Liquidity, Taxes and Yield Spreads between Tax-exempt and Taxable Bonds

Transcription

1 Liquidity, Taxes and Yield Spreads between Tax-exempt and Taxable Bonds Chunchi Wu Woongsun Yoo Abstract This paper proposes a dynamic pricing model for municipal bonds with the liquidity factor and time-varying risk premiums. We estimate the parameters of the model using the Kalman filter. We find that the estimate of the marginal investor s income tax rate from the generalized model is very close to the statutory corporate tax rate over the periods with different tax regimes. Ignoring the liquidity risk premium which is an important component of municipal yields, and failing to use the estimation method that efficiently captures time-varying features of risk premiums result in biased estimation of marginal investor s implicit tax rates. JEL Classification: G12, G32 Keywords: Kalman filter, liquidity, marginal tax rate, municipal bond yields School of Management, The State University of New York at Buffalo. chunchiw@buffalo.edu. Tel: (716) College of Business and Management, Saginaw Valley State University. wyoo@svsu.edu. Tel: (989)

2 1 Introduction How taxes affect asset returns is an important question for both researchers and practitioners. A large amount of literature has been devoted to the study of this subject, yet the effect of taxes is not fully understood. In particular, whether taxes are important for the pricing of bonds is an issue of ongoing debate and empirical evidence is mixed. While some studies have documented evidence that taxes affect the pricing of bonds (e.g., Green, 1993; Liu and Wu, 2004; Wang, Wu, and Zhang, 2008; Liu, Shi, Wang, and Wu, 2007; Longstaff, 2011; Ang, Bhansali, and Xing, 2010, 2014), others have either shown that taxes are irrelevant or simply ignored the tax effect in bond pricing (e.g., Green and Ødegaard, 1997; Duffie and Singleton, 1999; Longstaff, Mithal, and Neis, 2005). Unlike corporate or Treasury bond yields, municipal bond yields are tax-exempt. The effect of taxes can thus be inferred from the yields of municipal bonds relative to taxable bonds. Using a random intercept model to control for the risk of interest rates, Trzcinka (1982) estimates the marginal income tax rate from the relation between tax-exempt and taxable rates. He finds that the estimated tax rate is highly significant and close to the rate predicted by Miller (1977). Yawitz, Maloney, and Ederington (1985) derive a model for the relation between municipal and Treasury bond yields by accounting for the default probability of municipal bonds. Their model permits a joint estimation of default probability and the marginal investor s tax rate from municipal yield data. Using the municipal swap market data, Longstaff (2011) finds that taxes are an important asset pricing factor. In this paper, we propose a new model for estimating the marginal investor s income tax rate from the relation between tax-exempt and taxable bond yields. The model has two desirable features. First, it incorporates a liquidity factor in the pricing model of municipal bonds. The inclusion of the liquidity factor in the pricing model is strongly motivated by the recent evidence that liquidity is an important asset pricing factor for a wide range of financial assets (see, for example, Pástor and Stambaugh, 2003; Acharya and Pedersen, 2005; Lin, Wang and Wu, 2011; de Jong and Driessen, 2012; Dick-Nielsen, Feldhütter and Lando, 2012; Friewald, Jankowitsch, and Subrahmanyam, 2012; Acharya, Amihud and Bharath, 2013). Second, it permits time-varying risk and tax premiums. As both risk factors and systematic risks are time-varying, this dynamic pricing model can be suitably estimated by the Kalman filter. In Kalman filter estimation, systematic risks or loadings of risk 1

3 factors are formulated as the Markov process and estimated by the maximum likelihood. Anecdotal evidence suggests that taxes play a role in the pricing of tax-exempt bonds relative to taxable bonds. First, yields on corporate bonds are typically larger than those on municipal bonds of an equivalent rating and maturity. Taxes are a possible candidate to explain this yield difference as interest income is subject to taxes. Second, individuals hold a significant portion of both municipal and corporate bonds and are likely to be the investor at the margin. Individual ownership for both tax-exempt and taxable bonds has increased substantially over time. In 2014, holdings of corporate and municipal bonds by individual investors account for about 40% of the total outstanding amount, while holdings by insurance companies account for only 20% and holdings by banks and pension funds account for less than 10%. 1 The figures are even more astonishing in the tax-exempt bond market. Figure 1 shows that individuals surpassed institutional investors to become the dominant player in the municipal market after 1980, and their average market share has consistently been above 75% since early 2000s. This makes it very difficult to argue that tax-exempt pension funds are the marginal investor and taxes are irrelevant in the bond pricing as assumed by many studies on the term structure of defaultable bonds (see Duffie and Singleton, 1999; Longstaff et al., 2005). Consistent with this view, recent studies by Wang, Wu, and Zhang (2008) and Ang, Bhansali, and Xing (2014) have found that the taxes are an important determinant of municipal bond yields using the transaction data of Municipal Securities Rulemaking Board (MSRB). This paper examines the importance of liquidity risk and taxes in the pricing of municipal bonds using a parsimonious model similar to Trzcinka (1982). The model was initially used by Trzcinka (1982) to test the Miller (1977) hypothesis that the personal tax rate on bond interest equals the corporate tax rate at the margin. We generalize this model to incorporate the liquidity risk premium. This generalization is motivated by the recent literature that liquidity is an important pricing factor. The municipal bond market is highly illiquid and this has always been a great concern to investors. Intuitively, municipal bond yields should contain a significant liquidity premium to compensate investors. However, the liquidity premium has been ignored by past studies that attempt to retrieve the implicit marginal investor s tax rate from the relation between municipal and corporate bond 1 Source: Tables L.212 Municipal securities and L.213 Corporate and Foreign Bonds, Z.1 Financial Accounts of the United States released quarterly by the Board of Governors of the Federal Reserve System 2

