Credit rating and pricing: Poles apart

Blöchlinger, Andreas Article Credit rating and pricing: Poles apart Journal of Risk and Financial Management Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Blöchlinger, Andreas (2018) : Credit rating and pricing: Poles apart, Journal of Risk and Financial Management, ISSN 1911-8074, MDPI, Basel, Vol. 11, Iss. 2, pp. 1-26, https://doi.org/10.3390/jrfm11020027 This Version is available at: https://hdl.handle.net/10419/238874 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ Journal of Risk and Financial Management Article Credit Rating and Pricing: Poles Apart Andreas Blöchlinger 1,2,† ID 1Swisscanto Invest by Zürcher Kantonalbank, Hardstrasse 201, CH-8005 Zurich, Switzerland; [email protected] or [email protected]; Tel.: +41-44-292-4580 2University of Zurich, Rämistrasse 71, CH-8006 Zurich, Switzerland † Current address: Hardstrasse 201, CH-8005 Zurich, Switzerland Received: 3 March 2018; Accepted: 12 May 2018; Published: 23 May 2018 Abstract: Corporate credit ratings remove the information asymmetry between lenders and borrowers to find an equilibrium price. Structured finance ratings, however, are informationally insufficient because the systematic risk of equally rated assets can vary substantially. As I demonstrate in a Monte Carlo analysis, highly-rated structured finance bonds can exhibit far higher non-linear systematic risks than lowly-rated corporate bonds. I value credit instruments under a four-moment CAPM, between and within some markets there is no one-to-one relation between expected loss (rating) and credit spread (pricing). The linear CAPM beta is insufficient, buyers and sellers need also the same information on non-linear risk to have an equilibrium. Keywords: asset backed security (ABS); contingent convertible bond (CoCo); standard risk aversion; capital asset pricing model (CAPM); UBS crisis 1. Introduction The expected loss of a credit instrument comprises an assessment of default probability as well as loss expectation in the event of a credit default. The default risk is reflected in the rating assignments of the major credit rating agencies such as Standard and Poor’s, Fitch, and Moody’s. For instance, it is Moody’s intention that the expected loss rate associated with a given rating symbol and time horizon to be the same across obligations to ensure a consistency of meaning (see Moody’s Investors Service (2009), p. 6). The same rating assigned to bond obligations issued by a nonfinancial corporation, bank, insurance, sovereign, subsovereign borrower, or a structured finance obligation must imply the same expected loss. Originally, “Moody was in effect addressing the stability of the security’s credit spread,” Moody’s Investors Service (2009) (p. 6). The idea that each rating class translates into a rating-specific credit spread can also be found in modern finance (see, e.g., Jarrow et al. (1997), Figures 5 and 6). Ratings reduce the knowledge gap, or “information asymmetry,” between borrowers (sellers) and lenders (buyers) by providing an ordinal assessment about the expected loss. I will demonstrate that the advent of structured finance obligations and other credit derivatives has completely broken down the monotone relation between expected loss (rating) and credit spread (pricing). Even if the rating truly conveys an unbiased, powerful estimate for the expected loss of an underlying obligation, 1 ratings alone are informationally insufficient for the pricing of collateralized debt obligations (CDOs) and other credit instruments. 2 I will show that in equilibrium an investment-rated structured finance 1 I will assume that ratings represent powerful, unbiased forecasts even though there is evidence that agency ratings are not the most powerful predictors (see e.g., Blöchlinger and Leippold (2018)). Nowadays, new test statistics allow an easy validation of the unbiasedness of default forecasts (see Blöchlinger (2017)). 2 CDOs were at the heart of the 2007–2008 financial crisis. The CDO is the prototypical structured finance security and is a type of structured asset-backed security (ABS). A CDO is a promise to pay investors in a predefined sequence, based on the J. Risk Financial Manag. 2018,11, 27; doi:10.3390/jrfm11020027 www.mdpi.com/journal/jrfm J. Risk Financial Manag. 2018,11, 27 2 of 26 obligation can have a significantly higher credit spread than a subinvestment-rated corporate bond due to systematic risk. That is, rating and pricing can be far apart since there is no longer a one-to-one relation between expected loss and systematic risk for structured finance obligations. 3 However, a market in which potential buyers know the true rating but cannot correctly judge the systematic risk attracts sellers offering credit derivatives of good rating quality but of low systematic risk quality which can in the end lead to a market breakdown. Thus, I will demonstrate that a necessary precondition for a credit market to have an equilibrium at all is to have symmetric information between issuers and investors on rating and systematic risk. The beta according to the CAPM of Treynor (1962), Sharpe (1964), Lintner (1965), and Mossin (1966) based on the mean-variance criterion of Markowitz (1952) reflects the systematic risk of an asset in the form of a scaled correlation with the market. 4 However, as highlighted by Embrechts et al. (1999), correlation is only a linear risk measure and of limited suitability for measuring dependence. In a stylized market, I will demonstrate that the non-linear systematic risk of credit derivatives is highly substantial and even more important than the linear CAPM beta. I thereby offer another explanation for the high expected return of highly-rated debt securities given the low CAPM beta.5 According to the criticism of Krugman (2009), financial economists rarely ask the seemingly obvious question of whether asset prices make sense given real-world fundamentals. Instead, they ask only whether asset prices make sense given other asset prices. The central insight of asset pricing is that in the absence of arbitrage there exists a risk-neutral measure Q equivalent to the real-world probability P (see Harrison and Kreps (1979)). Hence, to find the equilibrium value of any derivative you can assume a risk-neutral world without making statements about the real world. However, to be in equilibrium, market participants need first a consensus on all pricing-relevant attributes. The paper of Collin-Dufresne et al. (2012) is a case in point relevant to the pricing of CDOs. The contribution of Collin-Dufresne et al. (2012) “is to investigate the relative pricing across the stock option and CDO markets” and they provide a model “to jointly price long-dated S&P 500 options and tranche spreads on the five-year CDX index.” Summers (1985) once characterized financial economists with a parable about “ketchup economists” who “have shown that two-quart bottles of ketchup invariably sell for exactly twice as much as one-quart bottles of ketchup,” and conclude that the ketchup market is perfectly efficient. Collin-Dufresne et al. (2012) use the model of Duffie et al. (2000) in which return dynamics under the risk-neutral measure are specified and they emphasize the importance of “catastrophic” risk-neutral jumps for the pricing of highly-rated securities. Collin-Dufresne et al. (2012) remain silent on whether these “catastrophic” jumps are also a real-world phenomenon or induced by risk aversion and therefore “only” a phenomenon in the risk-neutral world. I derive a four-moment CAPM under standard risk aversion to characterize the quality of a credit instrument by four fundamental real-world factors (i.e., three systematic risk factors in addition to the rating factor) to provide an elementary relation between risk and price. I show that aversion to fat tails primarily affects the pricing of senior debt securities whereas variance aversion has a proportionally higher impact on equity. Unlike equity, the systematic risk of senior debt is dominated cash flows the CDO collects from the underlying pool of assets. The CDO is “sliced” into “tranches”. Each CDO tranche receives the cash flow in sequence based on its priority/seniority. In a Financial Times article by Jones (2008), structured finance was considered “the single most important invention in finance, if not economics, in the past few decades.” 