A New Approach to Estimating Loss-Given-Default Distribution
AA NEW APPROACH TO ESTIMATING LOSS-GIVEN-DEFAULT DISTRIBUTION
MASAHIKO EGAMI AND RUSUDAN KEVKHISHVILIA
BSTRACT . We propose a new approach to estimating the loss-given-default distribution. More precisely,we obtain the default-time distribution of the leverage ratio (defined as the ratio of a firm’s assets over itsdebt) by examining its last passage time to a certain level. In fact, the use of the last passage time is partic-ularly relevant because it is not a stopping time: this corresponds to the fact that the timing and extent ofsevere firm-value deterioration, when default approaching, is neither observed nor easily estimated. We cal-ibrate the model parameters to the credit market, so that we can illustrate the loss-given-default distributionimplied in the quoted CDS spreads.
Key Words: Loss given default; asset process; intensity; credit risk; last passage time; CDS spread.JEL Classification: G32.Mathematics Subject Classification (2010): 60J70.1. I
NTRODUCTION
There exists a large body of literature regarding the loss-given-default distribution as the information aboutthis distribution is important for credit risk management. One example is the estimation of loan-loss reserves inthe banking industry. The estimation of loan-loss reserves should reflect current conditions of the credit market;however, predetermined 40% recovery rate is usually assumed in CDS spread quoting. Hence we cannot overem-phasize the effectiveness of a model that provides the loss-given-default distribution implied from the credit marketconditions. The model to be presented in this paper, with minimal and standard assumptions, captures the dynam-ics of the asset value close to default, while the default is treated as an unexpected event. By accomplishing these,the model can well describe default events observed in the real world.We derive the distribution of the ratio of the firm’s total assets over its total debt upon default. We shall hereafterrefer to this ratio as the leverage ratio and to the dynamics of this ratio as the leverage process . We focus on thisratio and its distribution at default since, based on this corporate-level loss distribution, one can calculate lossrates for each individual debt obligation having distinct characteristics; for example, senior or subordinated, andwith or without collateral. Note that we will examine the distribution of the leverage ratio at default, which is thecorporate-level recovery rate of the asset, and accordingly the loss-given-default is obtained by (1-recovery rate).We shall use the term of loss-given-default where no confusions should arise since it is this term that is mainlyused in the literature.While the firm-value approach proposed in Merton (1974) has been successful in estimating the default proba-bility, one has some difficulties in calculating the loss given default under this method: the firm’s default is definedas the first passage time of the leverage process to a certain point, say c , and hence the value of the leverage ratio This version: September 3, 2020. The first author is in part supported by Grant-in-Aid for Scientific Research (C) No.18K01683, Japan Society for the Promotion of Science. a r X i v : . [ q -f i n . R M ] S e p MASAHIKO EGAMI AND RUSUDAN KEVKHISHVILI upon default is necessarily equal to c . But this level c may not be consistent with the empirical research. Toovercome this difficulty, we consider the last passage time of the leverage process to another threshold level α > c .Let us denote this time by L α . To model the loss-given-default distribution, we explicitly incorporate the fact thatafter the last passage time L α , business environment surrounding the company severely alters and the firm is notable to recover to normal operations. As is well known, the last passage time is not a stopping time and this isthe reason why L α is appropriate for modeling loss-given-default distribution: market observers (such as regularshareholders) do not know whether L α has occurred or not and do not see when this switch in the dynamics of theasset value takes place. The information about the time L α may be known only to the company insiders. Therefore,the last passage time is relevant in the sense that it accounts for the reality that market observers are facing.After the firm’s leverage process passes the level α for the last time, it is appropriate to switch the modelparameters so that they reflect deteriorated business conditions of the company. Accordingly, we shall work, aftertime L α , with a transition semigroup (of the diffusion) that incorporates the fact that the leverage ratio wouldnot return to the level α . At this stage, we also consider the intensity-based approach to identify the defaulttime ξ which plays a major role in our model for the loss-given-default distribution. By suitably choosing anintensity process, we explicitly derive the distribution of the leverage ratio at the time of default. In identifying thedefault time, the use of the intensity-based model after time L α is appropriate because the market observers do notthereafter know the true state of the firm and would regard default as an unexpected shock. Finally, to be consistentwith the credit market, we set the triggering threshold α in such a way that the CDS spread implied by our modelis consistent with the quoted spread in the market.In summary, we obtain the distribution of the leverage ratio of the firm at the time of default. The assumptionsof our model are minimal and standard: we do not use jumps or additional risk factors. The key element is the lastpassage time L α which we cannot observe and after which the default is represented as an unexpected shock as inthe intensity-based approach.1.1. Literature Review.
As Doshi et al. (2018) and many other studies point out, it is a standard practice toassume a constant recovery rate (approximately 40%) of the debt upon default. In contrast to this standard assump-tion, the empirical literature has documented the evidence of time-varying realized recovery rates. This in turnindicates the importance of stochastic models for the loss given default.Gambetti et al. (2019) analyze determinants of recovery rate distributions and find economic uncertainty to bethe most important systematic determinant of the mean and dispersion of the recovery rate distribution. For theiranalysis, they use post-default bond prices of 1831 American corporate defaults during 1990-2013. Even thoughrecovery rates vary with firm’s idiosyncratic factors, Gambetti et al. (2019) state that the impact of systematicfactors related to economic cycles should not be underestimated. For the summary of studies related to cross-sectional and time-variation of recovery rates, we refer the reader to Gambetti et al. (2019).Altman et al. (2004) provide a detailed review of how recovery rate and its correlation with default probabilityhad been treated in credit risk models. They also discuss the importance of modeling the correlation between re-covery rate and default probability. Altman et al. (2005) analyze and measure the relationship between default andrecovery rates of corporate bonds over the period of 1982-2002. They confirm that default rate is a strong indicatorof average recovery rate among corporate bonds. Acharya et al. (2007) analyze data of defaulted firms in the U.S.during 1982-1999 and find that the recovery rate is significantly lower when the industry of the defaulted firm isin distress. They discover that industry conditions at the time of default are robust and important determinants
