B Granado

M Gatti

J Denoulet

M Rayrole

In flight real-time adaptation

Avionics is one kind of domain where prevention prevails. Nonetheless fails occur. Sometimes due to pilot misreacting, flooded in information. Sometimes information itself would be better verified than trusted. To avoid some kind of failure, it has been thought to add an new kind of embedded monitoring that enable adaptation.

Introduction

It is well known that avionics is a very restricting domain for obvious safety reasons. Along with miniaturization comes the idea of integration. More functionality on one spot requires a good management of privacy and congestion on shared platforms. This is why determinism is one of the keywords of avionics works. This led to protocols like ARINC653 [START_REF] Spitzer | ARINC Specification 653, Avionics Application Software Standard Interface[END_REF] assuring that, multitask embedded programs respect a predictable policy applied by the operating system (OS). Another key protocol is ARINC664, which guarantees that multiple communicating systems efficiently share the network. These two protocols are pillars of the Integrated Modular Architecture (IMA) concept [START_REF] Paul | ARINC 653 role in integrated modular avionics (IMA)[END_REF]. IMA concept consists of multitask module hosting ARINC653 OS, interconnected with ARINC664 data network. Compared to federated avionics architecture, it considerably reduces the overall weight and power consumption for aircraft, reduces development expenses and design cycle times as well as maintenance costs. With the intention to step forward with this concept, the CORAC (The Council for Civil Aeronautics Research) develops a technological demonstration platform (PDT) called Extended Modular Avionic (AME) [START_REF] Corac | COnseil pour la Recherche Aéronautique Civile[END_REF]. Therefore, as partner of the project, we work on a project dedicated to monitor the system. In this paper, we use our embedded real time monitor to take real time in-flight decision to adapt the behavior of the digital processing. We first present our embedded SystemC simulator that can be embedded in a plane as a processing module to monitor data traffic generated by key avionic applications in order to detect suspicious behavior such as missing data, unexpected communication of simply incoherent data. Second we explain how, by exploring several architecture configurations, the simulator can help the IMA system to decide how it should be configured, how the processing resources should be assigned in order to efficiently manage specific or critical situations. In the next sections, we first introduce our method and its related tools, most notably the SystemC language and the modifications required to make this language compliant with avionic constraints. Next, we show how the simulator can be implemented in a QorIQ T2080-based board. We then introduce the use cases (monitoring and architecture exploration) to illustrate the benefits of our approach. Finally, we will conclude. In Figure 1, we present the principle of our method. We consider an avionic architecture featuring core processing modules (CPM) implementing several applications and generating data traffic and avionics switch modules (ASM) which route data packets to their destination CPM. As an example, CPM1 in Erreur ! Source du renvoi introuvable. features three partitions, each one hosting an application dedicated respectively to GPS, Speed, and Angle estimation. Through an ARINC664 communication End System, data generated by these applications are sent through several Virtual Links (VL) of the data network. While performing data traffic management, the ASM also implements a simulator that runs a timed model of the expected communication traffic, considering the OS and network parameters. The ASM is the privileged place to implement a simulator, since its CPU only manages message traffic and has available time. The simulator performs two types of verification: temporal consistency which checks whether communication occurs at the expected time, according to the system scheduling, and data consistency which analyses N consecutive data values to determine if their evolution is coherent or if we can assume an error has occurred. The simulator can also be used as an architecture exploration tool. It can model different application mappings on the system and test which one is the most efficient to handle specific scenarios. This could help the system to perform dynamic reconfiguration when it comes upon critical situations. To achieve such a goal, we have chosen the SystemC [START_REF] Initiative | IEEE standard SystemC language reference manual[END_REF] language as an appropriate candidate to model as well software (application) and hardware system (processors and communication modules) under time constraints (defined by ARINC653 and ARINC664). The next subsections briefly presents the SystemC language specifications, as well as SystemCASS, a SystemC simulation kernel we modified so that it can meet avionics requirements. We finally show the implementation of our SystemCASS simulator on a QorIQ T2080 design board.

Embedded simulator 2.1 Method presentation

SystemC

SystemC is a C++ class library based on object-oriented design concept (OOD) providing common Hardware Description Language (HDL) features. As such, it allows hardware description along with software development. Hardware behavior concurrency is simulated by the way simulation time is being managed by the simulator. Hardware components are modeled using the sc_module class and are interconnected to each other with sc_port class objects. Module internal registers are represented by sc_signals, and module behavior by processes, which can be described as functions triggered by the update of ports or signals that are registered in a sensitivity list. A SystemC program usually consists in an elaboration phase where all the elements of the described system are declared and assembled, and where all processes are listed. Then comes the simulation phase, which is initiated by the sc_start method, which is a function of the simulator. Finally, the cleanup phase ends simulation, by cleaning objects and structures. The role of a SystemC simulator is to manipulate the timestamp to simulate the concurrency of hardware behavior. It determines in which order processes must be executed, and when values of ports and signals must be updated. The Accelera Systems Initiative (ASI) provides an event driven simulator with the language library. The simulator operates according to Erreur ! Source du renvoi introuvable.. The simulation phase features three steps: Evaluation, in which the simulator checks which processes must be executed, according to their sensitivity list. The simulator then executes these processes. When this is done, the second step, Update, updates the values of ports/signals according to the previous processes executions. If signal or ports updates trigger a process sensitivity list again, then we go back to the evaluation step. When no process is triggered anymore, the simulation timestamp is updated in the Time Elapse step. The ASI simulator, as it is implemented, features memory dynamicity, which avionic constraints don't allow. Furthermore, process scheduling at each timestamp is dynamic and nondeterministic [START_REF] Schumacher | SCandal: SystemC analysis for nondeterminism anomalies[END_REF]. This doesn't affect the result of the simulation, but can be an issue in an avionic context, considering execution time.

SystemCASS

SystemCASS (SystemC Accurate System Simulator) [START_REF] Buchmann | Fast cycle accurate simulator to simulate event-driven behavior[END_REF] is a SystemC simulator that establishes a static scheduling of processes, which is made at the start of simulation, To do so, SystemCASS requires describing all component models as CFSM (Communicating Finite State Machine) using a CABA (Cycle Accurate Bit Accurate) abstraction level. Furthermore, a single clock must drive all modules. SystemCASS modules can include three types of processes: Transition: triggered by the clock rising edge, it sets the new values of registers, depending on their actual values as well as input port values. Moore/Mealy Generation: triggered by the clock falling edge, these processes set the new values of output ports, depending on register values only (Moore) or register and input port values (Mealy). When calling the sc_start method, SystemCASS creates depending graphs that generate the static scheduling of processes, which will be used throughout the simulation phase. This implementation ensures a deterministic behavior of the simulation. As a result, SystemCASS is more suitable to avionic constraints than a dynamic event driven simulator. As we use gcc compiler, SystemCASS original implementation featured dynamic memory allocation during the creation of the depending graph after the elaboration phase, and right before the simulation phase. So we worked to remove these dynamic allocations. To do so, we first used a static version of gcc compiler and second we identified in run-time all the encountered dynamic memory allocations and replaced it with static memory allocations.

To identify dynamic allocation we used gdb debug tool and a script that put breakpoints on malloc call, this script is:

set

Implementation

To validate our system, we designed a demonstrator based on two QorIQ T2080 design boards each featuring a PowerPC E6500 processor (Erreur ! Source du renvoi introuvable.). The first board assumes the role of a CPM module, running test applications which are supposed to transmit data to other CPM modules. The second board assumes the role of an ASM module. It performs data reception and runs the embedded SystemCASS simulator.

Figure 3: T2080 demonstrator

Each QorIQ T2080 board hosts the PolyORB Kernel (POK) operating system. POK is a partitioned operating system compliant with ARINC653 avionic standard [START_REF] Delange | POK, an ARINC653-compliant operating system released under the BSD license[END_REF]. POK ensures enforcement of safety and security requirements at run-time. It also provides some example of avionics applications. One of these applications is the Flight Management (see Figure 4) 3 Use cases

Data monitoring

Considering the predictability and determinism of applications software ruled by the protocol ARINC653 and their windows of communication in ARINC664, one can predict part of the aircraft data traffic. Some verification within the communication protocol already exist concerning the integrity of the data transport but none can analyze the content itself to determine whether one or another application is really supposed to send a value, or if a communication disappeared or if a value is simply incoherent. Obviously simulating the whole communication flow to determine if it is coherent would be too much time expensive in simulation. The idea is to target specific applications, or specific suspect behaviors (missing material, erroneous values) we could watch over during the flight. Knowing what we're looking for, we can then create a simplified functional timed model of applications as communication providers. On the basis of ARINC664 and ARINC653 configurations values (major frame, bandwidth allocation gap ..), we can predict communication by simulation and compare it with the real traffic to verify temporal as well as data consistency. The application is implemented as follows: on the CPM QorIQ board, POK runs the Flight Management application, which features three partitions (speed, angle and GPS) and generates the application data (Figure 5). POK's ARINC653 properties guarantee space partitioning (meaning that memory of partition is protected) and also guarantees time partitioning (meaning that only one partition at a time is executed). The execution of each partition is handled by a static scheduler (as we can see in Figure 6) and is defined by the system integrator. Each partition (P1, P2 and P3) has a set of execution windows (T1, T2, T3) and this set of windows is repeated in time (T4, T5, T6 and so on…) and at the same order, which guarantees that each partition has access to the system resources once in a MAF (Major Frame). Once that data is generated by POK, they are put in the Queuing Port or Sampling Port and are then sent to the End System with the order defined by the scheduler. Queuing Port can be seen as a buffer and the Sampling Port as a FIFO. The End System then encapsulates the data in an ARINC664 frame with the specification of the Virtual Link (BAG, Frame Size, Jitters) that has been defined by the system integrator (see Figure 7). A Virtual Link defines an unidirectional logical connection from one source End-system to one or several destination End-System(s). Each partition has a dedicated Virtual Link (VLi is dedicated to the data of the Partition i). The data consistency consists in analyzing the payload of the ARINC664 frame that contains data of each application (GPS, speed, angle). In order to do so, a verification of the physical variation law between two data values T and T+1 for each application is performed. For example, the verification of the value of the partition P1-SEQ1 and P1-SEQ2 is performed as shown in Figure 8.

Architecture exploration

Embedded simulation can also be used to help decide in real time how the system should be configured (i.e. what is the most efficient application mapping configuration) when critical situations occur and processing resources should only focus on the most essential applications.

