Predicting Municipal Bond Yields: Can Municipal Indices Predict
Individual Bond Yields?
by
Le Duong
University of Arkansas, Little Rock
and
Duane Stock
University of Oklahoma
205A Adams Hall Price College of Business
Norman, OK 73019
Email: dstock@ou.edu
September 14, 2009
Presented at University of Oklahoma Center for Financial Studies workshop. The
authors appreciate comments by seminar Jesus Salas, Louis Ederington, Chitru
Fernando, and Scott Linn.
0
Abstract
Recent studies have shown varying degrees of predictability in returns and volatility for
different types of financial instruments. Of course any predictability should be exploited by
those dealing in the instruments. To date, there has been little analysis of predictability in
municipal bond yields and volatility. We develop econometric models of municipal bond
indices and then test for any usefulness in predicting yields of individual bonds. The sample is
restricted to larger negotiated, general obligation yields that meet strict tests of liquidity and
quality of yield data. We apply different types of econometric models including ARMAGARCH and alternative logit models where, in one type of logit model, volatility helps
predict the sign of yield changes. Then we develop strategies to exploit the predictability. In
summary, we find there is evidence of useful yield predictability that produces beneficial
strategies. The predictability and strategies are robust to different specifications.
1
Introduction
Every year United States corporations and municipalities issue huge amounts of debt.
As of the first quarter of 2009, there was $6,772 billion of corporate debt and $2,669 billion
of municipal debt outstanding. If the issuer can save as little as a few basis points due to
yield predictability, the life- time costs can be reduced by millions. As an example, Lambert
and Pierog (2009) report that in the first half of 2009 alone, California sold $13.4 billion of
municipal bonds. Reducing costs by 10 basis points on these first half issues would have
saved $13.4 million a year. Any possible small yield predictability is particularly useful to
bond issues with flexible timing. For example, negotiated municipal issues and corporate
shelf issues have a good deal of timing flexibility.
The purpose of our study is to test the ability of simple econometric models of
municipal yield indices to predict yields of individual municipal bonds. Our results are
potentially useful to anyone buying or selling municipal bonds including mutual funds,
hedge funds, banks and individual investors. For convenience and quality of data, we use the
context of timing an issuance. We include alternative models of municipal indices where
some include both a yield index forecast and a volatility forecast whereas others may only
include a yield index forecast. Our tests are strong in that we test the models on out of
sample actual yields of specific bonds, at issuance and soon after, as opposed to testing the
ability to merely forecast the index itself for periods before and after actual issuance. We
note that forecasting a municipal bond index is very problematic due to non-synchronous
trading. It is important to not confuse our analysis with a test of optimal timing for a window
including dates before actual issuance.
2
More specifically, we utilize daily, weekly, and monthly index data to aid in timing
(trading) for forty large, negotiated, general obligation bonds. The econometric models for
timing bond issues will be helpful in two ways. First, the models will give predictions of
changes (or the sign thereof) in yield of indices and individual bonds. Second, some models
will simultaneously give conditional volatility of the yield changes. Analyzing both level and
volatility of yields can clearly enrich analysis of the timing decision.
With regard to
volatility, an obvious question is how much will variance of expected yields increase if the
planned issuance (transaction) date is delayed one or two periods? Furthermore, in one type
of logit model, volatility modeling can help predict the sign of yield changes.
Timing is of course complex. At some point(s) in time the issuer (trader) has to
consider all the data available and decide to issue (trade) or not within a certain time frame.
Even deciding a cutoff for when to stop and analyze available data is part of the process. To
permit usage of the best possible bond specific data, we stress analysis from the vantage
point of the close of the period immediately prior to actual issuance. We note that for dates
prior to issuance, it is impossible for a researcher to know the precise yield at which the
specific bond “could have” been issued. Yield indices are problematic as they may or may
not provide precise answers given the myriad of bond specific factors affecting yield and,
furthermore, non-synchronous trading is quite problematic.
Our primary question is as follows. Given all the information available at the close of
the day (week, month) before issuance, can an econometric model of an index predict actual
individual bond yields for the issuance day and, also, immediately following periods? What
alternative model(s) could the issuer have potentially used to execute superior issuance
timing? Furthermore, is predictability robust across different models? Our results clearly
3
suggest that econometric models such as those we estimate can be developed to help time
trading and issuance of individual municipal bonds.
Our question is in contrast to an ex post issuance performance examination of a broad
window both before and after issuance which answers an alternative question: Did the issuer
pick the minimum index (as opposed to bond specific) yield within that window? See
Kadapakkam and Kon (1989) where, using government bond indices as opposed to bond
specific yields, they tested for optimal timing of corporate shelf issues as compared to
nonshelf issues. Such an analysis is obviously important but develops no model for issuance
timing of a specific bond, does not consider volatility of yields, and does not consider actual
prices of an individual bond with numerous issuer and issue-specific characteristics.
The next section describes recent evidence on the benefits of developing models to
predict returns, yield changes, signs of yield changes, and volatility for different types of
instruments. Then we describe our data sources where we stress that although indices are
necessary to develop econometric models, precise and unbiased strategy testing issuance
timing (trading) uses yields of individual bonds at issuance and immediately after. The next
section estimates the models. Then we execute strategies to test the usefulness of the models
for daily, weekly, and monthly periods. The conclusion summarizes the research.
Predicting Returns, Yields, Signs of Yield Changes and Volatility
Recent advances in conditional distributions of financial time series have greatly
advanced modeling of expected returns of equities and bonds, yield changes and conditional
variance. Jones, Lamont, and Lumsdaine (1998) and, also, Baker, Greenwood, and Wurgler
(2003) suggest that yields are predictable. Cochrane (1999) lists return predictability as one
4
of the new facts of finance and maintains that bond yields and returns are predictable. 1
Papageorgiou and Skinner (2002) provide evidence of government bond yield predictability.
Cochrane and Piazzesi (2005) find that excess returns of government bonds are predictable.
