Trading Noise and Default Risk

Transcription

1 Trading Noise and Default Risk Qiqi Zou (This draft: Nov 2013) Abstract This paper provides empirical evidence of the impact of trading noise on default risk estimation. Using a large sample of 11,166 US stocks from 1990 to 2012, it is found that adjusting for trading noise has material impact on firms distance-to-default (DT D) estimation, in terms of both magnitude and relative ranking among firms. More specifically, ignoring trading noise will lead to downward bias in DT D estimation and upward bias in default risk estimation; such bias attenuates during economic downturns and among risky firms measured by idiosyncratic volatility and leverage. When calibrating DT D in a reduced-form model along with other covariates, the adjustment of trading noise is shown to be statistically significant for default prediction, but not for prediction of other types of exit. Keywords: trading noise, market microstructure, distance-to-default, probability of default, forward intensity. JEL Classification: G33, C22, C23 Qiqi Zou is with Department of Finance, National University of Singapore. qiqizou@nus.edu.sg. I am grateful to my supervisor, Professor Jin-Chuan Duan for his constant support, invaluable guidance, and continuous encouragement. I appreciate the insightful comments and advice from Robert Kimmel and David Hirshleifer. I also would like to thank Risk Management Institute, National University of Singapore, for providing a comprehensive dataset and computing infrastructure. The support from High Performance Computing (HPC) provided by NUS computer center is also greatly appreciated. 1

2 1 Introduction Accurate assessment of default risk is important to individual lenders, financial institutions and regulators, especially following the Asian financial crisis in and the global financial crisis in Over the decades of studying default measures, the literature has realized that the most effective default measures derive from models that utilize not only financial statements, but also market prices. According to Crosbie and Bohn (2003, p. 22), we believe that the best source of information regarding the value of a firm is the market. Indeed, variables constructed from market data such as stock return, market-to-book ratio, idiosyncratic volatility of stock returns, have been shown to be useful in corporate default prediction, see Shumway (2001) and Campbell at al (2008). Market prices, though, can be contaminated by trading noises. The classical models of default risk often abstract away from market microstructure effects and implicitly assume that the observed transaction prices reflect fair value. While the actual transactions are indeed the best source of information of a security s price, a more realistic assumption would capture the possibility that prices also move in response to the process of trading due to various market frictions. However, it is not clear in the literature whether adjusting this assumption would have a material impact on the estimation of default risk. This paper attempts to answer this question by investigating: (a) how different is the estimated default measure once we account for trading noise? (b) In what scenarios are the differences more severe? (c) Can the new measure bring material benefits to corporate default prediction? Our analysis proceeds in two steps. First, we study the credit risk measure, the distance-to-default (DT D), motivated by the structural approach pioneered by the work of Merton (1974) that applies Black and Scholes (1973) option pricing formula to price market value of debt. The underlying assumption is that debt and equity are viewed as contingent claims on firm value, and debt has a claim priority over equity. DT D measures the number of standard deviations by which the market value of assets exceed a measure of liabilities, which is commonly referred to as the default point. It is a variable that is better than simple financial leverage ratio in assessing default risk, because it incorporates a firm s business and industry risk as measured by volatility of asset growth. DT D has been intelligently applied by academic researchers as well as industry practitioners. It is widely agreed among market practitioners that the distance-to-default is a useful measure for assessing the credit risk of a non-financial corporation. The prominence of DT D is partly due to its successful commercial implementation by Moody s KMV, which is said to revive the practical applicability of structural models by implementing a modified structural model called the Vasicek-Kealhofer (VK) 2

3 model. The advantage of structural models is that the capital structure of a firm is modeled in a consistent framework, but strong assumptions on the dynamics of the firm s asset and other variables are also required. This leads to our second step where reduced-form approach is applied. Contrary to structural approaches that explicitly model the ability of the firm to pay its debt, reduced-form approaches model default probabilities through an econometric specification with many explanatory variables. While DT D is a key covariate for default prediction, additional covariates are also included to capture the source of risk that are not revealed by DT D. In the first step, we estimate DT D for each firm-month in two scenarios, namely ignoring trading noise and explicitly adjusting for trading noise. The main interest lies in the difference between the two measures, in terms of absolute difference and rank correlation. In the second step, we would like to know whether the adjustment of trading noise is significant after DT D is further calibrated in a reduced-form model. Moreover, beyond the indirect impact, i.e. through DT D, of trading noise adjustment on probability of default, we are also interested in the direct impact when the trading noise is entered as a new covariate. We find empirical evidence that ignoring trading noise will lead to downward bias in DT D estimation and upward bias in default risk estimation. Intuitively, DT D is a volatility adjusted leverage measure, and is inversely related with both asset volatility and firm s leverage. By removing the noisy part of observed equity value, the estimated asset volatility is lower and the new DT D is generally higher. It is found that the difference is larger in good economic condition than in bad one. This is because during economic downturns, firms are more likely to default and the upward bias in default risk estimation attenuates. As evidenced by Hilscher and Wilson (2013), the failure rate during recessions is almost three times higher during and immediately after recessions (2.4%) than in normal period (0.9%). Thus, the bias of over-estimation of default risk can be less severe in economic downturns, and we expect to see smaller impact of adjusting for trading noise during such periods. Similarly, for risky firms with high volatility and high leverage, we expect to see smaller differences in the two DT D measures, since the tendency of over-estimating risks is lower. On the flip side, the largest differences in DT D are expected to be seen in safest firms with low volatility and low leverage, and during good economic condition. Further, we provide additional evidence that the change in DT D estimation is not a parallel shift. It is found that the rank correlation between the two measures is less than one, and is lower in nonrecession period than in recession period. Linking this with the finding that the absolute change in DT D estimation is larger in non-recession period, we conclude the impact of trading noise adjustment is material in normal and good economic condition. Interestingly, same pattern is not observed 3