3 Liquidity and taxation both play a role in the pricing of debt securities. I, like other studies, will abstract from these two quality factors. Elton et al. (2001) show that the expected loss (rating) can explain less than a quarter of the variation in the credit spread. Besides rating, liquidity, and taxation, the systematic risk (see Gabbi and Sironi (2005); Chen et al. (2007); Longstaff et al. (2005); Coval et al. (2009a); Blöchlinger (2011)) is highly pricing-relevant. 4Coval et al. (2009a) argue that “unlike actuarial claims, whose default probability is unrelated to the economic state (β=0), bonds are economic assets and have positive CAPM betas ( β> 0)” (p. 638). The important implication of the CAPM is that the scaled correlation between individual asset returns and the market return defines the systematic risk and matters for pricing. The remaining risk is often assumed to be idiosyncratic, can be diversified away and commands no premium. 5This observation is called the low beta anomaly and was reported by Jensen et al. (1972). J. Risk Financial Manag. 2018,11, 27 3 of 26 by non-linear cokurtosis rather than linear covariance risk, i.e., I explain the “risk-neutral catastrophic jumps” affecting senior debt by kurtosis aversion. Instead of expressing the CAPM traditionally in terms of relative returns—which was also heavily criticized by Krugman (2009)—I express the equilibrium price in terms of the promised end-of-period face amount with the expected loss rate as a direct input. The four factors—expected loss rate, covariance, coskewness, cokurtosis—can be interpreted as quality attributes in the spirit of Akerlof (1970). In my model, the factors for each asset can simply be aggregated to obtain the corresponding factors of the portfolio.6 Finally, I will show in an empirical application that the huge losses at the trading desks of two Wall Street titans, namely Morgan Stanley and UBS, were basically realized with a CAPM beta of zero but nonetheless with considerable non-linear systematic risk. These “hedged” structured finance portfolios had no covariation with the market and were no economic assets ( β> 0) in the sense of Coval et al. (2009a) but unfortunately also no actuarial claims whose risk could be diversified away in large portfolios. Morgan Stanley’s and UBS’s beta-neutral structured finance portfolios were nevertheless treated as actuarial claims in the corresponding risk departments despite the portfolios’ inherent non-linear systematic risk. Even worse, they misinterpreted the small premium for non-linear systematic risk as CAPM alpha and therefore leveraged their positions. Thus, I provide empirical evidence that even “too big to fail” banks were exposed to huge risks without proper assessment about all quality attributes by considering only rating and correlation/beta. My findings are highly relevant since structured finance markets such as the market for asset backed securities (ABSs) improve the efficiency of resource allocation and help contain systemic risk by freeing up the banks’ balance sheets. Given the absence of symmetric information, my model offers an explanation why public ABS issuances remain low in the EU as reported by the BOE and ECB (2014). The low demand for ABSs is unfortunate since “ABS can support the transmission of accommodative monetary policy in conditions where the bank lending channel may otherwise be impaired” (p. 2). I proceed as follows: Section 2highlights the difference between linear systematic risk (CAPM beta), non-linear systematic risk and idiosyncratic risk. In Section 3I derive a simple four-moment equilibrium CAPM. Section 4demonstrates that a credit market under asymmetric information about non-linear systematic risk is in a disequilibrium. Section 5shows in an illustrative market that some credit products can have superb rating quality but also high systematic risk exposures or exactly the other way around. I also discuss the influence of counterparty risk on credit derivatives. Section 6investigates the empirical cases of Morgan Stanley and UBS. Finally, Section 7concludes. 2. CAPM Beta and Premium for Residual Risk Credit portfolio distributions are typically characterized by non-linear risks such as skewness and heavy tails. However, many papers on credit portfolio risk such as Coval et al. (2009a,2009b); Hamerle et al. (2009); Brennan et al. (2009) largely ignore the pricing of non-linear systematic risks. In their CAPM-like models, based on the credit portfolio framework of Merton (1974), an asset uncorrelated with the market portfolio (β= 0 ) is also assumed to be stochastically independent and therefore an actuarial claim. However, stochastic independence is too strong an assumption, the cash flows of a beta-neutral portfolio may still contain significant non-linear systematic risk which—unlike the risk of actuarial claims—cannot be diversified away in a 6 According to MacKenzie (2011), the market for CDOs would have been quite limited if participation in the market required the proper understanding of the inherent (systematic) risks. Ratings “black boxed” these complexities. Ratings permitted prices of different CDOs to be compared, both with each other and with more familiar credit instruments such as single-name corporate bonds, by comparing the credit spread offered by a given credit instrument to that offered by others with the same rating. This pricing-rating nexus was thus a convention in the sense of Young (1996) “economics of convention”: a way of turning uncertainty into a form of order that is stable enough to permit coordination and (non-rational, unsustainable, short-term) equilibrium. The subsequent realization of investors that there are further but unknown quality factors besides the rating rendered coordination impossible with no (rational, long-term) equilibrium as predicted by the model of Akerlof (1970). J. Risk Financial Manag. 2018,11, 27 4 of 26 large portfolio and must be priced. For instance, Coval et al. (2009a) resort to a kind of Black and Scholes (1973) model in discrete time in the spirit of Rubinstein (1976), but they make a crucially different assumption by changing the Gaussian assumption regarding the market portfolio: 7 Proposition 1 ( Black and Scholes (1973) in discrete time ) . If the market portfolio M and the payoff of asset P follow a bivariate log-normal distribution such that log M log P!∼N µM µP!, σ2 MσMσPρ σMσPρ σ2 P!!, where ρ is the correlation and if the von Neumann-Morgenstern utility function u(·) of the representative agent exhibits constant relative risk aversion with risk aversion coefficient λ then the price qP of end-of-period payoff P is given by qp=q0exp µp−λρσMσP+1 2σ2 P . The Radon-Nikodym derivative Z=M−λ/EM−λ induces the risk-neutral measure Q: log M log P!Q ∼N µQ M µQ P!, σ2 MσMσPρ σMσPρ σ2 P!!, where µQ P=µP−λρσMσP=log qP−log q0− 0.5 σ2 P and µQ M=µM−λσ2 M=log qM−log q0− 0.5 σ2 M are the shifted means under Q. The proof can be found in the Appendix. By linear projection of the logarithmic portfolio cash flow log P onto the logarithmic cash flow of the market portfolio log M , one obtains an additive, mean-square efficient decomposition into market and residual/idiosyncratic risk: log P−µQ P=βlog M−µQ M+σPq1−ρ2ν, log P−µQ P σP =ρlog M−µQ M σM +q1−ρ2ν, with ν∼N(0, 1), (1) with β=ρ σP/σM denoting the CAPM beta. The inclusion of further transformations of M such as polynomial expansions cannot improve the goodness-of-fit (see Hamilton (1994), p. 102). Due to normality, the systematic risk exposure is completely described by β . That is, the residuum ν follows a standardized normal distribution under the real-world measure P as well as under the risk-neutral measure Q , so there is no premium associated with this residual risk. Hence, the value of any financial derivative of P—the price EQ[g(P)]of any σ(P)-measurable payoff g(·)—can be written as: EQhEQ[g(P)|M]i=Z∞ −∞Z∞ −∞ g(ξ) ξσPp1−ρ2h  log ξ−µQ P−βlog m−µQ M σPp1−ρ2 dξfQ(m)dm, where fQ(·) is the probability density function (pdf) of the market factor M under the risk-neutral measure Q , h(·) is the probability density function of ν under P and Q . Under the assumption of Rubinstein (1976), fQ(·)and h(·)both correspond to the Gaussian pdf φ(·). 