Pierre Picard
email: pierre.picard@polytechnique.edu

Jean Pinquet
email: jean.pinquet@polytechnique.edu.

Optimal risk …nancing in large corporations through insurance captives

A captive is an insurance or reinsurance company established by a parent group to …nance its own risks. Captives mix internal risk pooling between the business units of the parent group and risk transfer toward the reinsurance market. We analyze captives from an optimal insurance contract perspective. The paper considers the vertical contractual chain that links …rstly business units to insurance captives or to "fronters" through insurance contracts, secondly fronters to reinsurance captives through the cession of risks and thirdly insurance or reinsurance captives to reinsurers through cessions or retrocessions. In particular, the risk cession by fronters to a reinsurance captive trades o¤ the bene…ts derived from recouped premiums and from the risk sharing advantage of an "umbrella reinsurance policy", against the risks that result from the captive liabilities. The optimal captive scheme depends on the price of coverage in insurance and reinsurance markets and on the parent group's corporate capital. Since these variables ‡uctuate across time, the analysis developed in this paper corroborates the intertemporal variability of captives activity.

Introduction

A captive is an insurance or reinsurance company established by a parent group to …nance its own risks. Captive insurance thus provide a risk management technique to …nance the retained losses of a corporation in a separate legal entity.

The captive insurance industry has its origin in the formation of mutuals and coinsurance companies in the 1920s and 1930s in the US. However the real growth of the captive industry began in the 1960s and the move by parent companies to establish their captives o¤shore, particularly in the Bermuda market. The 1970s and 1980s were a period of tremendous growth of the captive industry. The greatest stimulus was the cost or lack of availability of certain types of insurance cover in the commercial lines, particularly liability coverage, but many other considerations apply [START_REF] Best | European Captives -a Growth Market During a Challenging Time ?[END_REF] . A corporation may indeed create a captive for various reasons [START_REF] Banks | Alternative Risk Transfer. Integrated Risk management through Insurance, Reinsurance and the Capital Markets[END_REF] . Captives allow the parents to mitigate the ‡uctuations of insurance costs and to save costs by eliminating brokers fees, reducing transaction costs and avoiding costly insurance regulations. Captives can also access the capacity of reinsurance markets with lower wholesale prices and higher limits of coverage than in the retail market. Captives may provide the cover when the commercial market is unwilling to underwrite certain risks or when the price quoted is too high. They also allow parents to bene…t from better-than-average claim experience. Recapturing investment income and managing cash ‡ows is another frequently mentioned motive for creating a captive. Although taking advantage of tax deductibility is no more the main advantage of captives, tax advantages may still be gained by using captives, particularly when captives underwrite unrelated risks in addition to group business [START_REF] Bodo¤ | Capital allocation by percentile layer[END_REF] . A captive can also act as a focus for risk management activities, particularly by providing its parent with a data warehouse to support its risk prevention e¤orts. Finally, captives allow large corporations to pool their risks and to retain aggregate losses within their overall risk-bearing capacity, while o¤ering insurance with relatively small deductibles to their individual business units. On the negative side, regulation or parents'shortage of capital may sometimes dampen [START_REF] Best | European Captives -a Growth Market During a Challenging Time ?[END_REF] The world's total number of captives was 5,290 in 2009 according to Business Insurance's annual captive survey. Most major corporations now own one or several captives. There are several types of insurance captives, including single parent captive, group captive, rent-a-captive, protected cell captive and agency captive, most of them making their home o¤shore; see Banks (2004). [START_REF] Banks | Alternative Risk Transfer. Integrated Risk management through Insurance, Reinsurance and the Capital Markets[END_REF] Porat and Scordis (1998) argue that the creation of captives enhances the status of the risk manager and test the hypothesis that corporations with heightened manager-owner con ‡icts of interest are more likely to operate a single-parent captive. They …nd that captive creation increases with the size of the …rm, but decreases with Tobin's q (the ratio between the market value of the …rm and the replacement value of net assets), which is seen as a proxy for manager-owner con ‡icts. Smith (1986), Cross et al. (1988), Han and Lai (1991) and Lai and Witt (1995) focus on the tax deductibility issue of captive insurance premiums. Diallo and Kim (1989) stress the aforementioned advantages of captive creation as a way to reach a better monitoring of risk prevention activities of the parent's business units. [START_REF] Bodo¤ | Capital allocation by percentile layer[END_REF] Cross, Davidson et Thornton (1988) show that captive creation in the USA created value until 1978, the year after which most tax advantages disappeared. enthusiasm for captives [START_REF] Cross | Taxes, stock returns and captive insurance subsidiaries[END_REF] .

Captives mix internal risk pooling between the business units of the parent group (thereby improving the parent's self-insurance capacity) and risk transfer toward the reinsurance market. The objective of the present paper is to analyze this mechanism from an optimal insurance contract perspective. Two stylized facts will be in the background of our analysis. Firstly, cessions to captives are substitutes for contingency and loss provisions and therefore they are targeted at lower risk layers. Put it di¤erently, insurance and reinsurance captives work as expansion tanks of parent groups' self insurance capacity and they typically cede or retrocede higher risk tiers to reinsurance markets. In particular they can o¤er some form of "umbrella policy" to their parent group [START_REF] Cummins | Capitalization of the property-liability insurance industry : overview[END_REF] . The aim of this paper is to analyze how such expansion tanks and umbrella policies can improve the risk sharing between parent groups and insurance or reinsurance …rms. From a contract theory standpoint, captives provide a fascinating case where self-insurance and risk transfer are made of a vertical chain of contracts between fronters, reinsurance captives and retrocessionnaires. Secondly, it has been extensively documented that insurance markets are characterized by alternating periods of hard markets with relatively high prices and limited capacity and soft markets with relatively low prices and plentiful supply -see Cummins and Doherty (2002). Captives' activity reacts to this insurance underwriting cycle. In particular, casual observation suggests that parent groups have used their captives more intensely when corporate insurance became more expensive or even unavailable after shocks, particularly in the 1980's for liability lines or in the post-Katrina period. Conversely, parent groups may keep their captives dormant in soft market periods [START_REF] Diallo | Asymmetric information, captive insurer's formation, and manager's welfare gain[END_REF] . Furthermore, the activity of captives is also a¤ected by wealth e¤ects : a larger amount of corporate capital available in parent groups provides additional capacity for self-insurance and is thus likely to induce a more intense use of captives. Conversely, the activity of captives may slow down after a …nancial shock reduced the parents'self-insurance capacity [START_REF] Eeckhoudt | Economic and Financial Decisions under Risk[END_REF] . We also aim to understand how such variations in insurance prices and corporate capital may a¤ect the design of the optimal captive arrangement.

The paper is organized as follows. Section 2 presents the model. We consider a …rm made up of a number of decentralized business units that may be subject to insurable losses. Stabilizing the current cash ‡ow of each business unit improves the value of the …rm because it is costly to transfer cash between business units in response to pro…tability shocks and because, in addition, the marginal cost of external capital is increasing. Business units purchase insurance from market insurers and / or from a captive insurance company created by the …rm. Market insurers may also act as "fronters" : they partly or fully cede the underwritten risks to the captive, which then acts as a reinsurer and possibly retrocedes the risks to market reinsurers. Sections 3 and 4 respectively focus on insurance and reinsurance captives. We shall conclude that reinsurance captives dominate insurance captive for reasonable assumptions on cost parameters. Intuitively a reinsurance captive allows the parent to cumulate the advantage of an "umbrella policy" through its retrocession activity with the lower underwriting and claims handling costs of fronters. We analyze the vertical contractual chain that links business units to fronters or directly to the captive through insurance policies, fronters to the reinsurance captive through the cession of risks and the captive to reinsurers through cession or retrocession. The risk cession by fronters to the reinsurance captive trades o¤ the bene…ts derived from recouped premiums and from the risk sharing advantage of an umbrella reinsurance policy, against the risks that result from the captive liabilities. The optimal mechanism involves full cession of risks by the fronter to the captive up to an upper limit, possibly with partial cession over this limit. Furthermore, business units should have partial insurance at the margin in order to reach an optimal risk sharing with the captive. The optimal insurance or reinsurance captive scheme depends on various parameters, including the cost of capital in insurance markets, the reinsurance loading factor and the parent group's corporate capital. Variations in these parameters may trigger changes in the intensity of the captive activity. Numerical examples illustrate these price and wealth e¤ects. Section 5 concludes. All proofs are in the Appendix.

The model

We consider a corporation organized as a holding company, which is made up of n business units indexed by i = 1; : : : ; n. Each business unit is a subsidiary of the holding company with speci…c location and activity. Let i be the sum of initial endowment and cash ‡ows of unit i during the current period. For unexpected low values of i unit i may have to call up external capital or to delay pro…table investment projects. In both cases, insu¢ cient current cash ‡ows entail additional costs and lower current or future pro…ts. We assume that these induced losses occur at an increasing marginal rate because when current cash ‡ows are decreasing then either the cost of external additional capital is increasing or more pro…table projects have to be postponed. To capture these two e¤ects of current cash ‡ows on the value of the …rm, we write the contribution of unit i to the shareholder value of the holding company (i.e. its consolidated expected discounted pro…ts) as W i = B i ( i ), where B i is an increasing strictly concave function, i.e. B 0 i ( i ) > 0; B 00 i ( i ) < 0. The concavity of B i will provide a rationale for stabilizing the short-term income of unit i. [START_REF] Froot | Risk management, capital budgeting, and capital structure policy for insurers and reinsurers[END_REF] This argument rests on the implicit assumption that the holding company cannot transfer cash between business units during the current period. Remark 1 below shows how our model can be extended to a more realistic framework where intraperiod transfers between business units are feasible but costly. [START_REF] Froot | Risk management : coordinating corporate investment and …nancing policies[END_REF] The current cash ‡ows of business units may be a¤ected by losses that can be covered by insurance contracts. Insurance can be taken out either in the insurance market (say from local insurers 10 ) or from a captive created by the holding company. For the time being, we assume that the captive only acts as direct insurer of business units. The role of a reinsurance captive will appear in Section 4. Let w i be the sum of initial endowment and current cash ‡ows for unit i, excluding insurable losses and insurance …nancial transfers : w 1 ; : : : ; w n are deterministic parameters and P n i=1 w i may be viewed as the parent's group corporate capital. [START_REF] Gollier | Optimal insurance design : what can we do with and without expected utility?[END_REF] Let e

x i be the insurable losses of unit i. The variables e

x i , i = 1 : : : ; n are random and independently distributed over intervals [0;

x i ] with cumulative distribution functions i (x i ). They have densities ' i (x i ) 0 i (x i ) over (0; x i ] and point masses of probability at x i = 0.

