Rational blinders: is it possible to regulate banks using their internal risk models?∗ Jean-Edouard Colliard† June 20, 2012

Abstract Financial institutions use quantitative risk models not only to manage their risks, but also to communicate information. The Basel regulation in particular uses banks’ own estimates to make capital requirements more sensitive to each bank’s risks, and both the models and the regulation have been blamed for their over-optimism. I link over-optimism to a hidden information problem between a regulator and a bank who knows better which models are correct. If the regulator treats this problem as “model risk” and only uses tighter capital requirements (e.g. switches from Basel II to Basel III), a wider adoption of optimistic models to bypass the regulation and an increase of banks’ risks can follow. On the other hand, there is a cost of ensuring banks use adequate models, which increases with the extent to which internal models are used to compute finer capital requirements. Informational constraints thus make the case for a model-based regulation much weaker. More broadly, this paper shows how economic incentives can impact the development of new predictive models. Journal of Economic Literature Classification Number : D82, D84, G21, G32, G38. Keywords: Basel Accords, internal ratings, banking regulation, credit risk models. ∗

I’m grateful to Toni Ahnert, Christophe Chamley, Alexandre de Corni`ere, Gabrielle Demange, Thierry Foucault, David Martimort, Eric Monnet, Jean-Charles Rochet, Arthur Silve, Ernst-Ludwig von Thadden, Gabriel Zucman, participants to the 28th GdRE Annual International Symposium on Money, Banking and Finance, the 11th Society for the Advancement of Economic Theory Conference, the 26th Congress of the ´ European Economic Association, and to seminars at the Bank of England, the European Central Bank, Ecole Polytechnique, ESSEC, HEC Paris, Oxford University, Paris School of Economics, Sciences Po Paris, University of Amsterdam, University of Mannheim and University of Z¨ urich for helpful comments and suggestions. † Paris School of Economics. From September 2012 on: European Central Bank, Financial Research Division. Address: PSE, 48 Boulevard Jourdan, 75014, Paris, France. E-mail: [email protected] . Phone: (33) (0)6 61 52 93 35.

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1

Introduction

The recent crisis has highlighted an important discrepancy between the latest advances in economics and finance and the practice of financial institutions and regulators. The inability of the latter to take into account extreme risks is denounced precisely as new generations of models using fat-tailed distributions or extreme value theory flourish. Dowd, Cotter, Humphrey, and Woods (2008) illustrate the extent to which some older models used in practice were flawed: “25-sigmas events” happening several times in a row in August 2007 were supposed to occur once in every 10135 years! Since the regulation of financial institutions increasingly relies on internal models, it is urgent to understand why many agents have chosen internal models that were too optimistic regarding extreme risks, and how this can be avoided. The regulation of banks in particular relies heavily on internal models to compute capital requirements more in line with a bank’s risk: “because the most accurate information regarding risks is likely to reside within a bank’s own internal risk measurement and management systems, supervisors should utilize this information to the extent possible” (Federal Reserve System Task Force on Internal Credit Risk Models (1998)). Dan´ıelsson (2008), Rochet (2010) or Eichengreen (2011) argue that the Basel regulation, by allowing banks to use internal models to compute regulatory capital, has given them incentives to use optimistic models to increase leverage, or at least has not encouraged them to produce cautious estimates of risk. A recent study by Barclays Equity Research (Samuels, Harrison, and Rajkotia (2012)) shows that investors have lost faith in the risk measures produced by internal models, and suspect the choice of risk models is a source of bias1 . More specifically, over the 130 equity investors surveyed, more than a half do not trust risk weightings, 80% think the way the banks’ risk models work is a significant driver of major differences between European banks’ risk weightings, and more than 80% think model discretion should be removed (Samuels, Harrison, and Rajkotia (2012), Figures 13, 19 and 1 respectively). The first part of this paper analyzes this problem formally. I consider financial intermediaries with limited liability, competing both to attract investors and to lend to final borrowers. Intermediaries have to choose a risk model which determines the regulatory capital they have to maintain. I model this situation as a hidden information problem in which a bank reports an internal model to the regulator, who chooses a capital constraint depending on the report. Adopting an overoptimistic model allows a bank to understate its risk and lend more. 1

See also the article “Investors lose faith in risk measures” by Brooke Masters in the Financial Times, 24.05.12. Many other recent events have eroded investors’ faith in internal risk models, see for instance ““Whale” makes a big splash on risk models” by Pablo Triana, Financial Times, 28.05.12.

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In the current state of the regulation, banks face few ex-post penalties for reporting overoptimistic models for some types of risk. The regulatory response has been to tighten capital requirements and set aside provisions for “model risk”. But model risk arises when models may be wrong due to unbiased mistakes, not to bad incentives. The first part of this paper shows that such a regulatory tightening, e.g. the transition from Basel II to Basel III, can counter-intuitively increase the risk that a bank defaults (Proposition 2). When the regulation tightens, the supply of bank loans decreases and the interest rate on loans goes up. As a result, using optimistic models to bypass the regulation becomes more profitable, and the wider adoption of over-optimistic models can lead to an increase in the average risk of banks. Regulators are currently looking for ways to restore investors’ confidence in risk weightings based on internal models but, according to the first part of this paper, the regulator cannot substitute a “naive” regulatory tightening for a regulation solving the hidden information problem. The regulator’s task is complicated by the intermediaries’ limited liability and the fact that they can opt out of the mechanism and remain unleveraged, or choose Basel’s “standard approach”, both options being type dependent. I first study a backtesting mechanism where penalties punish intermediaries who had reported optimistic models when high levels of losses occur (such a mechanism is used for market risk models). I give sufficient conditions for the regulator to be able to implement the first-best outcome (Proposition 3), and study a difficulty specific to these internal models: they typically differ in their predictions about extreme levels of losses, thus revelation of information relies on penalties after high losses. But if the regulation is very sensitive to the model reported, a bank with a very optimistic model is allowed a leverage so high that it can be already in default when it should be punished. I give conditions under which this prevents the regulator from implementing the first-best (Proposition 4), in which case a trade-off appears between the cost and the risk-sensitiveness of the regulation. I then study an auditing mechanism that would be a smaller departure from the existing regulation of credit risk, and show that a similar problem arises: making more use of banks’ information increases incentives to lie, thus the second-best capital requirements are less risk-sensitive than the first-best, to save on auditing costs (Proposition 5). Ultimately, in both cases ensuring truthful revelation has a cost that reduces the desirability of a regulation based on internal models. When informational costs are too high, the second-best regulation can be a capital ratio that is computed using the regulator’s information but where banks’ internal models play little role. Let me illustrate in more concrete terms the issue of model uncertainty and strategic

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model selection with the following example. To compute the capital requirement to be held for a corporate/sovereign or bank exposure, a bank under the Basel framework can opt for the “advanced-internal ratings based approach”, in which case the capital requirement k will be computed as follows: k=

LGD × N

N −1 (P D) √ + 1−R

r

! ! R 1 + (M − 2.5)b × N −1 (0.999) − P D × LGD 1−R 1 − 1.5b (1)

where P D is the probability of default, LGD the loss given default, R a correlation coefficient, M the effective maturity, b a maturity adjustment coefficient, and N the cdf of a Gaussian distribution. The risk-weighted asset will be obtained by multiplying k by 12.5×EAD, where EAD is the exposure at default. With the regulator’s approval, the bank can use its own internal model to compute P D, LGD and EAD, knowing that P D also enters the formulas defining b and R. These parameters can be computed by the bank in several ways, but a large bank would typically use a Jarrow-Turnbull type model. Tarashev (2008) performs an interesting exercise by comparing for different classes of bonds the regulatory capital obtained with different academic models. On a sample of BBB-rated bonds for instance regulatory capital ranges from 2.3% to 2.8% depending on the model chosen. If a bank were to choose the most pessimistic model it would thus have to keep 22% more regulatory capital than if it chose the most optimistic model (see table 5 in Tarashev (2008))! Moreover, this difference is obtained only due to the different structures of these models; playing with their actual implementation, and typically the time-window considered to calibrate them, would give even more opportunities to obtain very optimistic measures. The example developed in this paper fits the regulation of credit risk well, but section 4 in particular could also apply to market risk, and to operational risk where model uncertainty may be even more severe. It must be noted that credit risk models are extremely difficult to backtest due to their time horizon (typically one-year) and the scarcity of available data. The formula in equation 1 is supposed to lead in the end to enough capital to cover losses with probability 0.999. If the regulator applied the same methodology as for market risk, he would ask the bank to report directly its Value-at-Risk at 0.1% and would then count the number of years in which a “violation” occurs. Such a violation is supposed to occur once in every thousand years, so that a liar reporting a risk ten times lower than it really is would expect to remain undetected for a hundred years. Advanced tests can be used of course, such as the one proposed by Lopez and Saidenberg (2000), but the problem of insufficient data cannot be completely bypassed, see also Kupiec (2002). Thus the regulator can hardly 4

backtest credit risk models but can check their methodology ex-ante, which is also a difficult task (see Jackson and Perraudin (2000)). Clearly, it is possible to use optimistic models, and there are incentives to do so: due to its limited liability and to externalities of a bank’s default, a bank will target a lower probability of survival than the regulator wishes. The results in the first part of the paper give clear-cut empirical implications about which models will be used. The main one is that an exogenous increase of demand for banks’ loans or other products should cause more banks to use more optimistic models, as measured for instance by the VaRs given by the different models for a common representative portfolio. More banks should also abandon the “standardized” approach of the Basel regulation for the “internal ratings based approach” allowing the use of internal models to compute capital requirements. Finally, since the implementation of Basel III leads to much heavier capital charges, the development and adoption of more optimistic models should be used to mitigate the additional costs of complying with the regulation. An implication of the second part is that good candidate models to bypass the regulation are those which are proven to be over-optimistic only when the situation is so bad that the regulator cannot be too harsh on banks, by fear of deepening an economic downturn. Interestingly, it has indeed taken the recent financial crisis to reconsider the validity of risk models, and the pace at which new regulations are implemented partly reflects concerns about the effects of increased capital requirements on economic activity. Finally, the hidden information problem I consider here also turns up in other forms of “regulation”. A rating agency estimating the creditworthiness of a firm or the risk of a pool of loans similarly has to rely partly on internal models chosen by an agent with an incentive to cheat. Inside a firm, the models a team or a desk use to manage risk may also determine the funds the desk manages and the compensation its members receive. With slight amendments, the model I develop here can be used to analyze this larger class of problems.

Related literature There is a huge literature on the regulation of banks and on the Basel framework in particular. Kim and Santomero (1988) and Rochet (1992) show that risk-based capital requirements are necessary to control banks’ risks without inducing inefficient choices of assets. The use of VaR for market risk and internal ratings for credit risk has been seen as a way to implement capital requirements responding finely to a bank’s risk. Dangl and Lehar (2004) for instance have analyzed the risk-taking of a bank under VaR based regulation. Several papers like Dan´ıelsson, Shin, and Zigrand (2004), Heid (2007) or Kashyap and Stein (2004) have crit5

icized the pro-cyclical effects of using risk-based capital requirements when the focus shifts from an individual bank to a market equilibrium. The first part of this paper also focuses on equilibrium effects but without the assumption that risk-based requirements stem from a correct representation of risk. This part can be linked to several models of competition between leveraged banks, for instance Herring and Vankudre (1987), Matutes and Vives (2000), Bolt and Tieman (2004). Closely related are also recent papers studying the strategic choice of banks between Basel’s “standardized approach” and the “internal ratings based” approach; these include Ant˜ao and Lacerda (2011), Hakenes and Schnabel (2011) and Ruthenberg and Landskroner (2008). The framework of this paper is in general simpler so as to allow for a discussion of the effects of a more risk-sensitive regulation, additional uncertainty on internal models and other variables that were not the focus of the aforementioned papers. Carey and Hrycay (2001) estimate the extent to which portfolio managers can use different methodologies to understate credit risk and game the regulation or increase their bonus, and conclude on the necessity to monitor the use of credit risk models. Jacobson, Linde, and Roszbach (2006) show on a sample of Swedish banks that similar risks are estimated differently by banks using different methodologies, leading to different estimates of economic capital. They also suggest that “given the fact that many supervisors will have an informational disadvantage in their relation with banks, internal models are likely to become instrumental in banks’ search for lower regulatory capital buffers (that meet their economic requirements)”. Feess and Hege (2011) are to my knowledge the only authors who have included this possibility of using internal risk models to bypass the regulation in a theoretical framework, but they mainly focus on banks’ choice between using internal models or not, not on the choice of one model rather than another. Several papers have considered the possibility of biased models for the regulation of market risk. An interesting difference with credit risk models is that market risk is evaluated on a daily basis, typically by the value at risk at the 99% level, such that after 100 days it is already possible to detect blatant over-optimism. Incentives not to use optimistic market risk models have been carefully provided in the Basel framework (and actually since the 1996 amendment to the Basel I capital accord), as studied theoretically by Lucas (2001) and Cuoco and Liu (2006). Empirical studies by Berkowitz and O’Brien (2002), P´erignon, Deng, and Wang (2008) and P´erignon and Smith (2010) show that VaRs reported for market risk are actually too conservative, implying that the penalty for under-reporting the VaR is probably too high and has an unwanted impact on the way banks evaluate their market risk. These papers show that banks respond to incentives to choose pessimistic market risk models - thus

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they should also respond to incentives to choose optimistic credit risk models. The second part focusing on how the regulator could elicit the revelation of the true model echoes papers like Chan, Greenbaum, and Thakor (1992), Freixas and Rochet (1998) and Giammarino, Lewis, and Sappington (1993) on the problem of fairly priced deposit insurance. In these papers information is revealed by using risk-based insurance premiums and capital ratios, a possibility I do not consider here as insurance premiums are not riskbased in most countries, and when they are it is in too crude a way to realistically assume that premiums could be based on the internal models used so as to give correct incentives. I discuss the links with these papers further in section 4.3. Another difference is that in my paper the hidden information is about “models”, or distributions of losses, which typically give similar predictions except for high levels of losses. This adds an interesting difficulty to the design of the optimal regulation: the regulator can use the observed level of losses as a signal on the agent’s information, but only in extreme cases. The auditing problem generalizes the one considered by Prescott (2004) and shows an incentive to bias the required capital ratios downwards and not only upwards, such that in the end capital ratios are not necessarily higher than in the first-best, but less sensitive to the intermediary’s type. This paper contributes to yet another strand of the literature, concerned with “markets for models/theories”. Banks in my paper are on the demand side of such a market. Examples include Hong, Stein, and Yu (2007), who study agents relying on partial models and shifting from one model to the other depending on their observations, and Cogley, Colacito, and Sargent (2007) who study rational learning of macroeconomic models with a feedback from learning on economic variables. Fewer papers look at situations where the demand for models is not directly derived from their predictive power only. Exceptions include Millo and MacKenzie (2009) who study the usefulness of simple risk management models for communicating and reducing complexity, and Ghosh and Masson (1994), who suggest that governments could in fact pretend to believe in economic models they know to be false so as to gain in bargaining power when meeting with other countries’ representatives. The remainder of the paper is organized as follows: section 2 describes the framework, solves the maximization program of an intermediary with a given model and given prices, and derives the optimal regulation under complete information. Section 3 studies the problem of a regulator whose only tool is model-sensitive capital requirements. Section 4 studies how the regulator could use backtesting or auditing to reveal the true model. Finally, section 5 discusses extensions to account for other possible incentives to develop credit risk models.

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2

Framework

2.1

Agents and assets

In order to study how market prices depend on the models chosen by financial intermediaries, and how these prices determine the incentives to choose a given model, I need to introduce at least three classes of agents: Borrowers need to finance risky projects. They have a demand for loans D(rL ) as a function of the gross interest rate rL . D is decreasing in rL , D(1) = +∞ and lim+∞ D(rL ) = 0. I denote rL (L) the inverse demand function. A random proportion t of borrowers will default, where t is taken from a distribution f (t, σ) with support over [0, 1], F (t, σ) being the cumulative. σ is a parameter of the distribution, a higher σ being associated with more default risk (more on this below). Finally, I assume that a defaulting loan yields 0 (failure of the borrower’s project)2 . Investors with a large initial wealth W can invest in a safe asset yielding the exogenous riskless rate r0 with certainty, or lend to financial intermediaries at a rate rD , but not directly to borrowers. Financial intermediaries can lend to borrowers, invest in the safe asset, and borrow M from investors at rD . They initially own K (equity) and are protected by limited liability. Finally, I assume that a debt contract between an investor and an intermediary cannot be made contingent on the intermediary’s subsequent choice of leverage or assets. All agents of a given type are homogenous, risk-neutral, and act as price-takers on a perfect competitive market. Finally, a benevolent regulator can set limits to intermediaries’ leverage and aims at maximizing social welfare. Throughout the paper a female pronoun refers to the regulator, and a male pronoun to an intermediary. Figure 1 sums up the market structure.3 2

As I do not wish to address the problem of adverse selection, default is assumed to be independent of the interest rate and the amount lent. Relaxing this assumption would make the analysis much more cumbersome without altering the main results. 3 All figures are in the text, the notations used are summed up in A.1.

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Figure 1: Market structure.

