The relationship between default risk and interest rates: An empirical study research insight

1 INTRODUCTION 5 2 LITERATURE REVIEW . 6 3 DATA AND VARIABLE DESCRIPTION 7 4 THE RELATIONSHIP BETWEEN INTEREST RATES AND AGGREGATE DEFAULT RATES 11

OCTOBER 2, 2009 THE RELATIONSHIP BETWEEN DEFAULT RISK AND INTEREST RATES: AN EMPIRICAL STUDY RESEARCH INSIGHT ABSTRACT AUTHORS Andrew Kaplin Amnon Levy Shisheng Qu Danni Wang Yashan Wang Jing Zhang Understanding the relationship between credit and interest rate risk is critical to many applications in finance, from valuation of credit and interest rate-sensitive instruments to risk management This study empirically examines the relationship between interest rates and default risk using firm level corporate default data in the United States between 1982 and 2008 We find significant negative contemporaneous correlations between the changes in short interest rates and aggregate default rates, with a particularly strong relationship around financial crises We also explore the explanatory power of interest rate variables in predicting default when conditioned on Moody’s KMV EDF™ credit measures In addition, we study the impact of changes in short rates, expected changes in short rates, interest rate slopes, and unexpected changes in short rates Conditional on the EDF credit measure, interest rates and default were not found to have any statistically significant correlation Our findings have a number of important implications for risk measurement and management Copyright © 2009, Moody’s Analytics, Inc All rights reserved Credit Monitor, CreditEdge, CreditEdge Plus, CreditMark, DealAnalyzer, EDFCalc, Private Firm Model, Portfolio Preprocessor, GCorr, the Moody’s logo, the Moody’s KMV logo, Moody’s Financial Analyst, Moody’s KMV LossCalc, Moody’s KMV Portfolio Manager, Moody’s Risk Advisor, Moody’s KMV RiskCalc, RiskAnalyst, RiskFrontier, Expected Default Frequency, and EDF are trademarks or registered trademarks owned by MIS Quality Management Corp and used under license by Moody’s Analytics, Inc ACKNOWLEDGEMENTS We are grateful to our MKMV Research colleagues for their generous comments All remaining errors are, of course, our own Published by: Moody’s KMV Company To contact Moody’s KMV, visit us online at www.moodyskmv.com You can also contact Moody’s KMV through e-mail at info@mkmv.com, or call us by using the following phone numbers: NORTH AND SOUTH AMERICA, NEW ZEALAND, AND AUSTRALIA: 866 321 MKMV (6568) or 415 874 6000 EUROPE, THE MIDDLE EAST, AFRICA, AND INDIA: 44 20 7280 8300 ASIA-PACIFIC: 852 3551 3000 JAPAN: 81 5408 4250 TABLE OF CONTENTS INTRODUCTION LITERATURE REVIEW DATA AND VARIABLE DESCRIPTION THE RELATIONSHIP BETWEEN INTEREST RATES AND AGGREGATE DEFAULT RATES 11 4.1 4.2 CONDITIONAL CORRELATION ANALYSIS USING FIRM-LEVEL DATA 12 5.1 5.2 Contemporaneous Correlations 11 Corr(Δr, DR) Over Time 12 Predictive Logistic Regression Model at Firm Level 12 Contemporaneous Logistic Regression Model at Firm Level 13 CONCLUSION 14 APPENDIX A ˆ SIMULATION PROCEDURE FOR CALCULATING SE( β ) 15 THE RELATIONSHIP BETWEEN DEFAULT RISK AND INTEREST RATES: AN EMPIRICAL STUDY INTRODUCTION Credit and interest rate risks are among the most important risks faced by financial institutions It is well known that the two risks are economically related, and understanding their relationship is important to many applications in finance For example, the values of callable corporate bonds of fixed coupons depend on both the interest rate dynamics and the issuers’ credit qualities Alternatively, financial institutions’ balance sheets include both credit and interest rate-sensitive instruments If interest rates (or credit quality) change unexpectedly, the resulting impact on credit quality (or interest rates) will help determine how the assets and liabilities line up, consequently determining the institution’s financial health Thus, the relationship between credit and interest rate risks plays an important role in both pricing instruments whose values are sensitive to both risks, as well as in managing an institution’s balance sheet Despite its importance, the exact nature of the relationship between credit and interest rate risk is not quite clear For example, consider the relationship between default and interest rate If the economy is in recession and the default rate is high, interest rates are often relatively low through the traditional central bank monetary policy of lowering rates in the hope of stimulating the economy When the economy improves, the central bank tends to raise rates Given that government rates often form the basis of the cost of capital faced by the companies, when the