Nataliya Klimenko
email: nataliya.klimenko@univ-amu.fr.

Mohamed Belhaj

Bruno Biais

Renaud Bourlès

Jean-Paul Décamps

Dominique Henriet

Christian Koziol

Thomas Mariotti

Jean-Charles Rochet

Guillaume Plantin

Stephane Villeneuve

Tailoring bank capital regulation for tail risk

Keywords: capital requirements, tail risk, recapitalization, incentive compensation, moral hazard JEL classication: G21, G28, G32, G35

The experience of the 2007-09 nancial crisis has showed that the bank capital regulation in place was inadequate to deal with "manufacturing" tail risk in the nancial sector. This paper proposes an incentive-based design of bank capital regulation aimed at eciently dealing with tail risk engendered by bank top managers. It has two specic features: (i) rst, it incorporates information on the optimal incentive contract between bank shareholders and bank managers, thereby dealing with the internal agency problem; (ii) second, it relies on the mechanism of mandatory recapitalization to ensure this contract is adopted by bank shareholders.

Introduction

One of the important roles of bank capital regulation is to restrain banks from excessive risk-taking in the context of the explicit and implicit government guarantees they may enjoy. However, the experience of the 2007-2009 nancial crisis shows clearly that capital regulation in place at the time failed to perform this role. This failure may be partially related to the fact that the capital regulation framework did not adjust in time to the substantial changes in bank business culture brought about by new techniques in nancial engineering, growing securitization and the development of complex nancial products. One of the direct results of these changes was the emergence of tail risk characterized by rare but devastating losses. There is now extensive empirical evidence that, by focusing on short-term performance, top management in banks was engaged in "manufacturing" tail risk [START_REF] Acharya | Manufacturing Tail Risk: A Perspective on the Financial Crisis of 2007-09[END_REF]) in order to generate immediate prots, without regard to the long-term consequences. It seems that risk management strategies in the period preceding the crisis were beyond the control of bank shareholders. In such a context, to deal eectively with such "manufactured" tail risk in the banking sector, bank capital regulation should have taken into consideration the internal agency problem, instead of focusing exclusively on the interests of bank shareholders.

This study proposes an incentive-based design of capital requirements which will induce shareholders to shape executive compensation in such a way as to prevent managers from engendering tail risk. To illustrate these proposals, we build a simple continuoustime model in the principal-agent framework, where a senior bank manager has a reversible choice between prudent and imprudent risk management strategies. An imprudent risk management strategy exposes the bank to tail risk characterized by infrequent but devastating losses, while a prudent risk management strategy implies no tail risk but generates a lower expected asset return. We assume that imprudent risk management allows the manager to collect private benets. Such a set-up corresponds to the double moral hazard problem, since it is not only the manager who may be interested in engaging in tail risk, but bank shareholders as well. To be able to prevent imprudent risk management in this context, incentive-based capital requirements should incorporate information about the optimal incentive contract between bank shareholders and bank manager. This feature dierentiates our capital regulation design from existing approaches, which deal with a bank as if it was a black box.

The existing capital regulation literature (see, for instance, Brattachariya (2002), Décamps et al. (2004), [START_REF] Decamps | The Three Pillars of Basel II: Optimizing the Mix[END_REF]) shows that portfolio risk can be prevented through the mandatory incentive-based liquidation rule. However, the liquidation rule appears to be ill-suited to tail risk: even in the absence of internal agency problems, it fails to prevent the bank from engaging in tail risk when bank asset value approaches the liquidation point. In fact since, in the neighborhood of the liquidation threshold, the bank default is more likely to be triggered by random uctuations of asset return rather than by infrequent large losses, the bank will engage in tail risk in order to increase the asset growth rate and to move away from the liquidation point. 1 To specically deal with tail risk, we design capital requirements in the form of the incentive recapitalization rule. Under the enforceable recapitalization rule, portfolio risk cannot cause bank failure. At the same time, engaging in tail risk will increase the likelihood of further mandatory recapitalizations and thus impose higher expected recapitalization costs on bank shareholders. In fact, it is the fear of bearing the additional recapitalization costs due to tail risk exposure that will induce bank shareholders to promote prudent risk management in their bank.

Although costly recapitalizations are studied in the liquidity management literature (Décamps et al. (2011), [START_REF] Rochet | Liquidity management and corporate demand for hedging and insurance[END_REF]) and the capital regulation literature (Peura and Keppo (2003), Milne and Walley (2002)), no one has previously explored the potential incentive eect of mandatory recapitalizations, to the best of our knowledge. However, we are not the rst to point out the need to consider the internal agency problem within a capital regulation design. [START_REF] Bris | Bank capital requirements and managerial selfinterest[END_REF] address this issue in the context of portfolio risk management, examining the impact of capital requirements on the eort choice of a self-interested risk-averse bank manager in a discrete time framework. They come to the conclusion that capital regulation, which does not take into account the internal agency problem, leads to a socially-unoptimal choice of the lower level of risk. Here, in the context of tail risk and with risk-neutral agents, our model shows, in contrast, that the bank will operate at the higher level of risk. Moreover, we explicitly show how to adapt the optimal design of capital regulation to take account of the internal agency problem, linking the literature on incentive capital regulation and the fast-growing literature on optimal contracts in a dynamic framework [START_REF] Sannikov | A Continuous-Time Version of the Principal-Agent Problem[END_REF], [START_REF] Biais | Large risk, limited liability, and dynamic moral hazard[END_REF], DeMarzo, Lidvan and Tchistyi ( 2011)).

The nature of the moral hazard problem studied in this paper situates it close to [START_REF] Biais | Large risk, limited liability, and dynamic moral hazard[END_REF]. In their model, the agent's eort aects a rm's exposure to a disaster risk, given that lower eort enables the manager to collect private benets proportional to the rm's size. However, they consider Poisson risk to be the sole type of risk the rm faces, whereas in our setting the bank asset value is also aected by Brownian risk. Both Poisson and Brownian risks are present in the model of DeMarzo, Lidvan and Tchistyi (2011), who design an optimal contract in a two-dimensional setting with Arithmetic Brownian Motion (ABM), when the manager can privately choose between two risk regimes aecting a rm's prot and also has the option of diverting part of the generated cash ow. In our model constructed in a setting with Geometric Brownian Motion (GBM), the manager controls an asset growth, rather than a bank's protability. Moreover, we restrict our analysis to the class of linear contracts so as to be able to integrate information on the structure of the optimal contract into the capital regulation design.

The rest of the paper is organized as follows. Section 2 presents the model. Section 3 explains the incentive eect of mandatory recapitalizations. In Section 4 we design an optimal mandatory recapitalization policy in a setting free of the internal agency problem. Section 5 presents an optimal mandatory recapitalization policy in the context of the internal agency problem. In Section 6 we discuss related regulatory policy issues. Section 7 concludes. All proofs are provided in Appendix B.

The model

We consider a risk-neutral bank protected by limited liability. The bank is nanced by a constant volume of insured deposits, D, and incurs a continuous payment rD to depositors,2 where r denotes a riskless interest rate. Bank assets continuously generate a cash-ow δx t , where x t denotes a publicly observable asset value. Thus, an instantaneous cash-ow received by shareholders after paying debt service is given by δx t -rD. 3The bank is run by a manager. The contract with the manager can be described by a triplet {x T , R(x), R T }, where x T denotes a contract termination rule,4 R(x) ≥ 05 for x > x T is an asset-based remuneration and R T ≥ 0 is a terminal pay-o delivered at the contract termination date. The manager is protected by limited liability and has no initial wealth which could be pledged.

The manager has a reversible discretionary choice between 2 risk management strategies: prudent and imprudent. Imprudent risk management allows for expected asset return rate µ and involves a bank in tail risk, which implies infrequent but large losses of bank assets. Large losses caused by tail risk materializing follow a Poisson process {N t } t≥0 with intensity λ. A large loss destroys a fraction (1 -α)x t of bank assets, where α ∈ (0, 1) is a constant coecient reecting the proportion of assets remaining after the large loss is realized. Prudent risk management strategy is free of tail-risk exposure but has a lower expected asset return rate, (µ -∆µ), where 0 < ∆µ < λ(1 -α). From now on we will assume that r > δ + (µ -∆µ).

Let u t ∈ {0, 1} be a control variable reecting risk-management strategy, where u t = 1 corresponds to prudent and u t = 0 corresponds to imprudent risk management. Bank asset value follows:6

dx t = (µ -u t ∆µ)x t dt + σx t dZ t -(1 -u t )(1 -α)x t dN t , (1)
where σ is the asset return volatility and {Z t } t≥0 is a standard Brownian process.

We assume that imprudent risk management allows the manager to collect private benets bx t dt. Private benets may represent proceeds from private trading on the nancial markets, given that tail-risk management strategies may boost equity growth in the short term. 7 However, materialization of tail risk, will make the choice of management technology veriable ex-post, so that the contract with the manager can stipulate that he will be red without any terminal pay if a large loss occurs. The threat of being red after the loss represents a maximum feasible punishment under the limited liability of the manager, allowing shareholders to minimize the cost of creating incentives.

Since large losses incurred by a single bank may inict negative externalities on the rest of the banking sector (especially when they lead to the bank's failure), we assume that the objective of the bank regulator is to induce prudent risk-management by using capital regulation tools. Capital regulation in our model is designed in the form of a mandatory recapitalization policy involving two regulatory parameters: mandatory recapitalization rule x R and recapitalization coecient s > 1, such that sx R is the target bank asset value after the mandatory recapitalization. Thus, bank shareholders should inject fresh equity capital (s -1)x R each time the bank asset value hits the mandatory recapitalization threshold x R . However, if the bank asset value suddenly falls below the recapitalization threshold, the bank should be liquidated by the regulator (i.e., incumbent shareholders will be expropriated of equity), since a sudden violation of capital requirements shows imprudent risk management. 8 As in Décamps et al. (2011), we allow for two types of recapitalization costs: proportional costs ξ 1 , which are imposed on each unit of capital raised, and lump-sum costs ξ 0 .

