Francesco Mascia

Sara B Pereira

Catarina C Pacheco

Paulo Oliveira

Jennifer Solarczek

Anett Schallmey

Robert Kourist

Véronique Alphand

Paula Tamagnini

PCC 6803 expressing the heterologous CYP450 monooxygenase CYP110D1

Introduction

Cyanobacteria have a great potential to be used as whole-cell biocatalysts for the light-driven oxidation [1][2][3][4] or reduction of a variety of substrates, with activities close to what is required by industry. [5][6][7][8] Therefore, cyanobacteria are an appealing solution to perform chemical reactions more sustainably, helping to mitigate the impact that chemical and pharmaceutical industries have on the environment. The potential of these microorganisms as biocatalysts is mainly related to their photoautotrophic metabolism, which allows them to grow using light as energy and CO2 as a carbon source. Moreover, their photosynthetic apparatus can be tapped to provide reducing equivalents resultant from the light-dependent water oxidation reactions to sustain the activity of endogenous or recombinant redox enzymes. [START_REF] Nakamura | Chemical Communications[END_REF][START_REF] Sengupta | [END_REF] Most of the reports demonstrating the feasibility of this engineering approach used the model unicellular cyanobacterium Synechocystis sp. PCC 6803, owing to its well-known physiology and the availability of established molecular and synthetic biology tools. [11][12][13][14] Among the various classes of redox enzymes, those belonging to the cytochrome P450 (CYP450) monooxygenase superfamily are interesting for several applications, being used as biocatalysts in the production of pharmaceuticals, fine or bulk chemicals, bioremediation agents, flavors, and fragrances. 3,15 CYP450s are known to hydroxylate non-activated C-H groups using molecular oxygen as a co-substrate, under mild conditions, and in a regio-and stereo-selective way, with the aid of two electron-carriers, e.g. a NAD(P)H-dependent ferredoxin reductase and a ferredoxin. 16,17 CYP450s are particularly suitable to be expressed in cyanobacteria, compared to heterotrophic microorganisms, as the co-expression of the CYP450 electron carriers to sustain the monooxygenase activity may not be required since the photosynthetic apparatus could transfer electrons derived from water oxidation to endogenous ferredoxins, which in turn could provide them to the enzymes. 18,19 Consequently, it will not be needed to exogenously supply stoichiometric amounts of sacrificial electron donors (e.g. glucose) for the cofactor regeneration. 2 Furthermore, molecular oxygen, the fundamental co-substrate for CYP450 biocatalysis, is produced insitu as a by-product of photosynthesis, thus overcoming air-liquid mass transfer limitations. 20 Steroid drugs are an important class of pharmaceutical molecules used as anti-inflammatories, immunomodulators, and physiological regulators, with an estimated market of more than 4 billion USD per year. [START_REF] Company | Corticosteroids Global Market Report[END_REF] The selective hydroxylation of steroid carbon backbones can alter their physicochemical properties, leading to novel bioactive features, e.g. the hydroxylation in position 15 of the androgenic hormone testosterone was demonstrated to improve its oral bioavailability without affecting liver metabolism. 22 This molecule can become a good candidate for the development of oral formulation, overcoming the conventional administration procedure of injecting testosterone undecanoate intramuscularly. [START_REF] Nieschlag | [END_REF] However, hydroxylation of steroids under synthetic chemistry conditions is often a challenging process, due to the difficulty of selectively oxidizing target non-activated carbon atoms. This may also require several synthetic steps and the extensive use of protecting groups, solvents, and toxic reagents, increasing the process costs and the environmental impact. 24 An alternative route to hydroxylate steroids is the use of highly specialized enzymes available in nature, such as a variety of hepatic 25 and microbial enzymes 26 that are able to selectively hydroxylate testosterone. In a previous study, a library comprised of 213 different bacterial CYP450 genes were expressed in Escherichia coli, and the resulting strains were screened for their ability to use testosterone as a substrate. 27 Twenty-four CYP450s were able to mono-hydroxylate testosterone with high selectivity and specificity, and among them, CYP110D1 from the filamentous cyanobacterium Nostoc sp. PCC 7120, was able to convert the substrate into 15β-hydroxytestosterone. 27 This study aimed at investigating the potential of cyanobacteria as sustainable whole-cell biocatalysts for the selective hydroxylation of the androgenic steroid testosterone. Since the heterologous expression of CYP450s 3,19 was successfully achieved in Synechocystis sp. PCC 6803, the gene alr4766 (encoding the enzyme CYP110D1 from Nostoc sp. PCC 7120) was introduced into this model cyanobacterial strain under the control of different regulatory elements and the engineered strains were tested for the conversion of testosterone into 15β-hydroxytestosterone. Subsequently, the conditions promoting this reaction without the addition of sacrificial electron donors were optimized, demonstrating the potential of engineered Synechocystis cells for the selective light-driven hydroxylation of testosterone compared to whole-cell heterotrophic biocatalysts or to chemical synthesis.

Experimental Organisms and culture conditions

The cyanobacterium Synechocystis sp. PCC 6803 substrain GT-Kazusa 28,29 and Synechocystis mutant strains generated in this work were cultivated in BG11 medium, 30 supplemented with kanamycin (Km, 50 µg mL -1 ) in the case of the mutants, at 30 °C and under a 12 hours light (30 μmol photons m -2 s -1 )/ 12 hours dark regime. Cultures were routinely maintained in 40 mL of BG11 in 100 mL Erlenmeyer flasks, with orbital shaking at 150 rpm. For cultivation on solid medium, BG11 was supplemented with 1.5 % (w/v) noble agar (Difco), 0.3 % (w/v) sodium thiosulfate and 10 mM TES-KOH buffer, pH 8.2. For cloning purposes, E. coli strains DH5α (Stratagene) or XL1-Blue (Agilent) were used. For protein expression purposes the E. coli strain C43 (DE3) was used. Cells were grown at 37 °C in LB or TB medium, supplemented with ampicillin (Amp, 100 µg mL -1 ), kanamycin (Km, 50 µg mL -1 ), chloramphenicol (25 μg mL -1 ), or tetracycline (10 μg mL -1 ).

Assembly of the CYP110D1 modules

The open reading frame alr4766 encoding CYP110D1 was amplified by PCR from genomic DNA of Nostoc sp. PCC 7120 (obtained from the Institute Pasteur, France) using primers NS-P-NdeI-FW and NS-P-HindIII-RV (Table S1). The PCR reaction mixture contained 1 U of Phusion R High-Fidelity DNA Polymerase (NEB), 1X reaction buffer, 200 µM of each deoxyribonucleotide triphosphate, 0.2 µM of each primer, and 150 ng of genomic DNA. The PCR profile was: 5 min at 96 °C, followed by 30 cycles of 1 min at 96 °C, 30 s at 48 °C and 45 s at 72 °C, and a final extension at 72 °C for 10 min. Thereby restriction sites NdeI and HindIII were introduced for subsequent cloning of the gene into vector pIT2-MCS, generating plasmid pIT2::CYP110D1.

Please do not adjust margins

Please do not adjust margins Generation of vector pIT2-MCS and cloning of vector pACYC::camAB has been described previously. 31 To assemble the CYP110D1 modules to introduce in Synechocystis, alr4766 was amplified from the vector pIT2 either with primers CYP110D1_RBS0032_SpeI_FW and CYP110D1_PstI_RV, or with primers CYP110D1_His-tag_RBS0032_SpeI_FW and CYP110D1_PstI_RV (Table S1). The latter pair was designed to introduce a hexahistidine tag at the 5'-end of the sequence; in all cases, the respective open reading frames were preceded by the RBS BBa_B0032 (Registry of Standard Biological Parts, http://parts.igem.org; Table S1). The PCR reaction mixtures (50 µL) contained 1 U of Phusion R High-Fidelity DNA Polymerase (Thermo-Fisher Scientific), 1X reaction buffer, 200 µM of each deoxyribonucleotide triphosphate, 0.5 µM of each primer, and 10 ng of template DNA. The PCR profile was: 1 min at 98 °C, followed by 10 cycles of 20 s at 98 °C, 20 s at 53 °C and 45 s at 72 °C, 20 cycles of 20 s at 98 °C, 20 s at 65 °C and 45 s at 72 °C and a final extension at 72 °C for 5 min. Each of the amplicons was cloned into pJ201 vectors harboring the synthetic promoters PpsbA2* or Ptrc.x.tetO2, 12 between the SpeI and PstI restriction sites. Later, the synthetic modules constituted by the promoter, RBS, and coding sequence were excised with XmaI and PstI restriction enzymes and cloned into the selfreplicative vector pSEVA251 (Standard European Vector Architecture repository). 32 The correct sequence of the assemblies was confirmed by sequencing (STAB-VIDA).

Generation of the Synechocystis mutants

The synthetic modules generated (see above) were used to transform wild-type Synechocystis as previously described. 12 Briefly, cells were grown in BG11 medium until OD730 of ∼0.5. At that point, cells were pelleted at 4190 g for 10 minutes and washed three times with 10 mL of 1 mM HEPES buffer, pH 7.5. The cells were finally suspended in 1 mL of 1 mM HEPES and 60 µL aliquots were mixed with 1 µg of purified plasmid DNA (in deionized water), in an electroporation cuvette. The electroporation was performed using a Gene Pulser TM (Bio-Rad), with the capacitor set to 25 µF and the resistor to 400 Ω, for a time constant of 9 ms with an electric field of 12 kV cm -1 . After the electric pulse, 400 µL of BG11 were added to the cells, which were then spread onto Immobilon-NC membranes (0.45 µm pore size, 82 mm, Merck Millipore) laying on solid BG11 plates. After incubation at 30 °C under continuous light for 24 hours, the membranes were transferred to solid BG11 plates supplemented with kanamycin and incubated in the same conditions described previously. Colonies were observed after 2-3 weeks. The intracellular presence of the replicative plasmids was confirmed by colony-PCR using specific primers for the gene and the plasmid backbone (Table S1). To that end, Synechocystis colonies were collected from the BG11 agar plate and suspended in 50 µL of sterile water. The PCR reaction mixture (20 µL) contained 1 U of GoTaq® G2 Flexi DNA Polymerase (Promega), 1X reaction buffer, 2 mM MgCl2, 200 µM of each deoxyribonucleotide triphosphate, 0.5 µM of each primer (Table S1), and 12 µL of the cell suspension as a template. The PCR profile was: 5 min at 95 °C, followed by 30 cycles of 30 s at 95 °C, 30 s at 55 °C and 70 s at 72 °C, and a final extension at 72 °C for 5 min.

