Counterparty Risk: some new Advances in ...

Counterparty Risk: some new Advances in Structural Modelling, with an Application to the US Automotive Industry Didier Cossin, Henry Schellhorn, and Xihua Tu IMD and Claremont Graduate University [email protected] [email protected] July 31, 2008 Abstract Classical credit risk models tend to model firms in isolation from the network of economic relationships they are embedded in. We proposed in an earlier article (Cossin and Schellhorn, Management Science, 2007) a model of counterparty risk in a network economy. While the latter model does not consider catastrophic contagion types of effect it seems to be a relevant theoretical bridge between classical credit risk models and contagion models. In this article, we develop an extension of that earlier model, which makes calibration much more practical and effective. More specifically, we show that using dividend data is misleading. This was verified in the detailed empirical analysis we conducted on the particular network of firms we considered, namely the US car industry, for which actual dividend data does not seem to reflect fundamental economic conditions. (Credit Risk; Contagion)

1

Introduction

Classical credit risk models tend to model firms in isolation from the network of economic relationships they are embedded in. There are now considerable advances in the modelling of financial contagion, a topic of particular relevance in the year when we are writing this article, when the financial system almost suffered collapse due to the sub-prime loan crisis. We proposed in an earlier article (Cossin and Schellhorn, Management Science, 2007) a model of counterparty risk in a network economy, which we call hereafter CS1. While this model does not consider catastrophic contagion types of effect it seems to be a relevant theoretical bridge between classical credit risk models and contagion models. In this article, we develop an extension of that earlier model, which we call CS2. The motivation in developing CS2 was to make calibration much more practical and effective. More specifically, we show that using market data is empirically superior to using accounting data. This was verified in the detailed empirical analysis we conducted on the particular network of firms we considered, namely the US car industry, for which actual dividend data does not seem to reflect fundamental economic conditions. We address counterparty risk in the context of structural credit risk models, which have been initiated by Merton (1974). Structural credit risk models typically capture some of the following economic features: variability of asset value or revenue, contagion of default from borrowers to lenders, strategic bankruptcy of the shareholder, cash-flow risk, tax benefit of debt and optimal capital structure, bankruptcy costs and 1