4 when we sort the firms by riskiness. Safest firms with low volatility and low leverage, which have the largest change of DT D in magnitude, do not have the lowest rank correlation. Rather, we observe a smile pattern of rank correlation by firms riskiness: the correlation is highest for safest and riskiest firms, and becomes lower as riskiness, measured by leverage and volatility, moves away from two end points. This indicates that ignoring trading noise is less harmful in terms of relative ranks for safest and riskiest firms. Rather, firms with risks in between will see greater rank distortion if we ignore the presence of trading noise in observed prices. When we calibrate DT D in a reduced-form model along with other covariates, the adjustment of trading noise is shown to be statistically significant for default prediction. Moreover, both the indirect impact of trading noise adjustment on default probability through the change in DT D estimation, and the direct impact of trading noise as a new covariate, are statistically significant in short run default prediction, but not in prediction of other types of exit, such as M&A. Such results confirm our conclusion that it is important to adjust for trading noise when making corporate default prediction. The significance of direct impact of trading noise adjustment can be due to hidden unobserved covariates that also affect default, such as the precision of market-based variables like market-to-book ratio and idiosyncratic volatility. Taken together, our results indicate that adjusting for trading noise can bring material benefits to DT D estimation, and thus corporate default prediction. This paper adds to the big literature of corporate default prediction. The Merton s (1974) model that is applied in this paper assumes a simple capital structure for the firm: debt plus equity, with homogenous class of debt that has zero coupon form. Default can only happen at the maturity of debt. Black and Cox (1976), the first of the so-called First Passage Models (FPM), relax Merton s assumption on default time and postulate that default occurs at the first time that the firm s asset value drops below a certain time dependent barrier. These two approaches in the structural frameworks lie the foundation for subsequent studies, for example, Geske (1977), Longstaff and Schwartz (1995), Leland and Toft (1996), Collin-Dufresne and Goldstein (2001). Reduced-form models of default originated from discriminant analysis which derives an ordinal ranking of credit scores. Altman s Z-Score (Altman 1968) is one of the successful models. Later development of reduced-form models include the use of response models to predict bankruptcy, see Ohlson (1980), Zmijewski (1984), and more recently Campbell et al. (2008). Intensity based approach has also been explored in recent years, where default is modeled as some unpredictable Poisson-like event. Duffie, Saita and Wang (2007) propose a doubly stochastic Poisson intensity model to describe default event, and the time dynamics of the state variables need to be specified. Duan, Sun and Wang (2012) propose a forward intensity approach, and restrictions on the high-dimensional state 4

5 variable process are relaxed. The forward intensity approach specifies an explicit dependence of instantaneous intensities that are only known in future, as a function of state variables at the time of prediction. The approach is practically tractable due to the decomposition of the pseudo-likelihood function, and it has been implemented by NUS Risk Management Institute (RMI), who releases daily updated default probability forecasts with various horizons for around 60,400 firms globally. All of these studies take as given the premise that observed equity value reflect the best guess of true equity value and do not factor in the fact that the market is contaminated by various market microstructure noises. However, the literature on market microstructure has long realized that microstructure noises can cause observed equity values being different from their equilibrium values. Such deviation affects not only the first moment estimation, but also variances and covariances that are crucial in statistical inferences of finance studies. The literature on microstructure noises are motivated from the fact of price discreteness observed in the market, i.e. exchange regulations require all prices to be expressed as a multiple of some minimum tick, historically $ 1 8. Gottlieb and Kalay (1985) first incorporate the assumption that observed prices are obtained by rounding underlying values to the nearest eighth, and show that discreteness increases observed price change variance relative to the underlying value innovation variance. Harris (1990) generalizes Gottlieb and Kalay s model to take into account the bid-ask spread, and shows that the biases in the standard variance and in the serial covariance estimators depend on the underlying value innovation variance and on the bid-ask spread. Later, various market imperfections and frictions are explicitly modeled into price formation. Hasbrouck (1993) decomposes actual transaction price into implicit unobservable efficient price and pricing error. The pricing error is viewed as impounding diverse microstructure effects such as discreteness, inventory control, the non-information-based component of the bid-ask spread, the transient component of the price response to a block trade, etc. He proposes that the standard deviation of the pricing error be the measure of market quality. The measure of trading noise studied in this paper shares similar spirit to Hasbrouck s idea, in that both capture how closely the transaction price tracks the efficient price. Huang and Stoll (1994) and Madhavan, et al. (1997) also develop structural models of price formation that that captures many market frictions, including intraday bid-ask spread, execution cost, autocorrelation and volatility patterns of transaction prices and quotes. Presence of trading noises is also shown to have material effect on sampling frequency, see Aït-Sahalia, et al. (2005a,b) and volatility estimation of stock prices, see Zhang (2011) and Bandi and Russell (2008). Parametric likelihood-based identification in the presence of noise is studied by Gloter and Jacod (2001a,b). In light of the nontrivial effect of trading noises, recent literature has started to incorporate trading 5