2.2

Model uncertainty

The intermediary chooses how much to lend based on his estimation of the probability distribution of defaults among borrowers. This estimation may come from different methods, but is typically given in large banks by an internal credit risk model. Due to the difficulty of backtesting internal models, there is important model uncertainty. Moreover, there is asymmetric information about models since an intermediary is likely to have more information than outsiders about which models are more reliable. This motivates the following assumptions on risk models: • M1: Let {F (., σ), σ ∈ [σmin , σmax ]} be a family of cumulative distributions over [0, 1], parameterized by σ, twice-continuously differentiable in both arguments. Denote f (., σ) the corresponding densities. The family of distributions has the monotone likelihood ratio property: ∀t0 , t1 , σ0 , σ1 with t1 ≥ t0 , σ1 ≥ σ0 ,

f (t1 , σ1 ) f (t0 , σ1 ) ≥ f (t1 , σ0 ) f (t0 , σ0 )

• M2: A given σ is randomly selected in [σmin , σmax ] according to some distribution Ψ(.), density ψ(.). Intermediaries observe σ before they take any decision, but σ remains hidden to the regulator. t, the proportion of defaulting loans, is drawn from the cumulative distribution function F (., σ). M1 amounts to assuming there exists a set of different plausible models indexed by σ, with enough models and parameterizations available for the family to be continuous and 9

twice differentiable. Moreover, models are ranked in terms of likelihood ratios: models with a low σ give risk estimates unambiguously more optimistic than models with a high σ. This assumption will play a role mostly in section 4. The family F (., .) can be interpreted as one model with different parameterizations, or as different models from different families, where each model is indexed by some σ. Finally, M2 means that intermediaries know the true value of the parameter σ while the regulator does not, thus an extreme form of asymmetric information.

2.3

The intermediary’s program

Taking the model chosen and prices as given, I derive the demand for funds and the supply of credit by a financial intermediary. In the next section I will endogenize the choice of a model and study the equilibrium of the market. Take r0 , rD , rL as given with rL ≥ rD ≥ r0 . In this setup it never pays off for a financial intermediary to borrow M > 0 and invest at the riskless rate r0 since investors necessarily ask for an interest rD ≥ r0 . Thus we must have either L = M + K with M possibly zero, or L = M = 0 (intermediaries invest their equity at the riskless rate)4 . Due to limited liability, an intermediary’s realized profit if he lends L and a proportion t
of borrowers do not repay can be written as max 0, rL (1 − t)L − rD M . The intermediary cannot reimburse his creditors if there have been too many defaults in his portfolio, that is if: rD t>1− rL

  K 1− = θ(L) L

(2)

θ(L) is the maximum proportion of sustainable losses, that an intermediary can bear without defaulting. It is of course inversely related to leverage L/K and to rD /rL . If he chooses L > 0, the intermediary’s expected profit is Z

θ(L)

Z (rL (1 − t)L − rD (L − K))f (t, σ)dt = rL L

0

θ(L)

(θ(L) − t)f (t, σ)dt 0

It will be easier in most proofs to work with θ instead of L, inverting equation 2, L is determined by θ as: L(θ) =

rD K rD − rL (1 − θ)

4

(3)

It is of course possible to have intermediaries holding non-zero reserves by assuming random deposit withdrawals, this would make the analysis more cumbersome without affecting the main results, except that for some parameters the regulation would not be binding.

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Thus, denoting π(θ, σ) the intermediary’s expected profit, we have π(θ, σ) = rL L(θ) × s(θ, σ)

(4)

with s(θ, σ) = F (θ, σ)Eσ (θ − t|t ≤ θ) Profit is thus the product of two terms. rL L(θ) are the revenues if all borrowers repay their debt. The second term is the probability that the intermediary survives, times the expected difference between the maximum proportion of defaults the intermediary can handle, and the expected proportion of defaults given that the intermediary survives. Assume that all borrowers between θ and 1 repay their debt: these repayments do not bring any profit to the intermediary but enable him to repay his own debt. Then all further repayments are profit. Thus the second term is the probability of survival times the expected proportion of “surplus” repayments, with the convention that this proportion is 0 if the intermediary defaults. I denote this quantity s(θ, σ). Finally the operator Eσ denotes an expectation according to the distribution f (., σ). Notice that the intermediary is supposed to compute expected profit according to the true model σ, not to the model reported to the regulator. There are several interpretations of this assumption: the intermediary could engage in a form of “double accounting”, have a model used to compute regulatory capital and a different model to forecast the returns of loans, or for instance run the same model with several parameterizations and report only the results from one of them to the regulator. He could also have a single model, but take into account that the model is biased, and here I make the extreme assumption that the intermediary is able to perfectly anticipate the forecasting mistake made by the model. Lastly, different models typically give different predictions about the tails of the distributions, on which there is by definition less data. But tail events are privately less relevant for the intermediary: for extreme realizations of losses the intermediary will default and does not care about forecasting mistakes made at this level (section 5.2 deals with this last issue in more detail). The intermediary will either invest all equity in the riskless asset and not borrow, or maximize π(θ(L), σ) in L, taking prices as given. As is detailed in the next subsection, the intermediary also faces a regulatory constraint on the ratio K/L, which has to be larger than some α and the intermediary’s program can be written as: max π(θ(L), σ) s.t. L ≤ K/α L

(5)

An increase in L expands the scale of operation, bringing more profit for a given proportion of 11

expected surplus repayments. But this proportion itself is increasing in θ and thus decreasing in L (we have θ0 (L) ≤ 0 and s01 (θ, σ) = F (θ, σ)): using less leverage means a lower probability of default and less debt to repay. I show in Appendix A.2 that the profit function is decreasing and then increasing in L as on figure 3. Thus only three choices make sense: (i) investing K in loans without using any debt (L = K), (ii) investing K in the safe asset without using any debt (L = 0), (iii) borrowing until the leverage constraint binds and investing everything in loans (K/L = α). Lemma 1 (Demand for funds and supply of loans). Let (M ∗ , L∗ ) be the profit-maximizing choice of the intermediary. There exists rL such that: • If rL ≥ rL then L∗ =

K , M∗ α

=

K(1−α) . α

• If rL < rL then M ∗ = 0. L∗ = K if rL ≥

r0 , Eσ (1−t)

and L∗ = 0 otherwise.

The value of rL and the proof are in Appendix A.2. The intermediary uses the maximum leverage allowed by the regulation if rL is high enough to compensate for the high risk of defaulting, and otherwise doesn’t borrow but invests in loans or in the safe asset depending on which one has the higher expected return.

2.4

Regulation under complete information

I first analyze the optimal capital ratio the regulator can set if she also knows σ. Without intermediation, the optimal amount of loans would maximize the sum of investors’ gains and the surplus of the (1 − t) surviving borrowers. The optimal L in this case is such that at the interest rate rL (L) investors would exactly break even if they could lend directly to borrowers: rL (L) = rLe =

r0 Eσ (1 − t)

(6)

When intermediation is necessary and for a given level of capital however, reaching the optimal level of loans may require a high leverage, and thus the possibility that an intermediary defaults. One of the traditional rationales for regulation is that banks are indebted towards small retail depositors unable or unwilling to monitor the bank’s riskiness. Without regulation, investors would expect banks to use infinite leverage and default with a probability close to one, such that they would not be ready to lend at finite interest rates. The regulator can improve on this situation by insuring deposits. For simplicity I do not consider the cases of partial insurance or insurance with a non-fixed premium, but only complete

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deposit insurance5 . Then we have rD = r0 since loans to intermediaries are now riskless, and investors’ welfare is constant. The regulator will consider the surplus of borrowers, the profit of intermediaries, and the cost of repaying losses to investors. I assume a deadweight cost c > 0 from taxation, such that each unit of repayment with taxpayers’ money costs 1 + c to taxpayers and gives 1 to investors. Under complete information, the regulator can use “model-sensitive” capital constraints K/L ≥ α(σ). It will be convenient to translate this constraint on the capital ratio into a constraint on the maximum sustainable losses of the intermediary: K/L ≥ α(σ) ⇔ θ ≥ 1 −

rD (1 − α(σ)) = θ(σ) rL

As θ(σ) reflects α(σ), the constraint faced by the intermediary is just θ ≥ θ(σ), and as shown in the previous subsection this constraint will be binding for a high enough interest rate rL . Expressed in terms of θ, the regulatory constraint means that the intermediary has enough capital to bear at least θ(σ) losses in his portfolio. We thus have the following objective function for the regulator, to maximize in θ for a given σ: θ

Z V (θ, σ)

=

Z (rL L(1 − t) − r0 (L − K))f (t, σ)dt + Eσ (1 − t)

r0 W + 0

Z −

!

L

rL (u)du − rL (L)L 0

1

(r0 (L − K) − (1 − t)rL L)f (t, σ)dt

(1 + c) θ

Z =

L

Z

1

r0 (W + K − L) + Eσ (1 − t) rL (u)du −c (r0 (L − K) − (1 − t)rL L)f (t, σ)dt | {z } 0 θ {z } {z } | | Safe asset Surplus from loans Deadweight costs

(7)

Both rL and L will depend on the level θ chosen by the regulator. To keep things simple, I assume demand to be very elastic, such that the effect of θ on rL can be considered as negligible. Otherwise increasing θ could lead to an increase in rL , making losses lower and less probable, which can lead to multiple local optima and an optimal θ increasing in σ only by parts. As a result: Lemma 2 (First-best). For a given level of capital K 6 : 5

If investors are partially or not insured but know the true risk, and can fully monitor the bank, then e in equilibrium rL = rL . But interestingly intermediaries can still choose over-optimistic models, see section 5.3. Having a risk-sensitive premium would not change the results of this section much, because it would not give incentives to choose a less optimistic model. Quite the opposite: in the few countries where the premium is risk sensitive, it is sometimes decreasing in the capital ratio of the bank, giving further incentives to underestimate risk. 6 The result depends on the assumption that K is fixed in the short-run, otherwise it would be optimal for the regulator to impose that intermediaries are entirely financed by equity. As shown in an extension in B.1,

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1. If D(rLe ) ≥ K the first-best is to set θ∗ = 1 (L = K) and let intermediaries invest in loans up to the point where rL = rLe . 2. If D(rLe ) < K, c is high enough and demand D(.) is elastic enough, V (θ, σ) is concave and the optimal regulatory threshold θ∗ (σ) = argmaxθ V (θ, σ) is increasing in σ. Moreover the first-order condition can be written as: rL (σ) =

c r0 + (1 − F (θ∗ (σ), σ)) (r0 − rL (σ)Eσ (1 − t|t > θ∗ (σ)) | {z } Eσ (1 − t) Eσ (1 − t) ≥0

The case for a model-based regulation here is straightforward: when the true model is more pessimistic (higher σ), there is less surplus to gain by expanding credit and more risks of default for a given level of θ, hence the regulator wants to restrict leverage more. Note that with the first-best regulation the interest rate rL is higher than rLe when D(rLe ) < K: the regulator implements a capital requirement that involves under-investment to decrease the costs to taxpayers.

2.5

Numerical example

Consider the following numerical example, to be kept throughout the paper to illustrate the main results. The proportion of defaults follows a Beta distribution with parameters a, b. a measures the slope at the origin of the distribution, b is inversely related to the fatness of the tail. Assume a is known to be 3.5, b is equal to 31.5 but many values are possible for this parameter, uniformly distributed from b = 13 up to b = 50 which will be our most optimistic model. The true model and the most optimistic one yield of course very different predictions about defaults, not only in terms of expected defaults but also in terms of extreme events, much more likely to occur with b = 31.5. Take σ to be 1/b so that assumption M1 is satisfied. In practice, banking regulators aim at capping the probability that each intermediary defaults, typically 0.1% in the Basel framework. Assume this is an approximate solution of the regulator’s program. Then θ∗ is easy to compute: since an intermediary will default with probability 1 − F (θ, σ) when the true model is σ, to ensure a probability lower than p that the intermediary defaults, θ∗ (σ) has to be such that F (θ∗ (σ), σ) = 1 − p. For this example the only thing I need here is some informational cost of levying capital (Myers and Majluf (1984)) that the regulator cannot suppress, in which case all the results are qualitatively unchanged. This is consistent with the implicit assumptions underlying the Basel framework and taken here as given, although recent papers like Admati, DeMarzo, Hellwig, and Pfleiderer (2011) argue that a more ambitious regulatory reform should aim at decreasing the gap between the costs of debt and capital first, thus allowing for much higher capital ratios.

14

to be easier to visualize I assume p = 0.05 (5% probability to default). Plotting the CDFs we can easily see θ∗ (1/31.5) and θ∗ (1/50) graphically on Fig. 2. Figure 3 plots the profit π(θ(L), 1/31.5) as a function of L, with rL = 1.1, rD = 1, K = 1 and when defaults follow the “true” Beta distribution with a = 3.5, b = 31.5. rL is below rLe in this example, thus investing K in loans is less profitable than investing in the safe asset. However for L large enough investing in loans with a high leverage becomes more profitable, as the intermediary exploits the government’s guarantee on its debt. Notice that if the regulator knows the true model and imposes L ≤ L(θ∗ (1/31.5)), the intermediary prefers not to use any leverage. But if the regulator falsely believes that the true model is b = 50 and imposes L ≤ L(θ∗ (1/50)), the intermediary chooses maximum leverage. This result illustrates the adverse selection problem associated to regulation based on internal models: intermediaries have incentives to pretend the true model is more optimistic than it really is, or to spend resources on lobbying the regulator to be allowed to use optimistic models. In this rather extreme example, if the intermediary is successful at convincing the regulator that b = 50 he will increase his expected profit by 10% and default with a 25% probability, five times higher than the regulator’s objective. 1 1-p

True CDF

Optimistic CDF

0.5 ΘH150L

ΘH131.5L

Figure 2: Cumulatives and minimum default points.

15

t

Profit

Profit ΠHΘHLL,ΣL

Profit Safe asset only

K

LHΘH131.5LL

LHΘH150LL

L

Figure 3: Profit as a function of loans.

3

Model choice and market equilibrium without contingent transfers

3.1

Equilibrium

I now go back to the general case to study how intermediaries choose their models in equilibrium when the regulator lets them free to choose any model that meets prespecified requirements or “industry standards”, without transfers or penalties contingent on the choice of a given model or on the realization of losses. The goal is to have a stylized view of the current regulatory framework for credit risk and show that when the use of the correct model is not warranted, a simple tightening of capital ratios can actually increase the number and severity of intermediaries’ defaults, because it can lead to a wider adoption of optimistic models. A possible application is the shift from Basel II to Basel III requirements, that should lead to changes in the models used by banks. To study the role of market prices it will be convenient to assume there is a representative intermediary taking prices as given, or equivalently a continuum [0, 1] of intermediaries. The time line of the game is as follows: T=0 the regulator specifies a rule linking any model σ to a capital ratio α(σ), and a number of requirements that an intermediary’s internal model has to satisfy to be used for regulatory purposes. These requirements define a set of models accepted by the regulator. 16

For simplicity assume all models with a positive probability to be the true model are accepted, such that this set of models is the interval [σmin , σmax ]. T=1 σ is drawn from the distribution Ψ(.), each intermediary observes σ and reports a model σ 07 . If σ 0 ∈ [σmin , σmax ] the regulator constrains the intermediary to choose K, L satisfying K/L ≥ α(σ 0 ). Otherwise the intermediary gets 0. T=2 rL , rD , M and L are simultaneously determined by competitive equilibrium conditions. D(rL ) must be equal to the aggregate supply of loans. An intermediary who has reported a model σ 0 chooses a supply of loans and a demand for deposits maximizing his profit taking rL and rD as given under the constraints K/L ≥ α(σ 0 ) and L ≤ M +K. Finally, investors must be indifferent at the margin between lending to intermediaries at rate rD or investing in the safe asset at rate r0 . T=3 a proportion t of borrowers default, where t is drawn from the distribution F (., σ). The capital ratio α(σ) links a bank’s model to capital requirements, exactly as the formula given in the introduction (equation 1) does: a model σ determines LGD and P D, which in turn determine b and R, which determine capital requirements. The α(.) could incorporate additional measures of the regulator, such as floors on the capital requirements, extra safety margins and so on. Notice also that the model can accommodate a number of measures taken by regulators to ensure models are not too biased: comparison with “industry standards”, required assumptions and properties of the model, reasonable performance of the model on historical data... all this enters the definition of the interval [σmin , σmax ]. But then a bank is free to choose among all models that can get the regulator’s approval, and no payment is asked for the use of a very optimistic model, no penalties are set. As a consequence, the situation is equivalent to a “delegation game” (Holmstr¨om (1977) and Alonso and Matouschek (2008)): everything is as if banks were offered an interval of attainable leverage ratios from which they can choose8 . This assumption fits the letter of the Basel Accords, except the requirement that the bank has used the model for internal purposes for several years before it can use it for regulatory purposes (as shown in section 5.2, this is unlikely to make a difference). Although these assumptions are meant to provide a stylized view of the advanced IRB approach, the 7

The assumption that different intermediaries know the same σ is not key to the main results but allows to define easily a competitive equilibrium. It is possible to consider that for each σ there are ψ(σ)dσ intermediaries, for instance if intermediaries have different monitoring technologies for their loans and need to use different risk models. See the Appendix B.2 8 A difference with the aforementioned papers being that there are several agents interacting with each other.