To do so, a predefined set of application mapping configurations should be stored in a library. When the system detects some incoherent execution or some major misbehavior (based on the data monitoring simulation, or other verification mechanisms), a reconfiguration procedure can be started (see Figure 10). The embedded simulator then runs the stored configurations to get performance profiles. A decision motor then selects the most appropriate configuration (whether it's the one who reaches the best performance, or simply the first configuration who meets a predefined performance requirement) Figure 10: Architecture exploration use case

The system can then be dynamically reconfigured to remap the application according to the simulated scheme.

Conclusion

In this article we have presented a new method featuring embedded simulation. We have implemented a SystemC-based simulator compliant with avionic requirements. The simulator allows the real time monitoring of ARINC664 communications. The goal of this monitoring is to check whether communication occurs at the expected time, according to the system scheduling, and to validate data consistency. The simulator also allows architecture exploration to determine the most efficient mapping of applications in order to handle specific situations We show how this simulator could be used to adapt the in-flight processing in real-time to react to critical events.

Figure 1 :

 1 Figure 1: Embedded simulation methodology

Figure 2 :

 2 Figure 2: SystemC flow

Figure 4 :

 4 Figure 4: POK Flight Management Application We used this application to run on the CPM QorIQ board. POK OS handles the flight management application (GPS, Speed and Angle) and at the same time handles the ARINC 664 End System module. On the ASM QorIQ board, POK handles the SystemCASS simulator to perform data monitoring or architecture exploration.

Figure 5 :

 5 Figure 5: Data Generation and Space Partitionning

Figure 6 :

 6 Figure 6: Partitionning Scheduling

Figure 7 :

 7 Figure 7: ARINC 664 Frame at the Output of the End System On the ASM QorIQ board, POK runs the embedded simulator. SystemCASS runs a SystemC module that analyzes the ARINC664 frames coming from the CPM board. It performs data and temporal consistency.The data consistency consists in analyzing the payload of the ARINC664 frame that contains data of each application (GPS, speed, angle). In order to do so, a verification of the physical variation law between two data values T and T+1 for each application is performed. For example, the verification of the value of the partition P1-SEQ1 and P1-SEQ2 is performed as shown in Figure8.

Figure 8 :

 8 Figure 8: Data ConsistencyOn the other hand, temporal consistency consists in verifying that the execution order of each partition is consistent with the scheduling defined by the transmitter part Figure9shows an example of the temporal consistency verification.

Figure 9 :

 9 Figure 9: Temporal Consistency Verification


Edouard Debonneuil
email: edebonneuil@yahoo.fr

Anne Eyraud-Loisel

Frédéric Planchet

CONDITIONS OF INTEREST OF A LONGEVITY MEGAFUND FOR PENSION FUNDS

Pension funds that handle retirement risk need to invest assets in a diversified manner and on long durations while managing interest rate and longevity risks. In the recent years, a new class of investment called a longevity megafund was described, that invests in clinical trials for solutions against age-related diseases. Using simple models, we here study the financial interest for pension funds of investing in a longevity megafund.

INTRODUCTION

FORTHCOMING ANTI-AGING SOLUTIONS

In the 19th and 20th centuries, infant mortality rates and young adult mortality rates have dropped so that the future of life expectancy is now to a large extent a matter of solutions to old age conditions [START_REF] Vallin | Espérance de vie: peut-on gagner trois mois par an indéfiniment?[END_REF].

In this beginning of the 21st century, a series of biomedical discoveries suggest that mortality rates may drop at old age as well. The series started with animal models of human aging and is now turning to humans. The lifespan of laboratory nematodes was extended circa ten times with one single gene change [START_REF] Ayyadevara | Remarkable longevity and stress resistance of nematode PI3K-null mutants[END_REF]. The lifespan of laboratory mice was extended by 70% through a combination of gene change and diet (Bartke, 2008). There is a vast range of successful methods in animals that are reproduced at the level of cells and tissues in humans and human trials are being discussed [START_REF] Barardo | The DrugAge database of aging-related drugs[END_REF]. While a different use of existing drugs possibly represents the fastest route to concrete human life extension [START_REF] Barardo | The DrugAge database of aging-related drugs[END_REF], the graft of bioprinted organs [START_REF] Ravnic | Transplantation of Bioprinted Tissues and Organs: Technical and Clinical Challenges and Future Perspectives[END_REF], Mir 2017) and the in vivo degradation of old tissues that the body naturally replaces by younger tissue [START_REF] Fahy | Apparent induction of partial thymic regeneration in a normal human subject: a case report[END_REF][START_REF] Ocampo | In Vivo Amelioration of Age-Associated Hallmarks by Partial Reprogramming[END_REF][START_REF] Mosteiro | Tissue damage and senescence provide critical signals for cellular reprogramming in vivo[END_REF], Mendelsohn 2017) are already being tested in specific clinical settings. Indeed, science is not only on its way to slow down human aging but also to restore youthful characteristics to the body once old.

One may then expect much better health at old ages, which in turn means lower mortality rates and longer lives. By how much? The life expectancy impact of curing diseases can be delicate because beyond the disease there is a largely unknown associated global burden on health [START_REF] Martin | Research on aging: the end of the beginning[END_REF][START_REF] Arias | United States life tables eliminating certain causes of death, 1999-2001[END_REF][START_REF] Guibert | Mesure de l'espérance de vie sans dépendance en France[END_REF]. Where models or parameters are to be chosen, psychological ceilings lead to underestimate parameters such as predicting an ultimate average life expectancy limit of 64.75 (Dublin, 1928) or 85 years [START_REF] Olshansky | In Search of Methuselah: Estimating the Upper Limits to Human Longevity[END_REF]). Keeping maximum human lifespan around the age of 115 is one of those imaginary ceilings because in the last century it evolved much more slowly than the median age of death. However, the latter evolution is not true when accounting for cause of death; and comparing effects of current interventions in various animal species and humans rather suggests a further increase of 30%, i.e. 150 years of maximal lifespan may be at reasonable reach based on ongoing developments (Ben-Haim 2017).

HOPE FOR PHARMACEUTICAL COMPANIES, FEAR FOR RETIREMENT SYSTEMS

When will such old-age solutions come? That is a key question for the pharmaceutical industry. The pharmaceutical industry currently suffers from a lack of profitability of investments in research and development due to very low success rates in clinical trials (Scannel 2012), in part due to the attempt to gradually adapt the methods for single, acute diseases to multiple, chronic diseases [START_REF] Thiem | Prerequisites for a new health care model for elderly people with multiple morbidities: results and conclusions from 3 years of research in the PRISCUS consortium[END_REF], Roman 2017) instead of targeting the underlying aging and regenerative processes as described above. As many biotech companies now carry that latter approach, various investors reports, books and conferences consider that current retired persons may be the first populations to benefit in large from such advances [START_REF] Mellon | Juvenescence: Investing in the age of longevity[END_REF][START_REF] Pratt | 5 Private Longevity Stocks to Watch: From Calico to Genescient[END_REF][START_REF] Casquillas | Human longevity : The giants[END_REF]. When essentially focusing on hair and skin aging, which were the main anti-aging industry drivers until recently, estimated sizes of the anti-aging market are USD 122 billion in 2013, 140 in 2015, 192 in 2019and 217 in 2021(Zion Market Research 2017, Transparency Market Research 2016).

When will such old-age solutions come? That is a key question for retirement systems also: they face the "longevity risk" that retirements must be paid longer than financially planned. Current retirement systems essentially stem from the middle of the 20th century, when most young workers were not expected to reach retirement age, and they depend on various mortality tables that are regularly updated but that empirically underestimate historical life expectancy trends [START_REF] References Antolin | Mortality Assumptions and Longevity Risk[END_REF][START_REF] Debonneuil | Do actuaries believe in longevity deceleration?[END_REF]). In the USA only, if deaths caused by cardiovascular diseases and cancer were eliminated, the fiscal imbalance of Social Security and Medicare programs may be as high as USD 87 trillion in present value [START_REF] Zhavoronkov | Evaluating the impact of recent advances in biomedical sciences and the possible mortality decreases on the future of health care and Social Security in the United States[END_REF]. May such imbalances be offset by investing retirement contributions in the burgeoning anti-aging market?

1.3. LONGEVITY MEGAFUND: THE SOLUTION?

The concept of "cancer megafund" was developed to face the current pharmaceutical industry crisis that particularly affects cancer research and development: instead of having investors bet high amounts on one risky drug development, the megafund is a financial structure that pools a large number of diversified drug developments and structures investments into debt and equity with low returns but sufficiently low risk (due to the strong diversification) to make investments attractive (Fernandez 2012). The debt part is useful to attract larger investments and to finance long term projects, which is needed for many biomedical developments. The concept has then been and is being studied for various health applications, from rare diseases to biomedical innovation in large, in London, the USA, Australia or Sweden (Swedish Agency for Growth Policy Analysis 2017) and even for non-health-related applications (Hull 2016). In parallel, several remarks have proposed the need for strong contextual attention to properly implement the megafund concept [START_REF] Boissel | The cancer megafund: mathematical modeling needed to gauge risk[END_REF][START_REF] Marko | The cancer megafund: determinants of success[END_REF][START_REF] Tenenbaum | The cancer megafund: a catalyst for disruptive innovation[END_REF][START_REF] Fagnan | Financing drug discovery for orphan diseases[END_REF][START_REF] Fagnan | Financing translation: Analysis of the NCATS rare-diseases portfolio[END_REF][START_REF] Lo | Can Financial Engineering Cure Cancer? [Video] Available on YouTube Lo AW and Naraharisetti SV[END_REF][START_REF] Lo | Can Financial Engineering Cure Cancer? [Video] Available on YouTube Lo AW and Naraharisetti SV[END_REF][START_REF] Yang | Cancer megafunds with in silico and in vitro validation: Accelerating cancer drug discovery via financial engineering without financial crisis[END_REF].

One of the applications of the megafund concept is the "longevity megafund" (MacMinn 2017, [START_REF] Kahn | A Simple Hedge for Longevity Risk using Research-Backed Obligations[END_REF][START_REF] Stein | A Simple Hedge for Longevity Risk and Reimbursement Risk Using Research-Backed Obligations[END_REF]. Such a megafund tracks the biomedical innovations that are likely to increase lifespan, as much as possible, and invests in the corresponding developments to become a sort of longevity risk tracker. Such has diverse benefits. By investing in a range of innovations that is wider than cancer therapies, a better diversification of risk is provided which is an essential feature of the megafund concept to make investments attractive. The longevity megafund can notably attract health providers to hedge the risk of reimbursing new, expensive drugs [START_REF] Stein | A Simple Hedge for Longevity Risk and Reimbursement Risk Using Research-Backed Obligations[END_REF]. By tracking potential solutions to age-related conditions, that correlate with longevity risk, it can attract large investments for the equity part of the megafund: annuity insurance providers and pension funds can invest to cover their longevity risk and this is what we study in this article. The concept of longevity megafund is particularly appealing for the risk management of longevity risk: instead of matching mortality risks of workers who are less than 65 year old with mortality risks of retired persons, it matches mortality risks of retired persons with mortality risks of retired persons. Also, having the income from therapeutic developments come before the treated persons live longer is an interesting feature to possibly pilot retirement benefits in advance.

