 2 i )δ σ i √ δ ; √ 2δ -1 √ 2δ   +(a i α) 2(µ i +σ 2 i ) σ 2 i S -2r σ 2 i s Φ   ln a i α S i s + (µ i + σ 2 i )(δ -1 2 ) σ i δ -1 2 , -ln a i α S i s -(µ i + σ 2 i )δ σ i √ δ ; - √ 2δ -1 √ 2δ     .
where p 1 is given in Proposition 1 and δ = t -s.

Henceforth we consider the case of two correlated stocks (i.e. n = 2) where the correlation coefficient between the two Brownian motions is ρ ∈] -1, 1[. We introduce the following key functions needed throughout the sequel.

Definition 1 Let β ∈ [0, π], σ 1 , σ 2 > 0, µ 1 , µ 2 ∈ R some constants and q : R + × [0, β] × R 3 + → [0, 1], p : R 2 + × [0, β] 2 × R 3 + → [0, 1] two functions defined by q(r 0 , θ 0 , t; M 1 , M 2 ) = e M 1 α 1 +M 2 α 2 +bt 2 βt × ∞ n=1 e -r 2 0 2t sin nπθ 0 β β 0 sin nπ θ β ∞ 0 re - r2 2t e µ 2 σ 2 r sin θ+α 1 σ 1 √ 1-ρ 2 r cos θ I nπ β rr 0 t drd θ, p(r, r 0 , θ, θ 0 , t; M 1 , M 2 ) = e M 1 α 1 +M 2 α 2 +bt+ µ 2 σ 2 r sin θ+α 1 σ 1 √ 1-ρ 2 r cos θ × 2r βt ∞ n=1 e -r 2 0 +r 2 2t sin nπθ 0 β sin nπ θ β I nπ β rr 0 t ,
where I n is the modified Bessel function,

α 1 = -µ 1 σ 2 + ρµ 2 σ 1 (1 -ρ 2 )σ 2 1 σ 2 , α 2 = -µ 2 σ 1 + ρµ 1 σ 2 (1 -ρ 2 )σ 1 σ 2 2 , and b = -µ 2 1 σ 2 2 -µ 2 2 σ 2 1 + 2ρµ 1 µ 2 σ 1 σ 2 2(1 -ρ 2 )σ 2 1 σ 2 2 .
Since in the sequel we consider the case n = 2, then F t is the filtration generated by the two Brownian motions W i , i = 1, 2.

If we still intend to express the joint conditional distribution of (Γ, Σ), the first step leads to express to probability that, at least, one title among the book of assets is impaired.

Proposition 5 Let t > 1 2 . The next period probability of recording an impairment on a given book of 2 correlated shares with acquisition dates t i and acquisition costs a i , t i ≤ s < t -1 2 , i = 1, 2 is given by

P s (Γ t > 0) = 2 i=1 1 -p 1 (a i , S i s ) -Q(S 1 s , S 2 s , δ)
where

Q(S 1 s , S 2 s , δ) = E 1 X≤A 1 ,Y ≤A 2 q r 0 , θ 0 , 1 2 ; σ 1 δ - 1 2 (A 1 -X), σ 2 δ - 1 2 (A 2 -Y ) s i =S i s , q is given in Definition 1, p 1 is given Proposition 1. Here, (X, Y ) t is a Gaussian vector N 0 R 2 , 1 ρ ρ 1 and r 0 = δ -1 2 1 -ρ 2 (A 1 -X) 2 2 -2ρ(A 1 -X)(A 2 -Y ) + (A 2 -X) 2 2 1 2 tan β = - 1 -ρ 2 ρ , β ∈ [0, π], tan θ 0 = 1 -ρ 2 A 1 -X A 2 -Y -ρ -1 , θ 0 ∈ [0, β] A i = ln a i α s i -µ i (δ -1 2 ) σ i δ -1 Proof. See Appendix B.1
The Q function previously introduced is going to be used to express the conditional distribution of (Γ, Σ) in 2.

Theorem 2 In the case of two correlated shares S 1 and S 2 , the joint c.d.f. P t (Γ t+1 ≤ y, Σ t+1 ≤ x) can be written as follows for all x, y ≥ 0:

i,j∈{1,2},i =j E J 1 ij (X, Z)φ ij (s 1 , s 2 , X, Y, Z) s i,j =S i,j t - αa i S i t 2µ i σ 2 i e - µ 2 i σ 2 i E J 2 ij (X, Z) φij (s 1 , s 2 , X, Y, Z) s i,j =S i,j t +E 1 X≤ ln x s 1 -µ 1 σ 1 φ 1 σ 2 1 -ρ 2 ln x -s 1 e µ 1 +σ 1 X s 2 e µ 2 +σ 2 ρX s 1,2 =S 1,2 t J 3 + Q(S 1 t , S 2 t , 1)J 4 + ∞ 0 β 0 E t 1 X≤A 1 ,Y ≤A 2 p r, r 0 , θ, θ 0 , σ 1 √ 0.5(A 1 -X), σ 2 √ 0.5(A 2 -Y ) s 1,2 =S 1,2 t J(r, θ)drd θ,
where Q, r 0 , θ 0 , A i (for δ = 1) and β are given in Proposition 5, (X, Y, Z) t is a Gaussian vector