17

framework could be applied to other situations: for instance an intermediary may wish to keep a good rating, and thus must prove to a rating agency that his probability of default is low, using internal risk assessments. Intermediaries’ choice at T = 1 and T = 2. Solving the model backwards, I first define more formally what is the equilibrium of the subgame starting at T = 1 when a given σ is realized: Definition 1 (Equilibrium with choice of a risk model). For an increasing α(.) and a given realization of σ, an equilibrium is a 5-uple (rL , rD , µl , µr , µs ) and a function h : (σmin , σmax ] → [0, 1] where a proportion h(σ 0 ) of intermediaries choose σ 0 , µl choose σmin and K/L = α(σmin ), µr choose K = L, µs choose L = 0 and invest K in the safe asset, and 1. Each intermediary’s choice given his model, rL and rD is a solution to the intermediary’s program of Lemma 1, the supply of loans by intermediaries is equal to D(rL ) and funds borrowed by intermediaries equal funds supplied by investors at an interest rate rD . 2. Investors are indifferent between lending to intermediaries and investing in the safe asset. 3. No intermediary has an incentive to choose a different σ 0 or change his investment strategy. This definition simply enlarges the standard concept of competitive equilibrium by requiring that no intermediary wants to choose a different model. Solving the equilibrium is easy when investors are fully insured: since they face no risk when they lend to intermediaries, it must be the case by condition 2 that rD = r0 . This equality implies that if rL ≥ rLe it always pays to borrow at least a little, whereas if rL < rLe an unleveraged intermediary prefers the safe asset to loans, so in both cases µr = 0. We know from Lemma 1 that, depending on rL , either an intermediary uses no leverage at all or his capital constraint is binding. Then for any σ ∈ (σmin , σmax ] we have h(σ) = 0 and only two strategies may be used by intermediaries: not borrowing and investing in the safe asset (proportion µs of intermediaries), or choosing the most optimistic model and using maximum leverage (proportion µl ). Thus only the minimum of the function α(.), α(σmin ), will matter. I denote α ¯ this minimum. The result depends on how strong the final demand for loans is. To see this, assume the demand function depends positively on a parameter η and consider a family of demand functions ˜ L), η ∈ R+∗ }: {D(rL ) = η D(r, 18

Proposition 1 (Equilibrium with insured investors). For given α ¯ and σ, starting at T = 1 there exists a unique equilibrium with choice of a risk model, in which µl (¯ α, σ) intermediaries choose the most optimistic model and maximum leverage and 1 − µl (¯ α, σ) intermediaries do ˜ L ), not borrow and invest in the safe asset. Assuming the demand function is D(rL ) = η D(r µl is increasing in η and decreasing in σ. Since the equilibrium is unique, I denote rL (¯ α, σ) the equilibrium interest rate on loans, and pd (¯ α, σ) the expected proportion of defaulting intermediaries in equilibrium, where:    r0 (1 − α ¯) pd (¯ α, σ) = µl (¯ α, σ) 1 − F 1 − ,σ rL (¯ α, σ)

(8)

Corollary 1. When µl (¯ α, σ) < 1, the expected proportion of defaulting intermediaries pd increases in η. See Appendix A.4 for the proof. This proposition illustrates the role of demand in giving incentives to choose an optimistic model: when all intermediaries use a very optimistic model, they are able to use a high leverage and the supply of loans is high, which lowers the interest rate on loans. This situation is an equilibrium if and only if the interest rate is not so low as to make it more profitable not to borrow, that is if demand is high enough. Conversely, if few intermediaries use leverage, the supply of loans will be low and the interest rate high, and to have an equilibrium the interest rate must be low enough so that it doesn’t pay to use even the most optimistic model, thus demand has to be low. As a consequence an increase in demand leads to a wider adoption of optimistic models and a higher risk in the banking sector.

The regulator’s choice at T = 0. The regulator anticipates that intermediaries will always choose either L = 0 or σmin . The regulator’s choice thus amounts to choosing α ¯ . The regulation here cannot be model-sensitive since the choice of σ 0 by intermediaries does not depend on the realization of σ, and the regulator may just as well choose the function α(.) constant and equal to α ¯ . The equilibrium level of rL and µl however depend both on α ¯ and σ. Then the objective of the regulator can be written as: R σmax

(γ(¯ α, σ) − cµl (¯ α, σ)δ(¯ α, σ))ψ(σ)dσ R K/α¯ = r0 (W + K − (K/¯ α)) + Eσ (1 − t) 0 rL (u)du

max

σmin

α ¯

γ(¯ α, σ) δ(¯ α, σ) =

R1

r (1−α) ¯ ¯ L (α,σ)

1− r0

(r0 ((K/¯ α) − K) − (1 − t)rL (¯ α, σ)K/¯ α)f (t, σ)dt

19

(9)

The regulator has to select a single α ¯ to solve the trade-off between the expectation of surplus γ(¯ α, σ) and the expected deadweight losses from taxation cµl (¯ α, σ)δ(¯ α, σ), in expectation over all realizations of σ. A necessary condition for an interior solution is: α ¯2 EΨ (Eσ (1 − t)rL (¯ α, σ)) = r0 − c × EΨ K



 d (µl (¯ α, σ)δ(¯ α, σ)) d¯ α

This condition is similar to the one obtained in Lemma 2: the expected interest rate on loans implemented by the regulator is distorted from the average (over σ) break-even interest rate due to the cost of public funds. An interesting difference is that the regulator needs to take into account that the proportion of intermediaries who will adopt optimistic models is affected by α ¯ and σ. The latter effect is in the right direction: when σ increases and risk is more severe, we know from Proposition 1 that µl decreases so that less intermediaries are at risk. The flexibility given to intermediaries thus has a positive effect. However it can also limit the extent to which the regulator can control risk, as shows the following: Proposition 2 (Counterproductive tightening). For a low enough elasticity of the demand for loans, tightening capital requirements increases µl (¯ α, σ). If µl (¯ α, σ) is low enough, this also implies that pd (¯ α, σ), the expected proportion of defaulting intermediaries, increases. Proof : the equilibrium is defined by µl and rL satisfying the following two equations: µl K = D(rL )¯ α   r0 r0 α ¯ = rL s 1 − (1 − α ¯ ), σ rL

(10) (11)

Equation 11 defines rL as the interest rate such that an intermediary is indifferent between choosing L = K/¯ α and L = 0. When α ¯ increases there are two effects: rL increases so that D(rL ) decreases, and for a given rL the product D(rL )¯ α increases. If demand is rigid enough the first effect is negligible, so D(rL )¯ α increases in equation 10 and µl has to increase. pd (¯ α, σ) is the product of µl and the probability that an intermediary with maximum leverage fails (equation 8). The effect of increasing α ¯ on the second term is negative: each intermediary has a lower leverage and the interest rate on loans increases. But when µl is small enough this negative effect will be smaller than the positive impact of α ¯ through the increase of µl .  More intuitively, there are three effects when the regulator increases α ¯ . First, an intermediary already using the most optimistic model will have less leverage than before, which decreases losses to taxpayers. When the true model is quite pessimistic and few interme20

diaries use the optimistic model this effect is small. Second, choosing the most optimistic model is less profitable because it allows less leverage. Third, since intermediaries have a tighter capital constraint, the supply of loans decreases and the interest rate rL goes up. This increase makes it more profitable to use the most optimistic model. When demand is rigid enough, the third effect is stronger than the second, so an increase in α ¯ leads more intermediaries to adopt the most optimistic model, which in turn increases risk. Thus a naive tightening of the regulation can counter-intuitively increase risk precisely in those states of the world where the true model is quite pessimistic and risk is already high, as on figure 4.

3.2

Discussion and empirical implications: market and regulation, from Basel II to Basel III

In the extreme case where investors are fully insured, there is no market discipline since intermediaries’ creditors no longer care about the probability that an institution defaults. Intermediaries then face important incentives to use the most optimistic model possible. The market still gives a natural counterweight to this effect however: when more intermediaries adopt optimistic models and use a high leverage, the interest rate on loans goes down and increasing leverage becomes less profitable. Thus a high proportion of intermediaries using over-optimistic models is possible only if the final demand for loans is high enough (Proposition 1): a model-based regulation with few penalties for choosing optimistic models is necessary to explain how over-optimistic models can be used in equilibrium, but a high demand for loans or low risk-free rates also have to be part of the story. Intermediaries not adopting the over-optimistic model and remaining unleveraged can be thought of as banks sticking to more traditional banking activities for which model uncertainty and risk are low. Proposition 2 shows that market and regulation are partial substitutes in limiting the intermediaries’ use of over-optimistic models. A tighter regulation restricts leverage, tends to increase the interest rate on loans, and thus incentives to use optimistic models and high leverage. Tightening the regulation can thus be counterproductive because it alters market counterweights. In particular, the strong increase of capital requirements with the transition from Basel II to Basel III gives incentives to develop and use the most optimistic models that the regulator will accept, so as to minimize the impact of increased capital requirements9 . The following figures illustrate this section. I use the same example as in section 2 and 9

According to several commentators, many banks have reacted to higher requirements exactly as they do in the model. See for instance the article “Banks turn to financial alchemy in search for capital” by Tom Braithwaite, Financial Times, 24.10.11, or “Fears rise over banks’ capital tinkering” by Brooke Masters, Patrick Jenkins and Miles Johnson, Financial Times, 13.11.2011.

21

add a demand for loans equal to D(rL ) =

η . (rL −1)ζ

On Figure 4, I plot for different choices of

α ¯ by the regulator the expectation over σ of the welfare, the volume of loans, the proportion of defaulting intermediaries, and the number of intermediaries with optimistic models (with η = 1, ζ = 1.1). We see that tightening the regulation leads to more intermediaries adopting the most optimistic model in this example, and as a result, for low levels of α ¯ , the default probability increases when regulation tightens, as expected from Proposition 2. Several regulators are currently considering the imposition of floors on certain risk weights (Samuels, Harrison, and Rajkotia (2012), p. 13), the α ¯ chosen by the regulator in the model can also be interpreted as such a floor. A prediction of the model is that higher floors will lead to the selection of more optimistic models (so that the floor will be binding), and possibly to an increase in the total risk of the banking sector. Proportion

L,W

1.0

Average welfare EΨ HWHΘ,ΣLL

30

20

Average default probability EΨ H pd L

0.8

Average volume of loans EΨ HLL

Intermediaries with the most optimistic model EΨ HΜL

0.6

10 0.4

Α*10%

20%

30%

40%

50%

60%

70%

80%

90%

Α

100%

0.2

-10 0.0

2.5%

5%

Α*

10%

12.5%

15%

Α

Figure 4: Expected welfare, volume of loans, intermediaries using the most optimistic model and default probability as α ¯ increases. The optimal α ¯ for the regulator can be identified on the figure, it is close to 8.2%. I assume now that the regulator selects this value of α and plot the same variables on Figure 5, but for the different realizations of σ instead of the expected value. As expected from Proposition 1, when the true risk parameter is higher less intermediaries try to bypass the regulation. An interesting implication is that the imposition of the same regulatory floor on risk weights in different countries should lead to different choices of models, in a way that flattens the average default probability across countries.

22

Proportion

L,W 14

1.0

Average default probability pd

13 0.8

Intermediaries with the most optimistic model Μ

12 0.6 11

Welfare WHΘ,ΣL 0.4 10

Volume of loans L 0.2 9

Σmin

0.0211746

0.0223492

0.0235238

0.0246984

0.025873

0.0270476

0.0282222

0.0293968

0.0305714

0.0

Σ

Σmin

Σmax

0.0211746

0.0223492

0.0235238

0.0246984

0.025873

0.0270476

0.0282222

0.0293968

0.0305714

Σ

Σmax

Figure 5: Welfare, volume of loans, intermediaries using the most optimistic model and default probability as σ increases. Finally, for the same optimal α and the median value of σ, I plot the same variables on Figure 6, letting η vary between 0.05 and 5. As expected from Proposition 1, an increase in demand leads more intermediaries to adopt an optimistic model, and risk rises accordingly. When demand becomes so high that all intermediaries already use maximum leverage, then an increase in demand only leads to a higher interest rate and less risk. An interesting implication is that the choice of over-optimistic models should be procyclical. Proportion

L,W

1.0 50

Welfare WHΘ,ΣL

0.8

40

Volume of loans L

0.6

Average default probability pd

0.4

Intermediaries with the most optimistic model Μ

30

20

0.2

10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Η

0.05

0.5

1

1.5

2

2.5

Η

Figure 6: Intermediaries using the most optimistic model, interest rate on loans and intermediaries’ default probability as regulation tightens. A natural question is of course why the regulator would let banks so much freedom. In practice the regulator defines a certain number of characteristics that an eligible model must have; it must also have been used by the bank for two years before it can serve the computation of regulatory capital; and finally the model is backtested and audited to check it meets “industry standards”. This prevents banks from using models totally off the mark, 23

but not from using models slightly over-optimistic. Backtesting for example does not often lead to the rejection of a model for credit risk given the low power of the tests. I see four reasons why regulators may emphasize the necessity to rely on sophisticated models more than the hidden information problem: (i) When Basel II was put in place, internal models were already in use and had no reason to be biased, hence regulators thought they could rely on them but neglected incentives to tweak the models in the future. (ii) The regulator can consider that the priority is to give incentives to use quantitative models to increase transparency, and that the market will penalize banks using unrealistic models. (iii) Banks’ defaults affect non-national investors, so that national regulators use their discretion in allowing more or less optimistic models to favor national banks, which is exactly what the Basel framework was supposed to avoid (Rochet (2010)). (iv) The next section shows that giving incentives to use the correct models is a difficult task, the regulator may prefer to deal with this problem by using more cautious capital requirements. But Proposition 2 shows that adding capital requirements to cover “model risk” is not sufficient, and can actually increase risk instead of reducing it. It is therefore necessary to develop a mechanism giving incentives to use the correct model.

4

Optimal regulation with hidden model

It is natural to ask how regulation could prevent financial intermediaries from picking optimistic models. Checking that the model reported by the regulated meets industry standards can only lead to a regression toward the most optimistic models when interest rates are high enough. What is needed is a mechanism inducing truthful revelation of the regulated institution’s private information. To focus on information problems and abstract from market equilibrium effects, I assume from now on that rL is given, or equivalently that demand is very elastic. Moreover I assume rL > r0 /Eσmax (1 − t) such that, even if the true model is the most pessimistic one, any capital requirement will be binding. As a consequence, when she sets α(σ) the regulator equivalently sets a constraint of the form θ ≥ θ(σ), which will be easier to work with. Assume the regulator wants to implement a leverage constraint dependant on the intermediary’s type and expressed as θ = θ(σ). The goal is to maximize under incentive compatibility constraints: Eψ (V (θ(σ), σ)) − Expected costs of the regulation where V is the social welfare function studied in section 2.4. In particular the best θ(.) the 24

regulator can implement is the first-best θ∗ (.). The difficulty is that an intermediary’s profit is decreasing and convex in θ. There are also a number of constraints for the regulator in addition to incentive compatibility. I will focus on two of them. The first one, that also motivates the need for regulation, is limited liability: by the time the regulator realizes a financial institution has been over-optimistic, it may be too late for punishment. Moreover, the regulated institution’s outside option is typically type-dependent (Jullien (2000)): for instance an institution could choose not to borrow at all and earn rL K(1 − Eσ (t)), or in the Basel regulation a bank can opt for the “standard approach”, not use any internal model and earn a profit that will depend on the true state of the economy. I assume that an institution can opt out of the mechanism and then get π ¯ (σ) if σ is the true parameter, with π ¯ 0 ≤ 0. I will study two different mechanisms in this section. The first one is equivalent to backtesting, used for market risk models. I also study the specific difficulties of using such a mechanism for credit risk models. I then propose a second-best auditing mechanism, much closer to the current regulation and thus easier to implement.

4.1

Backtesting internal models ex post

Sufficient conditions to reach the first-best Consider a mechanism where an intermediary observes the true model σ, announces some parameter σ 0 and faces the constraint θ ≥ θ(σ 0 ). Given the assumption rL > r0 /Eσmax (1 − t), the intermediary chooses θ = θ(σ 0 ), then suffers some level of defaults t in his portfolio and finally pays a transfer T (σ 0 , t) if t ≤ θ(σ 0 ). It will be useful to denote u(θ, t) the profit before transfers of an intermediary choosing θ when t defaults realize: u(θ, t) = rL L(θ)(1 − t) − r0 (L(θ) − K)

(12)

The regulator’s program is the following: max Eψ (V (θ(σ), σ) + cEσ (T (σ, u))) with:

(13)

θ(.),T (.,.)