Recapitalization costs can reect taxes, expert and registration costs of the new equity issue, as well as asset restructuring costs. 9 We denote K(x t ) the total costs shareholders incur when issuing new equity at time t

K(x t ) = (ξ 1 + 1)(s -1)x t + ξ 0 (2)
As we will show in the next section, recapitalization costs represent a driving force of the proposed incentive mechanism.

The incentive eect of recapitalizations

To illustrate the need for capital regulation in the above set-up and its value when implemented in the form of mandatory recapitalization policy, let us consider the optimal strategy of bank shareholders when there is no capital regulation.

To focus on the incentive eect of recapitalizations, we will abstract from the internal agency problem, analyzing risk management decisions in the owner-managed bank. Bank shareholders maximize equity value, by instantaneously deciding which risk management strategy to implement. This decision is driven by a trade-o between the instantaneous gain from imprudent risk management and the expected loss of equity value caused by tail risk realization. We introduce a second order dierential operator A u f (x) such that:

A u f (x) = 1/2σ 2 x 2 f (x) + (µ -u∆µ)xf (x) -rf (x), (3)
where u ∈ {0, 1} and f (x) is any contingent claim.

Then, the shareholders' maximization problem can be written as follows:

max ut∈{0,1} {A ut E(x t ) -(1 -u t )λ(E(x t ) -E(αx t )) + δx t -rD} = 0, (4)
8 Equity expropriation represents a maximum feasible penalty under the limited liability of bank shareholders, allowing for the most ecient design of the incentive mechanism. However, this penalty will never be applied ex-post, since the incentive-based recapitalization mechanism will completely eliminate imprudent risk management. 9 Note that liquidity injections realized by shareholders in order to ensure the continuity of debt service are assumed to be costless, since they do not aect asset value and thus do not require asset restructuring. In contrast, a recapitalization which leads to asset expansion will incur asset restructuring costs. where x t is given by (1) and E(x) denotes bank equity value in the absence of the internal agency problem.

Consequently, bank shareholders have an interest in implementing prudent risk management, so long as the expected negative jump of equity value caused by a realization of tail risk exceeds the instantaneous gain from imprudent risk management:

λ(E(x) -E(αx)) ≥ ∆µxE (x) (5)
Let x * u be a critical threshold which makes the incentive constraint (5) binding and let x S L < x * u denote a threshold:

x S L = -γ 2 1 -γ 2 r r + λ D ν , (6)
where

γ 2 < 0 is a root of 1/2σ 2 γ(γ -1) + µγ = r + λ and ν = δ r-µ+λ < 1.
Result 1 Without any regulatory control, the owner-managed bank will implement prudent risk management for x t ≥ x * u , will opt for imprudent risk management for x t < x * u and will be liquidated by shareholders when x t = x S L .

To see the intuition of the above result, consider rst the optimal choice between a costly recapitalization and bank liquidation. The fact that r > δ + (µ -∆µ) means that the return on a one-dollar investment in riskless security would be higher than that on a one-dollar investment in the bank asset portfolio. Then, whatever the eort strategy chosen, in the absence of regulatory control bank shareholders will never recapitalize the bank on their own and will strategically default at the optimal liquidation point. However, in the neighborhood of the liquidation point, a moral hazard problem emerges as a consequence of the conict between portfolio risk and tail risk. In fact, when bank asset value becomes relatively low, bank liquidation is more likely to be triggered by a continuous decline of bank asset value rather than by a sudden negative jump. As a result, the bank will "gamble for resurrection", optimally engaging in tail risk in order to increase asset growth and to move away from the liquidation point. Here, capital regulation is needed in order to induce the bank to maintain prudent risk management.

A capital regulation literature dealing with the asset substitution problem has shown that it is possible to prevent a bank from choosing a riskier portfolio by imposing an appropriate incentive liquidation rule (see, for example, Brattachariya (2002), Décamps et al. ( 2004), Koziol and Lowrenz (2012)). The next question is then: why can't we prevent the bank from engaging in tail risk in a similar way? Why do we need, instead, to resort to mandatory recapitalizations?

Let us consider the incentives of bank shareholders when the regulator imposes any arbitrary liquidation rule x L ≥ x L , where10

x L = -β 2 1 -β 2 D η (7)
where

β 2 < 0 is a root of 1/2σ 2 β(β -1) + µβ = r and η = δ r-µ+∆µ < 1.
A verication of necessary condition [START_REF] Bebchuk | The wages of failure: Executive compensation at Bear Stearns and Lehman 2000-2008[END_REF] shows that it doesn't hold when bank asset value approaches x L . Thus, in the neighborhood of the mandatory liquidation threshold the bank will engage in tail risk for the same "gambling-for-resurrection" reasons we described above.

Result 2 Any mandatory liquidation rule x L ≥ x L would be unable to prevent tail risk.

It follows from this result that a mere increase in capital requirements would be insucient to discourage banks from "manufacturing" tail risk. As pointed out by several recent studies (see, for example, [START_REF] Perotti | Capital Regulation and Tail Risk[END_REF]), new regulatory tools are required in order to maintain control over tail risk. Incentive-based mandatory recapitalization policy could be viewed as one of them. Indeed, the eect of the recapitalization rule will be dierent from the eect of the liquidation rule. A crucial point is that, under the mandatory recapitalization rule, asset return volatility cannot lead to the bank's failure. Thus, the mandatory recapitalization rule will eliminate a conict between portfolio risk and tail risk. At the same time, realization of tail risk under an imprudent risk management strategy will raise the likelihood of mandatory recapitalizations, thereby increasing the total expected recapitalization costs. It is the very threat of suering the additional recapitalization costs under tail risk exposure that will induce bank shareholders to promote prudent risk management in their bank.

Thus, in order to induce shareholders to promote prudent risk management, mandatory recapitalization policy should be designed in such a way that: (i) the expected loss of equity value caused by realized tail risk exceeds the instantaneous gain from imprudent risk-management; (ii) equity value at the mandatory recapitalization point is suciently high for shareholders to optimally prefer to recapitalize the bank rather than to be punished by equity expropriation. [START_REF] Bebchuk | Regulating Bankers Pay[END_REF] Capital regulation when there is no internal agency problem: a benchmark

In order to track the impact of the internal agency problem on capital regulation, we rst design the incentive mandatory recapitalization policy in a setting where the interests of the bank manager are perfectly aligned with the interests of the bank shareholders.

Let us formally dene the regulatory problem. The regulator is looking for the optimal combination of recapitalization rule x R and recapitalization coecient s > 1 which will prevent imprudent risk management at the bank for ∀x ≥ x R , maximizing bank social value. Bank social value can be computed as a sum of bank equity value and the market value of deposits net of any social costs. However, in contrast to mandatory liquidation or a public bail-out, all costs associated with mandatory recapitalizations will be entirely borne by bank shareholders. At the same, under the incentive recapitalization mechanism, the bank will never default, so that the market value of deposits will remain constant. As a result, if the bank chooses prudent risk management, maximization of bank social value will be equivalent to maximization of bank equity value. Thus, for any current bank asset value x t > x R , the regulatory problem can be stated as follows:

Max x R >0,s>1 E(x) ≥ 0 s.t. λ(E(x) -1 x≥x R /α E(αx)) ≥ ∆µxE (x) for ∀x ≥ x R , where E(x) is given by E(x) = E +∞ t e -r(τ -t) (δx τ -rD -1 {xτ =x R } K(x τ ))dτ (8)
Let ξ 1 denote a critical level of marginal recapitalization costs, such that for ξ ≥ ξ 1 a solution of the above problem does not exist: 11

ξ 1 = 1 -s β 2 s -1 λ -∆µ λ -β 2 ∆µ η + η -1 (9)
Proposition 1 For ξ 1 < ξ 1 , the regulator can prevent an owner-managed bank from en- gaging in tail risk, by imposing the optimal mandatory recapitalization policy {s * , x B R (s * )} which implies:

• (i) a recapitalization rule x B R (s) = (1 -s β 2 )λD + ξ 0 (λ -β 2 ∆µ) (1 -s β 2 )(λ -∆µ)η -(1 + ξ 1 -η)(s -1)(λ -β 2 ∆µ) (10) • (ii) a recapitalization coecient s * = arg min x B R (s).
It can easily be shown that bank equity value is decreasing on x R . Thus, for any given recapitalization coecient s > 1, the maximum equity value can be attained under the minimum feasible recapitalization rule x B R (s) > D, such that λE(x) = ∆µxE (x) at x = x R . 12 Note that E(x B R (s)) > 0. This makes our recapitalization mechanism enforceable, since shareholders will prefer to recapitalize the bank rather than to be expropriated of equity. We also verify that, faced with mandatory recapitalization rule x B R (s), shareholders will not undertake voluntary recapitalizations at any x > x B R (s), since recapitalization costs would always exceed the expected growth of equity value resulted from capital injections. 13 Then, given that bank equity value is decreasing on recapitalization rule x B R (s), a complete solution of the above regulatory problem will be delivered by s * = arg min x B R (s).

In order to illustrate the optimal mandatory recapitalization policy, we resort to numerical simulations. For the parameter set D = 1, r = 0.04, µ = 0.035, ∆µ = 0.005,

δ = 0.01, σ = 0.2, λ = 0.05, α = 0.7, ξ 0 = [0.1 × 10 -4 , 0.1 × 10 -3 ], ξ 1 = [0.01, 0.1], 14 x B R (s * )
varies in the range of (1.12, 1.20) and s * takes values between (1.01, 1.08). For example, for ξ 0 = 0.1 × 10 -3 and ξ 1 = 0.1 we obtain x B R (s * ) = 1.20 and s * = 1.03, which 11 For ξ 1 > ξ 1 the incentive condition never holds in the neighborhood of x R . 12 We show in Appendix B that, if the incentive constraint is respected for x ∈ [x R , x R /α), it will automatically be respected for any x > x R /α. 13 In practice, shareholders are unwilling to undertake voluntary recapitalizations not only because of the private costs they incur, but also for fear of sending a negative signal about the bank's nancial health. Making a recapitalization mandatory, however, might partially mitigate this signaling eect. 14 Empirical estimations realized for the set of U.S. rms provide the following values of average marginal issuance costs: 2.8% in Gomes (2001), 5.1% in Altinkiliç and Hansen (2000), 10.7% for small rms and 5% for large rms in Hennessy and Whited (2007). corresponds to the minimum capital ratio of 16.7% and post-recapitalization capital ratio of 19, 1% respectively.

