Hedging parameter risk

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Hedging parameter risk Arndt Claußen, Daniel Rosch, Martin Schmelzle ¨ PII: DOI: Reference:

S0378-4266(19)30003-2 https://doi.org/10.1016/j.jbankfin.2019.01.003 JBF 5484

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Journal of Banking and Finance

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19 September 2017 9 November 2018 3 January 2019

Please cite this article as: Arndt Claußen, Daniel Rosch, Martin Schmelzle, Hedging parameter risk, ¨ Journal of Banking and Finance (2019), doi: https://doi.org/10.1016/j.jbankfin.2019.01.003

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Hedging parameter risk Daniel Rösch† January 9, 2019

Abstract

Martin Schmelzle‡

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Arndt Claußen*

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The accurate measurement and effective control of financial risk are of crucial importance to risk managers and regulators. However, risk measures are potentially affected by errors in the estimation of model parameters from limited samples, leading to parameter risk. The key contribution of this paper is the formulation

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of a general framework to hedge this parameter risk. Applying the new framework to credit portfolio modeling, we highlight the importance of parameter risk,

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estimation methods, and diversification effects.

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Keywords: estimation error, parameter risk, hedging JEL classification: C13, G13, G32

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The authors would like to thank seminar participants at Leibniz Universität Hannover, Universität Regensburg, University of Technology Sydney, the 2015 Quantitative Methods in Finance conference

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(Sydney), the 2016 German Operations Research Society (GOR e.V.) working group meeting on Financial Management and Financial Institutions at Universität Augsburg, and the 2016 6th Workshop Banks and Financial Markets at Universität Wien for helpful and valuable comments. We are particularly grateful for the suggestions and comments of the editor Carol Alexander and an anonymous referee.

* Leibniz

Universität Hannover, Institute for Banking and Finance, Königsworther Platz 1, 30167 Hannover, Germany, email: [email protected] † Universität Regensburg, Chair of Statistics and Risk Management, Universitätsstraße 31, 93040 Regensburg, Germany, email: [email protected] ‡ Universität Regensburg, Chair of Statistics and Risk Management, Universitätsstraße 31, 93040 Regensburg, Germany, email: [email protected]

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Introduction

The precise assessment of financial risk with risk measures such as value-at-risk or expected shortfall is of visceral importance for successful risk management. However, risk measures are subject to estimation risk. If estimated from sample data they are essentially estimates of risk—not known with certainty. Treating an estimate as if it

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were the true risk measure oftentimes yields to an underestimation of the true risk. Thus, it is imperative for rational risk managers and prudential regulators to take account of this uncertainty when assessing risk model outcomes.

First efforts devoted to the issue of estimation errors on statistical measures of risk

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exposures in risk management are due to Jorion (1996). Since then, a steadily growing body of literature is emerging. More recently Cont et al. (2010) analyze how estimation uncertainties may affect the calculation of risk measures, study the robustness of risk measurement procedures and their sensitivity to changes in the data set. The impact

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and assessment of parameter and model uncertainty on capital adequacy and risk capital calculations is further studied by, e.g., Schaller (2002), Bao and Ullah (2004),

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Branger and Schlag (2004), Christoffersen and Gonçalves (2005), Pflug and Schaller (2010), Kerkhof et al. (2010), Tarashev (2010), Alexander and Sarabia (2012), Bion-

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Nadal and Kervarec (2012), Embrechts et al. (2013), Lönnbark (2013), Barrieu and Scandolo (2015), Embrechts et al. (2015), Fröhlich and Weng (2015), and Bignozzi

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and Tsanakas (2016a,b) among others. Whereas the meaning of parameter uncertainty is unequivocal, the notion of model uncertainty in the literature remains somewhat

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ambiguous. That is, the uncertainty in the estimates of parameters within a model might also be understood as model uncertainty (Cont, 2006). Regardless the point of view and even if it were possible to neglect model uncertainty issues due to, e.g., regulatory guidelines, the need to specify unknown parameters within a given model class remains an important issue.1 In essence, the literature provides orientation to 1 Therefore

the present paper is not concerned with the hedging of model risk as such.

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ACCEPTED MANUSCRIPT add some ‘conservatism’ to estimates from statistical inference or otherwise provision additional capital buffers to cushion for estimation uncertainties. Ignorance of possible errors in the estimation of risk model parameters comes with important consequences. However, even if financial institutions are fully aware of the possibly harmful impact of uncertain estimates, they may be reluctant to the idea of providing additional and costly risk capital.2 This study marks the first attempt

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in the literature to develop a framework that allows financial institutions to hedge parameter risk in place of fully provisioning for ‘conservative’ risk model parameters. Here, the term hedge is used in its broadest sense, as an offsetting position to potential losses, and parameter risk is the possibility for errors in the parameters. This framework

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enables the involved contract parties to uniquely determine the fair price for parameter risk protection. The underlying idea is reminiscent to a simple one period swap deal. The protection buyer pays an insurance premium upfront in exchange for a specific option-like cash flow or payoff contingent on the future loss of the protected portfolio.

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The range of parameter risk protection is confined by the statistical point estimate of the underlying risk measure and some additional conservatism. Instead of fully

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provisioning for this difference, the protection premium is derived from the statistical expectation of this difference which is only a tiny fraction of the full amount. The

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additional conditional risk capital thus provides an enlarged buffer against unfavorable events thereby greatly reducing potential negative effects on financial institutions’

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behavior. Such a protection becomes increasingly expedient for circumstances characterized by a high chance for parameter errors. This might be particularly pronounced

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for risk models where estimated risk measures, on average, approach the true, but unknown, risk measure from below, compare, e.g., Figlewski (2004) and Lönnbark

(2010). Typically, dependent on the risk measure under consideration, implications of parameter risk become worse with asymmetric distributions and higher tail cutoffs,

where a few, but severe, underestimations harm much more than many small ones. 2 Discussions

on the problems of agency issues from a regulatory or supervisory perspective include incentives to underestimate VaR (Cuoco and Liu, 2006), and decrease the quality of risk management systems (Daníelsson et al., 2002). How a strong and independent risk management is related to (excessive) risk taking is examined by Ellul and Yerramilli (2013).

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ACCEPTED MANUSCRIPT We illustrate our framework applied to credit risk modeling.3 First, we conduct a simulation study. Employing a common credit portfolio model allowing for closed form hedge premium pricing, we find that (i) a protection buyer (seller) has an incentive to engage in such a hedge, if the true parameters are underestimated (overestimated), respectively, (ii) hedge fees increase ceteris paribus with increasing parameter risk, and (iii) a protection seller engaging in more than one hedge contract may diversify

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parameter risk, and decrease its risk of extreme losses. Second, an application of our framework to historical one year default rates reveals that, subject to contract type and rating grade, a financial institution may pay protection premiums for about 460 to 1430 years in comparison to a one time provision of additional risk capital to

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the difference between the VaR using the estimates and the VaR for the conservative parameters. We find strong empirical support that in practical applications highly rated financial instruments are substantially more prone to parameter errors than lower rated instruments, and—as a result—are more costly in relative terms to be hedged

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against parameter risk.