∀σ, σ 0 , π(θ(σ), σ) − Eσ (T (σ, t)) ≥ π(θ(σ 0 ), σ) − Eσ (T (σ 0 , t)) (IC) ∀σ, π(θ(σ), σ) − Eσ (T (σ, u)) ∀σ, t, u(θ(σ), t)

25

≥π ¯ (σ)

(IR)

≥ T (σ, t)

(LL)

(IC) is a standard incentive-compatibility constraint, an intermediary has to be better off telling the truth about the model. (IR) requires that the mechanism gives each type of intermediary at least what he would get by opting out of the mechanism (e.g. keeping Basel’s standardized approach). (LL) is the limited liability constraint: the regulator cannot tax more than what the intermediary has earned. In particular it is impossible to “punish” a defaulting intermediary. The spirit of such a regulation is easy to understand: the regulator offers a profile of transfers T (σ, t) such that an intermediary reporting σ will be heavily taxed if the realized level of defaults is relatively unlikely given the model announced. The backtesting mechanism used for market risk models belongs to this class of mechanisms. In the extreme case where for each model σ there exist some levels of default that have positive probability if and only if σ is the true parameter (shifting support), the regulator could leave some money to the regulated only in those states and tax everything in all other states. More generally, due to the constraints (IR) and (LL) the regulator faces a complicated problem, and may have either to leave some surplus to intermediaries, not be able to implement θ∗ (.), or shut down types with the highest outside options. Remark 1. Any menu T (., .) satisfying (IC) and such that Eσ (T (σ 0 , t)) is twice differentiable in σ and σ 0 must be such that for any σ: ∂π(θ(σ 0 ), σ) ∂Eσ (T (σ 0 , t)) 0 =σ | = |σ0 =σ σ ∂σ 0 ∂σ 0 ∂ 2 Eσ (T (σ 0 , t)) ∂ 2 π(θ(σ 0 ), σ) 0 | ≤ |σ0 =σ σ =σ ∂σ 0 2 ∂σ 0 2 The proof follows straightforwardly from the first-order condition of (IC) and the local second-order condition. Incentive compatibility requires that at the margin reporting a lower σ 0 to increase leverage will be exactly compensated by additional expected penalties. Moreover, it must be the case that these penalties increase more rapidly than profit when the reported σ 0 decreases. This can be a problem since the reservation utility is type-dependent. Consider a simpler example with only two types σ1 , σ2 , σ2 > σ1 , two possible realizations of defaults t, t¯, t¯ > t, and Pr(t = t|σi ) = pi , p1 > p2 . To satisfy (IC) and bind (IR) the regulator can give π ¯ (σ1 )/p1 to a type reporting σ1 if t realizes, 0 otherwise, and π ¯ (σ2 ) to type σ2 irrespective of the realization. (IC) for type σ2 gives π ¯ (σ2 )/¯ π (σ1 ) ≥ p2 /p1 . If the outside option of type σ1 is much higher than the outside option of type σ2 and the likelihood ratios of the different states under both models are not different enough, then it is impossible to bind (IR) while satisfying (IC) and (LL). In other words, if profit decreases quickly in σ it has to be the case that the different models give predictions different enough, otherwise 26

some inefficiency appears or a rent has to be left to the regulated. Figure 7 gives an example where p1 = 0.5, p2 = 0.25 and π ¯ (σ1 ) = 1. In the first graph π ¯ (σ2 ) = 0.8 and it is possible to find two contracts such that (IC) holds and (IR) binds for both types, in the second graph π ¯ (σ2 ) = 0.4 and it’s no longer possible. The condition π ¯ (σ2 )/¯ π (σ1 ) ≥ p2 /p1 simply ensures that the two lines cross at a point with a positive payoff after t¯. The following proposition generalizes this idea to a continuum of types:

Payoff in t

Payoff in t

2.0

2.0

1.5

Type Σ1 gets Π@Σ1 D

1.5

Type Σ2 gets Π@Σ2 D 1.0

1.0

0.5

0.0

0.5

0.5

1.0

1.5

2.0

Payoff in t 0.0

0.5

1.0

1.5

2.0

Payoff in t

Figure 7: Example of separation when π ¯ (σ2 )/¯ π (σ1 ) > p2 /p1 , and non separation otherwise. Proposition 3. If, in addition to M1, F (., .) is log-concave in its second argument and π ¯ is log-convex, then with θ(.) = θ∗ (.) there exists a menu of transfers T (., .) satisfying (IC) and (LL) and such that (IR) is binding for every σ. It is possible in particular to use the following menu: ( T (σ, t) =

max(0, u(θ∗ (σ), t)) ∗

u(θ (σ), t) −

π ¯ (σ) F (a(σ),σ)

if t > a(σ) if t ≤ a(σ)

with a(σ) increasing and such that: F20 (a(σ), σ) π ¯ 0 (σ) = F (a(σ), σ) π ¯ (σ) With the proposed menu, if he reports model σ, an intermediary is taxed such such that he gets

π ¯ (σ) F (a(σ),σ)

if the realized level of defaults is less than a(σ), and gets zero otherwise. By

definition such a mechanism satisfies the limited liability condition. Moreover, if he reports truthfully the model σ the intermediary gets exactly π ¯ (σ) in expectation, thus the mechanism binds condition (IR). We only have to find a(σ) such that the incentive compatibility condition holds for all types. Under M1 we can induce truthful revelation with an increasing a(.): intermediaries announcing a low σ get a high payoff but only if the level of defaults 27

is under a very low threshold (which will be crossed only with a small probability if their report is truthful), intermediaries announcing a higher σ get a lower payoff but for higher levels of default. The two other assumptions ensure that this particular mechanism satisfies (IR) and (IC). Since F (., .) is decreasing in its second argument and π ¯ decreasing, the log-concavity of F (., .) in σ expresses the idea that the different distributions do not give too similar predictions as σ increases, and on the contrary the log-convexity of π ¯ implies that the outside option does not decrease too quickly as σ increases. The intuition is the same as in the binary example above. See Appendix A.5 for the full proof. Figure 8 gives an example, the parameters are the same as in the example of section 3.2 and rL is supposed fixed and equal to 1.15, on the left panel I plot the expected payoff an intermediary gets if the true parameter is σ and he reports σ 0 , for all values of σ 0 and different values of σ. By construction of the mechanism the maximum payoff is obtained for σ 0 = σ. On the right panel I show how this is achieved by reporting the payoff an intermediary gets when he reports the true σ and t defaults realize. Payoffs

Payoff 1.09

uHΘHΣminL,tL

1.05 1.08

uHΘH0.023L,tL uHΘH0.026L,tL

1.07 1.00

UHΣ',ΣminL

uHΘH0.029L,tL 1.06

uHΘHΣmaxL,tL

UHΣ',0.023L 0.95

UHΣ',0.026L

1.05

UHΣ',0.029L UHΣ',ΣmaxL

0.90

Σmin

0.023

0.026

0.029

1.04

Σmax

Σ'

1.03

0

0.25

Defaults t

Figure 8: Expected payoff for a given σ to report σ 0 (left), and payoff from reporting the truth depending on the level of defaults (right). The logic behind this result is simple: if risk is indeed low, then a financial intermediary is ready to pay high taxes or penalties if high default levels realize in his portfolio, because this event is unlikely. In principle, observing the level of defaults ex-post gives a powerful tool to the regulator to detect and punish the users of over-optimistic models. A first limitation is that the intermediary’s outside option should not be too sensitive to his type. If one interprets this outside option as the profit of a bank under Basel’s standardized approach, an implication is that a more risk-sensitive standardized approach can make the revelation of models in the advanced internal ratings based approach a more complicated task.

28

A negative result Remember also that credit risk models are typically difficult to backtest because they give similar predictions for not too extreme levels of default. As a consequence, the optimal menu of penalties may include transfers for levels of default above those at which a bank defaults itself. Since limited liability may prevent the regulator from taxing enough for high levels of default, it will be necessary to subsidize intermediaries with high default levels who announced very pessimistic parameters. This happens with the proposed mechanism if a(σ) > θ∗ (σ), and in particular when backtesting is difficult, in the following sense: Definition 2. Models σ ∈ [σmin , σmax ] are distinguishable only above tˆ if for any (σ, σ 0 ) ∈ [σmin , σmax ]2 and for any t < tˆ we have f (t, σ) = f (t, σ 0 ). This definition reflects the idea that for low levels of risk there is a lot of historical data to calibrate different models, such that they tend to deliver similar predictions, while for extreme levels of risk data is much more sparse. This fact is modeled in a stylized way here by assuming the different models are perfectly equivalent up to a given level of defaults. Now take any two models σ, σ 0 with σ < σ 0 such that θ(σ) ≤ θ(σ 0 ). Assume σ is so low that the regulator wants to implement θ(σ) < tˆ, in which case a bank using model σ will default for levels of losses that give no information on which is the true model. I prove that in this case it will be necessary to subsidize a bank using model σ after it defaults. By contradiction, assume it is not the case. Then for any t ≥ θ(σ) we have u(θ(σ), t) = T (σ, t) = 0. To bind the constraint (IR) for type σ we thus need: Z

θ(σ)

[u(θ(σ), t) − T (σ, t)]f (t, σ)dt = π ¯ (σ) 0

If he reports truthfully, an intermediary with type σ 0 will get π ¯ (σ 0 ). If he lies and reports σ he will get: Z

1 0

θ(σ)

Z

[u(θ(σ), t) − T (σ, t)]f (t, σ )dt = 0

[u(θ(σ), t) − T (σ, t)]f (t, σ)dt = π ¯ (σ) 0

where the second term is implied by f (t, σ) = f (t, σ 0 ) for t ≤ θ(σ) ≤ tˆ. Since σ < σ 0 we have π ¯ (σ) > π ¯ (σ 0 ), which violates incentive compatibility for type σ 0 , a contradiction. It is thus necessary to have T (σ, t) < 0 at least for some t > tˆ: to have incentive compatibility and bind the individual rationality constraints, it must be the case that a type with a low σ gets a positive payoff for some realizations of t that have a higher probability when the true model is σ than when it is a more pessimistic model. 29

Having to subsidize defaulting banks is of course a bad property of the optimal menu of penalties. It is politically difficult ex-post to use taxpayers’ money to give a subsidy to the shareholders of a defaulting bank, and the regulator may be unable to commit to such a mechanism. If the regulator is unwilling or unable to commit to a T (σ, t) < 0 for t > θ(σ), the above reasoning shows that she has to use transfers such that for any σ with θ(σ) < tˆ, an intermediary truthfully reporting σ 0 must get at least π ¯ (σ). Hence the following proposition: Proposition 4. If models σ ∈ [σmin , σmax ] are distinguishable only above tˆ and θ(σmin ) ≤ tˆ: • To satisfy (IC), (LL) and bind (IR) for all types, then for all models σ with θ(σ) < tˆ the regulator needs to set T (σ, t) < 0 for some t > tˆ. She must be able to commit to subsidizing the intermediary after he defaults. • If she is unable to commit, the regulator has to set T (., .) such that each type gets at least π ¯ (σmin ), the highest reservation value, in order to induce the truthful revelation by all types. Remark 2. If the regulator is unable to commit to subsidizing an intermediary after he defaults, if θ∗ (σmin ) < tˆ there is a trade-off between extracting the intermediaries’ surplus and how model-sensitive the regulation can be. When the realized level of defaults gives information about which is the true model, it is so high that an intermediary having reported an optimistic model defaults. Thus it is impossible to “punish” the use of optimistic models, the only possibility is to give a “bonus” for the use of pessimistic models. This “bonus” can be very costly here since all intermediaries have to get the reservation value of the most optimistic type. To avoid these costs, the only solution is to increase the capital requirements of the most optimistic types such that there is no θ(σ) below tˆ. But this means reducing the model-sensitiveness of the regulation compared to the first-best solution. Moreover, when backtesting is more difficult (tˆ is higher) model-sensitiveness has to decrease more.

4.2

Auditing internal models ex ante

Since it is difficult to backtest models ex-post, another option is to try to detect overoptimistic models ex-ante by searching for strange parameterizations, extreme assumptions, suspect time-windows, theoretical flaws and so on. In the current state of the regulation, internal models have to be audited before their approval by the regulator. This auditing procedure is meant to check that a bank’s model meets industry standards, that the model is 30

sophisticated enough without being grossly over-optimistic. This is what I have modeled by assuming any σ ∈ [σmin , σmax ] is admissible: there is a range of models meeting standards. Can the regulator also use auditing to encourage truthful revelation? Assume she has the following auditing technology: if an intermediary announces parameter σ 0 , the regulator hires auditors to check the model during H(σ 0 ) hours, each hour costing w > 0. Auditing will be assumed to be the search for mistakes in the model specification: if an intermediary knowingly uses an over-optimistic model he has to use a false assumption or a strange parametrization at some point, something that can possibly be uncovered by an auditor. If the reported model is the correct one (σ 0 = σ) then the auditors never find the model to be wrong. Otherwise there is a probability P (H(σ 0 )) that they find a mistake and declare the model to be wrong, where P 0 ≥ 0, P 00 ≤ 0, P (0) = 0, ∀H ≥ 0 P (H) < 1. Contrary to the classical works on costly state verification (Townsend (1979)), I assume the regulator cannot make any transfers conditional on the intermediary’s report. If made before the realization of losses such transfers would interact with the leverage constraint in a non-trivial way, if made ex post they would interact with limited liability and the problems studied in the previous subsection. More importantly, the goal here is to consider the smallest possible departure from the current regulation, where such transfers are not used. Since I assume no type 1 error when auditing a model10 , a regulator who wants all types of intermediaries to report truthfully can punish as much as possible an intermediary caught with a wrong model. Thus an intermediary reporting the model σ 0 when the true model is σ gets: ( U (σ 0 , σ) =

(1 − P (H(σ 0 )))π(θ(σ 0 ), σ) + P (H(σ 0 )) × 0

if σ 0 6= σ

π(θ(σ), σ)

if σ 0 = σ

By definition, such a scheme satisfies limited liability and individual rationality constraints. To satisfy incentive compatibility it must be the case that for any σ and σ 0 we have π(θ(σ), σ) ≥ (1 − P (H(σ 0 )))π(θ(σ 0 ), σ). In words, for any model σ 0 , H(σ 0 ) has to be set such that no type has an incentive to falsely report model σ 0 . The regulator tries to maximize in θ(.) and H(.) the social welfare V (θ(σ), σ) minus auditing costs subject to incentive compatibility: Z

σmax

max θ(.),H(.)

(V (θ(σ), σ) − w(1 + c)H(σ)) ψ(σ)dσ s.t. ∀σ, σ 0 π(θ(σ), σ) ≥ (1 − P (H(σ 0 )))π(θ(σ 0 ), σ) (14)

σmin

Since auditing is costly, the regulator wants to audit models just enough to elicit truth10

In practice it is of course possible that the regulator wrongly rejects a good model. In the Basel regulation the bank can typically revert to the standard approach and get π ¯ (σ) instead of 0, which makes it more difficult to ensure truthful reports. I abstract from this problem here, but taking this into account would make auditing even more costly.

31

ful revelation while minimizing the expected cost of auditing. The incentive compatibility constraint is difficult to study because U (σ 0 , σ) is not continuous at σ 0 = σ. Take a given value of σ, θ(σ) appears several times in the program. First, when σ realizes type σ has to be truthful. Second, there may be several other types σ 0 whose best deviation is to falsely report model σ. Denote: π(θ(σ 0 ), σ 0 ) π(θ(σ), σ 0 ) d(σ) = {σ0 ∈ [σmin , σmax ], σ ∈ m(σ0 )}

m(σ) = argminσ0

(15) (16)

m(σ) is the set of types who bind the constraint associated with σ, that is a set of “mimickers”, just indifferent between telling the truth and mimicking σ. Conversely d(σ) is the set of models for which σ is a mimicker: all models in this set are also potential deviations of type σ. First, notice that m(σ) = ∅ if and only if θ(σ) maximizes θ. If m(σ) is the empty set then it is optimal not to audit this model at all, if θ(σ) is lower than the maximum θ, the type with the maximum θ has a strict incentive to mimic σ. Conversely, nobody wants to mimic the type with the highest θ, hence it is never optimal to audit this model. Second, m(σ) has to be a set of null measure in [σmin , σmax ], otherwise it would be optimal to set θ(σ) equal to one, which would have a welfare cost in a state with probability ψ(σ)dσ but makes it possible to reduce distortions compared to the first-best on a set with positive measure. But if θ(σ) = 1 then m(σ) = ∅, a contradiction. Third, d(σ) has to be a set of null measure, since otherwise it would be optimal to set θ(σ) = 0, which would save auditing costs on the whole set d(σ). But if θ(σ) = 0 then d(σ) = ∅, a contradiction. For the exposition, consider a given σ, and assume m(σ) = σm , d(σ) = σd : each set is a singleton11 . Then σ appears in the incentive compatibility constraint defining H(σ), (1 − P (H(σ)))π(θ(σ), σm ) = π(θ(σm ), σm ), associated with the multiplier λ(σ), and in the constraint defining H(σd ), (1 − P (H(σd )))π(θ(σd ), σ) = π(θ(σ), σ), associated with the multiplier λ(σd ). The first-order conditions with respect to θ(σ), H(σ) and H(σd ) give: V10 (θ(σ), σ)ψ(σ) = λ(σd )π10 (θ(σ), σ) − λ(σ)(1 − P (H(σ)))π10 (θ(σ), σm ) λ(σ)P 0 (H(σ))π(θ(σ), σm ) = −(1 + c)wψ(σ) λ(σd )P 0 (H(σd ))π(θ(σd ), σ) = −(1 + c)wψ(σd ) Notice that the impact of a change in θ(.) on σm and σd does not appear: the total derivative 11

Otherwise it is necessary to sum the constraints over all elements in sets d(σ) and m(σ), but since they must have null measure the first-order condition is similar. The complete equation is given in Proposition 5.

32

of λ(σ)(π(θ(σm ), σm ) − (1 − P (H(σ)))π(θ(σ), σm )) with respect to θ(σ) is equal to −λ(σ)(1 − P (H(σ)))π10 (θ(σ), σm ), the term which appears above, minus λ(σ)(dσm /dθ(σ))(∂(π(θ(σm ), σm )− (1−P (H(σ)))π(θ(σ), σm ))/∂σm ). But since by definition σm minimizes π(θ(σm ), σm )/π(θ(σ), σm ) the latter term is null. By the same reasoning the effect of θ(σ) on σd can be neglected. Using the two constraints defining H(σ) and H(σd ), we have: dH(σd ) 1 dP (H(σd )) −π10 (θ(σ), σ) = × = dθ(σ) P 0 (H(σd )) dθ(σ) P 0 (H(σd ))π(θ(σd ), σ) dH(σ) 1 dP (H(σ)) π10 (θ(σ), σm )π(θ(σm ), σm ) = × = dθ(σ) P 0 (H(σ)) dθ(σ) P 0 (H(σ))π(θ(σ), σm )2

(17) (18)

which finally gives us the following first-order condition: 

 V10 (θ(σ), σ)ψ(σ)

  dH(σ ) dH(σ)   d ψ(σd ) + ψ(σ) = (1 + c)w  dθ(σ)   dθ(σ) | {z } | {z } ≥0

(19)

≤0

This condition can be interpreted as follows: the first-best θ(σ) would satisfy V10 (θ(σ), σ) = 0, however auditing costs impose two distortions. First, when the true model is σ, the intermediary must be given incentives to report σ and not σd . If θ(σ) is high, the profit from reporting σ is low, and thus σd must be audited more. If the regulator biases θ(σ) downwards compared to the first-best, she increases the intermediary’s profit if he tells the truth, and thus saves on auditing costs when the true model is σd . Biasing θ(σ) downwards means that V10 (θ(σ), σ) is positive. Second, when the true model is σm , the intermediary would like to report σ. To prevent him from doing so, σ has to be audited more. A way to save on auditing costs when σ is the true model is to bias θ(σ) upwards, thus reducing the profit from misreporting σ. Interestingly, which effect dominates depends on ψ(σd ) and ψ(σ). Notice that the auditing costs to ensure truthful revelation by type σ are effectively paid when σd is the true model, hence the downward bias is strong when ψ(σd ) is high. Conversely, when σ is the true model, the auditing costs come from incentive compatibility for type σm , the upward bias is strong when ψ(σ) is high. Another interesting consequence is that if a model σ has a high prior probability to be the true model, such that H(σ) will have to be paid often, in order to reduce auditing costs the regulator needs to look not at the incentives of type σ, but at the incentives of the type who would like to misreport σ. In other words, if a model is more likely than the others, it is necessary to make sure that, when this model is wrong, the intermediary does not want to use it.