4 yields. If liquidity is an important pricing factor of municipal bonds, failing to account for the liquidity premium will lead to biased estimates of the marginal income tax rate. Several studies have investigated whether liquidity is priced in the municipal bond market. Wang et al. (2008) and Ang et al. (2014) find that municipal bond yields contain three important components - tax, default risk and liquidity risk premiums. The empirical evidence documented in this paper that taxes and the liquidity factor are important determinants of municipal bond yields is consistent with their findings. However, the objective of this paper is quite different. While past studies are interested in the components of municipal bond risk premium, this paper focuses on the relation between tax-exempt and taxable bond yields and the issue of whether the marginal tax rate in the bond market equals the corporate income tax rate as suggested by Miller (1977). Our study suggests that investors are concerned with both taxes and liquidity risk when choosing between corporate bonds and municipal bonds. Investors require a liquidity premium and they are compensated for investing in illiquid municipal bonds. We present evidence from the relative yields of corporate and municipal bonds that liquidity is an important factor in municipal bond pricing, which is consistent with the finding of the existing literature (Wang et al., 2008; Green, Li, and Schürhoff, 2010; Ang et al., 2014), and show that ignoring the liquidity factor leads biased estimates of marginal income tax rate. The marginal income tax rates are estimated from a dynamic asset pricing model with timevarying risks. Results show that controlling for time-variation in risks and the effects of liquidity risk and taxes, the implicit tax rate borne out by the relative yield data is strongly consistent with the statutory corporate tax rate. The sample period in this paper spans over two major tax regimes, one is the high tax regime in the 1970 s when the highest corporate tax rate was 48% and the other is the low tax regime from 1994 to 2014 when the highest corporate tax rate was 35%. The implicit marginal tax rates retrieved from the relative yield data are remarkably close to 48% in the high-tax regime and to 35% in the low-tax regime. Results show that the estimates of the marginal investor s income tax rate are closely in line with the statutory corporate income tax rates under different regimes. This finding suggests that taxes are a very important factor for bond pricing and in determination of capital structure. Results also show that liquidity is an important pricing factor for municipal 3

5 bonds and it is important to control for the effect of the liquidity risk factor on relative bond yields when estimating the marginal tax rate. The magnitude of the estimated liquidity risk premium is of both economic and statistical significance. Over most of the sample period, the two-factor Kalmanfiltering model with tax rates and the liquidity factor explains the behavior of municipal bond yields relative to corporate bond yields very well. Parameter estimates are generally quite stable and highly significant. However, we also find that the relative yield relation between municipals and corporates becomes much more complicated in the post-crisis period from 2009 to This phenomenon is due to high uncertainty in the municipal bond market associated with the debacle of the monoline insurance industry which insures a large proportion of municipal bonds and the poor fiscal health of state and local governments. This paper makes several contributions to the literature. First, it provides strong evidence that liquidity is an important pricing factor of municipal bonds. It shows that ignoring the effect of liquidity risk leads to serious bias in the estimate of the marginal investor s income tax rate. This finding is quite intuitive. If liquidity is an important factor but its effect is ignored in empirical estimation, then the tax parameter in the model will have to adjust to accommodate this omitted variable effect. As a consequence, the estimated marginal tax rate will be biased due to the misspecification in the empirical model. Second, the dynamic bond pricing model that permits time-varying risk premiums produces consistent and unbiased estimates of parameters. The Kalman filter performs much better than the least squares regression and the Cooley-Prescott (1976) method. Using the Cooley-Prescott method, we replicate Trzcinka s (1982) results over the same sample period and confirm that the model with time-varying risk premium fits the yield data better. However, the Kalman filter method performs much better than the Cooley-Prescott method in capturing the effect of time-varying risks. Third, our model helps to explain the anomalous behavior of municipal bond yields relative to corporates after the subprime crisis and sheds more light on the recent structural change in the relationship between municipal and corporate bonds. Yields of municipal bonds are often higher than those of corporate bonds after the crisis, and these anomalies can be reasonably interpreted by our model which accounts for both credit and liquidity risk factors ignored by previous studies. 4

6 To capture the effect of structural shift in the municipal bond market after the subprime crisis, we introduce a credit risk factor in the state CDS market, a state deficit variable, and a bond insurer s probability of default as an additional control. Over this subperiod, while the contribution of liquidity risk to municipal bond yields appears to decline somewhat relative to other factors, the estimate of the marginal tax rate remains quite robust. The remainder of this paper is organized as follows. Section 2 discusses the methodology and models used for empirical tests. Section 3 describes the data, and Section 4 presents the empirical results. Section 5 provides the test of the alternative hypothesis as a robustness check. Finally, Section 6 summarizes major findings and concludes. 2 Methodology 2.1 The Cooley-Prescott (1976) estimation procedure Cooley and Prescott (1976) propose a model with time-varying parameters and develop the procedure to estimate parameters in this model. A number of studies have applied Cooley and Prescott s (1976) method to address economic and financial issues (e.g., Laumas, 1978; Mullineaux, 1980; Pesaran and Timmermann, 2002; among others). Trzcinka (1982) employs this estimation procedure to adjust the difference in the risk premiums between tax-exempt municipal bonds and taxable corporate bonds. He assumes that the differential risk premium changes over time, while setting the coefficient on corporate bond yields in the regression of municipal yields on corporate bond yields constant over time. In the following, we outline the Cooley-Prescott (1976) estimation procedure for his model. For a given rating and maturity R mt = R t β t (1) where R t = [ ] 1 R ct, βt = [ λ t β τ], and t = 1, 2,, T. Rmt is the municipal bond yield and R ct is the corporate bond yield at month t. λ t is the time-varying risk premium at month t and β τ is the coefficient on corporate bond yield which in theory is equal to one minus marginal tax rate. Suppose 5

