Mohamed Abdallah Ali

Mazhar Mughal
email: mazhar.mughal@esc-pau.fr

MICROFINANCE AND POVERTY REDUCTION: EVIDENCE FROM DJIBOUTI

Keywords: Microfinance, poverty, productive loans, Djibouti

Does access to microfinance improve household welfare?

We seek the answer to this question using data on 2,060 borrower and non-borrower households based in six major urban centers of Djibouti. We construct a composite index of multi-dimensional poverty and carry out estimations using a number of econometric techniques. Our results show that neither access to micro-credit nor its ostensibly productive use is significantly associated with poverty regardless of the duration of time since the loan was acquired. This holds both for access to, and the amount of micro-credit obtained. The results raise doubts on the effectiveness of Djibouti's microfinance programme.

Introduction :

Extreme poverty is endemic to the countries located in the Horn of Africa. With limited natural resources, a poorly developed industry and climatic conditions unfavourable to agriculture, Djibouti has experienced economic crises since the 1990s. The country was forced to conclude agreements with the International Monetary Fund (IMF) in 1996 and the World Bank in 1997. The deterioration of the Djiboutian economy was due, not only to domestic issues (proliferation of internal conflicts) but also to external difficulties (decline in development aid, increased regional competition from the port of Assab1 , influx of refugees from Somalia and Ethiopia). Between 1992 and 1996, the economic situation deteriorated sharply, and the GDP declined2 (Foch,2010). Since 2010 however, the economic growth rate has increased substantially from 3.5% in 2010 to 7% in 2018. This strong growth fuelled by investments in trade infrastructure has enabled the country to rank among the fastest growing economies in the region (World Bank, 2018).

Thanks to a strong government focus on poverty alleviation, Djibouti has been able to reduce extreme poverty from 42.2% to 21.3% between 200242.2% to 21.3% between and 201742.2% to 21.3% between (DISED, 2018)). A major social safety net has been set up to combat poverty.

The concept of microfinance has also been adopted as a vital tool to improve the socio-economic conditions of those segments of the population excluded from the traditional banking system. After some initial steps taken in the 1990s, the microfinance sector gained visibility in the country during the 2000s with the establishment of Social Fund for Development (FSD), the Djiboutian Social Development Agency (ADDS) and an authority managing micro-loans through Credit Unions (Caisse Populaire d 'Epargne et de Crédit). By 2016, one-third of Djibouti's households had access to microcredit mainly provided by one of the three principal microfinance institutions (MFI's). In all, these institutions provided a total of US$12.3 million in loans. Despite this impressive growth of microfinance, there is little evidence so far to show its effectiveness or its contribution to poverty reduction in Djibouti.

Microfinance is seen by some as a 'magic wand' against poverty, a means to solve all problems. However, evidence on the effectiveness of microfinance as a poverty reduction measure in developing countries is mixed. At one end of the spectrum, there are studies affirming that microfinance is a positive and effective mechanism (for example [START_REF] Mosley | Microfinance and Poverty in Bolivia[END_REF]Imai and al.;[START_REF] Foch | Djibouti, une nouvelle porte de l'Afrique ? L'essor du secteur portuaire djiboutien[END_REF][START_REF] Imai | Does Microfinance Reduce Poverty in Bangladesh? New Evidence from Household Panel Data[END_REF][START_REF] Akotey | Does Microcredit Increase Household Welfare in the Absence of Microinsurance?[END_REF]). In contrast, some studies argue that microfinance actually plunges the population further into poverty, affecting women in particular (for example, [START_REF] Coleman | The impact of group lending in Northeast Thailand[END_REF][START_REF] Coleman | Microfinance in Northeast Thailand: Who benefits and how much?[END_REF]Crepon and al, 2015;[START_REF] Seng | Rethinking the Effects of Microcredit on Household Welfare in Cambodia[END_REF]. Others have warned against considering microfinance as a miracle solution, claiming that it helps poor households only to a limited extent, and have advocated its use with 'cautious optimism' [START_REF] Banerjee | The miracle of microfinance? Evidence from a randomized evaluation[END_REF]Karlan and Zinman, 2009;[START_REF] Duvendack | High Noon for Microfinance Impact Evaluations: Reinvestigating the Evidence from Bangladesh[END_REF]. The divergent conclusions drawn from existing literature are possibly a reflection of the diverse contexts of the studies as they focus on different geographical areas and use different methodologies.

In this study we analyse the welfare aspects of microfinance in Djibouti, a country which has received little interest upto now. The significance of the country in this context lies not only in the relatively high penetration of micro-finance institutions but also in the high vulnerability of the country's population to climatic changes and geo-political uncertainty.

We employ a large household survey conducted in 2015 by the Department of Statistics and Demographic Studies (DISED) in the capital, Djibouti-city, and the five regional capitals to analyse whether the households which take micro-loans manage to come out of poverty, does this depend on the amount of loan acquired or the time since it was obtained. We construct a composite poverty index by combining household characteristics pertaining to agriculture, employment, livestock, transport, household assets and sanitation. We employ Probit and Tobit models to estimate the effect of microfinance on the household's wealth status, and control for potential selection bias with inverse probability weighting (IPW) and augmented inverse probability weighting (AIPW). We find that the evidence supporting beneficial effects of microfinance services in Djibouti so far is not robust.

The following section of the document summarizes the microfinance sector and the profile of poverty and vulnerability in Djibouti. Section 3 presents the data and methodology used. Results are presented and discussed in Section 4. Robustness measures are described in Section 5. The final section concludes and offers policy recommendations.

Microfinance and poverty in

Djibouti-an overview.

State of poverty in Djibouti.

Djibouti is considered a low-income country, ranked 172 out of 188 countries on the development index. More than 70% of the country's population lives in urban areas. Economic activity is dominated by the public and tertiary sectors, which contribute 70.5% of GDP, much of which is concentrated in the capital (Djibouti-city). The contribution of the primary sector (1.8% of GDP) remains marginal, due to unfavourable climate, limited water resources, poor agricultural and fishing potential. The secondary sector contributes 16.9% of GDP, lagging behind the tertiary sector and limited due to the unavailability of primary materials, high production costs and labour shortage (Figure 1). According to EDAM-IS (DISED, 2017), extreme poverty for the country as a whole is estimated at 21.3% (Table 1). There seems to be a persistent gap between the well-being of inhabitants in the capital, Djibouti-city and those in other regions (Table 1). The extreme poverty rate is estimated at 13.6% in Djibouti-city compared with 45.0% in the other regions. Overall, 35.8% of the population is unable to meet its daily needs, whether food or non-food. The Gini index of the country at 0.42 is among the highest in the Middle-East and North Africa (MENA) region.

- ------------------------------------------------------Insert Table 1 here - ------------------------------------------------------

Djibouti's microfinance sector

Microfinance emerged in Djibouti in 2008. In 2010 it extended into rural areas. The main objective was to offer financial services to poor households, thereby enabling them to become self-sufficient by setting up income-generating activities.

There are three main microfinance actors :

The first actor is the Djiboutian Social Development Agency (ADDS). Its role is to structure and institutionalize microfinance activities as well as providing financial and logistic support, through donors and partners, for the development of the sector in question. The second actor is the bank. The relations between commercial banks and microfinance actors are limited to current operations (e.g. deposits, withdrawals and transfers). However, commercial banks do not provide credit lines for microfinance

Model and Variable Description

The central hypothesis of our study is that access to microfinance is associated with lower poverty.

Since the data available is cross-sectional in nature, a comparison can be made between households that have benefitted from MFIs loans and those that have not. A positive effect of access to microfinance can be statistically obtained provided there are more recipients among poor households of contracted loans from microfinance institutions (CPECs) than non-recipients.

Our empirical model can be given as:

(1)

Where is the poverty indicator, measures the matrix of control variables for i households, is the variable of interest, i.e. access to microfinance. and represent the coefficients of the controls and the variables of interest respectively. Finally, is the error term that follows normal distribution.

Dependent variable

We construct a composite poverty indicator that captures various aspects of well-being such as agriculture, employment, livestock, transport, household assets and sanitation. The constituent indicators of the index are listed in Table A1 in the Appendix. The composite poverty index of Katsushi et al (2010) took into account, in addition to the various dimensions mentioned above, the dimensions of income and food security. In our study, however, we could not include these two dimensions due to the limitations of the survey design.

Households are grouped into five wealth categories, namely :(i) extremely poor, (ii) poor, (iii) moderately poor (iv) self-sufficient and (v) wealthy.

Households from the extremely poor, poor and moderately poor categories experience deteriorating housing situations. In addition, they live in dwellings that do not benefit from sanitary facilities with widespread overcrowding and cramped space. The main energy sources are solar panels and candles. Poor and moderately poor households own large quantities of livestock (camels, cattle and sheep). The principal means of transport used by these categories are mopeds and public transport (buses and others) while many travel on foot.

Self-sufficient and wealthy households have certain assets in common, such as access to electricity, water source used, fuel used in the kitchen, flooring material and the type of toilet. They diverge, however, in terms of waste water disposal and household size.

In terms of access to microfinance, 26% of households belong to the extremely poor category, 21% come from poor and moderately poor categories, whereas 18% and 14% of the households come from the self-sufficient and wealthy categories respectively (Figure 2). Source: Author's calculations using PREPUD 2015.

