Ambiguity, Information Quality and Credit Risk∗ Nina Boyarchenko University of Chicago, Graduate School of Business 5807 S. Woodlawn Ave Chicago, IL 60637 email:
[email protected] First version: August 14, 2008
April 13, 2009
Abstract This paper studies the implications of ambiguity for the credit spreads. We consider two ways of incorporating ambiguity into the Black and Cox [1976] model of credit spreads under incomplete information. We begin by examining the credit spreads in a setting where news about the asset value are of an uncertain quality. In this setting, ambiguity-averse investors act as if they take the worst-case assessment of quality. Thus, they react more strongly to bad news than to good news. As a result, changes in the information environment can lead to a widening of credit spreads and implied default intensities, even if firm fundamentals do not change. If the underlying asset value dynamics are also ambiguous, then the ambiguity-averse investors in the secondary debt market act as if the drift rate of the underlying asset value is lower than it actually is. We calibrate the two competing models to match the observed increases in default-swap spreads after August 9, 2007. Comparing the implications of the two models for default spreads, we find that ambiguity about the underlying asset value dynamics is better able to match the patterns observed in the data.
∗
I would like to thank Lars P. Hansen, Lubos Pastor, Pietro Veronesi and three anonymous faculty reviewers
and participants in the Finance Brownbag, University of Chicago Booth School of Business, for helpful comments on the previous drafts of this paper. All the remaining errors are mine.
1
1
Introduction
In the summer of 2007, the world financial markets entered into a severe liquidity crisis. On August 9, 2007, France’s largest bank BNP Paribas announced that it was having difficulties because two of its off-balance-sheet funds had loaded up on securities based on American subprime mortgages. But Paribas was not alone in its troubles: a month before, the German bank IKB announced similar difficulties, and the Paribas announcement was followed the next day by Northern Rock’s revelation that it had only had enough reserve cash to last until the end of the month. These and other similar announcements lead to a freeze of the credit markets as banks lost faith in each other’s balance sheets. The situation was particular surprising considering the market conditions shortly before the crisis began. At the beginning of 2007, financial markets were liquidity-unconstrained and credit spreads were at historical lows. Even as late as May 2007, it would have been hard to predict the magnitude of the response that the losses on subprime mortgages had generated. Compared to the total value of financial instruments traded worldwide, the subprime losses were relatively small: even the worst-case estimates put them at around USD 250 billion 1 . Further, for investors familiar with the instruments, the losses were not unexpected. By definition, the subprime mortgages were part of the riskiest segment of the mortgage market, so it was hardly surprising some borrowers would default on the loans. Yet, despite their predictability, the defaults had precipitated the current liquidity crisis that spread between the credit markets. As Caballero and Krishnamurthy [2008a] argue, the crisis was primarily caused by the rise in ambiguity that followed the subprime defaults. That is, although the value of the underlying assets did not necessarily change, the credit market participants were taken by surprise at how the complex credit derivatives were reacting and became uncertain about the quality of their investments. A prime example is the behavior of the credit default spreads (CDS) and the CDX index. Before July 2007, the popular copula models of defaults were able to price both the CDS spreads on the securities that compose the CDX index and the tranches of the index itself. 1 2
2
Source: Caballero and Krishnamurthy [2008b] The procedure for this is as follows: first, use the data on the CDS spreads to estimate the (no) default
probability function for each asset at a given point in time. Then use a chosen copula function and the estimated default probabilities to price the CDX tranches for different correlation settings. Finally, find the correlations
2
Starting with July 2007, the copula pricing model failed. In fact, Bear Stearns announced in June 2007 that it no longer understood how to price mortgage-backed securities. The rise of the complex credit derivatives in the previous five years proliferated the problem. Because investors had no prior experience with how these instruments would behave in a time of crisis, they did not know how to interpret the securities’ response to mortgage failure. The complexity of the instruments involved made it almost impossible to calculate the proper response to changes in the underlying. Thus, when even AAA-rated subprime tranches suffered losses, the resulting increase in ambiguity about the quality of investments in place lead market participants to make decisions based on their perceived worst-case scenario. The revelations by major banks about their exposure to subprime mortgages through off-balance sheet funds only served to reinforce the increase in ambiguity about signals from the major market participants. The current paper takes the Caballero and Krishnamurthy [2008a] argument – that the credit freeze was precipitated by a sudden increase in ambiguity – as given and considers the implications of an increase in ambiguity for credit spreads. We extend the model of Duffie and Lando [2001] by introducing ambiguity into the secondary debt market in two different ways. First, we assume that the participants in the secondary debt market do not know the information quality of the signal about the underlying asset value. This assumption generates an asymmetric response to positive and negative news, with negative news being interpreted by the agents as having the highest possible precision and positive as having the lowest possible. As an alternative specification, we also consider the case when investors are ambiguous about the asset value dynamics itself. To capture the Caballero and Krishnamurthy [2008b] observation that the increase in ambiguity has lead investors to evaluate investment under the worst-case scenario, we specify the debt holder’s aversion to ambiguity by a max-min expected utility optimization problem. The axiomatic basis for preferences of this type was first introduced by Gilboa and Schmeidler [1989] and extended to the case of intertemporal utility maximization by Epstein and Schneider [2003]. A rapidly growing literature studies the behavior of asset prices in the presence of ambiguity in dynamic economies. A substantial part of this literature considers investor ambiguity about the data-generating model. Anderson et al. [2003] derive the pricing semigroups associated that minimize the pricing error.
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with robust perturbations of the true state probability law. Trojani and Vanini [2002] use their framework to address the equity premium and the interest rate puzzles, while Leippold et al. [2008] consider also the excess volatility puzzle. Gagliardini et al. [Forthcoming] study the term structure implications of adding ambiguity to a production economy. This setting has also been used to study the portfolio behavior of ambiguity-averse investors and the implications for the options markets (see e.g. Trojani and Vanini [2004] and Liu et al. [2005]). The second strand in the literature, however, assumes that, although the agents in the economy know the “true” data-generating model, they face uncertainty about the quality of the observed signal about an unobservable underlying. Chen and Epstein [2002] study the equity premium and the interest rate puzzles in this set-up, and Epstein and Schneider [2008] consider the implications for the excess volatility puzzle. The portfolio allocation implications of this setting have also been studied extensively in e.g. Uppal and Wang [2003] and Epstein and Miao [2003]. However, none of these papers study the relationship between ambiguity aversion and the term structure of credit spreads. We begin by introducing ambiguity aversion using the simple setting of Epstein and Schneider [2008], who measure ambiguity by the size of the set of possible signal precisions. In this way, we can easily study the impact of an increase in the signal quality ambiguity on the term structure of credit spreads, in dependence on the boundaries of the set of possible signal precisions. The preferences underlying the max-min expected utility problem solved by our representative agent are of the recursive multiple priors utility type, which, in the terminology of Epstein and Schneider [2003], implies a “rectangular” set of relevant likelihoods and admits an axiomatic foundation. Using this setting to extend the model of Duffie and Lando [2001], we find that, under the max-min preferences, the ambiguity-averse participants in the secondary debt market will always choose the lowest signal precision in evaluating positive news about the corporate bonds and the highest signal precision when evaluating negative news. This is exactly in line with the observation of Caballero and Krishnamurthy [2008b] that the current credit crisis prompted investor to consider the worst-possible case. Notice also that this coincides with the result of Epstein and Schneider [2008] who find that, when evaluating signals about a company’s cashflows, equity investors interpret negative signals as very precise and positive signals as having 4
low precision. Hence, in our case, since the accounting signals received were negative, the lower bound of signal variance needs to decrease to generate changes in the credit spreads. Using the calibration of Duffie and Lando [2001] as the parameters of the economy before August 9, 2007, we find that, in order to generate the observed changes in the credit spreads, the lower bound of signal variance had to decrease to at least 1%. Further, the decrease in signal quality was asymmetric between the Aaa and Baa bonds, with the quality of information decreasing more for the Aaa bonds. Next, we consider the setting where investors know the signal precision but are unsure about the model for the underlying asset value dynamics. In this setting, participants in the secondary debt market will always choose the likelihood corresponding to the lowest possible expected drift rate. As before, we use the calibration of Duffie and Lando [2001] as the base case, and find that the ambiguity tolerance (that is, the size of the set of likelihoods considered plausible by the debt holders) of the debt holders has to increase substantially to match the observed changes in the credit spreads. To distinguish between the two sources of ambiguity, we consider their implications for the default spreads of financial institutions. In particular, we calibrate the model to the balance sheet data of several prominent financial institution and repeat the above exercise to find the implied changes in ambiguity following the BNP Paribas announcement. We find that, although ambiguity about signal quality is better able to match the changes in the 1 year default spread, changes in the ambiguity about the asset dynamics are better able to match the changes in the overall term structure of default spreads. Thus, we conclude that ambiguity about the underlying is more suitable in this setting.
