Default Risk Premia in Synthetic European CDOs

Transcription

1 Default Risk Premia in Synthetic European CDOs Thomas Weber Department of Economics University of Konstanz Germany May 31, 2008 Dipl. Wirt.-Math. Thomas Weber, Chair for International Finance, Department of Economics, Faculty of Law, Economics, and Political Science, University of Konstanz, Box D147, Konstanz, Germany; Fone: +49-(0) ; Fax: +49-(0) ;

2 Abstract Default Risk Premia in Synthetic European CDOs In CDOs, credit risk referring to large portfolios of loans or bonds is sold. Information about the risk-return profile of 215 differently rated tranches of 59 European CDO transactions is used to estimate an implicit risk aversion parameter for each tranche. This is done by a pricing kernel approach that assumes constant relative risk aversion. As a state defining variable the aggregate loss rate of all outstanding loans is used. Riskier tranches imply significantly lower relative risk aversion. The lower the tranche in the structure of the CDO, the more market frictions matter. When controlling for tranche risk, poor portfolio diversification induces significantly higher risk premia for the lowest two tranches of a transaction. Keywords: credit risk, risk premia, pricing kernel, CDOs. JEL-Classifications: G13, G12, G24

3 1 Introduction The market for securitisations of assets has seen tremendous growth over the last decade. The global issuance volume of Asset Backed Securities has risen from $ 271 bn in 1997 to $ 1600 bn in 2005 (HSBC, 2006) with European transactions accounting for ca. $ 30 bn (39 bn e) in 1997 and $ 450 bn (315 bn e) in The European issuance volume of Collateralised Debt Obligation (CDO)-transactions also increased sharply in recent years (17 bn Euro in 2002 to 57.8 bn Euro in 2005). In CDO transactions investors earn premia, usually paid as credit spreads on top of the risk-free interest rate, for absorbing potential default losses of a portfolio of loans or bonds. In a synthetic transaction the originator buys protection against potential losses of a credit portfolio from a Special Purpose Vehicle (SPV) which, in turn, buys protection by issuing credit linked notes to investors. 1 These credit linked notes, the tranches of the transaction, differ with respect to their risk profile and with respect to the promised credit spread. The risk profiles differ because of the strict subordination between tranches. Losses are first assigned to the tranche which is lowest in the hierarchy of payments (the lowest ranked tranche). Only if the losses assigned to this tranche have reduced its remaining face value to zero the exceeding losses are borne by the next tranche, and so on. The tranche ranked lowest is called First Loss Piece (FLP) and usually not rated by an external rating agency. The other tranches have an external rating. They and earn a credit spread which compensates the investor for the credit risk borne. Obviously, the credit spread is higher the higher the risk of a tranche, respectively the lower its ranking. But how does the link between risk and return exactly look like in this market? Are the risk premia determined differently for different tranches? Do certain characteristics of the reference portfolio correspond to additional premia for the tranches? How important is the position of a tranche within the respective transaction? This paper addresses these questions first in a theoretical manner deriving some hypotheses. Subsequently, the hypotheses are tested on a dataset of 59 synthetic European CDOs consisting of 215 differently rated tranches. In order to properly compute the risk premia a simulation model is used to derive the expected repayment distribution of all tranches conditional on the realisation of a systematic risk factor. This macro factor is defined as the aggregate loss rate of all outstanding loans. It drives the values of the reference portfolio of the transactions and thus the repayments of the tranches. Furthermore it is the state defining variable in the pricing model. Since 1 I.e. bonds only repaying the full face value if no or little losses occur in the portfolio and less when losses are higher. 1

4 originators as well as most investors in the CDO market are financial intermediaries 2 the aggregate wealth in this sector is assumed to depend on the aggregate loss rate. Assuming constant relative risk aversion, the risk-return profile of a given tranche and its correlation structure with the systematic risk factor imply a risk aversion parameter. For each tranche, this parameter is computed. Given the extreme differences in transferred credit risk in, say a AAA and a BB rated tranche it is assumed that different types of investors are attracted. Probably the risk aversion of an investor declines the riskier the tranche he invests in. This first hypothesis is confirmed in the data. Effects on a transaction level have to be analysed in the view of this result. When assessing the impact of a change in the expected portfolio losses on the average implied risk aversion of a transaction, the adjustments in the tranche structure are crucial. This leads to a somewhat counter-intuitive result: higher expected portfolio losses are not fully offset by an increase in loss protection for the rated tranches, which leads to, on average, riskier issued tranches that in turn imply a lower risk aversion parameter. On the other hand, higher expected portfolio losses also leave more room for asymmetric information. This should increase the spread to be paid on tranches and thus, ceteris paribus, the measured risk aversion parameter. This counteracting effect however, is only relevant for the lowest two of all rated tranches probably because their loss protection is not considered high enough to shield against information problems. The reasoning when discussing the impact of portfolio diversification is similar. Only the lowest two rated tranches of a CDO show a significant reaction to changes in diversification. Controlling for tranche risk, a poorly diversified portfolio corresponds to higher measured risk aversion. This indicates additional premia due to information problems and/or idiosyncratic risks. The paper proceeds as follows. The next section briefly reviews the relevant literature. In section 3 the hypotheses are derived and the simulation model is explained. Section 4 presents the dataset and the variables characterising the CDO transactions. In section 5 the computed risk aversion parameters of the different tranches are used to empirically test the hypotheses. Section 6 concludes. 2 Literature Review This paper is related to three strands of literature. Tranches of CDOs essentially are bonds subject to some default risk therefore there is a link to the bond pricing literature. Secondly, this paper uses the parameter of a pricing kernel function as proxy for the implicit risk 2 For evidence for the German market see Deutsche Bundesbank (2004). 2

5 aversion. Lastly there has to be a credit risk model underlying the simulation procedure - these models are discussed in Section 4.2. Risk premia of corporate bonds have been widely studied in the literature with different approaches. Elton, Gruber, Agrawal and Mann (2001) analyse the spreads of US corporate bonds over US treasury bills using market data from 1987 to They first deduct expected losses and a tax premium from these spreads and then regress the remainder on Fama-French risk factors finding evidence that the return on corporate bonds contains a significant premium for these factors. Huang and Huang (2003) calibrate several structural credit risk models to historical default rates and historical equity premia and find that for investment grade bonds (rated Baa or better) credit risk premia account for only 20 to 30% of the observed credit spread. Chen, Lesmond and Wei (2007) suggest an illiquidity premium as a further component of corporate bond spreads. Driessen (2005) uses six components to explain the credit spreads of bonds including a risk premium on market-wide changes in credit spreads and one component accounting for a risk premium on the risk of a default jump - the downward jump of the bond price in case the debtor defaults. He shows that the latter risk premium explains an economically significant part of Baa rated corporate bond returns but cuts back that statistical evidence is inconclusive about the existence of this premium. When excluding the default jump risk premium he finds that the model underestimates expected excess returns on corporate bonds and overestimates observed default rates. He, and others, refer to this and similar findings as the credit spread puzzle. Further he finds that default jump risk may not be diversifiable. Amato and Remolona (2003 and 2005) also analyse the credit spread puzzle. Both papers argue that due to the strong skewness of the default loss distribution idiosyncratic default risk can not be fully diversified in realistically sized portfolios - and therefore has to earn a premium. They define the spread ratio of a corporate bond as the ratio of the promised spread and the annualised expected loss of this bond. For European Aaa, Aa, A and Baa rated bonds they find the average spread ratio to be 210, 35, 6.7 and 1.6, respectively. They reason that the large size of these ratios provides evidence for significant risk premia associated with idiosyncratic default risk. The spread ratio defined by Amato and Remolona can be seen as an approximation of the relation between the risk-neutral and the physical default probability for a two state setting. In order to illustrate this interpretation let us look at a corporate coupon bond with one year to maturity. For each e invested this bond repays either (1+ risk free rate + spread) in the good state or nothing in the bad state. The alternative for the investor is to buy a treasury bond which repays (1+risk free rate) in both states. Assuming both bonds are issued at par, the spread earned in the good state exactly compensates for the loss of (1+ risk free rate) in the bad state. In a risk neutral pricing environment and for low default 3

6 probabilities the spread therefore roughly equals the risk neutral default probability. So the relation between spread and physical default probability approximates the relation between the risk-neutral and the physical default probability. 3 The pricing kernel to be estimated in this paper is a generalisation of the spread ratio from the two-state setup to a model with a continuous state space. Usually a CDO transaction refers to a relatively large portfolios of bonds or loans. Since each bond or loan can default there is a range of possible terminal values for the portfolio. Consequently the total repayment of the issued tranches will take values from this range. In general, the pricing kernel is defined to be the transition function that describes the relation between the physical and the risk-neutral density functions. Several papers have used option prices on stock indices to estimate risk-neutral densities and compare these to the estimated physical densities derived from time series data of the underlying indices. The approaches differ in the choice of the so called state defining variable. For example, Jackwerth (2000), Ait-Sahalia and Lo (2000), Rosenberg and Engle (2002) use approximations of total wealth in an economy such as stock index values or stock index returns as state defining variable. Other studies estimate the pricing kernel on states defined by not only a market index value but also further variables. Buraschi and Jackwerth (2001) and Coval and Shumway (2001), among others, add volatility and find evidence for a negative volatility risk premium. Brennan, Liu and Xia (2005) add further variables that describe the opportunity set of investments, the real interest rate, the maximum Sharpe ratio and the expected rate of inflation to the set of state defining variables and find that the lack of these variables in part explains some empirical problems of classic asset pricing models. 3 Note that this relationship also holds if some money is recovered in case of default: If the defaultable bond pays x% of the promised payment in the bad state then the spread equals (1 x%) times the risk neutral default probability and the expected loss equals (1 x%) times the physical default probability, so after cancelling out the interpretation of the spread ratio as relation between the risk-neutral and the physical default probability stays the same. 4