N   0 R 3 ,   1 ρ 0 ρ 1 0 0 0 1     and J 1 ij (X, Z) = 1 X≤A i   I 1 (a i , a j )1 Z≤ ln x∧(αa i ) s i -µ i σ i √ 0.5 -X + I 2 (a i , a j )1 ln a i -y s i -µ i σ i √ 0.5 -X≤Z≤ ln x∧(αa i ) s i -µ i σ i √ 0.5 -X   , J 2 ij (X, Z) = 1 X≤A i   I1(ai, a j )1 ln a i -y s i -µ i σ i √ 0.5 -X≤Z≤ ln x∧(αa i ) s i -µ i σ i √ 0.5 -X + I 2 (a i , a j )1 ln s i (a i -y) (αa i ) 2 σ i √ 0.5 +X≤Z≤ ln s i (x∧αa i ) (αa i ) 2 σ i √ 0.5 +X    , I 1 (a i , a j ) = -1 x≥α(a i +a j ) 1 y<a i -1 y≥a j 1 αa i ≤x<α(a i +a j ) , I 2 (a i , a j ) = 1 a i (1-α)≤y<a i 1 x≥(αa j )∨(a i -y) , i, j ∈ {1, 2}, i = j, J 3 = 1 x≥α(a 1 +a 2 ) [1 + 1 y≥a 1 1 x≥αa 2 + 1 y≥a 2 1 x≥αa 1 ] , J 4 = 1 x≥α(a 1 +a 2 ) [1 y<a 1 -1 a 2 ≤y<a 1 +a 2 ] , J(r, θ) = -I 2 (a 1 , a 2 )1 (r, θ)∈D 1 -I 2 (a 2 , a 1 )1 (r, θ)∈D 2 + 1 x≥(a 1 +a 2 -y)∨α(a 1 +a 2 ) 1 y<a 1 +a 2 1 (r, θ)∈D 3 (a 1 +a 2 -y) + 1 x<α(a 1 +a 2 ) 1 y≥a 1 +a 2 1 (r, θ)∈D 4 (x) + 1 0<a 1 +a 2 -y≤x<α(a 1 +a 2 ) 1 (r, θ)∈D 3 (x)-D 3 (a 1 +a 2 -y) , φ ij (s i , s j , X, Y, Z) = φ 1 σ j 0.5(1 -ρ 2 ) ln x -s i e σ i √ 0.5(X+Z)+µ i s j e σ j √ 0.5(ρZ+Y )+µ j , φij (s i , s j , X, Y, Z) = φ    1 σ j 0.5(1 -ρ 2 ) ln x -(αa i ) 2 s i e σ i √ 0.5(Z-X) αa i s i 2ρσ j /σ i s j e σ j √ 0.5(ρZ+Y -ρX)+µ j -2ρµ i σ j σ i    , D 1 = (r, θ) : a 1 αe -σ 1 r(ρ sin( θ)+ √ 1-ρ 2 cos( θ)) + a 2 αe -σ 2 r sin( θ) ≤ x, 1 ≥ e -σ 1 r(ρ sin( θ)+ √ 1-ρ 2 cos( θ)) ≥ a 1 -y αa 1 , D 2 = (r, θ) : a 1 αe -σ 1 r(ρ sin( θ)+ √ 1-ρ 2 cos( θ)) + a 2 αe -σ 2 r sin( θ) ≤ x, 1 ≥ e -σ 2 r sin( θ) ≥ a 2 -y αa 2 , D 3 (m) = (r, θ) : a 1 αe -σ 1 r(ρ sin( θ)+ √ 1-ρ 2 cos( θ)) + a 2 αe -σ 2 r sin( θ) ≤ m , ∀m ≥ 0. Proof. See Appendix B.2.
Notice that our method can be generalized to n > 2 if the joint density (X, X) is known, with X = (X 1 , . . . , X n ) is a n-dimensional drifted Brownian motion and X = ( X1 , . . . , Xn ) its element-wise supremum over a given period. To our knowledge, this density only exists for some particular correlation matrices, see [START_REF] Escobar | A note on the distribution of multivariate brownian extrema[END_REF]. The determination of such correlation structures is based on the method of images, see [START_REF] Keller | The scope of the image method[END_REF]. In our case, we note R the correlation matrix of the vector X and introduce the matrices A n , B n and C n that will needed to characterize the structures that can lead to the derivation of the density depending on the dimension n. The elements of these three matrices are given as follows:

A n (i, j) = -cos π 3 , j = i + 1 0 , else , and
B n (i, j) =    -cos π 3 , j = i + 1 -cos π 3 , i = n -2, j = n 0 , else C n (i, j) =    -cos π 3 , j = i + 1 -cos π 4 , i = n -1, j = n 0 , else.
Theorem 3 The type of correlation matrices R where the method of images can be used to find the joint distribution of (X, X) is provided next :

1. If n = 3, then there are only three cases : A 3 , C 3 and G 3 where

G(i, j) =    -cos π 3 , j = i + 1, i = 2 -cos π 5 , i = 2, j = 3 0 , else 2
F 4 (i, j) =    -cos π 3 , j = i + 1, i = 2 -cos π 4 , i = 2, j = 3 0 , else and G 4 (i, j) =    -cos π 3 , j = i + 1, i = 3 -cos π 5 , i = 3, j = 4 0 , else (
E n (i, j) =        -cos π 3 , j = i + 1, i = 4 0 , i = 4, j = 5 -cos π 3 , i = 3, j = 5 0 , else.
Let's note that in the case n = 2 we use the joint density of (X, X) to find P t (Γ t+1 ≤ y, Σ t+1 ≤ x), but the c.d.f of this vector is enough for calculations.

Remark 1 If we add a dependent share to a set of comonotonic stocks, we can easily derive analytical results as above. In fact, the triggering events of the comonotonic shares reduce to exploring a single random process driving these shares. Thus, the calculation in presence of a dependent share follows similar steps as detailed above while taking into account the correlation between the two Brownian motions.

Operational implementation

In this section, we illustrate the computation of the provisions and compare the closed form formulas to the simulated approximations. For the sake of readiness we consider the case where the insurer holds two assets, i.e. n = 2, for which we run Monte-Carlo experiments to quantify the quantities introduced in the previous sections. To this end, we adopt a naïve approach to implement the simulation scheme. Let S M,i t k be the realizations of the dynamics of the process (S t ) t≥0 over a discrete-time grid

t k , k = 0, 1, • • • , M, of length ∆t = t k -t k-1 = 1/M with S M,i t k = S M,i t k-1 exp µ i ∆t + σ i √ ∆tX k ,
where X k are independent realizations of a standard normal r.v. X. Accordingly, a realization of the maximum over [t + 1/2, t] can be approximated by

S M,i t = max S M,i t k , t k ∈ [t + 1/2, t + 1], k = 0, 1, • • • . (5.1)
Using this path approximation, we run standard Monte-Carlo approach to simulate the impairment PDD i t in Equation (2.2) for a given observation time t. More formally, let n = 2 then for i = 1, 2, we have the following approximation for the future impairment:

Σ M t N = S M,1 t N + S M,2 t N Γ M t N = (S M,1 t 1 -S M,1 t N ) + 1 S M,1 t N ≤αS M,1 t 1 + (S M,2 t 2 -S M,2 t N ) + 1 S M,2 t N ≤αS M,2 t 2
.

These are used to approximate the different quantities considered in the above sections and will be compared to the proposed formulas on a number of examples. The simulation experiments were performed on a mac-OS with an Intel Core i7 2.80 GHz processor.

5.1 Comonotonic shares. We consider two lines of shares with the same underlying security S 1 . The latter was acquired at two different dates t 1 and t 2 at cost a 1 and a 2 . This makes the impairment applicable considering the two shares separately but with the same underlying dynamics. Hereafter, we omit the accumulated impairment losses and thus the carrying amount is equal to the cost (purchase) amount. For illustration, we consider the impairment at terminal period t = 1 and the quantification from the initial time s = 0. The security has an initial market value S 1 0 and a given implied volatility σ 1 . The prolonged component of the impairment is characterized at a prefixed level α = 80%. Figures 1 and2 depict the evolution of the next-period impairment probability as well as the expected impaired amount as a function of some key parameters. Here, we used N = 20000 replications on a discretized time-grid t k = k∆t with M (100 and 1000) equidistant steps. Figure 2 illustrates the expected impairment in Proposition 3. We observe that when the risk-free return r is high the impairment loss will be approaching zero almost surely, see Figure 1 for the corresponding probabilities. This is of course related to the initial buffer of capital gains, which is assumed to be zero for the base case. If the initial impairment is not omitted, one should expect a less of an impact. In the other hand, a volatility increase leads to the rise of the impaired value and thus can reach 45% of the initial cost. Regarding the simulated approximations (in blue), we remark that those are fairly reproducing the exact results with respect to the exact computations. This comparison demonstrates that the Monte-Carlo estimation is a biased estimator (the exact value is inside the 0.95 confidence interval of the estimates for any M ). However, we remark that the estimate based on a discretely monitored prolonged constraint (barrier-type condition), see the approximation in Equation (5.1) converges to the continuous case (red line) as M increases. However, the convergence is very slow and the bias is larger than 5% of the true values (both for the c.d.f. and the expected value) even for 1000 time steps. Therefore, the use of the analytical formulas are very efficient as the number of required amount of sampling dates and simulations to achieve a descent approximation is high, see Figure 2 and Figure 1. In fact, this is due to the error coming from the discretization of the dynamics in Equation (4.3) and especially the approximation of the maximum in Equation (5.1). More precisely, there is a loss of information about all parts of the continuous-time path between sampling dates, which introduces a substantial bias of estimation. Moreover, the latter decreases very slowly as 1/ √ M , see [START_REF] Gobet | Weak approximation of killed diffusion using euler schemes[END_REF]. In order to enhance this bias we can use a discretization scheme based on the exact conditional law of the maximum. The technique is very efficient because only two times steps (t 1 = 1/2 and t 2 = 1) are required to simulate the equity path and its supremum is drawn based on the simulation of a one-dimensional Brownian bridge supremum between the sampled dates using its distribution. This technique outperforms the naïve Euler scheme presented above and reduces the bias as well as the convergence rate, see [START_REF] Gobet | Weak approximation of killed diffusion using euler schemes[END_REF].

