Do Rare Events Explain CDX Tranche Spreads?

• Author: SANG BYUNG SEO,JESSICA A. WACHTER
• DOI: http://doi.org/10.1111/jofi.12705
• Date: 01 October 2018
• Published date: 01 October 2018

THE JOURNAL OF FINANCE •VOL. LXXIII, NO. 5 •OCTOBER 2018

Do Rare Events Explain CDX Tranche Spreads?

SANG BYUNG SEO and JESSICA A. WACHTER∗

ABSTRACT

We investigate whether a model with time-varying probability of economic disaster

can explain prices of collateralized debt obligations. We focus on senior tranches of the

CDX, an index of credit default swaps on investment grade firms. These assets do not

incur losses until a large fraction of previously stable firms default, and thus are deep

out-of-the money put options on the overall economy.When calibrated to consumption

data and to the equity premium, the model explains the spreads on CDX tranches

prior to and during the 2008 to 2009 crisis.

The period from 2005 to September of 2008 witnessed a more than 100-fold

increase in the cost of insuring against economic catastrophe. This cost is

apparent in the pricing of derivative contracts written on the CDX, an index

of credit default swaps on investment-grade firms. During the 2005 to 2009

period, tranches on the CDX were actively traded, with investors purchasing

and selling insurance that would pay off only if a certain fraction of firms

represented by the CDX went into default. The most senior tranches were

structured to pay off only if corporate defaults became extremely widespread—

more so than during the Great Depression. These senior tranches, which can be

viewed as extremely deep out-of-the-money options, are nonredundant assets

whose prices uniquely reflect the probability that investors place on an extreme

state of the world.

While the costs of insuring senior tranches on the CDX were close to zero

through most of 2006 and 2007, fluctuations began to appear in late 2007,

culminating in sharply rising prices in the summer and fall of 2008. Ex post,

such insurance did not pay off—only a small number of firms represented by

the CDX index went into default. Yet the pricing of these securities suggests a

substantial, time-varying fear of economic catastrophe.

∗Sang Byung Seo is at the C.T. Bauer College of Business, University of Houston. Jessica A.

Wachter is at the Wharton School, University of Pennsylvania. We thank Pierre Collin-Dufresne;

Hitesh Doshi; Kris Jacobs; Mete Kilic; Stefan Nagel; Nick Roussanov; Christian Schlag; Ivan

Shaliastovich; Pietro Veronesi; Amir Yaron; and seminar participants at AFA annual meetings,

the Midwest Finance Association, Baruch College, the University of Houston, and the Wharton

School for helpful comments. We thank the Rodney L. White Center for research support. The

authors do not have any potential conflicts of interest to disclose as identified in the Journal of

Finance’s disclosure policy.

DOI: 10.1111/jofi.12705

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2344 The Journal of Finance R

The CDX and its tranches are an example of the structured finance prod-

ucts that proliferated in the period prior to the 2008–2009 financial crisis.1In

the years since, both academic literature and the popular press have deeply

implicated structured finance in the series of events beginning with the near-

default of Bear Stearns in March 2008 and culminating in the collapse of

Lehman Brothers later that year (Reinhart and Rogoff (2009), Gorton and

Metrick (2012)).2Yet, despite the centrality of structured products to the crisis

and its aftermath, there have been few attempts to quantitatively model these

securities in a way that connects them to the underlying economy.

In this paper, we investigate whether an equilibrium model with rare eco-

nomic disasters in the spirit of Barro (2006)andRietz(1988) explains the

striking behavior of senior tranches on the CDX before and during the crisis.

We start with a model for investor preferences and for aggregate consumption.

These are calibrated to aggregate consumption in the data, to the equity pre-

mium, and to stock market volatility. We calibrate cash flows on firms to match

the low default rates in the data. Preferences, consumption, and firm cash flows

are thus the basic building blocks of the model.

Based on these microfoundations, we build a model for the CDX from the

ground up. From cash flows and the pricing kernel, we solve for firm valua-

tions in equilibrium. A firm defaults when its value falls below a prespecified

boundary.3Firm values and default determine losses on credit default swaps,

which in turn determine losses on the CDX. Given a process for losses on the

CDX, we obtain payoffs on CDX tranches and therefore equilibrium prices. Our

resulting model can account for tranche prices, and in particular the prices of

senior tranches, both before and during the financial crisis.