7 3 Model and Hypotheses In a perfect capital market securitization design is irrelevant. More general, there would be no need for banks, either. But the market for corporate debt is imperfect. In particular, there are strong information asymmetries between bank and debtor. Private information of the debtor are revealed to the bank at some costs. In the end, the debtor has to bear these costs in form of fees or in form of a relatively high interest rate. If a bank wants to securitise a portfolio of debt by issuing a collateralized debt obligation (CDO) information problems are again important. The bank or originator has an information advantage over some outsiders whom she wants to sell (part of) the claims to her loan portfolio. This resembles the setting modelled by DeMarzo (2005). Pooling the claims now on the one hand destroys information since else separate signals about debtor quality would have been possible. On the other hand pooling has a positive diversification effect allowing the originator to issue low-risk tranches that are less sensitive to private information. De- Marzo (2005) shows that for large portfolios the positive diversification effect outweighs the disadvantages of information destruction. This implication of De Marzo s model is in line with the observation that in basically all multi-tranche CDOs there is strict subordination between tranches. 4 Strict subordination means that there is a strict ranking between tranches in the priority of repayments. Losses only reduce the repayments to a tranche if the principal values of all lower ranked tranches have been reduced to zero, i.e. if there will be no further payments to these lower tranches. For given tranche sizes, assigning the repayments in this way maximises the loss protection for the higher tranches. Further it minimises their exposure to idiosyncratic risks and thereby also their vulnerability with respect to information asymmetries. The most information sensitive part of the loss distribution is covered by the first loss piece (FLP). In synthetic transactions the FLP is commonly modelled by including a threshold in the swap agreement between originating bank and special purpose vehicle - and thereby ultimately investors. This way only the less information sensitive parts of the loss distribution are covered by outside investors buying the tranches. In sum, due to strict subordination the issued tranches of a single CDO differ with respect to information asymmetries as well as in the transferred credit risk. Consequently, investors buying different tranches will vary with respect to their ability to resolve information asymmetries and/or with respect to their attitude towards risk. For most transactions, the FLP 4 For example all 169 transactions analysed by Franke, Herrmann and Weber display strict subordination between tranches as do all 59 transactions analysed in this paper. 5

8 is relatively large compared to the expected loss of the underlying portfolio. 5 In combination with the diversification effect of securitising a portfolio of at least 40, mostly more than 100 loans or bonds this shields investors of most idiosyncratic risks. Given that additionally rating agencies resolve part of the remaining information problems I assume that investors buying different tranches can be distinguished best by their attitude towards risk. Thus, differences in transferred risk separate the market for CDO tranches. Due to the fewer risks transferred and to the smaller spread to be earned in higher and better rated tranches, these relatively safe tranches will attract investors that are more risk averse. In the following I compute the risk aversion implied by the risk-return profile of all rated tranches for a set of synthetic European CDO transactions. I do so for each tranche separately. In a market separated by the amount of risk transferred the measured risk aversion parameter should be relatively high for the higher tranches with good ratings and decrease for lower ratings, respectively lower relative tranche positions. So irrespective of the modelling procedure I state: Hypothesis 1: The measured risk aversion should decrease the riskier the tranche. 3.1 Pricing Model and Simulation Procedure I use a pricing model that considers the risk-return profile of the issued CDO tranches as well as their correlation structure with an underlying macro factor to compute the risk aversion parameter implied by the repayment structure of each single tranche. The macro factor driving the model is the aggregate loss rate in the economy. The pricing model only rewards a risk premium for systematic risk as it is based on a pricing kernel approach where the states are defined by the aggregate loss rate. Suppose that tranche i of some CDO transaction is purchased by a group of investors that display similar risk attitude. This is expressed by a group-specific pricing kernel function φ i = φ i (s). Since affiliation in this investor group is defined by the purchase of tranche i the same index i will be used for both tranche and investor group. s denotes the state defining variable. Then the forward pricing equation for this tranche can be written as F i = E Q i (P i ) = E(P i s) q i (s) ds s S = E(P i s) f(s) φ i (s) ds, (1) s S 5 Franke/Herrmann/Weber (2007) find for 169 European CDO transactions that in expectation the FLP covers more than 85% of all losses. With on average 88% probability no rated tranche of a given CDO suffers any loss. 6

9 where F i denotes the forward price of tranche i, P i denotes the payoff of the tranche that depends on the state s and f(s) denotes the physical density function of the state variable. The risk neutral measure that has to be applied here depends on the investor group/market segment. For investor group i the risk neutral measure is labelled Q i, and the corresponding density function q i is computed as q i (s) = f(s) φ i (s). The expected value under this risk neutral measure is labelled E Q i, and E(P i s) denotes the expected payoff of tranche i conditional on state s. 6 For illustration, φ i (s) can be interpreted as the standardised marginal utility of an investor representative for investor group i: φ i (s) = u i (s) E(u i ( )).7. For each market segment I assume constant relative risk aversion. For each tranche, I compute the parameter of relative risk aversion under which equation (1) holds. Note that this parameter is uniquely defined since the forward price F i on the left hand side is given. On the right hand side there is a monotonous relationship between risk aversion and E Q i (P i ). The higher is risk aversion, the lower is E Q i (P i ). This holds for each tranche of each transaction. In order to perform this computation, several issues have to be addressed. First, the aggregate loss rate as systematic risk factor has to be estimated. Second, this risk factor has to be linked to the state defining variable. For all market participants the same state variable will be used which can be seen as an element of market integration. This variable will be the aggregate wealth of the representative agent within the credit risk market. Finally, as the aggregate loss rate drives the performance of the various securitised debt portfolios, the expected tranche repayments for given aggregate loss rate levels have to be computed for each CDO tranche. For this step, a credit risk model is needed. Systematic Risk Factor: Aggregate Loss Rate The aggregate loss rate is the only systematic risk factor. It is defined as the percentage loss of all outstanding debt in the economy. It has a direct impact on the aggregate wealth of the agents in the debt market and it also influences the payments of the CDO tranches. The loss rate of loan n, l n is defined as zero if loan n does not default and as the (stochastic) loss given default (LGD) of loan n (LGD n ) in case of default. The aggregate loss rate in the economy, l, is defined as the weighted average loss rate of all loans: l = l(d) = d E(LGD d), (2) where d is the default rate in the economy, i.e. the face value-weighted fraction of all out- 6 Note that the law of iterated expectations has been used to obtain the last equation. Further note that the state conditional expected payoff E(P i s) is the same under the statistical measure as under the risk adjusted measure Q i. Given the state, these two measures coincide, they only differ in the weights of different states. 7 See, e.g., Poon and Stapleton, 2005 for the general formula. u i denotes the utility function for investor group i 7

10 standing loans defaulting over a given time horizon. First, I estimate the distribution of the default rate d. In contrast to the LGD the default rate will depend on the respective time horizon t. Accordingly, the loss rate will also depend on t. For time horizons longer than one year the default rate as well as the loss rate are to be interpreted as the value weighted fractions of cumulated defaults and losses. They are not annualized. Let d t and l t denote the (cumulated) default rate, respectively the (cumulated) loss rate in the economy over a t year horizon. E.g. if d 3 = 10% and E(LGD d 3 = 10%) = 60%, then 10% of all loans have defaulted over the course of three years causing a loss of l 3 = 6% of the aggregate face value 8. When estimating the distribution of d t, I rely on data publicly provided by Moody s in their cohort study of This study includes aggregate default and loss data on all corporate borrowers rated by Moody starting from For each year and for each of seven rating classes (Aaa to Caa) the cumulative default rates are given for all maturities between one and 20 years. Since I am only interested in the default rate of the economy as a whole and not in the default rates of specific rating classes I compute the weighted average of the seven rating specific cumulated default rates as a proxy. I do so for each year and for each maturity. For all maturities the default rates are estimated to be lognormally distributed. The exact method and results are given in Appendix A. From the same data the LGD is estimated to be a normal distribution (truncated between zero and one) with a mean of 60% an a standard deviation of 10%. The LGD and the default rate are positively correlated with ρ = 0.5. This is accounted for by computing E(LGD d t ) as [ ρ ln(d t ) µ ] t E(LGD d t ) = E(LGD) + Stdev(LGD) σ t. (3) Since E(LGD d t ) increases with d t and low for low d t the distribution of l t is more skewed than the distribution of d t, i.e. more skewed than lognormal. Graph for, say, 5y loss rate? State Defining Variable: Aggregate Wealth This paper follows the limits of arbitrage theory when identifying the state defining variable. This means the marginal/average market participant in the CDO market is invested in the debt market rather than a perfectly diversified representative agent. The two sides negotiating the price or spread of CDO tranches are the originating banks as protection buyer and investors as protection sellers. Since the originating banks commonly sell only a small fraction of their debt portfolios their wealth strongly dependends on the performance of the aggregate outstanding debt. The majority of investors in the CDO market are financial 8 It is assumed that at least at the beginning of a period face value and market value of the loans coincide. 8