In Table 1 and Table 2 we reported the CPU times of the simulations for N = 1000, 5000 and 10000 and 100 and 1000 numbers of time steps. We display the analytical (true) value for the expected impairment at the end of the first period (lower panel) as well as the probability of positive impairment (left panel). The choice of the number of paths is an important issue here. In fact, the number of steps is of little impact especially when looking at the variance of the estimations. The latter is almost equivalent for different time steps but decreases as the number of simulations increases. However, this bias reduction on the estimated quantities increases the computational time. For instance, the 10000 simulations scheme increases the accuracy up to ten times compared to 1000 scheme but does increase the computational time by the same rate.

In the left panel of Figure 4 in the Appendix C, we plotted the joint distribution of the aggregated equities and the corresponding impairment. We used the analytical formula given in Theorem 1 for n = 2 and different values of the input parameters. We observe that the probability to recognize an impairment next year together with a fall of the underlying equities does not exhibit a particular pattern: neither globally convex in x and y nor concave. This probability is very low for some levels of x and y, which is intuitively predictable. However, for such levels the simulation based estimation, see Figure 5 in the Appendix C, had difficulties to capture these joint probabilities as the variance of the estimation is very high. In fact, even if the number of simulated paths is sufficiently high, the scenarios triggering the events (the impairment together with the equities drop-down) are rare and thus make the estimation unstable and volatile.

5.2 Correlated securities. The case of correlated assets relies on semi analytical formulas that require further computations. Indeed, the results in Subsection 4.2, i.e. Proposition 5 and Theorem 2, involve not only some expectations but also multiple improper integrals. However, the underlying r.v. have a standard normal density and should not be of a particular concern for the implementation. Also, we should note that these formulas can be simplified for some particular forms of the correlation coefficient ρ. Indeed, when the latter can take on only the special values ρ k = -cos(π/k), k = 1, 2, • • • , the two functions q and p in Definition 1 can be determined in a more tractable form. In particular, the infinite sum involved in the definition of p can be written as a finite sum of bivariate normal densities. A similar finite sum formula can be derived for the terms involved in the joint density, see [START_REF] He | Double lookbacks[END_REF] for instance and Lemma 2 in Appendix B. Still, the analytical form involves some expectations. Therefore, we focus on the execution time needed to implement each of these quantities as well as the underlying bias.

For the sake of simplicity, we will focus on the above mentioned special case and let the correlation coefficient equal to ρ k = -cos(π/k). For instance, we consider the case where k = 5 and look at the c.d.f. Table 1: Computational Time (Probability). The calculation time and variance of the estimation for the impairment probability P s (Γ t > 0) in Proposition 2 and Proposition 5 with t = 1 and s = 0 for two shares (n = 2). The input parameters are as follows α = 80%, r = 1%. Comonotonic shares (left panel): S 1 0 = 100 and acquisition costs S 1 t 1 = 120, S 1 t 2 = 150 and σ 1 = 30%. Correlated shares (right panel): S 1 0 = 100, S 2 0 = 120, S 1 t 1 = 120, S 2 t 2 = 150 and σ 1 = 30%, σ 2 = 40%. in Proposition 5. In fact, as mentioned earlier the next-period expected impairment does not depend on the correlation between the two shares and the discussion made in the previous section also applies as one can only simulate comonotonic shares but with different initial values to recover the underlying result for the linearly dependent case. In Figure 3 we depict this c.d.f. for different discretization schemes (M = 100 and M = 1000). As we can see similar conclusions can drawn as for the comonotonic case.

Comonotonic

Regarding the execution time, see the right panel of Table 1, the semi-analytical can be computationally intensive. However, for small number of paths, N = 1000, the variance of estimate is 10 7 smaller than a naïve Monte-Carlo scheme variance with an equivalent execution time, i.e. 0.28 seconds.

Concluding Remarks

In this paper, we considered French impairment rules regarding equity securities held by an insurer.

Our objective was to go further than common practice which consists in considering an overall portfolio of equities as a single aggregated stock, since the impairment mechanism is complex and has to be considered on an individual basis. Under the Black-Scholes framework (geometric Brownian motion dynamics for the stocks), we derived closed form formulas for the joint distributions of the impaired amounts and the equities market values as well as the next-period impairment provisions. Formally, the results present analytical formulas for the comonotonic shares and semi-analytical forms in the linearly dependent case. The latter can be improved to obtain more tractable formulas depending on the correlation factor. These results enable to improve cash-flows projection models which are of massive use for both prudential and financial reporting purposes. Moreover, our results are the starting ground for future research on the optimal aggregation of equities. In fact, in future work we intend to study the construction of a proxy for representing a book of equities while ensuring the adequate representation of the distributional behaviour of both the aggregated net accounting values and market values. Figure 3: Distribution function. The probability P(Γ t > 0) in Proposition 5, for t = 1 and s = 0, as a function of the interest rate r (left) and impairment level α (right) for equities S 1 0 = 100, S 2 0 = 120 and S 1 t 1 = 120 and S 2 t 2 = 150.

Proposition 6 Let X t = µt + σB t , and M t = sup u∈[0,t] X u where B is a standard Brownian motion.

Then, the probability P(X t ≤ y, M t ≥ x) is given by:

             φ y -µt σ √ t if 0 ≥ x, e 2µx/σ 2 φ y-2x-µt σ √ t if 0 < x, y ≤ x, φ y -µt σ √ t -φ x -µt σ √ t + e 2µx/σ 2 φ -x -µt σ √ t if 0 < x < y.
In the other hand, the c.d.f. P (M t ≤ x) is given by:

   0 if 0 ≥ x, φ x -µt σ √ t -e 2µx/σ 2 φ -x -µt σ √ t if 0 < x.
where φ is the c.d.f of a standard Gaussian r.v.

We easily deduce

P(X t ∈ dy, M t ≤ x) = 1 0<x,y≤x 1 σ √ 2πt e -(y-µt) 2 2σ 2 t -e 2µx σ 2 e -(y-2x-µt) 2 2σ 2 t dy.
Another useful result is the following :

Proposition 7 Let α, β, w, c ∈ R and X is a standard Gaussian random variable, i.e. X ∼ N (0, 1) with distribution function φ. Then

E 1 X≤w e -cX φ(aX + b) = e c 2 /2 Φ w + c, b -ac √ 1 + a 2 ; -a √ 1 + a 2 (1.1)
where Φ(U, V ; θ) is the joint c.d.f. of two correlated Gaussian r.v.s U and V with a given correlation coefficient θ.