Our findings relate to a recent debate concerning the pricing of CDX tranches.

Coval, Jurek, and Stafford (2009) examine the precrisis behavior of CDX

tranches, pricing these tranches with a static options model that assumes cash

flows occur at a fixed maturity.4Relative to the options data, and assuming the

CDX itself is correctly priced, they find that senior spreads were too low in the

precrisis period.

Collin-Dufresne, Goldstein, and Yang(2012) question these conclusions. They

note that the pricing of the equity tranche (the most junior tranche) is sensitive

to the timing of defaults and the specification of idiosyncratic risk. Coval, Jurek,

and Stafford (2009) assume that default occurs at the five-year horizon, which

1See Longstaff and Rajan (2008) for a description of structured finance products.

2Salmon, Felix, Recipe for disaster: The formula that killed Wall Street, Wired Magazine,

February 23, 2009.

3One strand of the credit derivative literature models default as an exogenous event that is the

outcome of a Poisson process (Duffie and Singleton (1997)). Such models are known as “reduced

form.” A second strand, which we build on, assumes that default occurs when firm value passes

through a lower boundary. Such models are known as “structural.” Structural models typically

take the price process and the risk-neutral measure as exogenous. In our paper, both are derived

from investor preferences and beliefs.

4Specifically, if all cash flows occur at year five, then in principle CDX prices can be calculated

using state prices derived from five-year options.

Do Rare Events Explain CDX Tranche Spreads? 2345

need not be the case.5When payoffs occur at a horizon other than five years,

the static method of extracting state prices from options no longer applies.

Collin-Dufresne, Goldstein, and Yang(2012) therefore specify a dynamic model

of the pricing kernel. They find that this model comes closer to matching tranche

spreads prior to the crisis.

However, the results of Collin-Dufresne, Goldstein, and Yang (2012) point to

the limitations in both papers. They find that the dynamic model can only ex-

plain tranche spreads during the crisis if one incorporates some probability of

catastrophic economy-wide losses. But if the probability of an economic catas-

trophe is nonzero, the available options do not complete the market: Options

that are sufficiently deep out of the money do not exist.6In such an economy,

CDX senior tranches are nonredundant and no-arbitrage alone cannot deter-

mine their prices.

It is for this reason that we turn to an equilibrium model. Our model gives

risk-neutral probabilities of an economic catastrophe based on assumptions for

preferences and aggregate consumption. These are calibrated based on prior

studies without regard to the CDX. The model explains senior tranches on

the CDX by the same mechanism that allows it to explain the high equity

premium. A severe economic disaster raises the probability that multiple firms

go into default. This is precisely the event that affects the payoff on CDX senior

tranches. At the same time, this catastrophe affects investors by reducing their

consumption and raising their marginal utility. The price that investors are

willing to pay to insure against such an event is consistent with the premium

they demand to hold the aggregate stock market. This connection between the

equity premium and CDX senior tranches is tight: When we recalibrate the

economy to a lower equity premium, we find that implied senior tranche levels

fall far short of their levels in the data.

Like any complete-markets equilibrium model, our model offers implications

for virtually any asset class given assumptions on its cash flows. We investigate

implications for index options and for the term structure of government bonds,

as well as for the aggregate market, the risk-free rate, and of course the CDX

and its tranches. Because of the link between CDX tranches and options, we pay

particular attention to index option prices, using one-month implied volatilities

to determine, through the lens of the model, the time series of the latent state

variables. We find that while the model captures average implied volatilities,

it cannot jointly account for the low autocorrelation in implied volatilities on

index options and the high autocorrelation of the price-dividend ratio on the

5Assuming that defaults occur at the five-year horizon makes the junior tranches look more

attractive. The model spreads will thus be artificially low on the junior tranches, and, because

the model is calibrated to match the index, the model-implied spreads on the senior tranches will

be too high. Collin-Dufresne, Goldstein, and Yang (2012) emphasize the need for fat tails in the

idiosyncratic risk of firms to capture the CDX spread at the three-year, as well as the five-year,

maturity. Introducing fat-tailed risk also raises the spread of junior tranches in the model and

lowers the spread of senior tranches.

6Collin-Dufresne, Goldstein, and Yang (2012) treat the risk-neutral probability of a disaster as

a free parameter that undergoes an unanticipated and permanent shift.