11 intermediaries themselves 9, their wealth therefore showing a similar dependence. The other investors are specialised investors, e.g. hedge funds seeking arbitrage opportunities in the credit risk market. They are assumed to have invested a significant share of their assets in this market, too. Further, for the market of mortgage backed securities - similar transactions as CDOs - Gabaix et al (2007) find evidence for the limits of arbitrage hypothesis. Since all market participants have a strong exposure to the performance of risky debt I assume that terminal wealth is given by terminal wealth = initial wealth endowment + interest earned default losses Note the important role of the initial wealth endowment (W 0 ) here. If W 0 equals the face value of the outstanding debt then the equation describes the equity of an un-leveraged i.e. fully equity financed bank. Normally banks and the other market participants leverage their credit risk exposure. This corresponds to W 0 being significantly lower than the volume of outstanding debt. Let D 0 denote the volume of outstanding debt at time 0 and i D the interest rate to be paid by debtors. Further, let L 0 denote the liabilities of the representative agent, say a representative bank, and i L the interest rate to be paid on L 0. Then W 0 = D 0 L 0 and W = D 0 (1 + i D loss rate) L 0 (1 + i L ). So, besides of the loss rate the distribution of wealth depends on the interest rate difference i D i L and the initial leverage D 0 /W 0. In the following I assume an initial leverage of 10, i.e. a representative bank with an equity ratio of 10%. Further, I assume the interest rate difference to equal six times the expected loss rate meaning that the debtors, if they do not default, pay six times their expected default losses on top of the refinancing interest rate of the representative bank. 10 representative bank will be constructed risk free. 11 This rate is assumed to equal the risk free rate since the wealth for different time horizons are characterised in appendix A. Simulation Procedure The resulting distributions of terminal 9 For the German market see Deutsche Bundesbank (2004). 10 Note that this assumption is motivated by estimates from the bond market: Between January 2000 and December 2005 the average spread of a five year BBB bond over the risk free rate - measured by the average difference in yield to maturity of the IBOXX BBB corporate bond index over the IBOXX sovereign index for maturities 3-5 and 5-7 years - was 127 basis points. The average of Moody s and S&P prediction for the probability of default for a five year BBB bond is 1.95%. For an average loss given default of 55% this implies an expected loss of 1.07% over five years or roughly 21 basis points annually. This is close to one sixth of the average bond spread of 127 basis points. 11 DEALSCAN data show that banks rated AA- or better on average pay less than 10 basis points spread on their bonds. 9

12 Several methods to analyse the loss distribution of a loan portfolio and the implications on the cash flows of the different tranches of CDO transaction are available. Migration models use latent variables that drive rating changes and defaults. This paper follows the CreditRisk+ 12 framework and models default probabilities directly, without a latent variable. Here, the default probability of each loan or bond 13 depends on the realisation of several macro variables. The impact of each macro variable is given by the factor loading. For the subsequent simulations I use a simple version of this model with only one macro factor, the default rate in the economy, which is in turn deterministically linked to the aggregate loss rate through equations (2) and (3). For a t year loan with rating j the (cumulative) probability of default conditional on the default rate in the economy d t, P D j,t (d t ) is then given by P D j,t (d t ) = P D j,t d t E(d t ). (4) P D j,t denotes the unconditional default probability of the loan. 14 In this setting the default events are independent conditional on the macro factor. This means that given, for example, a scenario with twice as many defaults in the economy as expected, d t = 2 E(d t ), the default probability of every loan is modelled to be twice as high as its unconditional default probability. Note that since high (low) realisations of the default rate in the economy lead to high (low) default probabilities of all the loans, this procedure induces an implicit correlation of default events. This correlation, either implicitly generated like in this model or accounted for by correlating the latent variables as in most migration models is crucial for the loss distribution of a loan portfolio. 15 Appendix B presents a migration model and compares the default correlations of the different model types. Given equation (4) and the independence of defaults conditional on d t, the expected repayments for a t year CDO-tranche conditional on d t, E(P t i d t), are now computed using a Monte Carlo simulation. The superscript t is added to the tranche repayments P i to distinguish the respective maturity of the transaction. The advantage of directly modelling the t- 12 For a detailed description see either Gordy (2000) or Credit Suisse Financial Products (1997). 13 In the remainder of this chapter I will refer to portfolios of loans only. All results hold for bond portfolios as well with the same derivation. 14 Of course, P D j,t(d t) has to be truncated at one, but this would only become necessary for extremely high values of P D j,t. In general, CreditRisk+ defines the (cumulative) probability of default of a loan or bond conditional on the realisation x = (x 1,..., x K ) of a vector of macro variables as P D j,t (x) = P D j,t K k=1 ω jk(x k ). Here, ω jk denote the factor loadings. 15 For example, Franke, Herrmann and Weber (2007) use a lognormal and a gamma distribution to model the loss distribution of loan and bond portfolios. They use a latent migration variable approach to calibrate the parameters of these distributions using different assumptions with respect to the asset correlation of the loans/bonds. The results, e.g. the loss share and default probability of the sold tranches, vary only little between the lognormal and the gamma distribution. The variation given a change in correlation assumptions however is much stronger. 10

13 year cumulative default probability is that one need not worry about possible autocorrelations effects as in the migration models. 16 necessarily increase linearly in t - already accounts for this. A correct specification of P D j,t - that does not Equation (4) and the respective repayment rules implemented in the simulation consequently provide the link between the systematic risk factor aggregate loss rate and the tranche payoffs. For the tranches of a t year CDO transaction this connection is established by the t year cumulative default rate in the economy, d t. For each transaction the Monte Carlo simulation proceeds as follows for a given realisation of d t. First, the conditional (cumulative) default probability of each loan is determined using equation (4). Then given this default probability a loan-specific random variable determines whether this loan defaults or not. The loan-specific random variables of different loans are independent and identically distributed. For each loan k I use a random variable (η k ) uniformly distributed on the interval [0; 1], meaning loan k with rating j and t years to maturity will default if η k < P D j,t (d t ), otherwise it pays back as scheduled. 17 So in each run of the simulation the loans defaulting given a realisation d t of the default rate are drawn. Thirdly, for each loan defaulting over the lifetime of the transaction the loss given default (LGD) and the default time are subsequently determined in the Simulation. a time series of losses induced by the underlying portfolio. This yields Next, given this information, the repayments to the tranches are computed. Finally, discounting these repayments to the P issuance date of the transaction yields i (1+r) d t t. The average value of this term, taken from repeated runs of the simulation for the same d t converges to E( Pi (1+r) t d t ). For a detailed description of the simulation procedure see Appendix 8.3. Figure 1 exemplifies the relationship between the cumulative default rate in the economy and discounted tranche repayments per Euro invested. It can be interpreted in the following way: investors buying one of the tranches pay one (risk-free) Euro in exchange for the respective risky payoff that depends on the macro factor as depicted. For each tranche separately, the risk aversion parameter is computed from this relationship. 16 See, e.g., Lando (2002) for a study on the autocorrelation of rating migrations. 17 Note that this model is similar to the migration models presented in the appendix using equation (8). For a given realisation of M, obligor k defaults if and only if ɛ k < (ζ k M ρ)/ 1 ρ. So in both models one macro factor define a conditional default probability and an idiosyncratic variable determines whether the obligor defaults. The idiosyncratic variables are mutually independent. 11

14 Figure 1. Simulation results for the discounted expected repayments of five differently rated tranches from the CDO transaction London Wall , conditional on the six year cumulative default rate in the economy d 6. Note that there are disproportionally many data points with high default rates since these are the most interesting. Computing the parameter of relative risk aversion With the results of the simulation the parameter of relative risk aversion can now be computed from the pricing equation. First, equation (1) has to be modified. The loss rate l t is plugged in as systematic risk factor and aggregate wealth W t is plugged in as state defining variable. The superscript t is added to the forward price of the tranche F i, too: F t i = 1 l t=0 E(Pi t d t ) f(l }{{} t ) φ k (W t (l t ))dl t (5) =E(Pi t lt) Assuming a constant risk free interest rate r the forward price can be written as F t i = P 0 (i) (1 + r) t, where P 0 (i) denotes the issue price of tranche i. Usually all tranches are issued at par. After standardising the face value of the tranche to one, I discount on both sides of the equation (5). Due to the deterministic relationship between l t and d t given by equations (2) and (3) I can exchange the variable E(P i ) is conditioned on. Assuming constant relative risk aversion in the pricing of each tranche and plugging in the functional form of 12