The optimal recapitalization coecient results from the trade-o between two opposite eects generated by xed and proportional recapitalization costs: s * is increasing on ξ 0 and decreasing on ξ 1 . Fixed recapitalization costs encourage shareholders to raise a maximum of funds in order to postpone further mandatory recapitalizations, whereas proportional costs reduce recapitalization capacity. 15 The optimal recapitalization rule x B R (s * ) is increasing on both xed and proportional recapitalization costs. Indeed, given signicant recapitalization costs, the bank equity must be strong enough to persuade shareholders to choose costly recapitalizations over bank liquidation. Moreover,

x B R (s * )
is increasing on asset volatility, since higher σ exacerbates a trade-o between tail risk and portfolio risk, aggravating a moral hazard problem.

Capital regulation under the internal agency problem

Now we turn to the set-up which allows for the internal agency problem. The manager has a dierent perception of tail risk exposure from that of shareholders, since his objective deviates from the maximization of equity value. Thus, besides the instantaneous loss ∆µx t dt in terms of asset growth, the real cost of prudent risk management for shareholders will be increased by the amount of incentive compensation. In such a context, shareholders have to make two strategic decisions: (i) whether to promote prudent risk management in their bank and (ii) if so, how to create the appropriate incentives for the manager at minimum cost. We start by answering the second question, dening the optimal incentive contract with the manager. Then we incorporate a structure of the optimal incentive contract into the initial problem of bank shareholders and build the optimal incentive-based recapitalization policy which will induce shareholders to promote prudent risk management in their bank.

The optimal incentive contract

Assume there is some incentive recapitalization policy {x R , s} for which shareholders nd it optimal to promote prudent risk management in their bank for ∀x ≥ x R . Then, we need to nd an incentive contract which will induce the manager to maintain prudent risk management at a minimum cost for bank shareholders. We restrict our analysis to the class of linear contracts, looking for asset-based compensation in this form:

R(x) = w 0 + w 1 x, (11)
where w 0 ≥ 0 and w 1 ≥ 0.

We analyze the manager's incentives for prudent risk management. Given any contract {x T , R(x), R T }, the manager maximizes contract continuation value, W (x), which is contingent on bank asset value and represents the current expected value of total future gains from managerial position, including any private benets:

W (x) = E τ ∧τ T t e -r(τ -t) (R(x τ ) + (1 -u τ )bx τ )dτ + e -r(τ T -t) R T , (12)
where u τ ∈ {0, 1}, x τ follows (1) and τ T = inf{t ≥ 0 s.t x t ≤ x T }.

Imprudent risk management has an ambiguous eect on the manager's wealth. On the one hand, it increases contract continuation value due to higher expected asset return and private benets. On the other hand, the manager risks losing his position (and, consequently, the expected value of further payos) with probability λdt in a short period of time dt. Then, the manager's maximization problem can be stated as follows:

max ut∈{0,1} {A ut W (x t ) + R(x t ) -(1 -u t )(λW (x t ) -bx t )} = 0 (13)
The manager will choose prudent risk management as long as the expected loss of contract continuation value under tail-risk exposure exceeds the instantaneous gain from imprudent risk management. Then, the optimization problem of bank shareholders looking to minimize the costs of creating incentives for the manager, M C(x), can be stated as follows:

min x T ≥x R ,w 0 ≥0,w 1 ≥0,R T M C(x) = E +∞ t e -r(τ -t) (1 xτ =x T R(x τ ) + 1 xτ =x T R T )dτ s.t. λW (x) ≥ ∆µxW (x) + bx, ∀x ≥ x R (14)
We can have either x T = x R , which means that the manager will be replaced at the recapitalization point, or x T = ∅, which means that the manager will be allowed to keep his position forever if no loss occurs. In Appendix B we compare the minimum value of shareholders' costs under these alternatives. We show that, faced with x T = x R , the manager has to be provided with a terminal payo R T equal to the expected value of the further contract payos he could obtain from continuation. Conversely, letting the manager stay forever provided that no loss occurs would allow shareholders to avoid these payments.

Proposition 2 The optimal incentive contract which will induce prudent risk management at minimum cost implies 16

R(x) = δ η b λ -∆µ x ≡ w * 1 x (15)
The manager is never red provided no loss occurs, i.e., x T = ∅ and R T = ∅.

The total shareholders' costs associated with the implementation of the optimal incentive contract will coincide with the minimum incentive-compatible contract continuation value for the manager:

M C * (x) ≡ W * (x) = η δ w * 1 x + x R s -1 1 -s β 2 x x R β 2 (16
) 16 Note that w * 1 > 0, since λ < ∆µ(1 -α) < ∆µ.

It is easy to see that M C * (x) are increasing with the manager's ability to generate private gains on the nancial market, b. In practice, top executives are partially remunerated via equity-based compensation. In the context of our model, this would create a self-amplifying mechanism: by rewarding the manager with stock options and stocks, shareholders would increase what he stands to gain from the nancial markets. In order to induce prudent risk management, shareholders will have to increase a variable proportion of the incentive compensation, which will again raise risk-taking incentives and so forth. Thus, it would be useful to maintain control over the proportions of equity-based compensation within compensation packages, in order to make managers less sensitive to the short-term reactions of the nancial markets to their performance.

The optimal mandatory recapitalization policy

Given the optimal incentive contract with the manager, we are now in a position to address the initial problem of bank shareholders faced with a decision on whether to induce prudent risk management or to assume tail risk exposure. Given the internal agency problem, bank shareholders have an interest in promoting prudent risk management in their bank when the following incentive condition is satised:

λ(E W * (x) -1 x≥x R /α E W * (αx)) ≥ ∆µxE W * (x) + w * 1 x, (17)
where

E W * (x) = E +∞ t e -r(τ -t) ((δ -w * 1 )x τ -rD -1 {xτ =x R } K(x τ ))dτ (18)
In contrast to the benchmark case, the incentive condition of bank shareholders now incorporates information on the optimal incentive contract. Then, the regulatory problem in the context of the internal agency problem becomes:

Max x R >0,s>1 E W * (x) ≥ 0 s.t. (17) for x ≥ x R (19)
Let b denote a critical value of private benets such that for b ≥ b the solution of the above problem does not exist:

b = η ξ -ξ 1 + ξ (λ -∆µ), (20)
where ξ 1 is given by [START_REF] Bolton | Executive Compensation and Risk Taking[END_REF]. Indeed, for b < b < δ the incentive condition never holds in the neighborhood of x R , whereas for b > δ the optimal incentive contract would be too costly for shareholders.

Proposition 3 If ξ 1 < ξ 1 and b < b, the regulator can prevent the bank from engaging in tail risk, by imposing the optimal mandatory recapitalization policy {s * * , x A R (s * * )} which implies:

• (i) a recapitalization threshold x A R (s) = (1 -s β 2 )λD + ξ 0 (λ -β 2 ∆µ) (1 -s β 2 )(λ -∆µ)(1 - w * 1 δ )η -(1 + ξ 1 -η(1 - w * 1 δ ))(s -1)(λ -β 2 ∆µ) • (ii) a recapitalization coecient s * * = arg min x A R (s).
It can be shown that x A R (s) > x B R (s) for the same recapitalization coecient s, so that an incentive recapitalization policy which does not allow for the internal agency problem would be unable to prevent the bank from engaging in tail risk. This proves that the internal agency problem matters and should be taken into account by bank regulators when designing capital regulation.

It is worth noting that shareholders' costs of creating incentives for the manager can be reduced through internal random audits. If an audit uncovers imprudent risk management, the manager should be red without receiving any terminal pay. This will increase his risk of losing contract continuation value should there be imprudent risk management, allowing shareholders to reduce the amount of incentive compensation. If instantaneous random audit costs are not too high, random audits could increase shareholders' incentives to promote prudent risk management in their bank, enabling the regulator to reduce a stringency of capital regulation. Moreover, recall that true bank asset value must be observable in order to implement the optimal incentive contract. Since the manager may have the opportunity to manipulate nancial statements in order to enjoy higher rewards, random audits could also involve verifying bank asset value, which would discourage the manager from misreporting.

6 Discussion

The impact of bonus taxes on risk management

In the aftermath of the 2007-09 nancial crisis, several European countries (UK, France, Italy, Greece) introduced a tax on the performance bonuses of bank top management. One of the ocial purposes of this measure was to improve the risk-management culture in the banking sector. We now use our model to examine the eect of bonus taxes on shareholders' incentives to promote prudent risk management in their bank.

Let a tax rate τ be applied to the variable proportion of managerial compensation w 1 x which might be interpreted as a performance bonus. Assume rst that taxes are paid by bank shareholders. In this case, bonus taxes have no impact on the manager's incentives, so that bank shareholders can induce prudent risk management by using the same optimal incentive contract as in a tax-free world. However, total shareholders' costs of creating incentives will be increased by the amount of taxes and will be equal to M C(x) = (1 + τ )W * (x), where W * (x) is dened in [START_REF] Koziol | Contingent convertibles. Solving or seeding the next banking crisis[END_REF]. Consider now the alternative setting, where bonus taxes are paid by the bank manager. 17 To be motivated to maintain prudent risk management, the manager should have at least the same level of wealth after the tax levy as in a tax-free world. Then, the minimum incentive contract continuation value that should be oered to the manager will be given by M C(x) = 1/(1 -τ )W * (x).