24% of extremely poor households who received microcredits were able to use them for productive purposes, while 21%, 22%, 18% and 15% of households from poor, moderately poor, self-sufficient and wealthy categories were able to use their loans for the purpose of starting income-generating activities.

Variables of Interest

We use two indicators of access to microfinance:

(a) a binary variable indicating whether the household is a client of a microfinance institution or not and (b) a binary variable indicating whether a household took out credit for productive purposes or not.

(b) «Being a client » means that all household members had a savings account or an account with MFIs at the time of the survey.

The definition of "productive purposes" is based solely on the respondents own reported use of the loan.

Approximately 62% of borrowing households use microcredit offered by formal financial institutions (CPECs of Djibouti, the North the South), while the remaining 38% subscribe to microcredit from informal lenders (friends, traders, employers and others).

64% of borrowing households report that they use credit for productive activities, while 36% reported using their loans for non-productive activities (Table 2).

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------------------------------------------------------ Insert Table 2 here -------------------------------------------------------

Control variables

The control variables used in our econometric analysis relate to the socio-demographic characteristics of the household: age of household head, sex of household head, marital status, level of education, geographical regions, household size and dependency ratio.

34% of female-headed households acquired microcredit and 74% of married households took out loans from microfinance institutions. On average, households which took micro-loans have received more education: 36% of household heads granted credit have attained primary education, 53% have attained secondary level, while 11% reached a higher level. 25% of households benefitting from microcredit come from Djibouti city while the majority (75%) are from the interior regions, i.e the five regional capitals, Ali-Sabieh, Arta, Dikhil, Obock and Tadjourah.

Methodology

The empirical analysis is carried out as follows :

First, we use the Probit model to estimate the effect of access to microfinance and productive loans on poverty reduction. Next, we use the Tobit model to study the effects on poverty of the amount (in logarithm) of microcredit and the amount (in logarithm) of loans acquired for productive purposes.

In the third step, two techniques, inverse probability weighting (IPW) and augmented inverse probability weighting (AIPW) were used to take into account the possibility that households with access to credit may differ in observable characteristics from those without access to credit, thus causing selection bias. In addition, a wide range of robustness measures were carried out by estimating alternative specifications and models using different population sub-samples.

Results

Table 3 presents bivariate statistics for access to microcredit. We see that borrower and non-borrower households differ little in most of the economic, demographic and geographical features. The borrower households on average have a higher dependency ratio, suggesting a greater need for the working-age members to engage in income-generating activities.

The association between access to microcredit and poverty estimated through probit model (shown in Table 4) however, appears to be statistically significant at the 1% level. The marginal effects shown at the bottom of Column 1 show that a household having taken loan from an MFI is 8.5% less likely to be extremely poor compared with a non-borrowing household, keeping all other factors constant. In contrast, the relationship between acquiring a loan for productive purposes and extreme poverty is not found to be significant (Column 2).

Is the microcredithousehold wealth relationship suggested above specific to a wealth category?

Focusing on the moderately poor, self-sufficient and wealthy household categories, we find the association between household wealth and the incidence of acquiring a micro-loan to be insignificant (P-value=0.39; P-value=0.22 and p-value=0.45 respectively).The results for acquisition of productive loans are likewise insignificant.

-

------------------------------------------------------ Insert Table 3 here ------------------------------------------------------ Insert Table 4 here ------------------------------------------------------
Next we check if the household wealth status is associated in any way with the amount of microcredit the household obtains. Table 5 presents results for Tobit estimations for the relationship between the amount of loan and the incidence of poverty. The relationship is found to be insignificant (Column 1).

Likewise, the association between poverty and amount of productive loans acquired (Column 2) is found to be statistically insignificant (P-value = 0.283).

-------------------------------------------------------Insert Tables 5 &6 here -------------------------------------------------------

The lack of statistical significance of the povertymicrocredit relationship seen so far could be lined with the length of time since the loan was obtained. We explore this possibility by considering subsamples of beneficiaries who acquired loans less than or more than six months before the time of the survey. Results given in Table 6 suggest that time is not a significant factor as the relationship is found to be insignificant for both subgroups of beneficiaries.

Robustness measures

Matching estimations

It is possible that households who participate in the microfinance programme differ from those who do not. For instance, as mentioned above, the heads of borrowing households are on average more educated than those of non-borrowing households. We account for this potential selection bias by using two matching methods, namely Inverse Probability Weighting (IPW) and Augmented Inverse Probability Weighting (AIPW).

IPW improves over Propensity Score Matching (PSM) by allocating higher weightage to observations receiving an unlikely treatment. This reweighting allows higher weights to be assigned to individuals in the middle of the probability distribution and lower weights to the extremes (Wooldridge,2007). The second technique is the Weighted Augmented Inverse Probability estimator (AIPW), a 'doubly robust' method, with the properties of both the regression-based estimator and the IPW estimator, requiring either the propensity or outcome model to be correctly specified [START_REF] Cao | Improving efficiency and robustness of the doubly robust estimator for a population mean with incomplete data[END_REF].

Table 7 presents the IPW and AIPW estimates. The results of the two estimators are statistically and qualitatively similar. For both techniques, the results of access to microfinance are statistically insignificant as P-value=0.228 is above the 10% level. The results of access to productive loans too follow the same direction as previous results.

-

------------------------------------------------------ Insert Table 7 here -------------------------------------------------------

Multiple Hypotheses Testing

We conduct the Multi-Variance and Covariance tests (MANOVA) using the composite poverty indicator focusing on poor households. The results of the four statistics (Wilks' Lambda, Trace de Pillai, Laweley-Hotelling and Roy) reported in Tables 8 and9 show that the null hypothesis of equality of means for access to microcredit is statistically insignificant at the 10% level (P-value=0.513). The result of the test for access to productive loans is similar (P-value=0.718 > 0.10).

We also use the Bonferroni correction to test the statistical significance of the regression coefficients of our variables of interest. The method corrects the p-value in the case where several tests are performed simultaneously on the same data. The corrected coeficients remain insignificant as before (P-value = 0.513

for the access to loan model and 0.718 for the access to loans for productive purposes model).

- ------------------------------------------------------

Insert Table 8 & 9 here -------------------------------------------------------

Concluding remarks

In this study, we aimed at finding whether access to microfinance is associated with lower incidence of poverty. We used a household survey on the use of microfinance services carried out in 2015 in the major urban centers of Djibouti to seek answer to this question. We constructed a composite indicator for wealth status of the household and employed a number of empirical strategies.

We failed to find a robust association between microfinance loans obtained by Djiboutian households and their wealth status. Whether or not the household acquired a micro-credit, and whether it reported using the loan for productive purposes shows no significant association with its being poor. This is also true for the amount of loan the household obtained. The results are robust across specifications and econometric techniques employed. The lack of significant beneficial effect of microfinance found in the study adds to the growing literature questioning the effectiveness of microfinance as a tool for poverty alleviation.

The findings raise a number of questions for the policymakers:

What are the causes for this lack of positive impact? Is it due to the relatively short span of time since the programme had been launched ? To what extent does it result from the cost of borrowing ? The fact that the majority of the beneficiaries of microcredit disbursed does not come from the poorest segments of the society points in this direction. A better targeting of loan disbursement can therefore enhance the effectiveness of the programme but would require better training and awareness of the MFI staff on the ground.

Another possible implication of the study's findings is the need to extend the focus on incomegenerating activities. Sectors such as fishing and agriculture which create employment and lead to rapid increase in household income should be given preference.

Finally, the impact of microfinance on Djiboutian households could be enhanced by focusing on the skill development of the population. Higher financial literacy, better technical knowhow and professional skills can enable the poor to make good use of available opportunities with the help of small loans.

Figure 1 :

 1 Figure 1: Sectoral contributions to GDP (in %).

  do they cover refinancing. The last actor is the public authority, which has established a legal and regulatory framework for the further development of the sector. In addition, the state has taken the initiative of setting up a national microfinance reflection committee under the supervision of the central bank.At the national level, the microfinance sector is served by three major Credit Unions (Caisse Populaire d'Epargne et de Crédit), the CPECs. The first CPEC is in Djibouti city, the second in the southern region, based in Ali-Sabieh and the third in the northern region based in Tadjourah. These CPECs are incorporated as savings and credit cooperatives.In terms of distribution, the CPEC of Djibouti comes on top with 18022 members for which loans worth 1,619 million Djiboutian Francs (FD), about 74% of the total, have been disbursed (Figure2). The CPEC of Dikhil comes in second place with 1583 members in all categories, allocating an amount of 208 million FD (9 % of the total). The CPEC of Ali-Sabieh has 104 members with a micro-credit grant of 194 million FD (10 % of the total) of microcredit granted. The smaller CPECs of the North, notably those of Tadjourah and Obock, have 1,063 and 1,291 members respectively, and account for 91 million FD (4 % of the total) and 72 million FD (3 % of the total) in dispersed loans.

Figure 2 :

 2 Figure 2: Share of credit granted by region (in %).

Figure 3 :

 3 Figure 3: Access to microfinance by household wealth status.

Table 2 : Data Description

 2

	Variables	Definition

Source: Authors' calculations using PREPUD 2015.

