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

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

AUTHORS

Andrew Kaplin

Amnon Levy

Shisheng Qu

Danni Wang

Yashan Wang

Jing Zhang

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:

1 866 321 MKMV (6568) or 415 874 6000

EUROPE, THE MIDDLE EAST, AFRICA, AND INDIA:

44 20 7280 8300

ASIA-PACIFIC:

852 3551 3000

JAPAN:

81 3 5408 4250

TABLE OF CONTENTS

1 INTRODUCTION 5

2 LITERATURE REVIEW 6

3 DATA AND VARIABLE DESCRIPTION 7

4 THE RELATIONSHIP BETWEEN INTEREST RATES AND AGGREGATE DEFAULT RATES 11

4.1 Contemporaneous Correlations 11

4.2 Corr(Δr, DR) Over Time 12

5 CONDITIONAL CORRELATION ANALYSIS USING FIRM-LEVEL DATA 12

5.1 Predictive Logistic Regression Model at Firm Level 12

5.2 Contemporaneous Logistic Regression Model at Firm Level 13

6 CONCLUSION 14

APPENDIX A SIMULATION PROCEDURE FOR CALCULATING SE( β ˆ2) 15

1 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.1 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

1

See Longstaff and Schwartz (1995), Duffee (1998), and Collin-Dufresne et al (2001)

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 2 provides a review of the existing literature describing the relationship between credit and interest rate risk

• Section 3 discusses the data and variable specification

• Section 4 describes historical correlation findings using aggregated data

• Section 5 presents the results from regression specifications using granular data

• Section 6 provides concluding remarks

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 2 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 longer-maturity 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 non-callable bonds rated from AA to BBB and maturities ranging from intermediate to long-term They find the relation

6

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 co-integrated 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

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

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 1 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 1 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

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

1982 1984 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008

Time

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Default Rate Change in Short Rate Short Rate

NBER Contraction Periods

FIGURE 1 Realized Quarterly Default Rates: 1982Q1–2008Q2 Table 1 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 1 Sample Statistics for EDF and Default Rates

EDF Default

rates

Mean 1.68% 1.43%

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:

(1)

m m t m t m

r,3 = ,3 − −3 ,3 Δ

8

,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:

(2)

m t y t m y

Slope,10 −3 = ,10 − ,3

Where Slopet, 10y-3m is the difference in 10-year and 3-month CMT yield averaged for each quarter Slopet, 10y-2yis defined similarly The expected change in short rate is defined as:

(3)

m t m m t m

r

e Δ ,3 = ,3 ,6 − ,3

Where ft,3m,6mwas forward implied 3-month rate in 3 month It could be calculated from the following equality:

4 6 , 3 , 4 3 , 2

6

1

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:

(5)

m m t m m t m m

r Shock _ +3 ,3 = +3 ,3 − ,3 ,6

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 2 and Figure 3 present the histograms of expected quarterly changes in short rate (e r Δt,3m) and shocks in short rate ( ), respectively

3 ,3 _ t m m

Shock r+

TABLE 2 Summary of Statistics of the Key Interest Rate Variables

Δr 3m Δr 2y Δr 10y Slope 10y-3m Slope 10y-2y e Δr 3m Shock_r 3m

Min -0.0552 -0.0328 -0.0264 -0.0064 -0.0044 -0.0034 -0.0709 Max 0.0102 0.0164 0.0134 0.0396 0.0257 0.0241 0.0026 Mean -0.0012 -0.0011 -0.0010 0.0170 0.0085 0.0047 -0.0059 STD 0.0082 0.0079 0.0064 0.0118 0.0077 0.0044 0.0091

0 0.02

0.04

0.06

0.08

0.1 0.12

-0.0050 0.0000 0.0050 0.0100 0.0150 0.0200 0.0250 0.0300

Expected Change in Short Rate

1982Q3

FIGURE 2 Historical Distribution of e r Δt,3m

0 0.02

0.04

0.06

0.08

0.1 0.12

0.14

-0.0800 -0.0700 -0.0600 -0.0500 -0.0400 -0.0300 -0.0200 -0.0100 0.0000 0.0100

Shock in Short Rate

1982Q3, Q4

FIGURE 3 Historical Distribution of Shock _ rt+3 ,3m m

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 3 shows that at the 95% confidence level, all time series except r3mon are stationary

TABLE 3 Stationary Test at Quarterly Sample Frequency

r 3m Δr 3m Slope 10y-3m ΔSlope 10y-3m e Δr 3m Shock_r 3m

10