Indemnity schedules I i (x i ) and t i (x i ) specify the payment made to unit i respectively by the local insurer (say by insurer i 12 ) and by the captive, as a function of realized losses x i . I i (x i ) and t i (x i ) are non-decreasing functions de…ned over [0; x i ]. Let P i and k i be the corresponding insurance premiums paid by unit i respectively to insurer i and to the [START_REF] Froot | Risk management, capital budgeting, and capital structure policy for insurers and reinsurers[END_REF] See Miller and Orr (1966), Meltzer (1993) and Mulligan (1997) on the cost of being short liquid assets and having to cut investment or raise external funds. Following Froot, Scharfstein, and Stein (1993) we could derive this rationale for corporate hedging from an agency-based approach. See Chapter 6 in Tirole (2006) for further developments. [START_REF] Froot | Risk management : coordinating corporate investment and …nancing policies[END_REF] Such transfers may be di¢ cult on a short term basis either because of regulatory restraints or due to contractual provisions, e.g. in case of of restrictions on currency exchange or when international transfers require the approval of local partners when a business unit is a joint venture. Even if transferring money is feasible, it will usually induce transaction costs and it may perturbate the …rm's tax optimization or the dividend distribution strategy. Transferring money between business units in an unexpected way may also perturbate incentives mechanisms and it may exacerbate transitory liquidity problems thereby inducing additional costs. Hartzell, Titman and Twite (2006) …nd empirical support for both transaction costs and precautionary motives to explain why US multinational …rms hold cash, but they also …nd evidence consistent with tax costs associated with repatriating foreign income driving …rms to hold more cash. [START_REF] Froot | Risk management, capital budgeting and capital structure policy for …nancial institutions : an integrated approach[END_REF] The terminology "local insurer" is used only for the sake of clarity but such insurers may be subsidiaries of the same insurance company, and in practice this is often the case.

11 w 1 ; : : : ; w n are given but all that matters is P n i=1 w i because capital can be reallocated between business units at the beginning of the current period. See the role of variables k 1; : : : ; k n below. [START_REF] Han | The tax deductibility of premiums paid to captive insurers: A Risk Reduction Approach[END_REF] For notational simplicity index i thus simultaneously refers to a speci…c business unit and to its local insurer. captive. [START_REF] Hartzell | Why do …rms hold so much cash? A tax-based explanation[END_REF] We thus have i = w i x i + I i (x i ) P i + t i (x i ) k i for all i = 1; : : : ; n:

(

The risks underwritten by the captive may be ceded through a reinsurance contract which provides payment T (t 1 ; : : : ; t n ) to the captive as a function of the transfers made to the business units, with the cession of premiums Q to reinsurers.

Local insurers and the captive bear …xed underwriting costs. Operating in market i, i.e. in the area of activity of business unit i, requires incurring …xed costs f i for local insurer i and b f i for the captive (respectively when I i (x i ) > 0 and t i (x i ) > 0 with positive probability). [START_REF] Kreps | Riskiness leverage models[END_REF] Reinsurance …xed underwriting costs are denoted by f 0 . Insurance and reinsurance costs also include variable costs (claims handling costs and capital costs) which take the form of loadings at rate i for local insurer i and for reinsurers. We assume that claim handling costs are proportional to the amount of claims, while capital costs are proportional to the losses that are retained by the insurer. [START_REF] Lai | The tax deductibility of captive insurance premiums: an assessment and alternative perspective[END_REF] We break down the insurer i's loading factor by writing i = c i + k i , where c i refers to the claims handling cost, while k i correspond to capital costs. Because of competitive pressure, insurance premiums charged by local insurers and reinsurers just cover expected costs, including …xed underwriting costs and variable costs. We thus have

P i = (1 + c i + k i )E[I i (e x i )] + f i if i 2 A `(2) Q = (1 + )E[T (t 1 (
e x 1 ); : : : ; t n (e x n ))] + f 0 ;

(3) where A ` f1; : : : ; ng denotes the set of business units insured by local insurers (i.e. such that I i (x i ) > 0 with positive probability), with

P i = 0 if i = 2 A `.
The captive also incurs variable costs, which include claim handling costs proportional to the amount of …led claims at rate b c i and capital cost. The capital cost corresponds to the opportunity cost of transferring …nancial resources from the business units to the captive through insurance premium k i , which is re ‡ected by functions B i (:). For notational simplicity, we assume that the captive has no initial endowment. Its current pro…t is written as

0 = n X i=1 k i X i2A c [(1 + b c i )t i (x i ) + b f i ] + T (t 1 (x 1 ); : : : ; t n (x n )) Q; (4)
where A c f1; : : : ; ng refers to the set of business units insured by the captive (i.e. such that t i (x i ) > 0 with positive probability) [START_REF] Meier | Business cycles in insurance and reinsurance : the case of France, Germany and Switzerland[END_REF] . When 0 is low, the captive breaks even by calling up external capital at an increasing marginal cost. Similarly to business units, the contribution of the captive to the holding company's shareholder value is thus written as W 0 = B 0 ( 0 ), with B 0 0 ( 0 ) > 0; B 00 0 ( 0 ) < 0 [START_REF] Meltzer | The demand for money : A cross-section study of business …rms[END_REF] .

An insurance captive scheme fI i (x i ); P i ; t i (x i ); k i for i = 1; : : : ; n; T (t 1 ; : : : ; t n ); Qg is feasible if it satis…es (2) and (3). The holding company chooses a feasible insurance scheme that maximizes its expected consolidated shareholder value W , which comes additively from the business units and from the captive. The shareholder value may thus be written as

W = E " n X i=0 W i # = E " n X i=0 B i ( i ) # ;
where 0 and 1 ; : : : ; n are respectively given by ( 4) and (1).

Remark 1

The present modelling can easily be extended to the case where the holding company can transfer funds between business units during the current period, but such transfers induce transaction costs. Let m ij denote the transfer from unit i to unit j with transaction cost c ij per unit of transferred money and w.l.o.g. we assume that this cost is incurred by unit i. In other words, unit i pays (1 + c ij )m ij while unit j receives m ij . Now the net cash ‡ows of unit i are written as

b i = i + X j6 =i m ji X j6 =i (1 + c ij )m ij .
where i is de…ned as before for all i = 0; : : : ; n. The contribution of unit i (including the captive) to the shareholder value of the holding company is written as B i ( b i ). Given 0 ,. . . , n ; the optimal transfers maximize [START_REF] Meier | Business cycles in insurance and reinsurance : the case of France, Germany and Switzerland[END_REF] The holding company may …nd that it is optimal to recapitalize the captive by transfering money from business units that buy insurance only from local insurers. We thus may have k i 6 = 0 even if i = 2 A c . We may have A `\ A c 6 = ;, i.e. some business units may buy insurance at the same time from local insurers and from the captive. [START_REF] Meltzer | The demand for money : A cross-section study of business …rms[END_REF] The concavity of function B 0 ( 0 ) conveys the risk aversion of the holding company w.r.t. the captive's cash ‡ows, with the same capital market rationale behind this assumption as for business units. See Froot and Stein (1998) and Froot (2007) on …nancial institutions'risk management when ‡uctuations in internal cash result in ‡uctuations in costs of raising external funds.

n X i=0 B i ( b i ) = n X i=0 B i ( i + X j6 =i m ji X j6 =i (1 + c ij )m ij )
with respect to m ij 0 for all i; j; i 6 = j, with obviously m ji m ij = 0 for all i; j; i 6 = j for the optimal solution. Let U ( 0 ; 1 ; :::; n ) be the value function of this optimization problem. Since B i (:) is concave for all i; U ( 0 ; 1 ; : : : ; n ) is a multi-variable concave function. In what follows P n i=0 B i ( i ) may be replaced by U ( 0 ; 1 ; : : : ; n ) as the objective function of the holding company without any change in the arguments and the conclusions. Our results are thus robust to the possibility of implementing costly intraperiod transfers between business units. For the sake of notational simplicity we will use the separable objective function P n i=0 B i ( i ), but our modelling can be extended to such a broader setting.

Optimal insurance captive schemes

Corporations that use an insurance captive trade o¤ the bene…ts associated with an umbrella reinsurance policy against costs which may be larger for the captive than for local insurers. Obviously, when local insurer i is faced with larger variable and …xed costs than the captive, then business unit i should purchase insurance only from the captive. When this is not the case, the balance may tip in favor of the captive or the local insurer. A business unit may even purchase insurance from both sources.

Proposition 1 An optimal insurance captive scheme speci…es that all business units i such that (1 + )(1 + b c i ) 1 + i and b f i f i purchase insurance only from the captive, while other units may purchase insurance from the captive and/or from the local insurer. The captive cedes a part of the underwritten risk through a stop-loss reinsurance contract

T (t 1 ; : : : ; t n ) h P n i=1 (1 + b c i )t i D i +
, with D > 0 and x + = max(x; 0).

Let us start commenting on Proposition 1 with its second part. The captive should cede its aggregate loss P n i=1 (1 + b c i )t i (claims handling cost included) through a reinsurance stop-loss contract with deductible D. This is analogous to the optimality of straight deductible contracts in the case of insurance demand by a risk averse individual under constant loading. Through its reinsurance contract, the insurance captive provides an umbrella policy to the holding company, i.e. a cover for the aggregate loss P n i=1 (1 + b c i )t i , which may be more e¢ cient than independent coverage separately obtained by business units from local insurers.