Some first simulations on life settlements indicate that rightfully investing in a longevity megafund is a better hedge than existing longevity hedges with respect to longevity shocks and basis risk (MacMinn 2017). In this article, is to study the management of longevity risk by investing in a longevity megafund with a more pedestrian approach: the words "covering" or "hedging" (longevity risk) will be used indifferently because the goal is to manage longevity risk without a clear view on the characteristics of the megafund. In particular, due to a lack of historical data on a longevity megafund, the difficulty of knowing what type of health breakthroughs will come first and how well the megafund will capture associated financial returns the article will not claim to find optimal investments to perfectly correlate longevity risk with megafund returns. Instead different scenarios will be proposed to provide examples of longevity risk coverage.

IN WHAT CONDITIONS IS A LONGEVITY MEGAFUND INTERESTING TO HEDGE LONGEVITY RISK?

We note that, despite so many theoretical advantages, the literature contains little studies about the conditions in which the longevity megafund is interesting to hedge longevity risk. This article aims at investing such conditions at a deeper level.

FEASIBILITY OF HEDGING LONGEVITY RISK WITH A MEGAFUND

WHAT TYPE OF RETIREMENT SYSTEMS WOULD BENEFIT FROM INVESTING IN A LONGEVITY MEGAFUND?

Retirement systems are mainly composed of defined benefits plans, defined contributions plans, pay-as-you-go plans, and insurance retirement contracts. So far, to our knowledge the literature suggests that longevity megafunds may hedge longevity risk, without investigating what type of plans would best benefit from investing in a longevity megafund.

"Defined benefits" means that the pension fund handles investment choices and is entitled to pay benefits until the ends of pensioner lives. The benefits are typically defined as a percentage of the pensionable salary, for example 20% of the average salary over the last three years of work. Investing in a longevity megafund could well be a way for defined benefit pension funds to partially hedge their longevity risk: in case of strong longevity improvements, investment returns are greater so the accumulated capital can pay benefits longer.

In defined contribution plans, employees provide contributions during their working years and make investment choices to build their capital for retirement. Then longevity risk is at the level of the pensioners: if a pensioner uses the whole capital during retirement the pension plan does not need to pay anything more to the pensioner. In recent decades, a shift has occurred from defined benefit plans to defined contribution plans in order to avoid longevity risk. Since they do not directly carry longevity risk, at first sight defined contribution pension funds have no interest in investing in a longevity megafund. At second sight, they may well have an interest as they would then propose a capital that reduces the longevity risk of their pensioners: in case of strong longevity improvements, pensioners would have a greater capital.

Whole life deferred annuities, that are often provided by insurance companies, have an important nuance compared to defined benefit and defined contribution plans: after contributions are collected and invested, benefits are defined as a percentage of accumulated capital (the percentage is given by a mortality table). This is actually fundamentally different from defining benefits as a percentage of the pensionable salary because here a larger capital will not cover longer durations of benefits. Instead, it provides larger benefits! Longer and larger benefits: the longevity risk is doubled so deferred annuity providers had rather not invest in the equity part of a longevity megafund.

In pay-as-you-go pension plans there is no, or very little, investment: the contributions from workers directly pays the benefits to retired persons. In such a system, that is for example widely spread in France, the lack of investment makes the longevity megafund of no use.

In conclusion, from a first qualitative perspective the equity part of a longevity megafund makes sense for defined benefit and possibly defined contribution pension plans, but a priori not for pay-as-you-go schemes nor for providers of deferred annuities.

Our analysis is somewhat schematic as numerous retirement systems exist and risks can be transferred to stakeholders who have distinct characteristics. For example, insurers providing deferred annuities may transfer their longevity risk to reinsurers who might benefit from investing in a longevity megafund. Also, various practical aspects such as counterparty risk, basis risk and megafund returns -of course -may modulate the conclusions.

NEEDED MEGAFUND RETURNS TO HEDGE LONGEVITY RISKS

Following that schematic, qualitative analysis we here study the investments returns of a megafund that are needed to hedge the longevity risk of a defined benefit pension plan.

High returns are crucial to hedge longevity risk

Let us start with a very simple model to highlight the orders of magnitude that a longevity megafund should return to offset longevity risk. We consider a pensioner who pays 1 every year from age 25 (included) to age 65 (excluded). Its contributions are invested in funds that are expected to provide the annual return i=1.5%. At age 65, the expected capital is

(1 + 𝑖) 65-25 + (1 + 𝑖) 65-26 + ⋯ + 1 + 𝑖 = (1 + 𝑖) 65-25+1 -(1 + i) 𝑖 ≈ 55
The annual benefits are contractually defined based on that expected capital, to pay the benefits for 20 years for example: up to an expected pensioners lifespan of 85 years. However, let's consider that strong biomedical interventions take place and that pensioners live to 95-100. Benefits must be paid for 30-35 years instead of 20, which is a circa 50% greater duration!

To pay benefits on a 50% greater duration, the capital must be 50% greater. Using the formula with 𝑖 = 3.3% instead of 1.5% provides the needed 50% greater capital. It means that the defined benefit pension plan would have offset its longevity risk if all contributions had been invested in a longevity megafund that provides annual returns of 3.3%.

However, pension funds would likely not transfer all their assets into a longevity megafund -a new type of structure -but rather at most for example 20%. The megafund should therefore provide much higher returns so that, when diluted in 80% of returns at1.5%, the overall return is 3.3%. Since 3.3% = (1 -20%) × 1.5% + 20% × 10.5%, if pensioners are to live to ages 95-100 on average the megafund should provide 10.5% annual returns. Now, if only 10% of assets were invested in the megafund, it should similarly provide 19,5% annual returns. If 5% of assets were invested in the megafund, it should provide 37.5% annual returns! Offsetting the longevity risk of currently 45-year-old pensioners rather than 25-year-old pensioners is even more demanding: with a 20% or 10% investment in the megafund, required annual returns are 13.8% and 26.1%.

Put differently, if a longevity megafund cannot generate 10%-20% annual returns in case of material longevity increases, unless pension funds invest surprisingly large percentages of their assets in the megafund it does not seem to be a viable approach to hedge longevity risk. We then have to check if such a return level is possible?

High levels of returns can be produced

The original cancer megafund paper [START_REF] Jose-Maria | Commercializing biomedical research through securitization techniques[END_REF] provided elements to answer that question: it investigated historical drug development, costs, success rates and sell prices at various stages of development (preclinical, phase I, II, III, approval) in an area that is known to lack financial interest: cancer drugs. It synthetized the observations with two simple models that have the same ten-year annual return of 11.9%: one for blockbusters (drugs with annual revenues more than $1 billion, after year 10) and one for nonblockbusters. We therefore take this 11.9% revenue as a conservative basis for our models.

The blockbuster models is as follows: N drug development programs each require an initial investment 𝐶 0 = $200 million and generate 10 years later a revenue 𝑌 10 = $12.3 billion (present value of returns starting after 10 years) with a percentage 𝑝 = 5% of successful programs (no revenue from unsuccessful programs). The expected 10-year return on investment is then 𝜌 = 𝑝×𝑌 10 +(1-𝑝)×0 𝐶 0 ≈ 3.1 and the expected annualized return is:

𝑟 = ( 𝑝 × 𝑌 10 + (1 -𝑝) × 0 𝐶 0 ) 1 10 -1 ≈ 11.9%
The non-blockbuster model is the same with 𝐶 0 = $100 million, 𝑌 10 = $3.1 billion, and 𝑝 = 10%. It also leads to 𝜌 ≈ 3.1 and 𝑟 ≈ 11.9%..

In what follows, 𝜌 is what matters, or rather the derived equity annualized return 𝑖. 𝑟 is not precisely the return that matters to hedge longevity risk. A megafund must be structured into debt and equity (non-debt). That is because without the debt part it would be difficult to find enough investors to have enough drug development programs financed for the megafund risk to become small compared to returns, hence to be financially attractive. The debt part, composed of "research-backed obligations" (RBOs), provides fixed annual returns: it cannot hedge longevity risk. Therefore 𝑖, the annualized return of the equity part of the megafund, that provides gains in excess of RBOs, is the rate that matters to possibly hedge longevity risk:

𝑖 = (max( 𝜌 -(1 -𝛼) × 1.05 10 𝛼 , 0)) 1 10 -1
where 𝛼 is the percentage of investments of the megafund and where we supposed that the interest rate of the RBOs is 5%. Indeed, for 𝛼𝐶 0 invested in the equity tranche the megafund generates 10 years later a revenue of 𝜌𝐶 0 that first pays the RBOs investors who invested (1 -𝛼)𝐶 0 . If sufficient, the latter receive (1 -𝛼)𝐶 0 × 1.05 10 so that equity investors receive the reminder, 𝜌𝐶 0 -(1 -𝛼)𝐶 0 × 1.05 10 . This must be compared to the initial 𝛼𝐶 0 investment 10 years earlier. When the 10-year return of drug developments is 𝜌 = 3.1, having 𝛼 = 50% or 25% of investments dedicated to the equity tranche leads to 𝑖 = 16.4% or 𝑖 = 22.3%, respectively: the equity share can be used as a lever to obtain higher returns if there are enough successful drug development programs (at the cost of reducing returns if there are not enough).

Such returns depend of course very much on drug selection and the quality of management of the megafund [START_REF] Yang | Cancer megafunds with in silico and in vitro validation: Accelerating cancer drug discovery via financial engineering without financial crisis[END_REF], as well as contractual aspects of funds, such as fees and carried interest [START_REF] Phalippou | Investing in Private Equity Funds: A Survey[END_REF]. Also here were keep the simple assumption that the drug development length is 10 years, in practice numerous details of course need to be taken into account.

But to conclude on the question of interest of investing in a longevity megafund, yes, the equity part can provide sufficient returns to cover the longevity risk of having pensioners live to 100 instead of 85.

VISUALIZATION OF THE FINANCIAL EQUILIBRIUM DEPENDING ON LIFESPAN X.