recovery rates, term structure of interest rates, and dividend/reinvestment policy. We do not model capital structure, term structure of interest rates, nor bankruptcy costs. While constant interest rates and recovery rates are usual simplifications in a first stage of modelling, we stress that our model is limited to a network of firms that have a fixed, exogenously determined, level of debt. Moreover, not only the quantity of debt is fixed, but also the network of lending and borrowing is fixed, that is, firms have preferred lenders, and the amount borrowed from them does not change with time. This is the main assumption of both our CS1 and CS2 models: (A1) the network structure of debt payments is fixed. We call cash flow risk the risk of either early or delayed debt payments that do not result in the shareholders losing ownership of the company, as opposed to bankruptcy risk. In our model, bankruptcy is a decision driven by shareholders. Upon bankruptcy, shareholders lose their stake in the company, i.e., they stop receiving dividends. Bankruptcy costs are then incurred by the debtholders. Bankruptcy typically occurs when earnings are significantly and durably lower than debt payments. This bankruptcy option is thus an asset for shareholders and a liability for debtholders. Cash flow risk has a more ambiguous effect though, as neither shareholders nor debtholders desire to enter costly bankruptcies because of a temporary shortfall of liquidities. A firm’s cash account thus serves as a buffer, and mitigates cash flow risk. Our models apply to both trade debt and financial debt since they are cash flow based models. This is a helpful guide for the scope of their applicability. Counterparty credit risk is a particularly important concern in industries where outsourcing is important as external capital markets have substituted to internal ones, thus increasing exposure. Also, as mentioned earlier, our model is most relevant for firms with sizeable cash accounts and rigid debt structure. A typical application of our model is therefore trade credit between large manufacturing firms. A second typical application is debt in countries such as Germany and Switzerland, where banks and not capital markets are the main sources of funds, and banks are conservative. A third application is sovereign debt, where the identity of major lenders does not change substantially over time. There are three main differences between the original CS1 model and the CS2 model. When applying our CS1 model to the US car industry, we found that actual dividends are not a good input. One difficulty comes from the fact that dividends are announced quarterly, whereas our model assumes geometric Brownian motion. A related and even more important fact is that all the companies in our sample offer very stable dividends. This empirical fact rules out the use of dividends as an input for the CS1 model. More generally, it invalidates in practice flow models, like the Goldstein, Ju, and Leland (2001) model (henceforth GJL) on which some of our previous results were based. GJL explain the endogenous bankruptcy decision of the firm as the optimal response of shareholders who receive a volatile payout flow (payout=debt payment+dividends) but must pay a fixed debt flow, and stop paying when the volatile flow is substantially less than the fixed flow. Again, when dividends are as flat as debt payments, this explanation is no more valid. We contend that the GJL model is still valid if we make a difference between actual dividends, which are smoothed by management in order to satisfy industry analysts expectations, and real dividends, which are not disclosed. How then can we access these "real dividends" in order to calibrate the model? We argue that the market is in general efficient enough to uncover part of the truth. This pushes us to return to a "stock"-based model, like Leland (1994), which has the advantage of taking as input market data, namely bond and equity prices. The argument of quasi-independence that we developed for the CS1 model, which resulted in semi-closed form formulae for the price of debt and equity (referred to below as theorem 1), is still valid, and carries over mutatis mutandis to theorem 2. The reason is that dividends still have a role in our model, but real dividends are not directly observable. A second difference is that, while in CS1, recovery was assumed to be a percentage of EBIT including retained earnings (or reinvestments), in CS2 recovery is proportional to EBIT net of retained earnings. As a consequence, our formulae for the price of debt and equity are a direct generalization of Leland’s (1994) results to counterparty risk. 2

Third, we use maximum likelihood to estimate the parameters of our CS2 model. This approach was pioneered in credit risk among others by Duan (1994) and Ericsson and Reneby (2001). Structural form models are not necessarily very good at fitting data. For example, Eom, Helwege and Huang (2004) confirm that Merton’s (1974) model tends to underestimate spreads while other structural models (such as Longstaff and Schwartz (1995) and Leland and Toft (1996)) tend to overestimate spreads in higher risk firms. Nonetheless, structural models provide economic insights about the economic drivers of the spreads. This gives such models value beyond their fitting ability. Anderson and Sundaresan (2000) also document that (single-firms) structural models underestimate credit spreads, but less so when bankruptcy is strategic. The mathematical difficulty of generalizing strategic bankruptcy to many firms has been alluded to in Giesecke (2002) p.15. Mella-Barral and Perraudin (1997) and Goldstein, Ju, and Leland (2001) pioneered the description of credit risk in terms of flows. Anderson and Sundaresan (1996) argue that shareholders wish to avoid bankruptcies driven by lack of liquidities. A popular alternative to our model, within the family of structural models, is copula-based models (see Hull and White (2000), Frey and McNeill (2001) and Giesecke (2004)). Copula-based models also extend single-firm models to multiple firms. In these models dependency between defaults derives from the dependency between equity prices. These models do generally not offer a theoretical justification of this crucial link; practitioners usually calibrate them by optimizing the fit between observed credit spreads and model-generated credit spreads. In contrast, we explain this link by the flows of revenues, costs, and payments between the different firms. There are clearly other approaches to model credit risk interdependencies than structural models. While classical reduced-form models (e.g., Duffie and Singleton (1999) or Jarrow and Turnbull (1995)) do not address credit risk contagion, more recent models have attempted to address the issue of counterparty risk in a simpler setting: what happens to a company’s credit risk if the default process is conditional on another company’s credit situation. Jarrow and Yu (2001) solve this problem when the relationship is unidirectional: company A’s credit risk is impacted by company B’s credit risk, but B’s credit risk is not impacted by A’s. Their model can be generalized to looping effects, but with the loss of analytical tractability. A more tractable generalization to looping effects between two counterparties has been given by Collin-Dufresne, Goldstein and Hugonnier (2004). These looping or "feedback" effects are important, as showed among others by Egloff, Leippold, and Vanini (2004). We stress that our model handles these looping effects seamlessly. Davis and Lo (2001), Giesecke and Weber (2004), Frey and Backhaus (2003), Neu and Kuehn (2004), Egloff et al (2004) study the dynamics of default in a network of firms subject to both macro and microeconomic (i.e., counterparty-related) risks. These models give an explanation of the kurtosis of the distribution of loss given default, which is observed to be higher than what typical models would predict without counterparty risk. When the number of firms becomes large, network effects are dominated by macroeconomic effects (Frey and Backhaus (2003)). Giesecke and Weber (2004) use the theory of interacting particle systems to show among others that counterparty risk is inversely proportional to the degree of connectedness of the network of counterparties. This effect is less pronounced when macroeconomic risk is large. This extended abstract is structured as follows. In section 2, we describe the two models CS1 and CS2. For more details on CS1 we refer the reader to our earlier article. We then describe CS2 and propose a semi-closed form formula for debt, which take as input only EBIT, network relations, and equity prices. We then move to our empirical analysis. Section 3 provides a survey of the US automotive industry, and its counterparty relationships.