15 the pricing kernel, equation (5) transforms to 1 = P 0 (i) = F t i (1 + r) t = 1 l t=0 E(P t i l t) (1 + r) t f(w t(l t )) (W t (l t )) γ i d l t. (6) This means An investor pays 1 e today in exchange for the claim on a random future payment Pi t. Note that the distributions of P t i and E(Pi t d t) depend on the distribution of losses allocated to tranche i as well as on the credit spread to be paid on this tranche. Given the monotonously negative dependance of tranche repayments from the aggregate loss rate, an increase in γ i always depreciates the right hand side of equation (6). 18 Consequently, the γ i solving (6) is uniquely defined. Ceteris paribus more losses allocated to a tranche will decrease γ i, a higher credit spread will increase γ i. 3.2 Hypotheses For each tranche, I use equation (6) to determine the respective risk aversion parameter. These parameters may differ if the tranches belong to different market segments i.e. attract different types of investors demanding different risk premia. Further, variations in some portfolio characteristics or tranche specific features may cause additional premia. For example, information asymmetries and unsystematic risk are not accounted for in equation (6). Should they motivate a higher credit spread this would show in higher values for the measured risk aversion parameter. Average Portfolio Quality How does the average loan or bond quality of the portfolio underlying a transaction influence the risk aversion parameter for the issued tranches? It is problematic to answer this question only with respect to single tranches since for CDOs referring to portfolios of different quality originators may choose different CDO designs. In particular, the size and loss protection of a given tranche may be adapted to the quality of the underlying portfolio making an ad-hoc prediction on the above question difficult. Therefore I first investigate how the average loan/bond quality may influence the average risk aversion parameter of a CDO transaction. This average can either be defined as the weighted average of all tranche parameters. Or it can be defined as the coefficient γ CDO that, instead of equation (6) solves #of tr. i=1 w i P 0 (i) = #of tr. i=1 1 w i d t=0 E(P t i d t) (1 + r) t f(w t (d t )) (W t (d t )) γ CDO d d t. (7) 18 Due to the limited number of simulation runs for some tranches this monotonicity is not fulfilled in a strict sense but only approximately. 13

16 In both cases the weight of a tranche w i is defined as size i #of tr.. I refer to γ i=1 size CDO that solves i this equation for a given CDO transaction the transaction-γ. To isolate the effect of loand/bond quality I idealize and investigate two CDOs, CDO A and CDO B, each referring to a perfectly diversified portfolio. Then the loss rates of portfolio A and B are: 19 l A = l 1 a, l B = l 1 b, where l is the aggregate loss rate and a, b are constants. Without loss of generality I assume the loans in portfolio A to be of better quality: a < b. Further, I normalize the portfolio volume to one and let at Ai (at Bi ) and de Ai (de Bi ) denote the attachment and detachment point of tranche i of the respective CDO. Crucial for the tranches of the transaction is how these points are chosen given the quality of the portfolio. As a benchmark case I first assume that both CDOs have the same S&P rating structure of the tranches. 20 This corresponds to: Assumption 1. The corresponding tranches of the two CDOs display equal probability of default. Consequently, for both transactions the respective tranches cover the same quantiles of the loss distribution. I.e. for each transaction tranche number i is characterized by P rob(lossp F A < at Ai ) = P rob(lossp F B < at Bi ) = p i P rob(lossp F A > de Ai ) = P rob(lossp F B > de Bi ) = p i 1, where i 1 denotes the next higher tranche. Given this assumption there exist values l i and l i 1 of the loss rate such that p i = F 1 (l i ) and p i 1 = F 1 (l i 1 ), where F denotes the cumulative distribution function of the loss rate. Given l i and l i 1 the attachment and detachment points are at Ai = l i a, de Ai = l i 1 a and at Bi = l i b, de Bi = l i 1 b. Let spr Ai (spr Bi ) denote the promised spread for tranche i of CDO A (CDO B ), A i (B i ). Then the repayments per e invested i tranche i of CDO A conditional on l are: 1 + r + spr Ai if l < l i repayment per e invested = l i 1 l (1 + r + spr l i 1 l i Ai ) if l i < l < l i 1 0 if l > l i 1 and the analogous equation for tranche B i where spr Ai is replaced by spr Bi. Due to the equal repayment structure, both tranches A i and B i are in the same market segment and the same γ i should be applied in the respective pricing equations (6). Accordingly spr Ai = spr Bi, else at least one of the equations does not hold. 19 Perfect diversification means that portfolio losses are deterministically driven by the macro factors because all idiosyncratic risk is diversified away. For simplicity I assume a maturity of one year. The results are easily generalized to multi-year transactions. 20 The rating agencies differ with respect to the repayment characteristics defining their respective assessment. Moody s put the emphasis on the expected loss of the tranches whereas S&P uses the probability of default. 14

17 However, Franke, Herrmann and Weber (2007) find that the loss probability of the lowest rated tranche increases for lower average loan/bond quality of the reference portfolio. This contradicts assumption 1. Supported by these empirical findings for the lowest rated tranche I alternatively state: Assumption 2. The lower the average quality of the reference portfolio the higher the loss probabilities of the lower tranches. If tranche ratings are defined by the default probability this would mean that the lower tranches carry a lower rating given a lower average reference portfolio quality. Given that for basically all transactions the highest tranche is rated Aaa I assume this effect to diminish the better the position of the tranche within the transaction. If tranche ratings are more influenced by the expected loss the rating of the lower tranches may stay the same even if their loss probability increases, but only at the expense of an increase in relative size. Given Hypothesis 1 this leads to following relationship: A lower average quality of the reference portfolio yields a lower average risk aversion factors and a lower transaction-γ. Overall, depending on how the transaction design reacts to changes in the average quality of the securitised loans or bonds, assumption 1 versus assumption 2, two alternative Hypotheses can be stated: Hypothesis 2a: The γ i s of the issued tranches and therefore also the weighted average γ, and the transaction-γ are not influenced by differences in the average quality of the underlying portfolio (Assumption 1). Hypothesis 2b: Lower average portfolio quality corresponds to a lower weighted average γ, respectively to a lower transaction-γ. (Assumption 2). These two hypotheses do not take market imperfections into account. Yet, information asymmetries may be strong in some CDO transactions and only partially resolved by the rating agencies. The rating agencies signal on the portfolio quality may be more accurate the better the quality of the underlying portfolio because there is usually more ambiguity with respect to lower rated loans. For instance, the differences in predicted default probability between neighboring rating classes increase for lower ratings. For higher tranches however this effect should be smaller. On the one hand, their higher loss protection shields these tranches against idiosyncratic risks. On the other hand is the better rating of these tranches a more exact signal on the tranche quality than for lower tranches with weak rating. Taking information problems into account thus adds a third hypothesis: Hypothesis 2c: Lower average portfolio quality increases the weighted average γ, respectively 15

18 the transaction-γ due to stronger information asymmetries. For the tranche- γ i s this increase is stronger for lower tranches. Portfolio Diversification The effect of portfolio diversification on the risk premium of a given tranche also depends on the extent of frictions in the credit risk market. If there are little or no market frictions, all market participants can easily diversify their credit risks and information problems are small. This leads to: Hypothesis 3a (no market frictions): The risk aversion parameter of a tranche does not depend on the degree of diversification of the securitised portfolio. If market frictions do play a role the situation is different. Amato and Remolona (2003 and 2005) argue that sufficient diversification is harder to achieve in bond markets than in stock markets. Due to the asymmetric payoff distribution a lot of bonds are needed leading to non-negligible transaction costs when building a well diversified portfolio. Given that diversification is costly there may be a premium on idiosyncratic risks, too. This would, ceteris paribus, increase the measured γ i. 21 For highly rated tranches though the idiosyncratic risks have less influence due to the stronger loss protection. Additionally, more idiosyncratic risks in a poorly diversified portfolio leave more room for information asymmetries which also leads to a higher γ i. Similar to the information problems caused by lower average portfolio quality this effect should be stronger for lower tranches. Combined this motivates Hypothesis 3b (market frictions): For tranches referring to a well diversified portfolio the risk aversion parameter should be lower. This effect should be stronger for lower rated tranches with less loss protection. Maturity, dynamic transactions, CBOs vs. CLOs The analysed transactions differ with respect to some other characteristics. They have different maturities. The portfolio may consist of bonds of of loans. Some transactions refer to static portfolios, on other dynamic transactions the portfolio may change over time because it is replenished, i.e. the originator may replace maturing or prepaying loans by new ones subject to some rules ( replenishment criteria ) γ i is computed assuming that only systematic risk is rewarded a premium. A higher credit spread paid due to idiosyncratic risk would show in a higher γ i. 22 The replenishment criteria commonly refer to the portfolio quality at time of replacement and/or the losses up to date. In dynamic bond transactions active trading of the underlying portfolio may be allowed subject to similar restrictions. The different types of replenishment criteria, and how they are dealt with in 16