This e¢ ciency gain associated with the captive umbrella policy may be annihilated if the captive is subject to higher costs than local insurers, and the …rst part of Proposition 1 gives su¢ cient conditions for covering the risk through the captive to be optimal. These conditions are more easily interpreted if we observe that

(1 + )(1 + b c i )E[I i (e x i )]
is the captive's variable cost for risks that are ceded to reinsurers, while (1+ i )E[I i (e x i )] denotes the local insurer variable cost. If …xed underwriting costs f i and b f i are large enough for purchasing insurance simultaneously from the local insurer and from the captive to be suboptimal, then business units are split between two groups (i.e. A `\ A c = ; as represented in Figure 1) according to whether or not the bene…ts of a better risk sharing through an umbrella policy are larger than the increase in …xed underwriting costs (when b f i > f i ) and/or variable costs (when

(1 + )(1 + b c i ) > 1 + i ).
When the balance tips in favor of the risk sharing advantage, then the business units should be insured by the captive, while they should be insured by local insurers in the other case.

Figure 1

As an illustration of Proposition 1, observe that condition

(1 + )(1 + b c i ) < 1 + i is approximately equivalent to k i > b c i c
i for small loading factors. The right hand side of this inequality corresponds to di¤erences in claim handling costs between the captive and local insurer i and we may consider it as a structural e¢ ciency gap. The left hand side is the di¤erence between the savings on direct insurance capital cost if a risk is transferred from local insurer i to the captive and the reinsurance costs should this risk be ceded by the captive. We may then think of hard market situations where the underwriting capacity of direct insurers is limited because they meet di¢ culties in raising additional capital, which result in an increase in k i . For given state of reinsurance market (i.e. for unchanged) transferring the business units risk to the captive is then more likely to be an optimal choice than in soft market periods where k i would be lower. Put di¤erently, if local insurer i has experienced adverse underwriting conditions in year t, then it is likely to have a higher cost of capital k i in year t + 1, with more cessions to captives. This is all the more likely given that the adverse shocks have been speci…c for insurer i and did not come with a hard reinsurance market. [START_REF] Miller | A model of demand for money by …rms[END_REF] We now turn to the characterization of the optimal insurance coverage o¤ered by the captive and local insurers. Substituting T (t 1 ; : : : 3) and ( 4), the optimal insurance captive scheme maximizes

; t n ) = h P n i=1 (1 + b c i )t i D i + ; in (
W = E " B 0 n X i=1 k i min ( X i2A c (1 + b c i )t i (e x i ); D ) X i2A c b f i Q !# + n X i=1 E [B i (w i e x i + I i (e x i ) P i + t i (e x i ) k i )] ; (5)
with respect to k i ; P i and I i (:); t i (:) 0 for all i and D 0 subject to (2) and

Q = (1 + )E "( n X i=1 (1 + b c i )t i (e x i ) D ) + # + f 0 : (6)
Proposition 2 characterizes the optimal insurance captive mechanism when …xed underwriting costs lead each business unit to buy insurance either from the local insurer or from the captive, but not from both simultaneously.

Proposition 2 When A `\ A c = ;, an optimal insurance captive scheme is de…ned by

d 0 i ; d 1 i > 0 such that I i (x i ) = x i d 0 i + if i 2 A `; t i (x i ) = 0 if x i d 1 i and dt i (x i )=dx i 2 (0; 1) if x i > d 1 i if i 2 A c :
The optimal insurance coverage obtained from local insurers is a straight deductible contract, i.e. local insurer i fully covers the losses above the deductible d 0 i if i 2 A `. This is a well-known optimal insurance characterization under constant loading and there is nothing new here. As for the captive, it o¤ers coinsurance coverage to business units i in A c when losses exceed threshold d 1 i . As shown in the proof of Proposition 2, a positive optimal transfer t i (x i ) equalizes the expected marginal value of cash in business unit i and in the captive. This equality may be written as

B 0 i (w i x i + t i (x i ) k i ) = (1 + b c i )E[B 0 0 ( 0 ) j t i (x i )] (7)
where E[B 0 0 ( 0 ) jt i (x i ) ] denotes the expected marginal value of an increase in the captive pro…t conditionally on t i (x i ). [START_REF] Mossin | Aspects of rational insurance purchasing[END_REF] We have @E[B 0 0 ( 0 ) jt i ]=@t i > 0 because B 0 (:) is a concave function and an increase in t i decreases the captive expected cash ‡ows anything else given, because of the reinsurance deductible D. Di¤erentiating (7) and using B 00 i < 0 then gives dt i (x i )=dx i 2 (0; 1). Note that D = 0 would be optimal if = 0: We would have dt i (x i )=dx i = 1 in that case, which highlights the fact that the coinsurance arrangement between the captive and the business units arises from the reinsurance loading. The proof of Proposition 2 also shows that it would be optimal to cover losses of any size if b c i = 0, but because of the captive claims handling costs small losses should not be covered, i.e. we have d 1 i > 0. We thus have a clear-cut connection between the captive cost factors and the shape of its indemnity schedule: …rstly business units should receive partial coverage "at the margin" because the captive is subject to reinsurance costs and secondly small losses should be retained because of claims handling costs.

The optimal insurance captive scheme depends on all the parameters which have been introduced so far, including : local insurers capital cost b k i , claims handling costs c i ; b c i , underwriting …xed costs f i ; b f i ; f 0 , reinsurance loading and initial endowments w 1 ; : : : ; w n . The optimal scheme also depends on the probability distribution of losses e x 1 ; : : : ; e

x n and on the shape of value functions B 0 ; B 1 ; : : : ; B n . We are more particularly interested in the e¤ect of capital cost, reinsurance loading and initial endowments. Indeed the activity of the captive industry is highly cyclical and understanding how price and wealth variations result in more or less risk cessions to the captive may contribute to a better understanding of this cyclical activity.

Let us present numerical examples derived from a simpli…ed version of the model. Consider a symmetric version of our model, with n identical business units: n c units are insured by the captive and n `= n n c units purchase insurance from local insurers. We thus assume A `\ A c = ; because of …xed underwriting costs. For each business unit, an accident occurs with probability p, and in that case the loss x is uniformly distributed over an interval [0; x]. We also assume B i ( i ) = ln( i ) for all i = 0; 1; : : : ; n. Let c ; b c ; k ; f; b f ; P be respectively the claims handling costs, the capital cost, the …xed underwriting cost and the premium charged by local insurers. w 1 denotes the initial endowment of a business unit and k c (respect. k `) denotes the transfer to the captive from any business unit that is insured (respect. not insured) by the captive. Finally and D still denote the reinsurance loading and deductible. In all our derivations we assume n = 10; p = 0:1; x = 0:5; f = b f c = 0:005; f 0 = 0.20 When X 1 ; : : : ; X n are i:i:d:random variables with the same distribution as X, we write X n P n i=1 X i . Equations ( 2),( 3) and ( 5) are then rewritten as

P = (1 + c + k ) E[I(e x)] + f; Q = (1 + ) E (1 + b c ) (t(e x)) nc D + ; W = E h ln n c k c + (n n c )k ` n c b f c Q min n D; (1 + b c ) [t(e x)] nc o i +n c E [ln(w 1 e x + t(e x) k c )] + n l E [ln(w 1 e
x + I(e x) P k `)] :

where I(x) and t(x) are the indemnity schedules respectively o¤ered by local insurers and by the captive, with I(x) = (x d 0 ) + . For the sake of computational simplicity, we restrict attention to piecewise linear indemnity schedules o¤ered by the captive and we thus write t(x) = a(x d 1 ) + , with 0 < a < 1. W is maximized with respect to n c 2 f0; 1; : : : ; 10g; k c ; k `; d 0 ; d 1 ; a; D subject to the constraints that de…ne P and Q. Table 1 allows us to visualize the e¤ects of changes in cost parameters and wealth on the optimal insurance captive scheme. [START_REF] Schmit | Cost e¤ectiveness of risk management practices[END_REF] RRRH denotes the Risk Retention Rate measured at the Holding level. Results show that an all or nothing solution is optimal, i.e. all business units should purchase insurance from the captive or none of them should do it. When c = b c (no structural e¢ ciency gap), we know from Proposition 1 that a su¢ cient condition for purchasing insurance from the captive to be optimal is (approximately) k > . When k < , the optimal scheme trades o¤ the advantages of an umbrella policy o¤ered by the captive against the lower costs of local insurers. In the cases considered in Table 1, the captive dominates local insurers for values of the capital cost k that are lower than the reinsurance loading . Note that a 20% rate of return on capital and a ratio of capital to expected loss of 20% would correspond to k = 0:04. When c = b c = = 0:05 and w 1 = 1, transferring risks to the captive and reinsuring them with deductible D = 0:10 is more advantageous than purchasing insurance from local insurers even if their cost of capital loading is only k = 0:045 (and a fortiori if it is larger). The local insurer solution dominates the captive solution when k 0:04 and there is a jump in RRRH from 29:4% to 49:0% when self-insurance through the captive becomes optimal after a small increase in the cost of capital from 0:04 to 0:045. Increases in claims handling costs c and b c lead to larger deductibles either by local insurers or by the captive, which is in line with standard results in optimal insurance contract theory [START_REF] Schlesinger | The theory of insurance demand[END_REF] : for instance if = 0:05; w 1 = 1 and c and b c increase from 0:05 to 0:1, then d 0 increases from 0:07 to 0:11 if k = 0:03 and d 1 increases from 0:06 to 0:11 if k 0:047. distribution of the cumulated costs transferred to the insurance captive. We derived these values by numerical integration and solved the program with a heuristic algorithm. In this algorithm, units were 10 3 for k c and k `and 10 2 for the other variables. Computational di¢ culties were mainly about the risk transferred to the captive, because of the convolution law [t(e x)]

nc . Note that the optimal solution obtained for n c = n does not depend on the cost of capital k . Such a solution was compared with the optimal solution obtained for n c = 0, which depends on k . Dominant interior solutions with respect to n c 2 (0; n) were not found. [START_REF] Schlesinger | The theory of insurance demand[END_REF] The consequences of an increase in the loading factor on the optimal level of insurance coverage may be analysed by distinguishing a substitution e¤ect and an income e¤ect, as in the classical theory of consumer demand; see Schlesinger (2000). Table 1 also illustrates the consequences of an increase in the reinsurance loading factor. Consider the case where k 0:075; c = b c = 0:05; w 1 = 1 : if increases from 0:05 to 0:1, then D increases from 0:10 to 0:17, which is the standard direct e¤ect of an increase in the reinsurance loading, and simultaneously d 1 increases from 0:06 to 0:09 and a lowers from 0:90 to 0:88, which is an induced e¤ect : higher reinsurance costs lead the captive to provide less complete insurance coverage to business units. Finally, increasing the business units'initial endowments leads to more retention within the captive and the business units, because the ln(:) function entails the same kind of wealth e¤ect as the decreasing absolute risk aversion of standard expected utility theory. For instance, when c = b c = = 0:05, if w 1 increases from 1 to 2, then D increases from 0:10 to 0:16, d 1 increases from 0:06 to 0:13 and a decreases from 0:90 to 0:81 if the business units purchase insurance from the captive, while d 0 increases from 0:07 or 0:08 to 0:15 if insurance is obtained from local insurers, with an important increase in the retention rate in both cases.