Growth measurements (OD), chlorophyll a and dry weight quantification

For growth experiments, Synechocystis harboring the empty pSEVA251 vector or expressing the gene encoding CYP110D1 under the control of either PpsbA2* or Ptrc.x.tetO2 promoters were inoculated in 20 mL of BG11 supplemented with 50 µg mL -1 of kanamycin at OD730 of 0.5 and grown for 7 days under the cultivation conditions described above until an OD730 of ∼2 was reached. Afterward, cells were diluted back to OD730 of 0.5 and 200 µL were transferred to a 96-well microtiter plate that was kept under the same growth conditions. Growth was monitored every day for seven days, using a Synergy TM 2 (Agilent) microplate reader measuring the optical density at 790 nm as previously described. 33 For this experiment, three independent biological replicates and four technical replicates were performed. For routine measurements of the growth of Synechocystis cultures, the optical density was measured at 730 nm 34 using a Shimadzu UVmini-1240 (Shimadzu Corporation) and the chlorophyll a content as described previously. 5 The dry weight was also quantified. For that, cells were grown to an OD730 ∼3 and harvested by centrifugation at 4190 g, for 10 minutes at RT. After suspending the pellet in BG11 medium to reach an OD730=10, 2 mL samples were centrifuged as described above. The resulting pellets were dried overnight at 80 °C and weighed on an analytical scale. The cell dry weight was expressed in g L -1 . For this experiment, three independent biological replicates and two technical replicates were performed.

SDS-PAGE and Western Blot

To evaluate the expression of CYP110D1, Synechocystis mutant strains were inoculated in BG11 supplemented with 50 µg mL -1 of kanamycin at OD730 of 0.5 and grown for 7 days under the cultivation conditions described above until an OD730 of ∼2 was reached. Subsequently, these precultures were used to inoculate new cultures to an OD730 of 0.5, which were cultivated under the same conditions for 10 days, until an OD730 of ~3. At that point, cells were harvested, pelleted by centrifugation, and suspended in lysis buffer (10 mM HEPES pH 8.0, 10 mM EDTA pH 8.0, 0.5 % (v/v) Triton X-100, 2 mM DTT) supplemented with EDTA-free protease inhibitor cocktail (Roche), before being disrupted by sonication as described previously. 35 After removing cell debris by centrifugation (16000 g, 10 minutes, at 4 °C), proteins in the supernatant were quantified using the BCA protein assay (Thermo Fisher Scientific). For protein analysis, cell extracts were denatured in reducing SDS sample buffer [62.5 Mm Tris-HCl, pH 6.8, 2 % (w/v) SDS, 10 % (v/v) glycerol, 5 % (v/v) β-mercaptoethanol, 0.002 % (w/v) bromophenol blue] at 95 °C for 5 minutes. 10 or 20 µg of total protein extract were separated by electrophoresis on 12 % or 10 % (w/v) SDS-polyacrylamide gels. 36 The proteins were visualized with PageBlue TM staining solution (Thermo-Fisher Scientific) or blotted onto a nitrocellulose membrane with pore size 0.45 µm (Amersham TM Protan TM , GE Healthcare). The transfer of the proteins from the SDS gel to the nitrocellulose membrane was performed within Towbin buffer (25 mM Tris base, 192 mM L-glycine, 20 % methanol), applying 100 V for 1 hour. Membranes were probed using a mouse anti-6X-His-Tag (Invitrogen) and an HRP-conjugated anti-mouse IgG a (Sigma) antibodies with a dilution of 1:2000 and 1:5000, respectively. Immunodetection was performed using Clarity TM Western ECL Substrate (Bio-Rad), and This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins images were acquired with a ChemiDoc TM XRS + Imager (Bio-Rad). The relative signal intensity of the bands obtained was quantified using the Image Lab software (Bio-Rad). Data were statistically analyzed as described below. In this experiment, three independent biological replicates were performed.

Light-driven biotransformations with Synechocystis

Cells of Synechocystis mutants were first inoculated in BG11 supplemented with 50 µg mL -1 of kanamycin at OD730 of 0.5 and grown for 7 days under the cultivation conditions described above until an OD730 of ∼2 was reached. Subsequently, these were used to inoculate new cultures to an OD730 of 0.5 and cultivated under the same conditions for 10 days. Afterward, cells were collected by centrifugation (4190 g, 10 minutes, at room temperature) and suspended in antibiotic-free BG11 medium to a final OD730 of 5 or 10 (0.9 or 1.8 gCDW L -1 ). Part of the concentrated cell suspension was used for dry weight and chlorophyll a measurements, and the rest was used for biotransformation experiments. The reaction was initiated by mixing within a 10 mL screw-cap glass vial, 2 mL of cells with 25 µL of testosterone stock solution (40, 80, or 160 mM in DMSO) to reach a final concentration of 0.5, 1 or 2 mM. For each condition tested and every sampling time point, separated reaction vials were used. The reactions were incubated in a photoreactor, providing cool white light (6000-6500 K) with an intensity of 150 µmol photons m -2 s -1 and magnetic stirring. Reactions in the dark were performed by covering the vials in aluminum foil and keeping them under the same conditions described above. When required, aeration was provided by removing the cap from the vials. Sampling was performed by thoroughly mixing the respective cell suspension with an equal volume of ethyl acetate, drying the resulting organic phase in a vacuum concentrator (SpeedVac), and finally suspending the dried residue in acetonitrile:H2O (60:40). After centrifugation at 14000 g for 10 minutes at room temperature, the supernatants were stored at 4 °C until analysis.

HPLC analysis was performed with a Hitachi L-7000 HPLC system (Merck-Hitachi) with a single wavelength UV detector set at 240 nm, equipped with a LiChrospher 100 RP-18 LiChroCART 250 column (250 mm length, 4 mm inner diameter, 5 µm particles size) (Merck). Each sample (10 µL) was chromatographed with a constant acetonitrile concentration in water in a ratio 60:40 (v/v) for 7 min at a flow rate of 1 mL min -1 at 40 °C. For this experiment, three independent biological replicates and two technical replicates were performed.

Expression of CYP110D1 together with Pdx and PdR in E. coli

For activity tests of CYP110D1, the enzyme was co-expressed in E. coli together with redox partners putidaredoxin (Pdx) and putidaredoxin reductase (PdR) from Pseudomonas putida DSM50198. 38 E. coli C43(DE3)(pACYC::camAB)(pIT2::CYP110D1) cells were grown in 500 mL TB medium containing 25 μg mL -1 chloramphenicol, 10 μg mL -1 tetracycline and 0.1 % v/v trace element solution at 37 °C to an optical density at 600 nm (OD600) of 1. Then, expression was induced with isopropyl β-D-1 thiogalactopyranoside (IPTG, 200 µg mL -1 ), and δ-aminolevulinic acid (84 µg mL -1 ) was added as a precursor for heme synthesis. After 48 hours of expression at 30 °C, the cells were harvested by centrifugation (4424 g, 4 °C for 15 min) and washed once with 40 mM potassium phosphate buffer pH 7.4. Cell pellets were frozen at -20 °C until further use.

Testosterone bioconversion with E. coli cells

Frozen cells of E. coli C43 (DE3) (pACYC::camAB) (pIT2::CYP110D1), overexpressing Pdx and PdR together with CYP110D1, were suspended in 100 mM potassium phosphate buffer pH 7.4 to a final OD600 of 40 (corresponding approximately to 13.8 gCDW L -1 ). Wholecell reactions were carried out in a 1 mL scale at 30 °C and 200 rpm with 1 mM testosterone [from a 100 mM stock solution in 36 % (w/v) hydroxypropyl-β-cyclodextrin in deionized H2O] and 30 mM glucose for cofactor regeneration. After 1 hour, the reaction was extracted with the same volume of ethyl acetate and the solvent was evaporated. The dried residue was dissolved in 200 µL acetonitrile:H2O (60:40) and injected on an HPLC (Prominence, Shimadzu) equipped with a Nucleosil 100-5 C18, 250 x 4,5 mm column (Macherey-Nagel) at 40 °C. Acetonitrile:H2O (60:40) was used as a mobile phase with a flow rate of 1 mL min -1 . Detection of substrate and product was performed by UV absorbance at 254 nm. The substrate testosterone and the corresponding hydroxylation product (15β-hydroxytestosterone) were eluted at retention times of 5.9 min and 2.9 min. For this experiment, three independent biological replicates were performed.

Oxygen evolution measurements

The oxygen evolution rates were determined using a Clark-type O2electrode (Oxygraph; Hansatech Ltd.). Calibration was performed using sodium bisulfite and air-saturated water at 30 °C. Cells were grown as described above, harvested, and suspended in fresh BG11 medium to an OD730 of 10 (1.8 gCDW L -1 ). After incubating 1 mL of concentrated cells in the dark for 1 minute, the assay was started by illuminating the samples for 2 minutes with cool white light (6000-6500 K) with an intensity of 120 µmol photons m -2 s -1 . After this period 12.5 µL of DMSO or of 80 mM testosterone dissolved in DMSO, were added to the cell suspension, reaching a final substrate concentration of 1 mM. The decrease of the oxygen evolution rate upon the addition of the chemicals was expressed as a percentage of the maximum evolution rate measured during light exposure. For this experiment, three independent biological replicates and two technical replicates were performed.