7 Rubinstein (1976) derives sufficient conditions under which the option-pricing formula of Black and Scholes (1973) in continuous time applies also in discrete time. Rubinstein (1976) remarks that “since the time interval between dates can be made arbitrarily small in discrete time models, they are in this respect of greater generality [than continuous time models].” Fama (1970) proved that even though a risk averter maximized the expected utility from the stream of consumption over his lifetime, his choices in each period would be indistinguishable from that of a properly specified risk averse investor with a singe-period horizon. J. Risk Financial Manag. 2018,11, 27 5 of 26 A common failing when giving up the Gaussian assumption is to preserve—as you will see later—the easily rejectable assumption that the residual ν is independent from M so that ν is idiosyncratic, can be diversified away and has the same distribution under P and Q . Coval et al. (2009a); Brennan et al. (2009); Hamerle et al. (2009) all analyze the CDO market under assumed independence between M and ν . However, under non-normality, orthogonality only implies that E[νlog M]=E[ν]E[log M]= 0 (see Hamilton (1994), p. 74). In the following, I will consider not only covariance but also coskewness and cokurtosis risk because agents under standard risk aversion care about all three co-moments. For a general credit portfolio P , I will show that EνM26= 0 and that EνM36=0. I therefore extend the CAPM by two further statistical (co-)moments. 3. Four-Moment Valuation Model I consider a portfolio choice problem faced by (buy-and-hold) investors/individuals in a general equilibrium involving a finite number of agents and a finite number of assets. My ultimate goal in the following sections is not to find “the best” multi-period pricing model to fit observable market prices for daily mark-to-market valuation but to show within a simple, one-period model that the non-linear systematic risk of equally rated credit products can be vastly different, and if buyers and sellers cannot agree on systematic risk then this credit market has no equilibrium. Thus, to revive the market for structured finance obligations requires a parsimonious set of publicly available systematic risk figures (such as systematic covariance, coskewness and cokurtosis)—the credit rating alone is arguably only sufficient for the market of corporate bonds. I derive a four-moment CAPM, but not via the standard way in terms of relative returns such as Kraus and Litzenberger (1976)orHarvey and Siddique (2000), instead the equilibrium price is a function of expected loss rate and systematic risk contributions per unit notional. In other words, the risk metrics are directly measured per unit at risk. For the time being, I assume to be in a representative agent economy under standard risk aversion as defined by Kimball (1993). Later, I will extend the model for an asymmetric market. The triple (Ω,F,P) is the probability space and I have a representative von Neumann-Morgenstern maximizer of expected utility whose utility function u(·) exhibits the following properties: (a) positive marginal utility for wealth, i.e., non-satiety, or monotonicity u0>0, (b) decreasing marginal utility for wealth, i.e., risk aversion, or concavity u00 <0, (c) decreasing absolute risk aversion, (d) decreasing absolute prudence. The quartic utility function is compatible with non-satiation, risk aversion, decreasing absolute risk aversion, decreasing absolute prudence, with positive coefficients for odd powers and negative coefficients for even powers (see the Appendix Afor a proof): Lemma 1. Standard risk aversion implies u0>0, u00 <0, u000 >0, and u0000 <0. The expected utility of a F -measurable payoff X can be approximated by a quartic utility function via fourth-order Taylor series expanded at the point E[X] by assuming that the fourth moment of X and fourth derivative of u(·)indeed exist (see also Samuelson (1970)): E[u(X)]≈u(E[X]) +1 2! u00 (E[X]) Eh(X−E[X])2i +1 3! u000 (E[X]) Eh(X−E[X])3i+1 4! u0000 (E[X]) Eh(X−E[X])4i. For a higher order expansion, the series converges in the case of logarithmic and power utility functions if |X−E[X]|<E[X] , P almost surely (see, Jurczenko and Maillet (2006), p. 81). However, even for divergent Taylor series Hlawitschka (1994) shows that truncated expansions J. Risk Financial Manag. 2018,11, 27 6 of 26 provide “excellent approximations to expected utility for the purpose of portfolio selection” even though moments do not fulfill the axioms of coherent risk measures according to Artzner et al. (1999). Lemma 2. A quartic von Neumann-Morgenstern utility u(·) exhibiting standard risk aversion implies preference for right-skewness and preference for platykurtic distributions. Similar as in Brennan (1979), I derive the equilibrium prices in a one-period, one-good economy with a capital market with K+ 1 assets with end-of-period payoffs {Y0, ..., YK} as though there existed only identical representative agents, i.a., all N buy-and-hold investors have the same probability beliefs and the same utility function. The payoff Y0is assumed to be strictly positive and non-random. Proposition 2 (Symmetric equilibrium) . Today’s equilibrium price qk of payoff Yk at the end of the period for any k ∈{1, ..., K}can be expressed in terms of risk-free asset 0: qk q0 =Eu0(w) E[u0(w)] Yk Y0=EZYk Y0=EQYk Y0, (2) where the variable Z=u0(w)/E[u0(w)] has mean one and is positive under the assumption of non-satiation, i.e., u0> 0, Z therefore fulfills all requirements of a Radon-Nikodym derivative. The Radon-Nikodym derivative Z induces a measure change from the real-world measure Pto the risk-neutral measure Q, i.e., Q{A}=E[1AZ], (3) where 1Ais the indicator function of any event A ∈ F.8 The proof can be found in the Appendix. Without loss of any generality, I assume that the price of the risk-less asset q0 is expressed per unit notional, i.e., q07→ q0/Y0 , to approximate the equilibrium relation in (3) with the first four statistical moments and first four mathematical derivatives of u(·): Proposition 3 (4-moment CAPM) . Under the assumption of a representative von Neumann-Morgenstern expectation maximizers under standard risk aversion whose utility function u(·) is approximated by a fourth order Taylor series around the mean end-of-period wealth w , today’s equilibrium price qX under the pricing measure Qin (3)of any F-measurable financial derivative payoff X can be written as follows: qX q0 =µX−λββX−λγγX−λδδX, (4) with µX=E[X], and βX=E[(X−E[X]) (M−E[M])] Eh(M−E[M])2i γX= Eh(X−E[X]) (M−E[M])2i Eh(M−E[M])3i δX= Eh(X−E[X]) (M−E[M])3i Eh(M−E[M])4i, 8 The Radon-Nikodym derivative Z is also known as pricing kernel in finance, the measure Q is called the risk-neutral measure since if the representative investor were risk-neutral, i.e., u0(w) = const , the real-world or physical measure P would coincide with Q. J. Risk Financial Manag. 