6 noises to credit risk models. Duan and Fulop (2009) develop a particle filter-based MLE method to estimate the structural parameters of Merton (1974) s model in which the presence of trading noises is considered. Using 100 random sample of firms, they show that ignoring trading noise can lead to significant over-estimation of asset volatility. Huang and Yu (2010) adopt a Bayesian approach and uses Markov Chain Monte Carlo (MCMC) technique to estimate the structural credit risk model with microstructure noises, whose distribution is more flexible than iid normal. The thesis by Kwon (2012) assumes that noise follows a mean reverting process to capture short-term autocorrelation in stock prices. Her paper is based on the Black-Box model and applies particle filter algorithm for sequential estimation of asset value and the generalized Gibbs sampling method for model parameters estimation. This paper contributes to this line of literature by empirically apply the MLE method of Duan and Fulop (2009) to the estimation of DT D in Merton s (1974) model. The reduced form model used to calculate default probabilities follows the forward intensity approach of Duan, Sun and Wang (2012), where DT D is one of the key covariates. The rest of the paper is organized as follows. Section 2 reviews the particle filter-based MLE method of structural risk model of Duan and Fulop (2009), introduces the methodology of DT D computation and comparison, and briefly describes the three models studied under forward intensity approach of Duan, Sun and Wang (2012). Section 3 details the sources and variables of our data, and the construction of model input variables. Section 4 presents and discusses empirical results. The impacts of trading noise on DT D and default probabilities are analyzed. Section 5 concludes. 2 Methodology 2.1 Estimating trading noise The smoothed localized sequential importance re-sampling (SL-SIR) particle filter-based MLE method proposed by Duan and Fulop (2009) estimates the structural parameters of Merton s (1974) model in which the presence of trading noises is considered. Under the framework of Merton s model, debt and equity are viewed as contingent claims on firm value, so that option pricing techniques are used in valuing corporate liabilities. While the Merton s model has been under criticisms of being unrealistic, there is a trade-off between ease of implementation and realistic assumptions, especially when microstructure noise is added into. This paper abstracts away from the drawbacks of Merton s highly stylized models, and focuses on the implications of estimated trading noise parameter. 6

7 The state space representation in Duan and Fulop (2009) is as follows: First, the measurement equation links the observed equity value with the true value, with a multiplicative error structure for trading noise, so that the logarithm of observed equity value is lns τi = lns(v τi ;σ,f,r,t τ i ) + δν i (1) where S(V τi ;σ,f,r,t τ i ) is the true equity value from Black-Scholes option pricing model, S(V t ;σ,f,r,t t) = V t Φ(d t ) Fe r(t t) Φ ( d t σ T t ) (2) ( d t = ln(v t F )+ r+ σ2 2 σ T t ) (T t) V τi is asset value at time τ i, {ν i,i = 1,...,N} are i.i.d. standard normal r.v., and δ is the trading noise parameter. F is the face value of zero coupon debt maturing at T, r is the risk-free rate. Each stock is sampled at fixed frequency at time τ i, where i = 0,...,N. The transition equation specifies the process of underlying latent asset value. lnv τi+1 = lnv τi + (µ σ 2 ) h + σ hε i+1 (3) 2 where{ε i,i = 1,...,N} are i.i.d. standard normal r.v. Note that equation 3 is the discrete-form of a geometric Brownian motion that governs asset value V, with drift and volatility parameter, µ and σ, respectively. dv t V t = µdt + σdw t (4) The state space representation is non-linear and can be estimated using particle filter algorithms. Particle methods are a subset of the class of methods known as the Sequential Monte Carlo (SMC) methods, which use simulation techniques to provide particles approximately distributed according to posterior distributions given observable variables. Quantities of interest like moments and quantiles can be estimated using the particles. A commonly used particle filtering algorithm is the two step Bayesian bootstrap filter algorithm proposed by Gordon, Salmond and Smith (1993), where sampling and importance re-sampling step are repeatedly conducted as we move forward through 7

8 time. However, the disadvantage of this algorithm is that the sampler is not efficient, because the most recent and highly informative equity value S t are ignored in sampling M particles of asset value V (m) t. The SL-SIR method proposed in Duan and Fulop (2009) adopts the auxiliary filtering idea of Pitt and Shephard (1999). SL-SIR scheme performs particle filtering in an enlarged dimension so that the information of S t is incorporated. The scheme remains in the order M in evaluating density functions, instead of M 2 for an arbitrary sampler that is also a function of S t. The SL-SIR scheme is as follows: 1. Begin with V τ (m) i in the equal weight filtering sample. Draw a ν (m) i+1from N(0,1), compute ( Vτ i+1 (S τi+1,ν (m) i+1 ) and get the pair V τ (m) i,v τ (m) i+1 ), m = 1,...,M. ( 2. Compute the importance weight w (m) i+1 = f ( sample point V (m) τ i+1. ) V τ (m) i+1 V τ (m) i,θ ) Φ dτ (m) i+1 e δν(m) i+1, and assign π (m) i+1 = w(m) i+1 M k=1 w(k) i+1 to the 3. Construct a piecewise linear empirical distribution using the weighted sample (V (m) τ i+1,π (m) i+1 ), m = 1,...,M to obtain a new equal-weight sample of size M. After applying the SL-SIR scheme to state space representation of equation 1 and 3, the likelihood function of observing this sequence of equity values is evaluated. Finally, the likelihood function is maximized to get parameter estimates Θ = {σ,δ,µ}. Implementation of the SL-SIR scheme in this paper is on a rolling basis. For each firm-month, the previous one year s equity values, risk-free rates and face value of debt are used to estimate one set of parameters. Although the parameters are assumed to be constant during the preceding one year estimation period, we actually obtain time-varying parameters because of rolling procedure at each month. 2.2 Measuring Distance-to-default Under Merton s Model, the probability of default is defined as the probability that the firm s asset value falls below the face value of zero coupon debt: P(V t ) = P(V T < F) = Φ( log V F t (µ 1 2 σ 2 )(T t) σ ) (5) T t where Φ is the cumulative distribution function of standard normal distribution and DT D is defined as the negative of the term inside Φ, i.e., 8