Table 3 : Household characteristics by access to microfinance

 3

	Source: Authors' calculations
	Notes: Standard errors given in parentheses. *** p< 0.01; **p<0.05; *p<0.1

Table 4 : Micro-credits and households wealth-Probit estimation

 4

	Source: Authors' calculations
	Notes: Standard errors given in parentheses. *** p< 0.01; **p<0.05; *p<0.1

Table 5 : Result of tobit model main covariate: totals of amounts and productive loans.

 5

	Poverty (IBR=Poor)

Source: Authors' calculations Notes: Standard errors given in parentheses. *** p< 0.01; **p<0.05; *p<0.1

Table 6 : Effect of participation in the programme on poverty -Probit estimation

 6

	Source: Authors' calculations
	Notes: Standard errors given in parentheses. *** p< 0.01; **p<0.05; *p<0.1

Table 7 : Microfinance and poverty reduction-IPW and AIPW estimates Source

 7 Standard errors given in parentheses. *** p< 0.01; **p<0.05; *p<0.1

	Inverse-Probability	Access to microfinance	POmean	Access to productive loan		POmean
	Weight
		Withaccess	Withoutaccess	Withaccess	Withoutaccess With	Without	With	Without
						productive	productive	productive	productive
						loan	loan	loan	loan
	ATE	-0.029	-0.021	0.232***	0.228***	-0.225	-0.225	0.429***	0.429***
		(0.036)	(0.039)	(0.031)	(0.034)	(0.094)	(0.094)	(0.076)	(0.076)
	ATET	-0.024	-0.024	0.237***	0.237***	-0.225	-0.225	0.429***	0.429***
		(0.048)	(0.048)	(0.044)	(0.044)	(0.094)	(0.094)	(0.076)	(0.076)
	Observations	780	780	780	780	96	96	96	96
	Augmented IPW
	ATE	-0.024	-0.024	0.231***	0.231***	-0.223	-0.223	0.428***	0.428
		(0.042)	(0.042)	(0.038)	(0.038)	(0.094)	(0.094)	(0.076)	(0.076)
	Observations	780	780	780	780	96	96	96

: Authors' calculations Notes:

The truce signed in 1993 between Eritrea and Ethiopia, which allowed Ethiopia to access the sea via the port of Assab, severely affected the country's economic situation.

The country's GDP growth rate was -3.9% in 1993, -2.9% in1994 and -3.1% in 1995.

Appendix Table 1 : Poverty and inequality indicators in Djibouti 2017

Source: DISED (2018).


Fabrice Borel-Mathurin
email: fabrice.borel-mathurin@acpr.banque-france.fr

Julien Vedani
email: julien.vedani@milliman.com

Market-consistent valuation: a step towards calculation stability

Keywords: European regulation, life insurance, no-arbitrage opportunity, economic valuation convergence, market-consistency

In this paper we address some of the stability issues raised by the European life insurance regulation valuation scheme. Via an in-depth study of the so-called economic valuation framework, shaped through the market-consistency contract we first point out the practical interest of one of the El Karoui, Loisel, Prigent & Vedani (2017) propositions to enforce the stability of the cut-off dates used as inputs to calibrate actuarial models. This led us to delegitimize the argument of the no-arbitrage opportunity as a regulatory criteria to frame the valuation, and as an opposition to the previously presented approach. Then we display tools to improve the convergence of the economic value estimations be it the V IF or the SCR, using usual variance reduction methods and a specific work on the simulation seeds. Through various implementations on a specific portfolio and valuation model we decrease the variance of the estimators by over 16 times.

Introduction

Dating back to the the late 1990s and 2000s,the European Commission began creating a path to enforce the insurance European market in a direction ensuring more protection for the consumer thanks to common and harmonised rules to characterise the solvency of a firm (Solvency I then Solvency II Directives).

The regulatory framework considered consumer protection, based on market and commercial practices. This objective led to one of the two sides of the harmonisation. This article deals with the other side, the implications of defining a common framework for the economic/regulatory capital, which is the main ingredient to explicit the notion of solvency.

Solvency II being risk-based, the need for a comparable measure of the different exposures, be it on the asset or liability sides, led naturally to the concept of "fair value" which was developed in a parallel path (see in particular [START_REF] Thérond | About market consistent valuation in insurance[END_REF] and chapter 2 of [START_REF] Laurent | Modelling in Life Insurance: A Management Perspective[END_REF]). Indeed, since the seventies, the notion of "Market price" has been used and refined especially in areas where a transparent and fair trading price is known. For this notion to be adapted to more complex situations, especially when financial options are involved, maiden theoretical works [START_REF] Bachelier | The pricing of options and corporate liabilities[END_REF], [START_REF] Merton | Theory of rational option pricing[END_REF]...) developed the risk-neutral valuation principles framework allowing for those calculations. The accountant community also developed notions voluntarily in line through the definition of "fair value measurement" (see International Accounting Standards Board ( 2011)).

In practice, though estimating the value of an asset, especially if it is listed, is quite straightforward, it is not the case for insurance companies' liability side.Since life insurance contracts are not exchanged, assigning a fair price to them involves a complex model and its cash flows' projection, even in the Standard Formula case. This complexity and the regulatory necessity to use economic scenarios on the one hand and, on the other hand, assess regulatory valuations through Monte-Carlo estimations, renders the stability of inputs and the convergence character essential for its use as an efficient tool to quantify solvency and ease the work of European control authorities. These are the two main lines of this paper.

In Section 1 we review briefly the different historical and regulatory aspects which led to the Solvency II characteristics highlighting those with material impact on the market-(in)consistency. Through this Section we underline its need for more stability. Finally we focus on one approach developed in El [START_REF] El Karoui | Market inconsistencies of market-consistent european life insurance economic valuations: pitfalls and practical solutions[END_REF] to improve the stability of economic scenarios calibration inputs and lower the manipulability of the current practical scheme. But this method is often opposed by the no-arbitrage opportunity condition, a regulatory requirement for these valuation. This led us to recall this condition basics, operational requirements, and finally to show its incompatibility with the life insurance regulation in Section 2. According to the authors, these developments promote the real practical interest of the aforementioned approach.

In Section 3 we consider the convergence issue for the regulatory estimators. We present and implement some practitioners-oriented approaches to improve their various estimations, of Values of In-Force and Solvency Capital Requirements. In Section 4 we finally propose and implement an aggregated approach (still relatively easy to implement forpractitioners) which has led us to improve Value of In-Force estimation by more than 16×.

1 Market-consistent life insurance valuation in Solvency II -stability issues and first proposition

Mixing fair-value and regulatory valuation led the regulator (through the Solvency II directive -European Commission ( 2009)) to develop an insurance-adjusted valuation scheme based on financial market practices, the so-called market-consistency.

Market-consistency: financial stability and accounting information comparability

With Solvency II, the European Commission desired to reinforce the European insurance market so that solvency appreciation is made comparable across Europe, ensuring both a level playing field for the companies and the same consumer protection for the insureds and/or policyholders. In this regard, one of the biggest challenges was to ensure a comparable prudential balance sheet template could be populated across Europe notwithstanding the different national accounting norms. However, despite this objective to increase the level of harmonisation, the reserving mechanisms which are driving the estimation of technical provisions still largely depend on country-specific "Solvency I"-GAAP norms. Indeed as all taxes and social contributions are generally based on those national accounting norms, the historical/booking value of assets and liabilities then play the same role as in the Solvency I framework. Therefore, comparing the different values for technical provisions can be quite challenging for the insureds or supervisory authority throughout the whole EU market.

In parallel with the development of solvency directives, the notion of "Fair value" emerged through the discussion at the International Accounting standards Board (IASB) as a way to tackle those comparability issues. In a nutshell, fair value can be characterised in three points (see [START_REF] Thérond | About market consistent valuation in insurance[END_REF]):

1. Level 1: the price of the products is directly listed in an active market 2. Level 2: Parameters or prices can be inferred via a model (such as a financial option, and the implied volatility (IV) of the underlying asset) ("Mark to Market")

3. Level 3: The price is inferred from a model where interactions are necessary to estimate the figure as a consequence of the model ("Mark to Model")

Even if liabilities can actually be exchanged between insurance companies, this does not happen quite often and, even worse, there are no listed market for these trades, be it indexes or reporting of the effective transaction value. The only possible choice of the exposed alternative is the level 3: estimating the value of those via a probability-weighted average of future cash-flows, discounted with a Risk-Free Rate as outlined in the article 77 of European Commission (2009). With such a procedure, at least on a theoretical point of view, all technical provisions are made comparable independently of any national insurance sector's characteristics. As a consequence, the solvency appreciation with respect to own funds and solvency of any EU insurance company is done on the same terms.

However, experts are now backing up from fair valuation because of its inherent procyclicality. Despite the merits regarding comparability, this valuation methods tends to minimise the value of liabilities 3/26 when markets are bullish and over-estimate them in times of crisis. (see [START_REF] Laux | The crisis of fair-value accounting: Making sense of the recent debate[END_REF], [START_REF] Laux | Did fair-value accounting contribute to the financial crisis?[END_REF], Plantin et al. (2008a) and Plantin et al. (2008b)).