NEW APPROACH TO ESTIMATING LOSS-GIVEN-DEFAULT DISTRIBUTION 3 of recovery rates. Their results suggest that recovery rates are lower during industry distress not only because ofdecreased worth of firm’s assets, but also because of the financial constraints that other firms in the industry face.The latter reason is based on the idea that the prices at which assets of the defaulted firm can be sold depend on thefinancial condition of other firms in the industry.Doshi et al. (2018) use information extracted from senior and subordinate credit default swaps to identify risk-neutral stochastic recovery rate dynamics of credit spreads and study the term structure of expected recovery. Theirstudy is related to Schl¨afer and Uhrig-Homburg (2014) which also uses the fact that debt instruments of differentseniority face the same default risk but have different recovery rates given default. Doshi et al. (2018) use 5factor intensity-based model for CDS contracts allowing stochastic dynamics of the loss given default. They allowfirm-specific factors to influence the stochastic recovery rate. Their empirical analysis of 46 firms through thetime period of 2001-2012 shows that the recovery rate is time-varying and the term structure of expected recoveryis on average downward-sloping. They also find that industry characteristics have significant impact on CDSimplied recovery rates; however, they do not find the evidence that firms’ credit ratings explain the cross-sectionaldifferences in recovery rates. For the summary of the literature related to the relationship between default rates andrealized recovery rates, refer to Doshi et al. (2018).Yamashita and Yoshiba (2013) is an example of a study that uses stochastic collateral value process to incor-porate stochastic recovery rate into the model which assumes that a constant portion of the collateral value isrecovered upon default. Yamashita and Yoshiba (2013) use a quadratic Gaussian process for the default intensityand discount interest rate and derive an analytical solution for the expected loss and higher moments of the dis-counted loss distribution for a collateralized loan. They assume that default intensity, discount interest rate andcollateral value are correlated through Brownian motions driving Gaussian state variables.Cohen and Costanzino (2017) is an example of a study that extends a structural credit risk model and incor-porates stochastic dynamics of the recovery rate. Cohen and Costanzino (2017) extend a Black-Cox model byintroducing recovery risk. They separate asset risk driver from the recovery risk driver and model asset and recov-ery processes as correlated geometric Brownian motions. In their model, the asset risk driver serves as a defaulttrigger and the recovery risk driver determines the amount recovered upon default. Cohen and Costanzino (2017)explicitly compute prices of bonds and CDS under this framework. See also Kijima et al. (2009).Finally, we provide an example of a risk model that uses occupation time because our framework for the lossgiven default includes occupation time of a random process. Albrecher et al. (2011) distinguish between negativesurplus and bankruptcy and assume that probability of bankruptcy is a function of negative surplus. In their model,bankruptcy rate function is positive when the surplus is below zero and zero when the surplus is positive. In thisway, their model is based on the occupation time of the surplus process below zero.The rest of the paper is organized as follows: Sections 2.1 and 2.2 introduce the models for asset process anddefault time. Section 2.3 derives the distribution of the loss given default. Section 2.4 discusses the connectionbetween the firm-value and intensity-based modeling employed in our framework. Section 3 demonstrates theimplementation of our model when the asset process follows geometric Brownian motion with alternating driftand volatility parameters. Section 3.5 gives detailed description of the estimation procedure that is necessary forderiving the loss-given-default distribution implied in the credit market. Section 4 provides and discusses theestimation result for Campbell Soup Company as an example. For our analysis, the risk-free rate was obtainedfrom the website of the U.S. Department of the Treasury. The remaining data used for the estimation was obtainedfrom Thomson Reuters.
MASAHIKO EGAMI AND RUSUDAN KEVKHISHVILI
2. M
ODEL
Given the fixed probability space ( Ω , G , P ) , let G : = ( G t ) t ≥ and F : = ( F t ) t ≥ denote filtrations satisfying theusual conditions where F is the sub-filtration of G ( F ⊆ G ). We assume the filtration F is generated by the Brownianmotion involved in the dynamics of the firm’s asset process defined on ( Ω , G , P ) . Let r denote the constant interestrate. As the size of the firm’s debt relative to its assets determines firm’s default, in the sequel we focus on the ratioof the asset value over debt. By such standardization, we can set the threshold levels (related to default time) toconstants.2.1. Process V . As in the firm-value approach, the diffusion V represents the asset process. We assume that V isa nonnegative regular time-homogeneous diffusion, measurable with respect to F , whose dynamics are given by(2.1) d V t = µ ( V t , I ( t )) d t + σ ( V t , I ( t )) d W t , where W is a standard Brownian motion. Namely, the state space of V is ( , + ∞ ) . The process I is a right-continuous process defined as(2.2) I ( t ) = I { t < S } + I { S ≤ t < T } + I { T ≤ t < S } + · · · + I n − { S n − ≤ t < T n − } + · · · , where I i ∈ { , } , I =
0, and I i + = − I i for all i ∈ N with the sequence of F stopping times, T = S n = inf { t > T n − : ( V / B ) t ≤ α } , and T n = inf { t > S n : ( V / B ) t > α } .The level α > ( B t ) t ≥ involved in the definition of stopping times S n and T n is defined below. We assume that a strong solution to (2.1) exists for each i and that | µ ( x , i ) | + | σ ( x , i ) | ≤ C ( + | x | ) , x ∈ ( , + ∞ ) for each i with some positive constant C < + ∞ . As we are interested in modeling the loss-given-default distribution,we assume that for the process V , the left-boundary 0 is attracting and the right boundary + ∞ is non-attracting. Inthis case, lim t → ∞ V t = ( B t ) t ≥ satisfying B ∈ F and d B t = rB t d t withconstant growth rate r . The process B represents a certain amount of debt to be repaid and we let it be the sum ofshort-term debt and a half of long-term debt (as in Moody’s KMV approach) since it has been known that such levelis an appropriate default threshold for the firm’s assets. We define the leverage process (cid:0) VB (cid:1) t ≥ with (cid:0) VB (cid:1) = x > T c : = inf (cid:26) t ≥ V t B t = c with c = (cid:27) . The firm-value approach to credit risk analysis (including Merton-type models), assumes that the default time isexactly this first hitting time T c . This implies that the value of debt B , used as the barrier, would be repaid in full;i.e., B T c = V T c . However, in reality, this equality does not hold: V Tc B Tc = V , together with T c , may not be the right process for the purposeof estimating the loss given default. Accordingly, while we shall still in part rely on the firm-value model, we mustredefine the time (2.3) for our modeling purpose. NEW APPROACH TO ESTIMATING LOSS-GIVEN-DEFAULT DISTRIBUTION 5
Let α ∈ ( c , x ) with c =