This paper complements previous work on the role and impact of estimation risk

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and parameter uncertainty. Pioneering work on the impact of estimation risk on investment decisions and portfolio selection include Kalymon (1971), and Klein and Bawa

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(1976). Barry and Brown (1985) and Coles and Loewenstein (1988) study effects upon market equilibrium and asset pricing in the presence of uncertain parameters and

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estimation risk. Common approaches to address estimation risk are Bayesian methods like diffusive priors (Barry, 1974) or informative priors formed via asset pricing models

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(Pástor, 2000). Other prominent approaches include ‘robust’ estimation principles (Garlappi et al., 2007). The role of parameter uncertainty in portfolio choice, asset pricing and asset allocation is further investigated in, e.g., Barberis (2000), Veronesi (2000), Xia

(2001), Maenhout (2004), and Kan and Zhou (2007). Other studies pertaining to uncertainty in model parameters include option pricing (Green and Figlewski, 1999; Bunnin 3 Estimation errors of parameters are documented to be particularly pronounced in credit risk,

Crouhy et al. (2000) for instance find evidence that industry credit models yield estimates substantially less accurate in comparison to market risk models.

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ACCEPTED MANUSCRIPT et al., 2002; Psychoyios and Skiadopoulos, 2006; Alexander et al., 2009; Lindström, 2010), credit spreads (Korteweg and Polson, 2010), bond portfolios (Feldhütter et al., 2012), mortgage securitization (Rösch and Scheule, 2014), structural credit modeling (Rodríguez et al., 2015), and foreign exchange rates (Beckmann and Schüssler, 2016) inter alia. Further applications of uncertainty in the parameters to credit risk modeling include, e.g., Löffler (2003), Gössl (2005), Yamai and Yoshiba (2005), Tarashev (2010),

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and Bernard et al. (2017) among others. The majority of these studies, however, do not focus on risk management issues as such.

Our research is also related to the fast growing literature considering the accuracy and backtesting of risk predictions, see, e.g., Berkowitz and O’Brien (2002),

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Berkowitz et al. (2009), Escanciano and Olmo (2010, 2011), Boucher and Maillet (2013), Gouriéroux and Zakoïan (2013), Boucher et al. (2014), and Du and Escanciano (2017) inter alia. These studies mostly provide means to adjust risk model outcomes due to estimation risk and give orientation to derive risk measure buffers. Our work differs

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from this line of research in that we do not adjust estimated risk measures as such but provide additional safety for a prespecified hedge range. Yet, these methodologies

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developed therein might deliver further indications on how to chose an adequate hedge level for practical applications of our framework.

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The rest of the paper is organized as follows. Section 2 introduces our framework to hedge parameter risk. In Section 3, we illustrate the key concepts of our framework,

Theoretical framework

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2

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and discuss real world implications. Section 4 summarizes our findings and concludes.

Let (Ω, F , P) be an atomless probability space and L0 (Ω, F , P) a set of F -measurable P-a.s. finitely valued random variables on that probability space. Definition 1 (Risk of a security). Let M ⊆ L0 (Ω, F , P) be a convex cone, in which any random variable L(Y , θ) ∈ M, with cumulative (discrete) distribution function G(`, θ) = P(L(Y , θ) ≤ `), ` ∈ [0, 1], and probability density (mass) function g(`, θ), describes a loss of a risky security (e.g., loan, bond, stock, portfolio). Y is a random vector and θ is a vector 5

ACCEPTED MANUSCRIPT of model parameters. Then L(Y , θ) models the risk (e.g., market risk, credit risk) of the security. Quantitative tools to aggregate the risk of a security and to express the riskiness of financial positions in one key figure are risk measures. A risk measure is a function that maps risk to the real numbers. Therefore any function R : M → R is a risk

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measure. Statistical risk models often require a number of model parameters θ to be specified. In practical applications θ is unknown and must be determined, e.g., via expert judgments, calibration to market data or estimation from past observations. However, the inferred parameters do not necessarily match the true underlying model

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parameters under a classical statistical perspective.

Definition 2 (Estimates and parameter error). Given the parameters according to Defini

tion 1 are unknown, then any specified parameters are estimates θˆ and the distance

θ − θˆ

p , p ≥ 1, is a parameter error.

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Definition 3 ((Positive/negative) effective parameter error). For a given risk measure R,

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an estimate θˆ implies an effective parameter error if

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ˆ R(L(Y , θ)) , R(L(Y , θ)),

ˆ > 0, R(L(Y , θ)) − R(L(Y , θ))

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and is called positive effective parameter error if

otherwise it is called negative effective parameter error. All estimates lead almost surely to effective parameter errors. In practical appli-

cations, a negative effective parameter error—equivalent to an overestimation of the quantified risk of a security—may imply a reduced possibility of a default, which will be costly for a financial institution due to misallocation of capital. On the contrary, a positive effective parameter error—equivalent to an underestimation of the quantified

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ACCEPTED MANUSCRIPT risk of a security—might increase the possibility of a default because the financial institution will not have enough provisions to cover incurred losses. Definition 4 (Parameter risk). The possibility of an effective parameter error is called parameter risk. Remark 1. Knight (1921) defines the terms ‘risk’ and ‘uncertainty’ as random variation

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according to a known or unknown stochastic law, respectively, whereas the type of desired or undesired outcome remains unspecified. In Definition 4 we introduce parameter ‘risk’, since for the definition of a parameter error we assume the model is known with certainty.

Financial institutions are hurt from positive effective parameter errors, therefore

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recent literature calls to adjust for this particular form of parameter risk by add ons, compare, e.g., Tarashev (2010). However, if these parameters—adjusted for parameter risk—are employed and serve as a basis for the allocation of capital reserves, the provisions may be too costly. Instead of fully providing additional capital reserves, a

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financial institution may benefit from engaging in a hedge of parameter risk.

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Definition 5 (Payoff of parameter risk hedge). Given a risk measure R, an estimate θˆ is

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hedged to the level of θ h for

ˆ ≤ R(L(Y , θ h )), R(L(Y , θ))

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if a protection seller pays the protection buyer the (possibly discounted) payoff

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     ˆ 0 L(Y , θ) ≤ R(L(Y , θ))       ˆ ˆ ≤ L(Y , θ) ≤ R(L(Y , θ h )) pc =  L(Y , θ) − R(L(Y , θ)) if R(L(Y , θ))         ˆ  R(L(Y , θ h )) < L(Y , θ), c (R(L(Y , θ h )) − R(L(Y , θ)))

with c ∈ { 0, 1 }. The contractual parameter c in Definition 5 defines the shape of the payoff structure above the hedge level θ h . Choosing c = 1 the hedge contract is a call spread on the 7

ACCEPTED MANUSCRIPT underlying state variable L(Y , θ). Here, a protection seller offers to bear losses beyond ˆ but subject to a cap R(L(Y , θ h )) and, as such, is basically a certain threshold R(L(Y , θ)), selling mezzanine protection. For c = 0 the hedge contract resembles an up and out call option. After breaking the upper barrier nothing is paid to the protection buyer resulting in a much cheaper fee to be paid for this type of protection. Relaxing the restriction, c ∈ { 0, 1 }, allows for more diverse payoff shapes. However, we interpret c = 1

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and c = 0 as lower and upper bounds of protection above the hedge level, respectively. For both characteristics the compensating payment is restricted by R(L(Y , θ h )) −

ˆ R(L(Y , θ)). Thus, by effectively truncating the payoff the protection seller faces no unpredictable and heavy-tailed worst case scenarios.

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Remark 2. A severe economic state described by an extreme realization of Y could imply a large loss. In the case of c = 1, the contract will payoff regardless whether parameter risk is causal to the loss. Alternatively, a compensation for a possible parameter risk may not seem

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adequate (c = 0) and it is then preferable to liquidate the financial institution. For bearing the parameter risk from the protection buyer, the protection seller

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receives a fair hedge fee, or, equivalently, a fair hedge premium. This premium is the expectation of the payoff, E[pc ], taken under some appropriate probability measure

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(see Remark 3 below).