2018,11, 27 7 of 26 where M= 1 /K∑K k=1Yk denotes the averaged payoff of the market portfolio, βX captures the covariance risk of X with M , γX the coskewness risk, δX the cokurtosis risk. Under standard risk aversion, the premium λγ for systematic skewness risk is positive if M is left-skewed and negative if right-skewed. The premia for variance and kurtosis risk, λβ,λδ, are positive. The proof can be found in the Appendix. Note, µX , βX , γX , δX are computed under the physical risk measure P but do not depend on the degree of risk aversion or the form of the utility function u(·) . On the other hand, the risk premia λβ , λγ , and λδ depend on u0 , u00 , u000 , and u0000 . Rating agencies provide an assessment about the expected loss µX but are silent on the other physical risk metrics βX , γX , δX . Two important remarks are in order: First, since the equilibrium price in (4) is not expressed in relative returns like other capital asset pricing models such as Sharpe (1964); Kraus and Litzenberger (1976); or Harvey and Siddique (2000), the risk metrics must be expressed relative to the underlying notional amount to make them comparable across instruments. Since the bond price is by convention expressed as a percentage of nominal value, for comparative statistics it is necessary to express µX , βX , γX , and δX of a financial derivative Xalso relative to its notional amount, i.e., per unit at risk. Second, even if the relation between price and risk metrics is only approximately true in practice, the real-world risk statistics of a F -measurable payoff X —rating µX , linear systematic risk βX , and non-linear systematic risk γX , δX —with respect to a well-defined market portfolio (e.g., a credit default swap index like CDX or iTraxx) provide pricing-relevant information about the underlying credit quality. The absence of such publicly available risk metrics may result in a non-functioning market due to information asymmetry between buyer and seller of credit risks. The market for structured finance products may even collapse. 4. Credit Markets under Asymmetric Information I introduce a market under asymmetric information in the spirit of Akerlof (1970). The risk metrics µX , βX , γX , and δX in (4) play here the role of the underlying quality of a credit derivative with payoff X . I still assume homogenous probability beliefs and risk aversions. Formally, I have the probability space (Ω , F , P) , the sigma fields {B,S} and the risk-neutral measure Q induced by the four-moment CAPM kernel in (A8) . So far, I implicitly assumed that the buyer’s information set B and the seller’s sigma algebra S are equal and both equal to the naive field {∅,Ω} . Now, the buyer still starts with the naive information {∅,Ω} , but rating agencies make public the information to calculate the mean µX=E[X|S] of X under the seller’s information S ⊃ {∅,Ω} so that a potential buyer has then the sigma algebra generated by µXavailable for decision making, i.e., B=σ(µX). Definition 1 (Asymmetric credit market) . In an asymmetric market, the statistical moments conditional on the seller’s information S and conditional on the buyer’s information B differ. In a symmetric market, however, the sigma algebras B and S result in the same (scaled) conditional moments µX , βX , γX , and δX , e.g., βX=E[M(X−µX)|S]/V[M|S]=E[M(X−µX)|B]/V[M|B] , almost surely, and I have again the equilibrium in (4). Under asymmetric information at least one of the risk metrics is different under Band S. In simple words, in an asymmetric market some pricing-relevant quality attributes are known to sellers but unknown to buyers. Such a market cannot function: Proposition 4 (No equilibrium) . An asymmetric credit market in which sellers and buyers disagree on at least one of the four quality factors µX,βX,γX,δXhas no equilibrium. J. Risk Financial Manag. 2018,11, 27 8 of 26 Proof. To show that a market with such asymmetric information cannot work properly, I work with a proof by contradiction, i.e., I start with the assumption that there is nonetheless an equilibrium. 9 By pX I denote the compounded equilibrium price qX/q0 and by rX the risk premium (credit spread) of the positive cash flow X with P{X=0}< 1. The positivity assumption is without loss of generality because a payoff with negative outcomes can be split into a long and short position of two positive cash flows. Thus, the price of Xis given by: pX=µX−rXwhere pX=qX/q0and rX=f(βX,γX,δX). (5) From Proposition 3, I know that pX is indeed the equilibrium price under symmetric information and a four-moment CAPM when βX , γX , δX are known. However, now I assume that µX is known but at least one of the factors βX , γX , δX is only known to the seller ( S -measurable) but not known to the buyer (not B -measurable). In particular, I assume there is no B -measurable pricing function g(·) such that g(µX) = pX, but the buyer knows that rX≤µX. (6) The upper bound in (6) must be µX , otherwise an arbitrage opportunity would arise for a positive end-of-period payoff X must have a positive price with probability one. Now, conditional on the information generated by 1{rX≥µX−pX} and µX , the buyer of payoff X knows that the mean risk premium on offer at the assumed equilibrium price pXis given by: ¯ rX:=EQhrX1{rX≥µX−pX},µXi. The expectation is taken under the risk-neutral measure Q induced by the Radon-Nikodym derivative in (A8) to account for risk aversion. The price bidden p∗ X by the buyer given she knows µX and that the average risk premium on offer is ¯ rXis therefore given by: p∗ X=µX−EQhrX1{rX≥¯ rX},µXi. (7) Since ¯ rX=EQhrX1{rX≥µX−pX},µXi≤EQhrX1{rX≥¯ rX},µXi⇔p∗ X≤pX. However, the bid price p∗ X is always smaller than the assumed equilibrium price pX . I only have a strict equality if pX= 0 or else if rX is σ(µX) -measurable so that ¯ rX=rX . However, a positive end-of-period cash flow X with P{X=0}< 1 must have a strictly positive price to exclude arbitrage and if rX is a function of µX then the market is symmetric, i.e., the rating is sufficient for pricing. I have a contradiction that pX is the equilibrium price under asymmetric information. That is, there is no equilibrium price pXand therefore no risk-neutral risk measure Q. No trade takes place. Note, the rating can be sufficient for pricing in a four-moment CAPM. Technically speaking, in this special case, the factors βX , γX , δX in (5) are σ(µX) -measurable and the rating is informationally sufficient in order to have an equilibrium. Sufficiency may hold for the segment of corporate bonds, but in general there is no rating-pricing nexus. On the contrary, as I will show, the risk premium can even be negative for derivatives with high expected losses and significantly positive for structured finance obligations of high rating quality. Counterparty risk further complicates the quality assessment 9 Without loss of generality, I assume here that ratings provide the seller’s information about the expected loss so that buyers work under a non-biased mean, i.e., µX=E[X|S] . Such a rating bias may have played a part during the financial crisis in 2007/08. It is straightforward to show that under biased ratings, i.e, E[X|B]6=E[X|S], the market breaks down as well. J. Risk Financial Manag. 