19 For these characteristics the line of thought is similar (as for the portfolio diversification). In a frictionless market, they should have no effect. If they indicate market imperfections, they should influence γ i. Longer maturities may increase uncertainty with respect to the loss rate distribution of the portfolio and accordingly also to the repayment structure of the tranches. This uncertainty may allow for stronger information asymmetries and make itself felt in additional premia. Dynamic transactions may also display higher γ if the replenishment criteria are not strict enough. In loan transactions, investors may see a larger potential for adverse selection and moral hazard since often the collateralized loans refer to obligors that are less well known than the obligors in a CBO. Issuance Date Market sentiment as well as the attitude towards risk is subject to changes. This may cause the risk premium for credit risk transferred in CDO tranches to vary over time. On the other hand, learning effects may also play a role. Cuchra and Jenkinson (2005) analyse a broad set of securitisations over a time horizon similar to the one investigated here and find that investor sophistication increases over time. This suggests that market participants improve their understanding of the CDO structures leaving less uncertainty for example concerning the loss allocation or the general properties of the portfolio loss distribution. This should result in a decreasing mistrust premium over time since information asymmetries diminish. Given that the mistrust premia are included in the computed risk premia the latter should decrease over time. Alternatively, if the reduction over time in the perceived information asymmetries is small we would rather see risk premia that vary along with other measures of how the market values risk, for example the spreads of corporate bonds within a given rating class. This leads to two opposing hypotheses: Hypothesis 4a: Variations in the risk aversion parameter include a time trend due to learning effects and changes in perceived information asymmetries. Hypothesis 4b: Variations in the risk aversion parameter over time are better explained by changes in corporate bond spreads than by a time component. the simulation, are described in Appendix

20 4 Data Set and Variables The data set contains 59 multi-tranche CDOs issued by European originators, which are divided into 19 collateralised bond obligation- transactions (CBOs) and 40 collateralised loan obligation- transactions (CLOs). These CDOs comprise 236 differently ranked tranches, i.e. tranches that differ in their position in the loss allocation rules. For 21 of these tranches, with 20 rated Aaa and one rated Aa2, belonging to 19 different transactions the simulations yield no losses and therefore no value of γ k would solve equation (6), since they pay positive credit spreads. They are removed from the data set leaving 215 tranches. Each CDO comprises between two and five differently rated tranches: 23 six CDOs have two tranches, 20 have three, 22 have four and 11 have five. To distinguish the tranches of the same transactions the following variables are defined: Rating: The Rating of a tranche, measured as an integer. A Aaa rating corresponds to -1, Aa1 corresponds to -2 etc. The distribution of tranche ratings: Rating Aaa Aa A Baa Ba B Tranches log(e(an. loss)): The logarithm of the annualised expected loss of a given tranche. 24 Rank: The position of a tranche in the hierarchy of the transaction. The tranche with the highest loss protection and accordingly with the highest rating carries number 1. The next tranche number 2, etc. Highest: A dummy variable equal to one if the tranche is the highest in the hierarchy of the transaction. Lowest: A dummy variable equal to one if the tranche is the lowest in the hierarchy of the transaction, i.e. if the only loss protection comes from the first loss piece. Issuance Date The transactions were issued between 1999 and 2005: Year Transactions If several tranches are ranked pari passu in the loss allocation rules and only differ with respect to the currency or the type of interest payment - i.e. fixed or floating - I count this as only one tranche 24 This variable is strongly correlated (nearly 75%) to the variable rating. The expected loss is computed in the simulations. It displays a strong correlation (> 90%) to the expected loss computed using the approximation in Franke/Herrmann/Weber (2007). 18

21 49 transactions were arranged between 1999 and This accounts for more than half of the respective market for European synthetic CDOs. Between 2003 and 2005 there are 10 transactions in the data set accounting only for a small portion of the market. When accounting for the date of issuance two variables are defined: Date: The date of issuance, measured by an integer for each quarter (e.g. 1.Q 1999 ˆ= 1, 1.Q 2000 ˆ= 5). IBOXX-spread: The spread of the IBOXX BBB-corporate bond index over the IBOXX sovereign index, both taken for maturities of five to seven years. Over the analysed time period it varies between 58 and 250 basis points. Average Portfolio Quality Each obligor is either assigned an external rating by Moody s or an equivalent rating derived by mapping the internal rating system of the originator onto the external (Moody s) rating scale. For each transaction, the average loan rating is known, for some deals the distribution of loan ratings is also known. The rating agencies compute this average rating by estimating the weighted average default probability and then translating this into a rating. This gives one further variable: WADP: Weighted average default probability of the securitised portfolio. between 0.7% and 8% with a mean of 2.3%. It ranges Portfolio Diversification The number of loans/bonds n in the portfolio is known. In addition, the rating reports from Moody s provide a so-called diversity score (DS) for the portfolio. Moody s defines the DS as DS = m [ n k G min(1, F i / F ] ) k=1 i=1 Here, m denotes the number of different industry groups, n k the number of loans/bonds within industry group k, F i the par value of claim i and F the average par value of all loans. G(y) is a concave function. It increases with y up to a maximum of 5 for y = 20. It is constant for y > 20. Summing over industry groups to compute the DS implicitly assumes an asset correlation of zero for obligors of different industries. Moody s removed this assumption when developing a new measure, the adjusted DS (ADS). 25 The adjusted DS explicitly accounts for 25 See, e.g. Fender and Kiff (2004) for a discussion of DS and adjusted DS. 19

22 asset correlations of obligors from different industries, ρ inter. The pairwise asset correlations within one industry group is labelled ρ intra. This yields the following formula: ADS = n 2 n + ρ inter n(n 1) + (ρ intra ρ inter ) m k=1 n k(n k 1). When computing ADS, I assumed ρ intra to be 20% and ρ inter to be 2%. Only the number of different obligors enters the simulation, DS and ADS - with varying assumptions regarding ρ inter - are later used in robustness checks. Because of the vast variation of the diversification measures I use the logarithm of the variables introduced above: ln(obl): The log of the number of different obligors in the portfolio. ln(ds): The log of Moody s diversity score. ln(ads): The log of Moody s adjusted diversity score. Other Variables Further CDO-specific variables included in the regressions are Maturity: The total maturity of the transaction measured in whole-numbered years. It ranges between 2 and 8 years. Nearly half the transactions (27 out of 58) having a maturity of 5 years. Dyn: A dummy variable equal to 1 (0) in dynamic (static) transactions. 44 CDOs are dynamic, 15 are static. CBO: A dummy variable equal to 1 (0) in bond (loan) transactions. Interestingly, there is a positive correlation of 28% between the IBOXX-spread and the CBOdummy. The originators of loan transactions apparently time the market. They tend to issue the CLOs rather in times of low spreads in the bond market. This reduces the spread they have to pay on the issued tranches. For bond transactions higher spreads in the bond market have less effect, because of the simultaneity of purchasing bonds and issuing tranches. If for both, bonds and tranches higher spreads are paid, the net effect will be small. 20

23 5 Results and Discussion For each of the 215 tranches the respective γ i is computed. Further, for each of the 59 transactions the weighted average of the γ i s ( γ) for the issued tranches is computed as is the respective transaction-γ (γ CDO ) solving equation (7). The results for γ i vary between 2.6 and 10.2 with a mean of 4.37: Figure 2. Histogram for the tranche specific risk aversion coefficient γ i. The histograms for γ CDO and γ are depicted in figure 3. The average value for γ CDO is 4.05 and the standard deviation is 0.6. γ is more prone to outliers. Its average value is 4.75 and the standard deviation is 1.1. Figure 3. Left hand side: histogram for the transaction-γ (γ CDO ). Right hand side: histogram for the weighted average γ of the transactions, γ. Tranche Risk The results for γ i vary systematically for tranches of different risk. Irrespective of whether the risk is measured by the rating of the tranche or by the expected annual loss per e invested, 21

24 ratings all Aaa Aa1 to Aa3 A1 to A3 Baa1 to Baa3 Ba1 to B1 γ k - mean γ k - stdev observations E(an. loss in %) < x < x < x 1 > 1 γ k - mean γ k - stdev observations Table 1: Upper part: Results for the tranche specific risk aversion parameter γ i given different tranche ratings. Lower part: Results for γ i given different ranges of annualised expected loss per e invested. hypothesis 1 is confirmed, the risk aversion is lower for riskier tranches as can be seen from table 1. More risk averse investors seem to restrict themselves to Aaa and Aa rated tranches, respectively to tranches with low expected loss per e invested. When looking at the relative position of a tranche in the transaction, the results are similar. This can be seen from figure 3 where γ i is plotted against the variable rank. Figure 4. Risk aversion parameter vs. relative position of the tranche. The line depicts the fitted values of a regression of γ k on a second order polynomial of rank. The respective number of observations for positions one through five are: 59, 59, 53, 33,