Theorem 1 (Fair fee fc for a hedge of parameter risk). The fair fee fc = E[pc ] of a

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parameter risk hedge according to Definition 5 is given by f0 = f1 − (D − A) P(D < L(Y , θ)), ZD f1 = D − A − P(L(Y , θ) ≤ `) d`, A

ˆ and D = R(L(Y , θ h )). where A = R(L(Y , θ))

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(1) (2)

ACCEPTED MANUSCRIPT Proof. The payoff p1 corresponds to the difference of a long call with strike A = ˆ and short call with strike D = R(L(Y , θ h )). The contractual payment equals R(L(Y , θ)) p1 = (L(Y , θ) − A)+ − (L(Y , θ) − D)+ . Due to the linearity of the expectation operator we calculate E (L(Y , θ) − A)

+

i

Z

Z

1

1

Z

1

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h

(` − A) g(`, θ) d` = ` g(`, θ) d` − A g(`, θ) d` A A 1 Z 1 1 = ` G(`, θ) − G(`, θ) d` − A G(`, θ) A A A Z1 Z1 = 1−A− G(`, θ) d` = 1 − A − P(L(Y , θ) ≤ `) d`, =

A

A

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A

which follows by integration by parts. Subtraction leads to Equation (2). The payoff p0 ˆ if R(L(Y , θ h )) < L(Y , θ). Due to the linearity equals p1 without R(L(Y , θ h )) − R(L(Y , θ)) of the expectation operator Equation (1) holds.

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Remark 3. Identifying the appropriate probability measure underlying the expectation oper-

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ator in Theorem 1 is a delicate issue and requires a careful consideration of the characteristics of the payoff function in Definition 5. Let Q be the family of martingale measures equivalent

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to the statistical or ‘reference’ probability measure P. When the payoff from a parameter risk hedge can be spanned by the payoffs from liquid assets, then there exists a unique probability

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measure Q∗ ∈ Q on the set of market scenarios (Ω, F ) that is equivalent to P. In other words, the market would be complete. When the payoff under consideration is not within the span of

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traded instruments, pricing issues become more involved. That is, the market is not complete anymore and potentially gives rise to an infinite amount of martingale measures Q, each of which is free from arbitrage. To ensure that the expectation operator in Theorem 1 becomes unique, a number of approaches have been proposed to select a single pricing measure according to some optimality criterion. For instance, there are the minimal martingale measure, the minimal entropy martingale measure, or the Esscher transform to mention a few well-known candidates. We refer to Carr et al. (2001), Xu (2006), or Staum (2007) for excellent reviews on the pricing of contingent claims in incomplete markets. Though these considerations are 9

ACCEPTED MANUSCRIPT of pivotal importance to the pricing of parameter risk hedging in practice, the actual choice of an adequate pricing measure is deliberately not part of our theoretical hedging framework. The protection seller pays on average E[pc ] without knowledge about the parameter θ. However, in practical applications θ is unknown and fair fees according to Theorem 1 are not computable. Thus, we introduce a contractual fee for a parameter risk hedge.

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Definition 6 (Contractual fee fc∗ for a parameter risk hedge). The contractual fee fc∗ is defined as the expectation of the parameter risk hedge payoff for L(Y , θ h ) instead of L(Y , θ). fc∗ accords with fc in Theorem 1 where θ is replaced by θ h .

The Definition 6 implies three advantageous consequences. First, given a hedge

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ˆ to R(L(Y , θ h )) the contractual fee fc∗ is deterministic. Second, the range from R(L(Y , θ)) contractual fee fc∗ trivially equals the fair fee fc if the hedge level θ h matches the latent

θ. And, third, θ h is merely a conservative estimate (or, the result of an add on to an estimate) for θ and as result the hedge framework does not induce any new parameter

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risk.

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Remark 4. Equation (2) reveals similarities to ‘classical’ derivatives pricing. Initially considering θ, we obtain the fair fee previously discussed assuming the true parameters are known. In addition, considering θ h subject to specified contract terms instead of θ, we arrive

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ˆ ultimately yields to at what we label the contractual fee. In conclusion, replacing θ with θ,

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conventional call spread pricing. Remark 5. A protection seller gets into the unique position to directly diversify parameter

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risk across different hedge contracts. If a parameter hedge contract does (not) fulfill the condition

Z

D A

Z P(L(Y , θ h ) ≤ `) d` >

D A

P(L(Y , θ) ≤ `) d`,

(3)

then it follows from Equation (2) and Definition 6 that the fair fee f1 is greater than (less than or equal to) the contractual fee f1∗ and, as result, the protection seller receives on average less than (more than or equal to) what is paid. This condition, given by the areas 10

ACCEPTED MANUSCRIPT under the cumulative distributions functions ranging from A to D, may be interpreted as an underestimation of the true extreme losses which triggers the payout of a hedge contract. In real world applications the condition in Equation (3) cannot be verified for a specific hedge contract, since θ is essentially unknown. However, if a protection seller engages in more than one hedge contract and the contractual terms therein are not perfectly correlated, then there is a positive likelihood that some contracts will fulfill the condition in Equation (3) and

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others will not. Thus, there is a chance that the underestimation of the true risk, represented by the condition in Equation (3), is offset—at least to some extent—with an overestimation of the true risk of some other hedge contract. As result, a protection seller is also able to diversify the parameter risk across several protection buyers.

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In conclusion, the parameter risk hedge framework, in particular Definition 5 and Theorem 1, is independent from the model type, chosen hedge interval, and holds for any risk measure.

3.1

Hypotheses development

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Applied hedging of parameter risk: Empirical evidence

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Guided by the parameter risk hedging framework developed in Section 2, we next assess

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its implications when applied to real world applications. To this end, we formulate

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three key hypotheses:4

Hypothesis 1. “Financial decision makers have an incentive to hedge parameter risk.”

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Central to this hypothesis is the existence of parameter risk within real world applications, where the occurrence of positive effective parameter errors is disproportionately high. In these scenarios financial decision makers benefit from actively managing and hedging parameter risk. Hypothesis 2. “Higher parameter risk leads ceteris paribus to higher hedge fees.” The fee is the expectation of the future payoff (see Theorem 1), and thus, the fee increases c.p. if the 4 We

are very grateful to the anonymous referee for suggesting the development of testable hypotheses for empirical investigations.

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ACCEPTED MANUSCRIPT probability for a payoff increases or in cases where the payoff increases. These two cases are summarized in Definition 4 as parameter risk. Hypothesis 3. “Parameter risk can be diversifiable.” The rationale directly follows from Remark 5. Key to this argumentation is that the contractual terms in different hedge contracts are at least to some extent independent from each other.

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Hypothesis 1 is at the core of the following investigation. If the hypothesis is corroborated, the parameter risk hedge framework offers clearcut advantages compared to the provision of costly add ons to achieve conservative parameters and greater safety. Hypothesis 2 formulates the intuition that higher parameter risk drives hedging fees.

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We note, however, that the inversion is not necessarily true. Finally, the mild restrictions imposed on Hypothesis 3 are unlikely to impede welfare benefits for an economy. More specifically, it would require that two or more protection buyers hold the exact same underlying portfolio, use the exact same estimation method and, in consequence, assign the exact same contractual terms to their respective hedge contracts.

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An examination of these three hypotheses requires knowledge about the true pa-

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rameter. To account for this caveat, we conduct extensive simulation studies with known true and realistic parameter settings with respect to given model and risk types.