2018,11, 27 15 of 26 Table 4shows the repackaging of second loss CDO tranches from Table 3. Each CDO squared structure consists of n∈{2, 4, 6, 8} underlying CDO tranches {T2`:`=1, ..., n} on the asset side and a senior debt tranche and an equity tranche on the liability side. The more diversified the underlying asset pool, i.e., the greater n , the higher the possible leverage N for a given rating and the systematic risk of debt (equity) decreases (increases) with increasing n . However, a CDO squared debt tranche compared to a simple CDO tranche, such as the third loss tranche in Table 3, the expected payoff µ , systematic variance β , and skewness γ are lower, but the systematic kurtosis δ is higher. In a three-moment CAPM all CDO squared debt tranches in Table 4must have the higher price compared to the third loss tranche T3 in Table 3. However, in the four-moment CAPM, the pricing relation between simple CDO and CDO squared is also influenced by δ which is higher for the latter. The CDO squared example highlights the relevance of all four quality factors for pricing risky debt.10 Table 4. Four different CDO squared structures. Tranche nNotional NPayoff µ β γ δ First loss 4 0.3750 max nPn 1−N−N 1−N, 0o0.990978 3.8883 7.6913 8.7413 Senior A 1 −max n1−Pn N, 0o0.999219 0.6104 1.9976 3.2683 First loss 6 0.4167 max nPn 1−N−N 1−N, 0o0.990344 4.1447 8.1368 9.1597 Senior A 1 −max n1−Pn N, 0o0.999284 0.5788 1.9427 3.2291 First loss 8 0.4375 max nPn 1−N−N 1−N, 0o0.989991 4.2868 8.3818 9.3864 Senior A 1 −max n1−Pn N, 0o0.999311 0.5663 1.9241 3.2216 First loss 10 0.4500 max nPn 1−N−N 1−N, 0o0.989768 4.3766 8.5356 9.5273 Senior A 1 −max n1−Pn N, 0o0.999326 0.5596 1.9146 3.2193 Underlying CDO portfolio Pn=1 n∑n `=1T`20.994077 2.6560 5.5513 6.6843 The underlying asset portfolio with payoff Pn consists of n CDO mezzanine tranches. Each mezzanine tranche T`j , `∈{1, ..., n} , j= 2, has an attachment point aj−1= 0.06 and a detachment point aj= 0.08, and the underlying digital bond portfolio of each mezzanine tranche consists of 100 credit names. The payoff of the underlying asset portfolio Pnis therefore given by: Pn=1 n∑n `=1T`j, with T`j=1{d`≤aj−1}+d`−aj−1 aj−aj−11{aj−1<d`≤aj},d`=1−1 100 ∑100 k=1Y10(k−1)+`, where Yk denotes the binary payoff of digital bond k , and d` the default rate in the underlying bond portfolio of the CDO mezzanine tranche T`j . The variable N denotes the notional amount of senior debt, ( 1 −N) the notional amount of equity (first loss tranche). Due to linearity, the risk contributions µ , β , γ , and δ of the underlying CDO pool Pn are independent from the number of underlying CDOs n . However, due to non-linear payoffs of debt and equity, the risk contributions of the CDO squared tranches depend on n (and N ). The more diversified the underlying CDO pool, the higher (worse) is the quality of senior debt (equity). 5.5. Credit Linked Note (CLN) under Counterparty Risk The relation between rating and pricing can be completely turned upside down as I will demonstrate for credit linked notes (CLNs). The issuer of a CLN is not obligated to repay the notional amount in full if a specified event occurs. With the structuring of a credit linked note, I can create a payoff X with a low expected loss but high systematic risk contributions βX , γX , δX or the other way around, i.e., a payoff with low rating quality but of high quality with respect to systematic risk. Besides the risk metrics of the underlying payoff characteristic X , it is the credit quality of the issuing counterparty that matters for pricing. 10 Overall, I confirm the suggestion of Coval et al. (2007) that the appearance of CDO squared can be explained by its high systematic risk. All A-rated CDOs squared in Table 4have similar systematic risk qualities as A-rated CDOs in Table 2 which are both much higher compared to an A-rated single-name bond in Table 1. J. Risk Financial Manag. 2018,11, 27 16 of 26 In Table 5I create fourteen structured products. The first CLN replicates the payoff of an average bond portfolio so that the systematic risk contributions β , γ , δ are all equal to one and the expected payoff µ is 99%. CLN (2) and (3) replicate digital CDO tranches with 100 and 1000 underlying digital bonds in the asset pool. CLN (2) and (3) have far higher expected payoffs with 99.93% and 99.94% than CLN (1) with a mean payoff of 99%, but the systematic skewness and the systematic kurtosis are higher. The more diversified CLN (3) has even higher systematic risk contributions than the less diversified CLN (2). CLN (3a), (3b), (3c), (3d) take into account the issuer’s counterparty risk. In fact, CLN (3) can also be interpreted as an investment into a synthetic digital CDO tranche with the market portfolio as underlying asset pool. The systematic risk quality of such an investment is already below average regarding coskewness and cokurtosis risk (γ , δ> 1 ) . However, with the inherent counterparty risk involved in a synthetic CDO, its quality factors are even worse. The “opposite” payoff of CLN 3), i.e., the payoff 1{M<0.91}Yk is in effect a digital default swap (DDS) under counterparty risk k . Thus, the synthetic CDO plus DDS issued by the same counterparty k yields the payoff Yk , i.e., the digital bond of issuer k . CLN (4), (5) are structured products with no systematic variance and no systematic skewness risk, respectively. CLN (4), (5) demonstrate the importance of considering non-linear systematic risk, the linear CAPM beta alone is insufficient. The price of CLN (4) entails compensation for non-linear systematic risk and not CAPM alpha. Conversely, CLN (5) is not a negative CAPM alpha investment but offers protection against systematic skewness. Table 5. Rating and pricing of credit linked notes under counterparty risk. CLN Payoff µ β γ δ (1) 0.25 ∑4 i=11{Y1+(i−1)250=1}Y00.990000 1.0000 1.0000 1.0000 (2) 1{1/100 ∑100 k=1Y(k−1)10+1≥0.90}Y00.999296 0.5000 1.5631 2.5387 (3) 1{M≥0.91}Y00.999419 0.5219 1.9003 3.3286 (3a) 1{M≥0.91}Y10.998430 0.6582 2.0397 3.4419 (3b) 1{M≥0.91}Y251 0.996447 0.8825 2.2434 3.6019 (3c) 1{M≥0.91}Y501 0.990491 1.4593 2.7198 3.9675 (3d) 1{M≥0.91}Y751 0.972540 2.8610 3.7416 4.7387 (4) 1{M6=0.990}Y00.960958 – 1.4613 0.6652 (5) 1 −0.16 ×1{M=0.975}−0.84 ×1{M=0.976}0.992046 −0.6652 −0.1250 – (6) 1{M<1.000}Y00.953889 −4.3342 −0.1032 −1.0630 (6a) 1{M<1.000}Y10.952886 −4.1850 0.0857 −0.8574 (6b) 1{M<1.000}Y251 0.950886 −3.9445 0.3517 −0.5813 (6c) 1{M<1.000}Y501 0.944896 −3.3362 0.9447 −0.0073 (6d) 1{M<1.000}Y751 0.926877 −1.8730 2.1965 1.1796 Market M=1/1000 ∑1000 k=1Yk0.990000 1.0000 1.0000 1.0000 CLN (1) replicates the payoff of an average portfolio of single-name digital bonds from Table 1, CLN (2) is a binary that pays out nothing if the default rate in an average portfolio of 100 digital bonds exceeds 10%. CLN (3) pays out nothing in case the market loss exceeds 9% and one else. The credit quality decreases significantly under decreasing counterparty quality in CLN (3a), (3b), (3c) (3d). CLN (4) pays out nothing in case the market’s default rate is exactly 1% and one else, it shows no covariation with the market yet above average coskewness risk (> 1 ) . CLN (5) has no cokurtosis exposure. CLN (6) pays out one but in the best state of the world when there are no market losses. Its default probability is quite high with 4.61%, but its systematic risk exposure could not be better. CLN (6a), (6b), (6c) and (6d) are the same CLN as (6) but under increasing counterparty risk. Counterparty 0 is default protected. CLN (6) has the lowest rating quality among instruments with no counterparty risk, i.e., it has the highest expected loss. However, CLN (6) only defaults in the best state of the economy when there are no defaults in the market portfolio, i.e., in a state when a marginal payoff is least beneficial. Apart from rating quality, CLN (6) offers the highest possible quality and is in that respect quite the opposite of CLN (3) which has the best rating quality but under quite unfavorable systematic risk. CLN (6a), 6(b), 6(c), 6(d) take into account the counterparty risk of the CLN issuer. That is, those four CLNs unlike CLN (6) can also default in bad states when the CLN issuer is not capable to fulfill its obligation. J. Risk Financial Manag. 2018,11, 27 17 of 26 Remarkably, all four CLNs under counterparty risk still have negative betas yet their coskewness is positive, again a clear demonstration that rating and beta are insufficient for pricing. 