Atom economy calculation

The atom economy of the reaction was calculated as described previously. 39

Statistical analysis

The statistical analysis was performed using an unpaired t-test, using the software GraphPad Prism 5 (GraphPad Software Inc.).

Results and discussion

Expression of the heterologous CYP450 monooxygenase CYP110D1 in Synechocystis

Please do not adjust margins Please do not adjust margins Previously, the selective hydroxylation of testosterone into 15βhydroxytestosterone mediated by an E. coli co-expressing the CYP450 monooxygenase CYP110D1 (alr4766) from Nostoc sp. PCC 7120, and the electron carriers putidaredoxin (Pdx; camB) and putidaredoxin reductase (PdR; camA) from Pseudomonas putida DSM50198 was reported. 27 This reaction was sustained by reducing equivalents originating from the heterotrophic catabolism of glycerol, which was supplemented, in the reaction medium, as a sacrificial electron donor. Here, aiming at performing this hydroxylation using reducing equivalents and molecular oxygen originating from the photosynthetic oxidation of water, the model cyanobacterium Synechocystis sp. PCC 6803 (hereafter Synechocystis) was selected for the heterologous expression of CYP110D1. For this purpose, the gene encoding CYP110D1, in its native form or with the coding sequence of a hexahistidine-tag (Histag) at its 5' end, was inserted together with the ribosomal binding site (RBS) BBa_B0032 downstream of the synthetic constitutive promoters PpsbA2* or Ptrc.x.tetO2 (Fig. 1a), since this combination of regulatory parts has been previously successfully used in Synechocystis. 12 Each of these synthetic modules was cloned into the self-replicating vector pSEVA251, which was used to transform Synechocystis by electroporation. The presence of the synthetic modules in the mutant strains was confirmed by colony PCR (Fig. S1). Then, protein extracts were obtained from the different Synechocystis strains, and an SDS-PAGE gel stained with PageBlue revealed the presence of an additional band in Synechocystis expressing CYP110D1 under the control of Ptrc.x.tetO2 promoter, that was not present in the strain harboring the empty pSEVA251 (Fig. 1b-S2). Under the control of PpsbA2*, it was not possible to observe a distinct band. To further confirm the presence of the heterologous protein CYP110D1, a Western blot analysis using a primary antibody recognizing the His-tag was performed. This method allowed the detection of the tagged CYP110D1 even when under the control of PpsbA2* (Fig. 1c). Moreover, under the regulation of Ptrc.x.tetO2, the level of CYP110D1 was approximately 2-fold higher compared to PpsbA2* (Fig. 1d). This is in agreement with previous studies on the strength of these two promoters that are 6-(PpsbA2*) and 21-fold (Ptrc.x.tetO2) stronger than the cyanobacterial reference PrnpB. 12, 40

Light-driven biotransformation of testosterone mediated by

Synechocystis expressing CYP110D1

Once the presence of CYP110D1 within the mutant strains was confirmed, a bioconversion reaction was set up to evaluate the efficiency of this enzyme for hydroxylation of testosterone (androst-4-en-17β-ol-3-one). For this purpose, three reaction vials containing either (i) 2 mL of cell-free BG11 medium, (ii) 2 mL of concentrated culture (OD730=5; 0.9 gCDW L -1 ) of Synechocystis harboring the empty pSEVA251, or (iii) 2 mL of concentrated culture (OD730=5; 0.9 gCDW L - 1 ) of Synechocystis expressing CYP110D1 under the control of PpsbA2*, plus the substrate (0.5 mM testosterone), were incubated for 21 hours in a homemade photoreactor equipped with white LEDs (150 µmol photons m -2 s -1 ) and a stirring system, placed inside a growth chamber with controlled temperature at 30 °C (Fig. S3a). The HPLC analysis showed the appearance of a product peak only when Synechocystis cells expressing CYP110D1 were present in the This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins reaction, demonstrating that the conversion was indeed dependent on the presence of the enzyme (Fig. S3b). The product formed was identified unambiguously as 15β-hydroxytestosterone (15-beta, 17beta-dihydroxyandrost-4-en-3-one), having the same mass fragmentation spectra (Fig. S5), and retention time in both HPLC and GC analysis (Fig. S3b andS4), as the commercial standard of reference (provided by Steraloids Inc.) and by comparison with 1 H and 13 C NMR data reported in literature (Table S2 andS3; Fig. S6, S7 and S8). 37 The formation of 15α-hydroxytestosterone was excluded after comparison with the previously described 1 H spectrum 41,42 of this compound, and notably the chemical shifts of H17 and H19 as shown Table S3. Altogether, these results indicate that testosterone was selectively converted by Synechocystis wholecells harboring the heterologous CYP110D1 into the desired product. Interestingly, and consistently with what was previously observed with two other CYP450s (namely CYP79A1 and CYP77E1), CYP110D1 exhibited its biocatalytic activity without the need to co-express electron carriers and relying only on Synechocystis native ferredoxins, that could be reduced either by photosystem I or by endogenous ferredoxin reductases. 19

Testosterone hydroxylation by Synechocystis whole-cells harboring different modules for the expression of CYP110D1

After confirming the ability of Synechocystis whole-cells expressing CYP110D1 to hydroxylate testosterone into 15βhydroxytestosterone, several parameters were evaluated aiming at optimizing this process. To investigate whether the amount of enzyme was a limiting factor, Synechocystis cells expressing the untagged CYP110D1 under the control of the stronger promoter Ptrc.x.tetO2 were tested. As shown in Fig. 2a, the cells of this mutant converted testosterone 3.5-fold more efficiently, achieving 57 % substrate conversion, while those with PpsbA2* reached only 16 % conversion, after 21 hours. Consistently, the amount of product reached 0.23 mM and 0.06 mM for cells harboring the module under the control of Ptrc.x.tetO2 and PpsbA2*, respectively (Fig. 2b). However, the sum of substrate and product was lower than expected when the biotransformation was catalyzed by cells expressing CYP110D1 under the control of the stronger promoter Ptrc.x.tetO2 (also noticeable in Fig. 3, Fig. 4 and Fig. S13). As incubating the purified product in light with cells harboring the empty pSEVA251 did not cause any change in the product concentration after 21 hours (Fig. S9), this might be due to the generation of small amounts of side-products or to the product itself being accepted as a substrate by CYP110D1 and be slowly consumed over time. This has been previously reported for other CYP450 monooxygenases, e.g. with Synechocystis expressing the CYP450 CHX from Acidovorax sp. CHX100, the product cyclohexanol was further oxidized to cyclohexanone, but at a slower rate. 3 Expressing CYP110D1 under the control of the stronger promoter led to a minor decrease in growth and chlorophyll a content compared to the strains harboring the empty pSEVA251 or expressing the same enzyme under the control of PpsbA2* (Fig. S10 andS11). This decrease might be due to a small metabolic burden caused by CYP110D1, either directly interfering with cellular processes or generating toxic reactive oxygen species through uncoupling reactions. 43 Once we verified that the His-tagged version of CYP110D1 was active (data not shown), the strains expressing this enzyme variant under the control of the two different promoters were also tested and compared. After 21 hours, conversions of 44 % and 12 % were achieved with cells expressing CYP110D1 under the control of Ptrc.x.tetO2 and PpsbA2*, with the former having a 3.7-fold higher conversion (Fig. S12). These results are in line with the higher intracellular accumulation of untagged CYP110D1 and His-tag CYP110D1 under the control of Ptrc.x.tetO2 observed previously (Fig. 1b andS2). Overall, the tagged version had approximately 25 % less activity than the native enzyme, therefore Synechocystis expressing the untagged CYP110D1 under the control of Ptrc.x.tetO2 was selected to be used in the next experiments.

Please do not adjust margins

Please do not adjust margins

Optimization of the cell density in the reaction mixture Subsequently, the correlation between the reaction mixture cell density and the efficiency of testosterone conversion was investigated. For that, Synechocystis cells expressing CYP110D1 under the control of Ptrc.x.tetO2 were grown under standard conditions and concentrated until OD730 5 or 10 (0.9 or 1.8 gCDW L -1 ), before being used in the bioconversion assays. The results obtained showed that doubling the cell concentration had a proportional effect on the conversion rate, which raised from 27 % to 52 % already after 8 hours (Fig. 3a). Consistently with the higher conversion rate, the product formation also increased from 0.12 mM to 0.22 mM (Fig. 3b). After 21 hours of incubation, conversions of 53 % and 79 % were observed for reaction mixtures at OD730 5 or 10 (0.9 or 1.8 gCDW L -1 ), respectively (Fig. 3a), corresponding roughly to an increase of about 1.5-fold. This smaller difference could be attributed to the depletion of a reaction component, slowing down the biotransformation after a certain time point. Further cell density was not evaluated, as it was already known that it would lead to a selfshading effect, thus decreasing the overall efficiency of the system. 1

Light-dependence of testosterone hydroxylation

As the aim of this study was to perform the biotransformation using the reducing power provided by the water oxidation mediated by the photosynthetic electron flow, we evaluated if the reaction was indeed light-dependent. When Synechocystis cells expressing CYP110D1 under the control of Ptrc.x.tetO2 were kept in the dark for 21 hours the bioconversion of testosterone dropped abruptly compared to cells exposed to light (Fig. 4a). This result is corroborated by the analysis of product formation (Fig. 4b), indicating that reducing power and/or the molecular oxygen required to fuel the reaction is mainly provided by the photosynthetic electron flow. Nevertheless, 16 % of the substrate was converted under dark conditions (Fig. 4), suggesting that other sources of reducing equivalents, like the storage compounds glycogen or polyhydroxybutyrate, can be catabolized for this purpose. Please do not adjust margins Please do not adjust margins 15β-hydroxytestosterone (mM) in the reaction mixtures after 21 hours incubation. Reactions were performed by incubating 0.5 mM testosterone in 2 mL reaction volume, with cells concentrated to OD730=10 (1.8 gCDW L -1 ) in BG11 medium, at 30 °C, in closed vials. Error bars represent the standard error of the mean (SEM) of 3 biological replicates and 2 technical replicates (*** p-value < 0.001).