9 DT D t = logv t F + (µ 1 2 σ 2 )(T t) σ T t (6) DT D t measures the number of standard deviations that the asset value is away from default point. In practice, it is well known in econometrics literature that µ cannot be estimated with good precision due to the nature of diffusion models. To see this, in equation 3, µ is multiplied by dt, which is very small (for daily frequency in one year, dt = 1/250 = 0.004), but σ is multiplied by dt, which is larger (here dt = 0.063). To reduce sampling errors, an alternative form, also discussed by Duan and Wang (2012), is used from now on: DT D t = logv t F σ T t (7) Since DT D will be further calibrated in reduced-form models, the alternative form, i.e., assuming µ = 1 2 σ 2, is not as harmful as the sampling errors in the estimated µ. The initial maturity of debt is set to 2 years, and gradually declines to 1 year, which is consistent with the standard KMV assumptions that the time to maturity T t is 1 year. Under the KMV implementation, the default point F, is proxied as the sum of the short-term debt and one half of the long-term debt. However, this proxy is not meant for financial firms as it ignores other liabilities that is usually quite large in financial firms. Thus, adjustment for other liabilities is needed in order to estimate reasonable parameters for both financial firms and non-financial firms. Following Duan and Wang (2012), the default point F is defined as: F = short term debt long term debt + ς other liabilities (8) Combined with the parameters estimated from SL-SIR scheme, we now have {σ,δ,ς}. Distanceto-default for each firm-month observation can be calculated as in equation 7. Denote this distanceto-default by DT D(δ), and the distance-to-default without considering trading noise simply by DT D. We would like to study how different is DT D(δ) from DT D, and denote DT D i,t = DT D(δ) i,t DT D i,t (9) for firm i and month t. The effect of trading noise on distance-to-default is indirect, as the noise parameter is not entered directly into the equation 7. This makes the study on DT D interesting. At best, we can have 9

10 some guesses, and empirical evidence would be helpful. To explore the answer, we look at effects from both firm specific characters and from macroeconomic changes. They are not necessarily independent with each other, since changes of firm specific attributes, like volatility and leverage, could be results from changes in macroeconomic condition. There are many choices of macroeconomic variables. During recession, there is a significant decline in activity across the economy, and so does business activity. Graham, and Harvey (2009) provide evidence that the 2007 financial crisis has had a severe impact on credit constrained firms, leading to deeper cuts in planned R&D (by 22%), employment (by 11%), and capital spending (by 9%). The inability of these firms to borrow externally has caused many firms to cancel or postpone attractive investment projects, leading to lower expected profitability and DT D. Stock market is also an indicator of economic environment because changes in stock prices reflect investor s expectations for the future of the economy. The volatility of stock market return is another dimension of risks in near future. Credit market activities such as commercial banks lending should also be considered because they reflect the risk appetite to businesses in general. Market sentiment such as index of consumer expectations is important, since it indicates future consumer spending or investment. For index return, its volatility, and consumer sentiment, the respective upper 20 percentile and lower 20 percentile are extracted to see whether there are significant difference in DT D. The two portfolios are then further separated into different industries, and the effect of economic changes to different industries is also studied. To assess whether the new estimate, DT D(δ) is a parallel shift from DT D, we use Spearman s ρ rank correlation coefficient and Kendall s τ rank correlation coefficient between DT D(δ) and DT D as test statistics. Both rank correlation coefficient lies between -1 and 1, and equals 1 if the agreement between the two rankings is perfect; 1 if the disagreement between the two rankings is perfect. 2.3 Probability of Default (PD) in forward intensity approach While distance-to-default alone is informative about default risk, researchers have long realized that it must be used together with other variables to have a high default prediction power. Trading noise can affect PD from two possible aspects. The first is its indirect effects on distance-to-default: substituting the new measure DT D(δ) will result in a slightly different PD. Another aspect is its direct effect on PD: trading noise can be regarded as a new covariate in the reduced-form credit risk models. The rationale of its direct impact are two folded. First, the trading noise covariate can be considered as an adjustment term correcting for the pricing errors in the observed equity value. 10

11 Second, it can enter with economic meaning. For example, firms with smaller trading noise may have higher market liquidity that provides funding to the firm at low cost, thereby reduce default probability. While this aspect is not modeled in structural models, we have more freedom to test this in reduced form models. This paper applies the forward intensity approach by Duan, Sun, and Wang (2012) and studies both indirect and direct effect of trading noise on PD. The forward intensity approach is a reduced form credit risk model where the probability of default is computed as a function of different input variables available at current time. A firm s default is signaled by a jump in a Poisson process determined by its intensity. The forward intensity approach specifies an explicit dependence of instantaneous intensities that are only known in future, as a function of state variables at the time of prediction. The forward intensities are usually specified as exponentials of a linear combination of input variables. Equation 10 specifies firm i s forward density at t, with input variables Y i,t, forward horizon τ, and coefficient vectors β(τ). In this way, PD for different horizons can be computed based on only current information. The pseudolikelihood function in Duan, Sun, and Wang (2012) is shown to be decomposable to independent components, making the implementation feasible. In particular, the model can be parallelized for multiple prediction horizons. h i (τ) = exp(β(τ) Y i,t ) (10) Three models are studied. The first model (Model 1) is the original forward intensity approach in Duan, Sun, and Wang (2012), where covariates include common variables and firm-specific variables. Common variables include stock index return, which is the trailing one-year simple return of S&P500 index, and short term interest rate. Firm-specific variables are transformations of liquidity, profitability, relative size, market evaluational/future growth opportunities, and idiosyncratic volatility. Liquidity is the funding liquidity of a firm and is measured as the ratio of cash and short-term investments to total assets; profitability is measured as the ratio of net income to total assets; relative size is measured as the logarithm of the ratio of market capitalization to the economy s median market capitalization; idiosyncratic volatility follows Shumway (2001) s measure of the unsystematic component of a firm s total risk associated with the stock return. Liquidity, profitability, and relative size are transformed into level and trend components. Level is computed as the one-year average of the measure, and trend is computed as its current value minus its one-year average. Such transformation is important as it captures different dimensions of current value for each variable. The second model (Model 2) uses all the covariates from the first model, plus an additional covariate 11