7 that β t obeys the following process β t = β P t + U t (2) β P t = β P t 1 + V t (3) where βt P = [ λ P t β τ], β P t 1 = [ λ P t 1 β τ], Ut = [ u t 0 ] and Vt = [ v t 0 ]. The Ut and V t are identically and independently distributed normal variables with mean of zero and variance-covariance matrix given by cov(u t ) = (1 γ)σ 2 Σ U (4) cov(v t ) = γσ 2 Σ V (5) [ ] 1 0 where Σ U = Σ V = and 0 γ It follows from (2) and (3) that and (1) can be rewritten as where β = βt P +1 and µ T +1 t = R t U t R t β t = β P T +1 s=t+1 T +1 s=t+1 V s + U t (6) R mt = R t β + µ t (7) V s. It can be also shown that cov(µ) = σ 2[ (1 γ)p + γq ] σ 2 Ω(γ) (8) where P is a T T diagonal matrix with p ii = R i Σ U R i and Q is a T T matrix with q ij = min(t i + 1, T j + 1)R i Σ U R j. Finally, the full model can be written as R m = Rβ + µ (9) 2 When γ = 0, the model becomes the classical regression model. Otherwise, it becomes similar to the Kalman filter model. When γ = 1, it is the model with the time-varying intercept term that follows the random walk. 6

8 where R m is T 1 column vector of R mt, R is T 2 matrix, and β is 2 1 column vector which is [ λ P T +1 β τ]. From (8), the distribution of Rm is R m N ( Rβ, σ 2 Ω(γ) ) Based on the above distribution, the log likelihood function can be written as L ( R m ; β, σ 2, γ, R ) = T 2 ln 2π T 2 ln σ2 1 2 ln Ω(γ) 1 2σ 2 ( Rm Rβ ) Ω(γ) 1 ( R m Rβ ) (10) By maximizing the log likelihood function with respect to β and σ 2, one can obtain the estimators conditional on γ as follows B(γ) = ( R Ω(γ) 1 R ) 1 R Ω(γ) 1 R m (11) s 2 (γ) = 1 T ( Rm RB(γ) ) Ω(γ) 1 ( R m RB(γ) ) (12) We construct the concentrated likelihood function by substituting (11) and (12) in (10): ( L c Rm ; γ ) = T 2 (ln 2π + 1) T 2 ln s2 (γ) 1 ln Ω(γ) (13) 2 By maximizing (13) subject to 0 γ 1, we can obtain ˆγ. Further, by plugging ˆγ into equations (11) and (12), we can get estimates of β and σ 2. Though the Cooley-Prescott (1976) method accommodates the presence of stochastic parameters, there are several concerns. First, the maximum likelihood estimates of γ and σ 2 are not consistent (see Pagan, 1980; Swamy and Tinsley, 1980; Tsurumi and Shiba, 1981). Second, this method is not designed so much for fitting time-varying coefficients as for testing the stability of parameters (see Roll, 1972; Laumas and Mehra, 1976; Cooley and Decanio, 1977). Third, this model assumes a strong variance-covariance structure which depends on γ. γ is the key parameter in variancecovariance structure and thus a critical element to draw inferences. Unless γ has sound asymptotic properties and consistency, the estimation may result in unreliable inferences. 7

9 2.2 Kalman filter estimation The Kalman filter method provides a more general framework for modeling time-varying parameters. By obtaining the conditional distribution of time-varying coefficients recursively and using the updating procedure where the prior mean is adjusted by the prediction error given the model and data, the Kalman filter method generates efficient and consistent estimates. Using this method to estimate the marginal tax rate from the relation between municipal and corporate yields, one can express the time-varying parameter model of municipal bond yields as the Gaussian linear state-space model. The measurement equation is expressed as R mt = R t β t + u t (14) where R t = [ ] 1 Ft 1 Ft 2 Ft k 1 R ct, βt = [ λ t βt 1 βt 2 βt k 1 R mt is the municipal bond yield, R ct is the corporate bond yield, and Ft 1, Ft 2,, Ft k 1 factors at month t. λ t is the time-varying risk premium at month t, β 1 t, β 2 t,, β k 1 t coefficients of k-1 risk factors, β τ t β τ t ], and t = 1, 2,, T. are k-1 risk are time-varying is the coefficient on corporate bond yield and is theoretically one minus marginal tax rate, and u t N ( 0, σ 2 u). The dynamic feature of βt is determined by the transition equation as β t = β t 1 + v t (15) and v t N ( 0, Q ). The Kalman filter is a recursive procedure for drawing an inference about β t. Given the data R mt = [ R m1 R m2 R mt ], an inference about βt can be drawn via Bayes theorem P ( β t R mt ) P ( Rmt β t, R mt 1 ) P ( βt R mt 1 ) (16) At t-1, the information on β t 1 is given by β t 1 R mt 1 N ( β t 1 t 1, Σ t 1 t 1 ) (17) 8