The rst conclusion that can be drawn from this analysis is that bonus taxes are inappropriate for dealing with excessive risk-taking in banks. Under both scenarios, they increase the real shareholders' cost of promoting prudent risk management and thus would lead to a situation requiring tougher capital regulation. 18 It is also easy to see that total shareholders' costs of creating incentives when taxes are paid by the manager would be higher than when bonus taxes are paid by the shareholders themselves. Thus, the lesser of two evils would be to collect bonus taxes from bank shareholders rather than from bank managers, which is consistent with the bonus tax policies adopted in UK and France in 2009-2010.

Implicit vs explicit regulation of managerial pay

There is now convincing empirical evidence that equity-based compensations have made bank executives willing to increase bank equity value at any price in order to reap gains [START_REF] Chen | Does stock option-based executive compensation induce risk-taking? An analysis of the banking industry[END_REF], [START_REF] Williams | Bank mergers, equity risk incentives, and CEO stock options[END_REF], [START_REF] Vallascas | CEO Remuneration and Bank Default Risk: Evidence from US and Europe[END_REF]).

In order to improve the risk-management culture in the banking sector, many academics and policymakers call for the explicit regulation of executive pay(Bebchuk and Spamann (2010), [START_REF] Bolton | Executive Compensation and Risk Taking[END_REF]). However, this proposal raises a range of concerns. The rst problem is that regulators do not dispose of all the information needed for the ecient design and enforcement of executive compensation policy. Second, it is still unclear what form the optimal incentive compensation structure should take. Finally, experience suggests that economic agents always nd a way to get around regulations if their incentives diverge from regulatory purposes. Thus, explicitly regulating executive pay without regulating shareholders' incentives would probably be a waste of regulatory resources. These reasons argue for implicit control over managerial incentives. The detailed design of managerial compensation should be left to bank shareholders; the role of the bank regulator is to ensure that shareholders have sucient incentives to promote prudent risk management in their banks.

Capital requirements and insurance protection

The last relevant question is whether capital requirements should be reduced if a bank acquires an insurance policy against tail risk. Actually, only the advanced approach of the Basel II capital requirements considers an insurance policy as a risk mitigation tool and authorizes banks holding such policies to operate with reduced mandatory capital. However, it seems that greater reliance on insurance protection may aggravate the problem of moral hazard. The point is that an insurance policy allows banks to transfer risk without tackling it at source, i.e., it helps to reallocate risks but cannot prevent their accumulation within the nancial system. Moreover, in the context of a systemic crisis, insurance companies themselves may experience serious nancial problems, 19 being therefore unable to provide loss coverage. Thus, even though recourse to insurance may be benecial for bank shareholders (i.e., it might be cheaper to buy an insurance policy than to create appropriate incentives for the manager), prudent risk management would be the only durable solution from the perspective of social welfare. Banks can be allowed to buy insurance protection against some external risks (like external fraud, hacking attacks, natural disasters), since an insurance policy will not promote moral hazard and risk accumulation in these cases. At the same time, regulators should induce banks to tackle the sources of internal risk. As have shown, this can be realized by means of the incentive mandatory recapitalization policy which allows for the internal agency problem.

Conclusion

This study is an attempt to rethink the approach to bank capital regulation in response to the huge incentive distortions revealed by the 2007-09 nancial crisis. We design a mandatory recapitalization policy, which deals with "manufactured" tail risk in banking. We show how, through the appropriate choice of mandatory recapitalization parameters, the regulator can induce bank shareholders to put in place an incentive compensation scheme that will deter bank managers from engaging in tail risk. We restrict our analysis to the class of linear contracts and derive the optimal incentive compensation in explicit form. It appears to be optimal oer the manager a contract of indenite duration. Otherwise, at the contract termination date, the manager would have to be granted a positive terminal payo equal to the remaining contract continuation value, which would impose additional costs on bank shareholders.

Since the internal agency problem makes it costly for bank shareholders to promote prudent risk-taking behavior by bank managers, shareholders should be required to maintain a larger stake in the game. This might be viewed as a justication for more stringent capital requirements for systemically important banks, which typically have both severe agency problems and a predisposition to large losses. However, as we have shown, merely increasing capital requirements would not prevent banks from engaging in tail risk. For this reason, we suggest implementing capital regulation in the form of the incentive-based recapitalization policy.

A.3.2. Recursive algorithm to compute unknown coecients i,1 (x) and C i,2 (x)

Consider the following non-homogeneous second-order ODE:

f (x) + q(x)f (x) = q 0 g(x) (A18)
where f (x) ∈ C 2 , q(x) ∈ C 0 , g(x) ∈ C 0 and q 0 is any arbitrary A general solution of the above equation is given as follows:

f (x) = a 1 f 1 (x) + a 2 f 2 (x) + q 0 f p (x), (A19)
where f 1 (x) and f 2 (x) are two solutions of the homogeneous equation, a 1 , a 2 are any arbitrary coecients and f p (x) is a particular solution which can be dened by using the method of the variation of parameters:

f p (x) = -f 1 (x) f 2 (x)g(x) W (f 1 (x), f 2 (x)) dx + c 1 + f 2 (x) f 1 (x)g(x) W (f 1 (x), f 2 (x)) dx + c 2 ,
(A20) where

W (f 1 (x), f 2 (x)) = f 1 (x)f 2 (x) -f 1 (x)f 2 (x) = 0
is a Wronskian and c 1 , c 2 are any arbitrary constants of integration.

Thus, (A19) can be rewritten as follows:

f (x) = k 1 f 1 (x)+k 2 f 2 (x)-q 0 f 1 (x) f 2 (x)g(x) W (f 1 (x), f 2 (x)) dx -f 2 (x) f 1 (x)g(x) W (f 1 (x), f 2 (x)) dx , (A21)
where k 1 = a 1 -q 0 c 1 and k 2 = a 2 + q 0 c 2 .

We can rewrite equations (A14) and (A15) in the form of (A18):

C i,1 (x) + 2(γ 1 + µσ -2 ) x C i,1 (x) = - 2λα γ 1 σ 2 C i-1,1 (αx) x 2 (A22) C i,2 (x) + 2(γ 1 + µσ -2 ) x C i,2 (x) = - 2λα γ 2 σ 2 C i-1,2 (αx) x 2 (A23) Let θ j = 1 -2(γ j + µσ -2 ), j ∈ {1, 2}
A general solutions of (A22) is given by:

C i,1 (x) = k i,0 • 1 + k i,1 x θ 1 + 2λα γ 1 σ 2 φ i (x), (A24)
where k i,0 , k i,1 are any arbitrary constants and

φ i (x) = 1 θ 1 C i-1,1 (αx) x dx -x θ 1 C i-1,1 (αx) x x -θ 1 dx (A25)
A general solutions for (A23) is given by:

C i,2 (x) = p i,0 • 1 + p i,1 x θ 2 + 2λα γ 2 σ 2 ϕ i (x), (A26)
where p i,0 , p i,1 are arbitrary constants and

ϕ i (x) = 1 θ 2 C i-1,2 (αx) x dx -x θ 2 C i-1,2 (αx) x x -θ 2 dx (A27)
Note that, for each x i = x L /α i , i = the following conditions should be satised:

E i-1 (x i ) = E i (x i ) (A28) E i-1 (x i ) = E i (x i ) (A29)
We denote

∆A i,1 = A i,1 -A i-1,1 = A 0,1 αλ δ i (A30) ∆A i,0 = A i,0 -A i-1,0 = A 0,0 λ r + λ i (A31) ∆C i,1 (x) = C i,1 (x) -C i-1,1 (x) = θ 1 x θ 1 -1 (k i,1 -k i-1,1 ) + 2λα γ 1 σ 2 [φ i (x) -φ i-1 (x)] (A32) ∆C i,2 (x) = C i,2 (x) -C i-1,2 (x) = θ 2 x θ 2 -1 (p i,1 -p i-1,1 ) + 2λα γ 2 σ 2 [ϕ i (x) -ϕ i-1 (x)
] (A33) Given (A13), equations (A28) and (A29) enable us to establish a link between coecients on contiguous regions:

C i,1 (x i ) = C i-1,1 (x i )- (1 -γ 2 )∆A i,1 x 1-γ 1 i -γ 2 ∆A i,0 x -γ 1 i + x i (∆C i,1 (x i ) + ∆C i,2 (x i )x γ 2 -γ 1 i ) γ 1 -γ 2 (A34) and C i,2 (x i ) = C i-1,2 (x i )+ (1 -γ 1 )∆A i,1 x 1-γ 2 i -γ 1 ∆A i,0 x -γ 2 i + x i (∆C i,1 (x i )x γ 1 -γ 2 i + ∆C i,2 (x i )) γ 1 -γ 2 (A35)
Given the recursive nature of (A34), we can rewrite coecient C i,1 (x i ) as a function of coecient C 0,1 :

C i,1 (x i ) = C 0,1 - (1 -γ 2 )a i,1 x 1-γ 1 i -γ 2 a i,2 x -γ 1 i + G i,1 (k i,1 ; p i,1 ; x i ) γ 1 -γ 2 (A36)
where

a i,1 ≡ i n=1 ∆A n,1 = A 0,1 αλ δ -αλ 1 - αλ δ i = A i-1,1 αλ δ (A37) a i,2 ≡ i n=1 ∆A n,0 = A 0,0 λ r 1 - λ r + λ i = A i-1,0 λ r + λ (A38) G i,1 (k i,1 , p i,1 , x i ) = x i i n=1 ∆C n,1 (x i ) + x γ 2 -γ 1 +1 i i n=1 ∆C n,2 (x i ), (A39)
where ∆C n,1 (x i ) and ∆C n,2 (x i ) given by (A32) and (A33) respectively.