interest rates increase, a firm must generate a higher rate of return on its assets to stay in business If the cost of capital is higher than the rate of return for a particular company, that firm will run into financial insolvency or bankruptcy In other words, the central bank is raising rates in an effort to slow the economy Therefore, we may conjecture that the relationship between default risk and interest rates is sensitive to some measure of where the economy is in the business cycle and/or other macroeconomic factors Moreover, co-movements between interest rates and default risk may exhibit different behavior whether analyzed contemporaneously or in a causal or predictive setting Given the potential ambiguity in intuition cited above, the main goal of this study is to analyze the empirical relationship between interest rates and default risk Moreover, the structure of the analysis focuses on understanding the dynamics within the context of risk management While we ultimately want to understand the relationship between credit risk, including the risk of default, migration and recovery, and interest rates Toward this goal, we analyze the relationship between interest rates and default rates using the Moody’s KMV public firm default database—the largest existing database of its kind We consider both contemporaneous and predictive relationships The predictive relationship focuses on the following question: Do interest rates provide information beyond Moody’s KMV EDF™ (Expected Default Frequency) credit measures that can be used in predicting default? The contemporaneous analysis provides insights into whether the correlation between interest rates and defaults should be modeled when measuring portfolio economic capital Specifically, we ask the following question: If we condition on the current term structure of interest rates, as well as on EDF credit measures, would the conditional distribution of future interest rates and defaults be correlated? In both cases we find no correlation between defaults and interest rates after conditioning on EDF credit measures It is worth noting the differences between our study and the existing literature Most of the related academic literature studies the relationship between credit spreads and interest rate The papers addressing the relationship between default and interest rates are relatively scarce and results can be contradictory For example, Fridson et al (1997) reported that on a quarterly basis during the period of 1971–1995, there was a moderate, significant positive correlation between default rates and real interest rate, and a strong positive correlation between default rate and lagged 2-year real interest rate We find negative correlation between changes in interest rates and default rates, with the correlations between changes in short rates and default rates being significantly negative This result generally was consistent with findings on the relationship between changes in credit spread and changes in interest rates documented in a few papers What is different and unique with this study is that our dataset allows us to perform firm-level regressions to test the impacts of interest rates on default conditional on EDF, whereas most previous analyses were performed at an aggregated level Our empirical findings have a number of important implications in practice The results suggest that the interest rate and default risk dynamics are more complicated than previously reported From the perspective of comprehensive risk modeling, this suggests that it is quite challenging, perhaps impossible, to specify a theoretical model that fully describes both the interest rate and default processes in a correlated manner with a single correlation parameter It may be more constructive to develop default risk model that captures the dynamic impacts of interest rate separately, as in the case of the Moody’s KMV EDF model This also suggests that once an accurate credit risk measure such as EDF is properly incorporated, interest rates and default risk become conditionally uncorrelated in the joint model, leading to a significant See Longstaff and Schwartz (1995), Duffee (1998), and Collin-Dufresne et al (2001) THE RELATIONSHIP BETWEEN DEFAULT RISK AND INTEREST RATES: AN EMPIRICAL STUDY decrease in computational complexity From the perspective of managing both interest rate and default risk, our results suggest that risk managers should be paying close attention to these dynamics, especially when hedging is involved This paper is organized in the following way • • Section discusses the data and variable specification • Section describes historical correlation findings using