Egorov, Hong, and Li (2006) suggest affine term structure models have potential for
predicting yields.2 Christoffersen, and Diebold (2006) note that researchers, such as White
(2000) and Pesaran and Timmerman (2004) have been successful in predicting the sign, if
not the magnitude, of asset returns.3 Green, Li, and Schurhoff (2008) find that lagged
municipal price changes are correlated with municipal yields. Also, conditional variance
(volatility) of returns for many types of securities has been clearly proven predictable. See
Andersen, Bollerslev, Christoffersen, and Diebold (2006) for a survey of volatility
forecasting. Given the extensive testing of predictability for other instruments, a test and
related application for the municipal bond market seems overdue.
Numerous authors have shown that even a small amount of predictability of
returns and volatility, even if not clearly statistically significant, can have a large impact on
financial strategy and decisions. Among others, Kandel and Stambaugh (1996) have shown
that weak return predictability can have a strong impact on portfolio decisions. Xu (2004)
finds that small levels of equity return predictability can be utilized to develop strategies with
both higher returns and less risk. Furthermore, Xu (2004) stresses that it is not the statistical
significance of return and volatility estimations that matter so much as does the profitability
1
We note that Cochrane (1999) and some others noted may tend to analyze data with lower frequency
than ours.
2
Some suggest that predictability of financial time series is due to time varying risk premia.
3
There is also considerable evidence that equity returns are predictable. For one recent example, see
Cohen and Frazzini (2008).
5
of any weak statistical significance.
Barberis (2000) finds that small levels of return
predictability affect asset allocation. Some may suggest that the ability to predict returns is in
violation of market efficiency but Marquering and Verbeek (2004) and Lewellen and
Shanken (2002) maintain this is not necessarily the case. In this context, rational asset
pricing implies time varying risk premia that lead to predictable patterns.
Our models of expected yields (yield changes) translate into useful strategies
that are an improvement over naïve, random timing for at least two reasons. First, the above
cited studies on equities, bonds, and other instruments suggest that any weak predictability
given by estimated mean and volatility equations leads to superior performance. Second, our
application is one where frequent transactions typically dictated by predictability and related
dynamic strategies will not occur. In this context, it has been known for decades that, for
example, equities sometimes exhibit small autocorrelation in returns. See, for example,
Neiderhoffer and Osborne (1966), Fama (1970, 1991), Conrad and Kaul (1989) and Lo and
MacKinlay (1999). More recent research suggests market microstructure may generate
autocorrelation. Nonetheless, investment strategies developed to exploit any autocorrelation
frequently find that any predictable autocorrelation in returns is frequently more than offset
by transactions costs needed to make numerous trades. Our research problem is one where
the issuer chooses to issue the bonds at only one certain time.
Of course, issuance
(transactions) costs thus occur only once where the level of costs are unaffected by the day
chosen to issue the bonds.4 Therefore, there is a greater likelihood that exploitation of any
4
Kadapakkam and Kon (1989) find that, ex post, corporate bond shelf issues seem to be better timed
than nonshelf. Adding on to our earlier comments, we note that their work does not analyze municipal bonds
nor does it report models of timing or volatility.
6
weak predictability in expected yield changes and volatility will lead to beneficial timing
strategies.
With regard to volatility, Fleming, Kirby, and Ostdiek (2001 and 2003) and Gomes
(2007), among others, predict volatility and demonstrate the economic gains from doing so.
For our application, we note that volatility modeling can be useful in more than one way.
Consider how volatility affects confidence of yield predictions. That is, if the model predicts
a decline in yields, thus encouraging a delay in issuance, it may be especially appealing to
delay if the expected decline is large and, also, the volatility is small as the issuer can be
relatively confident of a decline in yields. On the other hand, if the model predicts large
increases in yields and all the expected volatilities are small, it is very unappealing to delay
issuance as the likelihood of greater yields seems strong.5 In order to enhance our volatility
analysis and also develop a broad sample representing various environments, our sample of
bond issuances is developed from four distinctly different periods where yields tended to
have different trends (upward and downward) and levels (high and low) of volatility.
Volatility modeling can also be useful if it can forecast the level of yields. It is logical
to suggest that predictable volatility could lead to predictable risk premia, returns, and yields.
Thus, GARCH in the mean models may offer potential. More central to our research,
Chrisoffersen and Diebold (2006) show how predictability in the sign of returns and
volatility may well be interrelated. Any slight sign predictability can potentially be very
useful for bond issuers constantly asking themselves the question: Will the interest cost of
5
Also, we of course note that if a large increase in rates is predicted and expected volatility is high,
there is a strong likelihood rates will increase very dramatically.
7
the bond issue rise or fall if issuance is delayed one day, two days, one week, two weeks, one
month, etc.
After lengthy testing of alternative specifications and explanatory variables to best
answer our questions, we report the parsimonious models that worked best. The base model
is the ARMA-GARCH which predicts both changes in level of yields and volatility. In the
context of sign prediction only, we also report a simple logit model for sign prediction
(ARMA-Logit) that has no volatility component. Furthermore, we report an alternative logit
model (CD-Logit)6 suggested by Christoffersen and Diebold (2006) which uses volatility to
predict signs of yield changes. Alternative models are used to test robustness of any claims
of predictability.
Data Sources and Sample Selection
We are aware of controversies in claiming predictability in, for example, equity
returns. In one of the earlier studies, Lo and MacKinlay (1992) find equity predictability is
strong and economically significant. In more recent examples, Cochrane (2007) defends
return predictability in equities whereas Ang and Bekaert (2006) find that long run return
predictability is not significant. It is important that Ferson, Sarkissian, and Simin (2003)
describe how spurious regressions and data mining can lead to spurious predictability. Thus
we develop our sample and construct our tests to be as free as possible of bias.
6
We describe two alternative CD-Logit models later.
8
The dominant principles of our sample selection are 1.) quality of individual bond
yields and 2.) availability of indices that match characteristics of these specific bonds. Thus,
we match individual general obligation (GO) bonds and indices of GO bonds described in
more detail below.
Individual General Obligation (GO) Bond Yields
We examine forty large-issuance, negotiated, general obligation (GO), noncallable,
municipal bond issuances, given in Table 1, for ex post optimal issuance timing. We choose
negotiated issues as there is much more issuance flexibility than for competitive issues and
we can thus more credibly analyze the actual chosen issuance timing where the same can be
said of corporate shelf issues versus nonshelf issues. We only use bonds where the
Bloomberg financial system provides an issuance yield (yhp,issue) and a closing yield for
every trading day of the following four weeks. This requirement assures high quality of data
and also allows monthly analysis. Bloomberg issuance yield is not necessarily the closing
yield for the day but the yield at time of day when issued.