12 measuring the impact of trading noise on distance-to-default, i.e, DT D i,t defined in equation 9. A significant parameter of DT D i,t would imply the indirect effect of trading noise on PD through distance-to-default is significant. The third model also uses all the covariates from the first model but with two modifications. First, the original estimate of DT D is replaced with DT D(δ), for both level and trend transformations. Second, a new covariate that would capture the direct effect of trading noise on PD is added into. For simplicity, we just use the estimated trading noise parameter, δ i,t, and this variable would be referred to as NOISE(CURRENT) with the corresponding model being Model 3a. Following the level and trend transformation in the original model, we can also transform δ i,t into level and trend, which will be called as NOISE(AVE), and NOISE(DIF). The model will be Model 3b. A significant parameter of a noise measure would indicate significant impact of trading noise on PD directly. The study of trading noise s effect on PD is difficult, because we do not expect to see a large difference in PD after adjusting trading noise. However, a very small change in PD can still be significant. If the calibrated parameters for trading noise are statistically different from zero, we will be able to justify that trading noise has an additional role in predicting corporate default. Another common method in assessing default models is accuracy ratio derived from the cumulative accuracy profile (CAP). This methodology is not suitable in this paper due to two reasons. First, the accuracy ratio is computed purely from realized defaults. Realized defaults may not be a ideal model assessment tool since corporate default events can be intervened by external forces. For example, the accuracy ratio for financial firms during the financial crisis is not as high as in other periods, due to significant government intervention. Given that the original forward intensity approach has an accuracy ratio over 90% for 1-3 month default prediction, it is not expected that changing DT D to a new measure or adding trading noise as a new covariate will have a substantial improvement on accuracy ratio. Second, the accuracy ratio is limited in its ability to fully assess the accuracy of a default prediction model, because the ratio is derived from the CAP curve, which only depends on the rank of PDs, and ignores the difference between adjacent PDs. Also, the change in PDs for firms that have not defaulted can not be fully incorporated in the calculation of accuracy ratio. 12

13 3 Data We obtain the model input data for the US economy from NUS Risk Management Institute (RMI). The Credit Research Initiative (CRI) is a non-profit undertaking by RMI that releases daily updated PD forecasts for around 60,400 firms within the US, Canada, Latin American, Asian, African and European economies, as well as countries within the Middle East. This non-profit alternative can potentially counterbalance the for-profit credit rating agencies (Duan and Van Laere 2012), as rating agencies suffer from a conflict of interest problem and have been highly blamed for the financial crisis and the subsequent European sovereign debt crisis. The data sources of the CRI database are mainly Thomson Reuters Datastream and the Bloomberg Data License Back Office Product. Common factors such as stock index prices and short-term interest rates are retrieved from Datastream. Firm-specific data comes from Bloomberg s Back Office Product which delivers daily update files by region via FTP after respective market closes, and it includes daily market capitalization data based on closing share prices and includes new financial statements as companies release them. A major challenge, however, lies in choosing which financial statement to use when firms have multiple versions of financial statements within the same period, with different accounting standards, filing statuses, currencies or consolidated/unconsolidated indicators. The details of priority rule are described in the section 3 of the Technical Report on RMI CRI s website. The default events of CRI database come from a variety of sources, including Bloomberg, Compustat, CRSP, Moody s reports, TEJ, exchange website and news sources. A default event is recognized if one of the following happens: 1. Bankruptcy filing, receivership, administration, liquidation or any other legal impasse to the timely settlement of interest and/or principal payments; 2. A missed or delayed payment of interest and/or principal, excluding delayed payments made within a grace period; 3. Debt restructuring or distressed exchange, in which debt holders are offered a new security or package of securities that result in a diminished financial obligation. For trading noise and DTD calculation, the input variables for each firm-month are (Bloomberg field in bracket): current liabilities (BS CUR LIAB), long term borrowings (BS LT BORROW), total liabilities (BS TOT LIAB2), total asset (BS TOT ASSET), daily market capitalization based on closing prices for previous 1 year, and 1-year US Treasury constant maturity rate. The face value of debt is then approximated using equation 8 in section 2.2. The estimation of unknown fraction ς in equation 8 is also obtained from RMI database. We start DTD calculation only when we have a minimum of 250 days of valid observations for market capitalization, because we use the previous 1 year s data to estimate parameters in section 2.1. The final sample contains US listed and 13

14 delisted companies from Jan 1990 to Dec 2012, with 11,166 unique firms, 1,687,154 firm-month observations, among which 908,884 have parameter estimates. Macroeconomic variables are defined as follows: Recession indicator is the NBER based Recession Indicators for the United States from the Peak through the Trough. S&P500 return is from the original dataset and is used as a proxy of US stock market index return. Its lower 20 percentile and upper 20 percentile are used as low return and high return period, respectively. VIX is the CBOE volatility index. Its lower 20 percentile and upper 20 percentile are used as low VIX period and high VIX periods, respectively. Commercial and industrial loans at all commercial banks are used to classify data into bank loan decreasing period when the change from previous month is negative, and bank loan increasing period when the change is positive. Recession indicator, VIX, and loans are obtained from website of Federal Reserve Bank of St. Louis. The University of Michigan Consumer Sentiment Index is the index published monthly by the University of Michigan and Thomson Reuters. To compute the market liquidity measures described in section 2.2, our dataset is merged with CRSP daily stock file using identifier Ticker. Daily bid-ask spread, trading volume, Amihud ratio, and number of trades are calculated and the monthly average of daily value is used for each firmmonth observation. After merging, 8,965 firms and 1,127,932 firm-month observations remain. For PD calculation, we also need US Generic Govt 3-month yield as short term interest rate, net income (NET INCOME), cash (BS CASH NEAR CASH ITEM), marketable securities (BS MKT SEC OTHER ST INVEST) to construct the input variables in section 2.3. After all the variables are constructed, data cleaning is conducted in a 2-step procedure. First, to minimize the impact of outliers, we winsorize each of the above variables by a cap at 99.9 percentile value and a floor at the 0.1 percentile value. Second, missing variables are replaced under certain circumstances. The rule follows from RMI-CRI s technical report: For each firm, out of all the 12 firm specific variables, if less than or equal to 5 variables are missing during a particular month, we first trace back and use the most recent available values of these variables. The maximum look back period is 1 year. If this does not succeed in replacing all the missing variables, sector (financial or non-financial) median within US during that month will be used for the replacement. In this way, the replacement will have a neutral effect on the PD of the firm. It has been noted by the RMI-CRI team, however, that this treatment of missing values is not always meaningful and occasionally results in counter intuitive results in historical PD. We will also improve the way in handling missing values once a better solution is figured out. 14