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2

Models

The uncertainty is described by the filtered probability space (Ω, F, {F t }, Q). We call X the vector of state variables that are public knowledge, and summarize in assumption (A8) what this vector is composed of. X X generates the filtration {F X t }, which is a subfiltration of {F t }. While {F t } describes the information available to public investors, {F t } describes the information available to firm management. The risk-neutral measure Q is an equivalent measure under which asset prices, when discounted at constant interest rate r, are martingales in the public investor filtration FtX .

2.1

The Original (CS1) Model

We model the financial unit, or cash management unit, of each firm i, with i = 1..I. This financial unit receives (net) operational revenue Ri from the production unit, and redistributes Ci to the production unit as ”re-investment in production”. With this definition, revenue minus costs equals EBIT. The financial unit of the firm also receives financial and trade debt payments Pji from firm j (with j 6= i), pays debt Pij to firm j (with j 6= i), and dividends Di to shareholders. To balance revenue and expenses, the firm maintains a cash account. The equations of the cash account Z are thus: Zi (t) − Zi (0) = Ai (t) − Hi (t) where Ai is the total revenue, Hi the (total) expenses, given by: X Pji Ai = Ri + j6=i

Hi

= Ci + Di +

X

Pij

(1)

(2) (3)

j6=i

Revenue R, debt payments P , and non-debt expenses C + D are increasing processes. An example of a network is given in figure 1. The financial assumptions of the model are the following. They are described in more details in Cossin and Schellhorn (2007). (A1) The network structure of debt payments is fixed. (A2) Debt has infinite maturity. Interest is paid continuously. (A3) The cash account serves as a buffer and reduces the probability of bankruptcy. (A4) Firms maximize the value of equity by strategically declaring bankruptcy. (A5) The payout to investors is an exogenously determined proportion δ of total expenses, where: dδ i (t) = µi dt + σi dWiδ (t) δ i (t)

t ∈ [0, Ti1 )

(4)

(A6) Upon default times Tik , bankruptcy costs are incurred by the debtholders, who, alongside with new shareholders, inject an extra quantity of capital to restore the firm to its normal level of efficiency. The bankruptcy cost incurred at time Tik is equal to a constant wi times the present value of all future revenue of the firm, so that recovery rates are 1 − wi . (A7) Cash accounts are not appropriated upon default. (A8) The operational revenue and the value of the cash account of each firm is its private knowledge. (A9) The optimal default policy is sufficiently regular. 4