Proof. If one considers two correlated random variables X 1 and X 2 with correlation coefficient ρ such that X i ∼ N (µ i , σ i ), i = 1, 2 then the distribution of X 2 conditionally on X 1 can be written as follows:

X 2 |X 1 ∼ N µ 2 + ρ σ 2 σ 1 (X 1 -µ 1 ), (1 -ρ 2 )σ 2 2 .
Thus, we can write the joint distribution of X 1 and X 2 in the following form

P (X 1 ≤ x 1 , X 2 ≤ x 2 ) = E 1 X≤ x 1 -µ 1 σ 1 φ y -µ 2 -ρ σ 2 σ 1 (Xσ 1 + µ 1 -µ 1 ) σ 2 1 -ρ 2 ,
which can also be given in terms of the joint distribution Φ of two standard Gaussian r.v.'s with a given correlation coefficient. Therefore, we can write for any a, b, c and w

E 1 X≤w e -cX φ(aX + b) = e c 2 /2 E 1 X≤w+c φ(αX + b -ac) = e c 2 /2 Φ w + c, b -ac √ 1 + a 2 ; -a √ 1 + a 2 , (1.2)
which is the desired result.

A.1 Proof of Proposition 1. The cases x < 0 and x ≥ a i are obvious. Remark that on

{ Si t ≤ αa i }, a i (1 -α) ≤ a i -S i t . Hence, for x < a i (1 -α), P s (PDD i t ≤ x) = P s ( Si t > a i α) = 1 -P s (P t-1 2 ( Si t ≤ a i α)).
We then use the Markov property of S i and Proposition 6. For a i (1 -α) ≤ x < a i , we have

P s (PDD i t ≤ x) = E s (a i -S i t ) + 1 Si t ≤αa i ≤ x = E s E t-1 2 (a i -S i t ) + 1 Si t ≤αa i ≤ x .
Using the Markov property of S i , we get

E s 1 S i t-1 2 ≤αa i E t-1 2 a i -S i t-1 2 e µ i 2 +σ i B 1 2 1 sup u∈[0, 1 2 ] (µ i u+σ i Bu)≤ln a i α/S i t-1/2 ≤ x ,
where B is a standard Brownian motion independent of W . We firstly use Proposition 6 to compute the conditional expectation and secondly we remark that

S i t-1 2 = S i s e X δ-1 2 where X δ-1 2 = µ i (δ -1 2 ) + σ i δ -1 2 X
where X is a standard Gaussian random variable independent of F s , with distribution function φ(•). We easily obtain

p 1 (a i , S i s ) =1 -E s   1 X≤w φ   ln( a i α S i s ) -µ i δ -σ i δ -1 2 X σ i 1/2     + a i α S i s 2µ i σ 2 i e - 2µ 2 i σ 2 i (δ-1 2 ) × × E s   1 X≤w e - 2µ i σ i δ-1 2 X φ   -ln( a i α S i s ) + µ i (δ -1) + σ i δ -1 2 X σ i 1/2     , p 2 (a i , S i s ) =1 -E s   1 X≤w φ   ln( a i -x S i s ) -µ i δ -σ i δ -1 2 X σ i 1/2     + a i α S i s 2µ i σ 2 i e - 2µ 2 i σ 2 i (δ-1 2 ) × × E s   1 X≤w e - 2µ i σ i δ-1 2 X φ   -ln( (a i -x)S i s a 2 α 2 ) + µ i (δ -1) + σ i δ -1 2 X σ i 1/2     ,
This result could be enhanced as the expectations involved in the formulas of p 1 and p 2 can be explicitly computed using Equation (1.2).

A.2 Proof of Proposition 2. We remark that P s (Γ t > 0) = 1 -P s (Γ t = 0) = 1 -P s ( St > α max(a 1 . . . a n )) and we conclude as in the proof of Proposition 1 that P s (Γ t > 0) = 1-p 1 (max(a 1 . . . a n ), S s ), which is the desired result.

A.3 Proof of Proposition 3. We notice that the expected impaired amount for a single share S with an acquisition cost a i can be computed as follows

E s (P DD i t ) = a i P s ( S t ≤ a i α) -E s S t 1 St≤a i α .
The first term P s ( S t ≤ a i α) = 1 -p 1 (a i , S s ). The second term can be expressed, using the Markov property, in the following form:

E s S t-1 2 1 S t-1 2 ≤a i α E t-1 2 e µ 2 +σB 1 2 1 sup u∈[0, 1 2 ] µu+σBu≤ln a i α/S t-1/2
, which needs the density P X t ∈ dx, sup u∈[0,t] X u ≤ a . Now, using this density we can write that

E t-1 2 (. . . ) equals e r/2 φ    ln a i α S t-1 2 -µ+σ 2 2 σ 1/2    - a i α S t-1 2 2(µ+σ 2 ) σ 2 e r/2 φ    -ln a i α S t-1 2 -µ+σ 2 2 σ 1/2    .
Finally, using the fact that S t-1 2 = S s e µ(δ-1 2 )+σ δ-1 2 X , with δ = t -s and X a standard normal r.v. independent of F s , together with the identity in Equation (1.2) leads to the desired formula.

A.4 Proof of Theorem 1. To find the joint probability P t (Γ t+1 ≤ y, Σ t+1 ≤ x), we will need the following technical lemma (to prove it, we use Equation (1.2)).

Lemma 1 Let a > 0, β, γ and α be some constants, then for s, t ≥ 0 we have the following

E t 1 S t+s <a S -γ t+s φ(α + β ln S t+s ) = S -γ t e -γ(r-σ 2 2 )s × Φ ln a St -(r -σ 2 2 -γσ 2 )s σ √ s , α + β ln(S t ) + βs(r -σ 2 2 ) -βγσ 2 s 1 + β 2 σ 2 s ; - βσ √ s 1 + β 2 σ 2 s
.

Similarly, we have

E t 1 S t+s >a φ(α + β ln S t+s ) = E t [φ(α + β ln S t+s )] -E t 1 S t+s <a φ(α + β ln S t+s ) =Φ 0, α + β ln S t + βs(r -σ 2 2 ) 1 + β 2 σ 2 s ; - βσ √ s 1 + β 2 σ 2 s + Φ 0, α + β ln S t + βs(r -σ 2 2 ) 1 + β 2 σ 2 s ; βσ √ s 1 + β 2 σ 2 s -Φ ln a St -(r -σ 2 2 -γσ 2 )s σ √ s , α + β ln(S t ) + βs(r -σ 2 2 ) 1 + β 2 σ 2 s ; - βσ √ s 1 + β 2 σ 2 s .
Let n ≥ 1, we are concerned with the computation of the conditional joint c.d.f. of (Γ t+1 , Σ t+1 ):

P t (Γ t+1 ≤ y, Σ t+1 ≤ x) = E t P t+ 1 2 S t+1 ≤ x n , n i=1
(a i -S t+1 ) + 1 S t+1 ≤a i α ≤ y .