15 4 Empirical Results 4.1 Trading noise Table 1 Panel A shows the summary statistics of parameter estimates in the Particle-filter based MLE method for the Merton s model proposed by Duan and Fulop (2009). From the large sample of 11,166 US companies from Jan 1990 to Dec 2012, the estimated parameters are within similar ranges as in the 100 random sample from CRSP in Duan and Fulop (2009). The estimated asset volatility are stated per annum with mean of 0.39 and median of 0.31, which are quite reasonable. The trading noise parameter is multiplied by a factor of 100. It has a mean value of 0.63, and median of The differences of its mean and median can be explained by the cluster of estimates close to zero. Its 25 percentile is very close to the minimum of 1E-7, indicating that one quarter of the observations have very small trading noises. Figure 1 plots the histogram of estimated trading noise parameter, where we can see a peak at around zero, and a second peak at around 0.5. While it is possible that the one quarter of firm-month observations indeed have negligible trading noises, it is also possible that the estimated noise itself is noisy. As indicated by Duan and Fulop (2009), the noise parameter could also reflect other factors such as statistical sampling error, model-misspecification, etc.. Taking advantage of a large panel data, it is found that the time series standard deviation of noise parameter is indeed quite large, and when taking the moving average of preceding 12 months as the level of noise, the proportion of noises that are close to zero declines, with much smoother distribution. The interpretation of noise parameter is closely related to the specification of noise structure in equation 1. Rearranging equation 1, we have δϑ i = ln S τ i S(V τi ) (11) The right hand side is the percentage difference between observed equity value and true equity value, and ϑ i follows i.i.d. standard normal distribution. The noise parameter, δ, should be understood as the degree of dispersion of the percentage difference between observed equity value and true equity value, i.e. we should regard the right hand side as a random variable, and δ measures how much variation of its distribution. For one standard deviation of trading noise, a low δ indicates that the observed equity value tends to be very close to the true equity value, and a high δ indicates a large deviation of observed value from true value. With this in mind, it is clear that δ does not measure the return volatility of equity. This implies 15

16 that the trading noise measure is different from the market liquidity measures such as Amihud ratio, bid-ask spread, etc., which rely on the variation among the intra-day observed equity prices. On the other hand, trading noise should be correlated with proxies of market liquidity: More liquid firms will in general have less microstructure noises. Aït-Sahalia and Yu (2008) provides evidence in high frequency market that more liquid stocks based on financial characteristics have lower noise and noise-to-signal ratio. Table 2 Panel A shows the Spearman correlation coefficient of noise and commonly adopted market liquidity measures. Noise is positively correlated with Bid-ask spread and Amihud ratio, and negatively correlated with trading volume and number of trades. The results are consistent with prior literature. While all correlations are significant, the magnitude of correlation between noise and market liquidity measures are not large, with the highest correlation being 0.32 (Noise and Amihud ratio). On the contrary, the correlation among the four market liquidity ratios are very high; for example, the correlation between Amihud ratio and bid-ask spread is This shows that trading noise is different from the existing common market liquidity measures. Another interesting observation is from the Table 2 Panel B, that the correlation of trading noise with market capitalization (SIZE) and idiosyncratic volatility (SIGMA) is relatively high, and 0.34 respectively. Both SIZE and SIGMA are constructed purely from market data, and their correlations with trading noise are much stronger than correlations of trading noise with covariates from balance sheet. 4.2 Distance-to-default The second plot of Figure 1 is the histogram of DT D from the original model that ignores the presence of trading noise, as well as DT D(δ) that are calculated from models that explicitly adjusted for trading noise. Both distributions look like log-normal, but DT D(δ) is obviously different from DT D, with higher mean and less skewness. Since a lower DT D indicates higher default risk, this means that in general, ignoring trading noise will lead to more conservative estimation of DT D, or over-estimate a firm s default risk. When DT D is smaller than or equals to zero, we are predicting that the firm is not going to meet its debt obligation in future. From the graph, the frequency of such default event is much lower after adjusting for trading noise. The downward bias in DT D when ignoring trading noise can be explained by the higher volatility estimated, since asset movement due to microstructure noise is not disentangled from its volatility. By removing the noisy part of observed equity value, the estimated asset volatility is lower and the new measure is generally higher. 16