Influence of aeration and supplementation with inorganic carbon on testosterone hydroxylation

As all monooxygenases, CYP110D1 needs molecular oxygen as a cosubstrate to carry out its biocatalytic activity. Although the experiments described above were performed without aeration, with molecular oxygen being generated in-situ through photosynthesis, the hypothesis of whether this co-substrate availability may limit CYP110D1 activity was investigated. For this purpose, testosterone was incubated with Synechocystis expressing CYP110D1 under the control of Ptrc.x.tetO2 in either closed or open vials. Higher substrate conversion and increased product formation were obtained with the open vials (Fig. S13), confirming the importance of aeration for the bioconversion reaction. To verify whether molecular oxygen was indeed the reason for the increased activity observed, cultures of Synechocystis expressing CYP110D1 or harboring the empty pSEVA251 vector were concentrated to OD730=10 (1.8 gCDW L -1 ) and the amount of oxygen dissolved in solution was monitored with an oxygen electrode. Net oxygen evolution rate was measured before and after the addition of testosterone dissolved in DMSO (substrate) or DMSO only (control). The initial oxygen evolution rate decreased by about 25 % after the addition of the solvent or, independently of the presence of CYP110D1 (Fig. 5). These results suggest that the decrease is likely due to an effect caused by DMSO. Furthermore, the net oxygen evolution rate detected indicates that O2 is not a limiting factor in the hydroxylation of testosterone under the conditions tested. A possible reason for the increased substrate conversion could be the availability of CO2 for cells with aeration. As previously reported for different cyanobacterial strains, including Synechocystis, in closed systems under light, CO2 can be depleted rather quickly. [START_REF] Allahverdiyeva | Proceedings of the National Academy of Sciences[END_REF][START_REF] Lindberg | [END_REF][47] CO2 fixation is a major electron sink, and under high-light and carbon limiting conditions an overexcitation of the photosynthetic apparatus may occur, eventually leading to the production of reactive oxygen species and photoinhibition. 48 In a previous study, the addition of NaHCO3 in the reaction mixtures resulted in a 25 % increase in the specific activity of Synechocystis expressing the operon AlkBGT, encoding an alkane monooxygenase and its electron partners. 2 Here, the addition of 50 mM NaHCO3 to closed vials significantly increased testosterone bioconversion and product formation after 8 hours, reaching 83 % bioconversion, compared to 57 % and 50 % in standard BG11 and buffered BG11, respectively (Fig. S14). Therefore, supplementing inorganic carbon or applying aeration may be ways to prevent photodamage and to maintain the electron flux needed for the activity of heterologous monooxygenases.

Influence of substrate concentration on testosterone hydroxylation

The influence of the substrate concentration on the light-driven testosterone hydroxylation catalyzed by Synechocystis expressing CYP110D1 under the control of the Ptrc.x.tetO2 promoter was evaluated. For this purpose, cells concentrated to OD730=10 (1.8 gCDW L -1 ) were incubated in open vials with 0.5, 1, or 2 mM of testosterone.

After 21 hours, it was possible to observe 90 %, 90 %, or 61 % conversion of testosterone, respectively (Fig. 6a). Full conversion was never observed, probably due to the substrate hydrophobicity. In fact, a small fraction of it adhered to the vials' glass as a white precipitate above the reaction mixture, preventing it from getting in contact with the cells. Despite the lower % of conversion observed Please do not adjust margins Please do not adjust margins with 2 mM after 21 hours, it is highly probable that a higher % of conversion will be achieved by extending the reaction time (Fig. 6a).

In order to determine the maximum catalytic capacity of our whole-cell biocatalyst, the initial specific activity was measured. By comparing this value for the substrate concentrations tested, a 21 % increase (from 0.85 to 1.03 U gCDW -1 ) was observed when doubling the testosterone concentration from 0.5 mM to 1 mM. However, a further increase in the substrate concentration to 2 mM did not significantly change this parameter, indicating that the reaction reached a plateau (Fig. 6c). Moreover, the steady specific activity observed with 2 mM excludes the possibility that testosterone had toxic or inhibitory effects on our whole-cell biocatalyst under the conditions tested. An analogous effect was observed with the reaction productivity (Fig. 6d). Nevertheless, the highest product titer was achieved with 2 mM substrate after 21 hours, reaching 0.97 mM, with a 72 % yield (Fig. 6b).

As mentioned above, with 0.5 mM substrate it was possible to observe a slight decrease in the product concentration after 21 hours compared to the previous time point (Fig. 6b), supporting our hypothesis that the product itself may be accepted as a substrate by CYP110D1. As shown previously, this problem can be overcome by applying a two-liquid phase system to sequester the product before it could be further converted. 3 In this study, the application of a twoliquid phase system was investigated by testing, as an organic overlay, the solvents: di-isononyl phthalate (DINP) or dodecane that were previously shown to be well tolerated by Synechocystis 3, 49 or ethyl acetate. However, DINP and dodecane failed to efficiently solubilize testosterone and ethyl acetate showed toxic effects on the cyanobacterial cells. Therefore, the possibility to use a two-phase liquid system was not further explored.

Testosterone hydroxylation by photoautotrophic or chemoheterotrophic whole-cell biocatalysts

The hydroxylation of testosterone is generally considered a complex reaction. 24 Up to date, only a few examples of whole-cell biocatalysis This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins were reported for the 15β-hydroxylation of this substrate, all carried out using heterotrophic bacteria like Escherichia coli or Pseudomonas putida expressing the enzymes CYP109B1 or CYP106A2. [50][51][52] In these works, although using high cell concentrations and high loads of sacrificial electron donors, low substrate conversions, and sometimes not optimal selectivity were reported (for details see Table 1). [50][51][52] In order to compare our engineered Synechocystis expressing CYP110D1 with a heterotrophic whole-cell biocatalyst, an E. coli strain expressing the same CYP450 together with putidaredoxin and putidaredoxin reductase was also generated and tested. For this purpose, 1 mM testosterone was incubated with the aforementioned E. coli strain, concentrated to OD600=40 with 30 mM glucose as a sacrificial electron donor. A substrate conversion of 42 % was achieved after 1 h (Fig. S15 andS16), indicating a faster reaction rate compared to Synechocystis. Nonetheless, the maximum specific activity was consistently lower, with 0.51 U gCDW -1 for E. coli 1.03 U gCDW -1 for Synechocystis (Table 1). Therefore, the higher conversion rate observed with E. coli is most probably due to a higher whole-cell biocatalyst loading, which was 13.8 g L -1 for E. coli and 1.8 g L -1 for Synechocystis. Since the reaction catalyzed by our photoautotrophic strain is light-dependent, higher cell concentrations cannot be reached without addressing the light penetration issue. Therefore, the development of bioreactors with e.g. shorter light pathways such as flat-panel and/or internal illumination systems are mandatory to overcome this challenge. 53,54 Moreover, testosterone hydroxylation sustainability was also evaluated according to the principles of green chemistry. 39 The use of water as a sacrificial electron donor instead of glucose increased the reaction atom economy 55 from 61 % to 94 %, leading to a minimal waste production and facilitating the downstream processing. Photobiocatalytic testosterone hydroxylation was also considerably more sustainable than the chemical synthesis, not only in terms of waste production and atom economy. The chemical hydroxylation occurs in six synthetic steps, using 3β-androstenedione as a precursor and employing strong acids and bases, and toxic solvents such as dichloromethane. This process also requires derivatization with protecting groups to direct the hydroxylation selectivity, stoichiometric amounts of reagents and high energy inputs. 24 In contrast, testosterone hydroxylation with engineered Synechocystis was carried out in a single step, using DMSO (a non-toxic compound) as the only solvent, under mild conditions, with renewable and biodegradable feedstocks, i.e. cyanobacterial whole cells.

Conclusions

In this study, we demonstrated the potential of engineered Synechocystis cells for the selective light-driven hydroxylation of the steroid drug testosterone, using water as the sole sacrificial electron donor. Expressing CYP110D1 under the control of a stronger constitutive promoter (Ptrc.x.tetO2) resulted in higher protein levels and substrate conversion. Reaction parameters were also optimized, and the highest percentage of bioconversion was obtained by adjusting the cell density to OD730=10 (1.8 gCDW L -1 ) and by keeping the vials open (passive aeration). Under these conditions, the maximum specific activity of about 1 U gCDW -1 was reached with 1 mM of substrate. This value is 2-fold higher than the one obtained for E. coli expressing not only the CYP110D1 but also a putidaredoxin, a putidaredoxin reductase, and using glucose as a sacrificial electron donor. Furthermore, the reaction sustainability was improved, being the atom economy close to 100 %. Beyond the already promising results obtained with our photoautotrophic biocatalyst, there is still room for improvement. For instance, a more efficient electron supply could be achieved by Please do not adjust margins Please do not adjust margins using other Synechocystis-based chassis (e.g. Δflv1/Δflv3), 5 and/or by fusing the CYP450 with a ferredoxin, which will redirect reducing power towards the enzyme in a more efficient way. 56 The poor solubility of testosterone in aqueous media can be addressed by using other solvents/co-solvents, or with in-situ substrate supply via a two-liquid phase system. 3,16,51 Overall, we generated a greener and efficient cyanobacterial whole-cell biocatalyst for the challenging selective hydroxylation of steroid drugs, that could represent a valid alternative to the poorly selective and highly polluting chemical synthesis, or biocatalysis performed by heterotrophic microorganisms.