1. For the purpose of estimating the distribution of the leverage ratio upon default, wedefine the last passage time of level α > c by(2.4) L α : = sup (cid:26) ≤ t < ∞ : V t B t = α (cid:27) with the convention that the supremum of the empty set is zero. We then redefine the passage time T c as(2.5) T c : = inf (cid:26) t ≥ L α : V t B t = c with c = (cid:27) . Note that, in our setup, the leverage process ( VB ) t ≥ may hit c = L α . In reality, the asset process isunobservable and the leverage process may hit the level c multiple times before default. Our definition of T c in(2.5) does not exclude this possibility. Our reasoning is the following: the firm-value approach is based on theasset process which is estimated according to the option-theoretic approach using stock price data. Therefore,stock price fluctuations affect the asset value and large stock market turmoil may cause it to drop to such a lowlevel that the leverage process VB hits c = T c in the sense of (2.3) would occur; however, this does notmean a default in a practical and real sense. In contrast, the passage time to the level c = L α , which isour T c in (2.5), is important because it incorporates the fact that the leverage process will never recover to level α .While our model does not regard T c in (2.5) as a default time because it does not produce any loss-given-defaultdistribution, the notion of T c is still important because empirical research (regarding default probabilities, CDSspreads, etc.) based on the firm-value approach relies on a passage time. For this reason, we also consider the time T c in (2.5) when estimating necessary parameters in our model; specifically, in Steps 3 and 4 in Section 3.5.To model the loss-given-default distribution, we should explicitly incorporate the fact that after the last passagetime L α of a certain level α > c , business relationship that surrounds the company severely alters. Lost businessopportunities, shrinking market share, and worsening trade terms would not allow the firm to come back to thenormal situation. Therefore, we need another time clock that starts ticking at time L α , after which the leverageprocess does not return to α . For our purposes, it is convenient to work with a new positive diffusion process YB that starts at time L α at point α and does not recover to level α before converging to the left boundary 0.The technical tool that fits our objective is given in Meyer et al. (1972). The semigroup of the process VB aftertime L α is obtained by conditioning the process not to hit α before converging to the left boundary. We illustratethis point. Let F = F ∞ . Following Meyer et al. (1972), for any positive random variable R measurable withrespect to F , we define(2.6) F R : = { A ∈ F : ∀ t ≥ , ∃ A t ∈ F t s.t. A ∩ { R < t } = A t ∩ { R < t }} . For the co-terminal (and co-optional) random variable L α , the family ( F L α + t ) t ≥ is a right-continuous filtration.Due to our setup in (2.1) and (2.2), for t ≥ L α , the pair of drift and volatility parameters of V t is ( µ ( x , ) , σ ( x , )) .According to Theorem 5.1 in Meyer et al. (1972), (cid:16) V L α + s B L α + s (cid:17) s > is a strong Markov process with respect to thefiltration ( F L α + s ) s > . For any x , y ∈ ( , α ) , its transition semigroup is given by P t ( x , d y ) = P x (cid:16) V t B t ∈ d y , L α = (cid:17) P x ( L α = ) . (2.7)Hence ( P t ) is a transition semigroup of the process VB conditioned not to hit α before lifetime. In particular, thistransition semigroup incorporates the fact that the diffusion never reaches level α . MASAHIKO EGAMI AND RUSUDAN KEVKHISHVILI
Since we are interested in the process VB only after the time L α , we can simplify some notations: for y ∈ ( , α ) ,we set µ ( y , ) = µ ( y ) and σ ( y , ) = σ ( y ) . Let us denote, by s ( y ) , the scale function of VB that is associated with µ ( y ) and σ ( y ) . By using the semigroup (2.7),for y ∈ ( , α ) , the infinitesimal drift parameter of the process (cid:0) YB (cid:1) t ≥ is computed as(2.8) µ ( y ) − r − s (cid:48) ( y ) s ( α ) − s ( y ) σ ( y ) , while the volatility parameter turns out to be the same as σ ( y ) .Given the fact that the firm-value approach does not provide appropriate information about the loss-given-defaultdistribution, it is natural to consider another mechanism that determines the loss distribution upon default. Thisapproach reflects the fact that company’s business conditions radically change after its asset value, that is, thepresent value of its future cash flow prospects, goes below a certain level. This time of drastic change in ourmodel is captured by L α after which we consider the process YB . Note that the parameters of the process YB maynot be easily estimated from the company’s stock prices since the deteriorated trade or business conditions of thecompany may not be sufficiently reflected in the stock market.2.2. Default time ξ and time τ . In this subsection, we define a random time that is suitable for the purpose ofestimating the loss given default distribution. Let us first introduce an exponential random variable J defined on ( Ω , G , P ) with rate η =
1, independent of the filtration F . If necessary, the probability space can be enlarged tosupport this random variable. We let λ : R + (cid:55)→ R + be a nonnegative piecewise continuous function and define themoment inverse of integral functional by(2.9) ν ( t ) : = min (cid:18) s : (cid:90) s λ (cid:18) Y u B u (cid:19) d u = t (cid:19) . Following Bielecki and Rutkowski (2002, Sec.8.2), we assume that λ satisfies(2.10) (cid:90) ∞ λ (cid:18) Y s B s (cid:19) d s = ∞ P − a . s .. If we recall that(2.11) (cid:18) YB (cid:19) t ≥ = (cid:18) VB (cid:19) t ≥ L α in distribution , the random variable in (2.9) has the same distribution as min (cid:16) s : (cid:82) L α + sL α λ (cid:16) V u B u (cid:17) d u = J (cid:17) . We define τ : = ν ( J ) andmodel the default time ξ as(2.12) ξ : = L α + τ . It follows from (2.9) that P ( τ = ) = P ( τ < + ∞ ) =
1. We also see that P ( τ > s ) > s ≥
0. Moreover, with (2.12), we have P ( L α < ξ < + ∞ ) =
1. In our model, the time L α is treated as a specialtime in the sense that the firm goes into a different economic state. This is captured in (2.12) where the new clock τ starts at time L α . We have constructed the time τ using the process that never returns to α , this feature beingrepresented in (2.7).Note that we shall write (2.11) as (cid:0) YB (cid:1) t ≥ d ∼ (cid:0) VB (cid:1) t ≥ L α hereafter. NEW APPROACH TO ESTIMATING LOSS-GIVEN-DEFAULT DISTRIBUTION 7
We explain the implication behind this modeling of the default time ξ . Associated with the random time ξ , wedefine the jump process H by H t : = { ξ ≤ t } for t ≥
0. Let H : = ( H t ) t ≥ be the filtration generated by the process H , that is, H t = σ ( H u : u ≤ t ) and set H = H ∞ . We assume that G = F ∨ H . Following (2.6), let us define G L α : = { A ∈ G : ∀ t ≥ , ∃ A t ∈ G t s.t. A ∩ { L α < t } = A t ∩ { L α < t }} , (2.13) H L α : = { A ∈ H : ∀ t ≥ , ∃ A t ∈ H t s.t. A ∩ { L α < t } = A t ∩ { L α < t }} . Recall that F L α is defined by replacing R with L α in (2.6). We obtain the respective filtrations for t ≥ G = ( ˆ G t ) = ( G L α + t ) , ˆ F = ( ˆ F t ) = ( F L α + t ) , and ˆ H = ( ˆ H t ) = ( H L α + t ) by replacing L α with L α + t in the definitions(2.6) and (2.13).Then, by the definition τ = ν ( J ) , we have P [ τ > t | ˆ F t ] = P (cid:20) min (cid:18) s : (cid:90) L α + sL α λ (cid:18) V u B u (cid:19) d u = J (cid:19) > t (cid:12)(cid:12)(cid:12)(cid:12) ˆ F t (cid:21) = P (cid:20) (cid:90) L α + tL α λ (cid:18) V u B u (cid:19) d u < J (cid:12)(cid:12)(cid:12)(cid:12) ˆ F t (cid:21) = P (cid:20) (cid:90) t λ (cid:18) Y u B u (cid:19) d u < J (cid:12)(cid:12)(cid:12)(cid:12) ˆ F t (cid:21) = e − (cid:82) t λ ( YuBu ) d u = e − Γ ( t ) , where Γ ( t ) : = (cid:82) t λ (cid:16) Y u B u (cid:17) d u is an ˆ F -adapted right-continuous increasing process and it represents the ˆ F -hazardprocess of τ under P . Recall that the infinitesimal drift parameter of the process ( YB ) t ≥ is given by (2.8) and thevariance parameter is σ ( · ) .Note that the construction of the random time τ is similar to the canonical construction of the default time in theintensity-based approach (see Proposition 5.26 in Capi´nski and Zastawniak (2017)). In particular, for any γ > E [ e − γ Γ ( t ) ] = E (cid:20) e − γ (cid:82) t λ ( YuBu ) d u (cid:12)(cid:12)(cid:12)(cid:12) Y B = α (cid:21) . Loss-given-default distribution.