We use a type of bond that is as homogeneous as possible such that an index
(described below) based on such bonds is a credible average representation of the individual
bond issues. In this context we note that Harris and Piwowar (2006) find that complex
municipal bonds have higher costs than simple bonds. General obligation bonds are known
for frequently being typically less complex and more homogenous in sources of debt service
than revenue bonds. Noncallable bond issues are more homogenous in that the value of call
features can vary substantially due to variation in call price, call protection period, etc. even
when matching maturity and credit quality. To appreciate the importance that callability has
9
upon yields we refer readers to Duffee (1998) who stressed it is very problematic that many
corporate bond indices mix both callable and noncallable bonds. His empirical research
found that a sample of callable bonds had very different sensitivity to changes in the level of
interest rates than an otherwise similar sample of noncallable bonds. We choose bonds with
high credit quality as we think they are more homogenous than lower credit quality.
Although the above restrictions severely restrict sample size, we feel the need for such
requirements to assure accuracy in yields and strategic testing, especially since municipal
bonds are well known for illiquidity.
As mentioned, our source for bond specific yields was the Bloomberg financial
information system. To be sure we proportionately include issues from various types of
interest rate environments, we classified the largest negotiated, GO, noncallable municipal
bond issuances in the period January 2000 to August 2006 available on Bloomberg into the
following continuous, non-overlapping four groups.
(1) Issued between January 2000 and October 2001. This was a period when
municipal bond interest rates were generally decreasing and rate changes exhibited
relatively low volatility.
(2) Issued between November 2001 and July 2003. This was a period when municipal
bond interest rates were generally decreasing and rate changes exhibited relatively
high volatility.
10
(3) Issued between August 2003 and July 2005. This is a period when municipal bond
interest rates were generally increasing and rate changes exhibited high volatility.
(4) Issued between August 2005 and September 2006. This was a period when
municipal bond interest rates were generally increasing and rate changes exhibited
relatively low volatility.
The above categories identify contrasting environments and also allow analysis of how
timing may be more important, for example, in a more volatile environment than a less
volatile environment. Then from each of the four groups above, we sorted municipal bond
issuance into two credit quality groups, i.e., AAA and AA. From each of the groups, we
chose the five largest bond issuances that also had closing yields available for every trading
day in the four weeks after issuance to allow monthly analysis. We chose the largest because
one may argue timing is more critical the larger the issue and larger issues tend to be more
liquid. The earliest issuance used was January 24, 2000 and the latest was August 14, 2006.
The Bloomberg system calls their yields individual HP (historical price) closing yields
and reports an average of yields across a sample of brokers and dealers (“price contributors”)
on new issues. Where available, yields are reported on a daily basis after issuance. For
examination of timing strategies, we feel post- issuance HP yields provide superior
information compared to using index yields. Studies of bond issuance using index data to
determine timing performance are not as precise as we wish for individual bonds.
However, when we analyze the timing of a particular issue, HP yields on individual
bonds are not available before issuance. Therefore we need some measure of yields before
11
issuance in order to develop predictions and strategies for timing the issuance of a specific
individual bond issue. Hopefully the particular index chosen will represent the individual
bonds well enough to estimate models accurately enough to credibly suggest usefulness.
Thus, the second type of yield data was a Bloomberg daily time series index of
noncallable general obligation bonds with the same maturity and credit rating as the
particular issue. The authors were fortunate Bloomberg provides such a good match to our
preferred bond features. Alternative indices that we found were not as good a match. For
example, consider the S &P National Municipal Bond Index where the maturity requirement
is only that bonds in the index have a maturity of more than one month where there is thus a
large variation in maturity. Furthermore, callable bonds are mixed with noncallable bonds
and the credit rating is anything BBB- or above. Using the better Bloomberg index data, we
estimate expected yield changes and volatility using data through and including closing
yields the day (week, month) before issuance (t-1).
As for individual HP yields, the
Bloomberg index yields are derived from market quotes, not matrix prices. In order to
construct their yield series, Bloomberg collects the daily closing of bid and ask quotes from
reliable dealers. For each maturity and credit quality, for example, one year AAA, they
average the bid and ask quotes of their sample bonds to derive a given index yield. The
Bloomberg index data begins on January 2, 1992 for AAA bonds and on June 6, 1994 for
AA bonds where we use index yields through September 2006. Using simple strategic
decision rules, we evaluate the actual issuance for timing. That is, was it wise to issue on the
date chosen or would it have been better to defer issuance? If our strategies show good
performance, then the index models estimated are useful for future applications. Any issuer
12
can easily imagine a point in time is t-1, estimate a model for future values of the index, and
decide the timing of an issuance (trade).
We focus on the window from the period (day, week, month) just before issuance to
immediate periods after issuance for the below reasons. First, it is not our purpose to forecast
indices. But, of course, we need indices to forecast future yields of individual bonds not yet
in existence. The cleanest and strongest test of our econometric models is the ability to
forecast issuance yields of individual bonds which can be quite heterogeneous.
Different studies have found the municipal market is quite heterogeneous where yields
for individual bonds depend upon numerous bond specific factors. The size of issue, relative
supply of municipal bonds within the state, relative demand for bonds within the state, bank
pledging requirements, state income taxation, and bank qualification may all affect yields.
See Kidwell, Koch, and Stock (1984) and Forbes and Leonard (1984). Furthermore, Harris
and (2006) and Hong and Warga (2004) find liquidity effects on yields are dependent upon
such things as credit risk and bond complexity, in addition to finding that costs are clearly
higher (lower) on complex (simple) bonds. Thus, using an index to forecast yields on bonds
with differing complexity is problematic. However, our sample selection and (noncallable,
equal maturity, equal credit quality) index match was made to minimize such problems. Of
course, it is impossible to know the yield at which a specific bond would have been issued in
periods prior to issuance. Our tests could be construed as strong ‘out of sample’ tests of the
econometric models of municipal bond indices. In this context, a basic question is whether
the indices we have chosen are robust and well enough matched to the individual bonds so as
to successfully forecast yields of the individual bonds.