33

From these two effects we can deduce important properties of the second-best regulation (θ∗∗ , H ∗∗ ): Lemma 3. Second-best capital requirements are such that: 1. θ∗∗ (.) is increasing. 2. θ∗∗ (σmax ) ≤ θ∗ (σmax ) and θ∗∗ (σmin ) ≥ θ∗ (σmin ). Proof of the first part (see Appendix A.6 for the detailed proof): intuitively, the main problem for a regulator facing a high σ is to prevent this type from deviating, and for a low σ to discourage other types to mimic σ. Take two models σ0 , σ1 with σ1 > σ0 , θ(σ1 ) < θ(σ0 ). Type σ0 knows profits will be higher from a given θ, and faces a tighter constraint if he tells the truth, hence he always has strictly more incentives to lie than σ1 and d(σ1 ) = ∅. Thus there is no reason to bias θ(σ1 ) downwards. Moreover since θ∗ (.) is increasing it’s more costly in terms of welfare to bias σ0 upwards. And finally more auditing costs can be spared by decreasing θ(σ0 ) than by decreasing σ1 , thus overall it cannot be the case that θ(σ1 ) < θ(σ0 ) at an optimum.  Proof of the second part: consider figure 9, and assume (θ∗∗ , H ∗∗ ) are such that θ∗∗ (σmax ) > ˜ H) ˜ that improve welfare θ∗ (σmax ) as on the black curve. We want to show there exists (θ, without increasing auditing costs. Since θ∗∗ is increasing, if θ∗∗ (σmin ) > θ∗ (σmax ) it is possible ˜ is closer to the to improve over θ∗∗ by setting θ˜ constant and equal to θ∗ (σmax ), such that θ(.) first-best and induces zero auditing costs. If θ∗∗ (σmin ) ≤ θ∗ (σmax ), there exists σ ¯ such that θ∗∗ (¯ σ ) = θ∗ (σmax ). Then we can set θ˜ constant and equal to θ∗ (σmax ) for σ ∈ [¯ σ , σmax ], and θ˜ = θ∗∗ otherwise. No type has an incentive to misreport a model in [¯ σ , σmax ] since it cannot allow a higher leverage. Thus it is no longer required to audit models in this interval. For σ<σ ¯ we can use the same auditing intensity H ∗∗ (σ), all types below σ ¯ still report truthfully because they face the same incentives as before, and all types above σ ¯ make more profit ˜ H) ˜ is both more than before by telling the truth and thus also report truthfully. Overall (θ, efficient and less costly than (θ∗∗ , H ∗∗ ), a contradiction. The symmetric reasoning shows we must have θ∗∗ (σmin ) ≥ θ∗ (σmin ). 

34

Θ

Θ­@ΣD

Θ­­@ΣD Hhypoth.L

Ž Θ@ΣD HimprovedL Σmin

Σmax

Σ

Figure 9: θ∗∗ (σmax ) ≥ θ∗ (σmax ) I summarize all the results on auditing in the next proposition: Proposition 5 (Model-sensitivity/auditing costs trade-off). The optimal regulation with costly auditing satisfies: 1. θ∗∗ is increasing in σ. Moreover θ∗∗ (σmax )−θ∗∗ (σmin ) is lower than θ∗ (σmax )−θ∗ (σmin ): the second-best regulation is on average less model-sensitive than the first-best regulation. 2. H ∗∗ is decreasing in σ. H ∗∗ (σmax ) = 0 and P (H ∗∗ (σmin )) < 1. 3.



  ∗∗ X  dH ∗∗ (σd ) dH (σ) ψ(σd ) + ψ(σ) V10 (θ∗∗ (σ), σ)ψ(σ) = (1 + c)w  dθ(σ) dθ(σ) σd ∈d(σ)

Remark 3. When auditing costs w tend to +∞, the optimal regulation features a constant θ∗∗ (.). This proposition shows that the more the regulator tries to use the information conveyed by models to compute capital requirements, the higher the incentives to misreport the model and thus the auditing costs are. As a result, the regulator should ask higher capital requirements than the first-best when the report is optimistic, and lower capital requirements than the first-best when the report is pessimistic. The first-order condition helps us to understand why: if the true model is very optimistic the intermediary does not have much incentives to lie, since by telling the truth he already gets a low capital requirement and a high profit. But 35

such a model is a tempting deviation for an intermediary with a high risk. To discourage misreporting, capital requirements for the most optimistic types have to be higher. Conversely, the possibility that a type may want to misreport a very pessimistic model is not a concern. But an intermediary who knows the true model is very pessimistic has high incentives to lie, and to decrease them it is necessary to lower capital requirements for such types. Because of these two effects, the second-best regulation makes less use of the models’ information than the first-best. Finally, when auditing costs are infinitely high even ensuring that the most constrained type does not want to mimic the least constrained one has an infinite cost, so that the regulator chooses a constant θ∗∗ (.) and does not use the models’ information at all.

4.3

Discussion and policy implications

This section concludes on a rather negative note. In principle the regulator could use the observation of the realized level of default to detect intermediaries using over-optimistic models, she could offer a menu of capital requirements and penalties inducing truthful revelation. This is what the regulator does for market risk, although the current regulation only counts the number of violations of the VaR, not the amount by which it is exceeded. A possible implication of this section is that if only large violations are informative then it is necessary to take into account the size of the violation, along the lines proposed by Colletaz, Hurlin, and P´erignon (2011). Credit risk however is likely to be different: different models typically yield different predictions for tail values only, and when high levels of default are reached it is quite possible that the institution will already be at risk, such that punishing over-optimism will be impossible ex-post, unless the regulator can commit to giving positive transfers to the shareholders of defaulting institutions. If this is impossible, it limits how sensitive to the intermediary’s model the regulatory constraint can be. Even when the true model is the most optimistic one, the regulated should not be allowed too high a level of leverage, otherwise we will be in the case of models distinguishable only above the intermediary’s default point. But if the regulation has to be less reactive to the intermediary’s report, using internal models for regulatory purposes is also less useful. Notice that for market risk the regulator wants regulatory capital to cover losses during 99% of the trading days, while for credit risk the implicit target is to cover losses during 99.9% of years. In the terms of the model, the θ∗ for market risk is much lower than the one for credit risk, so that it is tempting to consider market risk as corresponding to the case where backtesting can work (θ∗ > tˆ) and credit risk as the case where it cannot (θ∗ < tˆ). The idea of auditing models seems easier to apply in practice. But the same trade36

off appears, although for different reasons. If the purpose of a model-based regulation is to obtain finely risk-sensitive capital requirements, then capital requirements will also be model -sensitive. Then there are incentives to report a false model, and a costly auditing procedure is necessary to ensure truthful revelation. Proposition 5 shows that the secondbest regulation will always be less risk-sensitive than the first-best. Two effects play a role: intermediaries knowing the true model is pessimistic must be allowed not too low a leverage to have less incentives to misreport, and intermediaries knowing the true model is optimistic must be allowed not too high a leverage, otherwise mimicking them would be too profitable. If auditing costs are high, these effects may make the second-best regulation much less risk-sensitive, such that the use of a model-based regulation is an unnecessary complication. For prohibitively high auditing costs by definition the regulator cannot use the models’ information at all. The auditing procedure is close to the one studied in Prescott (2004), but in his paper the amount a bank can invest is fixed, thus a lower capital constraint does not allow to lend more, but only to get funding at a lower cost. As a result the type with the highest risk is always the one with the highest incentives to misreport and the schedule of second-best capital ratios is above the first-best, but not necessarily less risk-sensitive. Models of adverse selection in a banking context similar to the menu of transfers discussed above have been used to study deposit insurance premia. In Chan, Greenbaum, and Thakor (1992) a bank is offered either a low insurance premium and a high capital requirement, or a high insurance premium and a low capital requirement, the low-risk bank selects the first offer and the high-risk bank the second one. A similar mechanism could theoretically be used to solve the adverse selection problem on risk models. A problem with this solution however is that the demand for loans is inelastic in Chan, Greenbaum, and Thakor (1992), so that the regulator does not take into account that higher capital ratios and/or higher insurance premia will imply fewer loans, a concern that seems currently extremely important. Increasing capital requirements when the true σ is low (low risk) would decrease the amount of loans in the economy precisely when they have a higher social value. It is in principle possible to do better by using transfers to banks depending on the model they reported and on the level of defaults that realizes, which gives information about whether the model used is realistic or not. In Giammarino, Lewis, and Sappington (1993), the problem studied is a bit different: the regulator is assumed to be able to check the quality of the bank’s assets at no cost through auditing, but implementing a risk-based regulation may give incentives for the bank not to spend enough effort to increase the quality of its assets. But internal risk models are used precisely because it is assumed that the bank has more information about

37

its assets than the regulator can acquire through simple auditing procedures, hence the main problem seems to be adverse selection, not moral hazard, although it would be interesting to introduce the moral hazard element into the picture (see the Appendix B.4). An interesting extension of this section is to take into account that developing models is also long and costly. Assume for instance an intermediary has to pay a cost C in order to learn which one is the true model. The regulator has a new constraint linked to “information gathering” as in Cr´emer, Khalil, and Rochet (1998): she must make sure that the intermediary searches for the true model and pays the cost C. If C is high enough, more rent has to be left to intermediaries in the case of a menu of penalties. But if auditing is used, it may become necessary for the regulator to increase the risk-sensitiveness of the regulation. The intuition is the following: if auditing costs are very high, the second best θ∗∗ is almost constant, so incentives to lie are low and few auditing hours are required. But an intermediary reporting σmax gets Eψ (π(θ∗∗ (σmax ), σ)), while if he searches he gets Eψ (π(θ∗∗ (σ), σ)) − C. If θ∗∗ is exactly constant, the first option is necessarily better. If we take into account this new constraint, the regulator can no longer get truthful revelation and arbitrarily low auditing intensity. Thus, for high enough auditing costs for the regulator and research costs for the intermediary, the second-best regulation will be not to use internal models at all. Conversely a model-based regulation is useful when intermediaries are very efficient at developing models and the regulator has a cheap auditing technology.

5

Extensions: possible countervailing forces to overoptimism

5.1

Gradual adoption of new models

It is certainly not always the case that risk managers consciously choose risk models to bypass regulatory constraints. More plausibly, there is a process in which new models are developed, and since model uncertainty is hard to resolve, more “useful” models have a competitive advantage in the process. Either their users tend to favor useful models, or their “suppliers”, often specialized firms, realize that models both plausible and not too pessimistic are more likely to become popular. The equilibrium of section 3.1 can be seen as a steady-state of such a process. It is first useful to make the following remark: Remark 4. With insured investors and incomplete regulation, the strategies µl chosen by two intermediaries are strategic substitutes. 38

The proof follows from Proposition 1: incentives to choose the most optimistic model are higher when rL is higher, and rL is higher when less intermediaries choose the most optimistic model. This fact has interesting dynamic implications. Imagine the following process: at the beginning intermediaries use all available models in the same proportions, or invest in the safe asset only. In each subsequent period, each intermediary has the opportunity to choose a new risk model/a new strategy. Intermediaries are not capable of computing precisely which model is the best to use, so they tend to adopt models which seem widely used and profitable: if ni,t is the number of intermediaries using strategy i in period t, π(i, t) the profit made by an intermediary adopting this strategy in period t, and π ¯t the average profit in this period, I assume: ∀i, ∀t ≥ 0, ni,t+1 =

π(i, t) nt π ¯t

This process of “replicator dynamics” has the property that the total number of intermediaries stays constant, a strategy yielding more profit than the average is more and more adopted, and if the process converges to some distribution of strategies in the population then this distribution and the associated market prices form an equilibrium in the sense of Definition 1. I simulate such a process with the same parameters for the distribution of defaults and demand as in the illustration of section 3.1. At period 0, I assume 90% of intermediaries invest in the safe asset only, and the others are using in the same proportions 1000 models giving them values of θ between θ(σmin ) and 1. Figure 10 shows the evolution over 10 periods of the proportion of intermediaries choosing the most optimistic model, and the distribution of intermediaries over the different models available in period 10. The figure shows the proportion of intermediaries choosing all models between 2 and 1000, 24% of intermediaries choose the most optimistic one, and 72% invest in the safe asset only. Proportion

Proportion

0.25

0.0001 0.20

Distribution, t=10 0.00008 0.15 0.00006

0.10

0.05

0.00

0.00004

Intermediaries with the most optimistic model Μ

0.00002

2

4

6

8

10

Time

Σmin

Σmax

Time

Figure 10: Distribution of intermediaries using the different models under incomplete regulation after 10 periods, and use of the most optimistic model over time. 39

At the beginning, most intermediaries do not invest in the risky asset, hence rL is high and larger than rLe . It is thus extremely profitable to use the most optimistic models, and many intermediaries switch to them. This behavior increases the supply of loans and decreases the interest rate, such that in period 2 already the increase in the number of intermediaries using the most optimistic model is much smaller. rL continues to shrink gradually, but as rL decreases it becomes profitable to use low levels of leverage, and we obtain a process in two waves: forerunners rush to very optimistic models and the interest rate on loans drops, then these first models are gradually abandoned and replaced by more conservative ones. In the end the process will converge to an asymmetric situation with intermediaries using either maximum leverage or no leverage at all, which corresponds to the equilibrium of section 3.1.

5.2

The cost of bad forecasts

I have made the rather extreme assumption that an intermediary could choose a very overoptimistic model with no more cost than if he chose a more realistic one. In a more general framework with several types of final borrowers for instance, the model could have two different functions: assessing the intermediary’s risk and evaluating relative risks of defaults for two claims. It could be costly, or even impossible, to choose a very optimistic model to bypass regulation and at the same time allocate between the several types of borrowers as the true model advises. This would add a countervailing force giving incentives to stay closer to the true model and intermediaries would face a more complicated tradeoff. However, for this countervailing force to be of any importance, it must be the case that optimistic models are too optimistic for default levels below the maximum sustainable losses. If the model just underestimates the probability of extreme events and the intermediary will default for events less extreme, the forecasting mistake is privately irrelevant. Assume the true model is σ and the intermediary reports σ 0 , the interest rate on loans is multiplied if the level of realized defaults is t by (|f (t, σ 0 ) − f (t, σ)|), with  a decreasing function and (0) = 1, (d) ≥ 0 ∀d. In words, the return on each loan (conditional on repayment) is discounted, and the discount is higher when the probability of the realized default level was more badly forecast (because loans were not properly monitored, or losses were not properly hedged for instance). Equation 4 can be rewritten: 0

Z

π ˜ (L, σ , σ) =

θC (L,σ 0 )

(rL L(1 − t)(|f (t, σ 0 ) − f (t, σ)|) − rD (L − K)) f (t, σ)dt

0

40

where θC (L, σ 0 ) s.t. rL L(1 − θC (L, σ 0 ))(|f (t, σ 0 ) − f (t, σ)|) − rD (L − K) = 0 Assume the most optimistic model σmin and the true model σ are distinguishable only in the tail above θC (L, σmin ). Then θC (L, σmin ) = θ(σmin ) and the profit of the intermediary is exactly the same as without costs for forecast errors since below θC (L, σmin ) the optimistic model’s predictions are correct. If an intermediary chooses the most optimistic model, he will default for relatively low levels of losses in his portfolio, and he doesn’t care about forecasting errors concerning higher levels of losses. Thus, the incentive to adopt overoptimistic models is robust to the introduction of costs associated to forecasting errors if some models are available which are both optimistic regarding the probability of extreme events, and realistic everywhere else, which is precisely the fact highlighted in Proposition 4. This reasoning also shows that a less model-sensitive regulation is a way to ensure all types of intermediaries care about forecasting mistakes.