To easily visualize and think about the appropriateness of a longevity megafund, we here model mortality as a unique age of death x: we estimate the financial needs of a defined benefit plan as a function of x, we estimate the megafund returns as a function of x and we observe how returns compare to needs as a function of x.

Using Asset Liability Management (ALM) practice standards, we consider employees of different age tranches, each with different salaries and different returns as shown in table 1. For the sake of simplicity, we consider 4000 employees who all enter the company at age 25 and stay in the company until retirement at 65. From age 20 to 65, the collected and invested annual contributions lead to a total capital of 15 (1 + 𝑖 3 ) 15 ] ≈ 1 587 000

𝐶 = 4000 [6000 (1 + 𝑖 3 ) 16 -1 -𝑖 3 𝑖 3 + 4500 (1 + 𝑖 2 ) 16 -1 -𝑖 2 𝑖 2 (1 + 𝑖 3 ) 15 + 3000 (1 + 𝑖 1 ) 16 -1 -𝑖 1 𝑖 1 (1 + 𝑖 2 )
For the sake of simplicity, let us consider that during retirement benefits increase with age at the discount rate such that benefit duration equals retirement length: x-65.

Considering that benefits are defined so that the capital exactly pays the benefits if pensioners all live to age x=85, the first annual benefit is then (1 587 000/4000)/(85-65) = 19 836 per pensioner. Now comes longevity risk and longevity hedge. If pensioners live to age x, the longevity loss for the company is 19 836*4000*(x-85). For example, if pensioners live up to age 100 the loss is 1.2 billion for the pension fund. To offset such losses, the pension fund may have invested a percentage of its assets in the longevity megafund.

Let us consider that the percentage of investment in the megafund depend on the age tranche: for example, p1=20%, p2=15%, p3=10%.

The needed annualized return i of the megafund equity tranche is then such that the greater needed capital (which is proportional to the duration of benefits, see left part of the following formula) is obtained by partially investing contributions in the megafund (right part of the following formula):

𝐶 × 𝑥 -65 85 -65 = 4000 [6000 (1 + 𝑖3) 16 -1 -𝑖3 𝑖3 + 4500 (1 + 𝑖2) 16 -1 -𝑖2 𝑖2 (1 + 𝑖3) 15 + 3000 (1 + 𝑖1) 16 -1 -𝑖1 𝑖1 (1 + 𝑖2) 15 (1 + 𝑖3) 15 ]
where

𝑖1 = 𝑝 1 𝑖 + (1 -𝑝 1 )𝑖 1 , 𝑖2 = 𝑝 2 𝑖 + (1 -𝑝 2 )𝑖 2 , 𝑖3 = 𝑝 3 𝑖 + (1 -𝑝 3 )𝑖 3 .
The formula links pensioner lifespan 𝑥 and needed megafund annual return 𝑖.

Accordingly, Figure 1 shows the needed return 𝑖 that is numerically needed to pay pensioners living to age 𝑥 given percentages of investment in the megafund. If pensioners on average live to age 70 a full loss of longevity megafund (return of -100%) can be supported by the remaining investments of the pension fund (of course, better returns are always welcome!). If pensioners live to the expected age of 85, the longevity megafund equity tranche should provide the same average returns as other funds, 3-4% annually. If pensioners live to age 100, 15% or 20% or even 29% annual returns should be provided by the longevity megafund depending on the percentage of investments of the pension fund in the megafund; if pensioners live longer even greater returns are needed.Megafund returns as a function of x.

The link between megafund equity returns and longevity depends on how longevity changes occur and whether the megafund is able to capture them from a financial point of view. A strong longevity increase would typically arise from many health solutions to age-related conditions that affect a large number of persons. We here imagine two types of scenarios.

2.3.1. Linkage 1: discovery success rate: 10-year megafund returns increase linearly with lifespan A longevity scenario is that a number 𝑛 of health solution would each provide a given lifespan increase (for example 5 years of added lifespan to 10% of retired persons): the greater the number 𝑛 the longer the average lifespan, in a proportional manner. The link between the megafund and longevity is that as the number 𝑛 of meaningful health solutions increases the megafund should be able to capture them in its given resources. The link may be proportional: the proportion 𝑝 of successful drug development programs should approximately increase in a proportional manner (𝑝 is the ratio of successful programs over funded programs, it is closer to 0 than to 1) if the megafund managers are sufficiently able to distinguish important programs from less important ones. In that scenario, the 10-year return on investments, for drug development programs, 𝜌 = where the coefficient 𝜃 (±) and the noise 𝜀 remain to be chosen. For increasing lifespan, we take 𝜃 + = 0.2: if pensioners live to 100 it corresponds to a double of return on investments (𝜌 100 = 6.1) and a double of successes in blockbuster and non-blockbuster programs: on the one hand, it is difficult to imagine much higher financial returns so 0.2 may be high, on the on the other hand it seems that greater successes than 10% of blockbuster programs and 20% non-blockbuster programs may be achieved so 0.2 may be small. For decreasing lifespan, we consider that the biomedical and longevity link is reduced, as a decrease in lifespan would probably largely be driven by pollution, obesity or other aspects that negatively affect health, more than by a lack of biomedical innovation, so we arbitrarily take 𝜃 -= 0.1. We take 𝜀 as a uniform random variable between -1 and +1 such that if 𝑥 is around 85 or lower the link between pharmaceutical results and longevity isn't clear. 𝜀 can be seen as the "luck" of the fund; it includes whether the IP of therapeutic developments was sold earlier or later than planned and at a lower or higher price than planned.

Of course, the reader may view those numbers as adequate or not. With the link we established here, equity returns are:

Linkage 2: due to breakthroughs, annualized equity returns increase linearly with lifespan

The rate of discoveries is one driver for longevity and it is easy to link it with returns as done just above. However, longevity breakthroughs are to be possibly expected as well, such as tissue regeneration techniques that are currently in labs and that may reach large populations in the forthcoming decades. The link between longevity and returns is less predictable in that scenario because of the novelty of the situation; however, it can be expected to be strong because longevity is then strongly driven by one or a few treatments.

In the previous scenario (linkage 1) the annualized equity return has a concave shape due to the exponent 1/10. In this scenario (linkage 2), we remove that concave shape by considering that the annualized equity returns increase linearly with lifespan:

𝑖 = ( 𝑥 -𝑥 0 85 -𝑥 0 ) 𝑖 85 + 𝜀 (𝐿𝑖𝑛𝑘𝑎𝑔𝑒 2)
where 𝑖 85 = (max( 3.1+𝜀-(1-𝛼)1.05 10 𝛼 , 0)) 1 10 -1 is the return in expected conditions with respect to longevity: the return showed at age 85 in Figure 2. 𝑥 0 is the age at which 𝑖 would be expected to be zero and 𝜀 is a uniform random number. As Linkage 1 is related to the rate of success of moderate health improvements, we can consider it as a scenario for moderate longevity risk so for Linkage 2 we choose coefficients that align Linkage 1 and Linkage 2 around the age of 85: we take 𝑥 0 = 75 and 𝜀 between -20% and +20%. We can see the alignment in Figure 3.

Of note, since longevity may be driven from many discoveries, a small number of key discoveries, and many key discoveries (the combination of both), Linkage 1 is probably prudent.

COMPARISON BETWEEN NEEDED AND PROVIDED RETURNS: A POSSIBLE HEDGE

Figure 3 superimposes needed returns for the pension fund in the case of approximately 15% investment in the megafund (p1=20%, p2=15%, p3=10%) and provided equity returns by the pension fund, for both Linkage 1 and Linkage 2, in case it has 50% of investments for RBOs. The megafund seems to provide enough return, except in the prudent Linkage 1 case when longevity decreases. Such is of course the result of models. Other models or other parameters could lead to provide insufficient returns compared to the needed returns for the pension fund. However, we see that the sufficient return stems from the main assumptions that i) the returns are above 5% under current longevity trends, i.e. that the assumptions published for megafunds so far are not absurd, and ii) that the link between longevity and megafund returns are at least like Linkage 1. So, with that simple analysis, there is a sound possibility of hedging longevity risk by investing in a biomedical megafund. Of note, prudence is required. The models and conclusions above assume that the longevity fund is reasonably well handled and that things go reasonably well: if longevity related projects are missed so that longevity increases but the fund doesn't take a share of the corresponding financial market, a strong mismatch between needed and provided funds may occur. If the megafund bets on an insufficiently diverse set of therapeutic developments, it leads to a poor risk-adjusted profile: the risk is that none of them succeed, or that only a few of them lead to financial gains as other strategies are in competition on the same market. So, the megafund should definitely cover a very diverse range of therapies, and should not position itself on other therapies than competitors: in should in the contrary participate in various external developments to take a share of the overall market. In the worst case that can be imagined, investments would not only be associated with no gain, but with further costs linked to law suits.

On the other hand, various details may in practice improve the hedge compared to the model. Since megafund equity returns are expected to reflect biomedical improvements, it should reflect retirement needs in advance. Hence, if the defined benefit plan leaves some area of adjustment of future benefits, for example regarding the increase of benefits, megafund equity returns may be an input to choose benefit adjustments.

IMPACT OF INVESTING IN A MEGAFUND ON NEEDED OWN FUNDS

The above analysis shows that investing in a longevity megafund can in theory provide adequate returns over the years to cover longevity risk, when taking a prudent view compared to a central view. This seems good news for a pension fund because it suggests that large increases of longevity can be managed by having invested in a longevity megafund and holding a minimum amount of prudential capital to face risks, instead of requiring very large amounts of capital to face very large risks.

The above analysis was voluntary schematic for the sake of clarity. Longevity was described by the age x at which everyone dies whereas in practice mortality spreads over ages. Also, the analysis stopped at comparing needed versus provided annualized longterm returns, whereas a sufficient prudential capital is needed at all times to avoid bankruptcy.

Therefore, we adjust the previous analysis of a defined benefit pension plan to incorporate such aspects. After defining refined assumptions to model longevity, pension fund and megafund behaviors, we measure what level of prudential capital is needed today to pay retirement until the last death of current employees.

REFINED ASSUMPTIONS

Assumptions related to mortality

We consider the following model of annual mortality rates, for someone aged x in t years:

q x,t = 0.5 1 + e a-b(x-φt)
As studied by [START_REF] Debonneuil | Do actuaries believe in longevity deceleration?[END_REF], such a model leads to non-decelerating life expectancy trends and the parameters typically are a=10.5, b=0.1 and φ=23%. With these parameters a and b the model currently has expected behaviors: the current (t=0) life expectancy at birth is 84.4 years, it is slightly above the general population life expectancy (the 2015 life expectancy in the UK, France and Japan is respectively 81.2, 82.4 and 83.7) which is desired as it concerns workers and insured persons. With this model also, the current life expectancy at 65 is 22.1 years leading to the average age of 87.1 years, which is expectedly higher than 84.4 years as it doesn't consider people who died before the age of 65.