13
The second reason for focusing on time periods after issuance is that
predictions using index data before issuance are frequently plagued by non-synchronous
(stale) trading data which leads to spurious autocorrelation in yield changes and misleading
predictability of actual yields. The problems presented by non-synchronous trading are
potentially quite large for municipal indices given the low liquidity of municipal bonds.
Consider making decisions for, say, a window including two periods before actual issuance
in period t. A regression of yield changes for the appropriate index through t-2 would be
estimated and closing index yields for t-1, t, t+1, etc. predicted. However, Campbell, Lo and
MacKinlay (1997) illustrate that non-synchronous trading can create a false impression of
predictability in equity returns and price changes even when prices are statistically
independent. As one an example of the difficulty of using data with non-synchronous trading
problems, see Ahn, Boudoukh, and Richardson (2002) where they find that non-synchronous
prices in a stock index create autocorrelations that are quite problematic when comparing
spot and futures prices. Also, Kadlec and Patterson (1999) find that non-synchronous trading
often explains much of the autocorrelation in equity returns. More recently, Schotman and
Zalewsaka (2006) and Bernhardt and Davies (2008) show that non-synchronous trading has
a sizeable effect.
Non-synchronous trading problems are likely even larger for municipal
bonds than for equities. Using index yields to predict individual HP yields is not subject to
non-synchronous problems.
14
Estimations of Expected Change in Yield and Conditional Volatility
ARMA-GARCH Estimations as the Base Model
Daily Periods
We test robustness of predictability across alternative classes of models without
necessarily declaring a winner as our purpose is more to test effectiveness and then general
robustness across classes of models as opposed to declaration of the best. For each class of
model, we report the specifications that show the best results. Perhaps the most obvious for
our analysis is the ARMA-GARCH model which simultaneously predicts change in yields
and volatility. Later we do an ARMA- Logit estimation, where there is an equation to
estimate (only) the sign of yield changes. Finally, we also report alternative sign prediction
estimations (CD-Logit and CD-EWMA) as suggested by Christoffersen and Diebold (2006)
where the sign of yield change is solely a function of volatility.
We stress the prediction of sign changes as opposed to forecasts of expected (mean)
changes in yields. To forecast the complete distribution of expected price (yield) changes
may be too ambitious and is likely more than what is needed to be successful in timing
transactions. Christoffersen and Diebold (2006) stress that profitable forecasting is more a
result of forecasting market direction (sign changes) than it is forecasting mean (expected)
returns. In fact, they maintain that forecasting the sign of returns (yields) can be done with
surprising success and provide numerous studies to support their statement.
Using index data for an ARMA-GARCH estimation, we predict changes in yield
(Δyt,M) and volatility at time “t” for bonds of maturity M where Δyt,M = yt,M - yt-1,M . Our
ARMA mean equation estimation uses one autoregressive (AR) term and one moving
average (MA) term. In other specifications tested we employed a long list of additional
15
explanatory variables such as lagged level of yields, various measures of the slope of the
term structure, various forward rates implied by the term structure, etc. but none of these
additional variables were consistently significant. Furthermore, GARCH in the mean models
were not helpful. Thus, we use the following parsimonious model for Δyt,M where ε
t,M
represents the error term of the regression.
∆yt,M = a0,M + a 1,M Δyt-1,M + b1, M εt-1,M + εt,M
 t2,M  M  1,M  t21 ,M  M  t21 ,M
(1)
(2)
The following omits M from yt notation without loss of meaning.
For volatility (  t ,M ) estimation at time “t” we use a standard GARCH above where
2
volatility depends only on the most recent squared residual and lagged volatility. As for the
mean equation, we tested many more volatility specifications that included, for example, the
lagged level of yields, forward rates, slope of term structure, asymmetric volatility, etc. but
none of these additional variables were consistently significant.
Strategies to Test Usefulness of Models
Assume an investment banker is considering the best time to sell a negotiated,
noncallable, general obligation issue. Day (week,month) t-1 has ended (or it is very early on
day/week/month t) and perhaps because of other bond market activities or a myriad of other
reasons, it is important to decide whether to issue as soon as possible, i.e. the next period, or,
alternatively, the following period. As empirical results suggest, trends in yield changes tend
16
to last for three or more periods and our choice of issuance period depends on detecting a
trend that lasts for the two periods of the issuance window. In our system, the actual day of
issuance is day t.
The above econometric model has been estimated using index yields through close of
day (week, month) t-1 and issuers thus have index predictions for close of day (week, month)
of future periods. Our strategies take advantage of predicted index yield changes for the
close of these future periods.
Hopefully these predicted yield changes for the index will
give an indication of yields for the individual bond issues of our sample at future times and
thus confirm the usefulness of the index model estimation and related strategies.7 We now
introduce different strategies to test the usefulness of estimated index models for predicting
individual bond yields on the day (week, month) of issuance and days (weeks, months)
thereafter.
Numerous rules for a strategy can be suggested. For ease of initial illustration, let
us assume the choice is simplified and limited to only two days (weeks,months) and call this
strategy ARMA-GARCH-A. The issuer only knows close of index for t-1 and a prediction
for the index on day t. See Figure 1 for illustration of index versus individual HP yields.
A simple strategy would take advantage of any trend in yields suggested by estimated
index regressions. More specifically, regression results reported below show that the signs of
yield changes are typically the same for each of the three periods after t-1. That is, a
predicted increase (decrease) in yields from t-1 to close of t tends continues for the next
period.
7
Green, Hollifield, and Schurhoff (2007) suggest that an investment banker can buy in periods before
delivery and sell pieces forward ‘when issued’.
17
Thus our simple strategy is totally dependent upon the prediction for the index close of
day t. Specifically, if index yields are predicted to be lower by the close of day t, delay
issuance to t+1 to take advantage of the expected decline. Our estimate of the strategic
issuance yield in this case is the realized Bloomberg HP yield for t+1 which is yhp,t+1. The
gain (cost reduction) of the strategy is yhp,issue – yhp,t+1 where yhp,issue is the actual issuance
yield for day t reported by Bloomberg. The effectiveness of this strategy hinges on the hope
that the likelihood of a sizeable increase in yield on t+1 that is of greater magnitude than the
expected decline on t is small. Again, expected yield changes tend to be the same sign.