In short, some important features of an optimal insurance captive scheme emerge from these numerical results, where there is no structural e¢ ciency gap between local insurers and the captive. Firstly, because of the umbrella policy mechanism, using a captive may be pro…table to the …rm even if reinsurance costs exceed the savings on direct insurance capital cost. Conversely, for given reinsurance costs, if the local insurers'capital cost is low enough (which is likely to be the case in periods of soft insurance markets with large insurers'surplus), then the …rm may turn to local insurers and keep its captive dormant, even if it means reinvigorating the captive later, under more favorable market conditions. Secondly, changes in reinsurance loading lead to more or less risk cessions by the captive to reinsurers, and indirectly to more or less retentions in business units. Since, capital costs and reinsurance rates do not move in a perfectly synchronized way, there are arbitrage opportunities, which also explains the structural instability in the activity of the captive industry. Thirdly, because of a wealth e¤ect the larger the …rm's initial endowment, the larger the risk retention in the captive and in the business units, which reinforces the price e¤ect argument for the variability in the captive activity. When changes in price or wealth parameters make the use of the captive pro…table, there is a jump in the risk retention rate of the holding company, which illustrates the role of captives as expansion tanks for the parent's self-insurance capacity.

Fronting and reinsurance captives

Let us come back to our general model and assume now that the captive can also act as a reinsurer, with local insurers behaving as fronters for the business units. The cession of risks by insurer i to the captive is written as C i (I i ); K i where C i (I i ) refers to the transfer from the captive to insurer i as a function of losses I i , with 0 C i (I i ) I i and K i is the amount of premium ceded to the captive. [START_REF] Scordis | Conditions for captive insurer value :a Monte Carlo simulation[END_REF] With such cessions, the claims handling costs incurred by local insurer i are still proportional to the amount of claims I i (x i ) as before, but now capital costs are proportional to retained claims I i (x i ) C i (I i (x i )) since no capital allocation is required for ceded risks. Because of competitive pressures, ceded premiums are equal to the decrease in local insurers'capital cost induced by cessions. We thus have

K i = (1 + k i )EC i (I i (e x i )) (8)
The captive cedes or retrocedes risks through a reinsurance contract that speci…es a transfer to the captive T (t 1 ; : : : ; t n ; C 1 ; : : : ; C n ) as a function of the payments to business units t 1 ; : : : ; t n and to local insurers C 1 ; : : : ; C n , with C i = C i (I i ) for all i. We thus have Q = (1 + )E[T (t 1 (e x 1 ); : : : ; t n (e x n ); C 1 (I 1 (e x 1 )); : : : ; C n (I n (e x n ))] + f 0 :

The captive's pro…t is now written as

0 = n X i=1 k i X i2A c [(1 + b c i )t i (x i ) + b f i ] + X i2A `[K i C i (I i (x i ))]
+T (t 1 (x 1 ); : : : ; t n (x n ); C 1 (I 1 (x 1 )); : : :

; C n (I n (x n ))) Q; ( 10
)
where A c and A `still respectively denote the set of business units that are directly insured by the captive or by local insurers . An insurance-reinsurance captive scheme is now written as fI i (x i ); P i ; t i (x i ); k i ; C i (I i ); K i for i = 1; : : : ; n; T (t 1 ; : : : ; t n ; C 1 ; : : : ; C n ); Qg. It is feasible if it satis-…es (2),( 8) and ( 9). An optimal insurance captive scheme maximizes W = E[ P n i=0 B i ( i )], where 0 and 1 ; : : : ; n are respectively given by ( 10) and (1).

Proposition 3 shows that the optimal risk sharing mechanism takes a simple form when local insurers have lower claim handling costs and underwriting costs than the captive. Indeed using local insurers as fronters allows the parent group to cumulate the advantages of an umbrella policy through the retrocession of risks by the captive and the bene…ts of lower claim handling and underwriting costs, while the direct insurers capital costs can be recouped through ceded premiums. Proposition 3 is illustrated with Figure 2 : all the claims go through the local insurers who act as fronters and the captive only acts as a reinsurer.

Proposition 3 An optimal insurance-reinsurance captive scheme speci…es that t i (x i ) 0 for all i such that c i b c i (1 + ) and f i b f i , i.e. when variable claim handling costs and …xed underwriting costs are lower for local insurers than for the captive, then local insurers are fronters: the business units purchase insurance from local insurers only. The captive retrocedes a part of the underwritten risks through a a stop-loss reinsurance contract T (C 1 ; : : :

; C n ) P n i=1 ((1 + b c i )t i + C i ) D + , with D > 0.

Figure 2

From now on we will assume c i b c i (1 + ) and f i b f i for all i and thus A c = ? and A `= f1; : : : ; ng. The remaining part of the paper restricts attention to this case of a pure reinsurance captive. Using T (: : : ; C i ; : : :) = ( P n i=1 C i D) + , an optimal insurance scheme maximizes

W = E " B 0 n X i=1 (k i + K i ) min ( n X i=1 C i (I i (e x i )); D ) Q !# + n X i=1 E [B i (w i e x i + I i (e x i ) P i k i )] ; (11)
with respect to k i ; K i ; P i ; I i (:); C i (:) 0 for all i and D 0 subject to (2), ( 8), 0 C i (I i ) I i and

Q = (1 + )E " n X i=1 C i (I i (e x i )) D ! + # + f 0 : (12)
Proposition 4 An optimal reinsurance captive scheme is de…ned by I i 2 (0; +1); d 1 i > d 0 i > 0 for all i = 1; : : : ; n such that C i (I i ) = minfI i ; I i g;

I i (x i ) = 0 if x i d 0 i ; dI i (x i )=dx i 2 (0; 1) if d 0 i x i d 1 i + I i ; I i (x i ) = x i d 1 i if x i > d 1 i + I i :
Proposition 4 is illustrated with Figure 3. 24 I i is an upper limit on cessions by local insurer i to the reinsurance captive and d 0 i is a deductible for business unit i, with partial marginal coverage over [d 0 i ; d 1 i + I i ] and full marginal coverage above d 1 i + I i .

Figure 3

The intuition of Proposition 4 is as follows. Let us look …rst at the determinants of optimal cessions to the reinsurance captive. For a given retrocession deductible D and for given I j (:) for all j and C j (:) for all j 6 = i, increasing C i (I i ) when I i is in some interval [a; b] leads to larger ceded premiums K i and also to larger costs incurred by the captive either directly when P n j=1 C j < D or indirectly through an increase in the retroceded premium Q. The optimal cession scheme C i (I i ) = minfI i ; I i g trades o¤ the gains drawn from the ceded premiums and the risk sharing advantage of an umbrella policy against the increase in the captive risks.

A precise de…nition of the cession upper limit I i is provided in the proof of Proposition 4 but its intuition is rather straightforward. Let E[B 0 0 ( 0 )] be the expected marginal shareholder value of an increase in the captive pro…t and let E[B 0 0 ( 0 ) jC i ] denote this expected marginal value conditionally on the transfer from the captive to local insurer i being equal to C i . [START_REF] Smith | Analyzing the tax deductibility of premiums paid to captive insurers[END_REF] The proof of Proposition 4 shows that it is optimal to increase C i when E[B 0 0 ( 0

) jC i ] < (1 + k i )E [B 0 0 ( 0 )
] because in that case, given the local insurer's cost of capital k i , the positive shareholder value of larger recouped premium exceeds the negative shareholder value of larger captive liabilities. It is thus optimal to increase C i as much as possible, and thus to choose C i (I i ) = I i , when I i < I i , where I i is de…ned by

E[B 0 0 ( 0 ) jC i = I i ] = (1 + k i )EB 0 0 ( 0 ):
Hence C i (I i ) = I i is optimal when I i I i and the …rm should choose C i (I i ) = I i when

I i > I i .
The shape of the optimal cession treaty C i (I i ) = minfI i ; I i g a¤ects the shape of the insurance coverage provided by the local insurer to the business unit. When x i is not too large, we have I i (x i ) < I i . Then any increase in I i (x i ) is re ‡ected by an equivalent increase in the ceded risks C i (I i ). Hence when x i is increasing, the increase in I i (x i ) should reach an optimal compromise between providing additional coverage to business unit i and not inducing too much additional risks for the captive. This compromise leads to partial coverage "at the margin", i.e. a coinsurance mechanism dI i (x i )=dx i 2 (0; 1). Insurance coverage is triggered only when the loss is larger than the threshold d 0 i since, as [START_REF] Scordis | Captive insurance companies and manager-owner con ‡icts[END_REF] For the sake of graphical simplicity, in Figure 3 function I i (x i ) is drawn as if it were piecewise linear, which may not be the case. The same remark applies for I i (x i ) and C i (I i ) in the case of Figure 4 below. [START_REF] Smith | Analyzing the tax deductibility of premiums paid to captive insurers[END_REF] See the proof of Proposition 3 in the Appendix for explicit de…nitions of E[B 0 0 ( 0 )] and E[B 0 0 ( 0 ) jC i ]. In these expressions, expected values are taken with respect to transfers from the captive to all local insurers j 6 = i by taking into account the captive's retrocession operations. We have @E[B 0 0 ( 0 ) jC i ]=@C i > 0 because B 0 (:) is a concave function and an increase in C i decreases the captive pro…t anything else given. usual, because of transaction costs it is optimal to have zero coverage when losses are very low. When x i is large enough (x i d 1 i +I i ), we have I i (x i ) I i and then increasing I i (x i ) has no e¤ect on the captive earnings : it is only re ‡ected by an increase in the premium P i paid to the local insurer, without any change in premiums ceded to the captive. In that case, the optimal insurance contract then speci…es full coverage at the margin, i.e.