17 From Table 1 Panel B, the mean value of DT D = DT D(δ) DT D is 0.81, and median is 0.49, so the distribution of DT D is skewed to the right. Since DT D measures the number of standard deviations the asset value is away from default point, a median of 0.49 in DT D implies that on average, ignoring trading noise will results in an over-estimation of default risk by 0.49 standard deviation. We could also compute the percentage difference between DT D(δ) and DT D, which is also shown in Panel B. The mean difference is 18.39% whereas the median difference is 12.30%. The standard deviation of percentage difference is too high compared with the level difference, and the minimum and maximum of percentage difference is also very large, suggesting that percentage difference is not as stable as the level difference. An example of large percentage difference is when DT D = 1, DT D(δ) = 0.001, so DT D = 1.001, and DT D/DT D(δ) Similar phenomenon also happens when we change the denominator from DT D(δ) to DT D. From now on, we will focus on DT D, instead of the percentage change. Figure 2 presents the histograms of DT D under different economic environment according to five measures that discussed in section 2.2. The frequencies used to plot histograms have been normalized to one, in order to make reasonable comparison. Note that the plots are not the level of DT D, but the difference of DT D when we adjust for and ignore trading noises. A positive and large difference indicates that default risk is over-estimated when trading noise is ignored. From the plots, we can observe smaller DT D during economic downturns, i.e. recession, low stock index return, high VIX, decreasing bank loans, and low sentiment. This implies that the original model s downward bias on distance-to-default estimation attenuates in these periods. This is because during economic downturns, firms are more likely to default and the tendency of over-estimation of default risk is lower. In fact, during bad economic condition and the default risk is high, we are more worried about under-estimating of default risk, or negative DT D in our case. From the plot, it is quite obvious that the frequency of negative DT D is higher during economic downturns. In other words, when ignoring trading noise, we are more likely to under-estimate default risk when it should be high, and over-estimate default risk when it is actually low. Such direction of bias justifies the importance of adjusting for trading noise in accessing a firm s default risk. What variables can explain DT D? The first guess could be the magnitude of trading noise parameter. Actually, when we sort trading noise into 10 quantiles, from smallest to largest, we will not be able to observe a strictly monotonic relationship between noise level and DT D. This is because leverage and volatility are not controlled for. Table 3 displays the triple sort of DT D across idiosyncratic volatility, leverage, and noise. 125 portfolios are formed as the intersections of the 5 SIGMA groups, 5 Leverage groups, and 5 Noise groups. SIGMA is the the unsystematic component of a firm s total risk associated with the stock return, first employed by Shumway (2001). 17

18 Leverage is total liabilities divided by total asset. Noise is the estimated trading noise parameter δ i,t for firm i and month t. Prior to the triple sorting of the firm-month observations, data is divided into Non-Recession period (Panel A) and Recession period (Panel B). Each entry of Table 3 is the mean of DT D i,t in each portfolio. First, comparing Panel A and Panel B, we can observe that during recessions, the mean of DT D is smaller than non-recession period. This is consistent with our analysis that recessions will attenuate the downward bias of original DT D. However, smaller mean value of DT D may not be an indicator of better performance when ignoring trading noise, since it is possibly due to more observations with under-estimated default risks. Within each SIGMA and Leverage group, DT D generally increases with trading noise parameter, indicating greater bias in distance-to-default estimation for a firm with higher trading noise parameter. Across different SIGMA groups, it is very clear that DT D decreases as SIGMA increases. Similarly, across different Leverage groups, as seen in the last column of Table 3, DT D also decreases as Leverage increases. In sum, DT D is larger, therefore upward bias of default risks is greater, for firms with low idiosyncratic volatility, low leverage, high trading noise parameter, and during non-recessions. While distance-to-default is defined as volatility adjusted leverage as in equation 7, it does not make the sorting using SIGMA and Leverage unnecessary. First, the leverage and volatility that are used to compute DTD are not observable, as asset value is filtered and asset volatility is estimated. However, SIGMA and TL/TA are both computed directly using observable variables. Using observable values to sort will give us an intuitive understanding of important factors that drives DT D. Second, while distance-to-default is inversely related to both volatility and leverage by construction, it is less clear for the difference of DTD using two methods, i.e. DT D(δ) DT D, which is one of the main interest in this paper. Performing the sorting by volatility and leverage will help us to justify our prior guesses. Table 4 displays the mean value of DT D for each industry under different economic and market conditions. Comparing the values within each column, we observe that Financial, Utilities, Energy, and Consumer Noncyclical firms have a higher DT D than firms from other industries. Hence, adjusting for trading noise will favor these industries in default risk assessment, as they would be less riskier according to the new measure, DT D(δ). Under different economic conditions as illustrated by the same five macroeconomic variables as in Figure 2, the mean values of DT D are smaller across all industries when the economy is bad, consistent with our earlier results. An exception is Diversified firms, where we only have a small sample, and the t-stats are not significant. In general, our prior result for the aggregate market holds for each industry as well: In good economic conditions, the original DT D tends to over-estimate a firm s default risk, especially for Financial, Utilities, Energy, and Consumer Noncyclical industries. During economic downturns, the problem 18

19 of over-estimation is less severe, but the problem of under-estimation is intensified. Table 5 shows the rank correlation between DT D(δ) i,t and DT D i,t across different idiosyncratic volatility and leverage groups. All the firm-month observations are first divided into Non-Recession period and Recession period. For each period, data are then sorted into 25 groups by the intersections of the 5 SIGMA groups and 5 Leverage groups. For each sub-group, both Spearman s rank correlation coefficient (Panel A) and Kendall s τ rank correlation coefficient (Panel B) are shown. First, it is obvious that for both correlation measures, rank correlation is lower in non-recession period than in recession period. Linking this with the finding that DT D is larger in non-recession period, we can conclude the impact of trading noise adjustment is material in normal and good economic condition. Interestingly, same pattern is not observed when we sort the firms by riskiness, as proxied by SIGMA and Leverage. Safest firms with low volatility and low leverage, which have the largest DT D, do not have the lowest rank correlation. Rather, for both correlation measures, we observe some kind of smile pattern of rank correlation by firms riskiness: the correlation is highest for safest and riskiest firms, and becomes lower as riskiness, measured by leverage and volatility, moves away from two end points. This indicates that ignoring trading noise is less harmful in terms of relative ranks for safest and riskiest firms. Rather, firms with risks in between will see greater rank distortion if we ignore the presence of trading noise in observed prices. Taken together, the empirical evidence shows that adjustment of trading noise would have a material impact on distance-to-default estimation, in terms of both magnitude of DT D and relative ranks among firms. 4.3 Probability of default When DT D is further calibrated into a reduced form credit risk model, we can test the implications of trading noise on PD. Table 1 Panel C displays the summary statistics of all input variables in Model 3b where the distance-to-default variables are adjusted for trading noise and transformed noise covariates are added. The variables have similar and reasonable ranges compared to the data used in Duan, Sun and Wang (2012). The NOISE(AVE) and NOISE(DIF) are newly constructed covariates that are defined in section 3. Three models are tested using forward intensity approach, as discussed in section 2.3. Table 6 shows the parameter estimates and p-values of DT D and DT D in Model 2. For the forward default intensity, DT D is very significant within 2 years prediction horizon but not for longer horizons. This makes sense since the impact of trading noise is likely to decrease in the long run since market will correct the pricing error over time. On the contrary, all estimates in other exit 19