Robustness to a sizeable t+1 yield increase is determined by the success of execution
strategies given below.
Alternatively, if index yields are predicted to be the same or higher on day t, do not
delay but issue on day t which is, in fact, the actual day of issuance. It is important to note
that Bloomberg HP issuance yield is not necessarily the closing yield for the day but the
yield at time of day when issued.
Bloomberg does not report a closing yield for the
individual bond on day of issuance, t, but only the issuance yield (yhp,issue) which occurs
sometime between the open and close of day t. The idea is that hopefully the bond can be
issued before any increase in yield occurring before the close of the day. If the strategy
chooses day t, our gain calculation is zero as the strategy selects the same day (t) as the
actual issuance. Note that if yields rise early on day t, the effectiveness of this strategy may
be compromised. Robustness to an early day t increase in yield is determined by any success
of strategy executions given below.
We emphasize that all predictions of yield changes are necessarily done from the
index regression model. All issuance strategy yields use the superior realized bond specific
18
HP yields. Realized individual HP yields are a stronger test of the models and strategies and
are also a superior out of sample measure reflecting all unique aspects of the issue and issuer
while also being free of non-synchronous trading bias.
An alternative strategy to that given above is to choose to issue on the day with the
lowest predicted index yield. The results of such a strategy are practically identical to that
above given the persistent pattern of yield changes. We use the strategy above so that clearer
comparisons can be made between ARMA-GARCH and models which only predict the sign
(not magnitude) of yield changes.
Alternatively, a more complex longer term strategy could be used where we call this
strategy ARMA-GARCH-B. Here, as of close t-1, predictions are made for the index at
close t and t+1.
Issue on day t+2 if expected change in index yield is negative on both day t and t+1.
The gain (cost reduction) is computed as yhp,issue – yhp,t+2 where yhp,t+2 , the individual HP
yield at t+2, is the strategy yield.
Issue on day t if the expected index yield change at close of day t is positive or zero
and the expected index yield change at close of day t+1 is positive or zero. That is, issue the
bonds before the expected increase in index yields where index yields are expected to rise on
both subsequent days. The gain is zero as the strategy issuance day is the same as actual
issuance day.
The combination of yield changes for close t and t+1 may have mixed sign in some
cases. Here we recognize that the variance around a prediction approximately doubles when
the horizon is increased from one day to two days. Thus we proceed as below.
19
Defer issuance until day t+1 if the expected index yield change at the close of day t is
negative and the expected index yield change at close of day t+1 is positive or zero. The
gain is computed as yhp,issue – yhp,t+1 where yhp,t+1, the HP yield at t+1, is the strategy yield.
If the series of expected index yield changes is positive on day t and negative on day
t+1, we merely issue on day t (actual issuance) because there is no clear advantage due to the
forecasts.8 The gain is thus zero because the strategy day chosen is the actual issuance day.
We note that
A shortcoming of the above strategies is that they are mostly based upon expected
yield change and pay little attention to the expected volatility of interest rates where the
greater the expected variance (  t2 ,M ), the greater the realized yield could differ from the
2
expected. Of course, a conventional risk aversion view is that the greater  t ,M , the greater
the possibility that rates could increase significantly if delayed (even given an expected
decline) in which case delaying the issuance would be very costly. Thus the rule could be to
delay if
E [ Δyt+k ] < 0
,
and, furthermore,
E  yt  k 
Var (yt k )
1/2
 z
8
Cases where the net of positive and negative gains is negative are rare and the net is small. We chose
to do this because of the volatility risk from delaying is greater as number of periods increase.
20
where k = 0, 1,…. n. That is, an issuer may require that the absolute value of the ratio of
expected yields changes to variance has to meet a threshold, z. For example, greater expected
decreases (increases) accompanied by low expected volatility make it more appealing
(unappealing) to defer issuance.9
Our work produces a longer menu of choices than to merely issue on one of two days.
Issuance could also be, for example, a week or a month hence. Thus, we also provide
weekly and monthly issuance choices for those with longer planning horizons where the
same logic applies. The most theoretically appealing decision may be to defer issuance until
the future time when the ratio z is greatest across all given alternative horizons.10
Model Estimations and Strategy Results
For each AAA-rated bond, we estimate the equations, reported in Table 2, using index
yields from January 2, 1992 (origination) until the day before (t-1) the bond specific issuance
date. Similarly, for each (lower rated) AA-rated bond, we estimate equations using data from
June 6, 1994 to the day before the issuance date (t-1). 11 The forecasted levels and variances
of yield change from t to t+2 are given in Table 3 where the range of forecasted change in
(index) yield for close day t is 0.0500 to -0.0200. Successive forecasted changes in yield are
9
This is very similar to the Sharpe performance measure where here the performance is ex ante.
Greater predictability represented by large expected declines in yield and low volatility suggest strong ex ante
performance.
10
We assume that the issuer really needs to sell the bonds within a certain period of time. Eventually,
the issuer has to sell and thus may set a threshold to sell when yields meet a certain level.
11
. The residual and squared residual results are white noise for all 40 regression results.
21
almost always the same sign but are not always so and the magnitude of the change (for the
same bond) can vary considerably.
The last column reports forecasted variance where the maximum one day forward
variance is 0.0014 and the minimum is much smaller at 0.0002. Thus the difference in
volatility among different grades of bonds and epochs in our sample can be very large.
Generally, for a specific bond, there tends to be little change in variance from t to t+2. It is
important to note that, given the variance for each forward period is very similar, the risk of
delaying more than one period is roughly a simple multiple of the number of days delayed. 12
Of course, this means the risk adjusted expected benefit of delaying is much diminished if
successive expected yield declines are small.
With the above estimations, we can test the usefulness of the estimated models and the
subsequent strategies. Consider Table 4. Here the decision is to either issue on day t, the
actual issuance date, or t+1.