5.6. Distress-Contingent Convertible Bond (CoCo) My model allows the analysis of many other credit instruments such as bank deposit insurance and loan guarantees as an alternative to Merton (1977), catastrophe bonds, or credit default swaps (CDSs) under counterparty risk. However, to provide yet a final example, I will discuss distress-contingent convertible bonds or simply CoCos (see Duffie (2010)). Let ω∈[0, 1] denote the dilution factor, ω= 0 means that the CoCo is a pure write-down bond, existing shareholders experience no dilution at all, ω= 1 means that existing shareholders are wiped out completely as soon as the conversion level in the form of an attachment point a1 is triggered. Such a waterfall structure can be thought of as a convolution of first, second, and third loss tranche in a standard CDO structure. The first loss is borne by the equity holder, the second loss tranche is borne by the CoCo bondholder, and claims for the third loss tranche are divided between equity and CoCo bondholder, the fraction ω belongs to the CoCo bondholder and ( 1 −ω) to the equity holder. Consequently, by linearity the prices of equity and CoCo bond are the weighted average of a more standard CDO structure: Payoff equity: a1T1+(a3−a2)(1−w)T3 a1+(a3−a2)(1−w) Payoff contingent convertible bond: (a2−a1)T2+(a3−a2)w T3 (a2−a1)+(a3−a2)w, where T1 , T2 , T3 are the payoffs of a first, second, and third loss tranche of a standard CDO structure, a1 , a2 , a3 are the attachment points. Due to linearity, the systematic risk contributions as well as the mean payoff of equity and CoCo can be obtained from the corresponding CDO tranches as listed in Table 6. The higher the dilution w , the higher is the systematic risk of equity but the lower the systematic exposure of the CoCo. In other words, the lower the dilution ω , the less “toxic” is equity. Under ω= 1, the rating quality of the CoCo is slightly better than that of a BBB-rated corporate bond in Table 1, yet its systematic skewness and kurtosis are even higher than that of a BB-rated corporate bond. Similarly, the expected loss of the CoCo is clearly lower than that of an average corporate bond portfolio, yet its systematic risk exposure is considerably worse. Table 6. Refinancing of bank assets with equity and contingent convertible bond. Complete Write down w=0/ CoCo Bondholders Have No Claim to 3rd Loss Tranche: Tranche Payoff µ β γ δ Equity a1T1+(a3−a2)(1−w)T3 a1+(a3−a2)(1−w)0.901228 9.4124 8.6546 8.1667 Contingent convertible bond (a2−a1)T2+(a3−a2)w T3 (a2−a1)+(a3−a2)w0.994077 2.6560 5.5513 6.6843 Portfolio ∑3 j=1 aj−aj−1 a3−a0Tj0.916703 8.2864 8.1374 7.9196 Complete Dilution of Equity w=1/Equity Holders Have no Claim to 3rd Loss Tranche: Tranche Payoff µ β γ δ Equity a1T1+(a3−a2)(1−w)T3 a1+(a3−a2)(1−w)0.836149 15.2050 13.0637 11.5560 Contingent convertible bond (a2−a1)T2+(a3−a2)w T3 (a2−a1)+(a3−a2)w0.997257 1.3677 3.2111 4.2832 Portfolio ∑3 j=1 aj−aj−1 a3−a0Tj0.916703 8.2864 8.1374 7.9196 The equity holder bears the first 6% of losses (first loss tranche T1 with attachment a0= 0 and detachment point a1= 0.06). The contingent convertible bondholder bears the next 2% of losses (second loss tranche T2 with detachment point a2= 0.08), the fraction w is the payoff of the third loss tranche T3 with detachment point a3= 0.12 that goes to the contingent convertible bondholder and ( 1 −w) to the equity holder. More senior debt holders are only affected if the losses exceed 12%. The underlying asset pool consists of 100 digital bonds. The total notional amount of contingent convertible bonds is (a2−a1) + (a3−a2)w , the total notional amount of equity is a1+ (a3−a2)( 1 −w) . As a consequence, the higher w , the lower (higher) the systematic risk exposure β , γ , δ of contingent convertible bonds (equity) per unit notional and the higher (lower) the expected payoff µ. J. Risk Financial Manag. 2018,11, 27 18 of 26 6. Empirical Cases As pointed out by Collin-Dufresne et al. (2012), “traders in the CDX market are typically thought of as being rather sophisticated. Thus, it would be surprising to find them accepting so much risk without fair compensation.” Longstaff and Rajan (2008), Li and Zhao (2012) show that CDX tranches are consistently priced under risk-neutral models and conclude that these securities are “reasonably efficiently priced.” I provide two empirical counterexamples that even professional participants in the market for structured finance obligations, namely Morgan Stanley and UBS, were seemingly not aware of the inherent non-linear risk of apparently hedged CDO portfolios. In particular, both banks neglected the coskewness and cokurtosis risk of their trading portfolios which were basically uncorrelated with the market portfolio. Both dealer banks thought that their structured finance portfolios were of higher quality than they actually were, so I doubt whether they were really fairly compensated for their unintentional risk taking. 6.1. Morgan Stanley According to Lewis (2011), the fixed-income trader Howie Hubler at Morgan Stanley bought insurance on BBB-rated CDOs by paying the CDS spread on a notional amount of roughly USD 2 bn. To offset his running cost he sold protection on AAA-rated CDOs by receiving the insurance fee on a notional amount of around USD 18 bn. 11 In effect, Morgan Stanley synthetically constructed a leveraged structured finance portfolio with a notional N = USD 16 bn by investing ( 1 + 1 / 8 )N = USD 18 bn into AAA-rated CDO tranches and by shorting 1 / 8 N = USD 2 bn of BBB tranches. As listed in Table 3, such a long-short strategy (of second- and fourth-loss tranches) is roughly beta-neutral and therefore virtually free of linear systematic risk but significantly exposed to non-linear systematic risk. From a systematic risk perspective in a two-moment CAPM, Hubler’s structure is as risk-free as Swiss or US Treasury bonds. Indeed, Hubler’s position “registered on Morgan Stanley’s internal report as virtually riskless,” Lewis (2011) (p. 207). Morgan Stanley managed to sell this leverage structure in July 2007 prior to expiration mainly to UBS with a loss of around USD 9 bn: “The other, bigger, buyer was UBS—which took $2 billion in Howie Hubler’s triple-A CDOs, along with a couple of hundred million dollars’ worth of his short position in triple-B-rated bonds. That is, in July, moments before the market crashed, UBS looked at Howie Hubler’s trade and said, "We want some of that, too." [...] traders at UBS who executed the trade were motivated mainly by their own models—which, at the moment of the trade, suggested they had turned a profit of $30 million.” Lewis (2011) (p. 215/216) In the first half of 2007, buyers of CDOs were gradually realizing that there are unknown quality factors in the sense of Akerlof (1970). UBS was arguably the last buyer before the market broke down completely: “In the second quarter of 2007 [...] The UBS leadership continued to be optimistic and the Investment Bank went on purchasing highly rated subprime paper while other banks were quickly unloading their positions” Straumann (2010). 6.2. UBS UBS incentivized its business divisions with an UBS-specific economic value added approach of Ospel-Bodmer (2001) which is fundamentally based on a two-moment CAPM. In the framework of Ospel-Bodmer (2001) there is no mentioning of non-linear risk, there is just an alpha and a CAPM 11 “[T]he premiums on the supposedly far less risky triple-A-rated CDOs were only one-tenth of the premiums on the triple-Bs, and so to take in the same amount of the money as he was paying out, he’d need to sell credit default swaps in roughly ten times the amount he already owned” (Lewis (2011), p. 206). J. Risk Financial Manag. 2018,11, 27 19 of 26 beta, so the compensation for non-linear systematic risk is falsely interpreted as economic value added, economic profit, or CAPM alpha. As seen above, UBS took over some CDO risks from Morgan Stanley by buying a beta-neutral long-short CDO portfolio. The conventional thinking that only beta measures systematic risk seems to have been deeply ingrained at UBS that ultimately reported net losses of 18.7 bn. In its shareholder report, UBS (2008) called its long-short CDO portfolio “Amplified Mortgage Portfolio” Super Seniors (AMPS): “these were Super Senior positions where the risk of loss was initially hedged through the purchase of protection on a proportion of the nominal position (typically between 2% and 4% though sometimes more)” (p. 14). 