20 intensity are not significant for all horizons. This supports our idea that the impact of trading noise is on default prediction not for other exit prediction. The negative sign before DT D indicates that firms with greater DT D, as often observed in firms with low SIGMA, low Leverage, and high Noise, will have a lower probability of default than the one ignoring trading noise. In other words, ignoring trading noise will cause upward bias in estimated PD. Figure 3 plots the parameter estimates of trading noise in the forward default intensity function. The upper figure compares the coefficients of NOISE(CURRENT) and NOISE(AVE) in Model 3a and Model 3b respectively. Similar pattern and trend are observed for the two parameters, and the differences are quite small. As horizon increases, the parameter estimates approach zero, implying that the impact of trading noise on PD diminishes as horizon increases. Indeed, as we can see from table 7, parameters for a longer horizon are no longer significant. The parameter of NOISE(DIF) in Model 3b seems very volatile, and it becomes insignificant much earlier than NOISE(AVE) and NOISE(CURRENT). Table 7 also shows that the trading noise covariates are significant in forward default intensity but not in forward other exit intensity. This sheds light on the usefulness of adjusting for trading noise in default prediction but not for other types of exits, such as mergers and acquisitions. In section 2.3, two rationales for the inclusion of trading noise are discussed, one as an adjustment term for the pricing errors and another with economic meaning that smaller trading noise may indicate a firm s ability to obtain funding from the market at lower cost to fulfill debt obligations. If the later dominates the prior effect, then the estimated parameter should be positive, i.e. higher trading noise will increase default probability. However, Figure 3 shows that the parameters for NOISE(CURRENT), NOISE (AVE), and NOISE(DIF) are all negative within 3 years horizon. This would imply that the adjustment effect for pricing errors dominates the funding effect, if there is any. Overall, the findings on PD reveals that the adjustments of trading noise, both indirectly through DT D, and directly from entering as new covariates, are significant for default prediction. Therefore, it is necessary to adjust for trading noise for default prediction, especially in short horizons. 5 Conclusion In this paper, we investigate the impact of adjusting for trading noise on default risk measures. We ask: (a) how different is the estimated default measure once we account for trading noise? (b) In what scenarios are the differences more severe? (c) Can the new measure bring material benefits to 20

21 corporate default prediction? We first demonstrate the existence of trading noise in majority of US firms from 1990 to On average, the magnitude of trading noise parameter is 0.4%-0.6%, indicating that when noise is one standard deviation in either direction, the average percentage deviation of the observed stock price from the true price is 0.4%-0.6%. The magnitude of such deviation is shown to be correlated with market liquidity measures such as bid-ask spread, Amihud ratio, trading volume, and number of trade. Second, ignoring trading noise will lead to downward bias in DT D estimation and upward bias in default risk estimation. After adjusting for trading noise, the change in DTD is 0.49 on average, i.e., ignoring trading noise will decrease our estimation of DTD by 0.49 standard deviation. During economic downturns and for risky firms with high volatility and high leverage, firms are more likely to default and the upward bias in default risk estimation attenuates. On the flip side, the largest differences in DT D are seen in safest firms with low volatility and low leverage, and during good economic condition. This phenomenon is more obvious for financial, utilities, energy, and consumer noncyclical industries. Further, we the change in DT D estimation is not a parallel shift, because the rank correlation between the two measures is less than one, and lower in non-recession period than in recession period. Interestingly, we observe a smile pattern of rank correlation by firms riskiness: the correlation is highest for safest and riskiest firms, and becomes lower as riskiness, measured by leverage and volatility, moves away from two end points. After we substitute the new DTD measure adjusted for trading noise and recalibrate in the forward intensity model, changes in PD are small. However, the parameter of DT D is negative, indicating its downward adjustment of PD. Overall, the adjustment of trading noise is shown to be statistically significant for default prediction. This holds true whether the direct or indirect impact of trading noise is entered into, and the results shed light on the significance of adjusting trading noise on default prediction, but not on other types of exits. There are a lot to be explored in this paper. The effect on PD can also be studied using some classification of the data, since the change in PD could be large for certain sub-samples. Default prediction accuracy is another dimension, but the challenge is the lack of a proper criterion that can fully reflect changes in PD. Other than cross-sectional differences of trading noise parameters, the time series variation of noises would also be interesting. For example, how does a sudden increase in noise affect a firm s default risk? What does the trend of trading noise tell us? In sum, many areas in this literature are still unexplored, and we look forward to finding more insightful results. 21

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25 Figure 1: Histogram of noise and DTD This figure plots the distribution of estimated trading noise parameter estimates (δ i,t for firm i and month t) in the Particle-filter based MLE method for the Merton s model proposed by Duan and Fulop (2009). The histogram is normalized such that total frequency is 1. The lower plot is the DT D distribution when we adjusted for trading noise compared with the original DT D that ignores trading noise. For each firm, parameter estimates start only when there are minimum of 250 days of valid observations for market capitalization. After that, each month s parameters are re-calculated on a rolling basis. In calculating DT D, the face value of debt is approximated using equation 8 in section 2.2. We use a large sample of US listed and delisted companies from Jan 1990 to Dec Data includes 11,166 firms and 1,687,154 firm-month observations, among which 908,884 observations have parameter estimates. 25