Insurance market-consistency -from regulations to practice

Since the only available technique to evaluate liabilities is the mark to model one, an insurance company needs to describe its asset-liabilities management strategy for which the following are needed:

1. a unique risk-free rate spot curve; 2. a methodology to produce future cash flows reflecting unequivocally the policyholders' behaviour; 3. economic scenarios deriving from the initial risk-free rate describing the future evolution of the assets;

4. a set of rules defining the interactions between the scenarios' tables and the policies to browse each sample path's scenario to ultimately produce cash flows out of any policy.

The only intrinsic part of this process is the determination of the risk-free rate spot curve. The regulatory framework from the Directive to the ITS define what is the basic risk-free interest rate term structure According to Article 77a of the Solvency II Directive the relevant risk-free interest rate term structure should be based on relevant financial instruments traded in deep, liquid and transparent (DLT) markets. This provision is further specified in recital 21, Article 1(32), ( 33) and (34), and Articles 43, 44 and 46 of the Delegated Regulation European Commission (2015b). The identification of the relevant financial instruments is based on this DLT assessment which is a list, for each currency, of the maturities for which the market of the relevant financial instrument is considered DLT including the identification of the last maturity for which rates can be observed in DLT markets (section 7.B refers to the determination of the Last Liquid Point (LLP)). On top of those elements, the determination of the Ultimate Forward Rate (UFR), recently provided by EIOPA1 is key. Indeed, EIOPA provides the EU insurance market a basic risk-free rate (RFR) curve which goes up to 150 years, thanks to interpolation from the LLP to the 60th year for Euro, via Smith Wilson extrapolation and interpolation.

To go from the basic RFR to the one used for calculation, one needs to add the effect of the Credit Risk Adjustment on the one hand and, on the other hand all the relevant Long Term Guarantee package items which can distort the produced interest rate curve such as the volatility adjustment (VA) or the Matching Adjustment (MA). Only with all such information taken into account, insurance companies or their information providers will set up their calibration process to produce their scenario tables. The potential large variation only due to calibration does induce great sensitivity to the final value of both BEL and own funds. This illustrates the principle from Thérond (2016) "one fair value principle, several valuation techniques"

A first proposition for deeper calibration data stability

From most operational practices, the main issue (though not only) of the valuation scheme is the fact most insurance companies use different calibration inputs for their model. This leads to both an instability of valuations in space (the space of potential inputs at a fixed date) and in time (linked to the high volatility of these market inputs). A particular aspect of these problems is linked to the end-of-year regulatory valuation date. This date is the accounting closure date for most listed companies and the financial markets undergo numerous trades for the sake of accounting optimisation. In particular, this leads to irrational market changes ("turn-of the year effect", well-known by traders, see e.g. [START_REF] Ritter | The buying and selling behavior of individual investors at the turn of the year[END_REF]), and it is very hazardous to pick market data up during this period being asset prices or other IV. In addition, having the life insurance whole market regulation depend on one single date renders easier possible manipulations (even the Libor rate can be more or less manipulated, see in particular [START_REF] Abrantes-Metz | LIBOR manipulation?[END_REF]).

For such reasons we have focused on a specific tool proposed by El [START_REF] El Karoui | Market inconsistencies of market-consistent european life insurance economic valuations: pitfalls and practical solutions[END_REF] to improve the fairness of prices obtained through insurance regulation valuations, decrease the manipulability of the scheme and add some stability through inertia to the interest rates models calibration data: the use of monthly-averaged IV matrices for calibration.

In subsection 2.2.3 of their paper, El Karoui et al. shape an adapted market-consistent constraint for interest rate models calibration (under the RFR yield curve) on different swaption log-normal IV matrices, in particular averaged matrices so as to insert inertia in calibration data (lower time-dependency).

• The IV averaged on the whole month of October 2017. Only the receiver swaptions of maturity 5, tenors 1 to 10, and the receiver swaptions of tenor 5, maturities 1 to 10.

• The IV averaged on the whole two months of October and November 2014. Only the receiver swaptions of maturity 5, tenors 1 to 10, and the receiver swaptions of tenor 5, maturities 1 to 10.

• The IV as of end-of-year 2014. Only the receiver swaptions of maturity 10, tenors 1 to 10, and the receiver swaptions of tenor 10, maturities 1 to 10 (v2).

• And the IV as of end-of-year 2014. Only the receiver swaptions of maturity 5, tenors 1 to 10, and the receiver swaptions of tenor 5, maturities 1 to 10 (v2).

The four calibrated then simulated tables embed 1,000 economic scenarios without any use of variance reduction techniques, obtained from the same simulation seed. The interest rates model considered is a Libor Market Model, often used by French practitioners at that time. They are used to valuate three standard saving products2 economic own funds3 . Results are summarized in table 1. This implementation has shown the high volatility of European economic valuation scheme. One can see a gap of about 140% between the end-of-year v1 calibration and the October 14' averaged IV calibration for portfolio 1. This is a clear example of the locality in space and time of market-consistency. 2 Insurance market-consistency and the no-arbitrage condition

The El [START_REF] El Karoui | Market inconsistencies of market-consistent european life insurance economic valuations: pitfalls and practical solutions[END_REF] proposition to improve the stability of market data used for insurance interest rates models calibration is often questioned by practitioners, in particular due to its non-respect of the no-arbitrage opportunity condition. Indeed, the averaged volatility matrices do not fit the yield curve date (end-of-year) which a priori leads to market prices arbitrage opportunities. In this Section we have chosen to demystify the no-arbitrage opportunity assumption, in particular as soon as the life insurance economic valuation is considered. After a short recall of the theoretical basics of this assumption, we focus on its various implications on financial market practices to better underline its non-adaptability to the insurance market. This leads us to show the limits of some regulation requirements and finally to refute the legitimacy of this counter-argument to the El Karoui et al. proposition.

Short reminder on risk-neutral theory and the no-arbitrage condition

Here we quickly recall the historical developments and underlying assumptions below the no-arbitrage principle.

Arrow and Debreu (see [START_REF] Debreu | Theory of value: An axiomatic analysis of economic equilibrium[END_REF], [START_REF] Arrow | The role of securities in the optimal allocation of risk-bearing[END_REF] and [START_REF] Arrow | Existence of an equilibrium for a competitive economy[END_REF]) determine the optimal probability measure to balance prices in dynamic arbitrage-free markets. Their work offered a solution to the set of supply/demand equations from the seminal [START_REF] Walras | Éléments d'économie politique pure[END_REF] which first led to the idea of optimal price on an economy where all supply and demand act rationally, under perfect competition.

In parallel to the economic theory, the mathematical finance theory evolved from [START_REF] Bachelier | The pricing of options and corporate liabilities[END_REF], to shape the conditions for the existence and unicity of a risk-neutral probability. It exists under the no-arbitrage condition (it is an equivalence) and is unique if the market is complete (all derivatives can be replicated by a linear strategy of simple assets).

Under these simple axioms, [START_REF] Bachelier | The pricing of options and corporate liabilities[END_REF] and [START_REF] Merton | Theory of rational option pricing[END_REF] develop risk-neutral models to formalise the stocks evolution (Geometric Brownian) and derivative prices at all time. Then many different models are developed for all sorts of financial assets (see [START_REF] Brigo | Interest rate models-theory and practice: with smile[END_REF], [START_REF] Cont | Financial modelling with jump processes[END_REF],...).

The no-arbitrage condition induces the non-existence of any 0-cost market strategy that ensures a strictly positive gain. Nobel price Milton Friedman talked of no free lunch. Under the complete market assumption, the risk-neutral measure is the only valuation probability measure that ensures the noarbitrage condition.

In practice, it is highly important to keep in mind that any use of a risk-neutral model for pricing (e.g. a derivative) assumes the valuated asset is well hedged (through market completeness) and only supports 6/26 a risk purely associated to its underlyings and not to the general market. This is why the risk-neutral models only focus on underlying's volatilities and assume a risk-free averaged evolution of all assets. Out of this scheme the risk-neutral valuation has no reality. It is (for example) not used in corporate finance and for banking investment strategies, but only for derivative pricing.4

2.2 The no-arbitrage condition in financial markets vs. in Solvency II

A financial market prerequisite

In practice every financial market presents limited arbitrage opportunities at all times. These are limited and quickly cancelled as soon as the market is liquid enough. Indeed, if enough buyers and sellers are active, well-informed and logical, perfect competition is close and, as the number of financial agreements rises, the situation converges towards fair prices and arbitrage reduction. This is an essential point to ensure the confidence of investors and of the market. On the 23 April 2013, after a fake tweet of Associated Press announcing two explosions in the White House, a quick arbitrage opportunity occurred in the Dow Jones. The index decreases by 1% but two minutes later recovers to its initial level (see Figure 1). Such an issue in the transparency of information barely happens, in particular on such a liquid market. And the fast recovery, almost exact, eases investors concerning the index fair price. The no-arbitrage condition is an absolute prerequisite to efficient market deals and financial markets well-being.

An insurance regulator requirement

From the Official Journal of the European Union volume 58 (2015) The consistency with the relevant risk-free interest rate term structure for both calibration and simulation is fixed and verified in practice but the consistency with market prices is not clear, apart when considering that in fact the texts means IV. However, the second condition needs further discussions at this stage of our presentation.