12 That is, UBS was long senior CDO tranches and hedged its position by shorting first or second loss CDO tranches. These long-short strategies are roughly beta-neutral as can be seen in Table 3and explain the majority of losses at UBS (2008): “As at the end of 2007, losses on these AMPS trades contributed approximately 63% of total Super Senior losses” (p. 14). However, even with the benefit of hindsight, UBS seems to have misunderstood the principal root cause of their losses. UBS (2008) states that its losses were primarily of idiosyncratic and not of systematic nature: “Trading losses: Insufficient accounting for the risk of divergent movements between previously correlated asset classes or instruments (basis risk)” and “insufficient attention to idiosyncratic risk factors (i.e., the risk of price change due to unique circumstances of a specific security, as opposed to the overall market)” (p. 30). The fact that the remaining risk was systematic and not idiosyncratic should have been obvious because a full hedge by insurance via CDS on the exactly same underlying (in UBS vocabulary a NegBasis trade), was more costly than beta-neutral hedging (AMPS trade). 13 If the remaining risk of a beta-neutral portfolio were indeed purely idiosyncratic there would have been no risk premium: “The cost of hedging through a NegBasis was approximately 11 bp, whereas the cost of hedging through an AMPS trade was approximately 5–6 bp. The reasons for the differential pricing of hedging strategies that from a risk metrics perspective were deemed equivalent appears not to have been closely scrutinised,” UBS (2008) (p. 30). In other words, the cost of full protection against systematic risk was around 0.10%, roughly half of the premium was for linear systematic risk and the other half for non-linear systematic risk. However, with their beta-neutral AMPS portfolio, UBS paid only around 0.05% for protection against linear systematic risk and left the non-linear systematic risk unhedged. Already an increase in the market price of kurtosis risk later resulted in a significant mark-to-market loss. Nonetheless, UBS (2008) considered its positions as completely hedged: “Once hedged, either through NegBasis or AMPS trades, the Super Senior positions were VaR and Stress Testing neutral (i.e., because they were treated as fully hedged, the Super Senior positions were netted to zero and therefore did not utilize VaR and Stress limits)” (p. 30). 14 Given UBS’s ignorance of non-linear systematic risk, UBS was hardly fairly compensated and it is not surprising that UBS was considered “the biggest fool at the table” (p. 215) according to Lewis (2011).15 12 “AMPS provide a platform for hedging the credit spread exposure from UBS holdings in long synthetic and cash assets. Typical trades would be that UBS buys protection on a specified percentage of market value losses in a specified reference pool of ABS assets (CMBS, CDO, CLO) or to buy protection between two predetermined levels” UBS (2008) (p. 45). 13 “A negative basis trade is a transaction in which UBS holds a highly rated (generally Super Senior AAA) structural financial asset hedged with a credit default swap on the exact same asset out to full legal maturity” UBS (2008) (p. 14). 14 UBS (2008) “considered a Super Senior hedged with 2% or more of AMPS protection to be fully hedged. [...] [T]he long and short positions were netted, and the inventory of Super Seniors was not shown [...]. For AMPS trades, the zero VaR assumption subsequently proved to be incorrect as only a portion of the exposure was hedged [...], although it was believed at the time that such protection was sufficient” (p. 30). 15 At the end of 2008 UBS’s structured finance portfolio had to be sold at a huge discount to a special purpose vehicle called “SNB StabFund”. The equity (first loss tranche) was injected by UBS, the debt capital (second loss tranche) was provided by the Swiss National Bank (SNB). In effect, the bail-out of UBS by SNB was achieved by a CDO squared structure since the asset pool of “SNB StabFund” consists of CDOs. J. Risk Financial Manag. 2018,11, 27 20 of 26 7. Conclusions I offer an explanation in the spirit of Akerlof (1970) for the fall of the structured credit market after the financial market crisis in 2007/08 and why we still see problems resuscitating this market (see, e.g., Segoviano et al. (2015)). The systematic risk of two credit instruments with the same rating can vary substantially, in particular the non-linear systematic risk, but the systematic risk—unlike the rating—is not readily available to the average investor. That is, structured credit ratings are informationally insufficient for pricing even if ratings provide unbiased, powerful estimates on expected losses. However, a market in which potential buyers cannot correctly assess all pricing-relevant attributes of a product can attract sellers offering inferior goods, in particular, credit products of good rating quality but low systematic risk quality. The presence of market participants who are willing to offer inferior goods tends to drive the market out of existence. I propose to assess the quality of a credit instrument with end-of-period payoff X by four statistical (co-)moments: mean µX, covariance βX, coskewness γX, cokurtosis δX, where the three latter metrics are expressed with respect to a well-diversified portfolio M . Currently, rating agencies offer an assessment only about µX but are silent about βX , γX , δX . In other words, besides the rating information to compute the expected payoff µX=E[X|S] under the issuer’s information S , I suggest making also public the systematic risk, i.e., the contribution of X to the variance of M as well as the contributions to the third and fourth central moment of M (always conditional on the seller’s information S ). I show that a market in which potential buyers are ignorant about at least one of the four quality attributes has no equilibrium. Making public the seller’s information about µX , βX , γX , δX , not only µX , results in a symmetric market with an equilibrium. Such holistic quality assessments for more complex credit instruments can then be benchmarked against simple single-name bonds. As I illustrate, an investment-rated structured finance obligation can have worse systematic risk qualities than a subinvestment-rated corporate bond. With additional assumptions (i.a., the well-diversified portfolio M is the market portfolio), I offer a simple and straightforward four-moment CAPM that combines the quality attributes of a credit instrument µX , βX , γX , and δX into an equilibrium price g(µX , βX , γX , δX) . The variables µX , βX , γX , and δX are real-world risk metrics independent from any preference assumptions. To arrive at the pricing function g(·), I assume having a representative agent under standard risk aversion, inter alia. The fact that single-name bonds, structured finance securities and other credit products carry systematic risk contributions β,γ,δthat can be so different from a pricing standpoint casts significant doubt on whether some credit markets can really smoothly function with only the information provided by rating agencies about the expected payoff µ . The corporate bond market is possibly homogenous enough but other credit markets—in particular CDOs—certainly not. I illustrate that systematic risk cannot solely be measured by a linear CAPM beta since an asset can be negatively correlated with the market but can still be heavily exposed to non-linear systematic risk. I also demonstrate that counterparty risk of credit derivatives—such as a credit linked note, synthetic CDO, or default swap—has a significant impact on the overall product quality, in particular the product’s systematic risk. Finally, by considering two empirical cases, namely Morgan Stanley and UBS, I show that even big dealer banks were unaware about inherent non-linear systematic risk of seemingly hedged structured finance portfolios. That is, credit quality was inadequately assessed only based on rating and correlation. The small compensation for non-linear systematic risk was wrongly interpreted as value creation or CAPM alpha and was therefore hardly a fair risk premium for the huge losses that later materialized. Author Contributions: All analyses are done and the paper solely written by the author. The views expressed do not necessarily reflect the views of Zürcher Kantonalbank. Conflicts of Interest: The author declares no conflict of interest. J. Risk Financial Manag. 