If we follow this strategy, we would not issue on the actual
issuance day (t) in 11 of the 40 cases. The actual dollar gain/loss from following this timing
strategy is also estimated in Table 4. The gain on par value (100) from delaying until t+1 in
selected cases for a specific bond issue is computed as
gain = (yhp,issue – yhp,t+1) (Di) (Pi)
where Di is duration of bond i and Pi is issue price relative to par.13 Of the 11 bonds where
t+1 is chosen, 8 show a gain for delaying, two show a loss, and one shows no gain. This
suggests the strategy is usually profitable.
12
See Andersen, Bollerlev, Christoffersen, and Diebold (2006).
13
For example, Pi is 100.5 if the bond sells at 100.5% of par.
22
The last column is the gain adjusted for the potential risk to waiting. The computation
is
gadj

gain
t Pi
where rag is risk adjusted gain and g is algebraic gain per $100 par. This is analogous to a
crude Sharpe measure.
Now consider the more complex strategy of ARMA-GARCH- B in Table 5. If this
strategy is used, issuance does not occur on the actual issuance day (t) in 29 cases. This
greater disagreement with actual issuance day is to be expected given the larger menu of
choices. Of these 30 cases, the gain is positive in 14 cases, negative in 7 cases, and zero in 9
cases again suggesting the strategy can be beneficial.
ARMA-Logit Sign Estimation and Strategy
As our second class of models, we consider logit as an alternative econometric
technique to predict only the direction (sign) and not the level of index yield changes. The
logit model specification for the probability (Pr) of an increase in index yields is
Pr( yi  1 x i ,  )  e
 xi' 
/ (1  e
 xi' 
)
where the dependent variable, yi, is now the probability of an increase in index yield at close
of day t compared to t-1, the xi’s are the independent variables in the mean equation in Table
2, and β’s are the coefficient estimates of the same table. Of course, it is more appealing to
delay issuance if the probability of an increase (decrease) is low (high). This model estimate
is a probabilistic statement on the direction of index yield change where one could formulate
23
a strategy based on the strength of the probability. For example, one may consider a delay
in issuance only if the probability of an increase (decrease) is less (more) than, for example,
50%. Or, one can be more demanding with the strategy and delay only if the probability of
an increase is less than 40%. A weakness of this alternative (logit) procedure is that, in
contrast to a GARCH process, there is no simultaneous estimation of volatility and we could
find no econometric procedure for doing so.
The results of Table 6 clearly suggest the strategy is useful. As in earlier strategies, at
close t-1, forecasts are made for sign change of the index close at t, t+1, and t+2 where
forecasts for t+1 necessarily utilize a forecast for t and forecasts for t+2 utilize forecasts for
both t and t+1. This table presents the gains from the following strategy. Defer issuance until
t+2, if the probability of an increase for both t+1 and t+2 is less than 0.50. Issue day t if the
estimated probability of an increase in index yield for both day t+1 and t+2, is more than
0.50. Issue on t+1 if the probabilities of an increase in index yield for t+1 is less than 0.50
and the probability an increase for t +2 is more than 0.50.14 As before, the gain is calculated
from the difference between yhp,issue and the HP yield (y hp, t+1 or y hp , t+2 ) chosen by the
strategy. Specifically, the strategy advocates a delay of issuance until after day t in 30 cases
where there is a positive gain in 17 cases, a negative gain in 6 cases, and zero gain in 7 cases.
Table 7 differs from Table 6 only in that we use a 0.60 threshold for a decline instead of
0.50. The results are 19 gains, 8 losses, and 2 zero gains again suggesting the strategy is
beneficial.
14
This strategy is chosen as the risk due to volatility increases with delay.
24
Christoffersen and Diebold Sign Estimation and Strategy: EWMA and GARCH
Our third class of models is that given by Christoffersen and Diebold (2006) who
maintain that it is possible to have sign dependence even without mean dependence in the
ARMA-GARCH context given in the first class of models above. Importantly, it is volatility
dependence that produces sign dependence. Furthermore, even if sign predictability occurs
in concert with mean dependence, CD (2006) suggest that volatility dependence is the
dominant factor in sign prediction of gains (returns). Briefly, an increase (decrease) in
volatility may reduce (increase) the probability of a positive sign. As CD suggest, a logit
model approximates the distribution of μ/σt. Let ∆yt be the index yield change on day t and
then define the “positive index yield change” indicator as It=1 if ∆yt > 0 and It=0 otherwise.
It is forecasted using a model of the form

It  F 
 t

  et

where F( ) is a monotone function with a left limit of zero and a right limit of one, µ is the
expected index yield change, and σt is a forecast of yield change volatility.
F( ) is
determined by
F ( x) 
exp( x)
1  exp( x)
which produces the logistic regression (logit) model. We refer the reader to CD (2006) for
further details on this type of modeling.
25
The μ estimate is the mean change in index yield for the previous 20 periods. 15 This
roughly estimates any possible trend in yield suggested by index behavior.16 Green, Li and
Schurhoff (2008) suggest there are trends in municipal yields. Then, σt is estimated as an
exponential weighted moving average (EWMA) as popularized by RISKMETRICS and used
by CD. We label this model CD- EWMA where the strategy is based on the probability of an
increase in index yields on close of day t being less than 0.50. That is, if the probability of an
index yield increase for close day t is less than 0.50, issue on day t+1, after the expected
decrease in index yields at close of day t. Again, yields typically trend the same way for a
number of periods after t. Otherwise, issue on day t, the actual day of issuance. Gains are
calculated as before. Table 8 shows the results where, if not issued on day t, there are 12
gains and 9 losses. The CD-EWMA methodology does not readily provide for a longer
window. CD (2006) also suggest that volatility could also be estimated as a GARCH process
where this model is labeled CD- GARCH.17 Table 9 shows gains in 10 cases and losses in 9
for t+1 issuance.
15
Of course, bond returns are functions of yield changes. Numerous interest rate models such as Cox,
Ingersoll and Ross (1985), Vasicek (1977), and Hull and White (1990) suggest interest rates may have trends
due to mean reversion processes. Mean reversion leads to mean dependence.
16
Any trend in the index could be due to things such as mean reversion, sluggishness in yield changes
as in Green, Li and Schurhoff (2008), and non-synchronous trading.