The no-arbitrage condition implications for the insurance market

As soon as a different yield curve but similar market IV are considered, new option prices are deduced and there exist arbitrage opportunities with financial markets, see Figure 2 below, for such differences in swaption prices as of end-of-year 2017.

At first, one could assume the insurance regulator does not mean comparing finance and insurance markets prices in its view of "no-arbitrage opportunity" in practice. Another possibility is that it only considers no-arbitrage on the (at least European) insurance market. But in practice there are still many arbitrages, such as for example, the swaption prices assessed using the European companies economic scenarios tables. Such options are used to calibrate the stochastic interest rates models which should lead to similar prices when simulating, but the maturities/tenors used to calibrate are not fixed by the regulator so that there exist many discrepancies among companies' choices even in the same country. In addition, different models and different calibration algorithms are used, leading to different market option prices and therefore arbitrage opportunities directly embedded in the various companies economic scenarios tables.

8/26

Figure 2: EIOPA swaption prices with the regulatory RFR yield curve vs. true prices (using the Euroswap yield curve) -matrix 20y×20y as at end-of-year 2017

In fact the only possibility to apply comparability of prices on the insurance economic values market would be to force the companies to use the same economic scenarios table .   Even if it were possible, actuaries are dealing with a market on which no one buys or sells products. In this context, prices obtained are never submitted to the supply and demand criteria and, unlike liquid financial markets, cannot converge towards fair prices.

Finally it must be admitted that the no-arbitrage condition has not the same implications here as for financial markets and practices. The "no-arbitrage opportunity" condition only states a valuation scheme where some market data and regulation choices are mixed to display more or less homogeneously obtained values. The fact this condition cannot be implemented for the insurance market should not be used as an excuse to legitimize insurance sector's practices. These conclusions do not invalidate the insurance valuation scheme entirely, but aim at partly disconnecting its current implementation from the theoretical and practical financial market. This last framework is very complex and must be wisely balanced and understood when used as a theoretical and practical justification for insurance solvency regulation choices. 3.1 Monte-Carlo estimators convergence -a general issue in the Solvency II directive

One economic scenario leads to one valuation of NPV . If one considers an economic scenarios table of S scenarios one obtains a Monte-Carlo estimator of the V IF through,

V IF = 1 S ∑ S s=1
NPV s . The Monte Carlo estimators convergence results from the Law of Large Numbers, without any additional model specification assumption. But the main issue with every Monte-Carlo valuation approach is its time complexity. The higher the variance of the underlying variable (NPV here), the higher the number of scenarios needed to provide an efficient estimator convergence.

V IF estimation therefore requires a high number of economic scenarios and corresponding NPV assessment (because the NPV variable has in general a very high variance). This is easily seen through the 0 < p < 100% asymptotic confidence interval that can be estimated for the V IF estimator,

CI p V IF = V IF ± √ V[NPV ]q 1+p 2 [N (0,1)] √ S
, with the usual unbiased variance estimator,

V [NPV ] = 1 (S-1) ∑ S s=1 NPV s -V IF 2 .
To increase the estimator convergence speed (and decrease the confidence interval length), it is possible to increase S but it is also possible to try to decrease the variance of the variable of interest, NPV here. We present two possibilities to do so in subsection 3.2.

In reality two different estimators are of great importance concerning the Solvency II regulation. Along with the V IF estimator, the Solvency Capital Requirement (SCR) denotes the regulatory capital. According to the regulation the latter must be assessed at least every end-of-year and it can be calculated through two different approaches, Internal Model or Standard Formula (see in particular [START_REF] Devineau | Risk aggregation in Solvency II: How to converge the approaches of the internal models and those of the standard formula?[END_REF] which depicts the way both are theoretically implemented and the assumptions under which both approaches converge). In this paper we focus on the standard formula approach, which is the most widespread estimation method used. This approach relies on the estimation of marginal risk capitals (stock risk, interest rates risk, longevity risk, lapse risk etc.), assessed as the difference between the standard (central) economic own funds and the marginally (instantaneously) shocked own funds. Assuming there exist I different shocks, any marginal SCR (denoted by SCR i , ∀i ∈ (risk 1 , ..., risk I )) is obtained as,

SCR i = OF -OF i = V IF -V IF i with OF i (respectively V IF i ) the i-marginally shocked own fund (respectively V IF),
10/26 estimated by,

SCR i = V IF -V IF i .
These estimated sub-capitals are then aggregated through an elliptic approach in order to assess SCR, the estimated standard formula SCR.5

In subsection 3.3 we address the estimator SCR i convergence issue and focus on a specific methodological aspect, important to respect, that enables a quicker convergence of the estimated sub-capitals and finally of the estimated standard formula SCR (see Appendix 1).

VIF estimator convergence -variance reduction methods

Two different easy techniques can be used to decrease the NPV variance and increase the V IF estimator convergence. One of these approaches is often well-known and used by actuarial practitioners, the antithetic variables. But the second one is as easy to implement, generally very efficient, though not used by practitioners, namely control variate. It is very important to note that these two techniques can be implemented together to improve the convergence even more.

Antithetic variables

The antithetic variables (AV) methodology is a well-known variance reduction approach, often used in finance and simulation theory in general (see e.g. [START_REF] Paskov | Faster valuation of financial derivatives[END_REF] or [START_REF] Glasserman | [END_REF]).

The idea behind most variance reduction techniques as is the case here is to speed up the Monte Carlo estimation of the interest variable average by considering an alternative variable with same mean but lower variance.

In practice each economic scenario can be summarised by a set of random standard Gaussian outcomes, those used to simulate its underlying economic variables (stock indexes, yield curves, spreads,...). Let ε be this sample of outcomes so that one can see any valuated NPV as a closed function of ε, which we simply note below (when necessary) NPV (ε).

One finally estimates, with evident notations (using NPV (ε s ) instead of NPV s ),

V IF = 1 S ∑ S s=1 NPV (ε s
). The application of antithetic variables to V IF estimation consists in considering S even (S = 2N below), and, denoting by (ε s ) s∈ 1;N the N first scenarios random outcomes vectors, in using ε s = -ε s-N ∀s ∈ N + 1; S for the N last scenarios random outcomes vectors.

In practice, all conditions sum up to CoVar [NPV (ε); NPV (-ε)] ≤ 0 be always verified whichever asset-liability management (ALM) model or economic conditions are considered. This leads to,

V V IF AV = V 1 S ∑ S s=1 NPV (ε s ) = V 1 S ∑ N s=1 NPV (ε s ) + NPV (-ε s ) = 1 2S V [NPV (ε) + NPV (-ε)] = 1 2S (V [NPV (ε)] + V[NPV (-ε)] + 2CoVar [NPV (ε); NPV (-ε)]) ≤ 1 2S (V [NPV (ε)] + V [NPV (-ε)]), and, as V [NPV (ε)] = V [NPV (-ε)](recall ε is a standard Gaussian outcomes vector), V V IF AV ≤ 1 S (V [NPV (ε)]) = V 1 S ∑ S s=1 NPV (ε s ) = V V IF .
Finally the variance of the AV-V IF estimator is lower than the one of a standard V IF estimator, that would have been estimated using S independent random outcomes vectors (S independent economic scenarios). This justifies the use of antithetic variables as a V IF estimator variance reduction. The AV estimator converges faster (and, in particular, the confidence interval length decreases).

Control variate

Control variate (CV) is another variance reduction methodology, also very well-known in finance and simulation theory (see e.g. [START_REF] Hull | The use of the control variate technique in option pricing[END_REF] or again [START_REF] Glasserman | [END_REF]), that can be easily adapted to our methodology.

The underlying idea of CV is to find a variable Y , also function of ε, whose exact average is known and which satisfies CoVar [NPV (ε);Y (ε)] = 0.

In practice such variables are easy to find for insurance NPV . The NPV are highly linked to the movements of the economic variables embedded within the economic scenarios their calculation relies on. Finally, one can select a portfolio of assets which cash-flows are highly correlated to the net present values of margin cash-flows. If these assets are not too complex, their true expectancy at t = 0 (their prices) is straightforward and finally the portfolio true averaged value at t = 0 is known. In this example Y is the portfolio net present value of future cash flows, which fully depends on the considered economic scenarios and therefore on its ε vector. This type of portfolio variable, highly correlated with the insurance NPV has already been investigated and is the base of the replicating portfolio theory (see e.g. [START_REF] Schrager | Replicating portfolios for insurance liabilities[END_REF], [START_REF] Devineau | Replicating portfolios: calibration techniques for the calculation of the Solvency II economic capital[END_REF], [START_REF] Pelsser | The difference between lsmc and replicating portfolio in insurance liability modeling[END_REF]).

The implementation of CV still relies on the use of an alternative variable of interest. Instead of NPV (ε), the practitioner may use a variable with the same average but a lower variance, the CV-NPV (ε), denoted by NPV CV (ε) and defined as,

NPV CV (ε) = NPV (ε) + c (Y (ε) -E [Y (ε)]), with c a constant. One verifies, E [NPV (ε) + c (Y (ε) -E [Y (ε)])] = E [NPV (ε)] = V IF, and, V [NPV (ε) + c (Y (ε) -E [Y (ε)])] = V [NPV (ε)] + c 2 V [Y (ε)] + 2c.CoVar [NPV (ε);Y (ε)] .
And if c is chosen to minimise the variance, that is,

c * = -CoVar[NPV (ε);Y (ε)] V[Y (ε)] one verifies that, V [NPV (ε) + c * (Y (ε) -E [Y (ε)])] ≤ V [NPV (ε)] (obtained for c = 0).