2018,11, 27 21 of 26 Appendix A. Proofs Proof of Proposition 1. By assumption the marginal utility function of the representative investor is a power function, u0(w)=w−λ , or equivalently, the utility function exhibits constant relative risk aversion, λ=−w u00(w)/u0(w). Using the definition of Z in (3) and noting that the distribution of w−λ and M−λ (with M=c w , where c is a scaling factor) are also log-normal, I can rewrite the equilibrium relation in (2) as follows: qk q0 =E"M−λ EM−λYk#, with a log-normally distributed pricing kernel Z: Z=M−λ EM−λ=M−λeλµM−λ2 2σ2 M. (A1) By assumption the portfolios M and P follow a bivariate log-normal distribution with correlation coefficient ρ: log M log P!∼N µM µP!, σ2 MσMσPρ σMσPρ σ2 P!!. (A2) Regressing log Monto log PI have: log M=µM+ρσM σP (log P−µP)+e, where log P and e are two independent Gaussian variables with the variance of e given by σ2 M1−ρ2 , a property resulting from linear projection. Since Z in (A1) is a positive variable with mean one under P, it has the properties of a Radon-Nikodym derivative. Hence, the moment generating function of the bivariate variable in (A2) under the martingale measure Qinduced by Zcan be written as follows: EQhet1log P+t2log Mi=Ehet1log P+(t2−λ)log M−log E[M−λ]i =e(t2−λ)µM−µPρσM σP+1 2σ2 M(1−ρ2)(t2−λ)2−log E[M−λ]·Eet1+(t2−λ)ρσM σPlog P =e(t2−λ)µM−µPρσM σP+1 2σ2 M(1−ρ2)(t2 2−2t2λ+λ2)+λµM−1 2λ2σ2 M ·et1+(t2−λ)ρσM σPµP+1 2[t2 1σ2 P+2t1(t2−λ)ρσMσP+(t2 2−2t2λ+λ2)ρ2σ2 M]. Let t1=t2= 0 and take advantage of the fact that EQhe0 log P+0 log Mi= 1, so that all terms in the exponent neither involving t1nor t2must sum up to zero, to obtain: EQhet1log P+t2log Mi=et1(µP−λρσMσP)+t2(µM−λσ2 M)+1 2t2 1σ2 P+1 2t2 2σ2 M+t1t2ρσMσP. (A3) However, (A3) is the moment-generating function of a bivariate Gaussian distribution. Hence, under the equivalent martingale measure Q , P is log-normally distributed with parameters µP− λρσMσPand σ2 P. Altogether, I have derived the following equilibrium relation: log PQ ∼NµP−λρσMσP,σ2 P, and log PQ ∼Nlog qP−log q0−1 2σ2 P,σ2 P, where the second expression follows from the martingale property EQ[P/Y0]=qP/q0. J. Risk Financial Manag. 2018,11, 27 22 of 26 Proof of Lemma 1.By decreasing absolute risk aversion, I have −u00 u00=−u0u000 +(u00)2 (u0)2<0, which requires that u000 >0 since u0>0. Conversely, decreasing absolute prudence implies u0000 <0, −u000 u00 0=−u00u0000 +(u000)2 (u00)2<0, since u00 <0. Proof of Proposition 2. The decision problem of such a representative agent can be written as a maximization of the expected utility: max c0,a0,...,aK{v(c0) + E[u(w)]}=max c0,a0,...,aK(v(c0) + E"u K ∑ k=0 akYk!#), (A4) subject to the constraint: w0=c0+ K ∑ k=0 akqk. (A5) where w is end-of-period wealth, the variable w=∑K k=0akYk is the the end of period wealth, Y0 is a constant or risk-free asset, respectively, v( . ) the utility function defined over initial consumption c0 , and u( . ) the utility function defined over end-of-period wealth, w0 is today’s initial wealth to be consumed and invested, ak is the number of units of financial claim k purchased, qk the price of claim k> 0, and Yk is the risky payout at the end of the period. Given the constraint in (A5) , I can rewrite the end-of-period wealth as follows: w=w0−c0−∑K k=1akqk q0 Y0+ K ∑ k=1 akYk. The first order conditions for a maximum in (A4) are therefore given by: v0(c0) = 1 q0 Eu0(w)Y0 Eu0(w)Yk=qk q0 Eu0(w)Y0, for k=1, ..., K, (A6) where the primes denote differentiation. The first order conditions (A6) can be written as: MRSk,0 = ∂ ∂akE[u(w)] ∂ ∂a0E[u(w)] =E[u0(w)Yk] E[u0(w)Y0]=Eu0(w) E[u0(w)] Yk Y0=qk q0 . In equilibrium the marginal rate of substitution MRSk,0 between asset k and the risk-free asset 0 must equal the quotient of their prices. It follows from the market clearing conditions and the identical characteristics of investors that c0=C0/N , w0=W0/N , w=W/N , where C0 , W0 , W represent the aggregates of current consumption, current wealth (for current consumption and to be invested for future consumption), and end-of-period wealth. By rearranging the first order conditions in (A6) , I obtain the desired equilibrium prices. J. Risk Financial Manag. 2018,11, 27 23 of 26 Proof of Proposition 3. A theoretical justification of the mean-variance-skewness-kurtosis analysis is to consider a fourth-order polynomial utility specification. In this case (and by the assumption that the fourth statistical moment exists, i.e., Ew4<∞ ), the expected utility ordering can be translated exactly into a four-moment ordering. Thus, let me first assume that the representative investor’s utility u(w)is a quartic polynomial function of end-of-period wealth w: u(w) = a0+a1(w−µ) + a2(w−µ)2+a3(w−µ)3+a4(w−µ)4, with µ=E[w] then u0(µ) = a1 , u00(µ) = 2 a2 , u000(µ) = 6 a3 , u0000(µ) = 24 a4 and I can write the marginal utility function in (2) expanded around the expected end-of-period wealth: u0(w) = u0(µ)+u00 (µ) (w−µ)+1 2!u000 (µ) (w−µ)2+1 3!u0000 (µ) (w−µ)3=:P3(w), where P3(w) denotes by definition the third order Taylor expansion of the marginal utility u0(w) around the mean end-of-period wealth µ. The mean marginal utility is therefore given by: Eu0(w)=u0(µ)+1 2!u000 (µ)Eh(w−µ)2i+1 3!u0000 (µ)Eh(w−µ)3i=E[P3(w)]. (A7) Since u0(w) = P3(w) by the assumption of a quartic utility function u(w) , I obtain an exact form of the Radon-Nikodym derivative Z: Z=u0(w) E[u0(w)]=P3(w) E[P3(w)]=u0(µ)+u00 (µ) (w−µ)+1 2! u000 (µ) (w−µ)2+1 3! u0000 (µ) (w−µ)3 E[P3(w)] =1 E[P3(w)]E[P3(w)]−1 2!u000 (µ)Eh(w−µ)2i−1 3!u0000 (µ)Eh(w−µ)3i | {z } =u0(µ) +1 E[P3(w)]u00 (µ) (w−µ)+1 2!u000 (µ) (w−µ)2+1 3!u0000 (µ) (w−µ)3 =1−λβK N w−µ Eh(w−µ)2i−λγK N (w−µ)2−Eh(w−µ)2i Eh(w−µ)3i−λδK N (w−µ)3−Eh(w−µ)3i Eh(w−µ)4i. The second line follows from (A7) , the last line follows by definition from the risk premia λβ , λγ , λδ : λβ:=−N K u00 (E[w]) E[P3(w)]Eh(w−E[w])2i, λγ:=−N K 1 2! u000 (E[w]) E[P3(w)]Eh(w−E[w])3i, λδ:=−N K 1 3! u0000 (E[w]) E[P3(w)]Eh(w−E[w])4i. Since w= 1 /N∑K k=0Yk is an affine transformation of M= 1 /K∑K k=1Yk and Y0 a constant, I finally obtain: Z=u0(w) E[u0(w)]=1−λβM−E[M] Eh(M−E[M])2i−λγ(M−E[M])2−Eh(M−E[M])2i Eh(M−E[M])3i −λδ(M−E[M])3−Eh(M−E[M])3i Eh(M−E[M])4i. (A8) J. Risk Financial Manag. 2018,11, 27 24 of 26 If the representative investor has a quartic utility function then u0(w)/E[u0(w)]=P3(w)/E[P3(w)] is an exact equality. However, more generally, if I approximate the utility function u(w) by a fourth order Taylor series around the expected end-of-period wealth then I can approximate the Radon-Nikodym derivative Z by P3(w)/E[P3(w)] . To obtain the pricing formula qX/q0=E[X Z] in (4) with Z given in (A8) note that with V= (M−E[M])m , m∈{1, 2, 3} , µV=E[V] , µX=E[X] , I have the equality E[X(V−µV)]=E[(X−µX)V]. As seen in Lemma 1, standard risk aversion implies u0> 0, u00 < 0, u000 > 0, and u0000 < 0. The risk premium λγ for skewness risk is positive if the third central moment of the wealth distribution is negative, i.e., if skewed to the left. The premia for variance and kurtosis risk, λβ and λδ , must be positive when u(·)exhibits standard risk aversion.16 References Akerlof, George A. 1970. 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