I i (x i ) = x i d 1
i as in standard insurance contract with constant loading. We have assumed so far that the capital loading is proportional to the expected indemnity cost incurred by local insurers. Although proportional loadings are usual in the literature on optimal insurance contracting, this is not a very satisfactory assumption when corporate insurance demand is at stake, particularly for multinational corporations that may incur large risks, and it may even be regarded as completely unacceptable when the focus is on insurance capital requirements. Indeed the capital costs involved by insurers' risk management and by regulatory constraints may be small or even negligible for small risks that do not a¤ect the solvency of the insurers, while they strongly increase when insurers face large size liabilities. This is more adequately modelled by assuming that capital costs depend on insurers'liabilities in a non-linear way.

Let us now write insurer i's capital cost as E[Z i (I i (e x i ))], where function Z i (:) is such that Z 0 i > 0 and Z 00 i 0. So far we had Z i (x i ) k i x i , but now assuming more generally that Z i (:) is a convex function allows us to consider the case where small risks do not involve any signi…cant e¤ect on insurers'required capital, while an increase in the exposure to large risks would result in a large increase in capital loading. Under this assumption, the premium charged by local insurer i is written as

P i = (1 + c i )E [I i (e x i )] + E [Z i (I i (e x i ))] + f i ( 13
)
instead of (2). [START_REF] Tirole | The Theory of Corporate Finance[END_REF] The premiums which are ceded by local insurer i to the reinsurance captive include the decrease in the capital cost, which goes down from

E[Z i (I i (e x i ))] to E[Z i (I i (e x i ) C i (I i (e x i )
))] when the local insurer's risk exposure goes down from I i (e x i ) to I i (e x i ) C i (I i (e x i )). We thus have

K i = E[C i (I i (e x i ))] + E[Z i (I i (e x i ))] E[Z i (I i (e x i ) C i (I i (e x i )))]; (14)
instead of (8).

26 Equation ( 13) re ‡ects the empirical observation that capital requirements increase more than proportionally with the size of the insurer's liabilities, and that the highest loss layers have the strongest e¤ect on the allocation of capital among lines of business. This will be the case if the insurer allocates capital to lines of business based upon their contribution to the total …rm's VaR (99%) loss scenario or if Tail Value at Risk (TVaR) is used as a basis for allocating capital. In such cases, capital is mainly allocated to perils that contribute to the worst scenarios -see Kreps (2005) and Venter (2006). Bodo¤ (2009) describes a formulation of VaR capital that allocates capital to all losses rather than only to extreme losses in the tail of the distribution, but in Bodo¤'s formulation capital is allocated disproportionally to severe losses. A di¤erent approach would consist in assuming that insurance prices are given by the expectation of costs (particularly capital costs) with respect to a distorted probability as in the approach of Wang (1996, 2000).

Proposition 5 Assume that the local insurer's cost of capital is E[Z i (I i (e x i ))] with Z 0 i > 0 and Z 00 i 0. Then an optimal insurance scheme is de…ned by I i 2 (0; +1); d 1 i > d 0 i > 0 for all i = 1; : : : ; n such that

C i (I i ) = I i if I i I i ; 0 < dC i dI i < 1 if I i > I i ; I i (x i ) = 0 if x i d 0 i ; dI i (x i )=dx i 2 (0; 1) if x i > d 0 i ; with I 0 i+ (d 1 i ) > I 0 i (d 1 i ) if I i (d 1 i ) = I i .
Proposition 5 is illustrated with Figure 4. Assuming that the marginal cost of capital is increasing provides a di¤erent picture of the optimal cession mechanism. Now the equality between expected costs and expected bene…ts of an increase in ceded premiums is written as

E[B 0 0 ( 0 ) jC i ] = [1 + Z 0 i (I i C i )]E[B 0 0 ( 0 )];
where the variable marginal cost of capital Z 0 i (I i C i ) has been substituted to k i . There exists an upper limit on indemnities I i given by

E[B 0 0 ( 0 ) jC i = I i ] = [1 + Z 0 i (0)]E[B 0 0 ( 0 )
] such that the risk is fully ceded to the reinsurance captive when I i is lower than I i , but there are partial cessions "at the margin" when I i exceeds I i . As in the case of constant marginal cost of capital, the optimal cession scheme C i (I i ) results from the trade-o¤ between on one side the gain drawn both from the ceded premiums and from the better risk sharing provided by an umbrella policy and on the other side the increase in the captive costs. When I i < I i the balance tips in favor of increasing the cession even with the smallest capital marginal costs Z 0 i (0) ceded to the captive and in that case C i (I i ) = I i is optimal. When I i > I i , it is optimal to increase the cession C i (I i ) up to the level where the expected gain in the …rm's marginal value

[1 + Z 0 i (I i C i )]E[B 0 0 ( 0 )
] derived from ceded premiums matches the expected loss due to larger indemnity payment

E[B 0 0 ( 0 ) jC i ], hence a marginal cession rate dC i dI i = Z 00 i (I i C i ) Z 00 i (I i C i ) + @E[B 0 0 jC i ] @C i 2 (0; 1); where = E[B 0 0 ( 0 )].
The optimal indemnity schedule involves partial insurance at the margin dI i (x i )=dx i 2 (0; 1) because an increase in coverage either involves an equivalent increase in the risks incurred by the captive when I i (x i ) < I i and also an increase in the local insurer's cost of capital which is only partially recouped through premium cessions when I i (x i ) > I i .

Figure 4

Propositions 4 and 5 are illustrated by examples derived from the same basic assumptions as in Section 3 : 10 symmetric business units, B i ( i ) = ln( i ) for all i = 0; 1; : : : ; 10; and each business unit facing the risk of an accident with probability p = 0:1, with losses e x uniformly distributed over [0; x] and x = 0:5. We restrict attention to piecewise linear indemnity schedules. In the case of a constant marginal cost of capital (Proposition 4), we thus assume that I 0 (x) is constant and equal to a 2 (0; 1) when x 2 [d 0 ; d 1 + I ]. [START_REF] Venter | Adapting banking models to insurer ERM[END_REF] Increasing cessions to the captive trades o¤ the advantages of an umbrella policy obtained through the stop-loss reinsurance policy against the costs of the captive's liabilities. Table 2 con…rms that ceteris paribus (i.e. for given values of c ; and w 1 ) , the higher k , the higher I because a higher capital cost makes risk cessions more pro…table. We also observe that increasing from 0:05 to 0:1 (with k = 0:03; 0:04 or 0:05; c = 0:05 and w 1 = 1) leads to higher values for D, once again this the direct e¤ect of a higher reinsurance loading, but it also leads to lower values for I because of an induced e¤ect of the cost of retrocessions on optimal cessions. The e¤ects of changes of c and w 1 are as expected: when k = 0:03; 0:04 or 0:05; increasing c from 0:05 to 0:1 and keeping = 0:05 and w 1 = 1 or increasing w 1 from 1 to 2 and keeping c = = 0:05 leads to larger d 0 : these are standard price and wealth e¤ects associated with changes in the loading factor or initial endowment. w 1 d 0 a I k D RRRH 0:03 0:05 0:05 1 0:07 0:92 0:04 0:088 0:07 38:2% 0:04 0:05 0:05 1 0:07 0:87 0:07 0:088 0:09 46:4% 0:05 0:05 0:05 1 0:06 0:77 0:07 0:088 0:09 46:4% 0:03 0:1 0:05 1 0:11 0:93 0:04 0:088 0:07 50:2% 0:04 0:1 0:05 1 0:11 0:87 0:06 0:088 0:08 55:8% 0:05 0:1 0:05 1 0:11 0:88 0:12 0:088 0:09 61:6% 0:03 0:05 0:1 1 0:07 0:94 0:03 0:088 0:10 36% 0:04 0:05 0:1 1 0:08 0:99 0:04 0:088 0:11 42% 0:05 0:05 0:1 1 0:09 1 0:05 0:088 0:12 47:7% 0:03 0:05 0:05 2 0:12 0:73 0:07 0:179 0:13 65:6% 0:04 0:05 0:05 2 0:11 0:69 0:12 0:179 0:15 76:1% 0:05 0:05 0:05 2 0:10 0:68 0:16 0:179 0:16 81:7%

The case of a convex cost of capital is illustrated by assuming Z(I) = k I+0:05I

2 ; I 0 (x) = a if x 2 [d 0 ; d 1 ] and I 0 (x) = b if x > d 1 (or equivalently if I(x) > I ) with 0 < a < b < 1.
The marginal cession rate is denoted as C 0 (I) = c, with 0 < c < 1 when I > I . Results are in Table 3, with expected e¤ects of parameter changes. In particular, an increase in the capital cost parameter k stimulates the risk cessions to the captive by increasing both I and c.