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This procedure essentially allows us to assess the sensitivity of a specific model to parameter risk and provide us with the necessary confidence to approach additional

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challenges from real world application without knowledge about the true parameter. We do not claim that these hypotheses are true for all models and all markets under

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consideration. However, our results below provide counterexamples rebutting the alternative hypotheses. 3.2

Incentives, hedge fees and diversification

To investigate our hypotheses, we focus on credit portfolio modeling. This focus is substantiated by the finding that in credit risk parameters and resulting risk measures are—on average—underestimated (Gordy and Heitfield, 2010). More specifically, Tarashev (2010) illustrates the importance of dealing with parameter risk within the asymp12

ACCEPTED MANUSCRIPT totic single risk factor (ASRF) model. This model has clear economic interpretations and offers closed form solutions for parameter risk hedge premiums. Underpinning the Basel internal ratings-based (IRB) approach (BCBS, 2006), the ASRF model is well known to academics and practitioners. Thus, we consider the ASRF model an ideal candidate to explore our Hypotheses 1–3.

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Asymptotic single risk factor model and test environment Losses on a homogeneous credit portfolio are assumed to be driven by a systematic risk factor, and each asset represents an infinitesimal share of this portfolio. Idiosyncratic risk disappears owing to full diversification. The distribution of credit portfolio loss in a given period

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equals    Φ −1 (π) − √ρ Y   , L(Y , [ρ, π]) = Φ  p  1−ρ

i.i.d.

Y ∼ N (0, 1),

(4)

where π ∈ (0, 1) is an unconditional probability of loan or bond default, ρ ∈ (0, 1) is the

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asset (return) correlation, Y is a standard normally distributed systematic risk factor, and Φ is the standard normal distribution function (with Φ −1 denoting its inverse).5

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Given the analytical tractability of the Gaussian ASRF, hedge premiums are straight-

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forward to derive and can be expressed in closed form.6 Corollary 1 (Parameter risk hedge in the Gaussian ASRF). Equation (2) from Theo-

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rem 1 for the Gaussian ASRF is the difference of two bivariate standard normal cumulative

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distribution functions

h i     E p1 = Φ2 −Φ −1 (A) , Φ −1 (π), % − Φ2 −Φ −1 (D) , Φ −1 (π), % ,

(5)

with p % = − 1 − ρ,

  ˆ , ˆ π]) A = R L(Y , [ρ,

and

5 For

  D = R L(Y , [ρh , πh ]) ,

simplicity, we skip the notion of a nonzero recovery rate. Section C of the Internet Appendix, we additionally consider non-Gaussian generalizations to the original Gaussian ASRF to analyze the model sensitivity of hedge fees. 6 In

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ACCEPTED MANUSCRIPT where Φ2 (·, ·; %) denotes the bivariate standard normal cumulative distribution function with correlation parameter %.   Additionally P D ≤ L(Y , [ρ, π]) in Equation (1) is   h i h i  % Φ −1 (D) + Φ −1 (π)   . E p0 = E p1 − (D − A) Φ  √  ρ

(6)

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Proof. Using Equation (30c) in Andersen and Sidenius (2005).

In the sequel, we consider the common value-at-risk (VaR) measure, also underlying the IRB approach,

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   Φ −1 (π) − √ρ Φ −1 (1 − α)   . VaRα (ρ, π) = Φ  p  1−ρ

As test environment, we consider a large homogeneous credit portfolio with true asset correlation of ρ = 20%, which lies well in between the lower and upper bounds

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found in the Basel Accords. The unconditional probability of default (π) is set to one percent, roughly corresponding to historical one year default rates of Moody’s Ba rated

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corporates (compare Table 2 below).

To embed parameter risk in our analysis, we sample credit losses according to

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Equation (4) for time horizons of T = 15 and T = 30 periods. Thus, for 15 and 30 years we observe realized losses (i.e., default rates) and estimate the parameter vector

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ˆ Again, the ASRF models offers the benefits of an analytical solutions for ˆ π]. θˆ = [ρ, ˆ (Duellmann et al., 2010). the estimates θˆ with corresponding standard errors σˆ (θ)

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Thus, we avoid any errors resulting from iterative or Monte Carlo based estimation procedures. However, due to the limited number of observations for each repetition the estimates will result in positive or negative parameter errors almost surely.7 For this illustration, we define the hedge parameter as the sum of the estimate and its standard error weighted by a factor κ = 75%. Although the ASRF is a highly nonlinear model, the value-at-risk with confidence level set at 99.9%, in line with the Basel

Accord, is monotonously increasing in θˆ for the considered parameter setup. Hence, 7 Section

A in the Internet Appendix illustrates the resulting errors for this and other scenarios.

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ACCEPTED MANUSCRIPT ˆ meets the requirement of the inequality in Definition 5. For the chosen θ h = θˆ + κσˆ (θ) each repetition out of 104 samples, we calculate the contractual fee f ∗ and true hedge fee f according to Corollary 1 for c = 0 and c = 1. These premiums are deterministic with respect to a chosen hedge range, however, the hedge range is stochastic because ˆ it depends on the random realization of the estimates θˆ and standard errors σˆ (θ). Therefore, for each single realization, we obtain different fair and contractual hedge

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premiums.8

Incentives and hedge fees The distribution of estimates and resulting hedge premiums enables an in depth analysis of Hypothesis 1 and Hypothesis 2. Figure 1 shows

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the realization of the distributions for the VaR sorted in ascending order and the corresponding fair fees and contractual fees for the two contract types c ∈ { 0, 1 }. In the left panel, we plot the distributions of the hedge range given by the difference of the lower attachment point A fixed by the VaR at the estimates as well as the upper detachment point D confined by the VaR at the hedged parameter. The vertical, dashed

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gray lines indicate the intersections where the estimated and hedged VaR equal the

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true VaR. These crossing points separate the simulated distributions in each subplot into three different areas labeled (I), (II), and (III).

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(I) A ≤ D ≤ VaR(θ). From a hedging perspective, the product is beneficial, since the

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true risk is clearly underestimated (intersection from D to VaR(θ)). (II) A ≤ VaR(θ) ≤ D. This case describes a situation where the true risk is—on

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average—slightly overhedged. Without the hedge product, however, the true risk would remain underestimated.

(III) VaR(θ) ≤ A ≤ D. The true risk is overestimated. In the right panel, we plot the contractual fee fc∗ in comparison to the resulting fair

fee fc for the two contractual types c ∈ { 0, 1 }. The passage from case (I) to (II) occurs, if 8 During

our case studies, we focus on the essentials of the parameter risk hedge framework and refrain from discussing possible influences due to risk preferences of economic agents. That is, we consider zero risk premiums and undiscounted payoffs.