25 The average γ i is highest for the tranches with the highest loss protection within a given CDO (rank 1) and decreases for tranches with rank two and three. Surprisingly, the average γ i for tranches with rank four or five and 3.89, respectively - is higher than that of tranches with rank three, Also the results in table?? show that for lower ratings, respectively higher expected losses the influence of tranche risk is lower. To further analyse the impact of the variables rating, log(e(an. loss)) and rank on γ i, I run several regressions. The results are displayed in table 2. The first three columns show the univariate effects. In order to capture nonlinear effects in rank I add (rank) 2. As shown in column 4, this increases the explanatory power, yet replacing rank and (rank) 2 by the variables highest and lowest and combining these with either rating or log(e(an. loss)) yields the highest adjusted R 2. These results shown in the last two columns provide evidence that there is an additional premium for the lowest issued tranche. 26 The positive and significant coefficient of lowest is the first hint an the influence of market frictions. Even though hypothesis 1 is confirmed in general, the lowest tranche deviates. The investors buying this tranche are the first to be hit by losses and therefore have to be concerned most about idiosyncratic risks and information asymmetries concerning the underlying portfolio as well as the exact loss allocation rules within the CDO. This may motivate the added premium. The potential influence of these market imperfections will reviewed when addressing hypotheses 2c and 3b. Average Portfolio Quality With respect to the average quality of the reference portfolio, three hypotheses were stated addressing the risk aversion coefficient on the transaction level. The opposing hypotheses 2a and 2b that both do not take market frictions into account differ with respect to the assumed change in the loss protection of the tranches given variations in the average portfolio quality. The plot of the weighted average γ, γ, of the transactions against the WADP depicted in figure 5 shows that higher WADP correspond to lower values of γ. This rejects hypothesis 2a and supports hypothesis 2b. 26 It is difficult to translate a higher γ i into additional spread since this relation strongly depends on the loss distribution. Lowering γ i by 0.6 for example for the transaction Promise G ( Promise K ) corresponds to a decrease in promised credit spreads by roughly 70 (120) basis points. 23

26 Regressing the tranche specific risk aversion γ i rating (0.000) (0.044) log(e(an. loss)) (0.000) (0.000) rank (0.000) (0.000) (rank) (0.000) highest (0.000) (0.000) lowest (0.000) (0.000) adj. R Table 2: This table displays the coefficients (Newey-West heteroscedasticity adjusted p-values in brackets) of OLS-regressions explaining the tranche-γ i with tranche specific variables. Figure 5. The weighted average γ of the CDOs plotted against the weighted average default probability (WADP) of the reference portfolio. The line depicts the fitted values of a linear regression. Regressions of either γ or the transaction-γ, γ CDO, confirm hypothesis 2b, too. In both cases the coefficient is significantly negative and the R 2 is close to 11 %, as shown for γ CDO in 24

27 the first column of table 5. Above results have shown higher tranche risk corresponds to lower risk aversion implied by the tranche payoff distribution. The reasoning in hypothesis 2b is based on the assumption that for lower quality portfolios with higher WADP the loss protection is higher, but not high enough to fully protect the tranches from additional risk. Therefore the issued tranches are on average more risky leading to lower γ CDO, γ. Figure 6 plots the loss protection of the Aaa- and Aa rated tranches against the WADP, respectively the loss protection of the lowest rated tranche - which equals the size of the first loss piece (FLP) - against the WADP. Linear regressions of the tranche credit support on WADP yield constants significantly greater than zero. This means the credit support increases less than proportionally given an increase in WADP which can also be seen from the lines depicting the fitted values of the respective linear regressions in figure 6. Consequently the default probabilities of the tranches increase given an increase in WADP backing the assumption made for hypothesis 2b. Figure 6. The credit support is plotted against the WADP of the reference portfolio. On the left hand side this is done for all tranches rated Aa or better. On the right hand side for all lowest tranches of the respective transactions. The credit support of the lowest rated tranche equals the size of the first loss piece (FLP). In both plots the data point with the highest WADP, WADP= 8.2% is omitted. Including it does not alter the results. Consequently, the effect of W ADP on the risk aversion coefficient should show on the tranche level, too. The net impact of a higher WADP should be a more risky tranche and a lower γ i. This impact can be decomposed when multivariately regressing the tranche γ i on the 25

28 subsets transactions all tranches upper tranches mid tr. lowest Observations WADP in % (0.028) (0.000) (0.001) (0.000) (0.000) (0.000) (0.000) (0.024) FLP/credsup (0.000) (0.000) (0.000) (0.012) (0.017) adj. R Table 3: The first two columns display the coefficients (Newey-West heteroscedasticity adjusted p- values in brackets) of OLS-regressions explaining the transaction-γ. The other columns show the respective coefficients when explaining the tranche-γ i. The upper tranches are defined as all tranches that have at least two lower ranked tranches beneath with respect to the loss allocation rules. The mid tranches are of rank two or lower and there exists exactly one lower ranked tranche. The lowest tranches are the first rated tranches to be hit by losses. The adjusted R2 is shown in the last row. When accounting for loss protection, the size of the first loss position FLP is used on the transaction level and the tranche specific credit support credsup is used on the tranche level. WADP of the portfolio and the credit support (credsup) of the tranches. The regression results are shown in columns three and four of table 3. A higher WADP decreases γ i whereas a higher credit support lowers the tranche risk and thereby increases γ i. Put differently, both a higher WADP and a lower credit support increase tranche risk and thereby decrease γ i. The same reasoning applies on the transaction level, the respective results are given in the second column of table 3. Including market imperfections changes the situation. Hypothesis 2c states that a higher WADP should aggravate information problems showing in higher γ s. Further, this increase should be stronger for lower tranches. On a transaction level, this hypothesis is rejected. Columns three and four of table 3 show that for all tranches combined the hypothesis is also rejected. Looking at different subsets of tranches however gives a more differentiated picture. Columns five and six show the results when regressing γ i on WADP and credsup only for upper tranches where at least two lower ranked tranches were issued in the CDO. The coefficients show that here higher WADP and lower credsup lead to a stronger decrease in γ i than for the average of all tranches. Column seven shows the result for the mid tranches, tranches of rank two or lower, where exactly one lower ranked tranche was issued. Here, the respective coefficients are smaller. Finally, the last column shows the result for the 59 lowest tranches of the transactions. For these tranches the coefficients are smaller and less significant and the explanatory power is weaker. For all tranche groups a higher risk decreases γ i, probably by attracting less risk averse investors. The smaller regression coefficients for mid and lower tranches indicate that this in- 26

29 subsets transactions all tr. upper mid tranches lowest tranches Observations log(ds) (0.000) (0.000) (0.19) (0.21) (0.097) (0.030) rating of lowest tranche (0.004) (0.845) (0.23) log(e(an. loss)) (0.000) (0.037) (0.000) (0.046) (0.000) (0.063) adj. R Table 4: The first two columns display the coefficients (Newey-West heteroscedasticity adjusted p- values in brackets) of OLS-regressions explaining the transaction-γ. The other columns show the respective coefficients when explaining the tranche-γ i. The upper tranches are defined as all tranches that have at least two lower ranked tranches beneath with respect to the loss allocation rules. For the mid tranches there exists exactly one lower ranked tranche. The lowest tranches are the first rated tranches to be hit by losses. The adjusted R2 is shown in the last row. For the lowest tranches regressed in the last two columns the rating of the lowest tranche coincides with the tranche rating. fluence is counteracted by the rising information asymmetry given higher tranche risk leading to additional premia. A Further effect may be that even for low WADPs and correspondingly little risk in the highest tranches a (relatively high) minimum spread has to be paid. This tends to increase the γ k of the highest tranche when WADP is (very) low. Portfolio Diversification Two opposing hypotheses were stated with respect to the influence of portfolio diversification. The regression results displayed in table 4 confirm hypothesis 3b. The first two columns show the results when regressing the transaction-γ,γ CDO. A higher diversity score significantly reduces γ CDO. On the one hand this hints at additional premia for idiosyncratic risks and information asymmetries. On the other hand, the originator may design the CDO differently given a better diversified portfolio. In order to control for this effect, the rating of the lowest tranche is added in the regression. Column two shows that a higher rating of this tranche increases γ CDO. This is not surprising since a higher tranche rating corresponds with higher γ i. Plugging in the weighted average rating respectively the weighted average expected annual loss of all tranches as a control variable yields similar results, yet the explanatory power is weaker. This result is not shown. Column two also shows that when controlling for changes in the tranche rating the impact of log(ds) is only slightly smaller. Thus, market frictions do play a role. 27

30 When analysing the effect on a tranche level I control for the pure risk effect by adding the logarithm of the expected annualised loss log(e(an. loss)) as control variable. I again differentiate the tranches into upper, mid and lowest tranche. Columns three and four show that diversity score and rating of the lowest tranche have no impact on γ i when looking at all tranches or the upper tranches only. They do not add explanatory power compared to the univariate regressions on log(e(an. loss)). 27 Columns five and six reveal that log(ds) is weakly significant when regressing only the mid tranches and that it adds a little explanatory power. When regressing only the lowest issued tranches log(ds) is significant and it adds more explanatory power. For these tranches log(ds) is more significant than log(e(an. loss)). These results are displayed in the final two columns. On a tranche level, including the rating of the lowest issued tranche does not add explanatory power when log(ds) and log(e(an. loss)) are already included. For mid and lower tranches it thus was dropped from the regressions. When controlling for the annualised expected loss as a proxy for tranche risk, better portfolio diversification lowers the risk premium only for mid and lower tranches. For these tranches this provides evidence for the pricing of idiosyncratic risk and/or additional spreads paid due to information problems. For mid tranches the effect is only weakly significant. The higher the tranche within the structure of a CDO the smaller is this effect. Maturity The estimated risk aversion parameter tends to be higher the longer the maturity of the transaction. This holds true for transaction averages as well as tranche specific numbers. Of course, default probabilities rise with longer maturities - but this is accounted for in the simulation procedure. The higher estimated γ therefore hint at additional premia paid for longer transactions. Due to the highly illiquid market for CDO tranches most investors pursue buy and hold strategies. The additional premia may be demanded as compensation for higher uncertainty associated with longer transactions. 28 When regressing all tranche γ i on several explanatory variables, the coefficient for maturity always is significant and adds explanatory power. It varies around 0.3 depending on the other variables included. The results are shown in table 5. In columns three through eight again higher, mid and lower tranches are differentiated. The size of the coefficient is higher for higher tranches but the added explanatory power is substantially larger for lower tranches. This may be explained by the higher variance of γ i for higher tranches. Issuance Date 27 The adjusted R 2 in the univariate regressions γ i is 0.24, respectively 0.19, so practically the same as when including the diversity score and the rating of the lowest tranche. These results are not shown. 28 The estimates for the weighted average default probability tend to be less accurate for longer time periods. 28