10 Prior to observing R mt, the best inference for β t is given by (15) and the conditional distribution in (17). Thus, the conditional distribution of β t given R mt 1 is β t R mt 1 N ( β t 1 t 1, Σ t 1 t 1 + Q ) (18) which represents the prior distribution P ( β t R mt 1 ) in (16). To obtain the posterior distribution of β t after observing R mt which is the left-hand side of (16), one needs the density function P ( ) ( ) R mt β t, R mt 1. To obtain the density function P Rmt β t, R mt 1, the error in predicting Rmt at t-1 can be defined as e t = R mt ˆR mt = R mt R t β t 1 t 1 (19) From (14), equation (19) becomes e t = R t ( βt β t 1 t 1 ) + ut (20) and the distribution of e t is given by ( ) ) e t β t, R mt 1 N (R t βt β t 1 t 1, σ 2u (21) Substituting (18) and (21) into (16) gives the posterior distribution of β t ( ( ) β t e t, R mt 1 N β t 1 t 1 + Σ t t 1 R t σ 2 u + R t Σ t t 1 R 1et t, ( ) ) (22) Σ t t 1 Σ t t 1 R t σ 2 u + R t Σ t t 1 R 1Rt t Σ t t 1 The mean and variance in (22) can be written as ( ) β t t = β t t 1 + Σ t t 1 R t σ 2 u + R t Σ t t 1 R 1 ( ) t Rmt R t β t t 1 (23) Σ t t = Σ t t 1 + Σ t t 1 R t ( σ 2 u + R t Σ t t 1 R t ) 1Rt Σ t t 1 (24) Through the recursive procedure, where the prior mean is corrected by the prediction error and using 9

11 (15), (23), and (24), one can estimate the time-varying coefficients for each time t. 3 Data 3.1 Bond yield average data Municipal and corporate bond yield data are from Salomon Brothers Analytical Record of Yields and Yield Spreads for the period from January 1970 to September This is the same period used by Trzcinka (1982). We choose the same sample period and data as used by Trzcinka (1982) to compare his marginal tax rate estimates based on the Cooley-Prescott (1976) method with those we obtain after controlling for the effect of liquidity and using the Kalman filter method. Salomon Brothers Analytical Record of Yields and Yield Spreads provides monthly yield averages for new issues by rating and maturity at the beginning of each month. Ratings are prime (Aaa), good (Aa to high A) and medium (A to high Baa) for municipal bonds and Aaa, Aa and A for corporate bonds. Maturities are 30-, 20-, and 10-year for municipal bonds and 10-year and over 10 years for corporate bonds. Corporate bond yield averages are divided further into utility bond yield averages and industrial bond yield averages by corporate sector. For the period from June 1994 to August 2014, both municipal and corporate yield data are collected from the Bloomberg system. Bloomberg provides daily municipal bond yield averages by rating and maturity. We collect 12 different municipal bond yield averages which are combinations of four different ratings (AAA, AA, A and BBB) with three different maturities (30-, 20- and 10-year) and use monthly averages for the analysis. Among several different corporate bond yield averages provided by the Bloomberg system, 3 we select Bank of America-Merrill Lynch corporate bond yield series because these series are categorized by both rating and maturity and are the longest series of data starting from June 1994 is the earliest month when both municipal and corporate bond 3 S&P corporate yield average, Citi industrial & utility bond yield average, and Barclays US aggregate corporate yield average are alternative corporate bond yield series available from Bloomberg. 4 To our knowledge, Moody s yield averages are the longest series of bond yield data available from Moody s provides yield averages for both municipal and corporate bonds by four ratings (i.e., Aaa, Aa, A and Baa) and two maturities (20-year and 10-year for municipal bonds, and long-term and intermediate for corporate bonds). Yet, 10-year municipal bond yield series are available from August 1977 and intermediate corporate bond yield series are available from June

12 yield averages are available from the Bank of America-Merrill Lynch yield series. We collect eight different corporate bond yield indexes which are a combination of four different ratings (AAA, AA, A and BBB) with two long-term maturities (10 to 15-years and over 15 years). Figure 2 plots the time-series of municipal and corporate bond yields. In both panels A and B, we select two maturities for municipal bond yields, 30-year and 10-year, and plot these series by rating. The sample period is from January 1970 to September 1979 in panel A and from June 1994 to August 2014 in panel B. To see the difference in yields between tax-exempt municipal bonds and taxable bonds, we plot 30-year municipal yields with long utility yields in panel A and with over 15 years corporate yields in panel B. 10-year municipal yields are paired with utility yields of the same maturity in panel A while with 10 to 15-years corporate yields in panel B. Taxable bond yields are always higher than tax-exempt municipal yields until 2007, while the gap between taxable yields and tax-exempt yields become smaller after For some months after 2007, municipal bond yields are higher than the yields of corporate bonds with the same rating and similar maturity. Table 1 provides the summary statistics for municipals and corporates by rating and maturity, and each panel in Table 1 represents different periods as indicated at the top of each panel. In every panel and for both municipal and corporate bonds, yields of bonds with a better rating are generally lower than those with a lower rating. In addition, yields of tax-exempt municipals are in general lower than those of taxable corporates. For instance, in panel A, municipal yields are lower than utility/industrial yields, when 30-year municipals are compared with long utilities/industrials in terms of maturity, and prime with Aaa, good with Aa and medium with A in terms of rating. The average yields across different ratings for 30-year municipals are smaller than those for long utilities and industrials by 2.68 and 2.32 percentage points, respectively. The same pattern applies to panel B, while the gap between municipals and corporates becomes smaller. The average yields across all ratings for 30-year municipals are smaller than those for over 15-years corporates by 1.82 percentages. This pattern however does not hold for A and BBB rating municipals and corporates in panel C, which reports the summary statistics for the after-crisis period. Yields of A and BBB 30- years municipals are greater than those of A and BBB over 15-years corporates in panel C. Another pattern is that both mean and standard deviation of municipal bonds yields are more stable relative 11