In a similar way we can express C i,2 (x i ) as a function of C 0,2 :

C i,2 (x i ) = C 0,2 + (1 -1 )a i,1 x 1-γ 2 i -γ 1 a i,2 x -γ 2 i + G i,2 (k i,1 ; p i,1 ; x i ) γ 1 -γ 2 (A40)
where

G i,2 (k i,1 ; p i,1 ; x i ) = x γ 1 -γ 2 +1 i i n=1 ∆C n,1 (x i ) + x i i n=1 ∆C n,2 (x i ) (A41)
Equating the right parts of (A36) and (A24) evaluated at x i , as well as the right parts of (A40) and (A26), we can set

k i,0 = C 0,1 (A42)
and

p i,0 = C 0,2 (A43) for i = 1.. + ∞.
Then, for ∀i = 1.. + ∞, k i,1 and p i,1 will be given by the following system of equations:

P i =          G i,1 (k i,1 ;p i,1 ;x i ) γ 1 -γ 2 + k i,1 x θ 1 i = - (1-γ 2 )a i,1 x 1-γ 1 i +γ 2 a i,2 x -γ 1 i γ 1 -γ 2 -2λα γ 1 σ 2 φ i (x i ) G i,2 (k i,1 ;p i,1 ;x i ) γ 1 -γ 2 -p i,1 x θ 2 i = - (1-γ 1 )a i,1 x 1-γ 2 i -γ 1 a i,2 x -γ 2 i γ 1 -γ 2 + 2λα γ 2 σ 2 ϕ i (x) (A44)
A.3.2. A choice of C 0,1 and C 0,2

We make a guess that

C 0,1 = 0, C 0,2 = r r + λ D -νx L x L -γ 2 (A45)
Let E α 0 (x) denote the bank equity value when α = 0, i. e., when a large loss will lead to a complete destruction of bank asset. It follows ODE:

1 2 σ 2 x 2 E α 0 (x) + µxE α 0 (x) -(r + λ)E α 0 (x) + δx -rD = 0 (A46)
Let E λ 0 (x) denote bank equity value when λ = 0, i.e., there is no tail risk exposure. It follows ODE:

1 2 σ 2 x 2 E λ 0 (x) + µxE λ 0 (x) -rE λ 0 (x) + δx -rD = 0 (A47)
We are going to verify that, given C 0,1 and C 0,2 dened in (A45), the following limit conditions are satised:

• (i) lim α→0 E 0 (x) → E α 0 (x) • (ii) lim λ→0 E 0 (x) → E 0 (x)
A general solution for E α 0 (x) will be given by

E α 0 (x) = a α 0 x γ 1 + b α 0 x γ 2 + νx - r r + λ D, (A48)
Given condition at x L and no-bubble condition for x → +∞, we have

a α 0 = 0, b α 0 = r r + λ D -νx L x L -γ 2 (A49)
Thus, limit condition (i) is satised.

A solution for E λ 0 (x) will be given by

E λ 0 (x) = a λ 0 x γ1 + b λ 0 x γ2 + δ r -µ x -D, (A50)
where γ1 > 0, γ2 < 0 are the roots of characteristic equation

1/2σ 2 γ(γ -1) + µγ = r and a λ 0 = 0, b λ 0 = D - δ r -µ x L x L -γ 2 (A51)
Given that lim λ→0 γ 2 → γ2 , it can be easily shown that condition (ii) is satised as well.

A.3.3. The optimal shareholders' liquidation rule

In order to avoid time consistency problem, the choice of the optimal shareholders' liquidation rule should be realized for x ∈ [x 0 , x 1 ), where equity value follows:

E 0 (x) = r r + λ D -νx L x x L γ 2 + νx - r r + λ D, (A52)
Then, from ∂E 0 (x) ∂x L = 0, we get

x S L = -γ 2 1 -γ 2 r r + λ D ν (A53) A.3

.4. A summary of the recursive algorithm

Given the couple {C 0,1 , C 0,2 }, the sequence of coecients {C i,1 (x), C i,2 (x)}, i = 1.. + ∞, can be dened according to the following iterative procedure:

• for each i, use C i-1,1 (x i ) and C i-1,2 (x i ) (which are already known) to dene φ i (x i ) and ϕ i (x) according to (A25) and (A27) respectively;

Replacing equity value into (B4), we get:

(E J (x * ) -ηx * u + D) (λ(1 -α β 2 ) -β 2 ∆µ) x x * u β 2 + (λ(1 -α) -∆µ)ηx > 0 (B5)
By Lemma 1, we λ(1 -α β 2 ) -β 2 ∆µ < 0, while λ(1 -α) -∆µ > 0 by the initial assumption. Then, using the fact that (B5) is binding at x * u , we conclude that

E J (x * u ) -ηx * u + D > 0.
Then, a rst derivative of the left part of (B5) is increasing on x, so that condition (B5) will hold for ∀x > x * u . Since x * u is unique, u t = 0 for x < x * u and equity value on this region will be dened as described in Appendix A.3. Then, by construction, x S L will be the optimal liquidation rule.

Proof of Result 2

Consider any arbitrary liquidation threshold x L . Assume that the bank will never switch to the imprudent risk-management strategy for x ≥ x L . Then, bank equity value would follow:

E(x) = (D -ηx L ) x x L β 2 + ηx -D (B6)
Let us show that, given bank equity value (B6), incentive condition (5) doesn't hold for ∀x L > x L . For x ∈ [x L , x L /α) a large loss would trigger bank liquidation, so that incentive condition (5) can be rewritten as follows:

λE(x) ≥ ∆µxE (x) (B7)
Thus, we have:

(λ -β 2 ∆µ) (D -ηx L ) x x L β 2 + (λ -∆µ)ηx -λD ≥ 0 (B8)
Let f (x) denote the left-hand side of (B8). Then, we have:

lim x→x L f (x) = ∆µ(ηx L (β 2 -1) -β 2 D) < 0, (B9)
for ∀x L > x L , where x L is given by

x L = β 2 β 2 -1 D η (B10)
Therefore, condition (B7) doesn't hold for ∀x L > x L .

Proof of Proposition 1

Consider the regulatory problem in the case of the owner-managed bank:

Max x R >0,s>1 E(x) ≥ 0 s.t. λ(E(x) -1 x≥x R /α E(αx)) ≥ ∆µxE (x) for ≥ x R
where E(x) is given by (A4).

For any s > 1, consider a minimum incentive-compatible recapitalization rule

x B R (s) such that λE(x B R (s)) = ∆µx B R (s)E (x B R (s)): x B R = (1 -s β 2 )λD + ξ 0 (λ -β 2 ∆µ) (1 -s β 2 )(λ -∆µ)η -(ξ 1 + 1 -η)(s -1)(λ -β 2 ∆µ) (B11)
Let us show that the pair s * = arg min x B R (s) and x B R (s * ) is a solution of the above maximization problem if

ξ 1 < 1 -s β 2 s -1 λ -∆µ λ -β 2 ∆µ η + η -1 ≡ ξ 1 (B12)
P1.1. Incentive-compatibility

For any arbitrary recapitalization coecient s > 1, let us check that x B R (s) ensures the incentive constraint of bank shareholders for ∀x ≥ x B R (s). Replacing equity value (A4) into the incentive condition, for

x ∈ [x B R (s), x B R (s)/α) we must have: -(λ-β 2 ∆µ) ξ 0 + (ξ 1 + 1 -η)(s -1)x B R (s) 1 -s β 2 x x B R (s) β 2 +(λ-∆µ)ηx-λD ≥ 0 (B13)
Since the above condition is binding at x = x B R (s) and its left-hand side is increasing on

x, it holds for ∀x ∈ [x B R (s), x B R (s)/α). For x ≥ x B R (
s)/α, we must have:

-(λ(1-α β 2 )-β 2 ∆µ) ξ 0 + (ξ 1 + 1 -η)(s -1)x B R (s) 1 -s β 2 x x B R (s)

P1.2. Feasibility

Condition (B12) ensures that x B R (s) > 0. Now, we verify that (i) it is optimal to recapitalize the bank at x B R (s), rather than to lose equity; (ii) given x B R (s), a voluntary recapitalization at any x > x B R (s) is unoptimal. First, we check that E(x B R (s)) > 0. Let x O R (s) denote a critical recapitalization threshold such that E(x O R (s)) = 0:

x O R (s) = (1 -s β 2 )D + ξ 0 (1 -s β 2 )η -(ξ 1 + 1 -η)(s -1) > D (B15)
In can be shown that x B R (s) > x O R (s), so that E(x B R (s)) > 0.

In order to show that there is no other recapitalization rule that could strictly increase equity value, it is sucient to verify the following condition for x > x B R (s): g(x) ≡ E(sx) -E(x) -K(x) < 0 (B16)

We have g (x) < 0, g (x) > 0 and g(x B R (s)) = 0. Thus, (B16) holds for x > x B R (s).

P1.3. Optimality

Let us show that {s * , x B (s * )} is the optimal recapitalization policy. Note that E(x) is decreasing on x R . Under ξ 1 < ξ 1 , x B R (s) is the optimal incentive-compatible recapitalization rule for given s > 1. Then, a solution of the maximization problem will be delivered by s * = arg min x B R (s).

Proof of Proposition 2

Assume there exists some incentive recapitalization policy {x R , s}, such that shareholders want to induce prudent risk management in their bank for ∀x ≥ x R . Then, the shareholders' maximization problem is reduced to minimization of the total expected amount of incentive managerial compensation: Under the no-bubble condition, a general solution of (B19) is given by:

W (x) = C 0 x β 2 + η δ w 1 x + w 0 r , (B20)
where C 0 is any constant depending on the boundary condition at x T . Let consider two alternative cases: x T = x R and x T = ∅.