31 subsets transactions all upper mid lowest Observations WADP in % (0.000) (0.005) (0.015) (0.004) (0.000) (0.000) (0.003) FLP/credsup (0.000) (0.000) (0.000) log(ads) (0.000) (0.000) (0.000) (0.000) (0.002) (0.018) rating of lowest tranche (0.000) (0.105) CBO-dummy (0.001) (0.000) (0.002) (0.005) maturity (0.000) (0.000) (0.000) (0.000) (0.000) IBOXX-spread (0.008) (0.008) (0.055) (0.057) (0.012) adj. R Table 5: This table displays the coefficients (Newey-West heteroscedasticity adjusted p-values in brackets) of OLS-regressions explaining the tranche-γ k. The upper tranches are defined as all tranches that have at least two lower tranches beneath with respect to the loss allocation rules. The mid tranches have exactly one tranche beneath. The lowest tranches are the first rated tranches to be hit by losses. The adjusted R2 is shown in the last row. The second row, the coefficients for FLP are shown in the first four columns referring to regressions on γ CDO. Columns five through eight in this row show the coefficients for the tranche credit support credsup when regressing γ i. Two opposing hypotheses were stated with respect to the effect of the issuance date. Here, clearly the IBOXX-spread plays a stronger role. When univariately regressing the transactionγ on the IBOXX-spread the variable is significant at the 1 % level and the adjusted R 2 is 11%, whereas the variables date and (date) 2 are insignificant and the adjusted R 2 of regressing the transaction-γ on either one of them or the two combined is practically zero. Further, the results displayed in table 5 show that in multivariate regressions of γ CDO as well as γ i including several other variables IBOXX-spread adds explanatory power. A higher IBOXX-spread increases the implied risk aversion. This is intuitive, higher spreads paid in the corporate bond should correspond to higher spreads in the CDO market that in turn show in higher γ s. In contrast, date and (date) 2 do not add explanatory power, these results are not shown. CLOs vs. CBOs 29

32 On average, bond transactions display higher estimates for γ CDO. The average transaction-γ is 4.58 for the 19 CBO transactions compared to 3.80 for the 40 CLOs. This would correspond to a coefficient of 0.78 when univariately regressing γ CDO on the CBO-dummy. However, bondand loan-transactions may systematically differ in the portfolio characteristics as measured by WADP and log(ads) and in the size of the first loss piece FLP. The results of multivariate regressions including these variables are shown in columns three and four of table 5. The coefficients when regressing the single tranche γ i are shown in columns five through eight of table 5. For the set of all 215 tranches as well as for the upper tranches the CBO-dummy is insignificant and close to zero when accounting for the afore mentioned variables. For mid and lower tranches however the coefficient is significantly positive as seen in columns seven and eight. This result - a still relatively high and significant coefficient with respect to the transaction-γ contrasting with lower and partially insignificant coefficients when regressing single tranche γ k s can be explained in a similar way as the influence of portfolio diversification: bond transactions tend to have fewer tranches, three on average, than loan transactions (average: 3.95). Further, their mid (lowest) tranche is on average rated A (BBB) compared to a BBB+ (BB/BB+) rating for respective tranche of the average CLO. These better ratings correspond to higher γ i and lead to higher transaction-γ s in CBOs. 30

33 6 Conclusion Analysing the risk-return profile of 215 CDO tranches from 59 different CDOs showed that the coefficient of relative risk aversion implied by the repayment structure of the tranches varies strongly. However, a good portion of this variation can be ascribed to tranche and portfolio characteristics. This holds both on a tranche and a transaction level. Most significantly, the measured relative risk aversion is lower for tranches with higher risk (risk effect). This indicates market separation, i.e. riskier tranches paying higher spreads attracting less risk averse investors. The CDO market as an aggregate therefore displays decreasing relative risk aversion. Given this finding and without looking at market frictions, any change in the underlying portfolio structure or the tranching of the CDO that increases the average tranche risk lowers the risk aversion implied by the overall transaction. For example a lower average portfolio quality combined with a less than proportional increase in tranche credit support causes this effect. Vice versa, in bond transactions the average tranche quality is higher, corresponding with higher implied risk aversion. Adding market frictions alters the situation, but only for mid tranches and in particular for the lowest issued tranches. Tranches that have at least two further rated tranches protecting them against losses are basically unaffected when changing the variables that may proxy for information problems and unsystematic risks. For lower tranches, market frictions counteract the risk effect as higher tranche risk aggravates the information problems. Lowering portfolio diversification leads to higher premia in these tranches when holding the ratings constant. This indicates additional premia for unsystematic risks. To quantify these premia and to specify the effects of information asymmetry on the tranche spreads is a task for future research. 31

34 References [1] Ait-Sahalia, Yacine and Lo, Andrew W., Nonparametric risk management and implied risk aversion, Journal of Econometrics, Elsevier, vol. 94(1-2), pages [2] Akhavein, Jalal D., Kocagil, Ahmet E. and Neugebauer, Matthias (2005): A Comparative Empirical Study of Asset Correlations Fitch Ratings, Fitch Ratings Structured Finance, taken from [3] Amato, Jeffery D. and Remolona, Eli M., The Credit Spread Puzzle, BIS Quaterly Review, December 2003, p [4] Amato, Jeffery D. and Remolona, Eli M., The Pricing of Unexpected Credit Losses, [5] Bliss, Robert R., and Nikolaos Panigirtzoglou, Option-implied risk aversion estimates, Journal of Finance 59, [6] Brennan, Michael J., Xiaoquan Liu and Yihong Xia, Option Pricing Kernels and the ICAPM, [7] Buraschi, A. and Jackwerth, J., The price of a smile: hedging and spanning in options markets, Review of Financial Studies 14, [8] Chen, L., D. Lesmond and J. Wei (2007):Corporate yield spreads and bond liquidity, Journal of Finance (forthcoming). [9] Coval, J.D. and Shumway, T., Expected Option Returns, Journal of Finance 56, no. 3., [10] Credit Suisse Financial Products, CreditRisk+: A CreditRisk Management Framework, London. [11] Cuchra, M. and T. Jenkinson (2005): Security Design in the Real World: Why are Securitization Issues Tranched? Discussion paper, Oxford University. [12] Delianedis, G. and R. Geske, The components of corporate credit spreads: Default, recovery, tax, jumps, liquidity, and market factors, Working Paper 22-01, Anderson School, UCLA [13] Deutsche Bundesbank, Instrumente zum Kreditrisikotransfer: Einsatz bei deutschen Banken und Aspekte der Finanzmarktstabilitt, Monatsberichte April, [14] Driessen, Jost, Is Default Event Risk Priced in Corporate Bonds?, The Review of Financial Studies, Vol. 18, No. 1. p

35 [15] Elton, E., M. Gruber, D. Agrawal, and C. Mann, Explaining the Rate Spread on Corporate Bonds, Journal of Finance, 56, [16] Fender,I. and J. Kiff (2004): CDO rating methodology: Some thoughts on model risk and its implications. BIS working paper no [17] Franke, Guenter and Krahnen, Jan Pieter; Default risk sharing between banks and markets: the contribution of collateralized debt obligations, forthcoming in: The Risk of Financial Institutions edited by Mark Carey and Rene Stulz [18] Gabaix, Xavier, Krishnamurthy, Arvind and Oliver Vigneron; Limits of Arbitrage: Theory and Evidence from the Mortgage-Backed Securities Market, Journal of Finance 62, [19] Gordy, Michael B.; A comparative anatomy of credit risk models, Journal of Banking and Finance 24, [20] Greene, William H., Econometric Analysis, fifth edition, Prentice Hall, New York. [21] Gupton, G.M., Finger, C.C.,Bhatia, M.; CreditMetrics-Technical Document. J.P. Morgan& Co. Incorporated, New York. [22] Hein, Julia, Optimization of Credit Enhancements in Collateralized Loan Obligations - The Role of Loss Allocation and Reserve Account. Discussion Paper, University of Konstanz. [23] HSBC, 2006: HSBC Global Research - Global ABS 2006 [24] Huang, Jing-Zhi and Huang, Ming, How Much of Corporate-Treasury Yield Spread Is Due to Credit Risk?: A New Calibration Approach; SSRN papers.ssrn.com [25] Jackwerth, Jens Carsten, Recovering Risk Aversion from Option Prices and Realized Returns, Review of Financial Studies, Oxford University Press for Society for Financial Studies, vol. 13(2), pages [26] Jobst, Andreas, 2002a. Collateralized Loan Obligations (CLOs) A Primer, Working Paper Series: Finance and Accounting, Goethe-University Frankfurt/Main Nr. 96. [27] Jobst, Andreas, 2002b. The Pricing Puzzle: The Default Term Structure of Collateralised Loan Obligations, CFS Working Paper No. 2002/14 [28] KfW, Europischer Verbriefungsmarkt. Home/Kreditverbriefung/Europaeischer Verbriefungsmarkt/index.jsp 33