With φ=23% the model reproduces current major longevity trends. Life expectancy at birth increases by 23% of year annually, which is the "best practice trend" of long lived countries [START_REF] Vaupel | Biodemography of human ageing[END_REF]. Current annual mortality improvements are q 70,0 -q 70,1 q 70,0 = 2.1% at age 70. For those currently aged 65, it leads to a remaining expected lifespan of 25.6 years, so an expected lifespan of 90.6; such is higher than 87.1 because improvements are expected during these years. For current newborns, it leads to an expected lifespan of 105.8; one shall note that even for newborns of the general population, if current life expectancy trends continue then lifespans of long-lived countries are expected above 100 [START_REF] Vaupel | Biodemography of human ageing[END_REF].

Future longevity trends are however unknown. For the sake of simplicity we suppose that φ is constant over time and lognormally distributed over scenarios of the future, with mean log(23%) and standard-deviation of 1: in such conditions, there is as much probability (50%) that probability φ is lower than 23% as greater than 23%. There is also a 7% possibility that φ>1. Over long periods of time, the latter corresponds to people enjoying better health and reduced mortality risks as time goes, for example because of the emergence and progressive generalization of tissue regeneration techniques. Some may find the latter 7% excessively high and others may find it excessively low [START_REF] Yang | Cancer megafunds with in silico and in vitro validation: Accelerating cancer drug discovery via financial engineering without financial crisis[END_REF]. That is why we will in particular consider four longevity scenarios: φ=0% (no improvement in the future), φ=23% (best practice trend), φ=46% (double trend) and φ=69% (triple trend). Of note, around 1950 Japan had a trend that was clearly above φ=100%, suggesting that high trends are possible when hygiene & biomedical knowledge as well as social contexts are adequate. Similarly, over the last decades, life expectancy has increased by more than φ=50% of year annually in countries like Malaysia, Philippines, Vietnam, Laos and Bangladesh [START_REF] Carbonnier | Asian and African Development Trajectories -Revisiting Facts and Figures[END_REF]. Of note also, during approximately 70 years after the microbial communications of Louis Pasteur, φ was around 30% in long-lived countries [START_REF] Vallin | Espérance de vie: peut-on gagner trois mois par an indéfiniment?[END_REF]; improving hygiene requires complex cultural, technological and urbanization changes, it may be that using anti-aging therapies once they are available is a faster process.

Assumptions related to population, contributions, investments and benefits

Populations. At time t=0 we consider 300 persons of age 20, 300(1 -q 20,0 ) employees of age 21, 300(1 -q 20,0 )(1 -q 21,0 ) employees of age 22, etc. until age 100. It provides a distribution of the population across ages that is roughly natural. Every year (t=1, t=2, etc.) the persons die according to mortality rates and 300 new persons aged 20 arrive. For the sake of simplicity, we do not model arrivals and departures at other ages. 20-34, 35-49 and 50-54 provide respective annual contributions of 3000, 4500 and 6000. Also, by default investments respectively provide i1=5%, i2=4% and i3=2% of expected annual returns, corresponding to three different funds that take different degrees of risk. For each of the three funds the volatility σ is given by a Sharpe ratio of 1.5 (aribrary number), a risk-free rate of 1% (arbitrary number) and a 2 by 2 correlation of 50% (aritrary number):

Contributions and investments. As in table 1, persons aged

σ k = i k -1% 1.5 〈σ k , σ l 〉 l>𝑘 = 50% k, l = 1,2,3
We consider a 2-by-2 correlation of returns of 50% between the three funds.

Benefits. Regarding contractual benefits, we consider that they have been defined by generation (generations of people contributing since age 20) so that the accumulated capital at age 65 match the remaining lifespan based on the mortality rates with φ=23%.

Since most actuarial tables have decelerating trends (Debonneuil 2010) actuaries often declare such benefits as prudent when building retirement products. As done in the above stylized models, for the sake of simplicity we suppose that the benefits increase with age like corresponding investment returns so the duration of benefits is equal to the remaining lifespan at age 65: we do not need to model annual benefits, rather the accumulated capital at age 65 and the remaining lifespan. For the computation of contractual benefits, we also take the simplifying approximations that mortality rates before age 65 at those at time t=0 and that investments are exactly 5%, 4% and 2% (without considering volatility). We will see in the results that these approximations have a good accuracy.

3.1.3. Assumptions related to the megafund A percentage of contributions (p1=20%, p2=15%, p3=10%) can be invested in the equity part of the megafund, whose return is correlated with longevity. In line with the previous approach, as a first possible linkage between megafund returns and longevity, the 10-year return of the whole megafund is linked with longevity as follows:

𝜌 𝜑 = 3.1 + 𝜃 (±) (𝜑 -23%) + 𝜀 (𝐿𝑖𝑛𝑘𝑎𝑔𝑒 1)
𝜀 is a random uniform number between -1 and 1 drawn once for each scenario of the future in order to represent that the correlation is not deterministic. In the formula, we replaced (x-85) by (𝜑 -23%) , therefore the coefficient 𝜃 (±) must be adjusted to the new scale. We do so using age 40 as a sound actuarial age: 𝜃 (±) was previously chosen to express the possibility of living to 100 instead of 85, i.e. 15 additional years brought by biomedical discoveries. When 𝜑 = 23%, people currently aged 40 are expected to live up to the age of 94.5 on average. When 𝜑 = 44.5%, people currently aged 40 are expected to live up to the age of 109.5: that is also 15 additional years. Consequently, the previous values of 𝜃 (±) are multiplied by 15 44.5%-23%

: 𝜃 (±) = 14 when 𝜑 > 23% and 𝜃 (±) = 7 when 𝜑 < 23%.

The annualized return of the megafund equity tranche in scenario 1 is then:

𝑖 = (max( 3.1 + 𝜃 (±) (𝜑 -23%) + 𝜀 -(1 -𝛼)1.05 10 𝛼 , 0)) 1 10 -1
By analogy with the previous approach also, a second potential linkage between the megafund and longevity is

𝑖 = 𝜑 23% [(max( 3.1 -(1 -𝛼)1.05 10 𝛼 , 0)) 1 10 -1] + 𝜀 (𝐿𝑖𝑛𝑘𝑎𝑔𝑒 2)
where 𝜀 is uniform between -10% and 10%.

In the two models of linkage we consider that 𝛼 = 50% of the megafund is structured in equity.

SIMULATIONS, PROBABILITY OF RUIN AND PRUDENTIAL CAPITAL

10,000 scenarios of the future were constructed using the software R. The code is in the supplementary materials. Each scenario draws the longevity trend 𝜑, the "luck" of the megafund 𝜀 and, for each future year, the annual returns 𝑖 𝑡 𝑘 , 𝑘 = 1,2,3 of the three funds (depending on the employee age). Other calculations within a scenario do not require drawing random numbers. A Cholesky decomposition of the covariance matrix 〈σ k , σ l 〉 is used to simulate correlated returns. At t=0, the defined benefit pension fund starts with its initial population of persons aged 20-65 and an accumulated wealth based on historical mortality rates (modelled with t=0 mortality rates) and exactly 5%, 4% and 2% past annual returns on investments. The defined benefit pension fund then either invests p1 p2 p3 in the megafund or does not invest in the megafund. We then simulate scenarios of the future where the longevity trend (𝜑), the megafund luck (𝜀) and the returns of funds (𝑖 𝑡 𝑘 ) intervene. In order to assess the right amount of prudential capital at t=0, we instead suppose no initial additional wealth, we allow wealth to become negative (instead of claiming bankruptcy) and for each scenario of the future we measure wealth after all benefits were paid. That measure is discounted using the investment composition of the scenario to represent the initial additional wealth that would have been needed and invested to provide the right amount to pay retirements in the future, without reaching bankruptcy. For the sake of simplicity, the discounting is performed along investments done, i.e. the wealth is divided by the value of an initial investment of 1 when invested in the different funds in the same proportion of contributions. In practice, long term investments choices and associated discounting may be optimized through the use of progressive utility (El Karoui 2014). In fact, considering the returns (𝑖 𝑡 𝑘 ) of the funds independent from longevity is certainly simplistic.

The needed prudential capital is then arguably defined as the 5% worst initially needed capital across scenarios of the future. This would mean that there is a 5% risk of not paying retirement benefits, however in case of strong risks the wealth provided by new business may be used to pay the older business to some extent, or other adjustments may be provided meanwhile. So, the risk of not paying retirements is arguably lower than 5% (arguably, as events can be worse than modelled and as reactivity can be questioned with respect to retirement systems).

RESULT: NEEDED PRUDENTIAL CAPITAL

Figure 4 shows the needed amount of additional initial wealth, expressed as a proportion of the initial wealth, depending on the future longevity trend. The first obvious result is that all curves go down on the right side: with these refined assumptions, under extreme longevity scenarios the hedge is partial when investing about p2=15% in the longevity megafund. The second obvious result is that the two linkage models lead to approximately the same need of prudential capital (the same bottom of curve) if the future longevity trend isn't three times the current one. The third results is that under such non-too-extreme longevity scenarios, much less prudential capital is needed when investing in the megafund. More precisely, the y-axis shows the actualized remaining wealth after paying retirement benefits for current employees, divided by the current wealth of the pension fund: "-1" means that 100% of additional wealth would be needed today, i.e. that the current wealth should be double. Each dot in the graph is the result of a scenario of the future. In black, no investment is performed in the megafund (p1=p2=p3=0). In the light color (light green), the pension fund invests in the megafund which behaves like Linkage 1. In the dark color (blue), the pension fund invests in the megafund which behaves like Linkage 2.

Of note, when not investing in the megafund (nor applying some other major longevity hedge mechanism), if the current longevity trend continues (𝜑 = 23%) an additional capital of circa 3.5% of the initial wealth may be considered. Such is due to the risk of the funds --the thickness of the black curve --and taking the average (the center of the black curve) instead of a VaR at 5% shows a perfectly neutral initial wealth --0 --indicating that the approximations used to compute benefits are accurate, as expressed above.

If the current longevity trend doubles however (𝜑 = 46%) then about 70% of additional initial wealth is needed. Such a strong additional need would be eliminated when investing in a longevity megafund, if the models described here hold well. If instead the trend triples (𝜑 = 69%) then about 200% of additional initial wealth is needed, which the megafund reduces to about 50% (60% if the megafund behaves like the linkage1 model, 40% if it behaves like the linkage2 model).