13 to those of corporate bonds yields. While municipal bond yields become smaller by only about 1 percentage, corporate bond yields become smaller by about 2 percentages from the 1970 s to the period from June 1994 to July The standard deviations of corporate yields are greater than those of municipal yields except for BBB municipals and corporates in the after-crisis period reported in panel C. Table 2 shows the ratios of the yields of tax-exempt municipals to taxable corporates paired by rating and maturity. These ratios should be equal to one minus marginal tax rate as predicted by Miller (1977), if tax-exempt municipals and taxable corporates are identical in every dimension except for tax status. If this is the case, ratios from the period in panel A should be 0.52 which is equal to one minus the corporate tax rates of 48%, while those from the period in panels B and C should be 0.65 which is equal to one minus the corporate tax rates of 35%. Though the ratios in panel A are smaller than those in other panels, neither of them is close to the ratios predicted by Miller (1977). The ratios are not stable across maturities and increase as maturity gets longer for all three subsample periods. The time-series of ratios are plotted in Figure 3. The time-series in the top of panel A is a replication of Figure 1 in Trzcinka (1982) for the ratios of prime municipals to Aaa utilities. Two other time-series plots in panel A are for the ratios of good to Aa and of medium to A. As shown in all three figures in panel A of Figure 3, the ratios become lower and less volatile after 1972 due to the State and Local Federal Assistance Act as Trzcinka (1982) documents. This change in both level and volatility of ratios provides evidence that equivalent ratings for municipal and corporate bonds do not imply the same risk premiums nor reflect changes in risk. The reason is that the effects of the State and Local Federal Assistance Act should not be observed in the figures, if risks of bonds are timely and precisely reflected in the rating of bonds. Evidence that the rating of bonds fails to capture their risks is observed in panel B as well. Before the start of the financial crisis in 2008, the ratios of municipals to corporates are stable relative to those after the crisis. The difference of the ratios for long-term bonds (i.e., the ratios of 30-year municipals to over 15-years corporates) and the ratios for intermediate-term bonds (i.e., the ratios of 10-year municipals to 10 to 15-years corporates) are quite small before the crisis, while the difference gets bigger after the crisis. Again, this pattern across all four different ratings after the crisis provides evidence that the rating 12

14 does not capture risk premium properly. Thus, to estimate marginal tax rates in a precise manner, adjusting the relative risk difference between municipal and corporate bonds is necessary even after matching municipals and corporates by both rating and maturity. 3.2 Market illiquidity measures Several illiquidity measures are used in this study to capture the effect of market illiquidity. In this section, we provide a brief explanation for each measure On-/off-the-run spread The on-/off-the-run spread is the difference between the 10-year constant-maturity Treasury yield obtained from the FRB and the 10-year generic Treasury yield collected from the Bloomberg system. We compute the on-/off-the-run spread for the period from August 1971 to December For this period, data available for measuring marketwide illiquidity is very limited. Among several available marketwide measures such as money market mutual funds flows and Hasbrouck s (2009) trading cost index, we select the on-/off-the-run spread as the marketwide illiquidity measure because on-/off-therun spreads better capture the future illiquidity condition in the bond market, while other measures do not closely reflect the illiquidity of the bond market. We take the monthly average of daily on- /off-the-run spreads as the monthly illiquidity measure. Panel A in Figure 4 plots the time-series of on-/off-the-run spreads Amihud (2002) and Pástor and Stambaugh (2003) measures We use corporate bond transaction data from both TRACE and the National Association of Insurance Commissioners (NAIC) transaction databases to construct marketwide illiquidity measures for the period from January 1994 to December In order to obtain longer and consistent measures of marketwide illiquidity, we employ TRACE and NAIC instead of using the Municipal Securities Rulemaking Board (MSRB) database which provides municipal bond transactions only from We construct the illiquidity measures of Amihud (2002) and Pástor and Stambaugh (2003). The Pástor-Stambaugh liquidity measure is converted to an illiquidity measure by adding a negative 13

15 sign. Panels B and C in Figure 4 plot the time-series of Amihud (2002) and Pástor-Stambaugh (2003) illiquidity measures, respectively. In both panels, the largest upward spikes occur around August 2008 when Lehman Brothers began to have troubles. The procedure for measuring these two illiquidity proxies is described in the Appendix A. 3.3 Municipal default proxies After the subprime crisis, a structural change occurs in the relationship between tax-exempt municipal and taxable corporate bond yields. To capture the change in the municipal bond market after the crisis which two-factor Kalman filter model cannot fully explain, we introduce a credit risk factor proxied by several variables such as US state CDS, US state deficit and the bond insurers probability of default. The following subsections provide how we construct each credit risk factor US state CDS We collect daily 5-year CDS spreads of 20 US states from the Bloomberg system. Out of all 50 US states, 20 states have CDS for their bonds traded and have sufficient observations for the long sample period. CDS spreads are daily midpoint spreads. We use the monthly average of daily CDS spreads. Panel A of Figure 5 plots the time-series of average state CDS spreads by state rating. CDS spreads of states with an AAA or AA rating are always smaller than those with an A rating. Except for a few months at the end of 2009, CDS spreads of states with an AAA rating are smaller than those with an AA rating US state deficits We collect revenue and expenditures of US state governments from the annual survey of state government finances provided by US Census Bureau to compute state deficit by year. State deficit is the difference between direct expenditure which is a state s expenditure for its own state and direct revenue which is a state s general revenue from its own sources. We scale each state s deficit by its GDP. US states GDP data are collected from the Bureau of Economic Analysis. Both state deficit and state GDP are annual. We interpolate the deficit scaled by GDP to fit monthly intervals of 14