Net Operational Revenue R1

Non-debt Expenses C1+D1

Cash account of firm 1 Z1

Debt payment P31


Debt payment P12

Debt payment P23 Cash account of firm 3 Z3

Cash account of firm 2 Z2




Figure 1: A network of 3 firms (A10) Revenue Ri is a doubly stochastic (Cox) process with intensity ν i given by: 1 (n¯ ν i − ν i )dt + mi dν i = mi ν i (0) = n¯ νi and jump sizes equal to

1 . n

r

νi dWik n¯ νi

(5) (6)

The long-run value of intensity ν¯i is a F∞ -measurable random variable.

The vector X of public information is thus: 

C D P M δ

  X=  

     

(7)

where M is a vector of macroeconomic variables. We call νˆ the market estimator of the long-run operational revenue rate, i.e.: ν i |FtX ] (8) νî (t) = E Q [¯ Since total revenue equals total expenses in steady state, the estimator α ˆ of the long-run total revenue rate solves: X α ˆ k λki (9) α ˆ i = νî + k6=i

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THEOREM 1 (Cossin and Schellhorn 2007). In steady state, the value of equity S, total bankruptcy T BC, and debt F are, for finite n, and t not a bankruptcy time: P P λij λij δ i (t) Ki (r − µi ) xi 1 j6=i j6=i +( − Ki )( ) ] + O( √ ) ˆ i (t)[ − Si (t) = α r − µi r r δ i (t) n · ¸ wi δ i (0) xi 1 [Ki (r − µi )]xi T BCi (t) = α ˆ i (t) + O( √ ) r δ i (t) δ i (0)xi − [Ki (r − µi )]xi n P λij 1 j6=i Fi (t) = α − T BCi (t) + O( √ ) ˆ i (t) r n

costs

(10) (11)

(12)

where: xi

=

Ki

=

r 1 σ 2i σ2i 2 + ) + 2rσ2i ] [µ − (µ − i σ2i i 2 2 P xi j6=i λij xi + 1 r

(13) (14)

In the limit, when n → ∞, our model offers thus the following characteristics: • cash flow risk is decoupled from strategic bankruptcy risk for both debt and equity • the magnitude of cash flow risk is the same for both debt and equity; what differentiates them is strategic bankruptcy risk • contagion risk stems from the updating of beliefs of market participants of the long-run total revenue rate.

2.2

The (CS2) Model

We assume that management smooths dividends so as to present a better picture of the health of the company to investors. The actual dividends offer Dact are the result of this smoothing operation, whereas the economic dividends are still called D like in our earlier model (see figure 2). The vector X of public information is thus:  act  D (15) X= P  M While in CS1, recovery was equal to (1 − wi ) times a percentage of the present value of EBIT including retained earnings, in CS2 recovery is equal to a constant (1 − ωi ) times the present value of EBIT net of retained earnings. As a consequence, our formulae for the price of debt and equity are a direct generalization of Leland’s (1994) results to counterparty risk. The resulting bankruptcy cost is denoted BCi . We define Vi as the value of the assets of the firm normalized by total revenue, i.e.: Vi =

Si + Fi + BCi α î 6

C

Dact

D

1−Σλij

Stochastic sharing

Queue B

Queue C

BC A

Random RandomArouting routing

Σλ

P

Queue A

ij

Figure 2: A firm where true dividends D are not directly observable but can be inferred from market value. The observable dividends are called Dact .