Fig. 1

 1 Fig. 1 CYP110D1 protein expression in engineered Synechocystis sp. PCC 6803 strains. (a) Schematic representation of the synthetic modules with the coding sequence under the control of the Ppsba2* or Ptrc.x.tetO2 promoter, the BBa_B0032 ribosome binding site, and without or with a sequence coding for a His-tag. Each module was introduced into pSEVA251 upstream the terminator λT0. (b) SDS-PAGE analysis of protein extracts isolated from Synechocystis mutants harboring the empty vector pSEVA251 or the synthetic modules encoding the CYP110D1 His-tag under the control of the Ppsba2* or Ptrc.x.tetO2 promoter. (c) Western blot analysis of protein extracts from Synechocystis mutants harboring the empty vector pSEVA251 or expressing His-tagged CYP110D1 under the control of the Ppsba2* or Ptrc.x.tetO2 promoter. (d) Relative densitometry analysis of CYP110D1 His-tag protein levels. Cells were cultivated in BG11 medium supplemented with 50 µg mL -1 kanamycin, at 30 °C and under a 12 hours light (30 μmol photons m -2 s -1 )/ 12 hours dark regimen, until an OD730 ∼3, harvested, suspended in lysis buffer and disrupted by sonication (for details see Experimental Section). 20 µg of proteins were loaded in each lane. M, Molecular mass standards are indicated on the left (kDa). Red arrowhead indicates the band corresponding to the heterologous protein CYP110D1. A single representative experiment of three biological replicates is shown. Error bars represent the standard error of the mean (SEM) of 3 biological replicates (*** p-value < 0.001).

Fig. 2

 2 Fig. 2 Bioconversion of testosterone into 15β-hydroxytestosterone by Synechocystis whole-cells expressing CYP110D1 under the control of the PpsbA2* or Ptrc.x.tetO2 promoter. (a) % of testosterone converted after 21 hours incubation. (b) Concentration (mM) of testosterone and 15β-hydroxytestosterone in the reaction mixture after 21 hours incubation. Reactions were performed in 2 mL reaction volume, with cells concentrated to OD730=5 (0.9 gCDW L -1 ) in BG11 medium supplemented with 0.5 mM testosterone, in closed vials, at 30 °C, with a light intensity of 150 µmol photons s -1 m -2 . Error bars represent the standard error of the mean (SEM) of 3 biological replicates and 2 technical replicates (*** p-value < 0.001).

Fig. 3

 3 Fig. 3 Bioconversion of testosterone into 15β-hydroxytestosterone by Synechocystis whole cells expressing CYP110D1 under the control of Ptrc.x.tetO2 promoter, with cells concentrated to OD730 of 5 or 10 (0.9 or 1.8 gCDW L -1 ) in the reaction mixture. (a) % of testosterone converted after 8 and 21 hours incubation. (b) Concentration (mM) of testosterone and 15β-hydroxytestosterone in the reaction mixtures after 8 and 21 hours incubation. Reactions were performed by incubating 0.5 mM testosterone in 2 mL reaction volume, in BG11 medium, at 30 °C, in closed vials, with a light intensity of 150 µmol photons s -1 m -2 . Error bars represent the standard error of the mean (SEM) of 3 biological replicates and 2 technical replicates (*** p-value < 0.001).

44

 44

Fig. 4

 4 Fig. 4 Bioconversion of testosterone into 15β-hydroxytestosterone by Synechocystis whole-cells expressing CYP110D1 under the control of Ptrc.x.tetO2 promoter, under light (150 µmol photons s -1 m -2 )or dark conditions. (a) % of testosterone converted after 21 hours incubation. (b) Concentration of testosterone and

Fig. 5

 5 Fig.5 Net oxygen evolution rate of Synechocystis strains harboring the empty pSEVA251 or expressing CYP110D1 under the control of Ptrc.x.tetO2 promoter, before (t0) or after the addition of 1.25 % (v/v) DMSO or 1 mM testosterone dissolved in DMSO (Sub). The measurements were carried out using a Clark-type O2-electrode, in 1 mL volume of concentrated cells OD730=10 (1.8 gCDW L -1 ) in fresh BG11, at 30°C, with a light intensity of 120 µmol photons s -1 m -2 . Error bars represent the standard error of the mean (SEM) of 3 biological replicates and 2 technical replicates (n.s., not significant, p-value > 0.05; ** p-value < 0.01; * pvalue < 0.05).

Fig. 6

 6 Fig. 6 Bioconversion of testosterone into 15β-hydroxytestosterone by Synechocystis whole-cells expressing CYP110D1 under the control of Ptrc.x.tetO2 promoter, incubated with different substrate concentrations. (a) % of testosterone converted at different time points. (b) Concentration of testosterone and 15βhydroxytestosterone in the reaction mixtures with different substrate concentration, at different time points. (c) Specific activity of the whole-cell biocatalyst expressed as µmol of substrate converted gCDW -1 min -1 after 2 hours of incubation. (d) Biotransformation theoretical productivity after 2 hours of incubation, expressed as µmol of product formed gCDW -1 h -1 . Reactions were performed incubating 0.5 mM, 1 mM, or 2 mM testosterone in 2 mL reaction volume, with cells concentrated to OD730=10 (1.8 gCDW L -1 ) in BG11 medium, in open vials, at 30 °C, with a light intensity of 150 µmol photons s -1 m -2 . Error bars represent the standard error of the mean (SEM) of 3 biological replicates and 2 technical replicates (n.s., not significant, p-value > 0.05; * p-value < 0.05; *** p-value < 0.001). When not visible, the error bars are smaller than the size of the symbol.

Table 1

 1 Results obtained for the 15β-hydroxylation of testosterone with several whole-cell biocatalysts.

	Strains	Enzymes expressed	Testosterone conversion	Cell density	Sacrificial electron donor	Maximum specific activity	Reference
	Synechocystis sp.	CYP110D1	67 % 1 mM;	∼1.8 gCDW L -1	Water	1.03 U gCDW -1	This study
	PCC 6803		86 % 0.5 mM;
			8 h
	E. coli C43 (DE3)	CYP110D1 + Pdx + PdR	42 % 1 mM; 1 h	∼13.8 gCDW L -1	5.4 g L -1 glucose (30 mM)	0.51 U gCDW -1	This study
	E. coli BL21 (DE3)	CYP109B1 + Fdx + FdR	79 % 1 mM; 9 h	∼6.5 gCDW L -1a	5 g L -1 glycerol 5 g L -1 glucose,	ND	52
	Pseudomonas putida S12	CYP106A2 + Fdx + FdR	72 h 11 % 0.1 mM;	ND b	(27 mM) 5 g L -1 glucose	ND	50
	E. coli BL21	CYP106A2 + Adx + AdR + ADH	40 % 0.25 mM; 26 h	∼4 gCDW L -1a	400 mM isopropanol	ND	51

a Calculated from the original data presented as OD600. b Not determined.

Acknowledgments

This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 764920 MSCA-EJD PhotoBioCat). C.C. Pacheco, P. Oliveira and S. B. Pereira acknowledge FCT -Fundação para a Ciência e a Tecnologia, I.P./Ministério da Ciência, Tecnologia e Ensino Superior for the Assistant Researcher contract CEECIND/00259/2017 (C.C.P.), the FCT investigator grant IF/00256/2015 (P.O.), and Junior Research contract DL57/2016 (S.B.P).

Conflicts of Interest

The authors declare no conflict of interest.


Greqam

Renaud Bourlès

Dominique Henriet

AND CREDIBILITY IN INSURANCE

Keywords: insurance, capital reserve, regulation, recapitalization

The aim of this paper is to analyze the need for capital and default regulation in insurance. Proponents of deregulation argue that these requirements are useless as insurers would hold enough capital as soon as the insured are fully informed about their default probability. Adding to the purpose the relationship between an insurer and her security holders (that is the issuance and dividend policy) we show that the second best capital reserve decided by the security holders is suboptimal whenever the return on cash inside the firm is smaller than outside. Because of limited commitment on recapitalization, disclosure of information may not be enough. Given these characteristics, State commitment to recapitalize could be an alternative regulation policy.

Introduction

Is capital regulation necessary, and, if so, which kind is the most accurate? In all countries that have insurance markets, regulation of insurance companies exists. The main motivation of such a regulation seems to be the protection of insurance buyers against the risk of insolvency of their insurers. This regulation generally takes the form of "technical" or "mathematical" reserves that insurers should at least carry in sufficiently liquid capital. These regulatory reserves are expressed as ratios of premium income and claims expenses. We want to focus in this paper on the necessity and the economic motivations of such regulation rules.

1 GREQAM.

2 University of Toulouse 1 3

Ecole Centrale Marseille

The first rationale for capital regulation seems to be the potential asymmetry of information between insurers and policyholders. The buyer of insurance pays a premium against the promise that she will receive a payment if specified random events occur. If the insurer does not hold enough reserves to fulfill this promise, the consumer is being cheated ex ante. This potential failure, itself, undermines the confidence on which the market is based. If the policyholders does not observe this default risk, the insurance market faces a typical "lemons problem" : uncertainty about product quality -here solvency -may drive high-quality firms out of the market. Regulation is then intended to make sure that only "good" firms (with low risk of insolvency) are in the market.

Proponents of deregulation however argue that capital requirement is not an appropriate tool to mitigate this adverse selection problem. As the problem arises because insured are not conveniently informed about the risk of insolvency, "disclosure of information" policy, that is the public provision of information about insurers' risk of insolvency, is sufficient to solve the adverse selection issue. Private incentives are then sufficiently high to induce companies to hold enough liquid capital to optimally reduce the risk of insolvency. Such a reasoning is however silent on other key actors of insolvency: shareholders or debt owners. Indeed, for a company to become insolvent, not only capital has to be insufficient to meet claims but it also has to be suboptimal to recapitalize (or impossible to issue debt). In the present paper we therefore want to focus on possible other reasons of capital regulation, beside the relation "insurer/insured" evoked above. Another bilateral relationship -between the insurer and her security holders -indeed seems to be of interest. It is now well established that agency problems may arise from the asymmetry of information between managers and shareholders. For example, managers can invest in inefficient projects that generate private benefits for them to the detriment of shareholders. Such an issue would therefore give security holders an incentive not to leave capital in the insurance company. The problem becomes clearer in presence of frictional capital market, if we assume that issuing new debt is costly. An interesting trade-off then arise between agency cost and recapitalization cost.