aggregated data • Section presents the results from regression specifications using granular data • Section provides a review of the existing literature describing the relationship between credit and interest rate risk Section provides concluding remarks LITERATURE REVIEW Numerous studies have examined the relationship between credit and interest rate risk in various contexts, from derivate pricing models and term structure modeling, to risk integration Most studies focus on the relationship between credit spreads and various interest rate variables As far as we know, Fridson et al (1997) is the only study that exclusively focuses on the correlation between real interest rate and default rate Using Moody’s quarterly default rate on high-yield bonds from 1971–1995, they find a weak positive correlation between default rate and nominal interest rates, a moderate positive correlation between default rates and real interest rate, and a strong positive correlation between default rate and lagged 2-year real interest rate They argue that interest rate level is the basis of cost of capital When the interest rate is high, the firm must generate higher rate of return in order to survive If the cost of capital is higher than the rate of return, the firm would run into financial insolvency or bankruptcy This indicates that there is a positive relationship between default rate and real interest rates Longstaff and Schwartz (1995) develop a simple approach to valuing risky corporate debt that incorporates both default and interest rate risk, and test its empirical implications Using the changes in the 30-year Treasury bond yield and the changes in the bond yield using Moody’s corporate bond database from 1977–1992, they find negative correlation between the two across combinations of industries and rating categories For example, a 100 basis point increase in the 30-year Treasury yield reduces Baa-rated Utility industry credit spreads by 62.6 basis points Duffee (1998) studies the correlation between the changes in 3-month Treasury bill yield, the changes in term structure slope (defined as the difference between 30-year and 3-month Treasury bond yields), and the changes in yield spread of corporate bond with data from 1985–1995 The changes in yield spread are constructed monthly from non-callable bonds rated from Aaa to Baa, maturities ranging from to 30 years He finds that an increase in T-bill yield corresponds with a decline in yield spreads for each combination of maturity and credit rating The relationship is stronger for longermaturity and for lower quality bonds The relation between yield spreads and slope is generally negative, insignificant for high quality bonds and significant for low quality bonds Duffee also test the correlation between callable bonds and interest rates using Moody’s and Lehman Brothers bond indices Callable bonds show stronger negative correlations than non-callable bonds Collin-Dufresne et al (2001) study the determinants of the credit spread changes using data of straight bonds issued by industrial firms in the Lehman Brother bond database from 1988–1997 The changes of credit spread are regressed over the change in the 10-year Treasury bond yield, the change in slope (defined as the difference between 10-year and 2-year Treasury bond yields), the convexity, the change in leverage, the change in asset volatility, the change in jump, the liquidity, and the individual firm’s stock return The regressions are performed for bonds in each unique combination of maturity and rating category They find significant negative correlations in the changes in interest rates, insignificant negative correlations in the convexity, and insignificant negative correlations in the change in slope for bonds with longer maturities, and insignificant positive correlations in the change in slope for bonds with shorter maturities Joutz et al (2001) study the dynamics of corporate credit spreads by examining how default and systematic risk measures influence corporate bond spreads for investment and non-investment grade corporate bonds over the 1987–1997 period The changes in credit spread are selected from Lehman Brothers bond indexes, or constructed from individual noncallable bonds rated from AA to BBB and maturities ranging from intermediate to long-term They find the relation between credit spreads and interest rates (level and slope) differ based on the maturity, credit ratings, and the sign of the relation changes based upon the time frame In aggregate, the results suggest that Treasury yields are positively related to credit spreads in the long run, but negatively related in the short run The relation between