THEOREM 2. In steady state, the value of equity S, total bankruptcy costs BC, and debt F are, for finite n, and t not a bankruptcy time: P λij 1 j6=i B (1 − pB (16) ˆ i (t)[Vi (t) − Si (t) = α i (Vi (t)) − Ki pi (Vi (t))] + O( √ ) r n 1 (17) ˆ i (t)ωi Ki pB BCi (t) = α i (Vi (t)) + O( √ ) n P λij 1 j6=i B Fi (t) = α (1 − pB (18) ˆ i (t)[ i (Vi (t)) + (1 − ω i )Ki pi (Vi (t))] + O( √ ) r n where the bankruptcy threshold Ki and xi are given by ( 14) and (13): Vi −xi ) Ki We remind the reader that our purpose is not to model the optimal capital structure of the firm. Thus, for simplicity, we do not model tax benefits, and the term T B in GJL’s equations is absent from our equations. The advantage of the CS2 formulation is that, even if δ is not observable, its effect is felt through V . Now, like in other option pricing models, the total value of the assets of the firm, namely α ˆ i Vi , is hard to observe. We develop in the next section a calibration procedure which allows to price debt based on equity price, revenue, and average dividend only. pB i (Vi ) = (

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2.3

Calibration

We use maximum-likelihood to estimate the firm asset volatility from stock data. For each firm i we first construct a time-series: Si (tn ) ; n = 1..N } α ˆ i (tn ) of stock prices normalized by unit of total revenue. We define the "call value" function fi by: P λij j6=i B (1 − pB fi (Vi ) = Vi − i (Vi )) − Ki pi (Vi ) r We observe that fi is strictly increasing. We mi for the drift of Vi in the physical measure, and define a parameter vector θi = (mi , σ i ). We denote by Φi (Vi (tn )|Vi (tn−1 ); θi ) the transition density of Vi in the physical measure. {si (tn ) ≡

Φi (Vi (tn )|Vi (tn−1 ); θ i ) = [ln(Vi (tn ) − ln(Vi (tn−1 )) − (mi − 12 σ2i (tn − tn−1 ))]2 1 1 p ) exp(− 2 2 2σi (tn − tn−1 ) Vi (tn ) 2πσi (tn − tn−1 )

By the Markov property, the likelihood function for Vi is:

L(si (t1 ), .., si (tN )|s0 , θ i ) = N Y

Φi (fi−1 (si (tn ); θ i )|fi−1 (si (tn ); θ i ); θ i )

n=1

1 0 fi (fi−1 (si (tn ); θ i ))

We approximate the drift µi of Vi in the risk-neutral measure by: time average of payout (19) time average of asset value As we can see in (19), dividends are part of the input of the calibration of CS2. However, the crucial difference between CS1 and CS2, regarding the impact of dividends, is that in CS2 we can safely use the average dividend rate and coupon rate over a long period of time. As a result, the negative impact of dividend smoothing is not felt in CS2. This is to be contrasted with CS1, where the values of securities at each time are functions of the contemporaneous values of those dividends, resulting in very flat security prices. An advantage of the CS2 assumption on recovery rates is that the recovery rate can be estimated separately from the parameters of the asset value process. For the estimation of the recovery rate, we followed Altman and Kishore (1996). and used a recovery rate of 40%. The methodology to determine the network routing parameters is explained in Cossin and Schellhorn (2007). µi = r −

3 3.1

Empirical Analysis of the U.S. Car Industry Data Extraction

We started by analyzing a the largest companies in the US supply chain (see figure 4), as identified in the Plunkett (2007) guide. The fundamental data came from Compustat; the interval was from 2002 to the 8