To study these mechanisms we build a dynamic model of insurance and analyze the dynamic of capital through the behavior of security holders. On the one hand, when the company is solvent, that is when assets are sufficient to meet claims, the security holders can either take dividend or issue new debt (or shares). On the other hand, if claims are too large for the current assets to cover it, the security holders choose whether to recapitalize the insurance company or to default.

When the capital market is frictionless, that is when issuing new debt is costless, the optimal strategy consists in taking dividend as long as it is possible -because of agency cost -and to recapitalize each time it is needed (provided the future value of the company is larger than the invested capital). However, if issuing new debt is costly, it can be optimal to leave some capital in the company. The optimal strategy can be, by the way, related to a well-known policy in inventory management: take dividend above a bottom limit, neither take dividend nor issue new debt if the ex-post (capital) stock is positive but below the limit and issue new debt in order to meet claims when the current reserve is insufficient. Taking into account the effect of default on policyholders, we show that the first best policy implies -when recapitalization is costless -no default (shareholders always recapitalize) but no capital reserve. Although this dynamic is a good candidate for a complete information competitive equilibrium, it appears to be hardly implementable. Indeed, it implies an exante commitment to recapitalize which is not credible. An efficient regulation would then consist in making this commitment credible or at least in guaranteeing that the company would always hold enough capital to continue operating. It therefore appears that State commitment to recapitalize can be a more efficient regulation than capital requirement.

Our work fits in the literature on capital reserve and solvency in insurance. Initiated by [START_REF] Borch | Is Regulation of Insurance Companies Necessary[END_REF] in a model where shareholders can only invest in capital during the first period, this literature has then developed in analyzing the optimal dynamic choice of capital. Munch and Smallwood (1981) and [START_REF] Finsinger | Reserve Levels and Reserve Requirements for Profit Maximising Insurance Firms[END_REF] for example analyze capital choices in a situation where the demand for insurance is elastic with respect to default risk. Both papers however assume that shareholders cannot recapitalize after claims are realized.

In a more recent paper, [START_REF] Rees | Regulation of insurance markets[END_REF] study a situation in which policyholders are fully informed of the default probability of their insurer. They show that, whereas an unconstrained insurer will optimally choose a corner solution (either zero or maximum), once the insured is informed about the probability of not being indemnified, the insurer's expected value is higher if it holds the maximum amount of capital. They however ignore the possibility of recapitalization when claims exceed assets. They indeed assume that contracts are not fully honored in these cases. Under this assumption insurers can commit on a default probability through their capital reserve. Being informed of the amount of capital their insurer holds, individuals can infer the probability of not being paid. Competition in insurance market then lead the companies to raise the maximum amount of capital. However, as we introduce recapitalization -that is the possibility to reinject capital when claims exceed assets -this mechanism no longer holds. Insurers then cannot commit on a default probability as they cannot commit on the behavior of their security holders. This creates a motive for an internal solution for capital reserve and therefore a room for capital regulation (minimal capital requirement or State guarantee) if this solution is suboptimal. Blazenko, Parker andPavlov (2007, 2008) analyze the concept of "economic ruin" by modeling a situation where new share can be issued in case of capital deficit. They however assume an exogenous dividend policy in the sense that a fixed return (the risk-free interest rate) is paid to shareholders whenever capital is positive and that insurer can continue operating with negative capital (debt). We however want to focus here on optimal (and therefore endogenous) issuance and dividend policy and we assume that shareholders has to recapitalize a company with negative capital if they want the company not to default. Finally, we want to focus on a different regulation scheme than Blazenko, Parker andPavlov (2007, 2008). They indeed consider a regulation that requires an immediate capital contribution to offset a capital deficit when we model regulation as capital requirement or State guarantee.

The sequel of the paper is organized as follows. In the next section (Section 2) we present the model of dynamic insurance and its implication on the optimal issuance and dividend policy under different setting (private/public optimum, costly/costless capital). Such an approach allows us to capture the need for capital regulation and to analyze in Section 3 the efficiency of two forms of regulation: capital requirement and State guarantee. Conclusions and directions for future research are eventually provided in Section 4.

The model

An insurance company has a portfolio of n policyholders. Each policyholder incurs a loss x i . We suppose that the random variables x i are IID and denote by x the total claim. We note f the distribution of claims, F its CDF and e its expectation.

Policyholders are completely insured. They value wealth through a von Neumann, increasing and strictly concave, utility function u.

We consider a discrete dynamic model with infinite horizon. The company holds a capital K ≥ 0 so that total assets at the beginning of the first period amounts to nπ + K where π is the premium payed by insured.

For sake of simplicity, we will assume that claims occur at the beginning of the period. Two cases are possible.

• In the first case, the total claim x is lower than total assets A = (nπ + K). Shareholders of the company can the recapitalize or take dividends for the following period. Let k(x) the amount of recapitalization (if negative , -k(x) is the amount of dividends).

For the following period, the insurance company begins with a new cash reserve that amounts to :

ρ(A -x + k(x))
, where ρ is the return on the cash left in the company.

• In the second case the total claim x is larger than A = (nπ + K) and the company is potentially insolvent. The security holders can either refuse to keep on operating -in that case insured are not fully indemnified and A is simply equally shared between them-or subscribe to an issue of new securities (shares or debt) to meet the claims. We suppose that there is a potential cost of issuance : 1 dollar of fresh capital in the company costs γ ≥ 1 to the security holders. This can be, for example, explained by transaction costs (see [START_REF] Gomes | Financing Investment[END_REF] for a justification and an evaluation of these costs).

Notice that there is no possibility of negative balance sheet. If x is larger than A the company must either stop or issue new shares or debt.

In the paper we describe the optimal policy of the insurer in alternative frameworks (private behavior, perfect information framework, and credible policy). We suppose that shareholders discount future with a discount rate δ. We make the assumption that δρ < 1. Inside cash has a lower return ρ than outside opportunities δ -1 .There is no direct incentive to leave cash in the company since it has a better return outside. For instance, firms' managers may commit inside cash to inefficient projects that generate only private benefits for them to the detriment of outside security holders (see La [START_REF] Porta | Agency Problems and Dividend Policies around the World[END_REF] for an overview of these problems).

The trade off between issuance and agency costs can either lead to solutions where capital reserves are optimally positive (if issuance costs are large) or to situations where no cash optimally remain in the insurance company.

Security holder's private policy

The frictionless case

In a frictionless world, fresh cash has no opportunity cost : γ = 1. We claim that, because of agency costs, the optimal policy is to maintain liquidity to zero and stop the activity when losses are too large. To show that, we are considering the optimal intertemporal decision of shareholders of a company starting with an initial capital K. In this part we consider that π is fixed. Notice that, in this section, we do not allow policyholders to leave their company. Let V * (K, π) the intertemporal optimal value of the company.

Using the Bellman principle, we have :

V * (K, π) = max I,k(x)≥x-A I [δV * (ρ (A -x + k(x)) , π) -k(x)] f (x)dx, ( 1
)
where I is the range of the values of x for which security holders decide to keep on operating.

A simple change of variables gives :

V * (K, π) = max I,H(x)≥0 I δV * (H(x), π) - 1 ρ H(x) + (K + nπ) -x f (x)dx (2)
In the above equation H(x) is the reserve decided for the next period by the security holders. As we do not allow the company to operate with negative cash reserve (short term debt), H(x)has to be positive.

Setting then arg max

H≥0 (δV * (H, π) -1 ρ H) = H * and max H≥0 (δV * (H, π) -1 ρ H) = δW * , the equation 2 becomes : V * (K, π) = max I I (δW * + K + nπ -x) f (x)dx (3)
By the envelop theorem we have, noticing I * as the optimal range of operating, V * K (K, π) = µ(I * ) (where µ is the measure associated with F ). This implies, as δρ < 1,that δV * (H, π) -1 ρ H is a strictly decreasing function of H, which in turn implies that H * = 0.

It is optimal to leave no cash reserve in the company. Even if δρ = 1,that is even when cash has the same return inside and outside, H * = 0 is an (now possibly non unique) optimal solution.

We can then deduce the optimal range I * . It is optimal to keep on operating until the integrand of 3 is positive. We have hence

I * = (0, b * (K)) , where b * (K) = K + nπ + .δW * .is the default threshold.
The optimal policy can therefore be depicted as follows.

- x A nπ + δW * 0 - - -

take dividends issue shares or debt default

If we note e K = E(x/x ≤ b * (K)), we can write the complete solution :

δV * (0, π) = δW * = δ F (b * (0))(nπ -e 0 ) 1 -δF (b * (0))
The value of the company holding K and pricing π is hence :

V * (K, π) = F (b * (K)) K + (nπ -e K ) + δF (b * (0))(nπ -e 0 ) 1 -δF (b * (0)) (4)
with e K , b * (K),solutions of :

       (1 -δF (b * (0))) b * (0) + δF (b * (0))e 0 = nπ F (b * (K))e K = b * (K) 0 xf (x)dx b * (K) = b * (0) + K (5)
On the following picture we represent in the plane (b, W )the functions

W = b 0 (b - x)f (x)dx, b = nπ + δW and W = b -e.
The intersection between the first two curves is the point b * (0), W * , the intersection between the two lasts gives W = nπ-e 1-δ which is the present value of the firm when it is systematically recapitalizes (never default). Obviously this value is smaller than W * .

[Figure 1 about here.]

As V * K (K, π) = F (b * (K)) and V * KK (K, π) = f (b * (K)) ≥ 0,
The value function is increasing and convex. The asymptote for large

K is K + (nπ -e) + δV * (0, π) = K + (nπ - e) + δF (b * (0))(nπ -e 0 ) 1 -δF (b * (0))
. The following picture gives the shape of V as a function of K.

[Figure 2 about here.]