In the above equation, the probability conditional on F t+ 1 2 can be decomposed in the following form:

P t+ 1 2 S t+1 ≤ x n , St+1 ≤ a 1 α, n i=1 (a i -S t+1 ) + ≤ y (1.3) + • • • +P t+ 1 2   S t+1 ≤ x n , a j α < S t+1 ≤ a j+1 α, n i=j+1 (a i -S t+1 ) + ≤ y   (1.4) + • • • +P t+ 1 2 S t+1 ≤ x n , S t+1 ≥ a n α . (1.5)
It is easily seen that the term (1.3) can be written as

P t+ 1 2 S t+1 ≤ a 1 α, 1 n ( n i=1 a i -y) ≤ S t+1 ≤ x n ∧ a 1 α ,
and similarly the term (1.4) as

P t+ 1 2 a j α < S t+1 ≤ a j+1 α, 1 n-j ( n i=j+1 a i -y) ≤ S t+1 ≤ x n ∧ a j+1 α .
Moreover, we use Markov property and Proposition 6 to write the term (1.5) in the following form

1 S t+ 1 2 ≥anα φ   ln x nS t+ 1 2 -µ 2 σ √ 0.5   + 1 S t+ 1 2 <anα 1 x n ≤anα a n α S t+ 1 2 2µ σ 2 φ    ln xS t+ 1 2 na 2 n α 2 -µ 2 σ √ 0.5    + 1 S t+ 1 2 <anα 1 x n >anα   φ   ln x nS t+ 1 2 -µ 2 σ √ 0.5   -φ   ln anα S t+ 1 2 -µ 2 σ √ 0.5   + a n α S t+ 1 2 2µ σ 2 φ   -ln anα S t+ 1 2 -µ 2 σ √ 0.5     . On { 1 n ( n i=1 a i -y) ≤ x n ∧ a 1 α}, the term (1.
3) is equal to :

1 S t+ 1 2 <a 1 α   φ    ln x n ∧a 1 α S t+ 1 2 -µ 2 σ √ 0.5    - a 1 α S t+ 1 2 2µ σ 2 φ    ln x n ∧a 1 α S t+ 1 2 -2 ln a 1 α S t+ 1 2 -µ 2 σ √ 0.5    -φ    ln n i=1 a i -y nS t+ 1 2 -µ 2 σ √ 0.5    + a 1 α S t+ 1 2 2µ σ 2 φ    ln n i=1 a i -y nS t+ 1 2 -2 ln a 1 α S t+ 1 2 -µ 2 σ √ 0.5       .
On { 1 n-j ( n i=1+j a i -y) ≤ x n ∧ a j+1 α} the term (1.4) is equal to :

1 S t+ 1 2 ≤a j+1 α   φ    ln x n ∧a j+1 α S t+ 1 2 -µ 2 σ √ 0.5    - a j+1 α S t+ 1 2 2µ σ 2 φ    ln x n ∧a j+1 α S t+ 1 2 -2 ln a j+1 α S t+ 1 2 -µ 2 σ √ 0.5    -φ    ln n i=j+1 a i -y (n-j)S t+ 1 2 -µ 2 σ √ 0.5    + a j+1 α S t+ 1 2 2µ σ 2 φ    ln n i=j+1 a i -y (n-j)S t+ 1 2 -2 ln a j+1 α S t+ 1 2 -µ 2 σ √ 0.5       +1 S t+ 1 2 ≤a j α 1 1 n-j ( n i=j+1 a i -y)>a j α -1 x n >a j α   φ   ln a j α S t+ 1 2 -µ 2 σ √ 0.5   + a j α S t+ 1 2 2µ σ 2 φ    ln -a j α S t+ 1 2 -µ 2 σ √ 0.5       +1 S t+ 1 2 ≤a j α 1 x n <a j α   φ    ln x n ∧a j+1 α S t+ 1 2 -µ 2 σ √ 0.5    - a j α S t+ 1 2 2µ σ 2 φ    ln x n ∧a j+1 α S t+ 1 2 -2 ln a j α S t+ 1 2 -µ 2 σ √ 0.5       +1 S t+ 1 2 ≤a j α 1 1 n-j ( n i=j+1 a i -y)≤a j α   φ    ln n i=j+1 a i -y (n-j)S t+ 1 2 -µ 2 σ √ 0.5    + a j α S t+ 1 2 2µ σ 2 φ    ln n i=j+1 a i -y (n-j)S t+ 1 2 -2 ln a j α S t+ 1 2 -µ 2 σ √ 0.5       .
We conclude the proof using Lemma 1 to compute the expectations conditional to F t .

B Correlated securities

In order to prove our results in the correlated case, we shall need the joint distribution of two correlated Brownian motions with drift and their suprema. The joint distribution is obtain by making the following change of variables x 2 = M 2 -σ 2 r sin θ and x 1 = M 1 -σ 1 r(ρ sin θ + 1 -ρ 2 cos θ) in Lemma 3 and Remark 4 from [START_REF] He | Double lookbacks[END_REF].

Proposition 8 Let X i = µ i t + σ i W i t , where µ i ∈ R, σ i > 0 and W i two correlated Brownian motions with correlation coefficient ρ ∈] -1, 1[. For x 1 ≤ M 1 , x 2 ≤ M 2 with M 1 , M 2 ≥ 0, we have : P(X 1 t ∈ dx 1 , X 2 t ∈ dx 2 , sup u≤t X 1 u ≤ M 1 , sup u≤t X 2 u ≤ M 2 ) = p(r, r 0 , θ, θ 0 , t; M 1 , M 2 )drdθ P( X1 0→t ≤ M 1 , X2 0→t ≤ M 2 ) = q(r 0 , θ 0 , t; M 1 , M 2 ),
where p and q are given in Definition 1 and

r = 1 1 -ρ 2 (M 1 -x 1 ) 2 σ 2 1 -2ρ (M 1 -x 1 )(M 2 -x 2 ) σ 1 σ 2 + (M 2 -x 2 ) 2 σ 2 2 , r 0 = 1 1 -ρ 2 M 2 1 σ 2 1 -2ρ M 1 M 2 σ 1 σ 2 + M 2 2 σ 2 2 , tan β = - 1 -ρ 2 ρ , β ∈ [0, π], tan θ = 1 -ρ 2 σ 2 (M 1 -x 1 ) σ 1 (M 2 -x 2 ) -ρ -1 , tan θ 0 = 1 -ρ 2 σ 2 M 1 σ 1 M 2 -ρ -1 , θ, θ 0 ∈ [0, β].
For some special correlation values, the function q may be written in a simpler form (see Corollary 2 in [START_REF] He | Double lookbacks[END_REF]). We easily obtain the following results4 :

Lemma 2 Under the assumptions of Proposition 8, if ρ n = -cos π n , we have : p(r, r 0 , θ, θ 0 , t; M 1 , M 2 )drdθ1 r≥0,θ∈ [0,β]