17
To estimate volatility, the mean equation is the ARMA (1,1) process used before.
26
Weekly and Monthly Estimations and Strategies
The horizon chosen for predicting index yields can of course be longer than a two or
three days. For example, the initial stages of planning a bond issue for a capital project that is
not urgent may well consider alternative weeks or months. It is noteworthy that CD theory
suggests results may be better for more intermediate weekly and monthly horizons.18 Thus,
we also consider weekly and monthly horizons. In these cases, the time series data is less
frequent and do not permit a credible GARCH estimation.19
The choice for issuance in weekly and monthly periods is parallel to daily periods. An
analyst observes the close of the index at the end of week (month) t-1 which is the period
before actual issuance. All the analyst knows is the closing index yield for t-1. However,
they do have predictions for close of future weeks (months).The prediction for closing index
value on week (month) t is from a model of index values. If the predicted index close is
higher or the same at end of week (month) t, issue on actual date (day) of issuance (at the
actual issuance yield, yhp,issue.) Hopefully this issuance occurs before much of the predicted
rise (for the week after t-1) occurs. Alternatively, if the predicted index close is lower for
week t, delay issuance to take advantage of expected lower yields. The length of the period
is problematic for weekly and monthly strategies because, for example, an actual issuance
could be early or late in the period after close of t-1. Any strategic issuance not in period t
(actual issuance period) is at the earliest possible daily close in the week (month) t+1.
CD (2006) refer to this as the sweet spot where the value of µ/σ is about 1.4. CD (2006) explain that
if μ/σ is close to zero, the probability cannot deviate much from 0.5. For large μ/σ, the probability cannot
deviate much from 1.0. Intermediate μ/σ have potential for much more sensitivity.
18
19
GARCH processes need large sample sizes because of the simultaneous likelihood estimation of two
equations.
27
As an example, assume end of t-1 is Friday, May 14. If t is strategically chosen, actual
issuance could be any day of the next week. Parallel to daily strategies, if the strategy
chooses t, the gain is zero. If the strategy chooses t+1, then issuance is chosen to be the close
of Monday May 24, 10 days later. If yields have risen between May 14 and May 24, the
strategy will not work well. Any success in our test results implies robustness to any
assumptions of this strategy.
Table 10 A gives results for weekly periods. Here CD-GARCH where there are 8
gains and 9 losses whereas Table 10 B,CD-EWMA, gives 10 gains and 10 losses for weekly
periods. Although not shown in a table, weekly results for an ARMA-Logit are better. If a
0.5 threshold is used, there are 16 gains and 4 losses and if a 0.7 threshold is used, there are
19 gains and 4 losses.
Table 11 gives CD-EWMA monthly period results of 15 gains and 7 losses where the
threshold is a 0.5 probability. Although not shown in a table, an alternative threshold of 0.7
gives 18 gains and 6 losses.
Conclusion
There have been strong advances concerning the predictability of returns and volatility
in recent years. Such research has been conducted on equities and different types of fixed
income instruments.
However, there has been little applied predictability research for
municipal bonds. As one example of the potential, given that the huge dollar amounts of
many issues, saving just a few basis points from better issuance timing can result in huge
savings. Our purpose is to test alternative econometric models for any ability to forecast and
time municipal bonds issuances or other municipal bond trading. We include models that
28
simultaneously predict both yields and volatility, models that predict yields (signs of
changes) only, and models that use volatility to predict yields (signs of changes).
We place strong restrictions on our sample to assure data quality. Only bonds that
tend to be very homogenous and demonstrate high liquidity are included. These bonds are
matched with an index that matches on both credit quality and maturity.
Our results suggest that there is predictability in the municipal bond market. Of
course any predictability should be tested for usefulness and we thus suggest strategies for
bond issuers. These strategies prove to be successful in that gains outnumber losses. This
usefulness seem robust to many alternative specifications.
29
References
Ahn, D.;J. Boudoukh, J. ; M. Richardson; and R. F. Whitelaw, 2002, Partial
Adjustment or Stale Prices? Implications from Stock Index and Futures Return
Autocorrelation”, Review of Financial Studies,15,655-689.
Ang, A. and G. Bekaert, 2007, “ Stock Return Predictability: Is It There?”, Review of
Financial Studies, 20, 651-707.
Andersen, T. G.; T. Bollerslev; P. F. Christoffersen, and F. X. Diebold, 2006 ,
Handbook of Economic Forecasting, North Holland Press, Amsterdam.
Cox, J.C., J.E. Ingersoll and S.A. Ross,1985, "A Theory of the Term Structure of
Interest Rates", Econometrica, 53: 385–407.
Baker, M., Greenwood, R. and Wurgler, J., 2003, “The Maturity of Debt Issues and
Predictable Variation in Bond Returns” Journal of Financial Economics, 70, 261-291.
Barberis, N., 2000, “Investing for the Long Run When Returns are Predictable”
Journal of Finance, 55, 225-264.
Bernhardt, D. and R. J. Davies, 2008, “The Impact of Nonsynchronous Trading on
Differences in Portfolio Cross-autocorrelations”, working paper, University of Illinois
(Champaign- Urbana).
Cochrane J.H., 1999, “New Facts in Finance”, Economic Perspectives, Federal
Reserve Bank of Chicago, Quarter 3, 59-78.
Cochrane, J. H. and M. Piazzesi, 2005, “ Bond Risk Premia”, American Economic
Review, 95, 138-160.
Cochrane, J. , 2008, “ The Dog That Did Not Bark: A Defense of Return
Predictability”, Review of Financial Studies, 21,1533-1575.
Cohen, L. and A. Frazzini, 2008, “Economic Links and Predictable Returns”, Journal
of Finance, 63, 1997-2011.
Conrad, J. and G. Kaul, 1989, “Mean Reversion in Short Horizon Expected Returns”,
Review of Financial Studies, 2, 225-240.
Egorov, A. V. ;Y. Hong; and H. Li , 2006, “Validating Forecasts of the Joint
Probability of Bond Yields: Can Affine Models Beat Random Walk?” Journal of
Econometrics 135, 255-284.
30
Fama, E. F. , 1970,
Work”, 25, 383-417.