12/26

Finally we obtain,

V V IF CV = V 1 S ∑ S s=1 NPV CV (ε s ) = 1 S V NPV CV (ε) ≤ 1 S V [NPV (ε)] = V V IF .
This justifies the use of control variate as a V IF estimator variance reduction. The CV estimator converges faster (and the confidence interval length decreases).

As a matter of fact, if CoVar [NPV (ε);Y (ε)] = 0 then c * = 0 and NPV (ε s ) CV = NPV (ε s ), the CV has no impact. In opposition, if NPV (ε) and Y (ε) are collinear/antilinear, then

E V IF is known up to a multiplicative coefficient E V IF = ± V[NPV (ε)] V[Y (ε)] E [Y (ε)] . Besides, in all generality Correl [NPV (ε);Y (ε)] ∈ ]-1; 0[ ∪ ]0; 1[,
and CV users aim to get Y such as this correlation is the closest to -1 or 1.

Once again, note that AV and CV can totally be implemented at the same time without full cannibalisation of their positive effects. The final AV+CV V IF estimator converges even faster.

Standard Formula SCR convergence -the simulation seed importance

In this subsection we address the standard formula SCR estimator convergence issue. We focus here on one single sub-risk SCR estimator but the point raised can be applied to all marginal SCR estimator, so that the final SCR estimator variance should decrease (see Appendix 1).

Recall the shape of our marginal SCR estimator associated to shock i,

SCR i = V IF -V IF i .
Under the notation of subsection 3.2 we can also express the estimator as,

SCR i = 1 S ∑ S s=1 NPV (ε s ) -1 S ∑ S s=1 NPV i (ε n s ), with n s ∈ N for all s ∈ 1; S , or, SCR i = 1 S ∑ S s=1 (NPV (ε s ) -NPV i (ε n s ))
. In practice we want to underline here the importance to conserve the same simulation seed in both the economic scenarios used to asses the standard V IF and those used to assess the shocked V IF. One therefore has (denoting by s.s when using the same simulation seed in our estimations, be it of OF, V IF, full standard formula or marginal SCR),

SCR s.s i = 1 S ∑ S s=1 (NPV (ε s ) -NPV i (ε s )).

This point is of great importance because one generally verify that CoVar

[NPV (ε); NPV i (ε)] > 0 so that, V SCR s.s i = 1 S V [NPV (ε) -NPV i (ε)] = 1 S (V [NPV (ε)] + V [NPV i (ε)])-2 S CoVar[NPV (ε); NPV i (ε)] ≤ 1 S (V [NPV (ε)] + V [NPV i (ε)]) = V SCR i .
This upper bound is generally satisfied in practice. We have here tested this point on a standard life insurance savings product, on a marginally shocked SCR, based on 2,500 economic scenarios, with results displayed in Table 2 (denoting by ρ the empirical correlation between the central and shocked NPV ), 13/26 The results show a direct application of the previous inequality. Whichever the shocked SCR, the Different seeds estimated SCR converges much slower than the Same seed estimated SCR (not the estimated V IF whose confidence intervals don't evolve). In particular, concerning the spread SCR, we obtain a decrease of the half-confidence interval by, at least, a coefficient 553 142 ≈ 3.9. If a different seed had been used, this means the number of scenarios that should have been used to evaluate the marginal SCR with the same efficiency would have been close to 2500 × 3.9 2 = 38025. This empirically justifies the use of the same seed, when estimating standard and shocked V IF, as a marginal SCR estimator variance reduction. Finally we assume the elliptically aggregated SCR convergence increases (we verify it empirically in Appendix 1).

Implementation -a control variate specific application to increase the V IF convergence speed

In this Section we present a relatively easy application of the methods from Section 3 using both variance reduction methods presented and the seed conservation impact to increase even more the V IF estimators convergence speed.

Implementation context and formalisation

All our work has been ignited by the empirical analysis of various ALM models. From a simplistic point of view, ALM models are complex accumulations of accounting functions with entity specific parameters (contract parameters, loading factors,...) and inputs (assets, liability model points,...). Both are regularly updated but embed a strong inertia due to various factors (including long-term contracts, top-management choices and long-term objectives,...).

Finally the way ALM models behave to an economic scenario random vector ε stays rather similar, whichever initial economic conditions and calibration data are chosen by the modelling team. At least on a limited time-lapse.

14/26

Still using the seed-keeping character, leading to correlated central NPV and shocked NPV i , we have tried to estimate the correlation of NPV α (ε) and of NPV β (ε), with NPV α and NPV β two different NPV variables associated to two different economic conditions (namely situations α and β ).

The proposed methodology is simply to use NPV α (and the true value of E [NPV α ], denoted by V IF α below), as a control variate for any other V IF one wants to estimate, here denoted as V IF β (estimated on NPV β outcomes).

The main issue here is to assess an efficient estimation of V IF α . This is why we only implement it one time though we test it on several different V IF β economic situations.

In the implementations below we have chosen to estimate this V IF α value using 50,000 economic scenarios so that the convergence of our estimator is good enough to be very close to the true V IF α . We abusively identify this estimator to the true V IF α below (see subsection 4.3 for a more precise development on the implications of the imperfect equality between this estimator and the true V IF α value). This V IF α is estimated based on the end-of-year 2017 regulatory-consistent economic conditions for all simulated economic quantity but nominal rates. Concerning nominal rates, we have used as the reference (calibration/simulation) yield curve the market Euroswap rates (not the regulatory EIOPA yield curve, used in Implementation 1 for the V IF β situation), and the calibration swaption IV matrix is the averaged 10 (maturities) x 10 (tenors) December 2017 matrix (not the standard end-of-year matrix, also used in Implementation 1 for the V IF β situation). This allows us to test very different economic conditions for V IF β . We assume any other economic situation chosen as our reference V IF α 's should have led to similar results6 , we have simply chosen one in all generality.

We obtain for 50,000 scenarios, using AV, and with a randomly chosen simulation seed,7

V IF α = E [NPV α ] ≈ E [NPV α ] = 22,005 ± 765.

Implementation of control variate using V IF α

Recall subsection 3.2.2 concerning control variate implementation. We want to improve the convergence of an estimator V IF β = 1 S ∑ S s=1 NPV β (ε s ) using a control variate, here NPV α outcomes, assumed well correlated with the NPV β (ε s ) s∈ 1;S , and knowing E [NPV α ] = V IF α (see 4.1 for more details).

Indeed, if we already know V IF α quite well, we also need NPV α and NPV β correlated (the most possible) outcomes. In the implementations below we therefore always consider the same simulation seed to search for a strong correlation between our NPV α and NPV β outcomes. Basically this means we have to assess couples NPV α (ε s ), NPV β (ε s ) s∈ 1;S . In addition we want to stay operationally realistic, with S low enough to be implementable. We have therefore chosen below S = 2,000 scenarios using antithetic variables.

We obtain for 2,000 scenarios, using AV, and with our fixed simulation seed,

V IF α = 1 2,000 ∑ 2,000
s=1 NPV α (ε s ) = 23,949 ± 3,666.

4.2.1 Implementation 1: V IF α economic situation vs. the standard regulatory economic conditions as of end-of year 2017

In this first implementation we use the full standard end-of-year 2017 regulatory-consistent conditions for the V IF β economic situation. We therefore try to assess the true regulatory V IF of our savings product as of end-of-year 2017. Compared to our V IF α situation, we therefore change the calibration swaption IV matrix, using the end-of-year 10x10 matrix, and we use the corresponding RFR to calibrate and simulate the 2,000 economic scenarios associated with the (ε s ) s∈ 1;2,000 .

We obtain,

V IF β = 1 2,000 ∑ 2,000
s=1 NPV β (ε s ) = 20,615 ± 3904. As expected, based on the couples NPV α (ε s ), NPV β (ε s ) s∈ 1;2,000 , we estimate,

Correl[NPV α (ε), NPV β (ε)] = 96.9%.
We have also,

ĉ * = - CoVar[NPV α (ε);NPV β (ε)] V[NPV α (ε)] = -1.03.
And eventually,

V IF CV β = 1 2,000 ∑ 2,000 s=1 NPV β (ε s ) + ĉ * (NPV α (ε s ) -V IF α ) = 18,609 ± 961.
We obtain a decrease of the half-confidence interval by a coefficient 3,904 961 ≈ 4.1. This means without using the CV such efficient estimation would have required close to 2,000 × 4.1 2 = 33,620 scenarios. In this second implementation we consider for V IF β the situation tested and favoured in El Karoui et al.

(2017) but adapted to end-of-year 2017. Compared to our V IF α situation, we change the calibration swaption IV matrix and use the averaged October-November 2017 v2 matrix (see subsection 1.3), and we use the RFR to calibrate and simulate the 2,000 economic scenarios associated with the (ε s ) s∈ 1;2,000 .