2
Model
In this section, we describe the model considered in this paper. We begin by describing the reference model used for pricing credit securities and then proceed to specify the model misspecification in the economy.
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2.1
Reference model
As the reference model, we consider a modified version of the economy in Black and Cox [1976]. In particular, denote by At = eZt the fundamental value of the firm’s assets. We assume that At evolves as a geometric Brownian motion: (2.1)
dZt = mdt + σdWt ,
t ≥ 0,
where σ ≥ 0 and dWt is the increment of the standard Brownian motion under the reference model probability law. Denote by AB the exogenous default boundary of the firm; that is, the firm automatically defaults if the value of its assets falls below AB . In economic terms, we can interpret this default rule as a technical default covenant. Since we assume that only the secondary debt holders fear model misspecification, explicitly modeling the liquidation decision of the equity holders does not add any relevant economic interpretation to the set-up. We assume further that, while the firm is in operation, the equity owners receive a constant fraction δ of assets as cash-flows. If the firm defaults, the assets are liquidated at the present value of discontinued cash-flows, δAτ /(r − µ), where µ = m + σ 2 /2 is the asset value growth rate, but a fraction α of assets is lost in liquidation. Finally, we assume that the equity owners observe perfectly the evolution of asset values, so that the information set at time t of equity owners is given by Ft = σ {Zs ; s ≤ t}. In the current setting, this assumption implies that there is no uncertainty about the firm being liquidated at the default boundary AB . For valuation of credit derivatives, we are interested in the behavior and beliefs of the participants in the secondary debt market. As in Duffie and Lando [2001], we assume that the participants in the secondary debt market receive imperfect signals Aˆ of the asset value at preset dates t1 , . . . , tn . More specifically, we assume that the signal observed by the debt holders is given by Y (t) ≡ log Aˆt = Z(t) + U (t), where U (t) is normally distributed and independent of Z(t). The participants in the secondary debt market understand that the firm will be liquidated if the firm’s asset value falls below AB , which is known to them. Further, in addition to the accounting signals, they observe at each t ∈ [0, +∞) whether the firm has been liquidated. Thus,
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the information set of the secondary debt holders is given by: Gt = σ {Y (t1 ), . . . , Y (tn ), 1τ ≤s : 0 ≤ s ≤ t} ,
(2.2)
for the largest K such that tn ≤ t and τ = τ (AB ) is the (random) hitting time of AB . As in Duffie and Lando [2001], we assume that equity is not traded in the public market and that the equity owners are precluded from trading in the public debt markets. This assumption allows us to keep the simple structure (2.2) for the information set of the secondary market and avoid solving the rational-expectations equilibrium problem with asymmetric information. We assume that the signal errors are serially uncorrelated, so that Ut ∼i.i.d N (u, σu2 ). We further assume that the agents believe the accounting signal to be unbiased, so that E eUt = 1. Thus, for a given level of the signal variance, σu2 , the quality of the signal is given by 1/σu2 and the mean error is u = −σu2 /2. Denote by Z (n) a history of the true realizations of the (log) asset value up to time tn and Y (n) the history of signals: Z (n) = (Zt1 , . . . , Ztn ) Y (n) = (Yt1 , . . . , Ytn ) and by bn (·|Y (n) ) the conditional probability of observing the history Z (n) : bn ·|Y (n) = P z (n) ∈ dz (n) , τ > tn |Y (n) . Then, applying Bayes’ rule, we have:
bn z
(n)
|y
(n)
P Z (n) ∈ dz (n) , τ > tn , Y (n) ∈ dy (n) . = P Y (n) ∈ dy (n)
Introduce two events: A = {τ > tn , Ztn ∈ dztn , Ytn ∈ dytn } B = τ > tn−1 , Ztn−1 ∈ dztn−1 , Ytn−1 ∈ dytn−1 . Then we can represent: bn z (n) |y (n) =
pY ytn
P (A|B) P(B) , P Y (n−1) ∈ dy (n−1)
|y (n−1) 7
or
(2.3) bn z (n) |y (n)
=
√ ψ ztn−1 − a, ztn − a, σ ∆t pZ ztn |ztn−1 pU ytn − ztn |ytn−1 − ztn−1 pY ytn |y (n−1) ×bn−1 z (n−1) |y (n−1) .
√ Here, ψ(z0 , x, σ t) is the probability, conditional on Z starting at some given level z0 at time 0 and ending at some level x at a given time t, that min{Zs : 0 ≤ s ≤ t} > 0; pU ut |utn−1 is the transitional density of Ut1 , Ut2 , . . .; pZ ztn |ztn−1 is the transitional density of Zt1 , Zt2 , . . .; pY ytn |y (n−1) is the conditional density of Ytn given Y (n−1) , a = log AB . After conditioning on Y (n) and also on survival to time tn , we have that the conditional density gn ·|Y (n) of Z (n) is given by: (2.4)
gn z
(n)
|y
(n)
bn z (n) |y (n) , =R (n) dz Rn bn z|y
where the region of integration is given by: Rn = {z ∈ Rn : z1 ≥ a, . . . , zn ≥ a} . Notice that, although we do not have an explicit formula for gn , we can use the recursive solution (2.3) to do the numerical integration recursively, one dimension at a time. In the rest of the paper, we focus on the special case of observing only one period of accounting signals. Since we are assuming that the signal errors are serially uncorrelated, this is equivalent to assuming that, after each accounting report, participants in the secondary debt market update the last “known” observation of Z to the new posterior mean. Denote: y˜ = y − a − u, z˜ = z − a, and z˜0 = z0 − a. Then, from Duffie and Lando [2001], we know that, conditionally on having observed a single noisy observation y at time t = t1 and on Z starting at some given level z0 at time 0, the time t distribution of z is given by: q β0 −J(˜ z0 z˜ y ,˜ z ,˜ z0 ) 1 − exp −2˜ π e σ2 t 2 2 , (2.5) g(z|y, z0 , t) = β β 1 2 exp 4β10 − β3 Φ √β2β − exp 4β20 − β3 Φ √−β 2β 0
8
0
where (2.6)
J(˜ y , z˜, z˜0 ) =
(2.7)
β0 =
(2.8)
β1 =
(2.9)
β2 =
(2.10)
β3 =
(˜ y − z˜)2 (˜ z0 + mt − z˜)2 + 2σu2 2σ 2 t 2 2 σu + σ t 2σu2 σ 2 t z˜0 + mt y˜ + σu2 σ2t z˜0 −β1 + 2 2 σ t 2 1 y˜ (˜ z0 + mt)2 + . 2 σu2 σ2t
Notice that we can interpret y˜ as the new information contained in the signal y. In particular, since we do not update the mean of the accounting error, u, and since the liquidation threshold a is given exogenously, y˜ measures the expected distance to the liquidation barrier, conditional on the signal y. Thus, the sign of y˜ indicates the “direction” of the news. If y˜ < 0, then the expected distance to the liquidation threshold is negative so the news contained in the signal is bad. Similarly, if y˜ > 0, then the signal carries good news as the expected distance to liquidation is positive.