“Efficient Capital Market: A Review of Theory and Empirical
Fama, E. F. ,1991, “Efficient Capital Market: II”, Journal of Finance, 46, 1575-1617.
Ferson, W.E; S. Sarkissian; and T. Simin, 2003, “ Is Stock Return Predictability
Spurious? “ Journal of Investment Management, 1, 1-10.
Fleming, J., Kirby, C. and Ostdiek, B., 2001, “The Economic Value of Volatility
Timing”, The Journal of Finance, 56, 329-352.
Fleming, J., Kirby, C. and Ostdiek, B., 2003, “The economic value of volatility timing
using “realized” volatility”, Journal of Financial Economics, 67, 473-509 .
Gomes, F. J. , 2007, “Exploiting Short-Run Predictability”, Journal of Banking and
Finance, 31,1427-1440.
Green, R. C.; D. Li and N. Schurhoff, 2008, “ Price Discovery in Illiquid Markets: Do
Financial Asset Prices Rise Faster Than They Fall?” , working paper, Tepper School of
Business, Carnegie Mellon.
Green, R. C.; D. Li and N. Schurhoff, 2007, “Dealer Intermediation and Price
Behavior in the Aftermarket for New Bond Issues”, Journal of Financial Economics, 86,643682.
Hull, J. and A. White, 1990, “ Pricing Interest-rate Derivative Securities”, Review of
Financial Studies, 3, 573-592.
Jones, C.M., Lamont, O., and Lumsdaine, R.L., 1998, “Macroeconomic News and
Bond Market Volatility”, Journal of Financial Economics, 47, 315-337.
Kadapakkam, P., and Kon, S., 1989, “The Value of Shelf Registration for New Debt
Issues”, The Journal of Business, 62, 271-292.
Kandel, S. and Stambaugh, R.F., 1996, “On the Predictability of Stock Returns: An
Asset Allocation Perspective”, Journal of Finance, 51, 385-424.
Kidwell, D; T. Koch, and D, Stock, 1984, "The Impact of State Income Taxes on
Municipal Borrowing Costs", National Tax Journal, 37, 551-562.
31
Forbes, R. W. and P. A. Leonard , 1984, “The Effects of Statutory Portfolio
Constraints on Tax-Exempt Interest Rates: Note”, Journal of Money Credit and Banking, 16,
92-99.
Harris and Piowar, 2006, “Secondary Trading Costs in the Municipal Bond Market”,
Journal of Finance, 61, 1361- 1397.
Hong, G. and A. Warga, 2004, “ Municipal Marketability”, Journal of Fixed Income,
14, 86-95.
Kadlec, G. B. and D.M. Patterson, 1999, “ A Transactions Data Analysis of
Nonsynchronous Trading, Review of Financial Studies, 12,609-630.
Lambert, L. and K. Pierog, 2009, “ U. S. Muni Bond Issuance Drops in 2009 First
Half”, Thomson Reuters, June 30, 2009.
Lewellen, J., and Shanken, J., 2002, “Learning, Asset-Pricing Tests, and Market
Efficiency”, Journal of Finance, 57, 1113-1145.
Lo, A. and A. C. MacKinlay ,1992, “ Maximizing Predictability in the Stock and Bond
Markets”,Working Paper, Sloan School, MIT and Wharton School.
Lo, A. and A. C. MacKinlay ,1997, The Econometrics of Financial Markets,
Princeton University Press, Princeton, NJ.
Lo, A. and A. C. MacKinlay ,1999, A Non-Random Walk Down Wall Street,
Princeton University Press, Princeton, New Jersey.
Marquering, W., and Verbeek, M., 2004, “The Economic Value of Predicting Stock
Index Returns and Volatility”, Journal of Financial & Quantitative Analysis, 39, 407-429.
Neiderhoffer,V. and Osborne, M.F. ,1966, “Market Making and Reversal on the
Stock Exchange” Journal of American Statistical Association, 61, December, pages 897916).
Papageorgiou, N. and F. S. Skinner, 2002, “Predicting the Direction of Interest Rate
Movements”, Journal of Fixed Income, 11, 87-95.
Pesaran, M.H. and A.G. Timmerman,2004, “ How Costly Is It to Ignore Breaks
When Forecasting the Direction of a Time Series?” International Journal of Forecasting,
20, 411-425.
32
Schotman, P. C. and A. Zalewsaka, 2005, “Nonsynchronous Trading and Testing for
Market Integration in Central European Emerging Markets”, Journal of Empirical Finance,
13,462-494.
Securities Industry and Financial Markets Association (Outstanding U.S. Bond Market
Debt, 2009).
Vasicek, O. ,1977, An Equilibrium Characterization of the Term Structure”, Journal
of Financial Economics, 5, 177-188.
Xu, Y., 2004, “Small Levels of Predictability and Large Economic Gains”, Journal
of Empiricial Finance , 11,65-77.
33
Figure 1
An analyst observes the close of the index at close t-1 and predicts the close for future
periods. All the analyst knows is the closing index yield for t-1 and the predictions for future
index closing yields. The prediction for closing index on day t is from an econometric model
of index data. If the expected (predicted) index close for t is lower than t-1 index close, a
downward trend is expected to occur for the window of issuance. So, the strategy is to delay
issuance until t+1. If the predicted index close is higher (or the same) for t than the t-1 index,
issue on day t, the actual day of issuance. Hopefully this issuance occurs before much of the
predicted rise at close of day t (and the issuance window) occurs. Bloomberg reports
issuance cost on day t (yhp,issue) where the actual time of issuance is sometime between open
of t and close of t. Bloomberg does not report a closing yield for the individual bond on day
of issuance, t, but, again, only the issuance yield (yhp,issue) which occurs sometime between
the open and close of day t.
Bond issued some
time on day t at
yhp,issue
(the Bloomberg HP
yield)
Index
Yields
Bloomberg
HP yields
t-1
t
t+1
close
close
t+2
close
close
Realized Closing
Index on t-1
Predicted
Closing Index on t
Predicted Closing
Index on t+1
yt-1
E(yt)
E(yt+1)
(nonexistent)
yhp,issue
yhp,t+1
yhp,t+2
(no closing day t yield
supplied)
34