In our model, the distribution of Y τ B τ | Y B = α provides the distribution of the true leverage ratio at default time. Let the loss rate for the overall debt B at default be denoted by K ( ξ ) . Then, inour model(2.14) K ( ξ ) d ∼ − Y τ B τ (cid:12)(cid:12)(cid:12)(cid:12) Y B = α . Recall that we have let B represent the sum of the short-term debt and one half of the long-term debt which is thecommonly used default barrier in the firm-value approach. Here we do not impose the restriction that K ( ξ ) > B .2.4. Connection between τ and T c − L α . Fix any y ∈ ( , α ) . According to (2.8), (2.9), (2.12) and TheoremIV.5.1 in Borodin (2017), U ( y ) : = E y (cid:104) Y τ B τ ≤ y (cid:105) , y ∈ ( , α ) is the unique continuous solution to the problem12 ( σ ( y )) U (cid:48)(cid:48) ( y ) + (cid:18) µ ( y ) − r − s (cid:48) ( y ) s ( α ) − s ( y ) σ ( y ) (cid:19) U (cid:48) ( y ) − λ ( y ) U ( y ) = − λ ( y ) y ≤ y , y ∈ ( , α ) . MASAHIKO EGAMI AND RUSUDAN KEVKHISHVILI
Here E y [ · ] denotes the conditional expectation E [ · | Y B = y ] . The solution to this ordinary differential equationwhen y approaches α , lim y ↑ α U ( y ) , provides the distribution of K ( ξ ) via (2.14). Note that for the process YB , α isan entrance boundary. Again, according to (2.8), (2.9), (2.12) and Theorem IV.5.1 in Borodin (2017), U γ ( y ) : = E y (cid:20) e − γ min ( s : (cid:82) s λ ( YuBu ) d u = J ) (cid:12)(cid:12)(cid:12)(cid:12) Y B = y (cid:21) , y ∈ ( , α ) is the unique continuous solution to the problem12 ( σ ( y )) U (cid:48)(cid:48) ( y ) + (cid:18) µ ( y ) − r − s (cid:48) ( y ) s ( α ) − s ( y ) σ ( y ) (cid:19) U (cid:48) ( y ) − ( λ ( y ) + γ ) U ( y ) = − λ ( y ) , y ∈ ( , α ) for γ ∈ R ++ . Note that by (2.9) the Laplace transform of τ is given by lim y ↑ α U γ ( y ) .We shall now ensure that our default time ξ = L α + τ is consistent with the time T c given in (2.5) since the latteris important in the empirical research. See the paragraph below (2.5). The Laplace transform of T c − L α does notdepend on the starting position of the process (cid:0) VB (cid:1) t ≥ which is denoted by the point x . If we know the form of theLaplace transform E [ e − γ ( T c − L α ) ] , a differentiation of the Laplace transforms provides us with the first and secondmoments of τ and T c − L α . We set(2.15) E [ τ ] = E [( T c − L α )] and E (cid:2) τ (cid:3) = E (cid:2) ( T c − L α ) (cid:3) . In summary, we have assumed that(1) the state space of the asset process V , defined in (2.1) and (2.2), is ( , + ∞ ) where the left boundary isattracting and the right boundary is non-attracting,(2) V B = x > α > c , and(3) (2.10), a technical condition for the intensity process λ ,and then introduced two types of times. One is T c , which is the first passage time of V to the boundary debt level B after time L α . The other is ξ = L α + τ , where τ is constructed using process YB and the intensity process λ through(2.9). Note that for this construction, we use the transition semigroup (2.7) that explicitly takes into account thefact that the diffusion ( YB ) t ≥ starting at α never returns to level α before converging to 0. However, these twotimes must be consistent to the extent that (2.15) holds. This is summarized in Table 1. Table 1
Comparison of our model with the conventional setting:
Default time Distribution of the final valueConventional T c = inf { t ≥ V t B t = c } V T c = B T c New approach ξ = L α + τ V ξ B ξ d ∼ Y τ B τ (cid:12)(cid:12)(cid:12)(cid:12) Y B = α
3. I
MPLEMENTATION
Let us assume that the firm value (market value of total assets) V follows a geometric Brownian motion on ( Ω , G , P ) . Let us set the drift and volatility parameters in (2.1)d V t = µ ( V t , I ( t )) d t + σ ( V t , I ( t )) d W t , NEW APPROACH TO ESTIMATING LOSS-GIVEN-DEFAULT DISTRIBUTION 9 as(3.1) µ v = µ ( v , ) (cid:54) = µ ( v , ) = µ v , and σ v = σ ( v , ) (cid:54) = σ ( v , ) = σ v , v ∈ ( , + ∞ ) with µ , µ ∈ R and σ , σ >
0. The debt process representing the default barrier in the firm-value approach is givenby B t : = B e rt , B ∈ F , t ≥ , where r > µ , µ , σ and σ are such that the left boundary 0 is attractingand the right boundary ∞ is non-attracting. In particular, we choose µ and σ so that µ − σ < V B = x > α > ( R t ) t ≥ given by R t = ln (cid:16) V t B t (cid:17) . We reset our previous definitions inregards to this process. By fixing a certain level of α , we reset the definition (2.4) as L α : = sup { t ≥ R t = ln ( α ) } . Similarly, (2.5) becomes T c : = inf { t ≥ L α : R t = } . On the set { t ≥ L α } , V t = V e ( µ − σ ) t + σ W t and hence(3.2) V t B t = V B e ( µ − σ − r ) t + σ W t and ln (cid:18) V t B t (cid:19) = ln (cid:18) V B (cid:19) + (cid:18) µ − σ − r (cid:19) t + σ W t . To avoid the triviality, we assume that µ − σ − r (cid:54) =