15

ACCEPTED MANUSCRIPT Hedge range, T = 15

0.4 0.35

(I)

(II)

3 ^ VaR(3) VaR(3h ) VaR(3)

(III)

(Contractual) fees, T = 15

#10 -4 (I)

(II)

f0$ f0 f1$ f1

(III)

2.5

0.3

Fee

Value-at-Risk

2 0.25 0.2

1.5

0.15 1 0.1 0.5 0.05 0 0.1

0.2

0.3

0.4 0.5 0.6 0.7 relative frequency of occurrence

0.9

1

0.1

Hedge range, T = 30

0.4 0.35

0.8

(I)

(II)

3 ^ VaR(3) VaR(3h ) VaR(3)

(III)

0.2

(I)

(II)

0.3

Fee

1.5

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Value-at-Risk

2 0.25

0.15

0.4 0.5 0.6 0.7 relative frequency of occurrence

0.8

0.9

1

(Contractual) fees, T = 30

#10 -4

2.5

0.2

0.3

CR IP T

0

(III)

f0$ f0 f1$ f1

1

0.1

0.5

0.05 0

0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 relative frequency of occurrence

0.8

0.9

1

0.1

0.2

0.3

0.4 0.5 0.6 0.7 relative frequency of occurrence

0.8

0.9

1

ED

M

Figure 1: Effects of parameter risk on hedge premiums. The left figure shows the distribuˆ (solid lines) and D given by VaR(θ h ) (dashed lines) in comparison to tions of A given by VaR(θ) the true VaR(θ) (horizontal lines) at confidence level α = 99.9% for θ = [ρ, π] with ρ = 20% and π = 1%. The right figure shows the contractual fee fc∗ in comparison to the resulting fair fee fc for the two contract types c = 0 (gray) and c = 1 (black). The solid and dashed lines plot the mean values, the dotted lines depict the corresponding 10% and 90% quantiles, respectively.

PT

fc = fc∗ . That is, in this particular case, when the upper hedge range D correspond to

CE

the true VaR, then the hedge premium—irrespective of the contractual type—naturally equals the fair fee.

AC

(I) The fair fee (expected payoff) is greater than the contractual fee. Recalling from the left panel that a parameter hedge is necessary, the protection buyer profits in two respects. Protection is needed, and the net present value for the protection buyer is positive.

(II) A parameter hedge is necessary, however, this comes at the cost of a slight overprotection. Thus, the net present value is negative. (III) Any unit of extra protection will disproportionately drive associated cost. 16

ACCEPTED MANUSCRIPT The three cases each make up roughly one third of the relative frequency of occurrences. Further, consistent with Hypothesis 1, we observe that in about two out of three possible scenarios hedging the parameter risk is beneficial for the protection buyer. Moreover, it is easily seen that hedging parameter risk is a very compelling alternative to provision extra add ons as safety buffers. For instance, in the case of T = 15 years the estimated VaR at α = 99.9% equals 10.17% whereas the hedged VaR is 14.53% which

CR IP T

results in a difference (or add on) of 4.36%. For this additional safety the protection buyer has to pay a fee of 0.491 basis points. Setting these two numbers into perspective, a protection buyer could pay the fee for 888 ≈

436 bp 0.491 bp

years instead of increasing the

risk capital once to account for the 4.36% in additional safety c.p. In the case of T = 30

AN US

years this would even translate to about 1341 years of paying a fee instead of provision for add ons. This is a simple consequence of the smaller estimation errors for longer estimation periods with more observational data available which in turn reduces hedge fees.

M

This closely relates to Hypothesis 2 which postulates that the hedge fee should c.p. increase with higher parameter risk. With T getting smaller, we observe a right shift

ED

of the three divisions in Figure 1. This shift implies, that for higher parameter risk a hedge becomes increasingly reasonable for the protection buyer.

PT

Furthermore, for the contractual type c = 1, the product is more vulnerable to parameter misspecifications viewed from both sides of protection, which becomes

CE

evident through the much steeper gradients in the right panel in comparison to c = 0. To provide an economic interpretation, we ascertain that sampled risk measures are,

AC

on average, underestimating true risks. Hence, risk measures themselves are subject to estimation risk which may threaten the solvency of individual institutions. One remedy is to provide additional risk capital to buffer unfortunate consequences resulting from parameter risk.9 However, risk capital is costly since this capital is not available for 9 However,

Hellmann et al. (2000) demonstrate that capital requirements, while putting bank equity at risk and inducing prudent bank behavior, at the same time adversely affect banks’ franchise values, hence stimulating gambling incentives. A similar argument is made by Keppo et al. (2010), who show that higher regulatory capital requirements may postpone recapitalization, thereby actually increasing the institution’s probability of default. Consequently, additional capital burdens to achieve a greater level of safety may negatively affect financial institutions’ behavior.

17

ACCEPTED MANUSCRIPT further investment or lending activities and consequently might hinder social welfare. To enable financial institutions to hedge parameter risk provides an enlarged capital buffer against adverse economic events thereby reducing potential harmful effects on institutions’ behavior. Diversification As demonstrated by the analysis presented in Figure 1, the product

CR IP T

translates the realized parameter risk of an underlying portfolio to a positive or negative difference of the fair fee fc to the contractual fee fc∗ . Moreover, as argued at the end of Section 2, a protection seller may diversify this risk by engaging in more than one hedge contract with different protection buyers. A requirement for such diversification

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is that the hedge contracts are at least to some extent independent of each other. Now, in order to address Hypothesis 3 we proceed as follows.

We extend the ASRF framework following Pykhtin and Dev (2002) to demonstrate this diversification effect. The systematic single risk factor Y in Equation (4) now turns into a sectoral risk factor. Thus Y j is composed of a super-systematic factor

M

Y ∗ , representing the overall macro-economy, and a sector j specific risk factor U j ,

ED

according to

PT

√ √ Y j = δ Y ∗ + 1 − δ Uj,

CE

where Y ∗ and U j are i.i.d. standard normal. This extension allows modeling credit risky portfolios covering different sectors and the linear dependence structure between

AC

the sectors is explicitly described by δ ∈ [0, 1]. If δ equals one, all sectors are perfectly correlated and are only driven by the super-systematic factor, while if δ equals zero, the super-systematic factor has no impact on the firms and they are only jointly affected by the sector-specific risk factor and become independent from each other. Next, we assume that the protection seller engages in two hedge contracts, covering two credit risky portfolios. These portfolios only differ in sectors, all other settings being equal. The true parameters of each portfolio is ρ = 20% and π = 1%. However, again these parameters are unknown and have to be estimated based on observable 18

ACCEPTED MANUSCRIPT (Contractual) fees, T = 15

#10 -4

Color / = 0%

Color / = 0%

Line Style f *1

/ = 50%

5

f0

/ = 75% / = 100%

f1

/ = 75% / = 100% 4

Fee

4

Fee

(Contractual) fees, T = 15

#10 -4

f *0

/ = 50%

5

6

Line Style

3

3

2

2

1

1

0

0 0.1

0.2

0.3

0.4 0.5 0.6 0.7 relative frequency of occurrence

0.8

0.9

1

0.1

0.2

0.3

0.4 0.5 0.6 0.7 relative frequency of occurrence

0.8

0.9

CR IP T

6

1

∗,1 ∗ Figure 2: Diversification risk. This 6figure shows the contractual + (Contractual) fees,of T = parameter 30 (Contractual) fees, T = 30 fee fc = fc #10 #10 6 Color Color Line Style Line Style 1 + f 2 for the two contract types fc∗,2 in comparison to the resulting fair fee f = f c = 0 (left c c / = 0% / = 0% f c f / = 50% / = 50% 5 5 graph) and c = 1 (right graph), for a protection seller engaging in two hedge contracts. f f -4

-4

* 0 0

/ = 75% / = 100%

* 1 1

/ = 75% / = 100%

4

4

Fee

3

3

AN US

Fee

losses for T = 15 years using the analytical estimation method outlined in Duellmann et al. (2010). For both contracts, the hedge parameter is the sum of the estimated 2

2

parameters and their standard errors weighted by κ = 75% and the VaR is calculated 1

1

at confidence level α = 99.9%. For each sample, we calculate the contractual and fair 0

0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 relative frequency of occurrence

0.8

0.9

1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

relative frequency occurrence fee for each contract. The protection seller receives fc∗ = fc∗,1 + fc∗,2 ofand has to pay on

M

average fc = fc1 + fc2 , where fc∗,i and fci is the contractual and fair fee of hedge contract

ED

i ∈ {1, 2}. We repeat this sampling procedure 104 times for δ ∈ { 0%, 50%, 75%, 100% } and Figure 2 summarizes the resulting fees with similar interpretation as in Figure 1.