credit spreads and the slope of Treasury term structure depends on credit quality, maturity, and time frame For intermediate investment grade bonds, the relation is positive in both the short and long run, but for long-term bonds the predominant relation is negative in the long run, and is statistically insignificant in the short run Similarly to Joutz et al (2001), Neal et al (2000) perform co-integration analysis on the correlations between the levels of credit spread and interest rates using Moody’s bond indexes from 1960–1997 They find that corporate rates are cointegrated with government rates and the relation between credit spreads and Treasury rates depends on the time horizon In the short-run, an increase in Treasury rates causes credit spreads to narrow This effect is reversed over the long run and higher rates cause credit spreads to widen Jarrow and Yildirim (2002) develop an analytic formula for valuing default swaps with correlated market and credit risk in the context of a reduced form model To illustrate the implementation of the model, they fit the model to use daily CDS prices of 22 firms from 8/21/00–10/31/00 With this data, they find positive correlations between instantaneous default rate and interest rate Lin and Curtillet (2007) take another look at the relationship between credit spreads and interest rates, and try to reconcile contradicting results from previous studies First, they argue the structural model of credit risk could imply either positive or negative relationship with interest rates depending on the assumption of the asset process Then, they present a way to break down credit spreads into components of default, downgrade, and liquidity, and show that previously documented overall negative correlation between credit spreads and interest rate may actually arise from the liquidity risk component rather than the default risk component They also show that previous documented positive correlation could be due to lead-lag relation by showing that the two-months lagged LIBOR rate changes and credit spread changes are positively related Furthermore, they argue that credit spreads widen around financial events, but fluctuate in a narrow band at other times In addition, they argue that there are no definite relationships between credit spreads and interest rates In summary, the majority of previous studies have found negative correlations between the changes in credit spread and the changes in interest rate Some find positive correlations between the levels of credit spread and interest rate, and positive correlations between default rate and real interest rate The significance levels of the correlations vary depending on the credit quality of issuers, bond maturities, credit sources, and time periods studied There is no consensus on the correlations between changes in credit spread and changes in interest rate slopes, or the lead-lag relations between changes in interest rate and changes in credit spread Moreover, the questions posed in our introduction about whether credit events and interest rates are conditionally correlated remain open DATA AND VARIABLE DESCRIPTION Moody’s KMV maintains the world’s largest default database, which records default events in public firms in the U.S and foreign countries from the 1970s to the present It records more than 8,000 publicly traded company defaults around the globe In the database, default is defined an event whereby any creditor suffers economic loss from missing payments, bankruptcies, distressed exchange, or liquidation events For this study, we focus on defaults associated with non-financial public firm defaults in the U.S from 1982 through the third quarter of 2008 To control data quality and avoid missing defaults that occur more often for small firms, we performed the analysis for large, non-financial firms only Large firms are defined as firms with annual sales greater than 300 million dollars In measuring aggregate default rates for a given period (a quarter or a year), we first find number of firms at the beginning of the period, and then use it to divide number of defaults during the period among these firms Taking the change in short rate as an example of an interest rate variable, we use the short rate change during the same period in which we measure the default rate when we analyze its contemporaneous relation with default rates We take a similar approach for firm level analysis; default for a firm is an indicator variable measuring whether the firm defaults during the given period In addition, interest rate variables are observable at the end of the period for contemporaneous analysis, or observable at the end of the previous