2.2
Default and credit spreads
We turn now to the implications of the above model for the term structure of credit spreads of the modeled firm. We begin by deriving the term structure of default swap spreads. As noted in Duffie and Lando [2001], this formulation is more convenient as it allows us to not make assumptions about the recovery rates on zero coupon bonds while allowing us to still recover the term structure of credit spreads. Let T be the maturity of the swap and denote by X the payment if default occurs before the maturity date T . Then, assuming that the default-swap annuity payments are made semiannually, we have: X =1−
(1 − α)δAB . (r − µ)D
Since the swap’s value at inception is 0, we have that the time t default-swap spread on a bond
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maturing at T is: (2.11)
2XE e−r(τ −t) 1τ ≤T |Gt h i. c(t, T ) = P k − ri 2 E 1 e 1 |Gt i=1 τ >t+ i 2
The implicit discount curve is then given by: Z(t, T1 ) = Z(t, Ti ) =
1 1 + c(t, T1 )/2 Pi−1 i) 1 − c(t,T j=1 Z(t, Tj ) 2 1 + c(t, Ti )/2
,
with the credit spread on a bond maturing at date Ti , ηi , defined by: (2.12)
Z(t, Ti ) = exp (−(r + ηi )(Ti − t)) .
We provide explicit expressions for Υ(T − t, At , AB ) ≡ E e−r(τ −t) 1τ ≤T |Ft and P (s) ≡ E [1τ >s |Ft ] in the Appendix. Notice that, by the Law of Iterated Expectations, we can express the default-swap spreads under the other information structures considered in terms of Υ and P . In particular, the default-swap spreads in the Duffie and Lando [2001] model can be expressed as: 2XE [Υ(T − t, At , AB ) |Gt ] c(t, T ) = Pk . − ri 1 2 E P t + e i i=1 2 |Gt For the remainder of the paper, we will use the above notation in discussing default-swap spreads.
2.3
Model misspecification
Assume now that the participants in the secondary debt market fear misspecification of the probability law described by ... That is, the representative participant in the secondary debt market believes that the signal Aˆt is related to the true realization At by a family of likelihoods Ξ. In the economy described above, there are two possible sources of model misspecification: the asset value dynamics and the dynamics of the accounting signals. We examine separately these two sources below. For now, consider however the general case, without specifying the source of ambiguity. In particular, assume that the representative investor in the secondary debt market believes that the signal Aˆt is related to the true realization At by a family of likelihoods Ξ, parametrized by some parameter a. 10
We assume that, unlike the equity owners, the secondary debt holders have the intertemporal version of the Gilboa and Schmeidler [1989] preferences axiomatized by Epstein and Schneider [2003]. In particular, keeping the assumption that the debt holders are risk-neutral (but ambiguity-averse), the representative agent’s utility from a consumption stream C is given by: Z +∞ −ru (2.13) U (Ct , t) = min E e Ct+u du|Ht . Ξ
0
Notice that, although in general we need to formulate the problem (2.13) recursively to maintain time consistency, Chen and Epstein [2002] show that, for the case of time-separable utility, the above representation holds. Since, in equilibrium, the representative participant in the secondary debt market will hold all the secondary market debt, it follows that the worst-case conditional probability minimizes the conditional expected payoff. Thus, in the presence of ambiguity, the time t credit default spread for a swap with maturity T is given by: (2.14)
2XEa [Υ(T − t, At , AB ) |Gt ] c(t, T ) = max Pk . − ri 1 Ξ 2 Ea P t + e i i=1 2 |Gt
To see this, recall that we can express the credit default spread as: 1 − Z(t, T ) c(t, T ) = Pk , 1 i=1 Z(t, t + 2 i) where Z(t, T ) is the time t price of a bond maturing at date T . Since the worst-case conditional probability minimizes the expected payoffs, it follows that we must solve: min Z(t, T ). Ξ
This in turn implies (2.14). Intuitively, under the worst-case scenario implied by ambiguity preferences, investors must be compensated more for the perceived threat of default. Proposition 2.1. Ceteris paribus, the presence of model misspecification leads to higher defaultswap and credit spreads. Proof. Consider first the default-swap spreads. Since the reference model probability law is always included in the set of admissible probability laws, it follows from (2.14) that the defaultswap spread in the presence of ambiguity is at least as high as that under the reference model. By the reverse of the argument above, higher default-swap spreads imply higher credit spreads. 11
Notice that a similar result exists for asymmetric information: ceteris paribus, asymmetric information leads to higher default-swap and credit spreads (see e.g. Lindset et al. [2008] for a proof of this result). Thus, for the same set of model parameters, the original Black and Cox [1976] model will have the lowest credit spreads, Duffie and Lando [2001] model will have higher spreads and the Duffie and Lando [2001] model with ambiguity will have the highest spreads. 2.3.1
Signal misspecification
Consider the case when the ambiguity in the economy is due to the precision of the accounting signals. That is, the set of possible likelihoods, Ξ, is parametrized by the signal quality var(Ut ) = σu2 ∈ σ 2u , σ 2u . The informational quality of the accounting signal is captured by a range of precisions 1/σ 2u , 1/σ 2u . Thus, in this case, the participants in the secondary debt market solve: 2
(2.15)
c(t, T ) =
2XEσu [Υ(T − t, At , AB ) |Gt ] max . P ri 2 ∈ σ2 , σ2 σu [ u u ] ki=1 e− 2 Eσu2 P t + 12 i |Gt
We have the following result: Proposition 2.2. The worst-case likelihood solving the problem (2.15) is given by: σ2 , y − u > a u 2 (2.16) σ ˜u = σ2 , y − u < a u
Proof. See the Appendix. Thus, participants in the secondary debt market treat positive signals (i.e. signals which are above the default threshold even after accounting for the mean signal error) as very imprecise and negative signals as very precise. Intuitively, since y − u − a measures the expected distance to default, a negative signal should be perceived as being very precise under the worst-case scenario interpretation implied by ambiguity preferences. On the other hand, a positive expected distance to default should be taken by the ambiguity-averse agents with a grain of salt: the worst possible scenario in this case is that the signal is very imprecise and the firm in fact is much closer to default than appears otherwise. This result is in line with the conclusion of Epstein and Schneider [2008], who find that, when evaluating signals about a company’s cash flows in an uncertain informational environment, equity investors believe negative signals to contain the 12
highest possible informational content and positive signals to have the lowest. Notice that this results differs fundamentally from the standard Bayesian learning. In the presence of multiple priors, the agent who uses Bayesian updating would update the probability of each prior using Bayes’ rule. Thus, no one prior would receive full probability weight with finite amount of data, unless the initial probability distribution over possible priors had the same property. Further, with Bayesian updating, negative and positive signals would not necessarily be perceived to have radically different informational quality. 2.3.2
Asset value model misspecification