Table 3 : The case of a convex cost of capital

Parameters

Variables

k c w 1 d 0 a b I c k
D RRRH 0:03 0:05 0:05 1 0:07 0:93 0:95 0:05 0:25 0:088 0:09 48:7% 0:04 0:05 0:05 1 0:07 0:85 0:97 0:06 0:58 0:088 0:10 52:3% 0:05 0:05 0:05 1 0:07 0:88 0:99 0:12 0:99 0:088 0:10 52:8% 0:03 0:1 0:05 1 0:11 0:93 0:95 0:05 0:19 0:088 0:08 57:9% 0:04 0:1 0:05 1 0:11 0:85 0:97 0:06 0:55 0:088 0:09 61% 0:05 0:1 0:05 1 0:10 0:82 0:99 0:11 0:99 0:088 0:09 61:1% 0:03 0:05 0:1 1 0:08 0:97 0:98 0:04 0:06 0:088 0:12 46:8% 0:04 0:05 0:1 1 0:08 0:94 0:95 0:05 0:09 0:088 0:13 52% 0:05 0:05 0:1 1 0:06 0:70 0:93 0:06 0:10 0:088 0:14 56:5% 0:03 0:05 0:05 2 0:10 0:65 0:91 0:09 0:20 0:179 0:15 75:6% 0:04 0:05 0:05 2 0:11 0:71 0:89 0:12 0:32 0:179 0:16 80:8% 0:05 0:05 0:05 2 0:11 0:72 0:99 0:17 0:99 0:179 0:16 81:9%

Conclusion

This paper has focused attention on the role of insurance and reinsurance captives as optimal risk sharing arrangements for large corporations with decentralized business units. We have modelled the role of captives as self-insurance devices that pool risks, transfer them to reinsurers and allow parent groups to o¤er optimal insurance coverage to business units, either directly or through fronters. Several conclusions emerge from this analysis. Firstly, reinsurance captives dominate insurance captives when standard direct insurers have lower claims handling and underwriting costs than captives: indeed, through a reinsurance captive the holding company may cumulate the bene…ts drawn from the lower costs of the fronters and the risk sharing optimality of an umbrella policy o¤ered by retrocessionnaires. Secondly, the optimal insurance scheme of decentralized business units involves deductible insurance contracts with partial coverage at the margin (at least when losses are in some interval) when business units are insured either directly by an insurance captive or through a fronter that cedes its risks to a reinsurance captive. Thirdly, when local insurers act as fronters, they fully cede their risks to the reinsurance captive up to a certain limit. Partial cessions above this limit are optimal under increasing marginal cost of capital. The reinsurance captive retrocedes its risk to the reinsurance market through a stop-loss contract, thereby o¤ering an umbrella insurance policy to the holding company.

Fourthly, the optimal insurance coverage as well as the optimal cession and retrocession operations depend on price parameters, and particularly on the capital cost of local insurers and on the loading factor of retrocessionnaires, as well as on initial endowments of the business units. Variations across time of these cost and wealth parameters a¤ect the attractivity of captive arrangements and may result in ‡uctuations in the activity of the captive industry. i ) 1 + i and b f i f i and that are insured (resp. not insured) by the captive under P 0 . Let C 2 be the set of other units, with C 0 [ C 1 [ C 2 = f1; : : : ; ng. Consider another feasible scheme P 1 = fI 1 i (x i ); P 1 i ; t 1 i (x i ); k 1 i for i = 1; : : : ; n; T 1 (t 1 ; : : : ; t n ); Q 1 g, with

I 1 i (x i ) 0; P 1 i = 0; t 1 i (x i ) t 0 i (x i ) + I 0 i (x i ); k 1 i = k 0 i + P 0 i if i 2 C 0 [ C 1 and I 1 i (x i ) I 0 i (x i ); P 1 i = P 0 i ; t 1 i (x i ) t 0 i (x i ); k 1 i = k 0 i if i 2 C 2 . Let 0 i (x i
) and 1 i (x i ) be the current cash ‡ows of business unit i respectively under P 0 and P 1 as a function of losses x i . We obviously have 0 i (x i )

1 i (x i ) from (1). Let for all i i (r i ) = I 0 i (x i ) I 0 i (x i ) + t 0 i (x i ) 2 [0; 1] if I 0 i (x i ) + t 0 i (x i ) = r i > 0:
Since I 0 i (x i ) and t 0 i (x i ) are non-decreasing, i (r i ) is well de…ned for all r i > 0. Let T 1 (t 1 ; : : : ; t n ) T 0 (:

: : ; (1 i (t i ))t i ; : : :) + X i2C 0 [C 1 (1 + b c i ) i (t i )t i : (15)
Using ( 3) and (15) gives

Q 1 = (1 + )E[T 1 (: : : ; t 1 i (e x i ); : : :)] + f 0 = (1 + )E[T 0 (: : : ; (1 i (t 1 i (e x i )))t 1 i (e x i ); : : : ; )] +(1 + )E X i2C 0 [C 1 (1 + b c i ) i (t 1 i (e x i ))t 1 i (e x i ) + f 0 Using (3) and (1 i (t 1 i (x i )))t 1 i (x i ) t 0 i (x i ) and i (t 1 i (x i ))t 1 i (x i ) I 0 i (x i ) for all i yields Q 1 = Q 0 + (1 + )E X i2C 0 [C 1 (1 + b c i )I 0 i (e x i ) Q 0 + X i2C 0 [C 1 (1 + i )EI 0 i (e x i ) = Q 0 + X i2C 0 0 [C 1 (P 0 i f i ): (16)
where C 0 0 C 0 refers to the set of business units that belongs to C 0 and that are simultaneously insured by the captive and the local insurer in P 028 . Let 0 0 (x 1 ; : : : ; x n ) and 1 0 (x 1 ; : : : ; x n ) be the pro…t of the captive respectively under P 0 and P 1 . Let C 0 2 be the subset of C 2 that gathers the units of C 2 that are insured by the captive (and also possibly by local insurers). Under P 1 the captive provides insurance to the business units in C 0 [ C 1 [ C 0 2 . Hence using ( 4), ( 15) and ( 16) gives 1 0 (x 1 ; : : :

; x n ) = n X i=1 k 1 i X i2C 0 [C 1 [C 0 2 [(1 + b c i )t 1 i (x i ) + b f i ] + T 1 (: : : ; t 1 i (x i ); : : :) Q 1 n X i=1 k 1 i X i2C 0 [C 1 [C 0 2 [(1 + b c i )t 1 i (x i ) + b f i ] + T 0 (: : : ; (1 i (t 1 i (x i )))t 1 i (x i ); : : :) + X i2C 0 [C 1 (1 + b c i ) i (t 1 i (x i )t 1 i (x i ) Q 0 X i2C 0 0 [C 1 (P 0 i f i ): Using …rstly (1 i (t 1 i (x i )))t 1 i (x i ) t 0 i (x i ) for all i, secondly k 1 i = k 0 i + P 0 i if i 2 C 0 [ C 1 with P 0 i = 0 if i = 2 C 0 0 and k 1 i = k 0 i if i 2 C 2 , thirdly t 0 i (x i ) 0 if i 2 C 1 and fourthly t 1 i (x i ) = t 0 i (x i ) if i 2 C 0 2 then gives 1 0 (x 1 ; : : : ; x n ) n X i=1 k 0 i X i2C 0 [C 1 [C 0 2 b f i + X i2C 0 0 [C 1
P 0 i + T 0 (: : : ; t 0 i (x i ); : : :)

X i2C 0 [C 0 2 (1 + b c i )t 0 i (x i ) Q 0 X i2C 0 0 [C 1 (P 0 i f i ) = n X i=1 k 0 i X i2C 0 [C 0 2 [(1 + b c i )t 0 i (x i ) + b f i ] + T 0 (: : : ; t 0 i (x i ); : : :) Q 0 + X i2C 1 (f i b f i ) + X i2C 0 0 f i = 0 0 (x 1 ; : : : ; x n ) + X i2C 1 (f i b f i ) + X i2C 0 0 f i 0 0 (x 1 ; : : : ; x n ):
Hence P 1 weakly dominates P 0 (it provides the same pro…t to each business units and no lower pro…t to the captive in all states of the world). P 1 strongly dominates P 0 if C 0 0 6 = ? or if f i > b f i for at least one i in C 1 : Given fI i (:); P i ; t i (:); k i for all i = 1; : : : ; ng the optimal reinsurance policy fT (:); Qg maximizes EB 0 ( 0 ), where 0 is given by (4), subject to (3). Using the concavity of B 0 (:) and the Umbrella Policy Theorem (see Gollier, 2000) shows that T (:) is a function of the aggregate loss of the captive

P i (1 + b c i )t i (x i ).
Using > 0 and a standard result in optimal insurance contract theory then yields T (t 1 ; : : : ; t n )

h P i (1 + b c i )t i D i + , with D > 0.

Proof of Proposition 2

Consider a given index i 2 f1; : : : ; ng. Let e y i P j6 =i (1 + b c j )t j (e x j ) and let F i (y i ) be the cumulative distribution function of e y i over R, with F i (0) = 0 and F i (y i ) = 1, where y i = P j6 =i (1 + b c j )t j (x j ). Choosing k j ; I j (:) and t j (:) optimally for all j 6 = i, and thus for an optimal distribution F i (y i ), the optimal insurance captive scheme maximizes

x i Z 0 D (1+ b c i )t i (x i ) Z 1 B 0 n X j=1 k j y i (1 + b c i )t i (x i ) X j2A c b f j Q ! dF i (y i )d (x i ) + x i Z 0 [1 F i (D (1 + b c i )t i (x i ))]B 0 n X j=1 k j D X j2A c b f j Q ! d (x i ) + x i Z 0 B i (w i x i + I i (x i ) P i + t i (x i ) k i ) d i (x i );
with respect to D; Q; k i ; P i and I i (:); t i (:) 0, subject to

P i = (1 + c i + k i ) x i Z 0 I i (x i )d (x i ) + f i if i 2 A `; (17)
P i = 0 if i = 2 A `; (18)
Q = (1 + ) x i Z 0 Z y i D (1+ b c i )t i (x i ) (y i + (1 + b c i )t i (x i ) D)dF i (y i ) d (x i ) +f 0 : (19)
Let and be Lagrange multipliers associated with (17) and (19) respectively and let

EB 0 0 Z x i 0 D (1+ b c i )t i (x i ) Z 1 B 0 0 n X j=1 k j y i (1 + b c i )t i (x i ) X j2A c b f j Q ! dF i (y i )d (x i ) + x i Z 0 [1 F i (D (1 + b c i )t i (x i ))]B 0 0 n X j=1 k j D X j2A c b f j Q ! d (x i ); EB 0 i x i Z 0 B i (w i x i + I i (x i ) P i + t i (x i ) k i ) d i (x i ):
The …rst-order optimality conditions on P i and I i (:) respectively give EB 0 i = and

B 0 i (w i x i + I i (x i ) P i k i ) (1 + c i + k i ) 0; = 0 if I i (x i ) > 0; (20)
if i 2 A `(and thus i = 2 A c ), which yields