1) Case 1: x T = x R . Under the boundary condition W (x R ) = R T , we get

W (x) = w 0 r + η δ w 1 x + R T - w 0 r - η δ w 1 x R x x R β 2 (B21)
Given that M C(x R ) = M C(sx R ) + R T in this case, the shareholders' optimization problem can be rewritten as follows:

min w 0 ≥0,w 1 ≥0,R T ≥0 M C(x) = w 0 r + η δ w 1 x + R T + η δ w 1 x R (s -1) (1 -s β 2 ) x x R β 2 s.t. (λ -∆µβ 2 ) R T - w 0 r - η δ w 1 x R x x R β 2 + (λ -∆µ) η δ w 1 x + λ w 0 r ≥ bx (B22)
The solution of the above problem is given by * 0 = 0, w *

1 = δ η b λ -∆µ , R * T = η δ w * 1 x R
The optimal value of M C(x) is given by:

M C * 1 (x) = η δ * 1 x + x R s (1 -s β 2 ) x x R β 2 (B23)
2) Case 2: x T = ∅. Note that, in this case, R T = ∅ and M C(x) = W (x), so that the shareholders' problem takes the following form:

min w 0 ≥0,w 1 ≥0 M C(x) = w 0 r + η δ w 1 x + η δ w 1 (s -1) (1 -s β 2 ) x R x x R β 2 s.t. (λ -∆µβ 2 ) η δ w 1 (s -1) (1 -s β 2 ) x R x x R β 2 + (λ -∆µ) η δ w 1 x + λ w 0 r ≥ bx (B24)
The couple w * 0 = 0 and w * 1 = δ η b λ-∆µ solves the above problem. The corresponding value of shareholders' costs is given by:

M C * 2 (x) = η δ w * 1 x + x R (s -1) (1 -s β 2 ) x x R β 2 ≡ W * (x) (B25)
Since M C * 2 (x) < M C * 1 (x), the optimal incentive-compatible contract will be dened by the triple Allowing for the internal agency problem between bank shareholders and the bank manager, the regulatory problem transforms to:

Max x R >0,s>1
E W * (x) ≥ 0 s.t.

λ(E W * (x) -1 x≥x R /α E W * (αx)) ≥ ∆µxE W * (x) for ∀x ≥ x R where E W * (x 0 ) is given by (A7).

For any s > 1, consider the minimum incentive-compatible recapitalization rule x A R (s)

such that λE W * (x A R (s)) = ∆µx A R (s)E W * (x A R (s)):

x A R (s) = (1 -s β 2 )λD + ξ 0 (λ -β 2 ∆µ) (1 -s β 2 )(λ -∆µ)(1 - w * 1 δ )η -(1 + ξ 1 -η(1 - w * 1 δ ))(s -1)(λ -β 2 ∆µ) , (B26) given that b < η ξ -ξ 1 + ξ (λ -∆µ) ≡ b, (B27)
where ξ 1 is dened by [START_REF] Bolton | Executive Compensation and Risk Taking[END_REF]. Let us show that the pair s * * = arg min x A R (s) and x A R (s * * ) is a solution of the above maximization problem.

β 2 +

 2 (λ(1-α)-∆µ)ηx ≥ 0 (B14) By Lemma 1, λ(1 -α β 2 ) -β 2 ∆µ < 0 and ∆µ < λ(1 -α) by the initial assumption. Therefore, (B14) holds for x ≥ x B R (s)/α.

  min x T ≥x R ,w 0 ≥0,w 1 ≥0,R T ≥0 M C(x) = E +∞ t e -r(τ -t) (1 xτ =x T R(x τ ) + 1 xτ =x T R T )dτ s.t. λW (x) ≥ ∆µxW (x) + bx (B17)where R(x τ ) = w 0 + w 1 x τ andW (x) = E τ ∧τ T t e -r(τ -t) R(x τ )dτ + e -r(τ T -t) R T (B18)Contract continuation value W (x) follows ODE:1/2σ 2 x 2 W (x) + (µ -∆µ)xW (x) -rW (x) + w 0 + w 1 x = 0 (B19)

x

  T = ∅, R(x) = δ η b λ -∆µ x ≡ w * 1 x, R T = ∅Proof of Proposition 3

The auxiliary result we get, while demonstrating this feature, is the design of equity value in a setting, where the underlying asset's value evolves according to jump-diusion process.

We do not explicitly model deposit insurance premium, since recapitalization mechanism will eliminate default risk. Otherwise, deposit insurance premium could be deducted from the expected asset return rate.

[START_REF] Acharya | Manufacturing Tail Risk: A Perspective on the Financial Crisis of 2007-09[END_REF] Note that δx -rD < 0 corresponds to liquidity injections from the "deep pockets" of bank shareholders as long as they interested in keeping the bank

alive.[START_REF] Bebchuk | Regulating Bankers Pay[END_REF] Since we are dealing with a stationary problem, a termination rule x T will determine the expected contract

duration.[START_REF] Bebchuk | The wages of failure: Executive compensation at Bear Stearns and Lehman 2000-2008[END_REF] We could consider a non-zero reservation wage for the manager in order to care about the existing competition for top managers in banking sector. However, the focus of this paper is on prudent risk management and not on value creation, which strongly depends on specic

managerial talent.[START_REF] Bebchuk | Pay without performance: The unfullled promise of executive compensation[END_REF] Expression (1) captures a trade-o between faster asset growth and asset safety, formalizing a socalled "search for yield" strategy of banks in the period prior to the crisis. Indeed, practices like aggressive subprime lending, abusive use of securitization, poor trading discipline, creative accounting might allow the bank to create the illusion of high performance in the short-run but will inevitably lead to large

losses in the long run.[START_REF] Belhaj | Excess capital, operational disaster risk and capital requirements for banks[END_REF] Bebchuk at al. (2010) estimate that the top managers of Bear Stearns and Lehman Brothers were able to realize about $2 bln, by unloading shares and options during 2000-2008.

Note that x L represents the optimal shareholders' closure rule, provided that the bank sticks to prudent risk management.

Décamps et al. (2011) point out the similar eects of recapitalization costs in the context of liquidity management.

We assume that bonus taxes are paid immediately after receiving compensation.

In fact, the eect of bonus taxes is equivalent to the eect produced by increasing b.

This happened to AIG, one of the biggest players on the world insurance market, bailed out by the Federal Reserve Bank and the U.S. Treasury in

Appendix A. Valuation of contingent claims

A.1. value in the benchmark case Let x R be any arbitrary recapitalization rule. Let K(x) denote the total costs shareholders incur when issuing new equity: K(x) = (ξ 1 + 1)(s -1)x + ξ 0 Equity value in the absence of the agency problem is given by

Solving a corresponding ODE

under the boundary condition

we obtain

where η = δ r-µ+∆µ < 1 and β 2 < 0 is a root of the characteristic equation

under the boundary condition

we obtain

A.3. Equity value in a jump-diusion framework Let x L be any arbitrary liquidation rule. Consider the case when only the imprudent risk-management technology is available and let us design equity value under permanent tail risk exposure.

A.3.1. General solution

Given x i = L /α i , i = 0.. + ∞, let E i (x) be bank equity value on the interval [x i , x i+1 ). a) First, we construct equity value on the interval [x 0 , x 1 ). Since a loss on this interval will lead to the bank default, equity value satises the equation:

Let γ 1 > 0 and γ 2 < 0 be the roots of

A general solution of equation (A8) will be given by:

where C 0,1 , C 0,2 are any arbitrary constants and

Note that, from initial assumptions r > δ + µ -∆µ and ∆µ < λ(1 -α), it follows that δ < r -µ + λ and, therefore, ν < 1.

b) On each interval

A general solution of the above equation is given by:

By substituting (A13) into (A12) and by equating identical terms, we get 4 equations which iteratively dene coecients

which will be the functions of k i,1 and p ;

• after solving the system P i , which is linear on k * i,1 and p * i,1 , compute coecients {C i,1 (x), C i,2 (x)} according to:

Example 1 Let us dene coecients C 1,1 (x) and C 1,2 (x). Given {C 0,1 , 0,2 }, we obtain

Then, given that k 0,1 = 0 and p 0,1 = 0,

Then, unknown parameters k 1,1 , p 1,1 can be uncovered from system P 1 dened in (A44), and coecients C 1,1 (x), C 1,2 (x) will be provided by expressions (A54) and (A55) respectively.

Appendix B. Proofs

Lemma 1 For any θ < α ∈ (0, 1) and ∆µ < λ(1 -α), it follows that λ(1 -α θ ) -θ∆µ < 0

Proof of Lemma 1

Given that ∆µ < λ(1 -α) and θ < 0, we have:

Proof of Result 1

Consider the optimal strategy of bank shareholders when there is nether internal agency problem nor capital regulation. Bank shareholders maximize equity value E(x), by choosing (i) a liquidation/recapitalization rule; (ii) risk management strategy, u t ∈ {0, 1}.

Let E J (x) denote bank asset value constructed in Appendix A.3. Consider the shareholders' liquidation rule derived in Appendix A.3.3:

Let u * denote risk management strategy such that:

u is a critical threshold given by:

Equity value constructed under {x S L , u * } is given as follows:

u } is the optimal shareholders' strategy in the absence of regulatory control.

Recall that by the initial assumption, r > δ + µ -∆µ. This means that a one-dollar investment in riskless security will bring more that a one-dollar investment in bank asset portfolio, so that recapitalization is never optimal.

Let us now show that u * is the optimal risk management strategy. First, we verify that u t = 1 for x > x * u . The corresponding necessary condition implies:

P3.1. Incentive-compatibility

For any arbitrary recapitalization coecient s > 1, let us check that x A R (s) ensures the incentive constraint bank shareholders for ∀x ≥ x A R (s). Replacing equity value (A7) into the incentive condition, for x ∈ [x A R (s), x A R (s)/α) we must have:

where H(s) denote:

Since condition (B28) is binding for x = x A R (s) and its left-hand side is increasing on x when w * < δ, it holds for ∀x ∈

s)/α, we must have:

P3.2. Feasibility

Condition (B27) ensures that x A R (s) > 0. For any given s > 1, we have

Thus, given x A R (s), shareholders will optimally prefer to recapitalize the bank rather than to be deprived of equity. At the same time, in order to ensure that, faced with x A R (s), shareholders will not recapitalize the bank at any x > x A R (s), we must have:

where W * (x) is a terminal payo to the manager. We have g W * (x) < 0, g W * (x) > 0 and g W * (x A R (s)) = 0. Thus, (B30) holds for any

x > x A R (s).

P3.3. Optimality

Since E W * (x) is decreasing on x R , x A R (s) is a solution of the regulatory maximization problem for any s > 1 and b < b. 20 Then, the choice s * * = arg min x A R (s) completes the solution of the maximization problem. 20 For b < b < δ the incentive condition never holds in the neighborhood of the recapitalization threshold, whereas for b > δ the optimal incentive contract would be too costly for shareholders.