However, if longevity trend is negative, a prudential capital is needed due to the fact to invest in the equity part of a megafund, that would not have been needed otherwise.

That is a reality if success rates of drug development programs are lower than historical pharmaceutical success. The structuring of the megafund may be optimized to reduce that risk, such as having a third tranche of the megafund that absorbs this risk.

These comments are based on specific longevity trends. In practice however, one does not know the future of longevity, so the prudential additional wealth shall cover a wide range of possible longevity trends. Using the lognormal distribution of longevity trends, our simulations suggest that with a 5% probability to have an insufficient initial wealth, the prudential capital would 8.5 times the initial wealth! Investing in the megafund would reduce it to 4 times under the Linkage 1 behavior, and to 1.7 times under the Linkage 2 behavior. With a 10% VaR, these numbers respectively become 4 1 0.5. One might have expected that longevity risk is handled with a prudential capital of 10% or 30% of the initial wealth, such is not the case here because we consider the type of breakthroughs that were described at the beginning of the article and that are burgeoning.

CONCLUSION

The current article studied the conditions in which investing in a longevity megafund can hedge longevity risk. With assumptions that such a megafund is reasonably well managed, it appeared that the hedge may work for defined benefit pension plans and with some degree if the change of longevity is too radical.

One might hope that in case of a very radical longevity change, retirement systems are thoroughly adjusted. However, adjusting retirement systems is obviously not easy, and that longevity hedge approach may be considered. The risk analysis performed at the end of this article suggests that the short-term prudential capital risk approach that is currently proposed by diverse regulators, such as the Solvency Capital Requirement in insurance, may not lead to the right order of magnitude of prudence.

The results we obtain depend very largely on models, and not on strong empirical evidence. Indeed, the megafund remains at this stage a theoretical concept. If the biomedical discoveries described at the beginning of the article truly extends human lifespan in a very significant manner, it may be important not to wait too much to develop this longevity megafund solution. It is therefore a good timing to perform more research on the predicted behavior of the megafund depending on its details. For example, aside mentioned potential refinements of actuarial aspects of the megafund, the pharmaceutical aspects are important. Also, a key dimension in the longevity megafund is intellectual property, and research should be performed on that aspect as well.

Fig. 1 -

 1 Fig. 1 -Needed megafund return depending on pensioner lifespan to perfectly hedge longevity risk. Three lines are displayed, they correspond to different percentages of investments in the megafund.

Figure 2

 2 Figure 2 describes the provided returns.

Fig. 2 -

 2 Fig. 2 -Equity returns provided by the longevity megafund, depending on pensioner lifespan, equity share 𝜶 of megafund investments, and a prudent view on other factors (𝜺 = -𝟏 instead of 0).

Fig. 3 -

 3 Fig. 3 -Available megafund equity returns (black and grey lines , prudent or best estimate) versus needed returns for the pension fund. The straight black lines use the second scenario ("Linear"), the curved grey lines use the first scenario.

Fig. 4 -

 4 Fig. 4 -Needed additional wealth depending on the future longevity trend -Excess initial wealth (or lack of, when <0) as a proportion of the initial wealth 2

Forthcoming longevity trend ϕ (was 0.23 in last decades).

Their work benefited partially from the financial support of the ANR project "LoLitA" (ANR-13-BS01-0011).

Moskalev A, Anisimov V, Aliper A, Artemov A, Asadullah K, Belsky D, Baranova A, de Grey A, Dixit VD, Debonneuil E, Dobrovolskaya E, Fedichev P, Fedintsev A, Fraifeld V, Franceschi C, Freer R, Fülöp T, Feige J, Gems D, Gladyshev V, Gorbunova V, Irincheeva I, Jager S, Jazwinski SM, Kaeberlein M, Kennedy B, Khaltourina D, Kovalchuk I, Kovalchuk O, Kozin S, Kulminski A, Lashmanova E, Lezhnina K, Liu GH, Longo V, Mamoshina P, Maslov A, Pedro de Magalhaes J, Mitchell J, Mitnitski A, Nikolsky Y, Ozerov I, Pasyukova E, Peregudova D, Popov V, Proshkina E, Putin E, Rogaev E, Rogina B, Schastnaya J, Seluanov A, Shaposhnikov M, Simm A, Skulachev V, Skulachev M, Solovev I, Spindler S, Stefanova N, Suh Y, Swick A, Tower J, Gudkov AV, Vijg J, Voronkov A, West M, Wagner W, Yashin A, Zemskaya N, Zhumadilov Z, Zhavoronkov A (2017): A review of the biomedical innovations for healthy longevity. Aging (Albany NY).;9(1):7-25


Elizabeth Craven (1750-1828) was a British woman of letters who spent several years on the Continent before travelling to the Ottoman Empire and Greece in 1785 and 1786. The narrative of her travel was published in 1789 as A Journey through the Crimea to Constantinople. Travelling through Europe and going as far as the Middle-East was quite uncommon, unexpected, and even unladylike in those days, especially for a married woman and a mother of seven children as was the case of Lady Craven. The story of this ladytraveller therefore immediately leads one to wonder why she undertook such a potentially dangerous journey almost on her own : what was she looking for ? Was she escaping something and, if so, what was it? Her story also raises the question of her impressions of the Orient : did her journey have any influence on her perception of the Middle East and its populations? Did this experience change her life and in what way ? This article will first try and determine Lady Craven's motives for travelling that far and will then examine her impression of the countries and people she discovered in 1785 and 1786 before finally questioning the ambivalent status of this transgressive lady. Elizabeth Craven started her tour to the Orient in the Spring of 1785. The first letters she sent to Charles-Alexander Margrave of Brandeburg-Anspach-Bayreuth and which she published three years later as A Journey through the Crimea to Constantinople, were posted in Paris in June 1785. Elizabeth Craven's trip lasted for about 18 months and took her to very varied and remote places : starting her tour in Paris, she crossed France, stopping in several towns such as Orleans, Avignon, Toulon, before reaching Italy where she visited Genoa, Florence and Venice. She then left for Vienna before undertaking a long tour of Russia through St Petersburg and Moscow in February 1786, which led her to Sebastopol in the Crimea. From there she took a boat to Constantinople, then visited Greece and returned to London across Eastern Europe 1

she later on published, the reader easily understands some of this lady's motives. She probably wanted first of all to explore the world, making the most of her young age -35 years old at the time and a good physical condition-to visit remote places : 'I may never have an opportunity of travelling again' she wrote from Bologna on November 15th 1785 2 . Undertaking this long journey was then a possibilityin those days the exclusive priviledge of mento visit the Middle East, Eastern Europe and to meet new people : 'I will see courts and people that few women have seen' 3 . As this quotation shows, Elizabeth Craven was fully aware that her initiative was rather uncommon for a lady and a unique opportunity in her life.

Yet this project --not unlike the Grand Tour 4 --could appear extremely risky and unseemly in the eyes of many, as shows this extract from a letter sent by Horace Walpole to Horace Mann, then a British Ambassador in Florence, expressing his concern about Lady Craven's general behaviour : 'She is very pretty, has parts and is good-natured to the greatest degree ;

has not a grain of malice or mischief (..) and never has been an enemy but to herself', (Walpole, Letters, IX, 27).

But, as Horace Walpole's letter subtly suggests, the reasons why in 1785 Lady Craven suddenly left Paris where she had lived for two years, probably also had to do with her past private life. In fact, one and the same reason led Craven to leave Britain in 1783, to settle in France for two years and finally to leave again, for the Middle East in 1785. Indeed, Lady Craven née Berkeley, was born in 1750. She was the third child of the Earl of Berkeley. She was married, aged only 17, to Baron William Craven, and had seven children with her husband. The couple split in 1781, as Lord Craven, who was himself unfaithful to his wife, considered that Lady Craven's extra-marital affairs had become unbearable. Lady Craven was then informed in the winter 1781 that her husband wanted her to leave their house. He nevertheless agreed to ensure her maintenance : '(…) her infidelity could no longer be overlooked even by the most indulgent husband (…) the Peer settled £1500 a year upon his 2 Elizabeth Craven, A Journey through the Crimea to Constantinople, 1789, 89. While in Bologna, where she expected the visit of her brother, Lord Berkeley, Elizabeth learnt that he would not come and spend the winter in Italy as was originally planned. She thus decided to continue her visit of Europe : 'I received about a week since a letter from Lord B----, informing me that he could not pass this winter in Italy ; so I immediately determined my course ; and it is North. Now I am on the wing (…) and I will make the best use of my time ; few as the months are I can allow myself to run about it, I will employ them, I hope, to my satisfaction and your amusement' (A Journey, 89). 3 Elizabeth Craven, A Journey, 89. 4 This journey to Italy, Greece and other countries where yound educated people sojourned after their studies to complete their study of humanities. In the present case, this Grand Tour was undertaken by a 35-year-old mother of seven who, being a woman, had not been educated at university. On page 152, Elizabeth Craven considers her journey as a tour : 'Notwithstanding all that has been said to deter me from continuing my tour'. spouse' 5 . Being utterly repudiated and the news being publicly known and widely circulated 6 , Lady Craven had little choice 7 but leave her country : 'Lady Craven, being turned away by her husband, and her reputation in England being such as to close society to her, decided to go abroad' 8 . She first moved to France : 'With her youngest son, Richard Keppel Craven, then four years old, she (…) settled herself in a house at Versailles' 9 .

Embarking on a long journey to the Middle East was therefore probably not only due to a desire for freedom and adventure but was also a way of escaping the social stigma of adultery, as even in Paris in 1785, she found that her reputation had been tarnished by rumours : Paris is not far enough from London. Generally speaking, then, moving further to the East was a way of ridding herself of her own past. Abroad, in Italy, Russia or Constantinople, she would be greeted and treated as the member of the British nobility that she was. And indeed, in Vienna she visited Emperor Joseph II 10 and was welcomed by Catherine II in person in St Petersburg 11 .