I i (x i ) = (x i d 0 i ) + , with d 0 1 > 0.
The optimality conditions associated with k i ; Q; D; t i (:)respectively give

EB 0 i = EB 0 0 ; (21) EB 0 0 = ; (22)
(1 + ) = B 0 0 n X j=1 k j D X j2A c b f j Q ! ; (23)
B 0 i (w i x i + t i (x i ) k i ) (1 + b c i )E[B 0 0 j t i (x i )] 0; = 0 if t i (x i ) > 0; (24)
for i 2 A c (and thus i = 2 A `) where

E[B 0 0 j t] D (1+ b i )t Z 1 B 0 0 n X j=1 k j y i (1 + b c i )t X j2A c b f j Q ! dF i (y i ) +[1 F i (D (1 + b c i )t)]B 0 0 n X j=1 k j D X j2A c b f j Q ! ; (25)
and

@E[B 0 0 j t] @t = (1 + b c i ) D (1+ b c i )t Z 1 B 00 0 n X j=1 k j y i X j2A c b f j Q ! dF i (y i ) > 0:
Di¤erentiating (24) gives

dt i dx i = B 00 i (w i x i + t i (x i ) k i ) B 00 i (w i x i + t i (x i ) k i ) (1 + b i ) @E[B 0 0 jt] @t t=t i (x i )
; and thus

dt i =dx i 2 (0; 1) if t i (x i ) > 0. ( 24
) implies t i (x i ) = 0 if x i 2 [0; d 1 i ] and t i (x i ) > 0 if x i > d 1 i where d 1 i is such that B 0 i (w i d 1 i k i ) = (1 + b c i )E[B 0 0 j 0]. Suppose d 1 i 0. ( 24
) would then give (21).

EB 0 i = x i Z 0 B 0 i (w i x i + t i (x i ) k i ) d i (x i ) = (1 + b c i ) x i Z 0 E[B 0 0 j t i (x i )]d i (x i ) = (1 + b c i )EB 0 0 > EB 0 0 ; hence a contradiction with

Proof of Proposition 3

Let P 0 = fI 0 i (x i ); P 0 i ; t 0 i (x i ); k 0 i ; C 0 i (I i ); K 0 i for i = 1; : : : ; n; T 0 (t 1 ; : : : ; t n ; C 1 ; : : : ; C n ); Q 0 g a feasible insurance-reinsurance captive scheme. Let C 0 (resp. C 1 ) be the set of business units such that c

I 0 i (x i ) I 0 i (x i ) + t 0 i (x i ) 2 [0; 1] if I 0 i (x i ) + t 0 i (x i ) = r i > 0 if i 2 C 0 [ C 1 ; i (r i ) 1 if i 2 C 2 :
Consider another insurance scheme P 1 = fI 1 i (x i ); P 1 i ; t 1 i (x i ); k 1 i ; C 1 i (I i ); K 1 i for i = 1; : : : ; n; T 1 (t 1 ; : : : ; t n ; C such that

I 1 i (x i ) I 0 i (x i ) + t 0 i (x i ) if i 2 C 0 [ C 1 ; C 1 i (I i ) C 0 i ( i (I i )I i ) + (1 i (I i ))I i if i 2 C 0 [ C 1 ; k 1 i k 0 i (1 + i )Et 0 i (e x i ) f i if i 2 C 0 ; k 1 i k 0 i (1 + i )Et 0 i (e x i ) if i 2 C 1 ; I 1 i (x i ) I 0 i (x i ); C 1 i (I i ) C 0 i (I i ); t 1 i (x i ) t 0 i (x i ); k 1 i = k 0 i if i 2 C 2 ; with P 1
i given by (2) for all i and Q 1 given by (3). Since P

1 i + k 1 i = P 0 i + k 0 i and I 1 i (x i ) + t 1 i (x i ) I 0 i (x i ) + t 0 i (x i ), we have 1 i (x i ) 0 i (x i
) for all i where 0 i (x i ) and 1 i (x i ) respectively denote the current cash ‡ows of business unit i under P 0 and P 1 as functions of losses x i . Let also T 1 (: : : ; t i ; : : : ; C i ; : : :) = T 0 (: : : ; t 0 i ; : : : ; C 0 i ; : : :)

X i2C 0 [C 1 b i t 0 i ; (26)
: ; x n ) = n X i=1 k 0 i X i2C 0 [C 1 [C 0 2 [(1 + b c i )t 0 i (x i ) + b f i ] + n X i=1 [K 0 i C i (I 0 i (x i ))] + T 0 (: : : ; t 0 i (x i ); : : : ; C 0 i (I 0 i (x i ) : : : ; ) Q 0 + X i2C 0 [C 1 [ b c i (1 + ) c i ]Et 0 i (e x i ) + X i2C 0 ( b f i f i ) + X i2C 1 f i = 1 0 (x 1 ; : : : ; x n ) + X i2C 0 [C 1 [ b c i (1 + ) c i ]Et 0 i (e x i ) + X i2C 0 ( b f i f i ) + X i2C 1 f i 1 0 (x 1 ; : : : ; x n ):
Hence in an optimal insurance scheme, any business unit i such that c i b c i (1 + ) and f i b f i should be directly insured by the local insurer only, possibly with reinsurance by the captive.

Given the transfer schedules t i (x i ) and C i (I i ), the captive bears a total random cost

P n i=1 [(1+ b c i )t i (x i )+C i (I i (x i ))].
The umbrella policy theorem then shows that the optimal reinsurance contract of the captive speci…es a transfer schedule T , which depends on this total loss, and using > 0 a standard result in insurance contract theory shows that it is a straight deductible contract.

Proof of Proposition 4

Part 1 : We …rst characterize functions C i (I i ). Consider a given index i 2 f1; : : : ; ng. Let

e z i X j6 =i C j (I j (e x j ))
and let G i (z i ) be the cumulative distribution function of e z i over R, with G i (0) = 0 and G i (z i ) = 1, where z i = P j6 =i C j (I j (x j )). Let i (I i ) be the cumulative distribution function of e I i I i (e x i ) over [0; I i ], with I i = I i (x i ). For K j ; C j (:) chosen optimally for all j 6 = i and k j ; I j (:) chosen optimally for all j, and thus for optimal induced distributions G i (z i ) and i (I i ), the optimal insurance scheme maximizes

Z I i 0 8 < : D C i (I i )) Z 1 B 0 n X j=1 (k j + K j ) z i C i (I i )) Q ! dG i (z i ) +[1 G i (D C i (I i ))]B 0 n X j=1 (k j + K j ) D Q !) d i (I i ) with respect to Q; D; K i ; C i (:) subject to K i = (1 + k i ) I i Z 0 C i (I i )d i (I i ); (28)
Q = (1 + )

I i Z 0 Z y i D C i (I i ) (z i + C i (I i ) D)dG i (z i ) d i (I i ) + f 0 ; (29)
0 C i (I i ) I i for all I i 2 [0; I i ]: (30)
Let and be Lagrange multipliers respectively associated with ( 28) and ( 29). The …rst-order optimality conditions associated with K i and Q are written as

EB 0 0 = = ; (31)
where

EB 0 0 Z I i 0 8 < : D C i (I i )) Z 1 B 0 0 n X j=1 (k j + K j ) z i C i (I i )) Q ! dG i (z i ) +[1 G i (D C i (I i ))]B 0 0 n X j=1 (k j + K j ) D Q) !) d i (I i ):
The …rst-order optimality conditions associated with D and C i (:) yield

(1 + ) = B 0 0 n X j=1 (k j + K j ) D Q ! ; (32) and
(1

+ k i ) E[B 0 0 j C i (I i )] 0; = 0 if C i (I i ) < I i ; (33)
where

E[B 0 0 j C] D C Z 1 B 0 0 n X j=1 (k j + K j ) z i C Q ! dG i (z i ) +[1 G i (D C)]B 0 0 n X j=1 (k j + K j ) D Q ! : (34)
We have

@E[B 0 0 j C] @C = D C Z 1 B 00 0 ( n X j=1 (k j + K j ) z i C Q)dG i (z i ) > 0:
Hence from (33), we have C i (I i ) = I i if I i I i and C i (I i ) = I i if I i > I i , where I i is given by (1 + k i ) E[B 0 0 j I i ] = 0: Part 2 : We now characterize functions I i (x i ). We know that there exists I i such that C i (I i ) = minfI i ; I i g. Choosing optimally k j ; K j ; I j (:) for all j 6 = i and C j (:) for all j, and thus for an optimal distribution G i (z i ), the optimal insurance scheme maximizes

x i Z 0 D C i (I(x i )) Z 1 B 0 n X j=1 (k j + K j ) z i C i (I i (x i )) Q ! dG i (z i )d i (x i ) + x i Z 0 [1 G i (D C i (I i (x i )))]B 0 n X j=1 (k j + K j ) D Q ! d i (x i ) + x i Z 0 B i (w i x i + I i (x i ) P i k i ) d i (x i );
with respect to D; Q; k i ; K i ; P i ; I i (:) 0, subject to

K i = (1 + k i ) x i Z 0 C i (I i (x i ))d i (x i ); (35)
Q = (1 + ) x i Z 0 Z y i D C i (I(x i )) (z i + C i (I i (x i )) D)dG i (z i ) d (x i ) +f 0 ; (36)
P i = (1 + c i + k i ) x i Z 0 I i (x i )d i (x i ) + f i ; (37)
The …rst-order optimality conditions on k i ; K i ; P i and Q yield

= = = EB 0 0 = EB 0 i ( 38
)
where multipliers ; and are respectively associated with (35),( 36) and (37) and

EB 0 0 = x i Z 0 D C i (I(x i )) Z 1 B 0 0 n X j=1 (k j + K j ) z i C i (I i (x i )) Q ! dG i (z i )d i (x i ) + x i Z 0 [1 G i (D C i (I i (x i )))]B 0 0 n X j=1 (k j + K j ) D Q ! d i (x i ); EB 0 i = x i Z 0 B 0 i (w i x i + I i (x i ) P i k i ) d i (x i ):
At any point where C i (I i ) is di¤erentiable, i.e. when I i 6 = I i , the …rst-order optimality condition on I i (:) is written as

B 0 i (w i x i + I i (x i ) P i k i ) + (1 + k i )C 0 i (I i (x i )) (1 + c i + k i ) C 0 i (I i (x i ))E[B 0 0 j C i (I i (x i ))] 0; = 0 if I i (x i ) > 0; (39)
where we use B 0 0 ( P n j=1 (k j + K j ) D Q) = (1 + ), which as before results from the optimality condition on D.