Unlike the discussion above, assume now that the signal quality is known (so that σu2 is fixed) but, instead, the participants in the secondary debt market fear misspecification of the firm asset dynamics itself. That is, the representative agent fears misspecifications of the asset value transition law that are sufficiently small so that they are difficult to detect in the data because they are obscured by the random noise, Ut . In particular, we consider the specification in (2.1) as the reference model for the asset value dynamics. The representative participant in the secondary debt market, however, assumes that the true asset value dynamics are of the form: (2.17)
dZt = (m + σh(m, t)) dt + σdWth ,
for all t ≥ 0 and some h(m) ∈ Ξ(m). That is, the representative investor in the secondary debt market recognizes that a whole class, Ξ, of means of the Brownian motion are statistically indistinguishable in the data. For the case Ξ 6= {0}, this implies that the filtering problem faced by the representative investor will lead to multiple likelihoods for the data and not the single likelihood implied by the standard Bayesian framework. Similarly to Leippold et al. [2008], we assume that the set of admissible disturbances, Ξ(m), is given by: 1 2 (2.18) Ξ(m) ≡ h(m) : h (m, t) ≤ η ∀ t ≥ 0 , 2 where η ≥ 0 is the ambiguity tolerance of the debt holder. The above assumption implies that, for finite values of the ambiguity tolerance, the discrepancy between the reference model-implied distributions and those implied by the h-likelihood is constrained to be statistically small. Recall
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that Anderson et al. [2003] define: (2.19)
1 (h)(m) = h2 (m, t) 2
to be the (local) relative entropy between the two probability laws and show this to bound model detection probabilities. Since we assume that the drift rate of the log asset value, m, is known to the debt holders, some interpretation of the above setting is in order. In this section, we assume that m is a parameter reported to the secondary market by the equity owners. Thus, although the participants in the secondary debt market directly observe m, they do not know for sure that the underlying asset value grows at the rate µ = m + 12 σ 2 , as reported by the equity owners. Instead, they assume that the asset value evolves according to one of the models parametrized by the set of admissible likelihoods Ξ(m). The underlying asset value then grows at the rate µh = m + σh(m) + 12 σ 2 for a given likelihood h(m) ∈ Ξ(m). Notice also that, in some situations, it might be plausible to allow the ambiguity tolerance of the debt holders to depend on the reported value m. For example, it is plausible to assume that η is (weakly) increasing in m. That is, if the equity owners report that the asset value is growing at a low rate, the secondary debt market will consider this to be more likely to be the true rate of asset value growth than if the equity holders report a high rate of growth. In the following, however, we derive the worst-case likelihood without imposing any assumptions on the form of η. Notice that, since the expected asset growth rate under the perturbed model is still independent of the asset value realization, the filtering solution given by (2.5) carries through. That is, for a given likelihood h(m) ∈ Ξ(m), the conditional distribution of z at time t is given by the expression in (2.5) but with m replaced by m + σh(m). With the above assumptions in place, we have that, in the presence of ambiguity about the fundamental asset value dynamics, the participants in the secondary debt market solve: (2.20)
2XEh [Υ(T − t, At , AB ) |Gt ] c(t, T ) = max Pk . − ri 1 h∈Ξ(m) 2 Eh P t + i=1 e 2 i |Gt
Proposition 2.3. The worst-case likelihood solving problem (2.20) is given by: (2.21)
p h∗ (m, t) = − 2η. 14
Proof. See the Appendix. Thus, in the presence of ambiguity about the asset value dynamics, the participants in the secondary debt markets choose the lowest possible drift rate for the underlying. Intuitively, because we consider the case m > 0, a lower drift rate implies that the firm is less likely to recover from a negative shock to the asset value. Thus, in following the worst-case scenario, ambiguity-averse investors compute the filtered asset value assuming the lowest possible drift rate. Notice that this result is in line with the results of Leippold et al. [2008] who find that, in the presence of ambiguity about the dividend growth dynamics, ambiguity-averse investors always choose the lowest admissible rate.
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Numerical Example
3.1
Base calibration
We begin examining the implications of the model described in Section 2 by considering the behavior of credit spreads for a set of calibrated parameters. As the baseline calibration in this section, we take the calibrated parameters of Duffie and Lando [2001]. In particular, we have: (3.1)
σ = 5%,
r = 6% m = 1%,
δ = 5% α = 30% σu = 10%.
For the model in Duffie and Lando [2001], these parameters imply that, for an initial level of assets V0 = 100, the liquidation boundary is AB = 78, the initial par debt level D = 129.4 and the CDS payment in case of default X = 97.38%. Duffie and Lando [2001] argue that these are reasonable parameters based on the implied recovery rate of the (primary) debt at default: the model-implied 43.3% as a fraction of face value versus the historical 41% for all defaulted bonds monitored by Moody’s between 1920 and 1997. We take these parameters as given for now and consider calibrating them below. As the base case, consider the credit spreads in the Black and Cox [1976] model, plotted in Figure 1. As is well-known, models without either asymmetric information or time-varying default boundaries fail to produce plausible levels of credit spreads for bonds of short maturity, as the probability of default before maturity goes to zero as maturity decreases (see Figure 2).
15
Notice that, for a given maturity, the default-swap spreads in the Black and Cox [1976] model are decreasing in m. This is in line with our result in Proposition 2.3 that, in the presence of ambiguity about asset value dynamics, the representative investor in the secondary debt market chooses the lowest admissible value of the asset value growth rate in valuing defaultable securities. Notice further that ambiguity about the mean growth rate alone is not enough to generate non-zero credit spreads for bonds of short maturity: an information asymmetry is needed. Consider now the credit default spreads in the Duffie and Lando [2001] model. Figure 3 plots the dependence of the 1 year CDS spread on the growth rate of (log) asset value, m, and the signal error volatility, σu2 , conditional on observing a single period of signals a year after the last known value of assets. Consider first the upper panel. Consistent with the results of Proposition 2.2 and Proposition 2.3, the CDS spread is increasing in σu (since a positive signal is observed) and decreasing in m. In the lower panel, however, CDS spreads are decreasing in both σu and m, once again in line with the results of Proposition 2.2 and Proposition 2.3. Notice also that the spreads in after a negative signal are higher than after a positive one. Intuitively, even in the presence of ambiguity, the probability of default is larger after a negative signal than a positive one.