Salvatore Tringali

On the congruences a n ± b n ≡ 0 mod n k

Keywords: Congruences, exponential Diophantine equations. 2010 Mathematics Subject Classification. Primary 11A07, 11A99

We determine all triples (a, b, n) of integers with gcd(a, b) = 1 and n ≥ 1 such that n k divides a n + b n for k = max(|a|, |b|). In particular, for positive integers m, n we show that n m | m n + 1 if and only if either (m, n) = (2, 3), (m, n) = (1, 2), or n = 1 and m is arbitrary; this generalizes a couple of problems from the 1990 and 1999 editions of the International Mathematical Olympiad. Then we solve the same question with a nb n in place of a n + b n . The results are related to a conjecture by K. Győry and C. Smyth on the finiteness of {n ∈ N + : n k | a n ± b n } when a, b, k are fixed integers with k ≥ 3, gcd(a, b) = 1, and |a|, |b| not simultaneously equal to 1.

Introduction

It is a problem from the 1990 edition of the International Mathematical Olympiad (shortly, IMO) to find all integers n ≥ 2 such that n 2 | 2 n + 1. This is reported as Problem 7. 1.15 (p. 147) in [START_REF] Andreescu | Number Theory -Structures, Examples, and Problems[END_REF], together with a solution by the authors (p. 323), which shows that the only possible n is 3. On another hand, Problem 4 in the 1999 IMO asks to list all pairs (n, p) of positive integers such that p is a (positive rational) prime, n ≤ 2p and n p-1 | (p -1) n + 1. This is Problem 5. 1.3 (p. 105) in the same book as above, whose solution by the authors (p. 105) is concluded with the remark that "With a little bit more work, we can even erase the condition n ≤ 2p." Specifically, it is found that the required pairs are (1, p), (2, 2) and [START_REF] Carmichael | On the Numerical Factors of Certain Arithmetic Forms[END_REF][START_REF] Carmichael | On the Numerical Factors of Certain Arithmetic Forms[END_REF], where p is an arbitrary prime.

It is now fairly natural to ask whether similar conclusions can be drawn in relation to the more general problem of determining all pairs (m, n) of positive integers for which n m | m n + 1. In fact, the question is answered in the positive, and even in a stronger form, by the following proposition, which represents the main contribution of the present paper:

The author is funded from the European Community's 7th Framework Programme (FP7/2007-2013) under Grant Agreement No. 276487 (project ApProCEM).

(iv) (a, b, n) = (2ε, ε, 3) or (a, b, n) = (ε, 2ε, 3) for ε ∈ {±1}. Proposition 1.1 is proved in Section 2.
For what it is worth, let us be explicit and observe, with the notation as in the above statement, that the result yields a solution of the IMO problems which have originally stimulated this work in the case where a ≥ 1 and b = 1. More specifically, the next corollary is immediate (we omit the obvious proof): (iii) (a, b) = (ε, ε) for ε ∈ {±1} and n is any positive odd integer ≥ 3.

Corollary. Let m, n ∈ N + . Then n m | m n +1 if and only if either (m, n) = (2, 3), (m, n) = (1,
(iv) (a, b, n) = (3ε 1 , ε 2 , 2) or (a, b, n) = (ε 1 , 3ε 2 , 2) for ε 1 , ε 2 ∈ {±1}. (v) (a, b, n) = (2ε, -ε, 3) or (a, b, n) = (-ε, 2ε, 3) for ε ∈ {±1}.
For the notation and terminology used throughout but not defined, as well as for material concerning classical topics in number theory, the reader should refer to [START_REF] Hardy | An Introduction to the Theory of Numbers[END_REF]. In particular, we write R for the ordered field of real numbers, P for the set of all (positive rational) primes, Z for the ordered ring of integers, N for the subsemiring of Z of nonnegative integers, and N + for N \ {0}. For a, b ∈ Z we denote by gcd(a, b) the greatest common divisor of a and b if a 2 + b 2 = 0, and we set gcd(a, b) := ∞ otherwise. Lastly, for c ∈ Z \ {0} and p ∈ P we use e p (c) to mean the greatest exponent k ∈ N such that p k | c, while we let e p (0) := ∞.

We will make use at some point of the following result, which belongs to the folklore and is typically attributed to É. Lucas [START_REF] Lucas | Théorie des Fonctions Numériques Simplement Periodiques[END_REF] and R.D. Carmichael [START_REF] Carmichael | On the Numerical Factors of Certain Arithmetic Forms[END_REF] (the latter having fixed an error in Lucas' original work in the 2-adic case).

Lemma 1.3 (Lifting-the-exponent lemma). For all x, y ∈ Z, ℓ ∈ N + and p ∈ P such that p ∤ xy and p | xy, the following conditions are satisfied:

(i) If p ≥ 3, ℓ is odd, or 4 | x -y, then e p (x ℓ -y ℓ ) = e p (x -y) + e p (ℓ).
(ii) If p = 2, ℓ is even and e 2 (xy) = 1, then e 2 (x ℓy ℓ ) = e 2 (x + y) + e 2 (ℓ).

In fact, our proof of Proposition 1.1 is but the result of a meticulous refinement of the solutions already known for the problems mentioned in the preamble. Hence, our only possible merit, if any at all, has been that of bringing into focus a clearer picture of (some of) their essential issues.

The study of the congruences a n ± b n ≡ 0 mod n k has a very long history, dating back at least to Euler, who proved that, for all integers a, b with gcd(a, b) = 1 and a > b ≥ 1, every primitive prime divisor of a n -b n is congruent to 1 modulo n; see [2, Theorem I] for a proof and [2, §1] for the terminology. However, since there are so many results related to the question, instead of trying to summarize them, we just refer the reader to the paper [START_REF] Győry | The divisibility of a nb n by powers of n[END_REF], whose authors provide an account of the existing literature on the topic. The paper also characterizes, for fixed a, b ∈ Z and k ∈ N + , the sets (respectively, a nb n ) for k = max(|a|, |b|) does not appear to be considered neither in [START_REF] Győry | The divisibility of a nb n by powers of n[END_REF] nor in the references therein.

R + k (a, b), respectively R - k (a,
On another hand, it is suggested in [START_REF] Győry | The divisibility of a nb n by powers of n[END_REF] that R + k (a, b) and R - k (a, b) are both finite provided that a, b, k are fixed integers with k ≥ 3, and |a|, |b| are relatively prime but not simultaneously equal to 1; the authors point out that the question is probably a difficult one, even assuming the ABC conjecture. Although far from providing an answer, Propositions 1.1 and 1.2 in the present paper prove in this respect that, under the same assumptions, R + k (a, b) and R - k (a, b) are finite for all sufficiently large k, and indeed for k ≥ max(|a|, |b|).

Proofs

For the sake of exposition, we premise a couple of lemmas.

Lemma 2.1. Let x, y, z ∈ Z and ℓ ∈ N + such that gcd(x, y) = 1 and z | x ℓ + y ℓ . Then xy and z are relatively prime, q ∤ x ℓy ℓ for every integer q ≥ 3 for which q | z, and 4 ∤ z provided that ℓ is even. Moreover, if there exists an odd prime divisor p of z and ℓ such that gcd(ℓ, p -1) = 1, then p | x + y, ℓ is odd and e p (z) ≤ e p (x + y) + e p (ℓ).

Proof. The first part is routine (we omit the details). As for the second, let p be an odd prime dividing both z and ℓ with gcd(ℓ, p -1) = 1; also, considering that z and xy are relatively prime (by the above), denote by y -1 an inverse of y modulo p and by ω the order of xy -1 modulo p, viz the smallest k ∈ N + such that (xy -1 ) k ≡ 1 mod p; cf. [5, §6.8]. Since (xy -1 ) 2ℓ ≡ 1 mod p, we have ω | 2ℓ. It follows from Fermat's little theorem and [START_REF] Hardy | An Introduction to the Theory of Numbers[END_REF]Theorem 88] that ω divides gcd(2ℓ, p -1), whence we get ω | 2, using that gcd(ℓ, p -1) = 1. This in turn implies that p | x 2y 2 , to the effect that either p | xy or p | x + y. But p | xy would give that p | x ℓy ℓ , which is however impossible by the first part of the claim (since p ≥ 3). So p | x + y, with the result that ℓ is odd: For if 2 | ℓ then p | 2x ℓ (because p | z | x ℓ + y ℓ and y ≡ -x mod p), which would lead to gcd(x, y) ≥ p (again, using that p is odd), that is to a contradiction. The rest is an immediate application of Lemma 1.3.

Lemma 2.2. Let x, y, z ∈ Z such that x, y are odd. Then x 2 -y 2 = 2 z if and only if z ≥ 3, x = (2 z-2 + 1)ε 1 and y = (2 z-2 -1)ε 2 with ε 1 , ε 2 ∈ {±1}.
Proof. Since x and y are odd, x 2y 2 is divisible by 8, i.e. z ≥ 3, and there exist i, j ∈ N + and ε ∈ {±1} such that i + j = z, xy = 2 i ε and x + y = 2 j ε. It follows that x = (2 j-1 + 2 i-1 )ε and y = (2 j-1 -2 i-1 )ε, and then either i = 1 or j = 1 (otherwise x and y would be even). The rest is straightforward.

We are ready to write down the proof of our main results. Furthermore, e 3 (n) = 1, and hence n = 3t for some t ∈ N + with gcd(6, t) = 1. It follows that t 2 | α t + 1 for α = 2 3 . Suppose by contradiction that t ≥ 2 and let q be the least prime divisor of t. Then another application of Lemma 2.1 gives 2e q (t) ≤ e q (α + 1) + e q (t), and accordingly 1 ≤ e q (t) ≤ e q (α + 1) = e q (3 2 ), which is however absurd due to the fact that gcd(3, t) = 1. Therefore t = 1, i.e. n = 3, and putting all together completes the proof (once checked that 3 2 | 2 3 + 1 3 ). (5)

Proposition 1 . 1 .