= p(x 1 , x 2 , t; M 1 , M 2 )dx 1 dx 2 1 x 1 ≤M 1 ,x 2 ≤M 2 , q(r 0 , θ 0 , t; M 1 , M 2 ) = q(M 1 , M 2 , t), where p(x 1 , x 2 , t; M 1 , M 2 ) = e α 1 x 1 +α 2 x 2 +bt n-1 k=1 g + k (x 1 , x 2 , t) + g - k (x 1 , x 2 , t) , q(M 1 , M 2 , t) = n-1 k=0 Q + (M 1 , M 2 , 2kπ n + θ) -Q -(M 1 , M 2 , 2kπ n -θ) , g + - k (x 1 , x 2 , t) = + - 1 2πt exp    - 1 2t   1 1 -ρ 2 n M 1 -x 1 σ 1 -ρ n M 2 -x 2 σ 2 + ξ 1 2 + M 2 -x 2 σ 2 -ξ 2 2      , Q + -(M 1 , M 2 , 2kπ n + -θ) = exp B 0 + B 1 ξ 1 + B 2 ξ 2 + ( B 2 1 + B 2 2 2 + b)t × × Φ ξ 2 + B 2 t √ t , -1 -ρ 2 n (ξ 1 + B 1 t) + ρ n (ξ 2 + B 2 t) √ t , ρ n , ξ 1 = -r 0 cos 2kπ n + -θ , ξ 2 = r 0 sin 2kπ n + -θ , tan θ = - 1 -ρ 2 n ρ n -M 1 σ 2 M 2 σ 1 , r 0 = 1 1 -ρ 2 n M 2 1 σ 2 1 -2ρ n M 1 M 2 σ 1 σ 2 + M 2 2 σ 2 2 , B 0 = a 1 M 1 + a 2 M 2 , B 1 = a 1 σ 1 1 -ρ 2 n , B 2 = -(a 1 σ 1 ρ n + a 2 σ 2 ) = -α 2 σ 2 , b = - α 2 1 σ 2 2 + α 2 2 σ 2 1 2(1 -ρ 2 n )σ 2 1 σ 2 2 + ρ n α 1 α 2 (1 -ρ 2 n )σ 1 σ 2 , a 1 = α 1 σ 2 -ρ n α 2 σ 1 (1 -ρ 2 n )σ 2 1 σ 2 , a 2 = α 2 σ 1 -ρ n α 1 σ 2 (1 -ρ 2 n )σ 1 σ 2 2 .
B.1 Proof of Proposition 5. We remark that

P s (Γ t > 0) = 1 -P s (Γ t = 0) = P s S1 t ≤ αa 1 + P s S2 t ≤ αa 2 -P s S1 t ≤ αa 1 , S2 t ≤ αa 2 = 2 i=1 p i (a i , S i s ) -P s P t-1/2 S1 t ≤ αa 1 , S2 t ≤ αa 2 .
Markov property, Proposition 8 and writing S 1 t-1/2 = S 1 s e µ 1 (δ-1/2)+σ 1 √ δ-1/2X with X ∼ N (0, 1) lead to the desired formula.

We need some technical results in order to find the joint law of Γ and Σ.

Lemma 3 Let x > 0. The c.d.f of the aggregate dynamics Σ of two correlated shares is given by

P t (Σ t+1 ≤ x) = E 1 X≤ ln x s 1 -µ 1 σ 1 φ 1 σ 2 1 -ρ 2 ln x -s 1 e µ 1 +σ 1 X s 2 e µ 2 +σ 2 ρX s 1 =S 1 t ,s 2 =S 2 t ,
where X is a standard Gaussian r.v. We denote this probability by f (S 1 t , S 2 t , x) = P t (Σ t+1 ≥ 0).

Proof. Since W 1 and W 2 are two correlated Brownian motions with correlation coefficient ρ, then W 2 = ρW 1 + 1 -ρ 2 W with W 1 and W two independent BM. Hence :

P t (Σ t+1 ≤ 0) = P t (S 1 t e µ 1 +σ 1 X + S 2 t e µ 2 +σ 2 ρX+σ 2 √ 1-ρ 2 Z ≤ x),
where X and Z are two independent standard Gaussian r.v. We integrate with respect Z and the conclusion holds.

Lemma 4 Let X i t = µ i t + σ i W i t , where µ i , σ i ∈ R and W i two correlated BM with correlation coefficient ρ ∈] -1, 1[. Then for M 1 > 0, x 1 , x 2 ∈ R P(X 1 t ∈ dx 1 , X 2 t ∈ dx 2 , sup u≤t X 1 u ≤ M 1 ) = h(x 1 , x 2 , M 1 )dx 1 dx 2 where h(x 1 , x 2 , M 1 ) = 1 x 1 <M 1 e - (x 1 -µ 1 t) 2 2σ 2 1 t -e 2 µ 1 M 1 σ 2 1 e - (x 1 -2M 1 -µ 1 t) 2 2σ 2 1 t 1 2πtσ 1 σ 2 1 -ρ 2 e - (σ 1 x 2 -ρσ 2 x 1 +(µ 1 σ 2 ρ-µ 2 σ 1 )t) 2 2σ 2 1 σ 2 2 t(1-ρ 2 )
.

Proof. Without loss of generality we consider

σ i = 1, i = 1, 2 (we then replace x 1 , M 1 , x 2 by x 1 σ 1 , M 1 σ 1 , x 2 σ 2 ). Since P(X 1 t ∈ dx 1 , X 2 t ∈ dx 2 , sup u≤t X 1 u ≤ M 1 ) = ∂ 2 ∂x 1 ∂x 2 P(X 1 t ≤ x 1 , X 2 t ≤ x 2 , sup u≤t X 1 u ≤ M 1 )
and W 2 = ρW 1 + 1 -ρ 2 W with W 1 and W two independent BM, we can write for ρ = 0

5 P(X 1 t ≤ x 1 , X 2 t ≤ x 2 , sup u≤t X 1 u ≤ M 1 ) =1 x 1 ≥M 1 P(X 2 t ≤ x 2 , sup u≤t X 1 u ≤ M 1 ) + 1 x 1 <M 1 P(X 1 t ≤ x 1 ∧ 1 ρ (x 2 -1 -ρ 2 W t + (µ 1 ρ -µ 2 )t), sup u≤t X 1 u ≤ M 1 ).
The derivative of the first term is 0. For the second term, we integrate with respect to the law of W t and then we use Proposition 6. Finally the derivative with respect to x 1 and x 2 leads to the desired result.

We easily deduce the following :

Corollary 4 Under the assumptions of Lemma 4 let R 1 , R 2 , x, w, M 1 > 0. Then P(R 1 e X 1 t + R 2 e X 2 t ≤ x, sup u≤t X 1 u ≤ M 1 ) = E   1 X≤ ln x R 1 ∧M 1 -µ 1 t σ 1 √ t φ 1 σ 2 t(1 -ρ 2 ) ln x -R 1 e σ 1 X √ t+µ 1 t R 2 e Xρσ 2 √ t+µ 2 t   -e 2 µ 1 M 1 σ 2 1 E   1 X≤ ln x R 1 ∧M 1 -2M 1 -µ 1 t σ 1 √ t φ 1 σ 2 t(1 -ρ 2 ) ln x -R 1 e σ 1 X √ t+2M 1 +µ 1 t R 2 e Xρσ 2 √ t+2 σ 2 σ 1 ρM 1 +µ 2 t   , 5 If ρ = 0, then P(X 1 t ∈ dx1, X 2 t ∈ dx2, sup u≤t X 1 u ≤ M1) = P(X 1 t ∈ dx1, sup u≤t X 1 u ≤ M1)P(X 2 t ∈ dx2) = 1x 1 <M 1 e -(x 1 -µ 1 t) 2 2t -e 2µ 1 M 1 e -(x 1 -2M 1 -µ 1 t) 2 2t 1 2πt e -(x 2 -µ 2 t) 2 2t dx1dx2 P(R 1 e X 1 t + R 2 e X 2 t ≤ x, sup u≤t X 1 u ≤ M 1 , e X 1 t ≥ w) = 1 w≤0 P(R 1 e X 1 t + R 2 e X 2 t ≤ x, sup u≤t X 1 u ≤ M 1 ) +1 0<w≤ x R 1 ∧e M 1   E   1 ln w-µ 1 t σ 1 √ t ≤X≤ ln x R 1 ∧M 1 -µ 1 t σ 1 √ t φ 1 σ 2 t(1 -ρ 2 ) ln x -R 1 e σ 1 X √ t+µ 1 t R 2 e Xσ 2 ρ √ t+µ 2 t   -e 2 µ 1 M 1 σ 2 1 E   1 ln w-µ 1 t-2M 1 σ 1 √ t ≤X≤ ln x R 1 ∧M 1 -2M 1 -µ 1 t σ 1 √ t φ 1 σ 2 t(1 -ρ 2 ) ln x -R 1 e σ 1 X √ t+2M 1 +µ 1 t R 2 e Xσ 2 ρ √ t+2ρ σ 2 σ 1 M 1 +µ 2 t     ,
where X is a standard Gaussian r.v.