PT

If δ equals one, both portfolios are solely driven by the super-systematic factor and result in the same default rates. Hence, all estimated parameters and hedge parameters

CE

are the same, leading to exactly the same fees. Therefore the figure equals the graph presented in the upper-right hand graph in Figure 1, with all values simply doubled.

AC

However, with δ < 1 the resulting default rates increasingly differ and, as a result, so do the corresponding hedge fees. To quantify and compare potential diversification benefits for different degrees of δ,

we consider perfect correlation with δ = 1 as a reference case (i.e., no diversification). The differences from the fair fees (dashed lines) to the corresponding contractual fees (solid lines) depicts the actual parameter error, i.e., the higher the difference, the higher the error. To aggregate the errors over the relative frequency of occurrences, we calculate the (absolute) areas Aδ confined by these two lines. Next, we compare the 19

ACCEPTED MANUSCRIPT Table 1: Diversification benefits. This table reports diversification effects of parameter risk. A −A(δ=1) The diversification benefit Dc is given by (δ<1) , where A is the (absolute) area between the A(δ=1) fair and contractual fees for two contract types c = 0, 1. Entries are in percentages. δ

D0

D1

0.75 0.50 0.00

−11.38 −25.73 −42.87

−11.37 −28.56 −49.03

The results of the diversification benefit Dc =

A(δ<1) −A(δ=1) A(δ=1)

CR IP T

resulting areas from the remaining δ and relate them to the base case given by δ = 1. are summarized in Table 1.

All six cases exhibit the potential for diversification effects. Even for a relatively high correlation with δ = 0.75, the average (absolute) deviances from fc to fc∗ diminish

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by more than 11% for both contract types. This gets more pronounced with decreasing values for δ up to the edge case with δ = 0, where we observe a reduction of premium differences of nearly 50%. Furthermore, for contract type c = 1 and the scenario under consideration, there seems to be more potential for diversification effects in comparison

M

to c = 0 with decreasing δ. This is important given that the fee is for c = 1 more sensitive to parameter misspecification as discussed above.

ED

In conclusion, we see with lower correlation between sectors that the difference of the fair fee and contractual fees decreases. As a result the protection seller can diversify

PT

the parameter risk across two protection buyers. Some degree of independence of the premium contracts may additionally emerge, if the protection buyer employs different

CE

modeling approaches, differ in implementation details or have differing data quality. The empirical evidence in support of Hypothesis 3 has the following economic

AC

implications. For one, the protection seller is able to improve his risk profile since by

virtue of diversification the overall risk of the portfolio is potentially smaller than the sum of the individual risks. Furthermore, in light of financial turmoils, enhancing the resilience of the financial system as a whole is of great concern great concern for prudential regulatory authorities and policy makers. The pooling of hedging activities may contribute to the reduction and control of systemic risks and thus potentially benefits the whole society. 20

ACCEPTED MANUSCRIPT Table 2: Summary statistics of one year corporate default rates. This table reports descriptive statistics for historical one year default rates in percent from the Moody’s annual default study 2015 (Ou et al., 2015). CV denotes the coefficient of variation defined as the ratio of the standard deviation to the mean. ø (#) denote the average (total) number of rated (defaulted) debt. The sample period is from 1985 to 2014.

3.3

A

Baa

Ba

B

Caa–C

– – – – – –

0.043 0.166 381.149 – – 0.687

0.048 0.114 237.021 – – 0.518

0.191 0.319 167.050 – 0.033 1.147

1.097 1.204 109.671 – 0.732 4.932

4.754 4.179 87.899 – 4.268 16.104

18.802 13.249 70.469 – 15.201 61.905

418 6

836 14

786 46

485 151

724 762

228 1,035

113 –

Real world application to historical defaults

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ø #

Aa

CR IP T

Mean SD CV Min. Med. Max.

Aaa

We now apply our framework to historical one year default rates from the Moody’s annual default study 2015 (Ou et al., 2015). The use of one year default rates is

M

supported by, e.g., the requirements of the current Basel Accords stating that “PD estimates must be a long-run average of one-year default rates for borrowers in the

ED

grade” (BCBS, 2006). We consider a 30 year time span ranging from 1985 to 2014. The number of rated companies documents an increase from 1,608 in 1985 to 5,340 in 2014.

PT

On average, we observe 3,590 rated companies per year. Table 2 provides descriptive statistics of the default rates for each rating grade.

CE

The mean of the historical one year default rates increases with rating grade, while the coefficient of variation (CV) decreases. Thus highly rated debt exhibits much

AC

greater uncertainty surrounding its default rates (and consequently might carry higher parameter risk) than lower rated debt. Based on this real world data set, we are now faced with a number of challenges

to cope with. In contrast to the asymptotic ideal of the ASRF in the simulation study above, real world portfolios are not perfectly fine grained. Therefore we consider a Gaussian finite pool model, where the loss LN of a homogeneous portfolio consisting of

21

ACCEPTED MANUSCRIPT N obligors is distributed as ! k X  N −i N P [LN ≤ k] = π(y)i 1 − π(y) φ(y) dy i −∞ Z

∞

i=0

where π(y) = Φ

√ Φ −1 (π)− ρy

√

1−ρ

with k ∈ 0, 1, . . . , N ,

! is the conditional default probability of each obligor and

φ(.) denotes the density function of the standard normal distribution.

CR IP T

Since we regard the finite pool model as the true model, we are not subject to model risk.10 However, we have no chance to gain any knowledge about the true model parameter. Therefore, we can only calculate the contractual fees and have no insights whether we over- or underestimate the risk of the portfolio. Remembering the results

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from the simulation study summarized in Figure 1, we know that we almost surely overor underestimate the true risk. From the perspective of a single financial institution, we have to keep in mind that we are looking at a single realization of the distribution now. More precisely, in two out of three possible cases, we may face an underestimation

M

of the true risk. Transferring this ballpark figure to the Moody’s data set means that in about four to five portfolios out of the seven available rating grades, we have to deal

ED

with an underestimation.

Additionally, in each rating grade, there is at least one year with zero defaults,

PT

which makes the asymptotic case inapplicable. Another data driven consequence is that we employ two different estimation methods, since the low number of default data

CE

for high rated risk buckets is too sparse for conventional statistical inferences. Guidance how to distinguish between low default and ‘normal’ default debt pools

AC

comes from, e.g., the FCA (2016) in BIPRU 4.3.95 “[. . . ] a firm’s internal experience of exposures of a type covered by a model or other rating system is 20 defaults or fewer [. . . ]”. For this reason, we apply the most prudent estimation principle detailed in Pluto and Tasche (2005) for the rating grades Aaa to A. For the remaining rating grades Baa to Caa–C, we apply the maximum likelihood estimation method outlined in Frey and McNeil (2003). 10 Section C of the Internet Appendix provides a sensitivity analysis of hedge fees for alternative specifications of credit risk models.

22

ACCEPTED MANUSCRIPT For the low default rating grades we infer the probability of default as an upper confidence bound for a given confidence level γ such that the maximum value of πˆ from the set of all admissible values of π fulfills the inequality 1 − γ ≤ P[No more then k defaults observed].