When I i (x i ) > I i , we have C 0 i (I i (x i )) = 0, and (39) gives

B 0 i (w i x i + I i (x i ) P i k i ) = (1 + c i + k i ) (40)
Since B 00 i < 0, (40) implies that I i (x i ) = x i d 1 i with

B 0 i (w i d 1 i P i k i ) = (1 + c i + k i ) (41)
. When 0 < I i (x i ) < I i , we have C 0 i (I i (x i )) = 1 and (39) gives

B 0 i (w i x i + I i (x i ) P i k i ) + (1 + k i ) (1 + c i + k i ) = E[B 0 0 j C i = I i (x i )];
which yields

B 0 i (w i x i + I i (x i ) P i k i ) = c i + E[B 0 0 j C i = I i (x i )]: (42)
EB 0 i = d 0 i Z 0 B 0 i (w i x i P i k i )d i (x i ) + Z I i +d 1 i d 0 i B 0 i (w i x i + I i (x i ) P i k i )d i (x i ) +[1 i (I i + d 1 i )]B 0 i (w i d 1 i P i k i ) = d 0 i Z 0 B 0 i (w i x i P i k i )d i (x i ) + Z I i +d 1 i d 0 i f c i + E[B 0 0 j I i (x i )]gd i (x i ) + (1 + c i + k i )[1 i (I i + d 1 i )]: If d 0
i = 0, we would have

EB 0 i = Z I i +d 1 i 0 f c i + E[B 0 0 j I i (x i )]gd i (x i ) + (1 + c i + k i )[1 i (I i + d 1 i )] < ; (44)
where the last inequality follows from E[B 0 0 j I i (x i )] < (1 + k i ) for all x i 2 [0; I i + d 1 i ). (44) would then contradict (38). We thus have d 0 i > 0.

Proof of Proposition 5

The proof is similar to the proof of Proposition 4. In Part 1, (28) is replaced by

K i = I i Z 0 [C i (I i ) + Z i (I i ) Z i (I i C i (I i ))]d i (I i ): (45)
Let and be Lagrange multipliers respectively associated with (45) and (29). The …rst-order optimality conditions associated with D and C i (:) still yield (32), while (33) is replaced by

A i (I i ; C i (I i )) 0 if C i (I i ) = I i > 0; = 0 if 0 < C i (I i ) < I i 0 if C i (I i ) = 0 < I i : ) ( l A i ∈

Business unit i

Business unit i ) ( x i

Table 1 :

 1 Some numerical derivations for optimal insurance captive schemes

i b c i ( 1 +

 1 ) and f i b f i and that are directly insured only by the captive (resp. directly coinsured by the local insurer and the captive) under P 0 . Let C 2 be the set of other business units, with C 0 [ C 1 [ C 2 = f1; : : : ; ng. Let i (r i ) =

gives 1 0

 1 (x 1 ; : :

  Figure 1: insurance captive

Table 2 :

 2 Optimal reinsurance captive schemes

		Parameters	Variables
	k	c

  Hence dI i =dx i 2 (0; 1) when x i 2 (d 0 i ;I i + d 1 i ) where d 0 i is such that B 0 i (w i d 0and we haveI i (x i ) = 0 if x i 2 [0; d 0 i ].Using (41) and (42) allows us to write

	Di¤erentiating (42) gives dI i dx i =	B 00	@E[B 0 0 jC] @C C=I i P i k i ) = c i + E[B 0 0 j 0];	(43)

i (w i x i + I i (x i ) P i k i ) B 00 i (w i x i + I i (x i ) P i k i ) i (x i )

2 (0; 1):

For example, in a special report on European Captive Insurance, A.M.[START_REF] Best | European Captives -a Growth Market During a Challenging Time ?[END_REF] observe that "some captives will be forced to close as they fail to comply with the new Solvency II Directive by the Oct.31, 2012 deadline"and that "the credit crisis has led some parent companies to tap their captives for cash", and concludes that "the (European) captive market will continue to develop in the face of its current challenges, although the growth will shift toward the formation of cells rather than the creation of pure captives".

In an umbrella policy the indemnity is a deterministic function of the aggregate loss incurred by the insured (see[START_REF] Gollier | Optimal insurance design : what can we do with and without expected utility?[END_REF]. This is what an holding company can manage through an insurance or reinsurance captive that will cede or retrocede the holding's aggregate risk to reinsurers.

[START_REF] Scordis | Conditions for captive insurer value :a Monte Carlo simulation[END_REF] use Monte Carlo simulation to identify conditions under which captives have a high probability of creating positive shareholder value, even without favorable tax treatment. They show that the main feature of a well-managed captive is ‡exibility : such a captive is incorporated in the least costly captive jurisdiction during a soft insurance market, but remains dormant until a hard insurance market.

This is illustrated by recent trends in captive markets : according to Business Insurance (2010, March

[START_REF] Froot | Risk management, capital budgeting, and capital structure policy for insurers and reinsurers[END_REF], in 2009 many parent companies have tapped their captives to enhance corporate capital.

For simplicity we refer to k i as a "premium" although it includes all forms of …nancial transfer from business unit i to the captive. Tax issues are completely ignored in this model, which makes this aggregation of …nancial ‡ows possible.

Direct insurers can spread such …xed costs among several customers (at least if there are other …rms operating in the same area of activity that purchase insurance from insurer i) while the captive only underwrites the risks of its parent's subsidiaries. Consequently, local insurers enjoy underwriting economies of scale by comparison with the captive, which suggests that we should have f i < b f i . However the captive insurer may be specialized in the type of risk run by the corporation, which may be the source of an information advantage over the local insurers, hence lower underwriting costs and thus possibly b f i < f i in such a case.

Risk cession by local insurers to the captive will be considered in Section 4.

The analysis of[START_REF] Weiss | US reinsurance prices, …nancial quality, and global capacity[END_REF] and[START_REF] Meier | Business cycles in insurance and reinsurance : the case of France, Germany and Switzerland[END_REF] suggest that the price of coverage in insurance and reinsurance markets do not move in a perfectly synchronized way. As observed by[START_REF] Scordis | Conditions for captive insurer value :a Monte Carlo simulation[END_REF], this creates an arbitrage opportunity, which allows …rms possessing insurance captives to shop across two markets, and also to treat the captive as a source of ‡exibility when the insurance market hardens.

In E[B 0 0 ( 0 ) jt i (x i ) ] the expected value operator is with respect to transfers from the captive to all local insurers j 6 = i by taking into account the captive's reinsurance. See the proof of Proposition 2 in the Appendix.

Assumption f = b f c = 0:005 guarantees that it is optimal to purchase insurance either from the captive or from a local insurer but not from both sources simultaneously. Assumption f 0 = 0 is made in order to suppress any discontinuity e¤ect when the captive is active (say with at least one business unit) and when it is not.

The value of each business unit and the insurance premiums have a closed form in our setting. This is not the case neither for the reinsurance premium nor for the value of the captive because of the intricate

In the reinsurance terminology, this is a non-proportional reinsurance mechanism. In the present setting, ignoring any underwriting commission, proportional reinsurance would correspond to the particular case where C i (I i ) = c i I i and K i = c i P i , with c i 2 [0; 1] the cession rate.

As in the simulations of Section 3, we assume f = 0:005; f 0 = 0 and w 0 = 0.

Note thatI 0 i (x i ) 0 if i 2 C 0 nC 0 0 :

The authors gratefully acknowledge the …nancial support from the AXA Research Fund (Chair on Large Risk in Insurance).

Appendix

Proof of Proposition 1 Let P 0 =fI 0 i (x i ); P 0 i ; t 0 i (x i ); k 0 i for i = 1; : : : ; n; T 0 (t 1 ; : : : ; t n ); Q 0 g a feasible insurance captive scheme. Let C 0 (resp. C 1 ) be the set of business units such that (1 + )(1 + b c where t 0 i = (1 i (I i )I i and C 0 i = C 0 i ( i (I i )I i with

For all i 2 C 0 [ C 1 , when direct insurer i pays indemnity I i to business unit i and receives C 1 i (I i ) from the captive under P 1 , he pays t 0 i = (1 i (I i )I i to business unit i and he receives C 0 i = C 0 i ( i (I i )I i from the captive under P 0 . We thus have

)E[T 0 (: : : ; t 0 i ; : : : ; C 0 i ; : : :) + f 0 ]:

Consequently,

Let C 0 2 be the subset of C 2 that gathers the units of C 2 that are directly insured by the captive (and also possibly by local insurers). Under P 1 only the business units of C 0 2 are directly insured by the captive. Let 0 0 (x 1 ; : : : ; x n ) and 1 0 (x 1 ; : : : ; x n ) be the pro…t of the captive respectively under P 0 and P [START_REF] Best | European Captives -a Growth Market During a Challenging Time ?[END_REF] We have

+T 1 (: : : ; t 1 i (x i ); : : : ;

+T 1 (:

Using ( 26), (27) and

is still given by (34). Note that

Hence we have 35) and (37) are respectively replaced by

and

while (36) is unchanged. Let ; ; and be the Lagrance multipliers, which are respectively associated with (46), (36) and (47). (38) still holds and (39) is replaced by

The remaining part of the proof then straightforwardly replicates the proof of Proposition 4. In particular, (42) still holds when 0 < I i (x i ) < I i , which gives dI i =dx i 2 (0; 1) in that case. When I i (x i ) > I i , we have

which also gives dI i =dx i 2 (0; 1). Function I i (x i ) has a non-di¤erentiable point when I i (x i ) = I i , which is reached at x i = d 1 i and a straighforward computation gives I 0 i+ (d 1 i ) > I 0 i (d 1 i ).