B.2 Proof of Theorem 2. The joint c.d.f

P t (Γ t+1 ≤ y, Σ t+1 ≤ x) = P t (a 1 -S 1 t+1 )1 S1 t+1 ≤αa 1 + (a 2 -S 2 t+1 )1 S2 t+1 ≤αa 2 ≤ y, S 1 t+1 + S 2 t+1 ≤ x ,
can be decomposed in the following form :

P t S 1 t+1 + S 2 t+1 ≤ x, S1 t+1 > αa 1 , S2 t+1 > αa 2 + i,j∈{1,2},i =j P t S 1 t+1 + S 2 t+1 ≤ x, Si t+1 ≤ αa i , Sj t+1 > αa j , S i t+1 ≥ a i -y +P t a 1 + a 2 -y ≤ S 1 t+1 + S 2 t+1 ≤ x, S1 t+1 ≤ αa 1 , S2 t+1 ≤ αa 2 .
Step 1 -Probabilities decomposition • The first term C 1 (S 1 t , S 2 t ) = P t S 1 t+1 + S 2 t+1 ≤ x, S1 t+1 > αa 1 , S2 t+1 > αa 2 may be written as :

1 x≥α(a 1 +a 2 ) P t (S 1 t+1 + S 2 t+1 ≤ x) -P t (S 1 t+1 + S 2 t+1 ≤ x, S1 t+1 ≤ αa 1 ) -P t (S 1 t+1 + S 2 t+1 ≤ x, S2 t+1 ≤ αa 2 ) + P t (S 1 t+1 + S 2 t+1 ≤ x, S1 t+1 ≤ αa 1 , S2 t+1 ≤ αa 2 ) .
Recall that S 1 t+1 + S 2 t+1 = Σ t+1 . Note that on {x ≥ α(a 1 + a 2 )},

P t S 1 t+1 + S 2 t+1 ≤ x, S1 t+1 ≤ αa 1 , S2 t+1 ≤ αa 2 = P t S1 t+1 ≤ αa 1 , S2 t+1 ≤ αa 2 .
Hence,

C 1 (S 1 t , S 2 t ) = 1 x≥α(a 1 +a 2 ) P t (Σ t+1 ≤ x) -P t (Σ t+1 ≤ x, S1 t+1 ≤ αa 1 ) -P t (Σ t+1 ≤ x, S2 t+1 ≤ αa 2 ) + P t ( S1 t+1 ≤ αa 1 , S2 t+1 ≤ αa 2 ) .
• The second term C 2 (S i t , S j t , a i , a j ) = P t S 1 t+1 + S 2 t+1 ≤ x, Si t+1 ≤ αa i , Sj t+1 > αa j , S i t+1 ≥ a i -y equals to :

1 y≥a i 1 αa j ≤x<α(a 1 +a 2 ) P t (Σ t+1 ≤ x) -P t (Σ t+1 ≤ x, Sj t+1 ≤ αa j ) + 1 y≥a i 1 x≥α(a 1 +a 2 ) P t (Σ t+1 ≤ x, Si t+1 ≤ αa i ) -P t ( Si t+1 ≤ αa i , Sj t+1 ≤ αa j ) + 1 a i (1-α)≤y<a i 1 x≥(αa j )∨(a i -y) P t (Σ t+1 ≤ x, Si t+1 ≤ αa i , S i t+1 ≥ a i -y) -P t (Σ t+1 ≤ x, Si t+1 ≤ αa i , Sj t+1 ≤ αa j , S i t+1 ≥ a i -y) .

• The last term C 3 (S 1 t , S 2 t ) = P t a 1 + a 2 -y ≤ S 1 t+1 + S 2 t+1 ≤ x, S1 t+1 ≤ αa 1 , S2 t+1 ≤ αa 2 may be written as :

1 x≥α(a 1 +a 2 ) 1 y≥a 1 +a 2 P t ( S1 t+1 ≤ αa 1 , S2 t+1 ≤ αa 2 ) + 1 x≥(a 1 +a 2 -y)∨α(a 1 +a 2 ) 1 y<a 1 +a 2 P t (Σ t+1 ≥ a 1 + a 2 -y, S1 t+1 ≤ αa 1 , S2 t+1 ≤ αa 2 ) + 1 x<α(a 1 +a 2 ) 1 y≥a 1 +a 2 P t (Σ t+1 ≤ x, S1 t+1 ≤ αa 1 , S2 t+1 ≤ αa 2 ) + 1 0<a 1 +a 2 -y≤x<α(a 1 +a 2 ) P t (a 1 + a 2 -y ≤ Σ t+1 ≤ x, S1 t+1 ≤ αa 1 , S2 t+1 ≤ αa 2 ).

Step 2 -Grouping of terms Remark that for the term C 2 (S i t , S j t , a i , a j ), we may use Markov property and Proposition 8. Hence we can write (consider for example the case i = 1 and j = 2) :

P t (Σ t+1 ≤ x, S1 t+1 ≤ αa 1 , S2 t+1 ≤ αa 2 , S 1 t+1 ≥ a 1 -y) =E t   1 S 1 t+ 1 2 ≤αa 1 ,S 2 t+ 1 2 ≤αa 2 P t+ 1 2   S 1 t+ 1 2 e X 1 1 2 + S 2 t+ 1 2 e X 2 1 2 ≤ x, X1 1 2 ≤ ln αa 1 S 1 t+ 1 2 , X2 1 2 ≤ ln αa 2 S 2 t+ 1 2 , X 1 1 2 ≥ ln a 1 -y S 1 t+ 1 2     , = ∞ 0 β 0 E t 1 X≤A 1 ,Y ≤A 2 p r, θ, σ 1 √ 0.5(A 1 -X), σ 2 √ 0.5(A 2 -Y ) s i =S i t 1 (r, θ)∈D 1 drd θ,
where X i t = µ i t + σ i B i t where B i are two correlated BM independent of F t+ 1 2 , and D 1 given before.