16 time-series of bond yields. Panel B of Figure 5 plots the time-series of average state deficits by state rating. On average, the deficit of US states is about 2.5% of its GDP during the period from 2010 to The ratio of deficit to GDP is the highest at the end of 2010 and decreases after Bond insurer s probability of default Most bond insurers suffer substantial losses during the subprime mortgage crisis. The downfall of bond insurers due to their tremendous losses from subprime mortgage market results in increasing the counterparty risk in the municipal bond market and affects the pricing of municipal bonds. To capture the structural change of the counterparty risk related to bond insurers in the municipal bond market after the crisis, we construct the implied probability of default for bond insurers. We compute individual bond insurer i s implied probability of default at day t as follow P D it = 1 exp( λ it ) (25) where λ it = CDS it /(1 Recovery it ) is default intensity of bond insurer i at day t. Following Chung, Kao, Wu, and Yeh (2015), we employ 5-year CDS spreads and recovery rates of bond insurers collected from the Markit database for CDS it and Recovery it, respectively. We calculate the average of implied probability of default across eight municipal bond insurers, 5 and use monthly average of daily implied probability as a default risk factor of monthly bond yields for the period after the crisis. 4 Empirical results 4.1 The high tax regime period: August December Regression of municipal bond yields The ratio of tax-exempt municipal yields to taxable corporate yields does not account for the difference in the risk premium existing between these two types of bonds. The simple regression of 5 Eight municipal bond insurers are Assured Guaranty Ltd., Ambac Financial Group, Inc., Berkshire Hathaway Assurance Corp., CIFG Assurance North America Inc., Financial Guaranty Insurance Company, Municipal Bond Insurance Association, Radian Group Incorporated and XL-CAPASS Capital Assurance. 15

17 municipal yields on corporate yields is a way to address this issue where the intercept term captures the difference in the risk premium. Though the risk premium may vary over time, we run a simple OLS regression with a constant intercept term first to see the extent that this adjustment affects the estimation of the marginal tax rate. In addition, we add liquidity risk premium to capture the difference between municipal and corporate bond yields in this simple OLS model. The OLS regression results provide an indication whether liquidity risk is relevant. We later estimate the model with time-varying intercept by the Cooley-Prescott (1976) and the Kalman filter methods. Table 3 reports the results of time-series regressions of municipal yields on corporate bond yields. There are two corporate bond yields which belong to two separate sectors, utility and industrial. For each regression, municipal yields are paired with corporate bond yields of comparable ratings and maturities. For example, we regress prime 30-year municipal yields on Aaa long utility or industrial yields. The coefficients on corporate yields, β τ, are one minus marginal tax rate by theory. β τ shows a wide range of variations. It is too big or too small in some of regressions to produce stable and consistent estimates of marginal tax rate. For example, in the regression of medium 30-year municipals on A long industrials, β τ is 0.643, which implies marginal tax rate of It is about 12% lower than statutory tax rate. On the other hand, β τ is and thus the estimate of marginal tax rate from this β τ is in the regression of prime 10-year municipal bond yields on Aaa 10-year utility bond yields. For all OLS regressions, intercepts are positive and most of them are significant at the 1% level, which is consistent with the findings of Trzcinka (1982). This reflects the differences in rating scale between municipal and corporate bonds. Moreover, municipalities are more opaque and less willing to disclose their information to the public than corporates are. Trzcinka (1982) indicates that credit rating agencies tend to pay attention to reflecting default risk correctly to ratings, while often ignoring the fact that there are other risk factors which might affect municipal yields. Thus, municipals often show relatively high average yields compared to after-tax corporate yields of the same rating and maturity. One of potential risk factors that might be specific to municipal bonds and affect the relative yields between municipals and corporates is liquidity risk. The municipal bond market is relatively 16

18 illiquid compared to the US Treasury or corporate bond markets (see Downing and Zhang, 2004; Wang et al., 2008; Fabozzi and Mann, 2012), and a number of studies document that liquidity risk is an important factor that affects corporate bond returns (see Lin et al., 2011, 2014; Acharya et al., 2013). To account for the liquidity risk of municipals and its impact on the relative yields between municipals and corporates, we regress municipal yields on their comparable pairs of corporates and on- /off-the-run spreads from 10-year maturity Treasury notes, which is a proxy for marketwide illiquidity. Results of regressions with liquidity are reported in Table 3. β τ and β L represent the coefficients on corporate yields and on-/off-the-run spreads, respectively. Out of 18 β L s, 15 are significant at the 1% level, which indicates that liquidity is indeed a significant and important risk factor that should be accounted for in the relation between municipal and corporate yields. The average β L in panel A which reports results of the regressions of prime municipals to Aaa corporates is and those in panels B and C are and 1.791, respectively. The sensitivity of municipal yields to illiquidity factor increases as rating decreases, showing that bonds with high credit risk tend to have higher liquidity risk. The average β L from the regressions of 30-year municipals on long-term corporate bonds is and those from 20-year municipals on long corporates and 10-year municipals on 10- year corporates are and 1.723, respectively, indicating that bonds with the longest maturity tend to have higher liquidity risk. After the liquidity factor is incorporated in regressions, the coefficients on corporate yields, β τ, also change. β τ decreases in 16 out of all 18 regressions after liquidity factors are included as additional explanatory variables. The average of β τ decreases by about 0.06 from in the models without the liquidity factor to in those with the liquidity factor, which represents about a 13% increase on average in the estimates of the marginal tax rate (from to 0.529). Overall, liquidity risk plays an important role in the relative yields of municipals to corporates. Liquidity risk is statistically significant in the model and changes the coefficients on corporates yields which, in turn, affect the estimates of marginal tax rates. Results suggest that liquidity risk should be accounted for when estimating marginal tax rates from municipal and corporate yields. All regression models in this section do not allow for time-varying risk premiums. In the following, we address this issue using the Cooley-Prescott (1976) and the Kalman filter methods. 17