Travelling and reporting this experience in letters may also have represented for Lady Elizabeth Craven an opportunity to tighten her bonds with the Margrave of Brandebourg whom she was to marry in 1791 12 . The tone of the letters she sent him often suggests that Elizabeth Craven wanted to seduce the Margrave, even though she conspicuously insisted on considering him as a brother : 'If I had possessed the invaluable blessing of having you for my real brother', wrote she 13 . Yet, interestingly enough, and despite the fact that she had an eye on the Margrave, Lady Craven embarked on her journey to the East in the company with Henry Vernon, Admiral Edward Vernon's grand-nephew. It is unsure whether the two travellers became lovers but this situation certainly did not help improve Craven's reputation as proves this cutting remark by Charles Pigott: 'She took up as a companion de Voyage the celebrated and veracious Henry Vernon. They visited divers and distant parts of 5 The Beautiful Lady Craven, The Original Memoirs, Introduction, xxv. 6 Queen Charlotte, George III's consort, was particularly disdainful to Elizabeth Craven : 'Queen Charlotte set her face sternly against the amiable and tolerant hostess' (The Beautiful Lady Craven, introduction, ix). 7 See also this far less benevolent account of Lady Craven's situation : «Our discarded B-r-n-ss being now at liberty, went beyond seas in search of liberty », Charles Pigott, 1792 part III, 160. 8 The Beautiful Lady Craven, The Original Memoirs, Introduction xxvi. 9 The Beautiful Lady Craven, xxvi. 10 A Journey, 108. 11 A Journey, 123. 12 The letters published in 1789 were all sent to the Margrave. Marrying a prince radically changed Lady Craven's situation. The Margrave was a prince and Elizabeth therefore bore the title of Princess Berkeley from then on. The couple remained however somewhat ostracised by some members of the aristocracy like Queen Charlotte, George III's wife who literally imposed a 'social veto' on her (The Beautiful Lady Craven, Introduction, ix). 13 A Journey (Dedication, iii). the world. Their adventures were wonderful. They breakfasted with the Empress of Russia, they dined with the Grand Signor, and supped with the Great Mogul' 14 . Horace Walpole commented on the situation in a rather tactless way as well in a letter he sent to Horace Mann: 'I suppose the adventures of a certain Lady and her cousin Vernon, which I have kept profoundly secret, will be public (…). From St Petersburgh the travellers went to Moscow (..) and thence by way of Cherson and Sebastopol to Constantinople where they arrived on April 20 and stayed for a few weeks' (16 March 1786) 15 . Leaving Europe and travelling abroad, then may also have represented an opportunity for Lady Craven to have a love affair far from the reproving judgement of the French and British nobilities 16 .

Crossing the Continent, Eastern Europe and reaching the Middle-East caused Lady

Craven to encounter people she had never met, to discover landscapes she had never seen and manners she was not used to. The accounts she gives of her experience are a wealth of information on her general perception of the unknown and her personal evolution in the course of this journey. As we shall see in the following lines, Lady Craven could have extremely admirative as well as very prejudiced views on what she discovered. The tone of her letters to the Margrave can be quite enthusiastic about her visits. In Hyères, for instance, after visiting the Golden Isles, she sent a letter to the Margrave promising to take him to this place, the landscape of which could have been imagined by a painter : 'I desire, if ever chance shoud bring you to Hyères, that you will ascend this hill, and examine the scenes around, towards the sea ; the islands of Portecroix and Pourquerolles are beautiful objects, and a peninsula called Gien, which is joined to the land only by a narrow road, forms a landscape worthy of a great master's pencil' 17 .

Turkey is another place of wonders which all astonish Elizabeth : 'The situation of Bursa is very beautiful, in a valley between two hills-and is now a very considerable town' (…) I had a very pleasant passage from Smyrna to Moudagnajust as the Tarleton arrived at the 14 Charles Pigott, part III, 160. 15 The Beautiful Lady Craven, xxxviii-xxxix. 16 It should be noted that Lady Craven dit not hide the fact that she was travelling in company with Henry Vernon, as shows a letter she sent from Constantinople to Robert Murray Keith, a diplomat, on 30th April 1786 : 'Mr. Vernon presents his compliments to you and we both hope to find you as well and as glad to see us, as we shall be to see you' (The Beautiful Lady Craven, The Original Memoirs, introduction, xxxix). But this letter was not addressed to the Margrave and was not published in 1789 as part of the Journey through the Crimea to Constantinople. The Margrave may then have been unaware that the pretty lady was in gallant company. 17 A Journey , 41.

entrance of the Straits, wind dropped entirely and we found ourselves in the midst of a large fleet, composed of vessels of every size and nation, that waited for a South wind to pass the Dardanels-luckily we did not lay a considerable time at anchor, a southerly breeze sprung up and our swift Tarleton left all the other sails' 18 . The letters sent from Greece also express her marvel at the area : 'We passed the western point of the islands of Mittelina and Ipsera, leaving to the left the island of Scioto the right that of Miconia (..). I saw but little remains of the temple dedicated to Bacchus' 19 . Leaving the area of Greece and Turkey was painful for Elizabeth whose letter 61 depicts the landscapes she saw in most emotional terms : 'Adieu, for the last time, from this beautiful and sublime situation, which must make all landscapes appear trifling to me for the future' 20 .

Towns and architecture also stun Elizabeth Craven as show her letters : the statues in Florence 21 , the cleanliness of the streets of this town 22 , the admirable setting of the city of Constantinople 23 , the majesty of Hagia Sophia's huge dome 24 , the spectacular beauty of the entrance to the Piraeus Port 25 : all fill her with wonder. Encountering unknown people and manners can also make her extremely admirative : the Italians' kindness 26 , the Turkish most elegant feminine outfits 27 are described at length, as well as all the private spaces that as a woman she can visit, like the baths and harems --places that no male traveller is allowed to enter : 'Craven is interested in the picturesque, in the private and domestic spaces', as notices Efterpi Mitsi 28 , which 'embodies a development in travel writing' 29 .

18 A Journey, 273. 19 A Journey, 244. 20 A Journey, 288. 21 'I think and dream of nothing but the statues from the time I leave them till I see them again', 78. 22 'this city is very clean and chearful ; the streets are paved with large pieces of rock, unlike any other pavement I have seen, and much smother ; fine statues and relievos in marble stand in the squares and angles of streets, without having any thing to protect them but the respect the very lowest people have for them', 83. 23 'Constantinople and the entrance of the Bosphorus by the sea of Marmora, is the most majestic, magnificent, graceful and lively scene the most luxuriant imagination can desire to behold', 210. 24 'The dome of st Sophia is extremely large and well worth seeing', 217. 25 'just below the town, standing unincumbered with other objects, the superb, the beautiful temple of Theseus', 254. 26 ''the Italians are very good-humoured', 84. 27 'You can conceive nothing so neat and clean to all appearance as the interior of this Harem ; (…) I am femmelette enough to have taken particular notice of the dress -which, if female envy did not spoil everything in the world of women would be graceful -it consists of a petticoat and vest, over which is worn a robe with short sleeves -the one belonging to the lady of the house was of sattin, embroidered richly, with the finest colours, gold and diamond -a girdle under that, with two circles of jewels in front, and from this girdle hangs an embroidered handkerchief -A turban with a profusion of diamonds and pearls, seemed to weigh this lady's head down', 244. 28 Mitsi, Efterpi, 21. 29 Mitsi, Efterpi, 24.

But prejudices and scorn are however far from absent from Craven's letters. The populations she encounters often find no favour in her eyes. The French appear ridiculous with their craving for English manners 30 , their ignorance and their absence of taste 31 . She also considers them rude, contrary to the Italians 32 and their hostility to the English, which she experiences in Toulon, shocks her 33 . Elizabeth Craven's negative judgment does not spare Russia, whose peasants get drunk during Carnival 34 and whose men can be extremely abusive to their wives 35 . In spite of their priceless monuments, the Greeks are, according to Craven, vulgar people whose dances are grotesque 36 and music atrocious 37 . In Naxos, Lady Elizabeth vents her disdain for the way young Greek girls look : 'The town of Naxos is a poor place ; we waited near four hours to see a Naxiote maiden dressed in her holiday clotheswhich are neither decent nor pretty' 38 .

Lady Craven fully discloses her deep orientalist 39 scorn for the locals once she is in Turkey.

In Constantinople, for instance, she insists that the pillars of mosque Hagia Sofia are not perfectly upright, that the decoration is poor and not really worthwhile. This tends to confirm, in her eyes, the absence of respect of those she calls the 'mahometans 40 for their God and his 30 'Everybody last year, that would be quite à l'anglaise at Paris, had to wait on them what they called a Jakay, a little boy with straight, lank, unpowdered hair, wearing a round hat -and this groom-like looking thing waited upon the mat dinner, and was frequently stuck up with three tall footmen behind a fine gilt coach -it was in vain for me to assert to some grave old French people that jockey meant riding-groom in a running-horse stable', 23. 31 'Ill taste prevails universally as to dress in France', 25. 32 'the Italians are very good-humoured which is more than I can say for their neighbours the French ; and they have likewise more natural civility to strangers', 84. 33 'I stopped in my way here at Toulon, and intended to look at the dockyards, but was refused, which surprised me very much (…). I could get no other reason assigned for the refusal but this -(…) nothing of English blood should ever be permitted to go into them', 36. 34 'I would never advise a traveller to set out from Petersburgh as I have, just at the end of the carnival ; he might with some reason suppose it is a religious duty for the Russian peasant to be drunk' », 139. 35 'There is a custom here which I think is abominable ; noblemen who are engaged to marry young ladies, make no ceremony but embrace them in the midst of a large company at a ball' ,137. 36 'In the evening the Athenian girls were invited to perform before me the ancient dance called Ariadne's dance -A more stupid performance as a dance I never saw', 264. 37 'The Greeks, in their parties upon the water, have generally a lyre, a fiddle and a guittar (sic) or two in the boats -with these instruments they make a horrid noise, each performer playing in a different key and if they sing, all in discordant tones (…)why then do not these Greeks find out they make nothing but discordant sounds when they sing or play -I confess it seems to me a very strange thing', 282. 38 A Journey, 245. 39 'The European encounter with the Orient, and specifically with Islam, strenghtened this system of representing the Orient (…) and turned Islam into the very epitome of an outsider against which the whole of European civilization from the Middle Ages on was founded' (Edward Said, 70). 40 This term, as Edward Said insists, is offensive : '(..) the polemic name 'Mohammedanism' given to Islam' (Edward Said, 60). 'Mohammedan' is the relevant (and insulting) European designation' (66).