We obtain,

V IF β = 1 2,000 ∑ 2,000
s=1 NPV β (ε s ) = 11,354 ± 4,336. As expected, based on the couples NPV α (ε s ), NPV β (ε s ) s∈ 1;2,000 , we estimate,

Correl[NPV α (ε), NPV β (ε)] = 92.9%.
We have also,

ĉ * = - CoVar[NPV α (ε);NPV β (ε)] V[NPV α (ε)]
= -1.10.

And eventually, 16/26

V IF CV β = 1 2,000 ∑ 2,000 s=1 NPV β (ε s ) + ĉ * (NPV α (ε s ) -V IF α ) = 9,219 ± 1,602.
We obtain a decrease of the half-confidence interval by a coefficient 4,336 1,602 ≈ 2.7. This means without using the CV such efficient estimation would have required close to 2,000 × 2.7 2 = 14,580 scenarios. 4.2.3 Implementation 3: V IF α economic situation vs. the regulatory conditions as of end of June (mid year) 2018

In this third implementation we use the full standard end of June 2018 regulatory conditions for the V IF β economic situation. We therefore try to assess the true regulatory V IF of our savings product as of end of June 2018. Compared to our V IF α situation, we therefore change the calibration swaption IV matrix, using the end of June 2018 10x10 matrix, and we use the corresponding RFR to calibrate and simulate the 2,000 economic scenarios associated with the (ε s ) s∈ 1;2,000 .

We obtain,

V IF β = 1 2,000 ∑ 2,000
s=1 NPV β (ε s ) = 38,470 ± 3,667. As expected, based on the couples NPV α (ε s ), NPV β (ε s ) s∈ 1;2,000 , we estimate,

Correl[NPV α (ε), NPV β (ε)] = 91.9%.
We have also,

ĉ * = - CoVar[NPV α (ε);NPV β (ε)] V[NPV α (ε)] = -0.92.
And eventually,

V IF CV β = 1 2,000 ∑ 2,000 s=1 NPV β (ε s ) + ĉ * (NPV α (ε s ) -V IF α ) = 36,682 ± 1,447.
We obtain a decrease of the half-confidence interval by a coefficient 3,667 1,447 ≈ 2.5. This means without using the CV such efficient estimation would have required close to 2,000 × 2.5 2 = 12,500 scenarios.

Final analyse based on our implementations results

The methodology presented and implemented in Section 4 may seem practically delicate to use but, as aforementioned, its unique complexity lies in the very precise estimation of one V IF (denoted by V IF α above, estimated by 50,000 scenarios here) 8 . Then, using control variate enables to take profit of the convergence knowledge assessed from this heavy estimation and improve any other V IF β valuation, as soon as the economic situation under which this valuation is made is not too far (in time and in space) from the V IF α situation.

Through our different implementations we have obtained, with our implementation choices, a re-8 Concerning the NPV α outcomes required to build the control variate part of the NPV CV β ( ĉ * NPV α (ε s ) -V IF α ) it seems also possible to use the same seed when estimating precisely V IF α as when estimating roughly (without using CV) V IF β so that the NPV α outcomes are already calculated when estimating the precise V IF α estimation and they do not need additional complexity. 17/26 duction of the estimators variance equivalent to using between 6 and 20× more economic scenarios. However one must note that this great improvement is also associated to the integration of a relatively small bias to the valuation linked to the fact that the true V IF α is not known in reality, we have simply an (good) estimation V IF α = 1 50,000 ∑ 50,000 s=1 NPV α (ε v s ) for some v s ∈ N, ∀s ∈ 1; 50,000 .9 Finally,

E V IF CV β = E NPV CV β (ε) = E NPV β (ε) + ĉ * NPV α (ε) -V IF α = V IF β + ĉ * V IF α -V IF α . The final bias of V IF CV β is therefore ĉ * V IF α -V IF α .
This bias is small and would be null if the estimator V IF α had converged. To reduce this bias it is therefore ideal to use the highest possible number of economic scenarios when estimating the true V IF α value. We consider the convergence improvement as relevant enough to prevail this small bias that can be controlled.

Basically, the underlying idea of this methodology is to obtain a footprint of the ALM model and its behaviour, at one moment in time. Then using this knowledge, the user knows how to adjust future ALM valuations. Under this comparison, it seems a priori necessary to keep the same ALM model. This is indeed ideal but in practice most ALM models behave similarly to similar economic scenarios. This means any two models would actually display grossly the same variations even if those are not exactly one-to-one. For example, when stock indexes rise the NPV rises, whichever ALM model is considered. It is impossible to generalise it to full scenarios that include stock indexes, interest rates (nominal/real/risky),... But we can assume a change in ALM model between the V IF α and V IF β may conserve Correl(NPV α (ε), NPV β (ε)) > 0.

Finally, we recall this approach would only lead to efficient results when regularly updated. Once again, the ALM model footprint is highly dependent on its underlying model points, Asset-Liability transition parameters and on many other points. We highly recommend, for the sake of efficiency, to update it at least yearly.

Conclusion

In this paper we have focused on the necessity to stabilise the European life insurance regulation valuation scheme based on the so-called concept of market-consistency. We have underlined the insurance-specific interpretation of this concept which originally tends to link the scheme to market finance practices and theory.

In section 1, we motivated the construction of the whole Solvency II Directive in its objective of enforcing financial stability or harmonising accounting norms to increase both transparency and financial resilience and as a consequence consumer protection. However, by doing so, different elements such as the Risk Free Rate or items from the Long term guarantee package -e.g. the volatility adjustmenthave material consequences on the levels of the interest rate curve and as outlined in El [START_REF] El Karoui | Market inconsistencies of market-consistent european life insurance economic valuations: pitfalls and practical solutions[END_REF] intrinsically contradicts the goal of market-consistency as far as the market prices and technical provisions best estimates are concerned.

In particular we have presented, through Section 2, some practical reasons why the no-arbitrage opportunity condition can never be respected through insurance valuation. This aspect is only theoretical and does not question the practical regulation implementation framework. We wanted here to further investigate the mathematical background, especially regarding the actual interest of the averaging approach introduced by El Karoui et al. (2017), in order to introduce stability in the calibration data used by practitioners, but also in reducing the manipulability of the valuation scheme.

In Sections 3 and 4 we have then addressed a second aspect of major importance in the stability of the considered valuation. We have proposed different approaches to improve the Solvency II Monte-Carlo estimators convergence by working on the simulation seeds and on the use of variance reduction techniques. Finally, in our implementations we have reduced the variance of our V IF estimators by more than 16× (but integrating a small -manageable -bias in estimation, see subsection 4.2 and 4.3) and the marginal (standard formula) SCR by up to more than 15× (in subsection 3.3). Note that these results have been obtained on a single ALM life insurance model and future investigations will be necessary to efficiently generalize this analysis.

Though we consider the simulation seed issue we must recall to practitioners the only question that must be addressed here is convergence. Indeed, though it is probable that for a given model there exist some seeds leading to estimations closer to their true value than others, it cannot be generalised that one specific seed is better than another. Therefore we do not address the seed choice problem but only promote the choice of the same simulation seeds for each calculations, to valuate standard formula SCR, use the control variate method presented in Section 4 but also to respect some stability in estimation bias and improve the comparability through time of economic valuations.

In future works we will test our methods on further ALM models and economic situations. We also intend to address deeper the similar simulation seed implications on valuation, propose a complete update of the El [START_REF] El Karoui | Market inconsistencies of market-consistent european life insurance economic valuations: pitfalls and practical solutions[END_REF] approach and test further sensitivities on economic valuations. Would those techniques be used on an industrial scale with all robustness checks being made, this would highlight and enhance the feasibility of forward looking analyses such as ORSA (or other type of estimations from the Pillar 2) which are highly time-consuming and resource-demanding.

Appendix 1 -impact of using similar simulation seed on aggregated SCR Let us now consider 2 different marginal SCR, for risks i and j ∈ (risk α , ..., risk I ) and denote by ρ i, j the elliptic aggregation correlation (users often choose the regulatory specifications which provide these correlations, see in particular European Insurance and Occupational Pensions Authority (2010)) between SCR i and SCR j . We consider below, SCR i, j = SCR 2 i + SCR 2 j + 2ρ i, j SCR i SCR j . 10

We aim at comparing V SCR s.s i, j and the standard V SCR i, j .

10 Basically the aggregated SCR is obtained by SCR = ∑ i, j∈(risk 1 ,...,risk I ) ρ i, j SCR i SCR j with ρ i, j = 1 when i = j.