19 4.1.2 Time-varying coefficients models Estimation using Cooley-Prescott (1976) To account for the time-varying nature of risk premium, we set up the model with a time-varying intercept following Trzcinka (1982) as equation (1) in section 2.1. Given the system of coefficient dynamics and the variance-covariance structure in (2) to (5), we estimate parameters in the model by rating and maturity using the Cooley-Prescott (1976) procedure. Municipals are paired with corporates of comparable ratings and maturities in the same way as in least squares regressions above. The results are reported in Table 4, which is a replication of Table II in Trzcinka (1982). β τ is the coefficient on corporate yields. γ and σ are estimates of parameters in the variance-covariance of the disturbance term in the model. The results are very close to those in Table II in Trzcinka (1982). 6 The average β τ is which implies a marginal tax rate of On average, the Cooley-Prescott (1976) method β τ is higher than that of the OLS regression. The estimates of γ decrease, as maturity decreases, which means that variations in municipal yields are less explained by the random walk as maturity decreases. As Trzcinka (1982) indicates, this pattern seems to be reasonable because it implies that the risk premium represented by intercept term accounts for less variation in short maturities and becomes less significant in explaining municipal yields than in long maturities. This pattern of γ and the intercept term is similar to the change in liquidity risk loadings from the least squares regressions with the liquidity factor, suggesting that the risk premium captured by time-varying intercept may include the liquidity risk premium which is ignored in Trzcinka s (1982) model. Though average marginal tax rate in Trzcinka (1982), which is estimated by Cooley-Prescott (1976), is close to the highest corporate tax rate of 48% and favors the Miller hypothesis, the estimates from the model without accounting for liquidity risk factor may be biased and happen to be close to the corporate tax rate. Further, the coefficients of corporate yields as well as the risk premium captured by intercept term may change over time. These patterns can be best captured by the Kalman filter method which is a more general way of estimating the model with time-varying parameters. 6 Estimates in Table II in Trzcinka (1982) and those in Table 4 show minuscule differences possibly due to different search algorithms employed when equation (13) is maximized. We use the algorithm based on golden section search and parabolic interpolation. 18

20 Estimation using the Kalman filter We next set up the model in which all coefficients are time-varying and follow random walks for the Kalman filter estimation. The model is R mt = βt τ R ct + βt L L t + λ t + u t, where R mt is municipal bond yields, R ct is utility or industrial bond yields, L t is on-/off-the-run spreads, λ t is the timevarying intercept term and u t N (0, σu). 2 It is assumed that each time-varying coefficient follows independent random walks. Table 5 reports the mean and standard deviation of βt τ, βt L and λ t estimates which are time-varying coefficients on corporate bond yields, illiquidity factor and risk premium, respectively. The coefficients of corporate yields β τ t are higher than those estimated from both least squares and Cooley-Prescott (1976) methods. In the models without the liquidity factor, Kalman filter estimates for β τ t For example, the average of β τ t are consistently larger than those from other estimation methods. from the model of prime municipals against Aaa corporates across maturity estimated by Kalman filter is while those by least squares regressions and Cooley- Prescott (1976) method are and 0.562, respectively. This tendency is also found in the model of good and medium municipals as well. After including liquidity as an additional risk factor, Kalman filter estimates for β τ t continue to be larger than those of least squares regressions. For instance, the average of β τ t from the model of prime municipals against Aaa corporates estimated by Kalman filter is while that by least squares regressions is Adding the liquidity risk factor as another component of state vector affects the estimates of β τ t. β τ t from the model with liquidity risk factor is consistently smaller than that from the model without liquidity risk factor, and this adjustment is consistent with the pattern in the results of least squares regressions. βt L is the coefficient on on-/off-the-run spreads used as a proxy for liquidity risk factor. In all 18 pairs of municipals and corporates which are comparable in terms of both rating and maturity, βt L is highly significant. As predicted from the results of least squares regressions, bonds with longest maturity tend to have higher liquidity loadings. β L t is higher for municipals with lower ratings, in general. For example, β L t from municipals with good ratings are lower than those with medium ratings of the same maturities. β L t for prime 30-year and 20-year municipals, however, are bigger than those for good 30-year and 20-year municipals, respectively. Low liquidity for municipal bonds with the highest rating and the longest maturity may explain high sensitivity of liquidity factor, β L t, in these 19