period for predictive analysis THE RELATIONSHIP BETWEEN DEFAULT RISK AND INTEREST RATES: AN EMPIRICAL STUDY For interest rate variables, we use yields of constant maturity treasury (CMT) to measure interest rates because the data could provide a maximal overlapping period with the default records we have Figure shows the Treasury yield (3-month, 2-year, and 10-year) and default rates for each quarter Treasury yields oscillate and decrease from above 10% in 1982, to less than 5% in 2007 Default rates for all public firms are low most of time Defaults peak around economic crisis in 1991 and 2001, with the highest quarterly default rate of 1.2% in 2001 The results in Figure are based on the default records of large, non-financial firms in the U.S in the Moody’s KMV default database Default rate is calculated as the number of defaults divided by the total number of firms recorded in the corresponding quarters Total observations are 185,564 firms-quarters Default Rate Change in Short Rate Short Rate 0.16 0.01 0.14 0.12 -0.01 0.1 -0.02 0.08 -0.03 0.06 -0.04 0.04 -0.05 NBER Contraction Periods Short Rate Default Rate / Change in Short Rate 0.02 0.02 -0.06 1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 Time FIGURE Realized Quarterly Default Rates: 1982Q1–2008Q2 Table shows sample statistics for EDF and Quarterly Default Rates for non-financial firms in the Moody’s KMV default database from 1982Q1 to 2008Q2 Default Rates are annualized TABLE Sample Statistics for EDF and Default Rates EDF Mean STD 1.68% 5.01% Default rates 1.43% 23.88% The interest rate variables used in our analysis include change in short rate, slope in interest rate term structure, expected change in short rate, and shock in short rate Change in the short rate is included since it is one of the primary variable the Federal Reserve manipulates in order to control the money supply If a significant correlation exists between interest rate and default risk, we speculate that the change in short rate would be the most sensitive variable to reflect it The change is defined as: Δrt ,3m = rt ,3m − rt −3m ,3 m (1) ,Where Δrt,3m is the difference in 3-month CMT yield between the last Thursday of each quarter and the first Thursday of the same quarter Δrt,2y and Δrt,10y are defined similarly Both the slope in interest rate and expected change in short rate are considered since they provide a forward-looking view of interest rate variables Because the interest rates are widely perceived to have a mean-reverting behavior, the slope in interest rate or the expected change in the short rate would reflect the trend of short rate in a near future, and might become one of the decision bases for money managers to take certain positions Thus, we are interested in its significance in default prediction conditioning on EDF The slope is defined as: Slopet ,10 y −3 m = rt ,10 y − rt ,3m (2) Where Slopet, 10y-3m is the difference in 10-year and 3-month CMT yield averaged for each quarter Slopet, 10y-2y is defined similarly The expected change in short rate is defined as: eΔrt ,3m = f t ,3 m , m − rt ,3m (3) Where f t , m , m was forward implied 3-month rate in month It could be calculated from the following equality: (1 + rt , m ) = (1 + rt ,3m ) (1 + f t ,3m , m ) (4) Unexpected shocks in the short rate are included in the study as well This variable is of particular importance since it allows us to analyze the conditional relationship between interest rates and credit states Unexpected shocks to the short rate are defined as: Shock _ rt +3m ,3m = rt +3m ,3m − f t ,3m , m (5) presents the summary statistics of key interest rate variables Statistics are computed for the yields of constant maturity treasury for each quarter from 1982Q1 to 2008Q2 There are a total of 105 observations Figure and Figure present the histograms of expected quarterly changes in short rate ( eΔrt ,3 m ) and shocks in short rate ( Shock _ rt + m ,3m ), respectively TABLE Min Max Mean STD Δr3m -0.0552 0.0102 -0.0012 0.0082 Summary of Statistics of the Key Interest Rate Variables Δr2y -0.0328 0.0164 -0.0011 0.0079 Δr10y -0.0264 0.0134 -0.0010 0.0064 Slope10y-3m -0.0064 0.0396 0.0170 0.0118 Slope10y-2y -0.0044 0.0257 0.0085 0.0077 THE RELATIONSHIP BETWEEN DEFAULT RISK AND INTEREST RATES: AN EMPIRICAL STUDY eΔr3m -0.0034 0.0241 0.0047 0.0044 Shock_r3m -0.0709 0.0026 -0.0059 0.0091 0.12 0.1 Prob 0.08 0.06 0.04 0.02 1982Q3 -0.0050 0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300 Expected Change in Short Rate FIGURE Historical Distribution of eΔrt ,3 m 0.14 0.12 Prob 0.1 0.08 0.06 0.04 0.02 1982Q3, Q4 -0.0800 -0.0700 -0.0600 -0.0500 -0.0400 -0.0300 -0.0200 -0.0100 0.0000 0.0100 Shock