3.2
Multiple Signal Periods
Consider now the following experiment: instead of observing just one period of signals, the investors observe several periods of signals. Further, instead of using the full history of signals to learn about the value of z after the last signal, the investors update z0 to the conditional mean after each signal. Figures 4-7 report the evolution of CDS for different histories for the three models under consideration and Figure 8 the corresponding evolution of the conditional mean. The signals are assumed to occur at a quarterly frequency (to match the frequency of earnings report releases). In Figure 4, the investors observe a history of positive signals. As expected, the CDS spreads are higher in the presence of ambiguity. In this particular case, the spreads are highest in the presence of ambiguity about the asset value dynamics. As we will see below, this is not always
16
the case. Because the investors observe several periods of positive signals, the spreads decline toward the end of the history. Notice also from the upper left panel of Figure 8 that the conditional mean of z increases over time in this case. The increase, however, differs across the three models. The conditional mean increases fastest for the Duffie and Lando [2001] model and least for the model with asset value dynamics ambiguity. Notice that these differences in the conditional mean are mirrored in the CDS spreads, with the CDS spreads decreasing the least for the model with asset value dynamics ambiguity. Consider now Figure 5. In this case, the investor observes a positive signal, two neutral signals and a negative signal. As expected, the CDS spreads increase after observing the negative signals. Notice also that, after observing the first neutral signals, the CDS spreads also increase. Intuitively, the neutral signal is perceived more negatively if it is observed after a positive signal. Notice also that, after observing the negative signal, the CDS spreads increase the most for the model with ambiguity about signal quality. From the lower left panel, we see that although the conditional expectation of z initially increases more in the presence of ambiguity about signal quality than in the presence of ambiguity about asset value dynamics, it also decreased more following the negative signal. Thus, in the presence of ambiguity about signal quality, CDS spreads seem to be more volatile than in the presence of ambiguity about asset value dynamics. In fact, we can see that the spreads under the model with ambiguity about asset value dynamics are the least volatile. Consider now Figure 6. In this scenario, the investors observe a history of a positive signal, a neutral signal, a negative signal and a neutral signal. We can now see qualitative differences in the behavior of credit spreads under the different models. In particular, while the CDS spread under the Duffie and Lando [2001] model and in the presence of ambiguity about asset value dynamics decreases after the second neutral signal (as it is perceived relatively good after the negative signal), the CDS spread continues to increase. Looking at the conditional mean of z in the lower right panel of Figure 8, we see that in the presence of ambiguity about signal quality, zt decreases for the last two periods and increases slightly under both the Duffie and Lando [2001] model and the model with ambiguity about asset value dynamics. FInally, consider Figure 7 where the investors observe a history of only neutral signals. The model with ambiguity about asset value dynamics has the highest CDS spreads out of the three 17
models considered. Further, unlike the other two models, the CDS spread fails to decline as the history extends. Notice that, under the Duffie and Lando [2001] model, observing a long history of neutral signals is not bad news: since the firm fails to default despite these bad realizations, it becomes much more likely that the signals are precise. This is also reflected in the behavior of the conditional mean of z; we can see from the upper left panel of Figure 8 that the conditional mean increases for all three models over time, but the change for the model with ambiguity about asset value dynamics is much smaller. The above observations imply that, overall, the model with ambiguity about asset value dynamics provides CDS spreads with the least variability over time. Notice, however, that this model implies higher CDS spreads during periods when the signal direction remains consistent. Since CDS spreads tend to be very persistent, and remain high during economic downturns, this implies that the model with ambiguity about asset value dynamics matches stylized facts about the data better.
3.3
Financial Institutions
We now turn our attention to the events of July/August 2007. As Caballero and Krishnamurthy [2008b] argue, when France’s largest bank BNP Paribas revealed that two of its off-balance-sheet funds had loaded up on sub-prime mortgage securities, it not only sent a negative signal about its own performance, but also increased the ambiguity present in the market as a whole. Similar revelations by IKB, a German bank, and Northern Rock, a British bank, reinforced the shock to the information quality that the credit market received. Because these revelations dealt with off-balance-sheet debt, banks reacted by losing trust in each other’s balance sheets, thereby freezing the credit markets. The effect of the Paribas announcement is well illustrated in the behavior of the default-swap spreads. In Table 2, we can see that the default-swap spreads for financial institutions increased dramatically immediately following the announcement. A similar effect is observed following the bail-out of Bear Stearns on March 16, 2008 (see Table 3). Both of these occurrences suggest a strong response of the credit spreads to deterioration in the market information quality. Notice that the revelation that banks were exposed to defaults in the subprime mortgage market through their off-balance-sheet debt increased ambiguity in the credit
18
markets in two ways. First, these revelations made financial reports seem less plausible, thereby increasing ambiguity about the information quality. The complexity of the securities involved, however, also lead to an increase in the ambiguity about the true “asset” value growth rates for financial institutions. Below, we calibrate the parameters of the Duffie and Lando [2001] model using balance sheet data for financial institutions and then set ambiguity levels to match the observed CDS spreads. We begin the calibration by identifying the firm’s asset value, At , with observations of book equity. Running a simple OLS regression on 1 quarter differences in the log book equity allows us to identify m and σ and, hence, µ. Following the model assumption that equity holders receive a constant fraction δ of assets as a cash-flow stream, we identify δ as the time series average of total earnings as a fraction of total assets. We use the last available (pre-August 2007) value of long-term debt as an estimate of D since, in terms of our model, D is the face value of infinitely-lived debt. Then, using the Duffie and Lando [2001] calibrated parameters for θ, r and α, we can determine C as the value that minimizes the distance between the modelimplied and calibrated D. Finally, as before, we assume that the initial value of signal volatility is σu0 = 10% and the initial value of η is 0. The results of this calibration and the corresponding quantities of interest are presented in Table 4. Notice that there are several major differences between the results of this calibration and the calibration of Duffie and Lando [2001]. First, the mean growth rate is much higher (around 3-5%) than their calibrated value of 1%. Further, the long-term bond yield is much smaller (ranging from 0.14% to 2% as opposed to 6%) despite the fact that the expected recovery as a fraction of face value in case of default is much lower as well: the Duffie and Lando [2001] parameters imply recovery rates of around 41-43% whereas our calibrated parameters yield recovery rates of at most 8%. The lower bond yield is due to the fact that the financial institutions are initially further away from the default boundary than the firm in the Duffie and Lando [2001] calibration; the lower expected recovery rate at default is due to the relatively high D value for the financial institutions. In the interest of brevity, we will use the parameters calibrated for a particular financial institution (Goldman Sachs) for the remainder of the discussion. The actual CDS spreads at three points in time - July 06, 2007 (before the start of the crisis), August 10, 2007(day after the Paribas announcement) and March 28, 2008 (after the Bear Stearns bailout) 19
- are presented in Table 5. We consider separately which level of σu would match the spreads, given the calibrated m, and which level of η would match the observed spreads, assuming that σu = 10%. We assume that the size of the set of admissible drift disturbances is parametrized as: η = ηm2 .
(3.2)
The results of this calibration are presented in Table 6. Consider first the calibrated signal quality. From Table 6, we can see that the (perceived) signal quality did not change much following the Paribas announcement, as the CDS-implied signal error volatility remained at around 40%. However, after the bailout of Bear Stearns, markets drastically increased their beliefs about the signal quality. Intuitively, after the initial shock to the system, market participants could still think that BNP Paribas just had a bad realization of the signal error. As time passed and the market received more negative signals, however, the market participants had to accept that these negative signals were likely to have a relatively high precision. Consider now the calibration of admissible drift misspecifications. We see that, while the set was not trivial even before the start of the crisis, the size of the set increased five fold after the BNP Paribas announcement. While η also increased after the Bear Stearns bailout, it was much less dramatic: only two-fold compared to the August 2007 level.