5.3

Other extensions

The model is flexible enough to allow for a number of other extensions, which can be used to analyze other incentives to adopt more or less optimistic models. Some of them are treated in more details in the Appendix B. A possible countervailing force to the selection of over-optimistic models would be of course to assume risk-adverse intermediaries, who wouldn’t take too much risk and thus wouldn’t choose the most optimistic models. This would probably not cancel incentives to be over-optimistic however, as the parallel with Gollier, Koehl, and Rochet (1997) suggests. It is also possible to assume some depositors will randomly withdraw their deposits before loans are reimbursed, or that banks have a “charter value” that limits their risk-taking. In all cases the choice of the intermediary would be made smoother, at the cost of additional complexity of the model, but the main results would not change. On top of the adverse selection problem considered in this paper, it would be interesting to consider the moral hazard problem of an intermediary having to choose in which assets or loans to invest before reporting risk estimates to the regulator, as suggested by Carey and Hrycay (2001): “investments might be focused in relatively high-risk loans that a scoring model fails to identify as high-risk, leading to an increase in actual portfolio risk but no increase in the banks estimated capital allocations”. Section B.4 sketches such an extension where intermediaries have incentives to focus on complex assets for which there is model uncertainty, even when they are on average riskier than simple assets. 41

It is also possible to consider what happens if investors are no longer insured. If they also know the true model and the regulator can commit on not bailing out defaulting institutions, investors charge higher interest rates to banks adopting more optimistic models. The financial structure of intermediaries becomes irrelevant and in equilibrium we have rL = rLe . Regulation then relies mainly on Basel’s third pillar: the regulator makes sure that banks provide investors with quantitative estimates of the risk they incur and that the methodology used is clear enough for investors to detect over-optimistic models. Investors ask higher interest rates to lend to banks using more optimistic models, market discipline then limits the incentives to use over-optimistic models and to use more leverage. This is optimal if the regulator cares only about protecting investors. But if a bank’s default has some external costs not borne by its creditors, then risk is too high. Thus another interpretation is that the possibility to freely choose the risk model on which regulation is based enables intermediaries and their creditors to entirely bypass the regulation and reach the level of leverage maximizing their joint profit. An interesting application is the case of an originator of securitized products (intermediary) who faces investors who can by law only invest in products with a sufficiently high grade. This model could be used to investigate the claim that many originators used the rating agencies to label their products “investment grade” and be able to sell them to regulated investors, while the latter may have been aware that the ratings were unrealistic but wanted to bypass the regulation (see for instance Pagano and Volpin (2009) and Bolton, Freixas, and Shapiro (2012)). Other possible mechanisms to reveal the intermediaries’ private information could be analyzed, in particular the regulator could use several reports from the different banks. I show in the Appendix B.2 that the first-best regulation satisfies Maskin monotonicity (Maskin (1999)) and how to adapt the canonical mechanism for Nash implementation, which is not trivial as there is an interaction between transfers and limited liability as in section 4. However the assumption made in section 3 that all intermediaries have the same information is to be interpreted as a simplifying assumption that allows to consider a representative bank (see footnote 7). I show in B.2 that the model can be rewritten to feature institutions with different σ, due for instance to different monitoring technologies such that they have to use models that are not directly comparable, which is one of the reasons why the regulator uses internal estimates and not a one-size-fits-all methodology. It is clear however that a bank’s knowledge about internal models has both an idiosyncratic and a common component, studying mechanisms along the lines of Cr´emer and McLean (1988) that the regulator could use to reveal the latter is an interesting topic for future research.

42

In the framework of section 3.1 it may seem that the regulator has a simple way to solve the problem: if all intermediaries report some σ 0 and the regulator observes the market value of the intermediaries or the value of some junior debt to be completely inconsistent with the reported risk estimate, then she could infer that intermediaries have lied and use this information to choose θ. The problem however is that this reaction will be anticipated by market participants in equilibrium, which can either make market signals much less informative or even destroy the possibility of a rational expectations equilibrium (see Bond, Goldstein, and Prescott (2010)). In the Appendix B.3 I show that the regulator can indeed use an intermediary’s market value to learn σ, assuming that the intermediary’s shares are priced by investors who know the true risk of the intermediary. This is not a very compelling case, because without informational asymmetries between a bank and its shareholders it would probably not be necessary to have a Basel type regulation in the first place, as high capital ratios would not be that costly. Conversely if the regulator tries to use the market price of junior debt, then no equilibrium exists when model uncertainty is too high. A last interesting extension would be to model the fact that externalities arising from an intermediary’s default are partly transnational, such that a national regulator can deliberately allow the use of optimistic models to favor domestic institutions. Rochet (2010) sees this as one of the main problems arising from the internal ratings based approach of Basel: it becomes impossible for agents outside the regulator-regulated relationship to check that the regulation is adequate, and the outcome is the race to the bottom in regulatory standards that Basel sought to prevent.

6

Conclusion

“Model-based” regulation is a sensible way to exploit banks’ better information about their own risk to compute capital ratios. This information however is private, and financial intermediaries cannot be expected to develop unbiased models if they face too many incentives to do otherwise. The choice of a risk model involves costs and benefits. In many cases the most “useful” model is also the correct one, but when a model has an “external use”, its popularity won’t depend on its validity only, but also on other characteristics. This paper shows how a regulation failing to give the proper incentives can lead to the wide adoption of over-optimistic risk models, and how tightening the regulation can even worsen the situation. This work is not to be interpreted as meaning that banks always choose internal models to maximize leverage, regardless of their plausibility; but rather as suggesting that the whole

43

process of elaboration/adoption of new models may be biased towards more “profitable” models. When one tries to improve on the best models already in use, it is tempting to relax the assumptions making the model too pessimistic first. The new model will be more general than the previous one and in this sense “better”, but if such incentives influence the whole process there can be a drift toward over-optimism. It is in principle possible to use a model-based regulation and ensure that financial institutions choose the correct models. This requires the regulator to be able to commit to charging important penalties for the use of over-optimistic models, which can be difficult, or to be proficient at auditing internal models. In both cases a more risk-sensitive regulation, for which using internal models would be the most relevant, makes it harder for the regulator to reveal an intermediary’s true model, for three reasons. First, a more risk-sensitive regulation gives more incentives to use slightly over-optimistic models as it will enable the intermediary to increase leverage. Second, if intermediaries are allowed to use more leverage it is more likely that they will default for high levels of losses. Since these high levels are the ones that enable the regulator to tell whether a model was too optimistic, it becomes more difficult to punish over-optimistic intermediaries. Lastly, if intermediaries default for lower levels of losses they care about less states of the world, and hence using a model that is over-optimistic about tail risk is less damaging for their profit. The conclusion is that the banking regulation should be either even more complex, or much simpler. A first possibility is to continue using internal models and take into account the hidden information problem. This option should be chosen if benefits from a risk-sensitive regulation are high (e.g. very heterogeneous banks), and if for the regulator it is much easier to audit banks’ risk models rather than developing some of her own (otherwise it would be better to use a regulatory model). Intermediaries must be very efficient at developing accurate models, and the regulator at checking their validity. The opposite solution is to simply abandon the use of internal models for regulatory purposes and have a regulator prompt to intervene when losses or capital ratios hit certain thresholds instead of trying to manage finely the banks’ risks ex-ante, as advocated in Dewatripont and Tirole (1994), Decamps, Rochet, and Roger (2004) or Rochet (2010). In both cases the issue of how banks select their model has to be addressed carefully by the regulation. Finally, although I developed the example of banks, there are other instances in which such a strategic use of models can take place. The regulation of insurance companies in the European Solvency II framework is comparable for what concerns risk model choice, with model uncertainty perhaps even more severe. Another interesting phenomenon along the

44

same lines is the use of internal models to measure the performance of employees, desks, and departments. More generally, many firms use internal models to convey information from one hierarchic level to another, to rating agencies, shareholders... Sibbertsen, Stahl, and Luedtke (2008) quote a report according to which some investors “tend to apply an across-the-board discount of about 20% to the published numbers” and present it as an evidence of model risk. But this negative discount cannot stem from honest and random mistakes about the true model, much rather from the suspicion that models are deliberately chosen to bias the reported information. Hence the relevant problem here may not be model risk, but “adverse selection of models”.

45

A

Appendix - Proofs

A.1

Notations

rL

gross interest rate on loans to borrowers

rD

gross interest rate on loans to intermediaries

r0

risk-free rate, normalized to 1

e rL

first-best interest rate on loans, equals r0 /(1 − E(t))

D(.)

demand for loans by final borrowers

rL (.)

inverse demand function for loans

L

amount lent by an intermediary/the representative intermediary

M

amount borrowed by an intermediary

K

capital owned by an intermediary

W

investors’ wealth

t

random proportion of defaulting loans

f (., σ), F (., σ)

family of pdf and cdf, parameterized by σ, modeling the proportion of defaulting loans

ψ(.), Ψ(.)

pdf and cdf from which the true σ is drawn

θ

default point, maximum proportion of defaults an intermediary can suffer in his portfolio

s(θ, σ)

expected proportion of surplus repayments in an intermediary’s portfolio

π(θ, σ)

expected profit of an intermediary

V (θ, σ)

social welfare

α(σ)

minimum capital ratio required from an intermediary reporting model σ

α ¯

minimum of the function α(.)

θ(σ)

minimum θ allowed by the regulation if the intermediary reports model σ

η

parameter of the demand function used in simulations

µl , µr , µs

proportions of intermediaries with max. leverage / investing K in loans / K in the safe asset

A.2

Proof of Lemma 1

For given prices rL , rD and a given constraint K/L ≥ α, the intermediary’s program if he invests in loans can be written as: max rL L(θ)s(θ, σ), s.t. θ ≥ 1 − θ

rD (1 − α) = θ rL

Notice in particular that α ≤ 1 and rL ≥ rD implies that θ¯ ≥ 0 and rD − rL (1 − θ) ≥ 0. It is easy to compute that s0 (θ, σ) = F (θ, σ). Then we have π10 (θ, σ) =

rL L(θ)(F (θ, σ)(rD − rL (1 − θ)) − rL s(θ, σ)) rD − rL (1 − θ)

46

(20)

Denoting G(θ) = F (θ, σ)(rD − rL (1 − θ)) − rL s(θ, σ), we have G0 (θ) = f (θ, σ)(rD − rL (1 − θ)) as both terms rL F (θ, σ) cancel out, thus G0 is always positive. This implies that π(θ, σ) is either decreasing and then increasing in θ, always increasing or always decreasing. Finally, we have π10 (1, σ) = • If rL ≥

rL K rD

(rD − rL (1 − E(t)) , limθ→1− rD π10 (θ, σ) = −∞. Using equation 6 we have: rL

e, (rD /r0 )rL

then

π10 (1, σ)

≤ 0 and π10 is negative for every θ, thus if he invests in loans the

intermediary chooses θ = θ. Notice that we also have rL (1 − E(t)) ≥ r0 , hence the intermediary prefers investing in loans to investing in the safe asset. e > r > r e then the intermediary chooses θ = θ or θ = 1, since profit is either first • If (rD /r0 )rL L L

decreasing and then increasing in θ, or always increasing. A comparison shows that he will choose θ = θ if and only if  rL ≥ rD

1 − E(t) − s(θ, σ) (1 − θ)(1 − E(t))

 = r1 (rD , θ)

e > r e > r profit is decreasing and then increasing in θ, but investing K in the safe • If (rD /r0 )rL L L

asset yields more than in loans. The intermediary chooses θ = θ over L = 0 if and only if rL ≥

r0 rD = r2 (rD , θ) rD s(θ, σ) + r0 (1 − θ)

e > r e = r the previous condition applies, except that the intermediary is indifferent • If (rD /r0 )rL L L

when he doesn’t borrow between investing in the safe asset or in loans. We have to compare the different thresholds for rL . First, we have: r1 (rD , θ) ⇔

θ

⇔

θ

e > (rD /r0 )rL

> E(t) + s(θ, σ) Z 1 Z θ ⇔ θ > tf (t, σ)dt + θF (θ, σ) − tf (t, σ)dt 0 0 Z 1 ⇔ θ(1 − F (θ, σ)) > tf (t, σ)dt θ

> E(t|t ≥ θ)

The last inequality is obviously false. Next, developing and rearranging r1 (rD , θ) and r2 (rD , θ), it is easy to show that e e r1 (rD , θ) > r2 (rD , θ) ⇔ r1 (rD , θ) > rL ⇔ r2 (rD , θ) > rL r0 (1 − θ) ⇔ rD > 1 − E(t) − s(θ, σ)

(21)

This last inequality may be true or false depending on rD . Thus we have two cases to consider and the conditions above prove the following:

47

e the intermediary chooses r ∗ = 0, when If rD > (r0 (1 − θ))/(1 − E(t) − s(θ, σ)): when rL < rL L e he chooses L∗ = K, when r > r (r , θ) he chooses θ ∗ = θ. Now if r r1 (rD , θ) > rL ≥ rL 1 D D ≤ L

(r0 (1 − θ))/(1 − E(t) − s(θ, σ)): if rL < r2 (rD , θ) the intermediary chooses L∗ = 0, if rL ≥ r2 (rD , θ) he chooses θ∗ = θ. This implies the proposition where I denote rL = max(r1 (rD , θ), r2 (rD , θ)).

A.3

Proof of Lemma 2

When demand is close to perfectly elastic rL does not depend on α, such that choosing α is equivalent to choosing θ = 1 −

r0 rL (1

− α). Moreover we can write: L(θ) =

r0 K r0 − rL (1 − θ)

L is obviously decreasing in θ. The first-order condition in θ gives us: V10 (θ, σ)

0



Z

= L (θ) rL Eσ (1 − t) − r0 − c

1

 (r0 − rL (1 − t))f (t, σ)dt

θ

= L0 (θ) (rL Eσ (1 − t) − r0 − c(1 − F (θ, σ))(r0 − rL Eσ (1 − t|t > θ))) = 0 Assumption M1 implies that a distribution with a higher σ dominates a distribution with a lower one in the sense of first-order stochastic dominance. As a result when σ increases Eσ (1 − t) and Eσ (1 − t|t > θ) decrease, (1 − F (θ, σ)) increases such that, since L0 (θ) ≤ 0, V10 (θ, σ) increases. Hence 00 (θ, σ) ≥ 0. We can then compute: V1,2

00 V11 (θ, σ) = L00 (θ)(rL Eσ (1 − t) − r0 ) − cL00 (θ)

Z

1

(r0 − rL (1 − t))f (t, σ)dt + cL0 (θ)(r0 − rL (1 − θ))

θ

L00 is positive, thus the first term may be positive since in general the regulator will allow less leverage e . Intuitively when the regulator reduces the leverage further the than what would lead to rL = rL

supply of loans decreases but at a declining speed, hence welfare losses due to credit restriction increase more slowly, which gives some convexity in θ to V . When costs are high enough however this effect can be compensated by the two other terms. By definition of θ we have r0 − rL (1 − t) ≥ 0 00 is negative and for t ≥ θ, such that the two other terms are negative. Hence if c is high enough V11 00 (θ, σ) positive, such that for every σ there is a unique maximum of V for θ = θ ∗ (σ), with θ ∗ V1,2

increasing.

A.4

Proof of Proposition 1 and Corollary 1

When rD = r0 , inequality 21 is equivalent for any θ to θ > E(t)+s(θ), which was proven to be wrong e ≥ r (r , θ(σ in the proof of Lemma 1. Thus we have rL 2 0 min )) ≥ r1 (r0 , θ(σmin )). Thus we know that

an intermediary will choose either L = 0 or θ = θ(σmin ). Thus in equilibrium a proportion µl of

48

intermediaries choose model σmin and L = K/¯ α, and the others invest only in the safe asset. If the interest rate on loans is rL , the supply of loans must equal the demand: µl

K = D(rL ) α ¯

(22)

It is impossible to have µl = 0 in equilibrium since this would require an infinite interest rate rL , at which supply would be positive. Hence there are two possibilities: if 0 < µl < 1 it must be the case that intermediaries are indifferent between investing only in the safe asset and choosing maximum leverage, in which case rL will be equal to r2 (r0 , θ(σmin )). This condition can be rewritten as:   r0 (1 − α) ,σ r0 α = rL s 1 − rL

(23)

The second possibility is to have µl = 1, in which case the left-hand side of equation 23 has to be lower than the right-hand side, or equal. Start with equation 23. The right-hand side is increasing in rL , for rL → +∞ it goes to infinity, ∗ for which there is and for rL = r0 (1 − α) it is equal to zero. Hence there is a unique value rL ∗ )(¯ equality. Then by using equation 22 we can compute µ∗l = D(rL α/K). If we find µ∗l ≤ 1 then we

have an equilibrium. Moreover it is unique: if we choose a higher µl then rL has to be lower, in which case the right-hand side in equation 23 becomes strictly lower than the left-hand side, which cannot be the case in equilibrium (in particular for µl = 1 we need exactly the opposite). If instead α = D(rL ). we find µ∗l > 1, it implies that µl = 1 is an equilibrium, and rL is determined by K/¯ Again this equilibrium is unique: if we decrease µ then rL will increase, and the right-hand side in equation 22 will become even greater. Assume now that we start the same reasoning with a higher σ. Due to assumption M1, s(., σ) is decreasing in σ, to have an equality in equation 23 we need a higher rL , which will give a lower D(rL ) in equation 23, and hence a lower equilibrium µl . If we start with a σ such that µl = 1 in equilibrium, an increase in σ will make it less interesting to invest in loans, which will either let µl unchanged or induce a switch from the case µl = 1 to the case µl < 1. Finally assume we do the same reasoning but we increase η. This does not affect the first step ∗ , but in the second step since demand is higher at this interest rate µ will in which we determine rL l

have to be higher as well. The corollary follows quite directly from equation 8 defining pd (¯ α, σ) as the product of µl (α ¯ , σ) and the default probability of an intermediary with maximum leverage. When µl (¯ α, σ) < 1, increasing η leaves rL (¯ α, σ) unchanged, µl increases and the second term of the product is unchanged, hence the product increases.