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V SCR i, j = V SCR 2 i + SCR 2 j + 2ρ i, j SCR i SCR j = E SCR 2 i + SCR 2 j + 2ρ i, j SCR i SCR j -SCR 2 i, j = V SCR i + V SCR j + 2ρ i, j E SCR i SCR j + SCR 2 i + SCR 2 j -SCR 2 i, j . We can easily assume, from the work above,

   V SCR s.s i ≤ V SCR i , V SCR s.s j ≤ V SCR j .
Concerning the +2ρ i, j E SCR i SCR j part it is complex to estimate a general comparison with the optimised +2ρ i, j E SCR s.s i SCR s.s j . We can assume ρ i, j ≥ 0 (true for most regulatory correlation factors) but it is probable that the comparability of both expressions depends on risks i and j... If ρ i, j = 0 it is clear that V SCR s.s i, j ≤ V SCR i, j . In any case we assume the efficiency of keeping the same simulation seed to reduce the variances should also lead to V SCR s.s i, j ≤ V SCR i, j . We have tested several ρ i, j > 0 values and several couples of risks i and j with following results obtained using bootstrap estimators assessed on 2,500 aggregated SCR outcomes, each one estimated on 2,500 standard bootstrapped -sampled with replacement (see e.g. [START_REF] Efron | Improvements on cross-validation: the 632+ bootstrap method[END_REF])outcomes among 2,500 i.i.d. outcomes NV P -NPV i . Using standard bootstrap it is possible to assess specific confidence intervals: CI bootstrap α% SCR = 2 Ê SCRq 1+α 2 SCR ; 2 Ê SCRq 1-α 2

SCR

The results are given in Table 3.

Table 3: Comparison between the approximated 2-risks aggregated SCR values -Same seed vs. Different seeds with 95% confidence interval Spread/stock SCR ρ i, j = 0.25 ρ i, j = 0.5 ρ i, j = 0.75 Different seeds 15,427 [14,482;16,032] 16,793 [15,766;17,460] 18,055 [16,936;18,778] Same seed 15,398 [15,184;15,545] 16,759 [16,525;16,920] 18,017 [17,765;18,180] Spread/RE SCR ρ i, j = 0.25 ρ i, j = 0.5 ρ i, j = 0.75 Different seeds 12,617 [11,746;13,180] 13,198 [12,234;13,820] 13,755 [12,689;14,455] Same seed 12,673 [12,472;12,799] 13,277 [13,066;13,411] 13,855 [13,635;13,997 8,526 [8,364;8,625] 9,091 [8,910;9,198] 9,623 [9,438;9,736] Finally the use of the same simulation seed seems to work very well on the 2-risks aggregated SCR but also on the 3-risks aggregated SCR we tested (see Table 4).

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Table 4: Comparison between the approximated RE/stock/spread risks aggregated SCR values -same seed vs. different seeds with 95% confidence interval 3-risks SCR ρ i, j = ρ i,k = ρ j,k = 0.25 ρ i, j = ρ i,k = ρ j,k = 0.5 ρ i, j = ρ i,k = ρ j,k = 0.75 Different seeds 16,400 [15,249;17,142] 18,371 [17,019;19,241] 20,150 [18,603;21,146] Same seed 16,426 [16,168;16,582] 18,417 [18,126;18,591] 20,213 [19,891;20,405]

Figure 1 :

 1 Figure 1: Arbitrage Opportunity in the Dow Jones -04/23/13 (source : The Telegraph)

3

  Economic valuation stability through estimators convergence accelerationEvery economic valuation requires economic scenarios tables (see El[START_REF] El Karoui | Market inconsistencies of market-consistent european life insurance economic valuations: pitfalls and practical solutions[END_REF] for more insights about economic valuation methodology and[START_REF] Vedani | Solvency assessment within the ORSA framework: issues and quantitative methodologies[END_REF] for its practical standard 9/26 and approximated implementation approaches). The practical methods to assess all type of values (be it asset, liability best estimate or own funds) are very similar. For this reason we focus our discussion on economic own funds (or Net Asset Values) and more precisely on the Value of In-Force (V IF), that is the part of own funds (OF = V IF + RNA [the Revalued Net Asset]) obtained through Monte-Carlo estimation of the averaged Net Present Value of future margins cash-flows (denoted by NPV below).

  4.2.2 Implementation 2: V IF α economic situation vs. an updated El Karoui et al. (2017) calibration approach

Table 1 :

 1 Comparison between the obtained economic own funds values -2014 results

	October 14' Oct. & Nov. 14' 12/31/14 v1 12/31/14 v2
	portfolio 1 16,898	15,614	7,046	10,000
	portfolio 2 12,826	12,283	9,517	10,000
	portfolio 3 12,553	12,073	6,050	10,000

  Insurance and reinsurance undertakings shall set assumptions on future financial market parameters or scenarios that are appropriate and consistent with Article 75 of Directive 2009/138/EC. Where insurance and reinsurance undertakings use a model to produce projections of future financial market parameters, it shall comply with all of the following requirements: a it generates asset prices that are consistent with asset prices observed in financial markets;

	supplementing directive Solvency II,
	some straightforward notification
	SECTION 3.1 -Methodologies to calculate technical provisions -Assumptions underlying the cal-
	culation of technical provisions -Article 22 (General provisions) paragraph 3 (European Commission
	(2015a)),

b it assumes no-arbitrage opportunity; c the calibration of the parameters and scenarios is consistent with the relevant risk-free interest rate term structure used to calculate the best estimate as referred to in Article 77(2) of Directive 2009/138/EC. [Solvency II]

Table 2 :

 2 Comparison between the approximated marginal SCR values -same seed vs. different seeds with 95% confidence intervals

	Spread SCR	Central V IF	Shocked V IF Marginal SCR ρ
	Different seeds	11,289 (± 369) -441 (± 408)	11,730 (± 553) ρ = -0.7%
	Same seed	11,289 (± 369) -448 (± 416)	11,737 (± 142) ρ = 94.2%
	Stock SCR	Central V IF	Shocked V IF Marginal SCR ρ
	Different seeds	11,289 (± 369) 3,789 (± 377) 7,500 (± 525)	ρ = 1.0%
	Same seed	11,289 (± 369) 3,831 (± 382) 7,458 (± 104)	ρ = 96.3%
	Real estate SCR Central V IF	Shocked V IF Marginal SCR ρ
	Different seed	11,289 (± 369) 8,724 (± 360) 2,565 (± 512)	ρ = 1.4%
	Same seed	11,289 (± 369) 8,618 (± 376) 2,671 (± 54)	ρ = 98.9%

The specifications for the determination of the UFR's level have been released in

("EIOPA-BoS-17/072") and can be retrieved on EIOPA's website: https://eiopa.europa.eu/Publications/Reports/Specification%20of%20the% 20methodology%20to%20derive%20the%20UFR.pdf

See the paper for more information about the portfolios tested.

The economic own funds valuation presents the same issues as for the economic liabilities Best Estimate as the market value of the Asset part in the economic balance sheet is obvious.

This is one of the major theoretical issue for its use in life insurance liabilities valuation, tools that are simply neither sold on complete markets nor hedgeable / hedged.

In reality the full aggregated SCR is a bit more complex but we chose to focus on an elliptic aggregation for the sake of simplicity.

The estimation actually depends on the choice of the reference date but the numerical evidence displayed in the article are here to illustrate the effect without any objective of exhaustivity rather than a thorough study of the reference date choice.

Note that in this section we focus on recent yield curves including negative interest rates. For this reason we use a Displaced Diffusion Libor Market Model (see in particular[START_REF] Joshi | A stochastic-volatility, displaced-diffusion extension of the libor market model[END_REF]) to model interest rates and calibrate these models on normal/Bachelier swaptions IV (see e.g.[START_REF] Hohmann | The new normal[END_REF]). These choices are realistic considering current actuarial practices.

It seems possible to use v s = s, ∀s ∈ 1; 50,000 (same seed) in particular when the number of simulations (50, 000 here) is high enough.

The standard and shocked OF estimators being estimated using different simulation seed/economic scenarios, the respective NPV are independent and so is their average.

The approximation obtained using the delta method on SR β ,i .

Appendix 2 -concerning solvency ratios

We have proposed above different approaches to both reduce the variance of the OF/V IF and the SCR. Now we also address the solvency ratio (SR = OF SCR or using our both approaches and implicit notations, SR CV = OF CV SCR CV,s.s ) issue. This ratio is in many ways the most important quantity used by practitioners and control authorities to evaluate and compare solvency among companies.As aforementioned, the variance of the OF is basically the one of the V IF (OF = RNA +V IF and the RNA is deterministic).

First, it appears natural to couple the approach on V IF (CV) and the one on SCR ( s.s ) while estimating, so that the user can obtain optimised estimators. Then, we still consider situation β as the valuation time economic situation, so that we use notation

In all generality it seems very difficult to theoretically prove our full-optimised estimator SR . However, we have obtained some first results considering a marginal SCR as the solvency ratio denominator. This leads to a simpler/marginal solvency ratio

(still considering risk i). We now compare the variances of an optimised

estimator.

Note first the three following points 11 , 11 We assume below, for the sake of simplicity, the aforementioned bias (see 4.3) impact is negligible.

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so that,

We can then use the well-known asymptotic delta-method approximation (see [START_REF] Stuart | Kendalls Advanced Theory of Statistics Arnold[END_REF], [START_REF] Elandt | Survival models and data analysis[END_REF]) for the variance of our marginal solvency ratio,

Let us now assume (weak assumptions, from our work),

In particular, CoVar OF CV β ,i ; OF CV β ≈ 0 12 . We therefore obtain,

CoVar OF β ,i ; OF β . 13 We can finally assume (up to our delta method approximations and other weak assumptions) that, with our CV approach we are supposed to decrease the marginal solvency ratio estimator variance,

Further developments (including at least empirical tests) will be required to better qualify our method impact on the solvency ratios precision.