 11 Let a, b, n be integers with gcd(a, b) = 1 and n ≥ 1. Then n k divides a n + b n for k = max(|a|, |b|) if and only if either of the following holds: (i) a, b are any coprime integers and n = 1. (ii) a, b ∈ {±1} and n = 2. (iii) (a, b) = (ε, -ε) for ε ∈ {±1} and n is any positive odd integer ≥ 3.

Proposition 1 . 2 .

 12 2), or n = 1 and m is arbitrary. Also, we use Proposition 1.1 to prove the following: Let a, b, n be integers with gcd(a, b) = 1 and n ≥ 1. Then n k divides a nb n for k = max(|a|, |b|) if and only if either of the following holds: (i) a, b are any coprime integers and n = 1. (ii) a, b ∈ {±1} and n is any positive even integer.

  b), of all positive integers n such that n k divides a n + b n , respectively a nb n (note that no assumption is made about the coprimality of a and b), and addresses the problem of finding the exceptional cases when R - 1 (a, b) and R - 2 (a, b) are finite; see, in particular, [4, Theorems 1-2 and 18]. Nevertheless, the related problem of determining, for fixed a, b ∈ Z with gcd(a, b) = 1, all positive integers n such that n k divides a n + b n

Proof of Proposition 1 . 1 . 1 )

 111 Let us suppose (by symmetry) that |a| ≥ |b| and n |a| divides a n + b n . The case n = 1 is trivial, so we assume n ≥ 2. Since on the one hand a and b are relatively prime, while on the other hand n ∤ a n + b n if |a| = 1 and b = 0, we then have |a| ≥ |b| ≥ 1. The case |a| = 1 is trivial too, and leads to points (ii) and (iii) in our claim. Hence, we suppose in the sequel that |a| ≥ 2 and |a| > |b| ≥ 1. (Considering that 4 | n 2 whenever 2 | n, it follows from Lemma 2.1 that n is odd and gcd(ab, n) = 1. Denote by p the smallest prime divisor of n. Again by Lemma 2.1, it is found that p | a + b and |a| -1 ≤ (|a| -1)e p (n) ≤ e p (a + b).(2)Also, a + b = 0 by equation (1), with the result that a + b = p r s, with r ∈ N + , s ∈ Z \ {0} and p ∤ s.(3)Then, equations (1) and (3) yield that 2|a| ≥ p r • |s| + 1. This implies by equation (2), since r = e p (a + b), that 3 r • |s| ≤ p r • |s| ≤ 2r + 1, which is possible only if p = 3 and r = |s| = 1. So, by equations (2) and (3), |a + b| = 3 and |a| = 2, to the effect that either (a, b) = (2, 1) or (a, b) = (-2, -1).

Proof of Proposition 1 . 2 . 4 )

 124 The case n = 1 is trivial, so suppose in the sequel that n ≥ 2. Since n | a nb n and gcd(a, b) = 1, this gives in the first place ab = 0; secondly, |a| = |b| only if a, b ∈ {±1}, and then if and only if either a, b ∈ {±1} and n is any even positive integer, or (a, b) = (ε, ε) for ε ∈ {±1} and n is any odd integer ≥ 3. Then by symmetry, we also assume for the remainder of the proof that |a| is greater than |b|, so that (to summarize) |a| > |b| ≥ 1, n ≥ 2, and n |a| divides a nb n . (With this in hand, write n as 2 r s, where r ∈ N, s ∈ N + and gcd(2, s) = 1. Then, equation (4) implies that α s + β s is divided by s |a| for α := a 2 r and β := -b 2 r (note that |α| ≥ |β| since |a| > |b|), which leads, by Proposition 1.1, to one of the following three cases. CASE 1: s = 1, viz n = 2 r with r ∈ N + . Obviously, n is even, and we get (by coprimality) that both a and b are odd, that is 8 | a 2b 2 . It follows from point (i) of Lemma 1.3 that e 2 (a 2 rb 2 r ) = e 2 (a 2b 2 ) + e 2 (2 r-1 ) = e 2 (a 2b 2 ) + r -1. (We apply Lemma 1.3 with x = a 2 , y = b 2 , ℓ = 2 r-1 and p = 2, where the notation is the same as in the statement of the lemma). Since (2 r ) |a| divides a 2 rb 2 r in view of equation (4) and our standing assumptions, then (|a| -1) • r ≤ e 2 (a 2b 2 ) -1.

Acknowledgments. I am grateful to Paolo Leonetti (Università Bocconi di Milano) for useful discussions, and to Carlo Sanna (Università di Torino) for his careful proofreading and for providing remarks that improved the readability of the paper.

Now, there exist u, v ∈ N + with u ≥ 2 and gcd(2, v) = 1 such that a 2b 2 = 2 u+1 v, with the result that |a| > 2 u/2 √ v. Hence, we get by equation ( 5), also taking into account that 2 x ≥ x + 1 for every x ∈ R with x ≥ 1, that

which is possible only if r = 1 and √ v < 2. Then 2 u/2 √ v < u + 1, with the result that 2 ≤ u ≤ 5 and v = 1 (using that v is odd). In the light of Lemma 2.2, all of this implies, in the end, that the conditions in equation ( 4), when n is a positive power of two, are satisfied only if a = (2 z +1)ε 1 , b = (2 z -1)ε 2 and n = 2, where ε 1 , ε 2 ∈ {±1} and z is an integer between 1 and 4; but now we need 2 z ≤ z + 1 by equation ( 5), so necessarily z = 1, i.e. a = 3ε 1 and b = ε 2 (and, in fact,

Then a 2 r = ε, which is impossible because |a| ≥ 2 by equation ( 4).

Then a 2 r = 2ε and b 2 r = -ε, to the effect that r = 0, and hence (a, b, n) = (2ε, -ε, 3) for ε ∈ {±1}.

Putting all the pieces together, the proof is thus complete.

Author information

Salvatore Tringali, Laboratoire Jacques-Louis Lions (LJLL), Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie (UPMC), 4 place Jussieu, boîte courrier 187, 75252 Paris (cedex 05), France. E-mail: tringali@ann.jussieu.fr


Wood is a porous, hygroscopic, anisotropic and non homogeneous biopolymer. It is classified as a viscoelastic material with mechanical properties depending on temperature and moisture content. The effect of creep is an important factor for the design and the durability of timber structures. Creep evolution results from the interaction between mechanical stress and moisture content variations. In this context, Gril [START_REF] Gril | Une modélisation du comportement hygro-rhéologique du bois à partir de sa microstructure[END_REF] proposed a formulation based on hygro-locks , a combination of rheological elements with their activation depending on the humidity level. Although this model accounted for several features of the mechano-soptive creep (the so-called ++, -and + effects) it failed to produce a realistic hygromechanical response for other type of loading, such as stress relaxation. Inspired by this concept, Husson [START_REF] Husson | Elastic response in wood under moisture content variations: analytic development[END_REF] proposed a new formulation based on hygro-locks introduced in the elastic response, using the following hereditary form characterized by a time synchronization between mechanical loading and moisture-dependent rigidity:

( ) ∫ - ′ ′ ∂ ∂ ′ = t t d t t t J t 0 max , ) ( σ ε
where J max (τ,t) designates the maximum of elastic compliance during the interval [τ,t]. The resulting stress-strain relationship defines a hygromechanical response represented by a mechanosorptive spring . An alternative description, directly based on the hygro-lock concept, is based on a discretisation (u 1 u N ) of the moisture content. The stress σ and strain ε of the spring are given by σ=σ 1 + σ N , ε=J 1 σ 1 + +J N σ N , where J 1 < <J N is a moisture-dependent compliance When the moisture content equals u n , any stress change is supported by the link n: dσ=dσ n , so that the instantaneous response of the spring is given by dε=J n dσ. When the moisture content increases from u n to u n+1 , the stress is transferred from link n to link n+1: δσ n+1 =σ n , δσ n .=-σ n , and the strain increase resulting from this internal load transfer is δε=(J n+1 -J n )σ n . This internal variable formulation is equivalent to the integral formulation when the continuous moisture-dependency J(u) is approximated by a discrete description. With both formulations it is possible to develop a thermodynamic approach based on the first principle, in order to put in evidence the capacity to store energy during drying phases.

( ) A similar hygro activation can be introduced in the viscoelastic behavior represented by a generalized Kelvin-Voigt or Maxwell model, where each spring is replaced by a mechanosorptive spring. These rheological models can be completed by hygro-expansion elements driven by moisture content level and expansion coefficient α. The case of a Kelvin-Voigt link is illustrated in Figure 1, where the springs have been replaced by a symbol of the mechanosorptive spring evoking the hygrolocks. Using the integral formulation, the total strain can be expressed as follow:

t σ ( ) t σ ( ) t ε η E 0 E , α ω
( ) ∫ ∫ ′ ′ ∂ ∂ + ′ ′ ∂ ∂ ′ = t t t d t u t d t t t J t 0 0 , ) ( α σ ε where ( ) , Jt τ
is a creep function given by: where J 0 =1/E 0 and J 1 =1/E are the instantanzous and delayed compliance, respectively, and η the viscosity coefficient of the dashpot. An incremental formulation of these equations has been successfully applied to predict the 1D response for mechanosorptive tests in L direction. However, the complex time-dependent relationship between stress, strain and moisture content, requires a formulation adapted to finite element implementation. A major obstacle inherent to the integral formulation is the need to store the whole loading history at each integration point. The use of the formulation based on internal variables should allow to overcome this obstacle. The time resolution allows a coupling with a heat and mass transfer algorithm and complex loading with time-dependent boundary conditions.

( ) ( ) () ( ) ( ) ∫∫ ′′ ′ ′ ′ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ ′ ′ ′ ′ ′ ′ ′ ′ ′ ′ - ′ ′ + ′ = ′ t t

  modelling of the mechano-sorptive behavior based on integral formulation or internal variables J.M. Husson 1 , J. Gril 2 , F. Dubois 1 , N. Sauvat 1

Figure 1 :

 1 Figure 1: A mechano-sorptive Kelvin Voigt model