4
Conclusion
In this paper, we have considered the implications of ambiguity for credit and default swap spreads. Using the Duffie and Lando [2001] model of credit spreads under incomplete information as a starting point, we introduced ambiguity in two different ways. First, we examined the behavior of default spreads in the presence of ambiguity about the information quality of the accounting signals that the participants in the secondary debt market receive. We find that, since the ambiguity-averse investors take the worst possible assessment of signal quality, they react more strongly to bad news than to good news. Thus, a deterioration of the information environment leads to a widening of credit spreads, even if the firm fundamentals do not change. 20
Further, even if the informational environment remains constant, ambiguity about signal quality causes an asymmetry in the response of credit spreads to negative and positive signals, with the increase in credit spreads following negative news larger than the corresponding decrease following positive news of similar magnitude. We continued by introducing ambiguity about the dynamics of the underlying asset value. In this setting, participants in the secondary market as if they believe the asset value growth rate to be the lowest feasible one. To distinguish between the two sources of ambiguity, we consider their implications for the default spreads of financial institutions. In particular, we calibrate the model to the balance sheet data of several prominent financial institution and repeat the above exercise to find the implied changes in ambiguity following the BNP Paribas announcement. We find that, although ambiguity about signal quality is better able to match the changes in the 1 year default spread, changes in the ambiguity about the asset dynamics are better able to match the changes in the overall term structure of default spreads. Thus, we conclude that ambiguity about the underlying is more suitable in this setting.
21
A
Technical Appendix
A.1
CDS spreads under perfect information
Denote: zt = log At , a = log AB . Then, under perfect information, the time t credit default swap (CDS) spread on a bond maturing at date T is given by: 2XΥ(T − t, At , AB ) , c(t, T ) = Pk −ri/2 P (t + i/2) e i=1 where: h i Υ(T − t, At , AB ) ≡ E e−r(T −t) 1τ ≤T |Ft (A.1)
= exp[−(zt − a)γ ]Φ + exp[−(zt − a)γ + ]Φ
(A.2)
! √ m2 + 2rσ 2 (T − t) √ T −t ! √ 1 a − zt + m2 + 2rσ 2 (T − t) √ σ T −t
1 a − zt − σ
−
P (s) ≡ E [1τ >s |Ft ] zt − a + m(s − t) −zt + a + m(s − t) −2m(zt −a)/σ 2 √ √ = Φ −e Φ σ s−t σ s−t √ m ± m2 + 2rσ 2 ± γ = , σ2
and Φ is the c.d.f. of the standard normal.
A.2
Proof of Proposition 2.2
Recall that, in the presence of ambiguity about signal quality, the default spread on a bond with maturity T is given by: 2
2XEσu [ Υ(T − t, zt , a)| Gt ] c(t, T ) = max . P ri 2 ∈ σ2 , σ2 σu [ u u ] ki=1 e− 2 Eσu2 P t + 12 i Gt Taking the derivative with respect to σu2 , we have: i ∂g R +∞ h Pk − ri 1 2 P t + 2XΥ(T − t, z, a) − c(t, T ) e i 2 dz i=1 2 a ∂c ∂σu (A.3) = . R Pk ri +∞ ∂σu2 e− 2 P t + 1 i g(z|y, z0 , t)dz i=1
2
a
22
Notice that: 2XΥ(T − t, z, a) − c(t, T )
k X i=1
( 2 σu
= 2XE
[ Υ(T − t, zt , a)| Gt ]
ri
e− 2 P
1 t+ i = 2 ) − ri 1 2 P t + e i i=1 2 2 − ri 1 2 Eσu P t + e i=1 2 i Gt
2XΥ(T − t, zt , a) − Pk 2 σ E u [ Υ(T − t, zt , a)| Gt ]
Pk
Since c(t, T ) < 1, this implies that 2XΥ(T − t, z, a) − c(t, T )
k X i=1
− ri 2
e
1 P t + i < 0. 2
Thus, to determine the sign of the derivative, we need to consider the sign of ∂g/∂σu2 . Recall that: b(z|y, z0 , t) , g(z|y, z0 , t) = R +∞ b(z|y, z0 , t)dz a where: (y − u − z)2 (z0 + mt − z)2 2(z0 − a)(z − a) b(z|y, z0 , t) = exp − − 1 − exp − . 2σu2 2σ 2 t σ2t Hence: ∂g ∂σu2
= = =
R +∞ ∂b dz ∂b/∂σu2 a ∂σ 2 − g(z|y, z0 , t) R +∞ u R +∞ bdz bdz a a Z +∞ 1 2 2 (y − u − z) g(x|y, z0 , t)dx g(z|y, z0 , t) (y − u − z) − 2(σu2 )2 0 1 g(z|y, z0 , t) (y − u − z)2 − Et [(y − u − z)2 ] . 2 2 2(σu )
Notice that: (y − u − z)2 = (y − u − a)2 − 2(y − u − a)(z − a) + (z − a)2 , so, substituting into (A.3), we obtain: i hn o Pk 2 − ri 1 σu 2 P t + i E 2XΥ(T − t, z, a) − c(t, T ) e (y − a − u) (z − a − E [z − a]) Gt t i=1 2 ∂c = P 2 k − ri 1 ∂σu2 2 Eσu P t + i=1 e 2 i Gt h n o i P 2 k − ri 1 σ 2 2 u 2 E 2XΥ(T − t, z, a) − c(t, T ) i=1 e P t + 2 i (z − a) − Et [(z − a) ] Gt + Pk ri − 2 σu E 2 P t + 12 i Gt i=1 e From the above, we can see that, if y − a − u > 0, the derivative is positive for all values of σu2 and the worst-case likelihood is given by σu2 = σ 2u . Similarly, if y − a − u < 0, the derivative is negative for all values of σu2 and the worst-case likelihood is given by σu2 = σ 2u . 23
A.3
Proof of Proposition 2.3
Recall that, in the presence of ambiguity about the firm’s asset dynamics, the default spread on a bond with maturity T is given by: 2XEh [ Υ(T − t, zt , a)| Gt ] c(t, T ) = max Pk . − ri 1 h∈Ξ(m) 2 Eh P t + e Gt i i=1 2 Notice that the choice of h is equivalent to the choice of m, with the feasible set appropriately defined. Thus, we can solve the above problem by maximizing over the set of feasible m’s. Taking the derivative with respect to m, we obtain: o R +∞ n Pk − ri ∂Υ ∂P 2 2X e − c(t, T ) i=1 ∂m ∂m g(z|y, z0 , t)dz a ∂c = − P ri k −2 h ∂m E P t + 21 i Gt i=1 e o ∂g R +∞ n Pk − ri 1 2 P t + e 2XΥ(T − t, z, a) − c(t, T ) i i=1 2 ∂m dz a + Pk − ri 1 2 Eh P t + i=1 e 2 i Gt s1 (m) + s2 (m) ≡ Pk . − ri e 2 Eh P t + 1 i Gt i=1
2
Consider the first summand in the derivative, s1 (m). We have: ∂Υ ∂m
∂P ∂m
! √ 1 a − zt ± m2 + 2rσ 2 (T − t) √ σ T −t ! r √ 1 a − zt ± m2 + 2rσ 2 (T − t) m T −t ± √ ± exp −(z − a)γ φ 2 + 2rσ 2 σ σ m T −t ! √ 2 + 2rσ 2 (T − t) γ+ 1 a − z + m t √ = −(z − a) √ exp −(z − a)γ + Φ σ T −t m2 + 2rσ 2 ! √ 2 + 2rσ 2 (T − t) − z − m γ− 1 a t √ +(z − a) √ exp −(z − a)γ − Φ σ T −t m2 + 2rσ 2 √ x + mt −x + mt x −2mx/σ2 −x + mt −2mx/σ 2 √ √ √ = t φ −e φ +2 e Φ σ σ t σ t σ t x −2mx/σ2 −x + mt √ = 2 e Φ . σ σ t ∂γ ± = −(z − a) exp −(z − a)γ ± Φ ∂m