36 [29] Krahnen, Jan Pieter and Wilde, Christian, Risk Transfer with CDOs and Systemic Risk in Banking. Conference Paper, Cambridge Endowment for Research in Finance. [30] Lando, David and Jens Christensen, Confidence Sets for Continuous-time rating transition probabilities. dlando/dlresearch.html [31] Liu, X., M.B. Shackleton, S.J. Taylor, and X. Xu, Closed form transformation from risk-neutral to real-world distributions, Working paper, Lancaster University. [32] Moodys, SPECIAL REPORT: Default Report Annual Default Study; Author: David T. Hamilton Accessible for (freely) registered users. Area Default Research Research [33] Rosenberg, Joshua V. and Engle, Robert F., Empirical pricing kernels, Journal of Financial Economics, Elsevier, vol. 64(3), pages [34] Schwarz, Robert (2005): Default Correlation: An empirical Approach In: Paper presented at the Scientific Conference Methody 2005, Ustron (Polen) [35] Standard & Poors, Global cash flow and synthetic CDO criteria, Standard & Poors Structured Research, March

37 7 Appendix 7.1 Loss Rate and Aggregate Wealth When using Moody s cohort study to estimate the distribution of the cumulative default rate d t, I first have to fix the weights for averaging the rating specific default rates. These weights vary slightly over time. In order to avoid a potential bias caused by these variations I use the same average rating-class proportions for all years (and maturities): rating class Aaa Aa A Baa Ba B Caa proportion 4.64% 13.04% 28.67% 24.66% 17.27% 10.02% 1.61% Table A1: Average rating class proportions used to compute a proxy for the default rate in the economy The weights represent the percentages of loans within a certain rating class with respect to all outstanding loans. Given these weights I compute one time series of cumulated default rates for each maturity. The data show that the lognormal distribution fits this sample of cumulative default rates for the maturities needed reasonably well, i.e. for t = 1,..., 8. So, I assume ln(d t ) N(µ t, σ t ) for all t and estimate the parameters by maximum likelihood. The estimates for mean and standard deviation of ln(d t ) and d t, respectively, are: t µ t σ t E(d t ) 1.24% 2.36% 3.51% 4.66% 5.73% 6.75% 7.65% 8.52% Stdev(d t ) 0.86% 1.29% 1.77% 2.19% 2.63% 3.01% 3.27% 3.49% Table A2. Estimates for the parameters µ t and σ t of the lognormal default rate distribution as well as the resulting mean and standard deviation for maturities t of one to eight years. Table A2 shows two things. The expected default rate rises slower than linear. Further, the relative width of the distribution, measured either by σ t or by the ratio Stdev(d t )/E(d t ), decreases over time. Given the distribution of losses, how does the distribution of total wealth look like and how does it evolve over time? I normalize the total volume of outstanding debt to 100%, so the assumed leverage of 10 corresponds to a starting wealth of 10%. Then the expected value of W t and its respective 95% and 99.9% quantile develop over time in the following way 35

38 t Wt max 14.46% 19.02% 23.35% 27.63% 31.68% 35.14% 38.82% 41.75% E(W t ) 13.71% 17.51% 21.13% 24.69% 28.07% 30.95% 34.02% 36.45% 95% Quantile 12.37% 15.49% 18.32% 21.18% 23.84% 26.02% 28.71% 30.70% 99.9% Quantile 7.56% 9.08% 9.94% 11.34% 12.22% 12.93% 14.99% 16.42% Table A3. Description of the distribution of aggregate wealth given an economy with total outstanding debt volume normalised to 1 = 100% and assuming a starting leverage of 10. Maximum possible wealth Wt max, expected wealth E(W t ) and the 95% and 99.9% quantiles, respectively, are stated for maturities t of one to eight years. Wt max is defined as the wealth in a scenario where there are no losses at all. The longer the time horizon, the more dispersed is the wealth distribution. This is obviously true in absolute terms since over a longer time horizon in good scenarios a higher interest difference accrues and in bad scenarios higher losses accumulate. Table A3 shows that in relative terms the wealth distribution is relatively stable for maturities of three years and longer. This is mainly due to the higher interest difference earned over a longer time horizon increasing the loss tolerance in combination with the time structure of loss rates. The wealth at a given quantile of the distribution roughly equals the same multiple of the expected value for all time horizons of three years and longer. Rows four and five of table A3 show for example that the 99.9% (95%) quantile of the wealth distribution equals roughly 45 (85) percent of the expected wealth for any maturity between three and eight years. 7.2 Comparing Credit Risk Models In contrast to the model implemented in this paper where the macro factor influence on single name default probabilities is modelled directly, Franke and Krahnen (2005), Krahnen and Wilde (2006) and Hein (2006) use an approach that depends on latent variables. In a Monte Carlo simulation these latent variables drive yearly rating-migrations of the underlying loans. This method approximately corresponds to the simulations implemented in valuation tools of Moody s and S&P. 29 In every run of the simulation and for each year of the transaction a stochastic migration variable is drawn for each loan in the portfolio. Depending on the quantile of the migration variable either the rating class of the loan remains unchanged or the loan is up- or downgraded by one or more rating notches. A downgrade to rating class D means that the loan defaults. The limits of these quantiles are calibrated to the historic rating migration matrices of the respective rating agency. 30 The migration variables are assumed to be independent over time but correlated between firms within the same year. This correlation 29 These approaches are, in turn, based on the CreditMetrics framework, see Gupton et al (1997). 30 See, e.g., for a description of the S&P evaluator. 36

39 structure is modelled by drawing the loan specific migration variables from the marginal distribution of a multivariate normal distribution whose covariance matrix possesses a certain structure reflecting the composition of the loan portfolio. In Franke/Krahnen and Hein this multivariate normal distribution is generated by a Cholesky transformation of previously independent migration variables. Krahnen/Wilde decompose each migration variable into several systematic factors and one idiosyncratic factor. By varying the factor loadings for the different systematic factors they can implement any correlation structure. In the easiest case - with one macro factor M - the latent migration variable y k of obligor k is defined as y k = M ρ + ɛ k 1 ρ. (8) All ɛ k are independent. M and ɛ k and thus also y k are standard normal. Obligor k defaults, if y k falls beneath a certain threshold ζ k. In this setting the macro factor M can best be interpreted as the average rate of return on corporate assets in an economy. Consequently, ρ is called asset correlation. Conditional on M rating migrations and defaults are independent. Default Event Correlations In the model used in this paper high (low) realisations of the default rate in the economy lead to high (low) default probabilities of all the loans leading to correlated default events. This correlation, either implicitly generated like in this model or caused by plugging some asset correlation into equation (8) of a latent variable model drives the width of the loss distribution of a loan portfolio. Generally, the correlation between the default events of two loans i, j with (unconditional) probabilities of default P D i and P D j is given by Corr i,j = P D i,j P D i P D j P D i P D i 2 P D j P D j 2 where P D i,j denotes the (unconditional) joint default probability of both loans. In this equation only P D i,j depends on the modelling assumptions. In the credit risk model used here the default probability of a given loan i is seen as a random variable - yet conditional on the realisations of the default rate in the economy d t they are constant: P D i (d t ) = P D i d t E(d t ). The joint default probability conditional on d t therefore also is constant: P D i,j (d t ) = P D i d t E(d t ) P D d t j E(d t ). To get the unconditional joint default probability P D i,j we have to compute the expected 37

40 value over all possible realisations d t : ( P D i,j = E(P D i,j (d t )) = E P D i d t E(d t ) P D j d ) t E(d t ) (9) d 2 t = P D i P D j E( (E(d t )) 2 ). (10) In comparison, from equation (8) the joint the joint default probability can be written as P D i,j = P D i,j = N ρ (N 1 (P D i ), N 1 (P D j )), where N ρ (.,.) denotes the cumulative distribution function of the bivariate normal distribution with correlation coefficient ρ and N 1 (.) denotes the inverse cumulative distribution function of the univariate normal distribution. For both models this yields default correlations that increase with rising P D i, P D j. This is in line with empirical observations. 31 For equally rated loans (i.e. P D i = P D j ) Figure A1 shows the relationship between default probabilities and default event correlations for the two models. Empirical results for the asset correlation to be plugged into the migration model vary. Akhavein et al (2005) use data from Fitch ratings on US corporates and retrieve five year inter- and intra-industry asset correlations of 4.56% and 7.85%, respectively. Five years is the average maturity of the CDOs analysed in this paper. Since equation (8) only considers one macro factor and therefore a uniform pairwise asset correlation, I fix ρ at 6%. 31 See, e.g. Schwarz (2005) for evidence from Austria for small and medium sized companies. 38