0. We have the normalized process1 σ d ln (cid:18) V t B t (cid:19) = µ − σ − r σ d t + d W t = m d t + d W t by setting(3.3) m : = µ − σ − r σ . Now by using (2.7) and (2.8), we see that the process X t : = σ ln (cid:16) Y t B t (cid:17) , t ≥
0, follows the dynamics(3.4) 1 σ d ln (cid:18) Y t B t (cid:19) = d X t = m coth ( m ( X t − α ∗ )) d t + d W t where α ∗ = σ ln ( α ) . The initial value of the process X is α ∗ , the property of which we shall describe here. Note that Y B = α . Notealso that since lim x ↑ coth ( x ) = − ∞ , when the process X approaches α ∗ from below, the drift approaches negativeinfinity so that the process shall never reach the point α ∗ from the region ( − ∞ , α ∗ ) . Mathematically, α ∗ is anentrance boundary of X and financially, the firm’s leverage ratio shall never recover to the threshold level α .3.1. Time τ and hazard rate λ . In this subsection, we construct the default time ξ . Select a certain level R d ∈ ( , α ) . Observe that(3.5) P (cid:18) Y t B t < R d (cid:12)(cid:12)(cid:12)(cid:12) Y B = α (cid:19) = P (cid:18) X t < σ ln ( R d ) (cid:12)(cid:12)(cid:12)(cid:12) X = α ∗ (cid:19) . To obtain a concrete result, we specify the function λ in (2.9) as λ (cid:18) Y t B t (cid:19) : = YtBt ≤ R d so that the integral in (2.9) represents the occupation time of the process ( YB ) t ≥ under the level R d . This definitionof λ is equivalent to the definition λ ( X t ) : = X t ≤ d with d : = σ ln ( R d ) . With this definition, (2.9) becomes, with τ = ν ( J ) ,(3.6) τ = min (cid:18) s : (cid:90) s X u ≤ d d u = J (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) X = α ∗ . Note that α ∗ > d >
0. Hence we identify the firm’s default once the leverage process (in its logarithmic form)spends a certain random time below level d . It is quite reasonable since, as in other intensity-based models, themarket observers, unaware of the firm’s state after a certain time (represented by L α in our model), accept defaultas a sudden shock.3.2. Laplace transform of τ . Let us indicate, by y ∈ ( − ∞ , α ∗ ) , an arbitrary constant initial value of the process X and write X α ∗ t = y + (cid:82) t m coth (cid:0) m (cid:0) X α ∗ u − α ∗ (cid:1)(cid:1) d u + W t . See (3.3) and (3.4). Proposition 3.1.
The Laplace transform of τ in (3.6) satisfies E [ e − γτ ] = lim y (cid:48) ↓ E (cid:20) exp (cid:18) − γ min (cid:18) s : (cid:90) s X u ≥ α ∗ − d d u = J (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X = y (cid:48) (cid:21) , where X t = y (cid:48) + (cid:82) t m coth (cid:0) mX u (cid:1) d u + W t with y (cid:48) = α ∗ − y.Proof. The Laplace transform of τ can be rewritten as follows: E [ e − γτ ] = lim y ↑ α ∗ E (cid:20) exp (cid:18) − γ min (cid:18) s : (cid:90) s X α ∗ u ≤ d d u = J (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X α ∗ = y (cid:21) = lim y (cid:48) ↓ E (cid:20) exp (cid:18) − γ min (cid:18) s : (cid:90) s X u ≥ α ∗ − d d u = J (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X = y (cid:48) (cid:21) . where y (cid:48) = α ∗ − y . This is because X α ∗ u ≤ d is equivalent to α ∗ − X α ∗ u ≥ α ∗ − d and for X t : = α ∗ − X α ∗ t , we haved X t = − d X α ∗ t = − m coth ( m ( X α ∗ t − α ∗ )) d t − d W t = m coth ( mX t ) d t + d W t where we used the fact that − W and W are both standard Brownian motions. (cid:3) We define(3.7) U γ ( y ) : = E (cid:20) exp (cid:18) − γ min (cid:18) s : (cid:90) s X u ≥ α ∗ − d d u = J (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) X = y (cid:21) for y ∈ ( , ∞ ) . Then, by Theorem IV.5.1 in Borodin (2017), U γ ( y ) is the unique continuous solution to the problem(3.8) 12 U (cid:48)(cid:48) ( y ) + m coth ( m · y ) U (cid:48) ( y ) − ( y ≥ α ∗ − d + γ ) U ( y ) = − y ≥ α ∗ − d , y ∈ ( , ∞ ) . Proposition 3.2.
The Laplace transform of τ is given by (3.9) E [ e − γτ ] = b ( b + coth ( | m | ( α ∗ − d ))) sinh ( | m | ( α ∗ − d ))( + γ ) ( b cosh ( | m | b ( α ∗ − d )) + b sinh ( | m | b ( α ∗ − d ))) , where b = (cid:113) + ( + γ ) m and b = (cid:113) + γ m . NEW APPROACH TO ESTIMATING LOSS-GIVEN-DEFAULT DISTRIBUTION 11
Proof.