24
Thus, the first summand can be expressed as: " ! # √ + 2 + 2rσ 2 (T − t) γ 1 a − z + m t √ exp −(z − a)γ + Φ s1 (m) = −2XEm (z − a) √ Gt 2 2 σ T −t m + 2rσ " ! # √ γ− 1 a − zt − m2 + 2rσ 2 (T − t) √ +2XEm (z − a) √ exp −(z − a)γ − Φ Gt σ T −t m2 + 2rσ 2 # " ! k X −(z − a) + mi/2 − ri m z − a −2m(z−a)/σ 2 2 p −2c(t, T ) e E Φ e Gt σ σ i/2 i=1
≤ 0 as γ − < 0. Consider now the second summand, s2 (m). We have: ∂g ∂m
= − =
1 g(z|y, z0 , t) [(z0 + mt − z) − Et [z0 + mt − z]] σ2
1 g(z|y, z0 , t) [z − a − Et [z − a]] . σ2
Thus, we can express the second summand as: "( m
s2 (m) = E
2XΥ(T − t, z, a) − c(t, T )
k X
e
− ri 2
i=1
# ) 1 P t+ i [z − a − Et [z − a]] Gt 2
≤ 0. Thus, ∂c/∂m is negative for all values of m and the maximum is obtained by taking the lowest admissible value of m. Hence, the worst-case likelihood is p h∗ = − 2η.
25
References E. W. Anderson, L. P. Hansen, and T. J. Sargent. A quartet of semigroups for model specification, robustness, prices of risk and model detection. Journal of the European Economic Association, 1(1):68–123, 2003. F. Black and J.C. Cox. Valuing corporate securities: Some effects of bond indenture provisions. Journal of Finance, 31(2):351–367, 1976. R. J. Caballero and A. Krishnamurthy. Collective risk management in a flight to quality episode. Journal of Finance, 63:2195–2230, 2008a. R. J. Caballero and A. Krishnamurthy. Musical chairs: a comment on the credit crisis. Banque de France, Financial Stability Review – special issue on liquidity, 11, 2008b. Z. Chen and L. G. Epstein. Ambiguity, risk and asset returns in continuous time. Econometrica, 70:1403–1443, 2002. D. Duffie and D. Lando. Term structures of credit spreads with incomplete accounting information. Econometrica, 69:633–664, 2001. L. G. Epstein and J. Miao. A two-person dynamic equilibrium under ambiguity. Journal of Economic Dynamics and Control, 27:1253–1288, 2003. L. G. Epstein and M. Schneider. Recursive multiple-priors. Journal of Economic Theory, 113 (1):1–31, 2003. L. G. Epstein and M. Schneider. Ambiguity, information quality and asset pricing. Journal of Finance, LXIII(1):197–228, 2008. P. Gagliardini, P. Porchia, and F. Trojani. Ambiguity aversion and the term structure of interest rates. Review of Financial Studies, Forthcoming. I. Gilboa and D. Schmeidler. Maxmin expected utility with non-unique priors. Journal of Mathematical Economics, 18:141–153, 1989.
26
M. Harrison. Brownian motion and stochastic flow systems. John Wiley and Sons, New York, 1985. M. Leippold, F. Trojani, and P. Vanini. Learning and asset prices under ambiguous information. Review of Financial Studies, 21:2565–2597, 2008. S. Lindset, A.-C. Lund, and S.-A. Persson. Credit spreads and incomplete information. Norwegian School of Economics and Business Administration Discussion Paper, 2008. J. Liu, J. Pan, and T. Wang. An equilibrium model of rare-event premia and its implication for option smirks. Review of Financial Studies, 18:131–164, 2005. F. Trojani and P. Vanini. A note on robustness in merton’s model of intertemporal consumption and portfolio choice. Journal of Economic Dynamics and Control, 26(3):423–435, 2002. F. Trojani and P. Vanini. Robustness and ambiguity aversion in general equilibria. The Review of Finance, 2:423–435, 2004. R. Uppal and T. Wang. Model misspecification and under-diversification. Journal of Finance, LIX:2465–2486, 2003.
27
θ
r
m
δ
α
σ
σu2
C
AB
D
X
C/D
Recovery
35%
7%
1%
5%
30%
5%
5%
8
78
129.4
97.38%
6.18%
43.3%
Table 1: Reference parameters for the Duffie and Lando [2001] calibration and associated base quantities
1 year
2 year
3 year
4 year
5 year
7 year
10 year
Bank of America
156.36
145.88
140.87
150.00
154.55
68.67
82.35
Citi Corp
184.48
183.70
185.60
184.47
184.26
140.50
116.19
Goldman Sachs
126.52
122.22
117.72
122.37
125.95
103.55
92.53
JP Morgan
174.24
189.36
201.67
182.35
173.85
75.85
80.25
Lehman Bros.
206.76
203.92
200.00
203.00
206.31
142.58
164.34
Merrill Lynch
118.31
118.04
112.79
119.20
123.71
90.00
95.03
Wells Fargo
243.75
263.89
274.07
256.19
244.96
94.77
109.60
Table 2: Change in default spreads after August 9, 2007, as compared to one month previous. Changes reported in percentage terms. Source: Bloomberg
1 year
2 year
3 year
4 year
5 year
7 year
10 year
Bank of America
37.08
59.67
73.39
69.24
59.47
68.81
76.80
Citi Corp
104.82
113.21
101.38
89.21
85.19
81.49
77.57
Goldman Sachs
124.19
137.29
133.16
128.87
96.81
135.75
141.42
JP Morgan
170.88
107.47
104.11
96.06
88.25
84.78
93.08
Lehman Bros
148.02
140.93
133.53
129.71
127.10
132.89
130.23
Merrill Lynch
-83.14
81.24
80.68
76.37
70.48
80.81
121.83
Wells Fargo
71.91
72.88
69.95
68.68
87.56
69.02
74.68
Table 3: Change in default spreads after Bear Stearns bailout, as compared to one month previous. Changes reported in percentage terms. Source: Bloomberg
28
m
µ
σ
δ
D
C
X
C/D
AB
Recovery
Bank of America
4.85
5.58
12.09
3.81
169317.00
2063.91
94.87
1.22
13045.69
5.13
Merrill Lynch
3.56
3.63
3.54
3.76
185292.00
240.97
99.24
0.13
2146.09
0.76
JP Morgan
3.42
4.21
12.57
2.22
172163.00
2420.22
93.35
1.41
29426.18
6.65
Citi Corp
3.99
5.16
15.33
3.88
507216.00
10268.65
91.72
2.02
61855.32
8.28
Wells Fargo
3.26
3.80
10.34
3.51
93830.00
939.54
94.82
1.00
7909.44
5.18
Goldman Sachs
5.93
6.08
5.57
5.03
167253.00
428.35
98.87
0.26
2142.60
1.13
Lehman Bros
3.54
3.61
3.73
3.61
100819.00
145.40
99.15
0.14
1350.59
0.85
Table 4: Calibrated parameters and quantities of interest for financial institutions. Calibration done using balance sheet data from COMPUSTAT. m, µ, σ, δ, X, C/D and Recovery reported in percentage terms.