in Short Rate FIGURE Historical Distribution of Shock _ rt + m ,3m The Augmented Dickey-Fuller (ADF) test is performed to determine whether the time series are stationary At a quarterly sample frequency, there are a total of 106 observations Table shows that at the 95% confidence level, all time series except r3mon are stationary TABLE P-value 10 r3m 0.4971 Stationary Test at Quarterly Sample Frequency Δr3m 0.0001* Slope10y-3m 0.0075* ΔSlope 10y-3m 0.0001* eΔr3m 0.002* Shock_r3m 0.0002* The ADF test is performed at the order of The corresponding regression equation was: ∇Yt = δYt −1 + θ1∇Yt −1 + θ ∇Yt − + ε t (6) t = δˆ SE (δˆ ) (7) The t-stat is: The null hypothesis is: the time series had a unit root The alternative hypothesis is: the time series is stationary A p-value of less than 0.05 means that at the 95% confidence level, the time series are stationary; otherwise, they are non-stationary THE RELATIONSHIP BETWEEN INTEREST RATES AND AGGREGATE DEFAULT RATES This section describes the relationship between interest rates and aggregate default rates, including contemporaneous correlations, and correlations over time 4.1 Contemporaneous Correlations Changes in interest rates can impact a firm’s cash flows through future income (e.g., consider the impact of Federal Reserve monetary policy) and default point (e.g., consider the ability of a firm to refinance its obligations at lower rates), and will therefore affect its default probability As described earlier, there are mixed views as to what the contemporaneous correlations is between credit spreads and changes in interest rates, or credit spread and level of interest rates Focusing first on aggregate default rates, the contemporaneous correlations of the paired variables Corr(Δr, DR), Corr(Slope, DR), and Corr(ΔSlope, DR) are presented in Table The null hypothesis is that correlation equals zero The alternative hypothesis is that correlation is not equal to zero P-value less than 0.05 represents significant correlations at the 95% confidence level It is evident that there were negative correlations between changes in interest rates and default rates, with the correlations between changes in short rates and default rates being significantly negative When interest rates decrease, default rates increase at the same timeAn intuitive explanation is that when default rate increases, the Fed decreases the short rate to prevent more defaults; the economy is essentially in a downturn TABLE Contemporaneous Correlations DR with Δr3m Δr2y Δr10y Corr Coeff P-value -0.2317 0.0169* -0.1318 0.1781 -0.0368 0.7082 Slope10y3m 0.0983 0.3159 Slope10y2y 0.25568 0.0082* ΔSlope10y3m 0.2367 0.0146* ΔSlope10y2y 0.2331 0.0162* These results are generally consistent with findings by a number of other authors In terms of the correlations between default rates and changes in interest rates, they are consistent with Longstaff and Schwartz (1995), Duffee (1998), Collin-Dufresne et al (2001), and Lin and Curtillet (2007) We also find positive correlations between slopes and default rates, and significant positive correlations between changes in slopes and default rates These results are in agreement with the correlations between changes in interest rates and default rates As can been seen in Table 4, the correlation default rates and changes in interest rates is more negative at shorter interest rates terms Decreases in shorter term interest rates would imply increases in slope; hence, the negative correlations between changes in (short term) interest rates with default rates will translate to positive correlations between interest rate slopes and default rates In addition, these results are generally consistent with findings in Collin-Dufresne et THE RELATIONSHIP BETWEEN DEFAULT RISK AND INTEREST RATES: AN EMPIRICAL STUDY 11 al (2001) and Joutz et al (1997) for shorter maturities bonds, but they contradict the findings in Duffee (1998) In particular, Collin-Dufresne et al (2001) find insignificant positive correlations between changes in corporate bond yields (maturity less than or equal to years) and changes in treasury yield slope10y-2y ; and insignificant negative correlations between changes in corporate bond yields (maturity longer than years) and changes in treasury yield slope10y-2y at monthly frequency Because our default records included missing payments, it was likely that our results represented a mixed result which is dominated by defaults on short-term liabilities 4.2 Corr(Δr, DR) Over Time We examine the correlations Corr(Δr, DR) over time using quarterly data The total time from 1982–2008 is divided into seven segments, partly to match the Federal Reserve Chairman’s terms