49

A.5

Proof of Proposition 3

Defining U (σ 0 , σ) the expected profit of an intermediary reporting σ 0 when the true parameter is σ, we have:

π ¯ (σ 0 ) F (a(σ 0 ), σ 0 ) (¯ π 0 (σ 0 )F (a(σ 0 ), σ) + a0 (σ 0 )f (a(σ 0 ), σ)¯ π (σ 0 ))F (a(σ 0 ), σ 0 ) F (a(σ 0 ), σ 0 )2 0 0 0 0 (a (σ )f (a(σ ), σ ) + F20 (a(σ 0 ), σ 0 ))¯ π (σ 0 )F (a(σ 0 ), σ) F (a(σ 0 ), σ 0 )2

U (σ 0 , σ) = F (a(σ 0 ), σ) U10 (σ 0 , σ) = −

(24)

The first-order condition gives for every σ: U10 (σ, σ) = 0 ⇔

F20 (a(σ), σ) π ¯ 0 (σ) = F (a(σ), σ) π ¯ (σ)

(25)

Notice first that the left-hand side is increasing in a (MLRP), and under the assumptions of the proposition the left-hand side is decreasing in σ and the right-hand side increasing. This ensures that if there exists a solution a(.) it is increasing in σ. Is it always possible to find such an a(σ)? Notice that whether π ¯ is the payoff of an intermediary not using any leverage or allowed some default point θ independent of σ, it can be written as π ¯ (σ) = rL KL(θ)s(θ, σ), with θ = 1 if no leverage is Rθ allowed. Moreover, s(θ, σ) can be rewritten as s(θ, σ) = 0 F (t, σ)dt. Thus we can rewrite equation 25 as:

Rθ 0 F (t, σ)dt F20 (a(σ), σ) = R0θ 2 F (a(σ), σ) 0 F (t, σ)dt

a(σ) can take values between 0 and 1, the right-hand side is negative. We have F20 (1, σ)/F (1, σ) = 0. If lima→0

F20 (a,σ) F (a,σ)

= −∞ we know there exists a value a to satisfy the required equality. If this limit

is some negative number −k, then we know that for every t in [0, 1] we have F20 (t, σ) ≥ −kF (t, σ). Integrating this inequality we find: Rθ

0 0 F2 (t, σ)dt Rθ 0 F (t, σ)dt

≥ −k

hence for a given σ the right-hand side is always lower than Since

F20 (a,σ) F (a,σ)

F20 (1,σ) F (1,σ)

and greater than lima→0

F20 (a,σ) F (a,σ) .

is increasing in a there is always a unique a satisfying the inequality for a given σ, and

hence there always exists a unique solution a(.) increasing and satisfying the first-order condition. We now have to show that the second-order condition is met. We can use equation 25 to replace F20 (a(σ 0 ), σ 0 )

in equation 24. After some rearrangements this gives us: U10 (σ 0 , σ) ≥ 0 ⇔

f (a(σ 0 ), σ) F (a(σ 0 ), σ) ≥ f (a(σ 0 ), σ 0 ) F (a(σ 0 ), σ 0 )

50

by the monotone likelihood ratio property we thus have U10 (σ 0 , σ) ≥ 0 ⇔ σ 0 ≤ σ, hence truthfully reporting σ globally maximizes U (., σ). Finally the set of families of distributions satisfying MLRP and the assumptions of the proposition is not empty, for instance a family of truncated Gaussian distributions differing in their means satisfies all the required properties when the variance of the underlying Gaussian distributions is 1 and the means are not too far from 5.

A.6

Proof of Proposition 5

The only part not proven in the text is that θ∗∗ (.) is increasing. To prove this we will have to use the following lemma: 00 (θ, σ) ≥ 0 and π 00 (θ, σ)π(θ, σ) − π 0 (θ, σ)π 0 (θ, σ) ≥ 0. Lemma 4. For any θ and σ we have π11 12 1 2

To prove the first part it is enough to differentiate π twice and get: 00 π11 (θ, σ) = rL Lf (θ, σ) − 2

2 L2 F (θ, σ) r3 L3 s(θ, σ) rL +2 L 2 2 r0 K r0 K

using the fact that π10 is negative and equal to: π10 (θ, σ) = −

2 L2 s(θ, σ) rL + rL LF (θ, σ) r0 K

00 (θ, σ) above is positive. shows that the the expression of π11

For the second part of the lemma, simple derivations and rearranging yields: 00 (θ, σ)π(θ, σ) − π 0 (θ, σ)π 0 (θ, σ) ≥ 0 π12 1 2 2 L2 (F 0 (θ, σ)s(θ, σ) − F (θ, σ)s0 (θ, σ)) ≥ 0 ⇔ rL 2 2

Remember s01 = F , thus we have to show that s0012 (θ, σ)s(θ, σ) ≥ s01 (θ, σ)s02 (θ, σ)

(26)

Since F (., .) has the MLRP property we have for σ1 ≥ σ0 : F (x, σ1 ) F (x, σ0 ) ≤ f (x, σ1 ) f (x, σ0 ) This can be rewritten as:

s0011 (x, σ1 ) s0011 (x, σ0 ) ≥ s01 (x, σ1 ) s01 (x, σ0 )

this implies in particular that s01 (x, σ1 )/s01 (x, σ0 ) increases in x. From which we deduce that for any

51

x, y, σ0 , σ1 with x ≤ y, σ0 ≤ σ1 we have: s01 (y, σ1 )s01 (x, σ0 ) ≥ s01 (y, σ0 )s01 (x, σ1 ) Integrating both sides with respect to x between 0 and y and rearranging we get: s01 (y, σ1 ) s0 (y, σ0 ) ≥ 1 s(y, σ1 ) s(y, σ0 ) which means that s01 /s is increasing in σ, which is equivalent to inequality 26 and implies the second part of the lemma. We can now prove the proposition by contradiction. Assume there are two models σ0 , σ1 with σ1 > σ0 , θ(σ1 ) < θ(σ0 ). The second part of the lemma implies that π10 (θ, σ)/π(θ, σ) is increasing in σ, which means that for any θ we have: π 0 (θ, σ0 ) π10 (θ, σ1 ) ≥ 1 π(θ, σ1 ) π(θ, σ0 ) this implies that π(θ, σ1 )/π(θ, σ0 ) is increasing in θ, from which we deduce that for any θ, θ0 with θ0 < θ we have: π(θ, σ1 ) π(θ, σ0 ) ≥ π(θ0 , σ1 ) π(θ0 , σ0 ) and finally this implies that for any model σ 0 such that θ(σ 0 ) < θ(σ1 ) we have: π(θ(σ1 ), σ1 ) π(θ(σ1 ), σ0 ) π(θ(σ0 ), σ0 ) ≥ > 0 0 π(θ(σ ), σ1 ) π(θ(σ ), σ0 ) π(θ(σ 0 ), σ0 ) such that d(σ1 ) = ∅, for any model that type σ1 wants to mimic, type σ0 has strictly more incentives to deviate to the same model. Now assume for clarity that m(σi ) = σmi , d(σ0 ) = σd0 , the result remains unchanged if these sets have zero or more than one element. The first-order conditions for models σ0 and σ1 give:   dH(σ0 ) dH(σd0 ) + ψ(σd0 ) = 0 − (1 + c)w ψ(σ0 ) dθ(σ0 ) dθ(σ0 )   dH(σ1 ) V10 (θ(σ1 ), σ1 )ψ(σ1 ) − (1 + c)w ψ(σ1 ) = 0 dθ(σ1 )

V10 (θ(σ0 ), σ0 )ψ(σ0 )

(27) (28)

00 ≤ 0, V 00 ≥ 0, it must be the case that V 0 (θ(σ ), σ ) ≤ V 0 (θ(σ ), σ ). Using Since we have V11 0 0 1 1 12 1 1

equations 27 and 28: dH(σ0 ) ψ(σd0 ) dH(σd0 ) dH(σ1 ) + ≤ dθ(σ0 ) ψ(σ0 ) dθ(σ0 ) dθ(σ1 )

(29)

since the second term on the left-hand side is positive, to contradict 31 it is enough to show the

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following: dH(σ0 ) dH(σ1 ) > dθ(σ0 ) dθ(σ1 ) using the equations 17 and 18, we need to show that: 1 π10 (θ(σ0 ), σm0 ) π(θ(σm0 ), σm0 ) 1 π10 (θ(σ1 ), σm1 ) π(θ(σm1 ), σm1 ) × × ≥ × × P 0 (H(σ0 )) π(θ(σ0 ), σm0 ) π(θ(σ0 ), σm0 ) P 0 (H(σ1 )) π(θ(σ1 ), σm1 ) π(θ(σ1 ), σm1 ) Notice that both sides are negative since π10 ≤ 0. Since θ(σ1 ) < θ(σ0 ) it must be the case that H(σ1 ) > H(σ0 ) and hence 1/P 0 (H(σ1 )) > 1/P 0 (H(σ0 )). Moreover, by definition of σm1 we have: π(θ(σm1 ), σm1 ) π(θ(σm0 ), σm0 ) π(θ(σm0 ), σm0 ) ≤ ≤ π(θ(σ1 ), σm1 ) π(θ(σ1 ), σm0 ) π(θ(σ0 ), σm0 ) Hence it is enough to contradict the assumption θ(σ1 ) < θ(σ0 ) to show that: π10 (θ(σ0 ), σm0 ) π 0 (θ(σ1 ), σm1 ) ≥ 1 π(θ(σ0 ), σm0 ) π(θ(σ1 ), σm1 )

(30)

00 ≥ 0, π 00 ≥ 0, we have to show that σ since according to Lemma 4 we have π11 m0 ≥ σm1 , that is the 12

mimicker of type σ0 is a more pessimistic type than the mimicker of type σ1 . To show this we use the definition of σm0 and σm1 : it has to be the case that σm0 has a higher incentive to mimic σ0 than does σm1 and conversely. Thus we must have: π(θ(σm0 ), σm0 ) π(θ(σm0 ), σm0 ) π(θ(σm1 ), σm1 ) π(θ(σm1 ), σm1 ) ≥ , ≥ π(θ(σ0 ), σm1 ) π(θ(σ0 ), σm0 ) π(θ(σ1 ), σm0 ) π(θ(σ1 ), σm1 ) from these two inequalities we deduce: π(θ(σ1 ), σm1 ) π(θ(σ0 ), σm1 ) ≥ π(θ(σ1 ), σm0 ) π(θ(σ0 ), σm0 )

(31)

Now assume σm0 < σm1 (to be contradicted). Using the second part of Lemma 4, we know that for any θ, σ the ratio π20 (θ, σ)/π(θ, σ) is increasing in θ. Thus we have for any σ: π20 (θ(σ1 ), σ) π 0 (θ(σ0 ), σ) ≤ 2 π(θ(σ1 ), σ) π(θ(σ0 ), σ) this inequality implies that π(θ(σ1 ), σ)/π(θ(σ0 ), σ) is decreasing in σ. If σm0 < σm1 we obtain: π(θ(σ1 ), σm0 ) π(θ(σ1 ), σm1 ) ≥ π(θ(σ0 ), σm0 ) π(θ(σ0 ), σm1 ) which contradicts equation 31. As a conclusion σm0 ≥ σm1 , thus inequality 30 is true and hence the first-order conditions 27 and 28 are not logically consistent, which implies that θ(σ1 ) cannot be lower than θ(σ0 ).

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B

Appendix - Extensions

This section is devoted to developing several extensions mentioned in the paper, in particular in 5.3.

B.1

Endogenous capital

Throughout the paper it is assumed that K is given and that intermediaries cannot issue new shares. There are two reasons for this assumption: first, it seems to fit well the current situation for banks, which are extremely reluctant to issue more shares for various reasons, one of them being that they probably fear sending a negative signal to the market; second, to incorporate equity issuance into the picture it is necessary to make assumptions about the information potential investors have: it seems extreme to assume they also know the true model, and if they don’t then equity issuance is a signaling game, complicated by a second adverse selection problem between the intermediaries and the regulator. In this section I sketch a simple extension to allow banks to issue new equity, and show that the main results of part 3.1 are unaffected. Assume that in order to increase its capital by one unit, an intermediary has to promise a return of r0 (1 + k), while buying back shares and thus decreasing capital by one unit will only bring r0 (1 − k). One can think of these payoffs as being the outcome of asymmetric information (Myers and Majluf (1984)): buying back shares signals that the intermediary’s assets are better than the market expects, and issuing new shares is a bad signal. k here is taken as given and should of course be endogenous in a more complete model, but the idea is to show that taking this into account would not change the type of equilibrium obtained in Proposition 1. The case k = 0 can be thought of as investors having perfect information about the intermediary’s assets. In this limit case the Modigliani-Miller theorem holds and a Basel-style regulation is not necessary at all, since intermediaries would be willing to have very high capital ratios (as advocated in Admati, DeMarzo, Hellwig, and Pfleiderer (2011), who acknowledge however the existence of a cost of equity due to asymmetric information). As the goal here is to study the Basel regulation and possible ways to mend it, we shall assume k > 0. For a given level of capital K, we still have that any intermediary will either choose to have no leverage, invest only in safe asset and get r0 K, or report the most optimistic model, choose maximal leverage and get rL (K/¯ α)s(1 − (r0 /rL )(1 − α ¯ ), σ). These two possibilities give a profit proportional to K, it is thus immediate to compute the marginal gain of increasing capital by one unit. ¯ and can choose to increase or decrease Assume all intermediaries start with some capital K, ¯ as given, we can compute the equilibrium as in Proposition 1. If we get a it. Taking K = K solution with µl < 1 it has to be the case that (rL /¯ α)s(1 − (r0 /rL )(1 − α ¯ ), σ) = r0 . Then, whether an intermediary chooses to report the most optimistic model or to invest only in the safe asset, ¯ by dK would bring r0 dK and cost r0 (1 + k)dK, thus there is no incentive to increase increasing K

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¯ by dK would imply a loss of r0 dK for a gain of r0 (1 − k)dK. Hence the capital, while decreasing K possibility to adjust the level of capital leaves the equilibrium unchanged. ¯ we find an equilibrium with µl = 1 and thus (rL /¯ Assume now that with K = K α)s(1 − (r0 /rL )(1 − α ¯ ), σ) > r0 . There are two possibilities. Either we have (rL /¯ α)s(1 − (r0 /rL )(1 − α ¯ ), σ) < r0 (1 + k) and there is no incentive to adjust capital, or intermediaries will issue new equity, so that rL we decrease until we reach an equilibrium with (rL /¯ α)s(1 − (r0 /rL )(1 − α ¯ ), σ) = r0 (1 + k). In all cases there is still a unique equilibrium with some proportion µl of intermediaries reporting ¯ we find an equilibrium with µl < the most optimistic model to the regulator. If with K = K 1, then again increasing α ¯ can lead to more intermediaries reporting the most optimistic model. One exception is the limit case where k = 0, in which case there can be multiple equilibria, in particular with a rigid demand a higher α ¯ can lead intermediaries to increase their capital, or more intermediaries to choose the most optimistic model, or both. Again, this is not a very natural case to consider for the regulation of banks: if capital is not costly for banks then there is no downside in imposing 100% capital ratios in the first place.

B.2

More on benchmarking

Heterogeneous intermediaries. In the first part of the paper there is a continuum of intermediaries who all know the true model σ, hence the idea that the regulator could use reports from several agents to learn the true model, for instance a kind of benchmarking mechanism. I study later in this section how this could be done. In practice however, financial intermediaries and banks in particular are likely to be heterogeneous, and a model adapted to bank A may be unadapted to bank B, depending on the specificities of their customers or how the banks monitor them. Section 4 is perfectly consistent with an economy in which there would be several banks with different types, drawn independently. In section 3 however, it is convenient to assume banks are homogeneous to have a representative bank and a simple market equilibrium. It is nonetheless possible to rewrite the model with several banks with independently drawn types, and I show that we can still define an equilibrium in this case and get results that are qualitatively similar to Propositions 1 and 2. Assume we still have a continuum of banks, the type of each one being drawn from the distribution Ψ(.), draws being independent. For each σ we will have a mass ψ(σ)dσ of banks for which the correct model is σ. The difficulty is that in practice different banks should have different models because they have different pools of customers, something which is difficult to include in the model as we would need to have several markets for different types of customers, possible issues of adverse selection if a customer’s type is private information and so on. To keep things simple, assume that the models differ only because some banks manage their loans better than others, but borrowers are homogeneous. Then borrowers will choose the bank with the lowest interest rate on loans, so

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that in equilibrium all banks that extend loans will have to ask the same interest rate rL . Then a bank for which the correct model is σ and constrained by the regulator to have L higher than some α will have exactly the same profit as in the original model. In particular it is still the case that a bank with a given σ will choose either to remain unleveraged and invest in the safe asset or in loans, or will report the most optimistic model and invest in loans. Moreover, with complete deposit insurance such that rD = r0 , a bank will never choose to have no leverage and invest only in loans. Thus for each type σ we need to find whether banks of this type choose the most optimistic model and maximum leverage or only invest their capital in the safe asset. The profit from using maximum leverage writes π(1 − (r0 /rL )(1 − α ¯ ), σ) = rL L(1 − (r0 /rL )(1 − α ¯ ))s(1 − (r0 /rL )(1 − α ¯ ), σ) and is decreasing in σ due to assumption M1, whereas investing in the safe asset brings a profit of r0 K independent of σ. Then for a given interest rate rL there are three possibilities: either π(1 − (r0 /rL )(1 − α ¯ ), σmax ) > r0 K, in which case all types of intermediaries choose to report the most optimistic model, or π(1 − (r0 /rL )(1 − α ¯ ), σmin ) < r0 K, in which case all types of intermediaries invest only in the safe asset, or π(1 − (r0 /rL )(1 − α ¯ ), σmax ) ≤ r0 K ≤ π(1 − (r0 /rL )(1 − α ¯ ), σmin ), in which case there exists σ ¯ such that for σ ∈ [σmin , σ ¯ ] a bank reports the most optimistic model, and for σ ∈ [¯ σ , σmax ] a bank invests only in the safe asset. There cannot be an equilibrium in which all intermediaries choose to invest in the safe asset only, because then rL would tend towards infinity and investing in loans would be profitable. In an equilibrium in which all banks report the most optimistic model the supply of loans is by definition K/¯ α and thus the interest rate rL (K/¯ α). To have such an equilibrium we need rL (K/¯ α)s(1 − (r0 /rL )(1 − α ¯ ), σmax ) > r0 α ¯ . If this is not the case then there exists σ ¯ and rL that simultaneously solve: Ψ(¯ σ )K = D(rL )¯ α   r0 (1 − α ¯) ,σ ¯ r0 α ¯ = rL s 1 − rL and in such an equilibrium all banks below σ ¯ will report the most optimistic model, all banks above will invest only in the safe asset. Thus Ψ(¯ σ ) is the number of banks choosing the most optimistic model, and plays exactly the same role as µl in the original model. Finally, it is still the case that when the demand for loans is rigid enough a higher α will induce a higher σ ¯ in equilibrium and thus more banks choosing to report the most optimistic model. The important results of section 3 thus do not depend on having homogeneous intermediaries.