CR IP T

Here, we infer the upper bound of a confidence interval, πγ , with the extension of the most prudent estimation principle to correlated default events in a multi-period setting. One restriction is that this estimation principle solely calibrates the probability of default. Thus we have to set ρ. In BCBS (2006) ρ is floored and capped by 12% and

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24%, respectively. That is why we determine the most prudent probabilities of default for these two values of ρ. To define the hedge range we apply the estimation principle to two different confidence levels of γ. As a proxy to an πˆ estimate, we consider a value of γ = 50% and πh is approximated by γ = 90% to add some conservatism to our estimate and serves as the hedge parameter. Table 3 summarizes the estimated π as

M

functions of the confidence level γ. The third and fourth column show the estimates for

ED

π corresponding to two correlation assumptions for each rating grade for two different levels of γ. Table 3 These two upper confidence bounds proxy the parameter estimate

PT

and hedge parameter. They also span the hedge range shown in column five and six for the VaR at confidence level α = 99.9%. The last two columns report the contractual

CE

fees fc∗ for a parameter risk hedge in basis points. Table 4 summarizes the results of the maximum likelihood approach (Frey and

AC

McNeil, 2003) for the rating grades Baa to Caa–C. The second to fifth column show the estimates for ρ and π for each rating class and their corresponding standard errors ˆ We hedge the estimates by a weighted standard deviation with κ = 0.75. ˆ and σˆ (π). σˆ (ρ)

Assuming that the estimator follows a normal distribution, the true parameter would then be less then or equal to this hedge level with a probability of 77.34%. The sixth and seventh column show the VaR for a confidence level at 99.9% at the estimates θˆ and the hedge level θ h . Finally, the eights and ninth column report the contractual fees fc∗ for a parameter hedge in basis points. 23

ACCEPTED MANUSCRIPT Table 3: Contractual fees for historical one year default rates for low default risk buckets. This table reports most prudent estimates for low default risk buckets Aaa to A. Upper confidence bounds for probability of default estimates for two confidence levels γ ∈ { 50%, 90% } are denoted by πγ . Entries are in percentages but the contractual fees fc∗ which are reported in basis points. Rating

ρ

π50%

VaR(π50% )

π90%

VaR(π90% )

f0∗ [bp]

f1∗ [bp]

12 24

0.057 0.056

0.082 0.112

1.770 3.540

2.655 4.425

0.118 0.070

0.151 0.155

Aa

12 24

0.057 0.061

0.088 0.117

1.435 2.632

1.675 4.306

0.017 0.168

0.039 0.315

A

12 24

0.060 0.065

0.099 0.131

1.196 2.632

1.794 4.426

0.068 0.195

0.114 0.370

CR IP T

Aaa

Baa Ba B Caa–C

14.779∗∗ 13.979∗∗∗ 23.459∗∗∗ 13.812∗∗∗

6.333 4.433 5.188 3.848

πˆ

ˆ σˆ (π)

0.195∗∗∗ 1.110∗∗∗ 4.932∗∗∗ 17.932∗∗∗

0.066 0.252 1.061 2.088

ˆ VaR(θ)

VaR(θ h )

f0∗ [bp]

f1∗ [bp]

3.435 11.546 43.232 60.965

5.471 15.464 51.934 67.983

0.245 0.393 0.780 0.805

0.443 0.766 1.650 1.477

and ∗∗∗ denote significance at the 10%, 5%, and 1% levels, respectively

ED

∗ , ∗∗ ,

ˆ σˆ (ρ)

M

ρˆ

Rating

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Table 4: Contractual fees for historical one year default rates. This table reports maximum ˆ of each risk bucket for the rating grades Baa to Caa–C. The VaR ˆ π] likelihood estimates θˆ = [ρ, ˆ are evaluated for a confidence level of at the estimate θˆ and hedge level θ h = θˆ + 0.75 σˆ (θ) α = 99.9%. Entries are in percentages but the contractual fees fc∗ which are reported in basis points.

In Table 3 and Table 4 the estimates πˆ are increasing with decreasing rating grades

PT

and compare well to the average default rates reported in Table 2. Due to the most prudent estimation principle, the probabilities of default for the first three ratings

CE

grades do not differ substantially and are increasing for the upper correlation level. For the rating grades Baa to Caa–C the asset correlation is also estimated and, with

AC

the exception of rating grade B, we observe that the asset correlations ρˆ are higher for higher rated risk buckets. For these rating classes the risk measures at the estimates

ˆ are strictly increasing with declining credit quality.11 However, we do not see VaR(θ) such behavior for the low default rating grades. The reasoning behind this is, that the value-at-risk in the correlated binomial model takes into account the absolute number 11 The

overall magnitudes of the risk measures may seem somewhat high, however, as noted above we do not consider nonzero recovery rates. On average, annual recovery is roughly about 38–42%, however, it is well documented that recovery rates tend to be quite volatile over time and are subject to the kind of provided bond collateral (secured, unsecured, subordinated), see, e.g., Altman et al. (2005).

24

ACCEPTED MANUSCRIPT of entities in each rating grade under consideration. Comparing Table 2, the highest rating grade has 113 entities, Aa rating grade has 418 entities and A has 836. With a greater number of entities idiosyncratic risk increasingly diversifies and consequently the risk measures decrease. However, if all three rating grades had the same number of entities, we would observe an increasing effect in the risk measures for these rating grades.

CR IP T

Given the sparsity of default events in the higher rated risk buckets, estimated standard errors are large in relation to the parameter estimates. By comparison, the ˆ Note, however, the pa(relative) standard errors for ρˆ are more pronounced than for π. rameter sensitivity in π of the hedge premiums is about twice than for ρ. Thus, there is

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no clearcut distinction between the importance of the parameters under consideration. For the resulting risk measures, the values for the hedge parameters are higher than the values for the estimates by construction. If a financial firm were to interpret the hedge level as conservative parameter estimates and fully provide capital reserves

M

to reduce the probability of (unexpected) losses, this would come at high costs, e.g., ˆ = 3.4351%. compare for the Baa rated risk bucket a VaRα (θ h ) = 5.4707% to a VaRα (θ)

ED

If, instead, the same firm engages in a parameter hedge, it could achieve the same security level for a periodic contractual premium payment amounting to f0∗ = 0.2448 bp

PT

or f1∗ = 0.4432 bp.12 The calculated contractual hedge fees—based on the hedge level determined by the parameter estimation errors—tend to result in higher absolute

CE

premiums with lower rating grades. We find, however, higher relative premiums fc∗ / VaR(θ h ) for higher rating grades. The relative premiums (in percent, c = 0) for the

AC

Baa through Caa–C rating grades for the VaR are 0.0447, 0.0254, 0.0150 and 0.0118, respectively. Thus, we conclude, highly rated risk buckets are more prone to parameter risk than lower rated risks buckets.13 The same observation also holds for c = 1. 12 Though

we regard A a low default risk bucket, we also estimate the corresponding maximum likelihood estimates (standard errors) yielding ρˆ = 20.0577% (13.2795%), πˆ = 0.0567% (0.0310%). The estimated correlation is well in between the 12% and 24% bounds we consider for the low default ˆ = 1.9139% to VaRα (θ h ) = 4.1866%. The upper hedge level estimations. The hedge range is from VaRα (θ) is now more than double than the risk measure at the estimate. The hedge premiums would yield to f0∗ = 0.3153 bp and f1∗ = 0.5276 bp. 13 This observation is in contrast to the findings from Section A in the Internet Appendix, where we assume the same (relative) error leading to higher relative fees for higher rating grades are smaller.