This division reflects our anticipation that the Federal Reserve’s money supply policy might play a role in determining the correlation Between 1982 and 2008, changes in interest rates and default rates mainly exhibit negative correlations, except during 96Q2 to 00Q2 when Greenspan was chairman Table displays the Pearson’s correlation coefficients and their p-values for the paired variable in each time region The null hypothesis is that the correlation is equal to zero The alternative hypothesis is that correlation is not equal to zero A p-value of less than 0.05 represents significant correlations at the 95% confidence level TABLE Corr(Δr, DR) Over Time Regions I II III IV V VI VII Time Duration 82Q1-87Q3 87Q4-91Q4 92Q1-96Q1 96Q2-00Q2 00Q3-04Q3 04Q4-06Q1 06Q2-08Q2 Volker Greenspan Greenspan Greenspan Greenspan Greenspan Bernanke -0.3712 0.0812 -0.3572 0.1593 -0.2269 0.3812 0.1577 0.5454 -0.7303* 0.0009 -0.3493 0.4974 -0.6167 0.0769 Fed Chairman Corr(Δr, DR) P-value CONDITIONAL CORRELATION ANALYSIS USING FIRM-LEVEL DATA While the correlations between interest rates and default rates are important for the understanding of the macroeconomic processes, understanding the correlations on a firm-level provides more definitive and comprehensive guidance on the assessment of single-obligor default risk and for the bottom-up analysis of risk and return of credit portfolios To get a better understanding of these dynamics we analyze how expected and unexpected shocks on interest rates and firm-level EDF credit measures relate to the firm level defaults 5.1 Predictive Logistic Regression Model at Firm Level The EDF measure is a single-name default probability based on the proprietary Vasicek-Kealhofer (VK) model of Moody’s KMV The VK model is developed under a contingent-claim theory, or a Merton-type, framework The model assumes the firm's equity is a perpetual option with the default point acting as the absorbing barrier for the firm's asset value When the asset value hits the default point, the firm is assumed to default The model calculated a distance-todefault (DD) value for each firm at the end of each month The DD value is then mapped to the EDF value using an empirical formula developed from the default database To understand if interest rates can help predict default, above and beyond the information in the EDF credit measure, we use a logistic regression to fit individual defaults on both EDF and expected shocks to interest rates The predictive regression included forward-looking interest rate variables available at the same time period as EDF was computed The classical logistic regression model was specified as following: For additional details on the EDF model and its performance, see Crosbie and Bohn (2004), Dwyer and Korablev (2007), and Dwyer and Qu (2007) 12 log Here, Pr [ Di (t ) = ] − Pr [ Di (t ) = ] = β + β1 * log EDFi (t − 1) + β * eΔr(t − 1) + ε (i ,t ) − EDFi (t − 1) (8) Di (t ) is the default indicator and equals if the firm i defaults during sample period ending at t, and [ ] otherwise; Pr Di (t ) = is the probability that firm i defaults during sample period ending at t In other words, t -1 corresponds to the time point that is one sample period before t In the case of quarterly sample frequency, t -1 is one quarter before t Notice that EDFi (t − 1) is the default probability for the sample period t-1 to t, measured at t-1 It is transformed by the logit function, so we expect β1 to be close to The transformation is conducted so that the order of the variable and the error is comparable across firms in the sample Similarly, eΔr (t − 1) is the expected change in the yield of 3-month Treasury bill, calculated at t-1 Since defaults are correlated cross-sectionally, the regression residual errors ε (i ,t ) are not independent To account for this coroelation, a simulation approach described in Appendix A is used when computing the standard error for the β estimate Table presents the regression results using the simulated standard error to test the significant level of β estimate There are 105 quarters (82Q1-08Q2) and total 185564 observations in the regression Asterisks (*) represents significance at the 95% confidence level When the regression is performed at an annual sample frequency, we obtain similar results For comparison, the Slope(t − 1) variable is used in a separate predictive regression Predictive regression results with Slope (t − 1) term are consistent with the results using the eΔr (t − 1) term TABLE Variables β0 β1 β2 5.2 Predictive Regression Results β Estimates (SE) Chi-squared Quarterly Frequency 1.1863 85.6832 (0.1282)* 1.3655 1330.5864 (0.0374)* -20.1898 3.3441 (33.7161)1 P-value