The same reasoning allows to write P t (Σ t+1 ≤ x, S1 t+1 ≤ αa 1 , S2 t+1 ≤ αa 2 , S 2 t+1 ≥ a 2 -y), P t (Σ t+1 ≥ a 1 +a 2 -y, S1 t+1 ≤ αa 1 , S2 t+1 ≤ αa 2 ), P t (Σ t+1 ≤ x, S1 t+1 ≤ αa 1 , S2 t+1 ≤ αa 2 ) and P t (a 1 +a 2 -y ≤ Σ t+1 ≤ x, S1 t+1 ≤ αa 1 , S2 t+1 ≤ αa 2 ) in a similar form with D 2 , D 3 (a 1 +a 2 -y), D 3 (x) and D 3 (x)-D 3 (a 1 +a 2 -y). Hence, we have P t (Γ t+1 ≤ y, Σ t+1 ≤ x) = i,j∈{1,2},i =j P t (Σ t+1 ≤ x, Si t+1 ≤ αa i )I 1 (a i , a j ) + P t ( S1 t+1 ≤ αa 1 , S2 t+1 ≤ αa 2 )J 4 + i,j∈{1,2},i =j P t (Σ t+1 ≤ x, Si t+1 ≤ αa i , S i t+1 ≥ a i -y)I 2 (a i , a j ) + P t (Σ t+1 ≤ x)J 3

+ ∞ 0 β 0 E t 1 X≤A 1 ,Y ≤A 2 p r, θ, σ 1 √ 0.5(A 1 -X), σ 2 √ 0.5(A 2 -Y ) s i =S i t J(r, θ)drd θ.
Step 3 -Final calculation Remark that P t ( S1 t+1 ≤ αa 1 , S2 t+1 ≤ αa 2 ) = Q(S 1 t , S 2 t , 1) with Q given in Proposition 5. Also, the conditional law of Σ, i.e. P t (Σ t+1 ≤ x), is given in Lemma 3. Since the calculation method is the same for all the other terms, we only detail here the probability P t (Σ t+1 ≤ x, S1 t+1 < αa 1 ). Thus, Markov property gives

P t (Σ t+1 ≤ x, S1 t+1 < αa 1 ) = E t   1 S 1 t+ 1 2 <αa 1 P t+ 1 2   S 1 t+ 1 2 e X 1 1 2 + S 2 t+ 1 2 e X 2 1 2 ≤ x, X1 1 2 < ln αa 1 S 1 t+ 1 2     ,             
, where Z is a standard Gaussian r.v. independent of F t+ 1 2 .

We conclude writing S 1 We also find

t+ 1 2 = S 1 t + µ 1 2 + σ 1 √ 0.
P t (Σ t+1 ≤ x, S1 t+1 ≤ αa i ,S 1 t+1 ≥ a 1 -y) = E   1 X≤A 1 , ln a 1 -y s 1 -µ 1 σ 1 √ 0.5 -X≤Z≤ ln x∧(αa 1 ) s 1 -µ 1 σ 1 √ 0.5 -X φ 12 (s 1 , s 2 , X, Y, Z)   s 1,2 =S 1,2 t - αa 1 S 1 t 2µ 1 σ 2 1 e - µ 2 1 σ 2 1 E   1 X≤A 1 , ln s 1 (a 1 -y) (αa 1 ) 2 σ 1 √ 0.5 +X≤Z≤ ln s 1 (x∧αa 1 ) (αa 1 ) 2 σ 1 √ 0.5 +X φ12 (s 1 , s 2 , X, Y, Z)    s 1,2 =S 1,2 t .
Remark that on I 1 , x > αa 1 , so :

P t (Σ t+1 ≤ x, S1 t+1 < αa 1 ) + P t (Σ t+1 ≤ x, S1 t+1 ≤ αa i , S 1 t+1 ≥ a 1 -y) =E [J 1 (s 1 , s 2 , X, Z)φ 12 (s 1 , s 2 , X, Y, Z)] -αa 1 S 1

t 2µ 1 σ 2 1 e - µ 2 1 σ 2 1 E J 2 (s 1 , s 2 , X, Z) φ12 (s 1 , s 2 , X, Y, Z) .
This completes the proof.

C Additional Figures

  . For any n > 3 there are at least 3 cases of correlation matrices : A n , B n , C n . (a) If n = 4 then there are a total of 5 cases : A 4 , B 4 , C 4 and F 4 , G 4 where

  b) If n = 5 then there are only three cases : A 5 , B 5 , C 5 . (c) If n = 6, 7 or 8, then there are 4 cases : A n , B n , C n and E n where

Figure 1 :

 1 Figure 1: Distribution function. The probability P s (Γ t > 0) in Proposition 1, for t = 1 (s = 0) as a function of the risk-free interest rate r (left), the level α (middle) and the volatility σ (right).

Figure 2 :

 2 Figure 2: Expected impairment. The conditional first moment E s (Γ t ) in Proposition 3, for t = 1 (s = 0) as a function of the risk-free interest rate r (left), the level α (middle) and the volatility σ (right).

Figure 4 :Figure 5 :

 45 Figure4: Joint distribution function. The joint probability P t (Γ t+1 < y, Σ t+1 ≤ x) in Theorem 1 (left) and Theorem 2 (right), for t = 0 with n = 2. The input parameters are as follows α = 80%, r = 1%. Comonotonic shares: acquisition values S 1 t 1 = 80 and S 1 t 2 = 90 and initial value S 1 0 = 120.

Table 2 :

 2 Computational Time (Expectation). The calculation time and variance of the estimation for the impairment probability E s [Γ t ] in Proposition 3 with t = 1 and s = 0 for two shares (n = 2). The input parameters are as follows α = 80%, r = 1%, S 1 0 = 100 and acquisition costs S 1 t 1 = 120, S 1 t 2 = 150 and σ 1 = 30%.

	N	1000 5000 10000
	Analytical		12.8967
	Simulation (M = 100)		13.4434
	Time (in seconds)	0.01	0.03	0.12
	Variance	0.689 0.151 0.036
	Simulation (M = 1000)		13.1812
	Time (in seconds)	0.06	0.34	1.98
	Variance	0.758 0.139 0.033

  5X (resp. S 2

												t+ 1 2	= S 2 t + µ 2 2 + σ 2
												
		 1	X≤A 1 ,Z≤	ln	x∧(αa 1 ) s 1 σ 1 √ 0.5 -µ 1	-X	φ 12 (s 1 , s 2 , X, Y, Z)  s 1,2 =S 1,2 t
	-	αa 1 S 1 t	2µ 1 σ 2 1 e -	µ 2 1 σ 2 1 E	  1	X≤A 1 ,Z≤	ln	s 1 (x∧αa 1 ) (αa 1 ) 2 σ 1 √ 0.5	+X	φ12 (s 1 , s 2 , X, Y, Z)	   s 1,2 =S 1,2

√ 0.5Y ) with X, Y two correlated standard Gaussian r.v's, that :

P t (Σ t+1 ≤ x, S1 t+1 < αa 1 ) =E t .

Règlement ANC n •

 available at https://www.anc.gouv.fr

The details on the French insurance regulation is available at https://www.legifrance.gouv.fr

The insurance company may compute a so-called "recoverable value" corresponding to the market value projected over the remaining expected holding period (with a maximum of 5 years) of the considered financial asset.

, δ = t -s.

Let's note that there is a mistake in Corollary 2[START_REF] He | Double lookbacks[END_REF] and t is sometimes missing

* This work has been supported by the BNP Paribas Cardif Chair "Data Analitycs and Models in Insurance".

Appendices

A Comonotonic securities

In order to prove our results, we shall need the joint distribution of a Brownian motion with drift and its supremum. This can be easily derived using [START_REF] Harrison | Brownian Motion and Stochastic Flow Systems[END_REF].