Let us set b and b as in the statement of the proposition. For notational simplicity, we assume m > m instead of | m | .When y ≥ α ∗ − d , (3.8) becomes U (cid:48)(cid:48) ( y ) + m coth ( m · y ) U (cid:48) ( y ) − ( + γ ) U ( y ) = −
1. According to Section IV.16.7and Appendix 2.12 in Borodin (2017). Its fundamental solutions are given by ψ ( y ) = √ ( mb y ) √ π b sinh ( m · y ) , φ ( y ) = √ π e − mb y √ ( m · y ) . Here ψ is an increasing solution and it satisfies lim y ↓ ψ ( y ) = (cid:113) π and lim y ↑ ∞ ψ ( y ) = ∞ because b >
1. Onthe other hand, φ is a decreasing solution and it satisfies lim y ↓ φ ( y ) = ∞ and lim y ↑ ∞ φ ( y ) =
0. Therefore, for y ≥ α ∗ − d we have U ( y ) = + γ + C φ ( y ) with C constant. We have eliminated the second fundamental solutionbecause the Laplace transform needs to be bounded when y → ∞ .When y < α ∗ − d , (3.8) becomes U (cid:48)(cid:48) ( y ) + m coth ( m · y ) U (cid:48) ( y ) − γ U ( y ) =
0. Its fundamental solutions are givenby ψ ( y ) = √ ( mb y ) √ π b sinh ( m · y ) , φ ( y ) = √ π e − mb y √ ( m · y ) . The limiting behavior of ψ and φ is the same as that of ψ and φ . Therefore, for y < α ∗ − d we have U ( y ) = C ψ ( y ) with C constant. Again, we have eliminated the second fundamental solution because the Laplacetransform has to be bounded when y →
0. Therefore, we obtain(3.10) U γ ( y ) = y ≥ α ∗ − d (cid:18) + γ + C φ ( y ) (cid:19) + y < α ∗ − d C ψ ( y ) . From the continuity of U γ ( y ) and U (cid:48) γ ( y ) at α ∗ − d , we obtain C = − √ π e mb ( α ∗ − d ) sinh ( m ( α ∗ − d )) ( b cosh ( mb ( α ∗ − d )) − coth ( m ( α ∗ − d )) sinh ( mb ( α ∗ − d ))) π ( + γ ) ( b cosh ( mb ( α ∗ − d )) + b sinh ( mb ( α ∗ − d ))) , C = b √ π ( b + coth ( m ( α ∗ − d ))) sinh ( m ( α ∗ − d )) ( + γ ) ( b cosh ( mb ( α ∗ − d )) + b sinh ( mb ( α ∗ − d ))) . Finally, we let y ↓ y ↓ C ψ ( y ) = C (cid:114) π = b ( b + coth ( m ( α ∗ − d ))) sinh ( m ( α ∗ − d ))( + γ ) ( b cosh ( mb ( α ∗ − d )) + b sinh ( mb ( α ∗ − d ))) . This proves the proposition in view of (3.7) and Proposition 3.1.Note that, recalling that J is an exponential random variable with rate 1, lim y ↑ ∞ U γ ( y ) = + γ = E [ e − γ J ] asrequired because min (cid:16) s : (cid:82) s X u ≥ α ∗ − d d u = J (cid:17) | X = y = J a.s. for sufficiently large y . (cid:3) Laplace transform of T c − L α . The following equations hold for the process ( R (cid:48) t ) t ≥ given by R (cid:48) t : = σ R t = σ ln (cid:16) V t B t (cid:17) : L α = sup { t ≥ R (cid:48) t = α ∗ } and T c = inf { t ≥ L α : R (cid:48) t = } in view of (2.4) and (2.5), respectively. The process R (cid:48) t = σ ln (cid:16) V B (cid:17) + mt + W t is a Brownian motion with driftstarting at σ ln ( x ) . Using (2.7) and (3.4), the process ( Z t ) t > = ( R (cid:48) t ) t > L α has the infinitesimal drift parameter m coth ( m ( z − α ∗ )) with z ∈ ( − ∞ , α ∗ ) and variance parameter 1. The distribution of T c − L α is the same as the distribution of T = inf { t ≥ Z t = } for the process Z starting at α ∗ . Note that α ∗ is an entrance boundary forthe process Z . Consider the process Z (cid:48) t = α ∗ − Z t . As in the proof of Proposition 3.1, we obtaind Z (cid:48) t = − d Z t = m coth ( mZ (cid:48) t ) d t + d W t . Set T (cid:48) = inf { t ≥ Z (cid:48) t = α ∗ } . Note that lim z ↑ α ∗ P ( T ∈ d t | Z = z ) = lim z (cid:48) ↓ P ( T (cid:48) ∈ d t | Z (cid:48) = z (cid:48) ) . The left-hand-sidecan be evaluated using the result in Proposition 3.9 in Egami and Kevkhishvili (2020). Thus, we obtain(3.11) E [ e − γ ( T c − L α ) ] = (cid:112) γ + m m sinh ( m α ∗ ) sinh (cid:16)(cid:112) γ + m α ∗ (cid:17) . This quantity and (3.9) are needed to match the moments as in (2.15).3.4.
Distribution of V ξ B ξ . As we have summarized in Table 1, V ξ B ξ d ∼ Y τ B τ (cid:12)(cid:12)(cid:12)(cid:12) Y B = α under P , where ξ in (2.12) is thedefault time. For an arbitrary level R < R d < α , we have, in view of (3.5), P (cid:18) V ξ B ξ ≤ R (cid:19) = P (cid:18) σ ln (cid:18) V ξ B ξ (cid:19) ≤ σ ln ( R ) (cid:19) = P (cid:18) σ ln (cid:18) Y τ B τ (cid:19) ≤ σ ln ( R ) (cid:12)(cid:12)(cid:12)(cid:12) Y B = α (cid:19) = lim y ↑ α ∗ P (cid:18) X α ∗ τ ≤ σ ln ( R ) (cid:12)(cid:12)(cid:12)(cid:12) X α ∗ = y (cid:19) = lim y (cid:48) ↓ P (cid:18) X τ ≥ α ∗ − σ ln ( R ) (cid:12)(cid:12)(cid:12)(cid:12) X = y (cid:48) (cid:19) , (3.12)with y (cid:48) = α ∗ − y where the last equation is due to the same argument as in the proof of Proposition 3.1. Note that P (cid:16) V ξ B ξ ≤ R (cid:17) = R ≥ R d . Therefore, we derive the distribution of X τ to prove the following proposition: Proposition 3.3.
The distribution of (cid:16) V ξ B ξ (cid:17) is given by P (cid:18) V ξ B ξ ≤ R (cid:19) = csch ( | m |· ( α ∗ − d ))( cosh ( | m | y )+ b sinh ( | m | y )) b + coth ( | m | ( α ∗ − d )) (cid:16) RR d (cid:17) b | m | σ , R ≤ R d , R > R d . where y = α ∗ − σ ln ( R ) and b = (cid:113) + m .Proof. Set b and y as in the statement of the proposition. For notational simplicity, we assume m > m instead of | m | . We deduce from Section 2.4 that for y > E y [ X τ ≥ y ] is the unique continuous solution to(3.13) 12 U (cid:48)(cid:48) ( y ) + m coth ( m · y ) U (cid:48) ( y ) − y ≥ α ∗ − d U ( y ) = − y ≥ α ∗ − d y ≥ y , y > . Recall that d = σ ln ( R d ) .Let y ≥ α ∗ − d . When (i) y ≥ α ∗ − d , (3.13) is further transformed into U (cid:48)(cid:48) ( y ) + m coth ( m · y ) U (cid:48) ( y ) − U ( y ) = − y ≥ y . For y ≥ y , we have U (cid:48)(cid:48) ( y ) + m coth ( m · y ) U (cid:48) ( y ) − U ( y ) = − U ( y ) = + C (cid:114) π e − mby sinh ( m · y ) with C constant. Here we have eliminated the second fundamental solution because the solution has to be boundedwhen y → ∞ . For y > y ≥ α ∗ − d , we have U (cid:48)(cid:48) ( y ) + m coth ( m · y ) U (cid:48) ( y ) − U ( y ) = U ( y ) = D (cid:113) π sinh ( mby ) b sinh ( m · y ) + D (cid:112) π e − mby sinh ( m · y ) with D , D constants. NEW APPROACH TO ESTIMATING LOSS-GIVEN-DEFAULT DISTRIBUTION 13
When (ii) y < α ∗ − d , we have U (cid:48)(cid:48) ( y ) + m coth ( m · y ) U (cid:48) ( y ) = U ( y ) = E with E constant. We have eliminated the second solution coth ( m · y ) because lim y ↓ coth ( m · y ) = ∞ if m > y ↓ coth ( m · y ) = − ∞ if m < y ≥ α ∗ − d >
0, we combine the above cases to obtain E y [ X τ ≥ y ] = (3.14) 1l y ≥ y (cid:18) + C (cid:114) π e − mby sinh ( m · y ) (cid:19) + y > y ≥ α ∗ − d (cid:32) D (cid:114) π sinh ( mby ) b sinh ( m · y ) + D (cid:114) π e − mby sinh ( m · y ) (cid:33) + y < α ∗ − d E . Due to the continuity of the solution and its first derivative at y and α ∗ − d , we obtain E = e bm ( α ∗ − d − y ) csch ( m · ( α ∗ − d ))( cosh ( my ) + b sinh ( my )) b + coth ( m ( α ∗ − d )) . Note that α ∗ − d − y = σ ln ( RR d ) , so that y ≥ α ∗ − d corresponds to R ≤ R d . In view of (3.12), by letting y ↓ R ≤ R d . Finally, let y < α ∗ − d . From the definition of ν in (2.9), E y [ X τ ≥ y ] = (cid:3) Estimation procedure.