1 year
2 year
3 year
4 year
5 year
7 year
10 year
07/06/2007
13.2
17.1
23.7
30.4
37.0
42.2
49.5
08/10/2007
29.9
38.0
51.6
67.6
83.6
85.9
95.3
03/28/2008
178.4
171.3
164.6
157.7
150.7
146.6
150.0
Table 5: CDS spreads (in bps) for Goldman Sachs at three points in time. Source: Bloomberg.
σu
m
η
07/06/2007
40.1
5.0
0.0123
08/10/2007
40.1
3.9
0.0586
03/28/2008
6.9
3.0
0.1221
Table 6: Ambiguity calibration for Goldman Sachs CDS spreads. σu and m reported in percentage terms. η is set to make the “ambiguity” m match the one calibrated from balance sheet data.
29
Black and Cox [1976] model
300
CDS spread (in bps)
250 200 150 100 50 0 10 8
0.04 6
0.03 4
0.02 2
τ
0.01 0
0
m
Figure 1: Credit default spreads in the Black and Cox [1976] model. Model parameter values are listed in Table 1.
30
Student Version of MATLAB
Black and Cox [1976] model
100 95
P(no default)
90 85 80 75 70 65 10 8
0.04 6
0.03 4
0.02 2
τ
0.01 0
0
m
Figure 2: Probability of no default in the Black and Cox [1976] model. Model parameter values are listed in Table 1.
31 Student Version of MATLAB
CDS spread (bps), 1 year maturity
Single Positive Signal 600 400 200 0 0.25
0.2
0.15
0.1
CDS spread (bps), 1 year maturity
σu
0.05
0
0.02
0.01
0.03
0.04
m Single Negative Signal
1000
500
0 0.25
0.2
0.15
0.1 σu
0.05
0
0.02
0.01
0.03
0.04
m
Figure 3: 1 year credit default spreads in the Duffie and Lando [2001] model. Upper panel: spreads conditional on observing a single positive signal (y = 4.4807); lower panel: spreads conditional on observing a single negative signal y = 4.4104). Model parameter values are listed in Table 1.
32
Student Version of MATLAB
Signal ambiguity
CDS spread (bps)
Duffie and Lando [2001]
300
300 200 100 10
4
5 τ
200
0
0
2 period
Dynamics ambiguity 150
100 10 4 5
3
CDS spread (bps)
CDS spread (bps)
250
400
0
1
400 300 200 10
4
5
2 τ
500
τ
period
0
0
2 period
Figure 4: Time evolution of CDS spreads under various models, given a history of good signals (y = [4.4807 4.4807 4.4807 4.4807]). Right panel: Duffie and Lando [2001] model with exogenous default boundary; upper left panel: Duffie and Lando [2001] model with signal quality ambiguity; lower left panel: Duffie and Lando [2001] model with ambiguity about asset value dynamics. Model parameter values are listed in Table 1.
Student Version of MATLAB
33
Duffie and Lando [2001]
CDS spread (bps)
Signal ambiguity
400
1000 500 0 10
4
5
300
τ
0
0
2 period
250 Dynamics ambiguity 200 150 10 4 5
3
CDS spread (bps)
CDS spread (bps)
350
1500
800 600 400 200 10
τ
0
1
4
5
2
τ
period
0
0
2 period
Figure 5: Time evolution of CDS spreads under various models, given a history of semi-good signals (y = [4.4807 4.4578 4.4578 4.4104]). Right panel: Duffie and Lando [2001] model with exogenous default boundary; upper left panel: Duffie and Lando [2001] model with signal quality ambiguity; lower left panel: Duffie and Lando [2001] model with ambiguity about asset value dynamics. Model parameter values are listed in Table 1.
Student Version of MATLAB
34
Duffie and Lando [2001]
CDS spread (bps)
Signal ambiguity
400
1000 500 0 10
4
5
300
τ
0
0
2 period
250 Dynamics ambiguity 200 150 10 4 5
3
CDS spread (bps)
CDS spread (bps)
350
1500
800 600 400 200 10
τ
0
1
4
5
2
τ
period
0
0
2 period
Figure 6: Time evolution of CDS spreads under various models, given a history of semi-bad signals (y = [4.4807 4.4578 4.4104 4.4578]). Right panel: Duffie and Lando [2001] model with exogenous default boundary; upper left panel: Duffie and Lando [2001] model with signal quality ambiguity; lower left panel: Duffie and Lando [2001] model with ambiguity about asset value dynamics. Model parameter values are listed in Table 1.
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Signal ambiguity
CDS spread (bps)
Duffie and Lando [2001]
350
300 200 100 10
4
5 τ
250
0
0
2 period
Dynamics ambiguity 200
150 10 4 5
3
CDS spread (bps)
CDS spread (bps)
300
400
0
1
500 400 300 10
4
5
2 τ
600
τ
period
0
0
2 period
Figure 7: Time evolution of CDS spreads under various models, given a history of bad signals (y = [4.4578 4.4578 4.4578 4.4578]).Right panel: Duffie and Lando [2001] model with exogenous default boundary; upper left panel: Duffie and Lando [2001] model with signal quality ambiguity; lower left panel: Duffie and Lando [2001] model with ambiguity about asset value dynamics. Model parameter values are listed in Table 1.
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Good history 4.475
DL01 Signal Asset
4.47
DL01 Signal Asset
4.468 4.466
4.465
zt
zt
Bad history 4.47
4.464 4.462
4.46 4.46 4.455
1
2
3
4.458
4
1
2
period Semi−good history 4.48
4.46
4.46
4.45
4.44
4.44
zt
zt
4
Semi−bad history
4.47
4.42
4.43
DL01 Signal Asset
4.42 4.41
3 period
1
2
DL01 Signal Asset
4.4 3
4.38
4
period
1
2
3
4
period
Figure 8: Time evolution for the conditional mean of the (log) asset value for different signal histories, under different models. “DL01” is the Duffie and Lando [2001] model; “Signal” is the Duffie and Lando [2001] with ambiguity about signal quality; “Asset” is the Duffie and Lando [2001] model with ambiguity about asset value dynamics. Upper right panel: good history (y = [4.4807 4.4807 4.4807 4.4807]); upper left panel: bad history (y = [4.4578 4.4578 4.4578 4.4578]); lower right panel: semi-good history (y = [4.4807 4.4578 4.4578 4.4104]); lower left panel: semibad history (y = [4.4807 4.4578 4.4104 4.4578]). Model parameter values are listed in Table 1.
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