The optimal strategy is the following : at the first period, shareholders take away all the money left if any, or reinject enough capital to meet claims. Then the liquidity is maintained to zero except when the total claim exceeds δF (b * (0))(nπ -e 0 ) 1 -δF (b * (0)) + (K + nπ). This value is exactly the present value of the firm computed with the "modified" discount factor δF (x * (0)) that takes into account the probability of default. This simply means that security holders refuse to put more money than the present value of future returns.

It is worth noticing that the value of a firm holding K and pricing π, V * (K, π) is larger than the one obtained for a company that would adopt the non optimal strategy I = [0, +∞), for which the value is simply equal to the NPV :

K + nπ -e 1 -δ .
With this optimal strategy from the point of view of security holders, the level of utility achieved by the consumer as long as he stays in the company is :

B(K, π) = (u(w -π) + δB(0, π)) F (b * (K)) + +∞ b * (K) u w + K -x n f (x)dx
The above result must be contrasted with the ones obtained by [START_REF] Rees | Regulation of insurance markets[END_REF] In their model default is exogenous : if assets A are at least enough to meet claims, then the insurer remains in business and receives a continuation value V that is the expected present value of being in the insurance business at the end of the first period. If claims costs turn out to be greater than assets,the insurer pays out his assets and defaults on the remaining claims, losing the right to the continuation value. In this framework, they find that it can be optimal, for the insurer, to put enough initial capital to avoid default, provided that insurance claims distributions belong to the class of "increasing failure rate" distributions on a bounded support. This result seems to be questionable since, in particular, there is no reason to assume that cash or assets must be put ex-ante and that ex-post recapitalization is impossible. This feature can, by the way, lead to accumulate ex ante a huge (and potentially infinite) level of capital up to the maximal value of total claims. It is as if there was an infinite cost of recapitalization and a zero cost of initial capital.

We obtain here a more nuanced result : default is endogenous and optimally decided when new capital needed is too large compared with the expected returns. This leads to a policy where permanent capital is useless. The only reason for permanent cash to be useful would be the case where recapitalization is costly.

In the following section we will introduce a (finite) cost of issuing new debt or shares.

Opportunity cost of recapitalization

In this second part we assume that capital market imperfections make issues of new shares (or new debt) costly : γ > 1. In particular, when cash reserve becomes negative security holders have to choose between issuing new shares and to stop operating. The important consequence is that this cost creates an ex ante incentive to some precautionary policy which takes the form of capital reserves. Intuitively, when γ is low, reserves can be maintained to zero . But when γ becomes larger, it turns to be optimal to hold some permanent strictly positive capital.

Suppose that shareholders can recapitalize when needed, and inject k(x) in the firm at a cost γ > 1. When k is negative, that is when shareholders take dividends, there is no opportunity cost.

The optimal value of the firm holding Kand pricing π becomes :

V (K, π) = max k(x)≥x-A, I min α(x)∈{1,γ} I [δV (ρ (A -x + k(x)), π) -α(x)k(x)] f (x)dx, (6)
In the above equation, α(x) is optimally fixed to γ for x such that k(x) ≥ 0 and to 1 elsewhere.

Setting, as previously, H(x) = ρ (A -x + k(x)) this equation becomes :

V (K, π) = max H(x)≥0,I min α(x)∈{1,γ} I δV (H(x), π) -α(x) 1 ρ H(x) + α(x) (K + nπ -x) f (x)dx (7) Therefore α * (x) = 1 if H(x) ≤ (A -x)and α * (x) = γ if H(x) ≥ (A -x) .
Let set

H * = arg max H≥0 δV (H, π) -1 ρ H and H * γ = arg max H≥0 δV (H, π) -γ ρ H.
The two levels of capital H * and H * γ have the following meanings : the optimal strategy consists in taking all cash reserves above H * if any, and to recapitalize up to H * γ when needed.

As previously, set : max δV (H, π) -1 ρ H = δW * and max δV (H, π)

-γ ρ H = δW * γ . Lemma 1 : 0 ≤ H * γ ≤ H * and W * γ ≤ W * Proof. see appendix
It is now easy to state the following proposition which gives the optimal strategy and the value of the firm.

Proposition 1 The optimal cash policy is given by two optimal thresholds a * (K)

≤ A ≤ b * (K) such that : if x ≤ a * (K), H(x) = H * : take money above H * if a * (K) ≤ x ≤ A, H(x) = ρ (A -x) : neither take money nor issue shares if A ≤ x ≤ b * (K), H(x) = H * γ = 0 : issue shares up to H * γ = 0 if x ≥ b * (K) the company defaults.
where a * (K), b * (K), and V (K, π), are solutions of :

V (K, π) = max a≤A≤b a -∞ [δW * + (K + nπ -x)] f (x)dx + A a δV (ρ (K + nπ -x) , π)f (x)dx + b A [δV (0, π) + γ (K + nπ -x)] f (x)dx
Proof. see appendix

Here, when issuing debt or shares is necessary, the security holders put just enough to meet claims : H * γ = 0. When the profit is large, security holders take away dividends and leave some "precautionary reserve" H * ≥ 0 . Intuitively, this capital reserve is larger as γ is large. Conversely, when γ is sufficiently low, this level can be maintained to 0. Indeed, the derivative of V w.r.t. K for K = 0 can be computed (thanks to the envelop theorem) :

V K (0, π) = F (a * (0)) + nπ a * (0) δρV (ρ (nπ -x) , π)f (x)dx +γ (F (b * (0)) -F (nπ))
Suppose now that the optimal value is H * = 0. This implies that a * (0) = nπ and b * (0) = B is defined by :

   V (0, π) [1 -δF (B)] = B -∞ [(nπ -x)] f (x)dx + (γ -1) B nπ [(nπ -x)] f (x)dx γB = δV (0, π) + nπ + (γ -1) nπ
The optimal reserve will be H * > 0,if the previous expression of V K (0, π) in which we take a * (0) = nπ is greater than 1 δρ . That is :

F (nπ) + γ (F (B) -F (nπ)) > 1 δρ
In this case the optimal policy can be depicted as follows.

-

x 0 - - take dividends default - - a * A b *
"wait" issue shares or debt This issuance and dividend policy can be related to the one defined in [START_REF] Décamps | Free Cash-Flow, Issuance Costs and Stock Volatility[END_REF] on banking market. The main difference is that, in their continuous framework, default never optimally occurs.

When γ is large enough the value function as a function of K has a concave part for small values of K.

Another interesting question concerns the variation of the default threshold b * (K)when γ increases. We know that :

b * (K) = δ γ V (0, π) + K + nπ.
It is easy to show that V (0, π) is decreasing with γ,so that we have the following proposition :

Proposition 2 When the opportunity cost γ increases, both the precautionary reserve H * and the probability of default increase.

Proof. see appendix

A simple consequence of this proposition is that the level of permanent capital is not a signal of better solvency risk of the insurer. The precautionary reserve is not aimed at diminishing the risk of insolvency but at diminishing the cost of recapitalization.

Optimal First Best reserve policy

In the previous section, when deciding the reserve policy, shareholders don't take into account the impact of this policy on the welfare of policyholders. A natural question would then be to find the optimal policy, (premium, dividends, reserves) under the constraint that the insured achieve a given per-period level of expected utility u. As a consequence, the premium is now endogenous. This provides the company a new instrument and introduces a trade off between capital reserve and premiums as in [START_REF] Bourlès | On the Emergence of Private Insurance in Presence of Mutual Agreements[END_REF]. In a perfect information framework, this situation (optimal policy under some reservation level for the insureds), can be considered as a good candidate for a competitive equilibrium. We examine here the case γ = 1.

The problem now is hence the following :

V F B (K, u) = max I,k(.),π I -k(x)f (x)dx + I δV F B (ρ (A -x + k(x)))f (x)dx s.c. I u(w -π)f (x)dx + I C u w + K -x n f (x)dx ≥ u (8)
That is :

V F B (K, u) = max I,H(.)≥0,π I (K + nπ -x) f (x)dx + I δV F B (H(x), u 0 ) - 1 ρ H(x) f (x)dx s.c. I u(w -π)f (x)dx + I C u w + K -x n f (x)dx ≥ u (9)
As previously set H F B = arg max δV F B (H, u)-1 ρ H and δW F B = max δV F B (H, u 0 )-1 ρ H And then :

V F B (K, u) = max I,π I (K + nπ -x) f (x)dx + I δW F B f (x)dx s.c. I u(w -π)f (x)dx + I C u w + K -x n f (x)dx ≥ u (10)
Let λ be the lagrange multiplier associated with the constraint. Optimization with respect to π gives :

(n -λu (w -π)) µ(I) = 0 Endogenously, shareholders provoke default when the present value of future income is less than the claims, that is when x is larger than K + nπ + δW SB , where W SB is the optimal second best value of the firm.

We hence have :

V SB (K, u) = K+nπ+δW SB 0 K + nπ + δW SB -x f (x)dx (13)
with π the largest value of the premium p such that,

U (K, p) = K+np+δW SB 0 u(w -p)f (x)dx + +∞ K+np+δW SB u w + K -x n f (x)dx ≥ u (14)
with :

δW SB = max δV SB (H, u) - 1 ρ H (15)
The difference with the First Best situation is that I is constrained to be the interval (0, K + nπ + δW SB ].This is due to the fact that the decision not to default must be ex ante credible.

Intuitively increasing K allows to decrease the probability of default which in turn increases the value of the contract for the policyholders and hence allows an increase of nπ. A natural question is to compare the return of this "investment" with its cost.

As π is implicitly defined by 14, we have :.

n ∂π ∂K = -n U K U π = +∞ b u w + K -x n f (x)dx + n u(w -π) -u w -π - δW SB n f (b) u (w -π)F (b) -n u(w -π) -u w -π -δW SB n f (b)
,

with b = K + nπ + δW SB .
As π is the largest value providing the level u, U π is necessarily negative, and then -n U K Uπ ≥ 0. We Have hence :

n ∂π ∂K ≥ (1 -F (b)) F (b) E u w + K-x n /x ≥ b u (w -π) .
As u is strictly concave and w + K-x n ≤ w -π when x ≥ b, we have :

n ∂π ∂K ≥ (1 -F (b)) F (b) .
Now differentiating 13 gives :

∂V SB ∂K (K, u) = F (b) + F (b)n ∂π ∂K ≥ 1.
This investment is profitable if this derivative is greater than 1 ρδ .