21 segments, as documented by Wang et al. (2008) who use the municipal bond transaction data from MSRB and report the number of transactions across different ratings and maturities. Overall, after accounting for the liquidity risk effect and estimating the model by the Kalman filter method, we obtain more consistent and stable estimates of marginal tax rate, which are about on average. This estimate of marginal tax rate is very close to the statutory tax rate of 48% for corporations that belong to the highest tax bracket during the period from 1970 to 1979 as predicted by the Miller hypothesis. The estimates of marginal tax rate are little smaller than the highest corporate tax rate. This may be due to the use of tax shelters by companies in the highest corporate tax bracket or due to the presence of small companies which are in the lower corporate tax brackets. 4.2 The low tax regime period: June August The pre-crisis period from June 1994 to July 2007 During the period from 1987 to 1993, US income tax rates were lower after the US Congress passed the tax reform act of 1986 and the top tax rate for corporations decreased from 46% to 35%. This corporate tax cut is an important event to test the validity of the model which accounts for the liquidity risk factor with time-varying coefficients. By comparing estimates from two different tax regimes, one can see whether this model generates consistent estimates of marginal tax rate close to the highest statutory corporate tax rate regardless of the tax regime. For the period from January 1970 to September 1979, we employ on-/off-the-run spreads as a proxy for marketwide illiquidity. More data are available to measure different dimensions of liquidity after this period. We employ Amihud (2002) and Pástor and Stambaugh (2003) illiquidity measures constructed from corporate bond transaction data. We select these two measures for two reasons. First, corporate bond transaction data are available to construct these two illiquidity measures to cover the sample period for municipal and corporate yields data from 1994 to We can consistently employ the same illiquidity measures through this period. Second, these two measures capture marketwide illiquidity conditions very well. In addition, these measures reflect the liquidity condition of bond markets quite well (see Ang et al., 2014) since these are constructed from corporate bond transaction data. Table 6 reports the mean and standard deviation of the β τ t, β L t and λ t estimated by the same 20

22 models reported in the previous section but for the sample period from June 1994 to July 2007 when the highest corporate tax rate was 35%. Again, βt τ is the coefficient on the corporate yields. βt L is the liquidity risk loading on Amihud s (2002) illiquidity measure (in panel A) and Pástor-Stambaugh s (2003) illiquidity measure (in panel B). λ t is the time-varying intercept term which captures risk premium. Municipals and corporates that have the same ratings are paired for the estimation. For maturity, 30-year and 20-year municipals are paired with over 15-years corporates, and 10-year municipals are paired with 10 to 15-years corporates. sample period from August 1971 to September 1979, β τ t Compared with those estimated from the from the recent period are bigger in both models with and without liquidity factor. Overall, the increase in estimates of β τ t seems to reflect the corporate tax cut as marginal tax rate and βt τ are inversely related. Though βt τ increases on average compared to that from the period of high tax regime, it is still not close to 0.65, which is equal to one minus the top corporate tax rate of 35% from the low tax regime period, without accounting for liquidity factor in the model. When we add Amihud s (2002) measure as a proxy for the liquidity factor in the model, the coefficients of the liquidity proxy, βt L, are significant and positive in every estimation except the one from the pair of BBB 10-year municipals and 10 to 15-year corporates. When we use the Pástor-Stambaugh (2003) measure, the results are similar. In all ratings except BBB, β L t is significant and positive. Again, in the period after the corporate tax cut, the liquidity risk factor continues to be an important factor in the relation between municipal and corporate yields. After the liquidity factor is added in the model, the mean of time-varying risk premium captured by λ t decreases, indicating that the liquidity risk effect is embedded in λ t when the liquidity factor is omitted. In addition, β L t become smaller as the maturity of bonds decreases in panel A, indicating that bonds with longest maturity tend to have higher liquidity risk, while the same pattern does not occur in panel B. After including the liquidity factor, β τ t becomes much closer to In panel A where we use Amihud s (2002) illiquidity measure as the proxy, the average estimates of β τ t is about The average βt τ is about in panel B where the Pástor-Stambaugh (2003) measure is used as a liquidity proxy, which is similar to that in panel A. Overall, estimates of marginal tax rate from the time-varying coefficient model with the liquidity factor estimated by the Kalman filter method are very close to the statutory tax rate of 35%, which strongly supports the Miller hypothesis. 21

23 4.2.2 Test of structural change around the subprime mortgage crisis in 2007 During the subprime credit crisis and the bankruptcy of Lehman Brothers, from July 2007 to September 2009, both municipal and corporate bond yields become more volatile compared to the pre-crisis period (see panel B of Figure 2). In addition, the relationship between municipal and corporate bond yields changes and becomes less predictable. In panel B of Figure 3, the ratio of municipal yields to corporate yields has increased since the onset of the financial crisis and becomes more volatile. Before the crisis, the difference in the ratios of municipal and corporate yields between long and intermediate maturities is moderate. The difference becomes bigger as the ratio of municipal yields to corporate yields increases. Though the period from June 1994 to July 2007 and the period after September 2009 belong to the same tax regime, the change in the behavior of relative yields between municipals and corporates suggests that additional risk factors need to be considered in the latter period. In order to examine whether a significant structural change exists in the relationship between municipal and corporate bond yields around the period of the subprime crisis and the collapse of Lehman Brothers, we employ F statistics test by Andrews and Ploberger (1994) and Chow (1960) s test. A brief description of test procedure suggested by Andrews and Ploberger (1994) is provided in Appendix B. Panel A of Table 7 reports breakpoints estimated from F statistics test for the time-series of ratios of municipal to corporate bond yields by each rating and maturity pair. The breakpoints reported are ones that minimize the residual sum of squares and in turn maximize F statistics. The residual sum of squares calculated from two regression models fitted separately for before and after the breakpoints are reported along with those calculated from the single regression model fitted for entire time-series without a breakpoint. The residual sum of squares from the regression with a break are much smaller than those from the single regression without a break, which strongly suggests that a structural change exists and thus running two separate regressions to the period before and after the break fits the data better. Breakpoints estimated are concentrated around the period from 2008 to 2010, supporting that the relationship between municipal and corporate bond yields changes after the crisis. Unlike Andrews and Ploberger (1994) s test, Chow (1960) s test assumes that the breakpoint is known and tests whether the structural change occurs after the known breakpoint. We use two- 22