Fix any point in time t = t .Step 1: We need to check the value of V B and fix an appropriate level 1 < α ≤ V B for the analysis. To obtain thecurrent value of V , we use Merton’s geometric Brownian motion asset model which is different from ourspecification (2.1) and (2.2). We estimate the diffusion parameter (we denote it by q ) of Merton’s modelby implementing the option-theoretic approach as in Lehar (2005). We use, for the estimation, the marketcapitalization of previous 130 days as well as interpolated daily debt values (the sum of short-term and onehalf of long-term debt) from the quarterly balance sheet. Using the diffusion parameter q , current equityand debt value B , we retrieve the present asset value V from the Black-Scholes equation (see eq.(2) inLehar (2005)). If the current level of V B is less than α , there is a possibility that L α has already occurredand the process will never return to α . To avoid this possibility, we choose α so that α ≤ V B .We check different levels of α and choose α in a way that the CDS spread implied by our model isconsistent with the market quoted CDS spread provided by Thomson Reuters. The market spread is saidto be calculated with the assumption of 40% recovery rate. In contrast, the value of the CDS spreadobtained from our model depends on the distributions of recovery rate and of default time. We calculatethe implied recovery rate IR (given that default would occur) using the result of Proposition 3.3:(3.15) IR : = (cid:90) R R · P (cid:18) V ξ B ξ ∈ d R (cid:12)(cid:12)(cid:12)(cid:12) V ξ B ξ ≤ (cid:19) . We calculate two types of implied spreads:(i) The first type is calculated using the recovery rate IR and default time distribution obtained from ourmodel.(ii) The second type is calculated using 40% recovery rate and default time distribution obtained from ourmodel.As mentioned above, the initial state of the company is represented by V B which greatly depends on theparameter q . As Step 2 describes, the estimated value of q is the lower bound for σ . Therefore, the currentcompany condition affects the implied recovery rate IR . We choose the level of α for which the second type of the implied spread is consistent with the quotedspread. It is because the quoted spread is based on the 40% recovery assumption. We check the consistencyof implied and quoted CDS spreads for each set of estimated parameters (Step 4) for different levels of α .For the steps below, we assume that α is fixed.Step 2: In this step, we estimate the parameters µ and σ in (3.1). Note that these parameters µ and σ are associatedwith the leverage process below level α . We use 5Y U.S. Treasury yield at current time 0 as the constantrate r . By setting R d = + β | α − | with 0 < β < , we estimate the parameters µ , σ , β by minimizing the following function f : f ( µ , σ , β ) = θ (cid:32) ∂ E [ e − γ ( T c − L α ) ] ∂ γ (cid:12)(cid:12)(cid:12)(cid:12) γ = − ∂ E [ e − γτ ] ∂ γ (cid:12)(cid:12)(cid:12)(cid:12) γ = (cid:33) + ( − θ ) (cid:32) ∂ E [ e − γ ( T c − L α ) ] ∂ γ (cid:12)(cid:12)(cid:12)(cid:12) γ = − ∂ E [ e − γτ ] ∂ γ (cid:12)(cid:12)(cid:12)(cid:12) γ = (cid:33) . (3.16) See (3.9) and (3.11). Here θ ∈ ( , ) denotes the relative weight of the two terms and must be specifiedbefore the minimization. The minimization of f corresponds to finding the parameters µ , σ , β such that(2.15) is satisfied approximately. For minimization of f , we set the following constraint: µ − σ < , σ ≥ ˆ q , where ˆ q denotes the estimated diffusion parameter in Merton’s model (see Step 1). Note that the firstcondition ensures that the left boundary 0 is attracting and the right boundary + ∞ is non-attracting andthe second condition comes from the fact that the company in financial distress has a larger volatility thanunder normal business conditions.Step 3: As we mentioned in the paragraph following (2.5), we use T c in (2.5) for this and the next step since thecredit market data used here are associated with the standard firm-value approach. We estimate the param-eter µ in a way that the 5-year default probability is consistent with the 5-year default probability providedby Thomson Reuters. Specifically, we use iteration. We simulate asset paths using the estimated param-eters in Step 2 together with some fixed level of µ to obtain T c and calculate 5-year default probability.Note that we simulate the path of the leverage process for 20 years ahead to obtain L α . For computationalsimplicity, we assume that σ = σ . As µ decreases, the associated 5-year default probability increases.Using iterative procedure by changing the level of µ , we choose an appropriate level of µ that provides5-year default probability consistent with the default probability from Thomson Reuters.Step 4: We check the validity of α and the estimated parameters by calculating 5-year CDS spread based on ourmodel and comparing it to the quoted CDS spread in the market. In general, the underlying credit riskbehind CDS refers to the risk of the whole company, not a particular debt obligation. We calculate theCDS spread in the following way. We assume the spread payments are made quarterly and simulate T c .We set the principal amount to $1. Using the constant rate r as the discount factor, we calculate the presentvalue of spread payments until T c or maturity, whichever is earlier and the present value of the paymentof $ ( − . ) (the recovery rate is assumed to be 40%) at the default time if it happens until maturity. Weestimate the CDS spread so that these two present values are equal. NEW APPROACH TO ESTIMATING LOSS-GIVEN-DEFAULT DISTRIBUTION 15
Step 5: Using the estimated parameters above, we find the distribution of V ξ B ξ according to Proposition 3.3. Thisprovides the distribution of the loss given default as in (2.14). We also obtain the Laplace transform of thetime τ from Proposition 3.2.Step 6: For further illustration, we also display the distribution of pre-default value of a corporate zero-couponbond. Let D ( s , T ) denote the price of a zero-coupon bond with maturity T at each point in time s for0 ≤ s ≤ T . We find the values of D ( s , T ) using zero-yield data of the company at current time t = D ( s , T ) with maturity T = T c and pick the values D ( T c − , T ) . This gives the distribution of the value of the zero-coupon bond right before default.4. E XAMPLE OF C AMPBELL S OUP C OMPANY
Let us estimate the loss-given-default distribution for Campbell Soup, hereafter denoted shortly as Campbell.For this, we choose February 28, 2020 as the current time and use the information available up to this time. Weuse the 5Y U.S. Treasury yield curve rate 0 .