prophet [START_REF]Sophia is extremely large and well worth seeing, but some of the finest pillars are set topsy-turvy[END_REF] . Laying her expert eyes on the Turkish women's physical appearance, she describes them as wearing too much make-up [START_REF]I have no doubt but that nature intended some of these women to be very handsome, but white and red ill applied, their eye-brows hid under one or two black lines[END_REF] , as obese [START_REF]I think I never saw so many fat women at once together, nor fat ones as fat as these', 263. 44[END_REF] , not elegant 44 , with black teeth, round shoulders [START_REF]teeth black by smoaking, and an universal stoop in the shoulders, made them appear rather disgusting than handsome[END_REF] , and awful skin wrinkled by too hot baths [START_REF]The Turkish women pass most of their time in their bath or upon their dress ; (..) the frequent use of hot-baths destroys the solids, and these women at nineteen look older than I am at this moment', 225. 47 'Turkish idleness (…) Turkish supineness (…) Asiatic splendour, superstition, and laziness[END_REF] . Elizabeth proves unable to consider these women as equal to her and incapable of imagining that these women could wish to experience the same degree of liberty as she does. She therefore totally approves of the Turkish practice of keeping women secluded in harems. Considering this tradition as a way of protecting women, Elizabeth even insists that the Turks treat women with respect and protect them from the hostile world : 'The Turks in their conduct towards our sex are an example to all other nations […] in Turkey likewise, women are perfectly safe from an idle, curious, impertinent public, and what is called the world can never disturb the ease and quiet of a Turkish wifeher talents, her beauty, her happiness, or misery, are equally concealed from malicious observers' (Craven, 232). This perception of Turkish women's situation and this incapacity of regarding them as human beings who might be willing to explore the worldas she herself was doing at the very moment when she wrote these lettersprove that Elizabeth never really identified with these women. Even though they were women, just like her, she apparently considered that the relative liberty she had conquered could not be extended to these foreign, non-White and non-Christian women : 'I repeat it, Sir, I think no women have so much liberty, safe from apprehension, as the Turkishand I think them, in their manner of living, capable of being the happiest creatures breathing' (Craven, 232).

Elizabeth is also harsh towards the Turkish males whom she regards with disdain, depicting them as lazy and even lethargic 47 --another orientalist cliché [START_REF]An Oriental lives in the Orient, he lives a life of oriental ease, in a state of Oriental despotism and sensuality, imbued with a feeling of Oriental fatalism[END_REF] --or criticising their 'ignorance' and lack of respect for the antique wonders their country is blessed with 49 .

Mosques are portrayed as dangerous places, constantly open, where any wrong-doer, hidden behind the traditional outfit, can come and commit any sort of evil act : 'Mosques are constantly open ; and I could not help reflecting that their mode of worship is extremely convenient for the carrying of a plot of any sorta figure, wrapped up like a mummy, can easily kneel down by another without being suspected and mutter in a whisper any sort of thing' 50 . The Tchouadar who accompanied her through her journey in Turkey is in his turn described as rude, annoying and arrogant by Elizabeth Craven : 'a Tchouadar who seems to promise me much more delay and inconvenience than his presence can make repay (…) If any travellers were to meet us, they would certainly take him for some Grand Seigneur, and that I am of his suite' 51 . At the end of her stay in Turkey, Elizabeth's letters simply refer to: 'my abominable tchouadar' 52 . Back in Europe, Elizabeth confesses in a letter to the Margrave that she is relieved to be once again 'with Christian people' 53 .

What did Lady Craven get then from this experience abroad, through so many different countries ? We have previously seen that this tour to the Orient was in part a way of ridding herself of the social pressure exerted on her after her repudiation for adultery in 1781.

It should besides be noticed that if her journey to the Orient lasted about a year and a half, her exile from Great-Britain lasted nine years, to which must be added the years she spent in Italy after the Margrave's death 54 . Lady Craven was in a way a fallen woman, a social outcast in her own country and this partly accounts for her becoming a travelling woman from 1783 until her death in 1828. But we have also noticed that Lady Craven was a very daring person who was attracted by the idea of travelling the world. Could it be considered that this journey to the Orient was decisive for Lady Craven's emancipation ? This long journey was the sequel to her departure from England in 1783 after her repudiation. It was a way of taking distance with her relatives, friends and the British society in general. It was also a way of proving to them all that even though she paid a high price for cheating on her husbandwho did exactly the same but was never stigmatised for thisshe was able to re-settle elsewhere, to travel the 49 'we returned to the Consul's, very much concerned at the excessive injustice and ignorance of the Turks, who have really not the smallest idea of the value of the treasures they possess, and destroy them wantonly on every occasion', 256. 50 A Journey, 217. 51 A Journey, 290. 52 A Journey, 300. 53 A Journey, 322. 54 The Margrave died in 1806.

But the most striking alteration that this tour to the Orient caused is the radical change in Lady Craven's status. She used to be an English aristocrat. After this tour she became a nomad until her death in Italy in 1828. A nomad, that is to say, someone whose life is always in-between, always at the junction of several places, languages, civilizations, social codes, activities and peoples. As Deleuze and Guattari put it : 'the life of the nomad is the intermezzo' 62 . From 1783 when she left Britain, until her death, Lady Craven was undoubtedly in a situation of territorial, social and ideological intermezzo or in-betweenness.

She became difficult to locate geographically as Horace Walpole noticed in a letter to Horace Mann : 'I thought her at Paris, and was surprised to hear of her at Florence' 63 . Socially speaking, her status was in-between respectability and scandal : unfaithful to the husband who cheated on her, she faced the public humiliation of her repudiation, re-married a prince, bore the title of Princess Berkeley and ended up owning a fortune in Italy. Ideologically, she spent her adult life encroaching limits, deliberately ignoring people's criticisms, leaving Britain, settling abroad, travelling on her own, writing, performing plays and losing contact with five of her children who reproached her with her attitude. Elizabeth Craven's difficulty to conform was thus reinforced and turned into an asset by this journey to the Orient which gave her a new status and the taste of living abroad.

Lady Craven never really blended in and the events of her life, more particularly this journey to the Orient, led her to develop a hybrid identity. Touring Europe to the Orienta male upper-class priviledgegave her the authority typical of learned male noblemen and thereby turned her into a sort of gender hybrid 64 : a woman with the experience of a learned male aristocrat and the first British woman to travel almost on her own to the Orient. Her hybridity was also geographical : born in Britain, she left her country for nine years from 1783 to 1792 and again from 1806 to 1828. Her exile, due to degrading reasonsher repudiation first and later on persistent accusations regarding her alleged unfaithfulness to her second husband-pushed her to the continent and to territories at the crossroads of the West and the East, with her journey to Constantinople first, then with her settlement in Naples 62 'la vie de nomade est intermezzo', Deleuze et Guattari, 471. 63 Horace Walpole, Letters, IX, 27. Walpole also wrote a worried letter to Elizabeth Craven in November 1786, expressing, then again, his concern : 'To my extreme surprise, Madam, when I knew not in what quarter of the known or unknown world you were resident or existent.' Horace Walpole, Letters, IX, 75-6. 64 'a fearless and self-disciplined narrator, a 'great traveller' who crosses over the gender divide', Efterpi, Mitsi, 29.

(Italy), a Greek-named town which once belonged to the Byzantine Empire. Lady Craven, a British aristocrat, an unfaithful and repudiated wife, an expatriate, who became a princess after marrying the Margrave, a writer, a playwright, an actress and most of all a daring traveller who challenged sexist prejudices and limits : a woman with a hybrid identity living in an ambiguous and transgressive in-betweenness. -Cazacu Mateï, Des femmes sur les routes d'orient, Genève : Georg, 1999.

Bibliography

  Elizabeth, A Journey through the Crimea to Constantinople in a series of letters from Elizabeth Craven to his serene highness the Margrave of Brandebourg, Anspach and Bareith, written in the year 1786, London : Robinson, 1789. -________, _________, The Beautiful Lady Craven, The Original Memoirs (1826), edited by Broadley and Melville, London : 1914, -________, _________, Modern Anecdotes of the Ancient Family of Kinkvervankotsdarsprakengotchderns, Anspach, 1787, -Montagu, Mary, Turkish Embassy Letters (1763), London : Pickering and Chatto, 1993. -Pigott, Charles, The Jockey Club, or a Sketch of the Manners of the Age, London : Symonds, 1793, -Walpole Horace, Letters, ed. Cunningham, London : Bickers, 1880. Analytical sources -Blondel, Madeleine, « Le récit de voyage féminin au XVIIIe siècle », Bulletin de la société d'études anglo-américaines des XVIIe et XVIIIe siècles, 17, 1983 (109-127).

world with whomever she liked, to plan her journey on her own, to deal with money, to face all sorts of perilous situationsshe crossed Russia in February using kibitkas 55 , she visited the Antiparos grotto and was the first lady to do so-, in a word, to live an independent, successful and highly transgressive life for a woman of the late 18th century. This tour to the Orient was therefore another proof that Craven was a bold and clever lady, but the journey undeniably contributed to making her even more independent. It should be noted that Lady Craven was actually the first English lady to undertake such a tour to the Middle East, since even though Lady Mary Wortley Montagu lived in the Ottoman Empire before Lady Craven 56 , she merely followed her husband who held a diplomatic position for a few years in Constantinople. Lady Craven's experience was therefore more resolute, risky and more genuinely emancipating than Lady Montagu's : 'Elizabeth Craven's account represent the exploits of an aristocratic and ambitious Englishwoman, travelling alone in regions where no other European woman had ever travelled before' 57 . But this journey to the Orient was also helpful in many other ways. It was an opportunity for Lady Craven to pursue a literary career.

On her return, Horace Walpole strongly encouraged her to publish her letters : 'Your Ladyship tells me that you have kept a journal of your travels (…). I conclude and hope, prepare it for the press (…)' 58 . Walpole insisted that this would be totally unique as no woman before her had undertaken such a long and remote journey : 'How proud I should be to register a noble authoress of my own country, who has travelled over more regions and farther than any female in print ! Your Ladyship has visited these islands and shores whence formerly issued those travelling sagas and legislators who sought and imported wisdom, laws, and religion into Greece' 59 . The publication of her letters gave Craven some additional literary fame,and credibility : although she had already written poems and books 60 , the publication of these letters probably helped her promote the theatrical performances she later held in the Margrave's estate in Anspach, then in Brandenburg House in London and certainly encouraged her to publish her personal memoirs many years after 61 . 55 Sledges. 56 Her Turkish Embassy Letters were published in 1763. 57 Mitsi, Efterpi, 21. 58 The Beautiful Lady Craven, li. 59 The Beautiful Lady Craven, li. 60 'I certainly did not send you, Madam, Lady Craven's verses, nor intend it, though they are extremely pretty', Horace Walpole, Letters, VI, 233. In 1779, Lady Craven had her novel Modern Anecdotes of the Ancient Family of Kinkvervankotsdarsprakengotchderns published and shortly after, her first play, The Miniature Picture was performed. 61 The Beautiful Lady Craven, ix.