A benchmarking mechanism. Imagine now that we keep the assumption that all intermediaries share the same information. Intuitively the regulator could use some kind of benchmarking: a bank could be forbidden to use a model if all other banks use a different one. A difficulty however is that such a mechanism will give rise to multiple equilibria, for instance collusive ones: all banks

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report the most optimistic model, and a bank who would like to report the truth will be identified as lying. Or, if for instance the regulator believes banks reporting models that are more pessimistic than the average, we could be stuck in an equilibrium where all banks report an overpessimistic model. What is needed is a mechanism designed by the regulator such that there is a unique equilibrium, in which all banks tell the truth about the model. The literature on Nash implementation (Maskin (1999)) addresses this class of problems, there is a small complication here due to the fact that banks may default, thus if the regulator uses transfers ex post they may come too late, if she uses transfers ex ante they will typically change the default point of banks and complicate the game. To solve this problem I propose the following adaptation of the canonical mechanism for Nash implementation: Assume the regulator wants to implement the θ∗ (σ) of Lemma 2. Each intermediary i is asked to send a message mi = (σ, θ, T, n) ∈ [σmin , σmax ] × [0, 1] × R × N, the message consists in the true model according to agent i, the θ to be implemented, a transfer T to be paid after losses are realized, conditional on i’s survival, and an integer. The rules are the following: 1. If ∀i, mi = (σ, θ∗ (σ), 0, n) then each intermediary gets the constraint θ(L) ≥ θ∗ (σ) and a null transfer. 2. If ∃i such that mj = (σ 0 , θ∗ (σ 0 ), 0, n) for all j 6= i and mi = (σ, θ, T, k) with σ 6= σ 0 , then: (a) if π(θ, σ)+T F (θ, σ) > π(θ∗ (σ 0 ), σ) and π(θ, σ 0 )+T F (θ, σ 0 ) ≤ π(θ∗ (σ 0 ), σ 0 ), then i will be constrained by θ(L) ≥ θ and receive T if he survives, while all the other intermediaries will be constrained by θ(L) = 1 and receive no transfer. (b) otherwise all intermediaries are constrained by θ(L) ≥ θ∗ (σ 0 ) and receive no transfer. 3. In any other situation, define i∗ = min{i|ki = max kj } and give the θ and the transfer i

j

proposed by i∗ to i∗ , and θ(L) = 1 and no transfers to the others. The intuition of such a mechanism is the following: if all intermediaries tell the same σ to the regulator, she believes them and implements the optimal θ∗ (σ). To prevent a collusive equilibrium, if all intermediaries but one announce the same σ, the regulator considers the message sent by the last intermediary, who can be seen as a “whistle-blower”: the whistle-blower claims that the true σ is different from the one reported by the other intermediaries, and to support his claim is ready for instance to be allowed to lend less, in exchange for some payment in the future if he survives. This is credible if this deal is profitable for the whistle-blower if he tells the truth, but not profitable if the other intermediaries tell the truth. If for any true model and wrong model reported by the intermediaries we can find such a credible message, then the only equilibrium where everybody sends the same message has all intermediaries telling the truth.

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Moreover, there are no other equilibria because of point (3): if we are in a situation where more than two different messages are sent, then the regulator picks the intermediary who has chosen the highest integer and gives him what he wants, or if several intermediaries announce the same integer the one with the lowest index among those. Of course a situation like (3) cannot be an equilibrium since each intermediary who does not have the highest integer would like to deviate. A situation like (2) cannot be an equilibrium either because again sending a third message and announcing a very high integer is a profitable deviation. The only thing we have to check is that for any true model σ, and for any model σ 0 6= σ reported by all intermediaries but one, a whistle-blower can find a θ and a transfer T that make his claim that the true model is σ credible. Assume that σ 0 < σ, that is all intermediaries announce an over-optimistic model. If the whistleblower sends the same message, θ∗ (σ 0 ) will be implemented and he will get π(θ∗ (σ 0 ), σ). Assume he tells the regulator that the true model is σ, agrees to face an infinitesimally higher constraint θ∗ (σ 0 ) + dθ, in exchange for a payment of dθπ10 (θ∗ (σ 0 , σ 0 ))/F (θ∗ (σ 0 , σ 0 )) if he survives. If the whistle-blower is wrong and the other intermediaries are right, then this proposal would give the whistle-blower: π(θ∗ (σ 0 ), σ 0 ) + dθπ10 (θ∗ (σ 0 ), σ 0 ) − F (θ∗ (σ 0 ), σ 0 )

dθπ10 (θ∗ (σ 0 ), σ 0 ) = π(θ∗ (σ 0 ), σ 0 ) F (θ∗ (σ 0 ), σ 0 )

and there would be no incentive to send this message. If the whistle-blower is right, then he would get: π(θ∗ (σ 0 ), σ) + dθπ10 (θ∗ (σ 0 ), σ) − F (θ∗ (σ 0 ), σ)

dθπ10 (θ∗ (σ 0 ), σ 0 ) F (θ∗ (σ 0 ), σ 0 )

and we need to check that this expected profit is higher than π(θ∗ (σ 0 ), σ), which can be obtained by sending the same message as the other intermediaries. This is true if and only if: π10 (θ∗ (σ 0 ), σ)F (θ∗ (σ 0 ), σ 0 ) ≥ π10 (θ∗ (σ 0 ), σ 0 )F (θ∗ (σ 0 ), σ) Using the expression of π10 (equation 20), this condition can be rewritten as: F (θ∗ (σ 0 ), σ 0 ) F (θ∗ (σ 0 ), σ) ≥ s(θ∗ (σ 0 ), σ) s(θ∗ (σ 0 ), σ 0 ) Remember that F = s01 , and due to assumption M1 we have the inequality given in 26: s0012 s−s01 s02 ≥ 0, which implies that F (θ, σ)/s(θ, σ) is increasing in σ. Since σ > σ 0 the inequality above is true, which shows that the whistle-blower indeed gains by proposing this deal. The reasoning is symmetric when σ 0 > σ, the whistle-blower will propose to get a lower θ in exchange for a fee that he will pay in case he survives, the proof is the same replacing dθ by −dθ.  The message of this extension is that, even if one thinks there is only one correct model that

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should be shared by all banks, making sure in practice that all banks report truthfully is not a trivial problem. In particular simple intuitive mechanisms like “imposing the most pessimistic model reported and punishing optimistic reports” are not good enough as they can lead to an equilibrium where all banks report an overpessimistic model. Even in the very simple case developed above, ensuring that the unique equilibrium is truthtelling requires a complicated mechanism and condition M1 is key to the feasibility of such a mechanism. Additional problems will occur in practice: a repeated game element (a whistle-blower may be punished by other banks), the interaction with market prices, the necessity to encourage research on new models. Designing regulatory mechanisms taking as inputs the reports of different banks is an interesting avenue for improving on the current regulation, but it is certainly not the case that the regulator could use a very simple mechanism to fully solve the adverse selection problem studied in this paper.

B.3

Use of market-based measures

The shareholders and bondholders of a given intermediary form their own beliefs about the quality of the intermediary’s assets and may not rely on risk estimates produced by the bank’s internal models. In the limit case where the market has perfect information about the bank’s assets, in other words knows the true model, can the regulator use market prices to detect the use of overoptimistic models? At first sight the answer seems to be yes: if an intermediary reports a model σ 0 so that the regulator will implement θ∗ (σ 0 ), but the market value of the intermediary is different from π(θ∗ (σ 0 ), σ 0 ), the regulator should realize that the intermediary lied and take some corrective measure. However this corrective measure should be anticipated by market participants, and the market value of the intermediary will incorporate the anticipated reaction of the regulator. Taking this into account, it is not obvious that the regulator can use market indicators to discipline the intermediary, as shown in Bond, Goldstein, and Prescott (2010). In this section I show that in the framework of section 4 the regulator can perfectly reveal the intermediary’s true model if she bases her reaction on the intermediary’s market value, while using the price of junior debt is possible if and only if F (θ∗ (σ), σ) is strictly monotone in σ over [σmin , σmax ], which will typically be wrong if [σmin , σmax ] is wide enough. These are not very encouraging results: the assumption that an intermediary’s shareholders know the true distribution and that the market price of the intermediary reflects this information in a timely and stable way is not extremely compelling (if it were true the Basel regulation may not be necessary in the first place). Junior debt may be held by more sophisticated agents spending more resources to estimate the probability that the intermediary defaults, and its price may be a more reliable signal on the quality of the intermediary’s assets. But precisely in this case the existence of a rational expectations equilibrium is not warranted. Moreover in both cases I consider only a simple setting in which rL is exogenous, there is no interaction between an intermediary and its competitors, something that

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would probably make the problem even more complicated for the regulator.

Junior debt. Assume the regulator asks the intermediary to issue a very small quantity of junior debt, whose holders are not covered by any explicit or implicit insurance, and get repaid if and only if the intermediary does not default. Denote rJ the interest rate on this type of debt, assume it is determined on a competitive market, investors know the correct model and the regulator can observe rJ . The game is the following: the intermediary learns the true model σ and reports a model σ 0 . The interest rate rJ is determined on the market such that investors are indifferent between investing in the safe asset or in junior debt, anticipating correctly the θ chosen by the regulator. Finally the regulator receives the report of the intermediary, sees rJ and updates his beliefs about the intermediary’s true model. Based on these beliefs he chooses θ to maximize E(V (θ, σ)|σ 0 , rJ ). In equilibrium an intermediary of type σ will send some report m(σ), the interest rate rJ will depend on the report and the true model, which we can write rJ (m, σ), and the θ chosen by the regulator will depend on the report and on rJ , which we can write θ(m, rJ ). Take as given σ and m. Equilibrium on the market for junior debt implies that: rJ =

r0 F (θ(m, rJ ), σ)

Consider now the regulator’s problem. It is common knowledge that she is supposed to choose θ(m, rJ ) in equilibrium. Thus she infers from the observation of rJ that σ has to be such that F (θ(m, rJ ), σ) = r0 /rJ . Since F (., .) is monotone in its second argument the regulator learns σ, and chooses θ(m, rJ (m, σ)) = θ∗ (σ). Thus we must have: rJ (m, σ) =

r0 F (θ∗ (σ), σ)

If rJ (m, σ) is strictly monotone in σ, that is if F (θ∗ (σ), σ) is strictly monotone, then different values of σ give different observations of rJ and the regulator can indeed infer σ from the observed rJ . Otherwise we have no rational expectations equilibrium: since it must be the case that investors correctly anticipate θ the regulator can infer σ from observing rJ and knowing the θ she is supposed to choose, but she also chooses θ = θ∗ (σ), in which case it is not true that the regulator can infer σ from seeing rJ . Remember that θ∗ (σ) is increasing in σ, while F10 ≥ 0, F20 ≤ 0, thus F (θ∗ (σ), σ) is not necessarily monotone. In other words under the first best regulation it is not necessarily the case that probability of default of a regulated intermediary is monotone in the risk of its assets. Consider an extreme example where model uncertainty is so important that when σ = σmin all loans are repaid with probability 1, while when σ = σmax it is certain that no loan is ever repaid. Then we have θ∗ (σmin ) = 0 and θ∗ (σmax ) = 1, so that F (θ∗ (σmin ), σmin ) = F (θ∗ (σmax ), σmax ) = 1: in both cases there is a zero probability that the intermediary defaults. In this sense a wide range of possible

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models is likely to destroy the possibility of a rational expectations equilibrium.

Market value. Consider now that the intermediary’s shares are publicly traded on a competitive market, and that shareholders know the true model. The reasoning is similar but instead of considering rJ (m, σ) we will consider E(m, σ) the market value of the intermediary. The latter is defined by: E = π(θ(m, E), σ) Again knowing the θ(m, E) she is supposed to choose, the regulator is able to use the expression above to infer σ from the observation of E, so that she will choose θ(m, E(m, σ)) = θ∗ (σ). Then we must have: E(m, σ) = π(θ∗ (σ), σ) but now since π is strictly decreasing in θ and in σ, it is true that E(m, σ) is strictly monotone in σ and the regulator can indeed infer σ from the observation of the intermediary’s market value. It is certainly possible for the regulator to use market-based observations to detect “lying” intermediaries. What this section shows is that it is not trivial. In particular, it is tempting to think in the framework of section 3 that when she observes the equilibrium outcome the regulator could easily learn the true σ by looking at the prices of intermediaries’ shares or bonds. It is true ex-post, but this simple intuition neglects the fact that if the regulator indeed reacts to such observations, then they are anticipated by market participants, which can make these prices less informative, even in the extreme case where market participants are perfectly informed.

B.4

Moral hazard

The paper only deals with an adverse selection problem: an intermediary has some assets and knows the correct model to estimate their riskiness, model that the regulator would like to use. This problem may be complicated by a moral hazard element, as Carey and Hrycay (2001) put it: “investments might be focused in relatively high-risk loans that a scoring model fails to identify as high-risk, leading to an increase in actual portfolio risk but no increase in the banks estimated capital allocations”. It is possible to incorporate moral hazard in the model to study such a question. This section develops a short example. Assume there are two types of assets, simple assets and complex assets. Simple assets are easy to recognize and there is no model uncertainty about them, the distribution of defaults in a portfolio of simple assets is given by F (., σs ). Complex assets are more difficult to analyze and two models are available. With probability q the correct model to analyze these assets is σ1 , and with probability 1 − q it is σ2 , the distribution of defaults is thus either F (., σ1 ) or F (., σ2 ). Assume σ1 < σs < σ2

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and moreover assume that M1 is still satisfied. Thus, due to model uncertainty, the regulator is not sure whether complex assets are better assets than simple assets or not (in particular these assets could suffer less losses on average, but there is uncertainty about whether they have important tail risk or not). Second, assume that an intermediary has a choice between two strategies: if he exerts no effort then with probability p the intermediary will have simple assets and with probability 1 − p complex assets, if he exerts an effort at cost ce > 0 it has complex assets for sure. Imagine for instance that an intermediary would normally sometimes face customers with specific needs for complex products, but can also spend resources trying to convince customers with no need for these products to take them, or to tailor simple products into more complex assets. Keeping the same notations as in the paper, we can focus on the case where: pV (θ∗ (σs ), σs ) + (1 − p)qV (θ∗ (σ1 ), σ1 ) + (1 − p)(1 − q)V (θ∗ (σ2 ), σ2 ) ≥ qV (θ∗ (σ1 ), σ1 ) + (1 − q)V (θ∗ (σ2 ), σ2 ) ⇔

V (θ∗ (σs ), σs ) ≥ qV (θ∗ (σ1 ), σ1 ) + (1 − q)V (θ∗ (σ2 ), σ2 )

In this case if the regulator could observe the choice of the intermediary she would make him choose not to exert any effort. Since σ1 < σs < σ2 this will be the case in particular if q is low enough, that is if there is a high chance that complex assets are actually riskier than simple assets. Now assume that the regulator backtests the models ex post as in section 4. There are two problems to solve: first there is an adverse selection problem, an intermediary with complex assets must reveal whether the true model is σ1 or σ2 , and on top of that there is a moral hazard problem, an intermediary must be given incentives not to try to have complex assets. Assume first we are in the case where models are easy to distinguish, as in Proposition 3. Without the moral hazard problem the regulator would offer menus of penalties such that an intermediary gets π ¯ (σi ), its outside option, when its type is σi . Is it compatible with incentives not to try to get complex assets? If an intermediary does not exert effort it expects to get p¯ π (σs ) + (1 − p)q¯ π (σ1 ) + (1 − p)(1 − q)¯ π (σ2 ), compared to q¯ π (σ1 ) + (1 − q)¯ π (σ2 ) − ce if it does. Moral hazard does not lead the regulator to change anything if the former quantity is larger than the latter, that is if: π ¯ (σs ) + (ce /p) ≥ q¯ π (σ1 ) + (1 − q)¯ π (σ2 ) Again if q is low enough this condition will be satisfied. Since in the adverse selection stage the intermediary will have no rent due to private information and complex assets are on average much worse than simple assets, there is no incentive to choose the former. Assume now that σ1 and σ2 are difficult to distinguish and that the regulator would like to implement θ∗ (σ1 ) and θ∗ (σ2 ) below the point at which it is possible to distinguish the two models, and moreover the regulator is unable to commit to subsidizing intermediaries after a default, as in Proposition 4 and Remark 2. Now, as shown in the proposition, if the intermediary has complex assets the regulator will have to design penalties such that the intermediary gets π ¯ (σ1 ) whether the

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true model is actually σ1 or σ2 . The intermediary will have incentives not to exert effort to get the complex assets if: p¯ π (σs ) + (1 − p)¯ π (σ1 ) ≥ π ¯ (σ1 ) − ce Since π ¯ (σ1 ) > π ¯ (σs ) now even when q is very low only a high cost effort will prevent the intermediary from getting the complex assets. When this is not the case, the regulator will have either to implement different θs, or to leave more surplus to the intermediary when it has simple assets. What we have here is a case where the intermediary gets an informational rent at the adverse selection stage when it has complex assets, so that there is an incentive to invest in assets for which model uncertainty is important, even when on average they are riskier than simple assets.

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