Figure 3: Accounting Data for Automotive Industry. All values in $millions. present. Bond data was collected from the Mergent NAIC database. In addition to data from Compustat and Mergent we collected company financial reports and contacted representatives from each firm’s investor relations department in an effort to estimate the industry’s exposure matrix. Finally, we gained valuable insights on where to locate the necessary data through discussions with Fitch Ratings. After collecting and analyzing the various data we found three manufacturers, two suppliers and one dealer with sufficient data for parameters to be estimated in the context of the CS1 model. Descriptive statistics on company fundamental data are listed in figure 3. Figure 3 also indicates the reason for a company’s exclusion from the model, if applicable. Of the various reasons for a company’s exclusion we observed two primary issues. First, many companies in our sample maintained a very rigid dividend policy over our sample period. Firms such as Visteon had no dividend variation and therefore had to be excluded. Even the firms which we did calibrate the model to had limited dividend variation and this made implementation difficult. The second issue was lack of bond trade data. As mentioned earlier our bond database was skewed toward larger Investment Grade names. Because of this we found limited trade data for some names, especially the smaller cap companies. This caused us to exclude many of the dealers and smaller cap parts suppliers. The companies with enough data for estimation are as follows: • Manufacturers: General Motors (GM), Ford Motor Company (F), Daimler (Formerly DCX now private) • Suppliers: ArvinMeritor (ARM), Lear (LEA) • Dealer: Sonic Automotive Inc. (SAH)

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Figure 4: Graphical Representation of Auto Industry Network. Bold represents a company in our study. Additional assumptions were made during the implementation. First, all preferred share dividends were ignored along with the impact of share repurchases and secondary offerings. Second, we made no attempt to determine whether a company had factored their accounts receivable, and if they had, whether it was factored with our without recourse. In general companies may sell their accounts receivable to accelerate their cash conversion cycle. These accounts are often sold to the major investment banks. These accounts may be sold with or without recourse. If the account is sold with recourse, it means that if the bank that purchased the account cannot collect the bank had the right to give the account back to the company in return for the money they paid. Recourse is in essence a return policy and it means that the credit risk of the account stays with the company. In this case accounts are often sold with only a small haircut ($0.95 on the dollar for example). When accounts receivable are sold without recourse the credit risk of the account is transferred as well. Because of this additional risk the amount a bank is willing to pay for an AR w/o recourse is far less. Regardless, in our situation it does not matter whether the accounts are factored since the credit costs of securitization affect both the debt and equity as a future decrease in revenue flow.

3.2

Results

We applied the methodology above to calibrate the (CS2) model to the following companies: Ford, GM, and Sonic. We split our data into two periods: the calibration period and the out-of-sample period. For each 10

company, these periods are different, due to different data availability. Because our model uses quarterly accounting data, for revenue, our data sample is limited. We focus our analysis on Ford, for which the maximum likelihood method above seems to give the best parameters. For the other companies, the maximum likelihood method seems to produce volatilities that are too low compared to simply minimizing the sum of squared errors between model debt and actual debt. For Ford, we chose 2002-2005 for the calibration period, and 2006 for the out-of-sample period. In both periods, equity prices are the input, and (model) total debt price the output. The difference is that the data in the estimation period serves to calibrate our parameters µi and σ i , while these parameters are held fixed during the out-of-sample period. We found the following parameters in the estimation period: µF ord σF ord

= 3.3% = 11.41%

We compare in (5) the (CS2) model total debt with the actual total debt price. The main observation, which is consistent across all the firms that we analyzed, is that actual total debt volatility exceeds model total debt volatility. It is too early in our analysis to determine whether incorporating the network significantly enhances the fit compared to a single-firm application of the Leland model. Indeed, we are currently working on determining what is the best proxy for total debt price, assuming that all the debt issued by Ford is a maturity. This is not as trivial an issue as it may look. Our formula for calculating total debt price, namely interest expense divided by yield a particular yield measure, seems to be very sensitive to market movements. Indeed the actual volatility of (total) debt is much superior to actual equity volatility. Another issue that needs more careful work is the level of fixed costs compared to variable costs, as explained in Cossin and Schellhorn (2007).

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Acknowledgments

We thank Fitch Corporation for its financial support. Special thanks go to the student team that performed the empirical analysis. The student team was composed of Tim Long, Nan Song, Vincent Thilly, Satjaporn Tungsong, and Joe Plotkin, team leader.

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References

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Millions

Total Debt Price for Ford 200,000 180,000 160,000 140,000 120,000 100,000 80,000 60,000 40,000 20,000 0

Actual Model

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 2002

2003

2004

2005

2006

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