25

ACCEPTED MANUSCRIPT Summarizing, we find strong empirical support that in practical applications highly rated financial instruments are more prone to parameter risk than lower rated instruments, and—as a result—are more costly in relative terms to be hedged against parameter risk. Further, we exemplify different estimation methods and highlight the significant effects a departure from the ASRF ideal might have on real world credit portfolios. Finally, our analysis suggests that the parameter risk hedging framework is quite

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robust to challenges from real world applications where the true model parameters inevitably remain hidden.

4

Conclusion

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The possibility that inferred model parameters deviate from the true model parameters— termed parameter risk—may significantly affect calculated risk measures. We introduce a framework allowing economic agents to hedge this parameter risk. It is easily applicable to arbitrary types of risk, is independent from the model type, chosen hedge

M

interval, and works for any risk measure.

ED

We show that the possibility to hedge potential parameter errors up to a certain level may drastically reduce costs in comparison to provisions. This may have important economic implications in light that additional capital requirements may influence

PT

institutional’s behavior and potentially provide adverse incentives. Further, we demon-

CE

strate that prospective protection sellers may get into the unique position to diversify parameter risk. This may have significant policy implications as the consequences from

AC

beneficial diversification might provide positive macro side effects to an economy as a whole.

For practical applications, parameter risk hedging hinges on institutions willing

to sell protection and are capable to steadily provide protection even during periods of high uncertainty surrounding the state of the economy. In case of privately owned enterprises such as commercial banks or hedge funds, prospective protection buyers may be subject to counterparty credit risk and loose protection when it is most needed. This might lead to cascading domino effects to be avoided at all costs. Other prospects 26

ACCEPTED MANUSCRIPT could be federal institutions or state funds who themselves are less vulnerable to get into financial distress. These considerations and the possible embodiment of a ‘parameter risk hedge funds’ form an important line for future research. The empirical analysis in this paper focuses on credit risk modeling. Future applications could include other loss categories like, e.g., market risk or operational risk. Additionally, it would be of great relevance to compare different estimation methods

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and survey feasible approaches to identify appropriate hedge levels. The framework may further be a promising addition to the toolset for model validation, discriminate between parameter and model risk or analyze effects of hedged parameters on stress

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testing.

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M

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PT

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CE

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AC

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ED

risk models with desk-level data’, Management Science 57(12), 2213–2227. Bernard, Carole, Ludger Rüschendorf, Steven Vanduffel and Jing Yao (2017), ‘How

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robust is the value-at-risk of credit risk portfolios?’, The European Journal of Finance

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measurement’, Journal of Risk 18(3), 1–24. Bignozzi, Valeria and Andreas Tsanakas (2016b), ‘Parameter uncertainty and residual estimation risk’, The Journal of Risk and Insurance 83(4), 949–978.

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ACCEPTED MANUSCRIPT Boucher, Christophe M., Jón Daníelsson, Patrick S. Kouontchou and Bertrand B. Maillet (2014), ‘Risk models-at-risk’, Journal of Banking & Finance 44, 72–92. Branger, Nicole and Christian Schlag (2004), Model risk: A conceptual framework for risk measurement and hedging, Working Paper, Goethe-Universität Frankfurt am Main.

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Bunnin, F. O., Y. Guo and Y. Ren (2002), ‘Option pricing under model and parameter uncertainty using predictive densities’, Statistics and Computing 12(1), 37–44.

Carr, Peter, Helyette Geman and Dilip B. Madan (2001), ‘Pricing and hedging in

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incomplete markets’, Journal of Financial Economics 62(1), 131–167.

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Crouhy, Michel, Dan Galai and Robert Mark (2000), ‘A comparative analysis of current credit risk models’, Journal of Banking & Finance 24(1–2), 59–117.

Cuoco, Domenico and Hong Liu (2006), ‘An analysis of VaR-based capital requirements’, Journal of Financial Intermediation 15(3), 362–394. Daníelsson, Jón, Bjørn N. Jorgensen and Casper G. de Vries (2002), ‘Incentives for effective risk management’, Journal of Banking & Finance 26(7), 1407–1425.

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ACCEPTED MANUSCRIPT Du, Zaichao and Juan Carlos Escanciano (2017), ‘Backtesting expected shortfall: Accounting for tail risk’, Management Science 63(4), 940–958. Duellmann, Klaus, Jonathan Küll and Michael Kunisch (2010), ‘Estimating asset correlations from stock prices or default rates—Which method is superior?’, Journal of Economic Dynamics & Control 34(11), 2341–2357.

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Embrechts, Paul, Giovanni Puccetti and Ludger Rüschendorf (2013), ‘Model uncertainty and VaR aggregation’, Journal of Banking & Finance 37(8), 2750–2764.

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Escanciano, J. Carlos and Jose Olmo (2010), ‘Backtesting parametric value-at-risk with estimation risk’, Journal of Business & Economic Statistics 28(1), 36–51.

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models’, Journal of Financial Econometrics 9(1), 132–161. FCA (2016), Prudential sourcebook for Banks, Building Societies and Investment Firms,

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ACCEPTED MANUSCRIPT Fröhlich, Andreas and Annegret Weng (2015), ‘Modelling parameter uncertainty for risk capital calculation’, European Actuarial Journal 5(1), 79–112. Garlappi, Lorenzo, Raman Uppal and Tan Wang (2007), ‘Portfolio selection with parameter and model uncertainty: A multi-prior approach’, The Review of Financial Studies 20(1), 41–81.

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enough?’, American Economic Review 90(1), 147–165.

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ACCEPTED MANUSCRIPT Kerkhof, Jeroen, Bertrand Melenberg and Hans Schumacher (2010), ‘Model risk and capital reserves’, Journal of Banking & Finance 34(1), 267–279. Klein, Roger W. and Vijay S. Bawa (1976), ‘The effect of estimation risk on optimal portfolio choice’, Journal of Financial Economics 3(3), 215–231. Knight, Frank Hyneman (1921), Risk, Uncertainty, and Profit, number 16 in ‘Hart,

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Lönnbark, Carl (2010), ‘A corrected value-at-risk predictor’, Applied Economics Letters

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17(12), 1193–1196.

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shortfall’, Journal of Banking & Finance 37(3), 847–853.

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ACCEPTED MANUSCRIPT Psychoyios, Dimitris and George Skiadopoulos (2006), ‘Volatility options: Hedging effectiveness, pricing, and model error’, The Journal of Futures Markets 26(1), 1–31. Pykhtin, Michael and Ashish Dev (2002), ‘Credit risk in asset securitisations: an analytical model’, Risk 15(5), S16–S20.

Finance 55(1), 179–223.

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Pástor, Ľuboš (2000), ‘Portfolio selection and asset pricing models’, The Journal of

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Tarashev, Nikola A. (2010), ‘Measuring portfolio credit risk correctly: Why parameter

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uncertainty matters’, Journal of Banking & Finance 34(9), 2065–2076. Veronesi, Pietro (2000), ‘How does information quality affect stock returns?’, The Journal of Finance 55(2), 807–837.

Xia, Yihong (2001), ‘Learning about predictability: The effects of parameter uncertainty on dynamic asset allocation’, The Journal of Finance 56(1), 205–246. Xu, Mingxin (2006), ‘Risk measure pricing and hedging in incomplete markets’, Annals of Finance 2(1), 51–71. 33

ACCEPTED MANUSCRIPT Yamai, Yasuhiro and Toshinao Yoshiba (2005), ‘Value-at-risk versus expected shortfall:

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A practical perspective’, Journal of Banking & Finance 29(4), 997–1015.

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