Proposition 4 Supposing W 0 , b 0 and π 0 are solutions of

           W = b 0 (b -x)f (x)dx b = nπ + δW u(w -π)F (b) + +∞ b u w - x n f (x)dx = u (16)
If

F (b 0 ) + (1 -F (b 0 )) E u w -x n /x ≥ b 0 u (w -b 0 n ) > 1 δρ (17)
it is optimal to hold positive permanent capital H SB > 0

Proof. Easily proven by supposing that H SB = 0 and minoring

∂V SB ∂K (0, u)by F (b 0 ) + (1 - F (b 0 )) E[u (w-x n )/x≥b0] u (w- b 0 n )
.

On the following picture we represent in the plane (b, W ), the curves

W = b 0 (b -x)f (x)dx and u w - b n + δW n F (b) + +∞ b u w - x n f (x)dx = u.
The latter is an increasing curve with a slope smaller than 1 δ , = 1 δ for W = 0 and an asymptote b = nπ + δW. The intersection of these two curves give the solution W 0 , b 0 ,and π 0 if H SB = 0. Obviously, W 0 < W F B ≤ W * meaning that the credible proof value of the company is lower than under the first best policy.

[Figure 3 about here.]

When the condition of the previous proposition is fulfilled, V (K, u) is a function of K with a slope greater than 1 δρ for K = 0, and a slope exactly equal to 1 δρ for K = H SB . For K going to infinity, the slope tends to 1as π tends to π (and hence ∂π ∂K tends to 0).

The asymptote being K + (nπ -e) + δW SB , the optimal credible capital reserve can be represented as follows.

[Figure 4 about here.]

Regulation

In this section, we intuitively analyze the possible impact of two possible regulation schemes: capital requirement (that is the minimal amount of capital insurers have to hold to continue operating) and State guarantee to recapitalize. As the lack of credibility on recapitalization has been show to be the main issue in reserve policy, the second option may be preferable.

The impact of minimal capital requirement

Let us first examine the possible implications of capital requirement on optimal capital reserve. Such a regulation rule (chosen in most countries) constraints the insurance companies to hold a minimal amount of capital K. In our setting, this implies that recapitalization is needed as soon as claims exceed assets minus this ceiling K (and therefore, more often than without regulation, for a given amount of initial capital). Moreover -under this scheme -when new shares (or debt) have to be issued, security holders need to build up capital reserve up to the required minimum (and no longer up to zero). As, the amount of capital shareholders are willing to inject are still bounded by the present value of future returns these two mechanisms that (i) increases the need for recapitalization and (ii) potentially reduce the value of company, may lead to a perverse effect of capital requirement through an increase in the probability of default. Lastly, it appears that capital requirement may reduce the potential amount of "precautionary reserve" (by imposing early recapitalization) that could -as shown in Proposition 2 -reduce the cost of recapitalization.

The exact implication of reserve requirement (in particular on the value of capital) however remains to be investigated and calls for further research. The analyze of the precise constrained program would for example allow us to discuss more precisely the impact of capital requirement on the value of the firm and to provide some interesting comparative statics on K. Our analysis moreover seems to call for the study of an alternative policy that would consist in fixing a minimum amount of capital above which security holders are prevented to take dividend but do not constrain them to recapitalize above zero (when claims exceed assets). Such a policy would create precautionary reserves without increasing the need for recapitalization.

A solution to the credibility issue: State guarantee to recapitalize

The questionable efficiency of capital requirement and the fact that the first best policy cannot be implemented because of a credibility issue lead us to consider an alternative form of regulation: State guarantee to inject capital. We have shown in this paper that default occurs when security holders are reluctant to inject enough capital for the company to keep on operating. This issue will therefore be solved if the State commits -in these cases -to buy enough shares for the capital reserve not to be negative. This however creates a typical moral hazard issue as it will then be easy for the shareholders to cheat on there capacity/willingness to add capital. They we then benefit from the capital injected by the government without bearing the costs. This issue however disappears if we assume that government can infer the value of the company and can be eased by assuming that the State conditions its intervention to a takeover of the company. Such a nationalization would lead to a null value of the company (from the point of view of shareholders). Therefore the call for State capital would only be optimal if the amount of needed capital exceed the value of the company.

Conclusion

We highlight in the paper to role of the relationship between an insurer and its security holders in the need for capital and default regulation. We show that beside the informational issue between an insurer and its policyholders, regulation can be needed to solve an issue of credibility on recapitalization. It indeed appears in our work that the first best policy is not credible as it induces recapitalization in situation where shareholders have no incentive to inject capital.

Contrarily to existing literature we moreover show that an interior solution for capital reserve can be optimally chosen if recapitalization is costly. When the capital market is frictionless, that is when issuing new debt is costless, the optimal strategy consists in taking dividend as long as it is possible -because of agency cost -and to recapitalize each time it is needed (provided the future value of the company is larger than the invested capital). However, if issuing new debt is costly, it can be optimal to leave some capital in the company. The optimal strategy then consists in (a) taking dividend above a bottom limit, (b) neither take dividend nor issue new debt if the ex-post capital reserve is positive but below the limit and (c) issue new debt in order to meet claims when the current reserve is insufficient.

Taking into account the effect of default on policyholders, we show that the first best policy implies no default but no capital reserve. This policy however appears to be hardly implementable as it implies an ex-ante commitment to recapitalize which is not credible. An efficient regulation would therefore consist in making this commitment credible and therefore State commitment to recapitalize may be a more efficient regulation than capital requirement.

Is left for future research to analyze more precisely the optimal regulation. It would for example be interesting to evaluate optimal capital policy under capital requirement that is if capital reserve are constrained to be above a given level. We would then be able to define more exactly the efficient regulation. An other extension of interest would consist in studying the value of a share (and not of the company). Such a variant of our model may create an incentive for positive reserve (even with costless capital) as recapitalization -that is the issuance of new shares -would reduce the value of an existing share.

Appendix

Proof of Lemma 1

We have

δW * = δV (H * , π) -1 ρ H * ≥ δV (H * γ , π) -1 ρ H * γ δW * γ = δV (H * γ , π) -γ ρ H * γ ≥ .δV (H * , π) -γ ρ H *
Summing up these two inequalities gives :

γ -1 ρ H * ≥ γ -1 ρ H * γ Moreover S(γ) = max H≥0 δV (H, π) -γ ρ H

Proof of Proposition 1

We have : This in turn implies that : In order to find H * and H * γ it is helpful to compute the first derivative of V Thanks to the envelop theorem we have :

V (K, π) = max
V (K, π) = max
V K (K, π) = I * α * (x)f (x)dx
Where we define α * (x) by :

α * (x) = 1 if x ≤ a * α * (x) = γ if x ≥ b * α * (x) = δρV K (ρ (A -x) , π) if a ≤ x ≤ b
This implies that V K (K, π) is smaller than γ,and then that δV K (K, π)is smaller than δγ.which is smaller than γ ρ This means that H * γ = 0. and hence that b * = A and W * γ = V (0, π) Under the optimal cash policy the company issues new shares only in case of negative cash, that is only when claims are greater than reserves A.

Then the set of equations characterizing the solution of the problem are : We know that : Thanks to the envelop theorem the derivative of V with respect to γ is hence :

V (K, π) =
V (K, π) = max
V γ (K, π) = b * (K)
A δV γ (0, π) + (K + nπ -x) f (x)dx.

In particular :

  is the supremum of decreasing affine (and hence convex) functions. It is hence a convex decreasing function, such that S (γ) =

H

  (x)≥0,I I δV (H(x), π) -α(x) 1 ρ H(x) + α(x) (A -x) f (x)dx (18) With α(x) = 1 if H(x) < ρ (A -x) and α(x) = γ if H(x) > ρ (A -x)take H * (x)the optimal policy and define :I 1 = {x, H * (x) < ρ (A -x)} , I 2 = {x, H * (x) = ρ (A -x)} , I 3 = {x, H * (x) > ρ (A -(H * (x), π)f (x)dx (20) + I 1 δV (H * (x), π) -γ ρ H * (x) + γ (A -x) f (x)dx(21)Clearly in I 1 , H * (x) = min(ρ (A -x) , H * ), and in I 3 , H * (x) = max(ρ (A -x) , H * γ ).

  π) + γ (K + nπ -x)] f (x)dx δV (ρ (K + nπ -a * ) , π) = δW * + (K + nπ -a * ) δV (0, π) + γ (K + nπ -b * ) = 0 W * = max H≥0 δV (H, π) -

  π) + γ (K + nπ -x)] f (x)dx

Fig

  Fig. 1.-On the suboptimality of "automatical" recapitalization

Set I = (0, b], when total loss is larger than b,then the security holders provoke default. The derivative of the Lagrangian w.r.t.b is :

It is easy to see that L(b) is a convex function with a minimum for b such that

This minimum is exactly δW F B . This value being essentially positive we can conclude that L(b)is always positive, and hence :

the optimal policy is again to maintain cash reserves to zero, and hence :

1 -δ , the next proposition holds.

Proposition 3 The First Best policy implies recapitalization in situations where it is not the (private) optimal behavior for security holders. Therefore, an insurer has a short term incentive to deviate from this First Best policy by provoking default. This makes the First Best allocation unstable even if the policyholders perfectly observe the level of reserve capital.

Credibility : second best policy

The optimal First Best policy cannot be non cooperatively implemented since there is no way for the shareholders to commit to recapitalize when losses are larger than the expected present value of future income. An interesting question would be to find a second best credible policy. In this second best world, capital allows to commit to some default probability.