89% (of February 28, 2020) as the constant rate r . The quoted 5-yearCDS spread for Campbell is 60 . V using the method described in Step 1 in Section 3.5. For the estimation, wedefined the debt as the sum of the following items: “Capitalized Leases - Current Portion”, “Short-term borrow-ings”, “Short-term borrowings - Balancing value”, “Operating Lease Liabilities - Curr/ST”, and one half of thelong-term debt. We used the sum of “Capital Lease Obligations”, “Long-term debt - Balancing value”, “TotalLong Term Debt & Lease - Balancing”, “Long-term debt”, and “Operating Lease Liabilities - Long-Term” as thelong-term debt. We obtained the diffusion parameter ˆ q = . . q , currentequity and debt values, we retrieve the present asset value V from the Black-Scholes equation as described in Step1. We obtain V B = . α , starting at 1 / . ≈ . < V B and increasing the denominator by 0 . θ = . f in (3.16), obtaining theparameters µ , σ and β as in Table 2. We selected the parameter vector that provides the lowest value of f as longas it has reasonably small standard errors. The estimated values of µ reported in Table 2 were calculated by theiteration as explained in Step 3 in Section 3.5. We conducted simulation 10,000 times for Steps 3, 4, and 6.Figure 1 displays the Laplace transform of the time τ provided in Proposition 3.2. Figures 2 and 3 displaythe cumulative distribution function P (cid:16) V ξ B ξ ≤ R (cid:17) and the conditional density function P (cid:18) V ξ B ξ ∈ ( R − ∆ R , R ] , R ≤ (cid:19) P (cid:18) V ξ B ξ ≤ (cid:19) with ∆ R = .
001 for different levels of R . Figures 2 and 3 are produced using Proposition 3.3. Recall that B representsthe sum of short-term debt and one half of the long-term debt and note that there is a possibility for V ξ B ξ >
1. Finally,Figure 4 shows the distribution of the zero-coupon bond with $1 face value and the maturity of 5 years (issued atcurrent time) right before the time T c . The calculation for Figure 4 is carried out according to Step 6 in Section 3.5.According to Table 2, the model-implied spreads (both Type I and Type II) do not change monotonically inrelation to the level α since the spreads are also influenced by other parameters µ , µ , σ , and β . As compared tothe quoted CDS spread of 60.2 bps on that day, our model provides for a higher spread (Type II) of around 80bps for each α . While further calibrations of α could produce spreads closer to 60.2 bps, the difference of thesenumbers is not significant based on the fact that we are working with a long-term credit spreads. More importantly,the arbitrarily fixed recovery rate of 40% (used in the convention) may distort appropriate formation of credit Table 2
Estimated parameters, implied spreads, and expected recovery rate IR together with quantilesof V ξ B ξ distribution for each α . Standard errors are given in parentheses. 5Y DP denotes 5-year default prob-ability obtained from our model. 5-year default probability obtained from Thomson Reuters is 7 . α µ -0.0176 (0.0696) 0.0027 (0.0620) -0.0517 (0.0597) 0.0190 (0.0074) 0.0197 (0.0014) 0.0165 (0.0017) 0.0197 (0.0096)ˆ σ β f µ -0.9820 -0.0800 -0.1600 -0.3220 -0.2800 -0.2610 -0.2700 spread. In fact, recovery rates should differ from company to company and this predetermined 40% recovery rateis an eyesore in rather efficient CDS markets. In the calibration we use the information available both in the stockand credit markets and the estimated recovery rate turns out to be higher: IR in (3.15) is around 80%. Accordingly,for each α , the implied spread of Type I (which uses our recovery distribution) indicates much lower credit spreads.We see from Table 2 that for α ranging from 1 / . / .
9, ˆ β is very close to c = σ is close to its lowerbound ˆ q (note that the parameters are displayed up to 4 decimal points). The occupation time of YB in such a smallinterval (with a reasonably large σ ) is not meaningful; therefore, α ranging from 1 / . / . α in the range of 1 / . ∼ / . NEW APPROACH TO ESTIMATING LOSS-GIVEN-DEFAULT DISTRIBUTION 17 (a) α = / . α = / . α = / . α = / . α = / . α = / . α = / . Fig. 1.
The Laplace transform of τ for each level of α . The graphs were produced basedon the result of Proposition 3.2 and the estimated parameters in Table 2. (a) α = / . α = / . α = / . α = / . α = / . α = / . α = / . Fig. 2.
The cumulative distribution function of V ξ B ξ for each level of α . The graphs dis-play P ( V ξ B ξ ≤ R ) for different levels of R based on the result of Proposition 3.3 and theparameters in Table 2. Note that P ( V ξ B ξ ≤ R d ) = NEW APPROACH TO ESTIMATING LOSS-GIVEN-DEFAULT DISTRIBUTION 19 (a) α = / . α = / . α = / . α = / . α = / . α = / . α = / . Fig. 3.
The conditional probability density function P (cid:16) V ξ B ξ ∈ d R (cid:12)(cid:12)(cid:12) V ξ B ξ ≤ (cid:17) of V ξ B ξ for differ-ent levels of α . This density is used for the calculation of the implied recovery rate IR in(3.15). The graphs display P (cid:18) V ξ B ξ ∈ ( R − ∆ R , R ] , R ≤ (cid:19) P (cid:18) V ξ B ξ ≤ (cid:19) with ∆ R = .
001 for different levels of R ,based on the result of Proposition 3.3 and the parameters in Table 2. (a) α = / . α = / . α = / . α = / . α = / . α = / . α = / . Fig. 4.
Distribution of the value of the zero-coupon bond D with 5-year maturity right be-fore default for each level of α . The left panels display the values of D ( s , ) for 0 ≤ s ≤ D ( T c − , ) using the parameters in Table 2. NEW APPROACH TO ESTIMATING LOSS-GIVEN-DEFAULT DISTRIBUTION 21
MASAHIKO EGAMI:
Graduate School of Economics, Kyoto University, Sakyo-Ku, Kyoto, 606-8501, Japan.Email: [email protected]
RUSUDAN KEVKHISHVILI:
Graduate School of Economics, Kyoto University, Sakyo-Ku, Kyoto, 606-8501,Japan. Email: [email protected]
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